Seismic Retrofitting Manual for

Highway Structures: Part 1 – Bridges

PUBLICATION NO. FHWA-HRT-06-032 JANUARY 2006

Research, Development, and Technology

Turner-Fairbank Highway Research Center

6300 Georgetown Pike

McLean, VA 22101-2296

Technical Report Documentation Page 

1. Report No.

FHWA-HRT-06-032 

2. Government Accession No. 

N/A 

3. Recipient’s Catalog No.

N/A 

5. Report Date 

January 2006 

4. Title and Subtitle 

SEISMIC RETROFITTING MANUAL FOR HIGHWAY 

STRUCTURES: PART 1 – BRIDGES 6. Performing Organization Code

N/A 

7. Author(s) 

Ian Buckle, Ian Friedland, John Mander, Geoffrey Martin, Richard Nutt 

and Maurice Power 

8. Performing Organization Report No. 

10. Work Unit No. 

N/A 

9. Performing Organization Name and Address 

Multidisciplinary Center for Earthquake Engineering Research 

University at Buffalo, State University of New York 

Red Jacket Quadrangle 

Buffalo, NY 14261 

11. Contract or Grant No. 

DTFH61-92-C-00106

13. Type of Report and Period Covered 

Final Report 

12. Sponsoring Agency Name and Address 

Office of Infrastructure Research and Development 

Federal Highway Administration 

6300 Georgetown Pike 

McLean, VA 22101-2296 

14. Sponsoring Agency Code 

 

15. Supplementary Notes: Contracting Officer’s Technical Representatives: 

James Cooper, P.E., HRDI-03, Wen-huei (Phillip) Yen, Ph.D., P.E., HRDI-07, John D. O’Fallon, P.E., HRDI-07 

16. Abstract 

This report is the first of a two-part publication entitled: 

Seismic Retrofitting Manual for Highway Structures: 

Part 1: Bridges 

Part 2: Retaining Structures, Slopes, Tunnels, Culverts and Roadways. 

Part 1 of this manual is based on previous Federal Highway Administration (FHWA) publications on this subject including 

Seismic Retrofitting Manual for Highway Bridges, published in 1995 as report FHWA-RD-94-052. Revisions have been made to 

include current advances in earthquake engineering, field experience with retrofitting highway bridges, and the performance of 

bridges in recent earthquakes. It is the result of several years of research with contributions from a multidisciplinary team of 

researchers and practitioners. 

In particular, a performance-based retrofit philosophy is introduced similar to that used for the performance-based design of 

new buildings and bridges. Performance criteria are given for two earthquake ground motions with different return periods, 100 

and 1000 years. A higher level of performance is required for the event with the shorter return period (the lower level earthquake 

ground motion) than for the longer return period (the upper level earthquake ground motion). Criteria are recommended 

according to bridge importance and anticipated service life, with more rigorous performance being required for important, 

relatively new bridges, and a lesser level for standard bridges nearing the end of their useful life. 

Minimum recommendations are made for screening, evaluation and retrofitting according to an assigned Seismic Retrofit 

Category. Bridges in Category A need not be retrofitted whereas those in Categories B, C and D require successively more 

rigorous consideration and retrofitting as required. 

Various retrofit strategies are described and a range of related retrofit measures explained in detail, including restrainers, seat 

extensions, column jackets, footing overlays, and soil remediation. 

17. Key Words 

Bridges, earthquakes, performance-based retrofitting, 

seismic retrofit categories, screening, prioritization, 

evaluation, retrofit strategies, measures, guidelines 

18. Distribution Statement 

No restrictions. This document is available to the public through 

the National Technical Information Service, Springfield, VA 

22161. 

19. Security Classif. (of this report) 

Unclassified 

20. Security Classif. (of this page) 

Unclassified 

21. No of Pages

656 

22. Price 

 

 Form DOT F 1700.7 (8-72) Reproduction of completed pages authorized 
PREFACE 

This report is a major revision of the Federal Highway Administration publication Seismic 

Retrofitting Manual for Highway Bridges, which was published in 1995 as report FHWA-RD94-052. This edition expands the coverage of the previous publication by including procedures 

for evaluating and retrofitting retaining structures, slopes, tunnels, culverts, and pavements, in 

addition to bridges. It is published in two parts as follows: 

Part 1: Bridges 

Part 2: Retaining Structures, Slopes, Tunnels, Culverts, and Pavements 

Whereas Part 1 maintains the basic format of the retrofitting process described in the 1995 

report, major changes have been made in this revision to include current advances in earthquake 

engineering, field experience with retrofitting highway bridges, and the performance of bridges 

in recent earthquakes in California and elsewhere. It is the result of several years of research with 

contributions from a multidisciplinary team of researchers and practitioners. 

In particular, a performance-based retrofit philosophy is introduced similar to that used for the 

performance-based design of new buildings and bridges. Performance criteria are given for two 

earthquake ground motions with different return periods, 100 and 1000 years. A higher level of 

performance is required for the event with the shorter return period (the lower level earthquake 

ground motion) than for the longer return period (the upper level earthquake ground motion). 

Criteria are recommended according to bridge importance and anticipated service life, with more 

rigorous performance being required for important, relatively new bridges, and a lesser level for 

standard bridges nearing the end of their useful life. 

Minimum recommendations are made for screening, evaluation, and retrofitting according to an 

assigned Seismic Retrofit Category. Bridges in Category A need not be retrofitted, whereas those 

in Category B may be assessed without a detailed evaluation, provided certain requirements are 

satisfied. Bridges in Categories C and D require more rigorous evaluation and retrofitting, as 

required. Various retrofit strategies are described and a range of related retrofit measures 

explained in detail, including restrainers, seat extensions, column jackets, footing overlays, and 

soil remediation. 

This manual comprises 11 chapters and six appendices as follows: 

Chapter 1 gives a complete overview of the retrofitting process including the philosophy of 

performance-based retrofitting, the characterization of the seismic and geotechnical hazards, the 

assignment of the Seismic Retrofit Category, and summaries of recommended screening 

methods, evaluation tools, and retrofit strategies. Topics in this chapter are described in greater 

detail in the following 10 chapters. 

Chapters 2 and 3 describe the characterization of the seismic and geotechnical hazards. 

ii
Chapter 4 presents two screening and prioritization methods, with examples of each method. 

Chapters 5, 6, and 7 describe six evaluation methods, of increasing rigor, for the detailed 

assessment of demand and capacity, using either a component-by-component approach or a 

system approach for a complete bridge. 

Chapters 8, 9, 10, and 11 describe retrofitting measures for bearings, seats, columns, piers, cap 

beams, column-to-cap joints, abutments, and foundations. Remedial techniques for hazardous 

sites are also addressed. 

Appendices A through D provide supplementary material on conducting site-specific 

geotechnical investigations, the evaluation of geotechnical hazards, fragility curve theory, and 

the calculation of capacity/demand ratios for bridge components. 

Appendices E and F present two examples illustrating the application of the component 

capacity/demand method (Method C) to multi-span concrete and steel highway bridges, 

respectively. 

A glossary and lists of abbreviations, symbols, and references are also included. 

It is noted that this manual was developed while the U.S. Department of Transportation was 

transitioning to metric units. As a consequence, example problems are presented in SI units. 

Future editions may however use Customary U.S. units to reflect the current movement in many 

State DOTs back to customary units. 

iii
iv

SI* (MODERN METRIC) CONVERSION FACTORS 

APPROXIMATE CONVERSIONS TO SI UNITS

Symbol When You Know Multiply By To Find Symbol 

LENGTH 

in inches 25.4 millimeters mm 

ft feet 0.305 meters m 

yd yards 0.914 meters m 

mi miles 1.61 kilometers km 

AREA 

in2 square inches 645.2 square millimeters mm2

ft2 square feet 0.093 square meters m2

yd2 square yard 0.836 square meters m2

ac acres 0.405 hectares ha

mi2 square miles 2.59 square kilometers km2

VOLUME 

fl oz fluid ounces 29.57 milliliters mL 

gal gallons 3.785 liters L 

ft3 cubic feet 0.028 cubic meters m3 

yd3 cubic yards 0.765 cubic meters m3 

NOTE: volumes greater than 1000 L shall be shown in m3

MASS 

oz ounces 28.35 grams g

lb pounds 0.454 kilograms kg

T short tons (2000 lb) 0.907 megagrams (or "metric ton") Mg (or "t") 

TEMPERATURE (exact degrees) o

F Fahrenheit 5 (F-32)/9 Celsius oC 

or (F-32)/1.8 

ILLUMINATION 

fc foot-candles 10.76 lux lx

fl foot-Lamberts 3.426 candela/m2 cd/m2

FORCE and PRESSURE or STRESS 

lbf poundforce 4.45 newtons N 

lbf/in2 poundforce per square inch 6.89 kilopascals kPa 

APPROXIMATE CONVERSIONS FROM SI UNITS 

Symbol When You Know Multiply By To Find Symbol 

LENGTH

mm millimeters 0.039 inches in 

m meters 3.28 feet ft 

m meters 1.09 yards yd 

km kilometers 0.621 miles mi 

AREA mm2

 square millimeters 0.0016 square inches in2 

m2 square meters 10.764 square feet ft2 

m2 square meters 1.195 square yards yd2 

ha hectares 2.47 acres ac

km2 square kilometers 0.386 square miles mi2 

VOLUME 

mL milliliters 0.034 fluid ounces fl oz 

L liters 0.264 gallons gal 

m3 cubic meters 35.314 cubic feet ft3 

m3 cubic meters 1.307 cubic yards yd3 

MASS 

g grams 0.035 ounces oz

kg kilograms 2.202 pounds lb

Mg (or "t") megagrams (or "metric ton") 1.103 short tons (2000 lb) T 

TEMPERATURE (exact degrees) o

C Celsius 1.8C+32 Fahrenheit oF 

ILLUMINATION 

lx lux 0.0929 foot-candles fc 

cd/m2 candela/m2 0.2919 foot-Lamberts fl

FORCE and PRESSURE or STRESS 

N newtons 0.225 poundforce lbf 

kPa kilopascals 0.145 poundforce per square inch lbf/in2

*SI is the symbol for th International System of Units. Approp e riate rounding should be made to comply with Section 4 of ASTM E380. 

(Revised March 2003)
v

TABLE OF CONTENTS

PAGE 

CHAPTER 1: SEISMIC RETROFITTING OF HIGHWAY BRIDGES ............................ 1 

1.1 General.............................................................................................................................. 1 

1.2 Background....................................................................................................................... 3 

1.3 Philosophy......................................................................................................................... 5 

1.4 Seismic Performance Criteria ........................................................................................... 6 

 1.4.1 Performance Levels .............................................................................................. 6 

 1.4.2 Earthquake Ground Motion Levels....................................................................... 9 

 1.4.3 Bridge Importance .............................................................................................. 11 

 1.4.4 Anticipated Service Life ..................................................................................... 11 

 1.4.5 Selection of Performance Level.......................................................................... 13 

 1.4.6 Retrofitting Process for Dual Level Ground Motions......................................... 13 

 1.4.7 Exempt Bridges................................................................................................... 16 

1.5 Seismic Hazard Level ..................................................................................................... 16

 1.5.1 Ground Motion Representation........................................................................... 16 

 1.5.2 Geotechnical Factors........................................................................................... 17 

 1.5.2.1 Soil Amplification of Ground Motion ................................................ 17 

 1.5.2.2 Geotechnical Hazards ......................................................................... 17 

 1.5.3 Determination of Design Response Spectrum .................................................... 20 

 1.5.4 Determination of Seismic Hazard Level............................................................. 21 

1.6 Performance-Based Seismic Retrofit Categories............................................................ 21 

1.7 Retrofitting Process for the Lower Level Ground Motion.............................................. 23 

 1.7.1 Screening and Prioritization for the Lower Level Ground Motion..................... 25 

 1.7.2 Evaluation for the Lower Level Ground Motion ................................................ 25 

 1.7.3 Retrofitting for the Lower Level Ground Motion............................................... 26 

1.8 Retrofitting Process for the Upper Level Ground Motion.............................................. 26 

1.9 Minimum Requirements for Upper Level Ground Motion............................................. 28 

1.10 Screening and Prioritization for Upper Level Ground Motion....................................... 31 

 1.10.1 General................................................................................................................ 31 

 1.10.2 Factors to be Considered..................................................................................... 32 

 1.10.3 Seismic Vulnerability Rating Methods............................................................... 32 

 1.10.3.1 Minimum Screening Requirements .................................................... 34 

 1.10.3.2 Seismic Inventory of Bridges.............................................................. 34 

 1.10.4 Indices Method.................................................................................................... 34 

 1.10.5 Expected Damage Method.................................................................................. 37 

1.11 Methods of Evaluation for Upper Level Ground Motion ............................................... 38 

1.12 Retrofit Strategies for Upper Level Ground Motion....................................................... 42 

 1.12.1 General................................................................................................................ 42 

 1.12.2 Selecting a Retrofit Strategy ............................................................................... 42 

 1.12.2.1 Objective of Retrofitting and Acceptable Damage............................. 42 

 1.12.2.2 Cost Considerations of Seismic Retrofit............................................. 43 

 1.12.2.3 Other Considerations and Non-Seismic Issues................................... 44 
vi

TABLE OF CONTENTS (Continued)

PAGE 

 1.12.2.4 Identifying and Evaluating a Retrofit Strategy ................................... 45 

 1.12.2.5 Do-Nothing and Full-Replacement Options ....................................... 47 

 1.12.3 Seismic Retrofit Approaches .............................................................................. 47 

 1.12.3.1 Strengthening ...................................................................................... 48 

 1.12.3.2 Improvement of Displacement Capacity ............................................ 48 

 1.12.3.3 Force Limitation.................................................................................. 49 

 1.12.3.4 Response Modification ....................................................................... 49 

 1.12.3.5 Site Remediation by Ground Improvement ........................................ 50 

 1.12.3.6 Acceptance or Control of Damage to Specific Components .............. 50 

 1.12.3.7 Partial Replacement ............................................................................ 50 

1.13 Retrofit Measures for Upper Level Ground Motion....................................................... 51 

 1.13.1 General................................................................................................................ 51 

 1.13.2 Seismic Retrofit Matrix....................................................................................... 52 

CHAPTER 2. SEISMIC GROUND MOTION HAZARD................................................... 59 

2.1 General............................................................................................................................ 59 

2.2 Basic Procedures for Characterizing Horizontal Seismic Ground Motion..................... 63 

2.3 Determination of Spectral Accelerations, Ss and S1 ....................................................... 63 

 2.3.1 USGS Probabilistic Ground Motion Maps ......................................................... 64 

 2.3.2 USGS Probabilistic Ground Motion Data Files from CD-ROM........................ 68 

2.4 Site Classes and Site Factors........................................................................................... 77 

 2.4.1 Description of Site Classes and Site Factors....................................................... 77 

 2.4.2 Considerations for Site Class F Soils.................................................................. 78 

 2.4.3 Effects of Site Class Variation Along a Bridge .................................................. 83 

 2.4.4 Effect of Depth-of-Motion on Site Classification and Site Factors.................... 83 

2.5 Developing Horizontal Ground Motion Response Spectra Using National 

 Ground Motion Maps and Site Factors........................................................................... 83 

 2.5.1 Two-Point Procedure for Constructing Response Spectra.................................. 83 

 2.5.2 Multi-Point Procedure of Response Spectrum Construction.............................. 85 

 2.5.3 Modification of Elastic Spectral Demand for Higher or Lower Damping ......... 86 

 2.5.4 Obtaining Peak Ground Acceleration for Ground Failure Evaluations.............. 88 

2.6 Developing Site-Specific Response Spectra of Horizontal Ground Motions................. 88 

2.7 Developing Vertical Ground Motion Response Spectra................................................. 91 

2.8 Developing Acceleration Time Histories........................................................................ 92 

 2.8.1 General Requirements for Time Histories .......................................................... 92 

 2.8.2 Methods of Selecting and Developing Time Histories....................................... 94 

 2.8.3 Requirements for Compatibility of Time Histories with the 

 Design Response Spectrum ................................................................................ 96 

 2.8.4 Number of Time Histories .................................................................................. 97 

2.9 Spatial Variation of Ground Motions ............................................................................. 98 
vii

TABLE OF CONTENTS (Continued)

PAGE 

CHAPTER 3. GEOTECHNICAL HAZARDS ..................................................................... 99

3.1 General............................................................................................................................ 99 

3.2 Soil Liquefaction............................................................................................................. 99 

 3.2.1 Liquefaction Hazard Description........................................................................ 99 

 3.2.2 Initial Hazard Screening for Liquefaction ........................................................ 101 

 3.2.3 Earthquake Ground Motions for Liquefaction Analysis................................... 104 

 3.2.4 Summary of Procedures for Detailed Liquefaction Evaluations ...................... 104 

 3.2.4.1 Evaluation of the Potential for Liquefaction Using 

 Simplified Method ............................................................................ 104 

 3.2.4.2 Numerical Modeling Methods and Laboratory Cyclic 

 Testing for Evaluation of Potential for Liquefaction........................ 105 

 3.2.4.3 Evaluation of the Potential for Liquefaction-Induced 

 Flow Slides and Lateral Spreads....................................................... 105 

 3.2.4.4 Other Evaluation Procedures ............................................................ 106 

3.3 Soil Settlement.............................................................................................................. 106 

 3.3.1 Settlement Hazard Description ......................................................................... 106 

 3.3.2 Initial Hazard-Screening for Settlement ........................................................... 106 

 3.3.3 Summary of Procedures for Detailed Settlement Evaluations.......................... 107 

3.4 Surface Fault Rupture ................................................................................................... 107

 3.4.1 Surface Fault Rupture Hazard Description....................................................... 107 

 3.4.2 Initial Hazard Screening for Surface Fault Rupture ......................................... 107 

 3.4.3 Summary of Procedures for Detailed Surface Fault Rupture Evaluations ....... 108 

3.5 Flooding ........................................................................................................................ 108 

 3.5.1 Flooding Hazard Description............................................................................ 108 

 3.5.2 Initial Hazard Screening for Flooding .............................................................. 109 

 3.5.3 Summary of Detailed Evaluations for Flooding ............................................... 109 

CHAPTER 4. SEISMIC RATING METHODS FOR SCREENING AND 

PRIORITIZATION ................................................................................................................ 111

4.1 General.......................................................................................................................... 111 

4.2 Seismic Rating Method Using Indices.......................................................................... 111 

 4.2.1 Calculation of Bridge Rank .............................................................................. 113 

 4.2.1.1 Vulnerability Rating (V)................................................................... 113 

 4.2.1.1(a) Vulnerability Rating for Connections, Bearings, and Seat 

 Widths, V1 ........................................................................ 114 

 4.2.1.1(b) Vulnerability Rating for Columns, Abutments, and 

 Liquefaction Potential, V2 ................................................ 118 

 4.2.1.2 Seismic Hazard Rating (E)................................................................ 126 

 4.2.1.3 Examples ........................................................................................... 126 

 4.2.2 Calculation of Priority Index Based on Indices ................................................ 126 
viii

TABLE OF CONTENTS (Continued)

PAGE 

4.3 Seismic Rating Method Using Expected Damage........................................................ 127 

 4.3.1 Background....................................................................................................... 127 

 4.3.2 Database Requirements..................................................................................... 133 

 4.3.3 Fragility Curves for Reference Bridges ............................................................ 133 

 4.3.4 Scaling Reference Bridge Fragility Curves to Account for Skew and 

 Three-Dimensional Effects ............................................................................... 135 

 4.3.4.1 Definition of Parameters: K and K skew D3 .......................................... 135 

 4.3.4.2 Scaling Relations for Damage States 3, 4, and 5 .............................. 135 

 4.3.4.3 Scaling Relations for Damage State 2 (Slight Damage)................... 138 

 4.3.5 Economic Losses .............................................................................................. 139 

 4.3.5.1 Total Economic Losses..................................................................... 139 

 4.3.5.2 Direct Economic Losses ................................................................... 139 

 4.3.5.3 Indirect Economic Losses................................................................. 141 

 4.3.5.4 Examples........................................................................................... 141 

 4.3.6 Calculation of Bridge Rank Based on Expected Damage ................................ 142 

 4.3.7 Calculation of Priority Index Based on Expected Damage .............................. 142 

CHAPTER 5. EVALUATION METHODS FOR EXISTING BRIDGES ....................... 147 

5.1 General.......................................................................................................................... 147 

 5.1.1 Summary of Evaluation Methods ..................................................................... 147 

 5.1.2 Demand Analyses ............................................................................................. 149 

5.2 Method A: Connection Details and Seat Width Checks Only..................................... 150 

 5.2.1 Method A1 ........................................................................................................ 150 

 5.2.2 Method A2 ........................................................................................................ 151 

5.3 Method B: Component Capacity Checks..................................................................... 152 

 5.3.1 Procedure for Method B.................................................................................... 152 

 5.3.2 Restrictions on Use of Method B...................................................................... 153 

5.4 Method C: Component Capacity/Demand Method ..................................................... 154 

 5.4.1 Approach ........................................................................................................... 154 

 5.4.2 Selection of Elastic Analysis Method............................................................... 157 

 5.4.2.1 Uniform Load Method ...................................................................... 157 

 5.4.2.2 Multi-Mode Spectral Analysis Method ............................................ 159 

 5.4.2.3 Elastic Time History Method............................................................ 160 

 5.4.3 Procedure for Method C.................................................................................... 160 

 5.4.4 Restrictions on Use of Method C...................................................................... 161 

 5.4.5 Examples........................................................................................................... 161 

5.5 Method D1: Structure Capacity/Demand Method ....................................................... 161 

 5.5.1 Approach ........................................................................................................... 161 

 5.5.2 Bridge Capacity ................................................................................................ 162 

 5.5.2.1 General.............................................................................................. 162 

 5.5.2.2 Calculation of Bridge Capacity Curve.............................................. 163 
ix

TABLE OF CONTENTS (Continued)

PAGE 

 5.5.3. Earthquake Demand.......................................................................................... 166 

 5.5.4 Capacity/Demand Spectrum ............................................................................. 168 

 5.5.4.1 Calculation of Bridge Capacity/Demand (C/D) Ratios .................... 168 

 5.5.4.2 Calculation of Bridge Response........................................................ 170 

 5.5.5 Procedure for Method D1 ................................................................................. 171 

 5.5.6 Restrictions on Use of Method D1.................................................................... 174 

5.6 Method D2: Structure Capacity/Demand (Pushover) Method..................................... 175 

 5.6.1 Approach ........................................................................................................... 175 

 5.6.2 Displacement Capacity Evaluation................................................................... 175 

 5.6.3 Demands ........................................................................................................... 176 

 5.6.4 Procedure for Method D2 ................................................................................. 176 

 5.6.5 Restrictions on Method D2 ............................................................................... 177 

5.7 Method E: Nonlinear Dynamic Procedure................................................................... 177 

 5.7.1 Approach ........................................................................................................... 177 

 5.7.2 Procedure for Method E.................................................................................... 178 

 5.7.3 Restrictions on Method E.................................................................................. 178 

CHAPTER 6. GEOTECHNICAL MODELING AND CAPACITY ASSESSMENT..... 179 

6.1 General.......................................................................................................................... 179 

6.2 Foundation Modeling.................................................................................................... 179 

 6.2.1 Soil-Foundation-Structure Interaction .............................................................. 179 

 6.2.1.1 Shallow Footings .............................................................................. 180 

 6.2.1.2 Piles................................................................................................... 181 

 6.2.2 Stiffness and Capacity of Foundation Components.......................................... 182 

 6.2.2.1 Shallow Bearing Footing Foundations.............................................. 183 

 6.2.2.1(a) Stiffness Parameters...................................................... 184 

 6.2.2.1(b) Capacity Parameters...................................................... 185 

 6.2.2.2 Pile Footing Foundations .................................................................. 191 

 6.2.2.2(a) Pile Cap – Lateral Stiffness and Capacity..................... 191 

 6.2.2.2(b) Pile-Head Stiffness – Lateral Loading.......................... 191 

 6.2.2.2(c) Pile-Head Stiffness – Axial Loading ............................ 203 

 6.2.2.2(d) Pile Group Analysis ...................................................... 205 

 6.2.2.2(e) Pile Group Analyses – Moment-Rotation Capacity...... 206 

 6.2.2.3 Drilled Shafts .................................................................................... 209 

 6.2.2.4 Abutments......................................................................................... 211 

 6.2.2.4(a) Abutment Capacity – Longitudinal Direction............... 211 

 6.2.2.4(b) Calculation of Best-Estimate Passive Force, Pp ........... 214 

 6.2.2.4(c) Abutment Stiffness – Longitudinal Direction............... 214 

6.3 Ground Displacement Demands on Foundations ......................................................... 217 

 6.3.1 Sources of Demand........................................................................................... 217 

 6.3.1.1 Earthquake-Induced Settlement........................................................ 217 
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TABLE OF CONTENTS (Continued)

PAGE 

 6.3.1.2 Liquefaction-Induced Lateral Spreads.............................................. 217 

 6.3.1.3 Slope Stability................................................................................... 220 

CHAPTER 7. STRUCTURAL MODELING AND CAPACITY ASSESSMENT .......... 221

7.1 General.......................................................................................................................... 221 

7.2 Load Path for Lateral Forces......................................................................................... 221 

7.3 Modeling Recommendations for Bridge Structures ..................................................... 224 

 7.3.1 Distribution of Mass ......................................................................................... 225 

 7.3.2 Distribution of Stiffness and Strength............................................................... 225 

 7.3.2.1 Substructures..................................................................................... 226 

 7.3.2.2 Superstructures.................................................................................. 227 

 7.3.3 In-Span Hinges.................................................................................................. 228 

 7.3.4 Damping............................................................................................................ 228 

 7.3.5 Permanent Ground Movement.......................................................................... 229 

7.4 Combination of Seismic Force Effects ......................................................................... 229 

 7.4.1 Seismic Loading in One Direction.................................................................... 229 

 7.4.2 Seismic Loading in Two or Three Orthogonal Directions................................ 230 

 7.4.3 Combination of Response Quantities for Biaxial Design................................. 231 

 7.4.4 Vertical Acceleration Effects............................................................................ 232 

7.5 Member Strengths......................................................................................................... 233 

7.6 Member Actions in Piers Using Capacity Principles.................................................... 234 

 7.6.1 Single Column Piers ......................................................................................... 234 

 7.6.2 Multi-Column Piers .......................................................................................... 235 

7.7 Strength Capacity of Bridge Members ......................................................................... 236 

 7.7.1 Flexural Strength of Reinforced Concrete Columns and Beams...................... 237 

 7.7.1.1 Expected Flexural Strength............................................................... 237 

 7.7.1.2 Flexural Overstrength Capacity ........................................................ 238 

 7.7.1.3 Flexural Strength of Columns with Lap-Splices in Plastic 

 Hinge Zones...................................................................................... 241 

 7.7.2 Shear Strength of Reinforced Concrete Columns and Beams .......................... 242 

 7.7.2.1 Initial Shear Strength, Vi................................................................... 242 

 7.7.2.2 Final Shear Strength, Vf.................................................................... 243 

 7.7.3 Shear Strength of Beam-Column Joints............................................................ 244 

 7.7.3.1 Maximum Beam-Column Joint Strength, Vji.................................... 244 

 7.7.3.2 Cracked Beam-Column Joint Strength, Vjf....................................... 244 

7.8 Deformation Capacity of Bridge Members................................................................... 245 

 7.8.1 Plastic Curvatures and Hinge Rotations in a Cantilever Beam......................... 245 

 7.8.1.1 Deflections and Plastic Curvature, φp .............................................. 245 

 7.8.1.2 Plastic Hinge Rotation, θp ................................................................. 247 
xi

TABLE OF CONTENTS (Continued)

PAGE 

 7.8.2 Characterization of Deformation-Based Limit States....................................... 247 

 7.8.2.1 Compression Failure of Unconfined Concrete ................................. 249 

 7.8.2.2 Compression Failure of Confined Concrete ..................................... 249 

 7.8.2.3 Buckling of Longitudinal Bars.......................................................... 250 

 7.8.2.4 Fracture of the Longitudinal Reinforcement..................................... 250 

 7.8.2.5 Low Cycle Fatigue of Longitudinal Reinforcement ......................... 250 

 7.8.2.6 Failure in the Lap-splice Zone .......................................................... 251 

 7.8.2.6(a) ‘Long’ or Confined Lap-splice, llap > ls......................... 251 

 7.8.2.6(b) ‘Short” and Unconfined Lap-splice, llap ≤ ls ................. 252 

 7.8.2.7 Shear Failure ..................................................................................... 252 

 7.8.2.7(a) Brittle Shear, Vi ≤ Vm ................................................... 253 

 7.8.2.7(b) Semi-ductile Shear, Vf < Vm < Vi ................................. 253 

 7.8.2.8 Joint or Connection Failure............................................................... 253 

 7.8.2.8(a) Weak Joint and Strong Column, Vji ≤ Vjh..................... 254 

 7.8.2.8(b) Semi-ductile Shear, Vjf < Vjh < Vji................................ 254 

 7.8.3 Neutral Axis Depth in Columns and Beams..................................................... 254 

CHAPTER 8. RETROFIT MEASURES FOR SUPERSTRUCTURES, BEARINGS, 

AND SEATS ............................................................................................................................ 257

8.1 General.......................................................................................................................... 257 

8.2 Bridge Decks and Girders............................................................................................. 257 

 8.2.1 Lateral Load Path Enhancement ....................................................................... 257 

 8.2.1.1 Strengthening of Deck to Girder Connection ................................... 257 

 8.2.1.2 Diaphragm Strengthening or Stiffening............................................ 258 

 8.2.1.3 Energy Dissipating Ductile Diaphragms .......................................... 259 

 8.2.1.4 Girder Strengthening......................................................................... 263 

 8.2.2 Providing Longitudinal Continuity................................................................... 264 

 8.2.2.1 Web and Flange Plates...................................................................... 264 

 8.2.2.2 Superstructure Joint Strengthening................................................... 265 

 8.2.3 Reduction of Dead Load ................................................................................... 267 

 8.2.4 Strengthening of Continuous Superstructures .................................................. 267 

8.3 Bearings, Anchorages and Pedestals............................................................................. 268 

 8.3.1 Strengthening of Existing Bearings .................................................................. 270 

 8.3.1.1 Expansion Bearings .......................................................................... 270 

 8.3.1.1(a) Sole Plate to Girder....................................................... 270 

 8.3.1.1(b) Bearing to Masonry Plate.............................................. 271 

 8.3.1.1(c) Masonry Plate to Substructure...................................... 275 

 8.3.1.2 Fixed Bearings .................................................................................. 276 

 8.3.1.2(a) Sole Plate to Girder....................................................... 276 
xii

TABLE OF CONTENTS (Continued)

PAGE 

 8.3.1.2(b) Masonry Plate to Substructure...................................... 276 

 8.3.2 Bearing Replacement........................................................................................ 276 

 8.3.2.1 Conventional Bearings.................................................. 276 

 8.3.2.2 Replacement with Seismic Isolation Bearings.............. 281 

 8.3.3 Strengthening of Superstructure to Substructure Connections ......................... 285 

8.4 Expansion Joint Retrofit ............................................................................................... 286 

 8.4.1 Bearing Seat Extensions ................................................................................... 286 

 8.4.1.1 Concrete Seat Extensions and Catcher Blocks ................................. 286 

 8.4.1.2 Pipe Extenders .................................................................................. 290 

 8.4.2 Restrainers......................................................................................................... 291 

 8.4.2.1 Longitudinal Joint Restrainers .......................................................... 292 

 8.4.2.1(a) Restrainer Cables .......................................................... 293 

 8.4.2.1(b) High Strength Bar Restrainers ...................................... 295 

 8.4.2.1(c) Bumper Blocks.............................................................. 298 

 8.4.2.1(d) Anchorage of Restrainers.............................................. 299 

 8.4.2.1(e) Design and Construction Issues.................................... 299 

 8.4.2.1(f) Evaluation of the Need for Restrainers 

 for Simply Supported Spans ......................................... 306 

 8.4.2.1(g) Restrainer Design Methods........................................... 310 

 8.4.2.2 Transverse Restrainers...................................................................... 316 

 8.4.2.2(a) General.......................................................................... 316 

 8.4.2.2(b) Shear Keys .................................................................... 316 

 8.4.2.2(c) Keeper Brackets............................................................ 317 

 8.4.2.2(d) Steel Pipe Restrainers ................................................... 317 

 8.4.2.2(e) Design Forces................................................................ 332 

 8.4.2.3 Vertical Motion Restrainers.............................................................. 332 

 8.4.3 Energy Dissipation Devices.............................................................................. 333 

 8.4.4 Shock Transmission Units ................................................................................ 333 

CHAPTER 9. RETROFIT MEASURES FOR SUBSTRUCTURE COMPONENTS .... 337

9.1 General.......................................................................................................................... 337 

9.2 Retrofit Measures for Piers ........................................................................................... 338 

 9.2.1 Reinforced Concrete Columns.......................................................................... 339 

 9.2.1.1 Column Replacement........................................................................ 339 

 9.2.1.1(a) Total Replacement ........................................................ 340 

 9.2.1.1(b) Partial Column Replacement ........................................ 340 

 9.2.1.1(c) Supplemental Columns ................................................. 345 

 9.2.1.2 Column Flexural Strengthening........................................................ 347 

 9.2.1.2(a) Concrete Overlays......................................................... 347 

 9.2.1.2(b) Added Reinforcement in Conjunction with 

 Steel Shell ..................................................................... 348 
xiii

TABLE OF CONTENTS (Continued)

PAGE 

 9.2.1.2(c) Composite Steel Shell................................................... 348 

 9.2.1.3 Column Ductility Improvement and Shear Strengthening ............... 352 

 9.2.1.3(a) Steel Jacketing .............................................................. 353 

 9.2.1.3(b) Fiber Composite Jacketing............................................ 364 

 9.2.1.3(c) External Prestressing Steel............................................ 371 

 9.2.1.3(d) Concrete Jacketing........................................................ 375 

 9.2.1.4 Supplemental Column Shear Walls .................................................. 378 

 9.2.1.5 Preservation of the Vertical Load Capacity of Columns .................. 379 

 9.2.1.6 Limitation of Column Forces............................................................ 380 

 9.2.1.6(a) Isolation Bearings ......................................................... 380 

 9.2.1.6(b) Flexural Strength Reduction ......................................... 381 

 9.2.2 Steel Columns, Frames and Compression Members ........................................ 382 

 9.2.2.1 Braced Frames .................................................................................. 382 

 9.2.2.2 Built-up Compression Members....................................................... 383 

 9.2.3 Concrete Wall Piers .......................................................................................... 387 

9.3 Retrofit Measures for Cap Beams and Column-to-Cap Beam Joints ........................... 388 

 9.3.1 Pier Cap Replacement....................................................................................... 390 

 9.3.1.1 Partial Replacement at a Joint........................................................... 390 

 9.3.1.2 Total Replacement ............................................................................ 390 

 9.3.2 Pier Cap Strengthening ..................................................................................... 390 

 9.3.3 Reduction of Pier Cap Forces ........................................................................... 394 

 9.3.4 Strengthening of Column and Beam Joints ...................................................... 395 

 9.3.5 Supergirders ...................................................................................................... 397 

CHAPTER 10. RETROFIT MEASURES FOR ABUTMENTS, FOOTINGS, AND 

FOUNDATIONS..................................................................................................................... 401

10.1 General.......................................................................................................................... 401 

10.2 Retrofit Measures for Abutments.................................................................................. 401 

 10.2.1 Approach Slabs ................................................................................................. 403 

 10.2.2 Anchor Slabs..................................................................................................... 405 

 10.2.3 Diaphragm Walls .............................................................................................. 408 

 10.2.4 Transverse Abutment Shear Keys..................................................................... 408 

 10.2.5 Transverse Abutment Anchors ......................................................................... 408 

 10.2.5.1 Soil Shear Keys................................................................................. 409 

 10.2.5.2 Wingwall Strengthening ................................................................... 411 

 10.2.5.3 Cast-in-Drilled Hole Piles................................................................. 414 

 10.2.6 Soil and Gravity Anchors.................................................................................. 414 

10.3 Retrofit Measures for Footings ..................................................................................... 418 

 10.3.1 Footing Replacement ........................................................................................ 418 

 10.3.2 Strengthening of Footings................................................................................. 418 

 10.3.3 Retrofit Design of Footings .............................................................................. 420 
xiv

TABLE OF CONTENTS (Continued)

PAGE 

 10.3.4 Limiting Forces Transmitted to the Footings.................................................... 433 

10.4 Retrofit Measures for Piles and Pile-to-Footing Connections ...................................... 433 

 10.4.1 Pile Type Considerations (New Piles) .............................................................. 435 

 10.4.2 Pile Tie-Downs ................................................................................................. 435 

CHAPTER 11. RETROFIT MEASURES FOR BRIDGES ON HAZARDOUS SITES 437

11.1 General.......................................................................................................................... 437 

11.2 Bridges Across or Near Active Faults .......................................................................... 437 

11.3 Bridges on or Near Unstable Slopes ............................................................................. 438 

11.4 Bridges on Liquefiable Soils......................................................................................... 438 

 11.4.1 Structural Retrofitting of Piled Foundations..................................................... 440 

 11.4.1.1 Simplified Analysis........................................................................... 443 

 11.4.2 Site Remediation using Ground Improvement ................................................. 448 

 11.4.3 Superstructure Improvements ........................................................................... 450 

APPENDIX A. GUIDELINES FOR CONDUCTING SITE-SPECIFIC GEOTECHNICAL 

INVESTIGATIONS AND DYNAMIC SITE RESPONSE ANALYSES........................... 453

A.1 Site-Specific Geotechnical Investigation...................................................................... 453 

A.2 Dynamic Site Response Analysis ................................................................................. 453 

 A.2.1 Modeling the Soil Profile.................................................................................. 453 

 A.2.2 Selecting Input Rock Motion............................................................................ 454 

 A.2.3 Site Response Analysis and Results Interpretation........................................... 454 

APPENDIX B. EVALUATION OF GEOTECHNICAL HAZARDS............................... 455

B.1 General.......................................................................................................................... 455 

B.2 Evaluation of Soil Liquefaction Hazard ....................................................................... 455 

 B.2.1 Field Exploration for Liquefaction Hazard Assessment................................... 455 

 B.2.1.1 Location of Liquefiable Soils ........................................................... 455 

 B.2.1.2 Location of Groundwater Level........................................................ 456 

 B.2.1.3 Depth of Liquefaction....................................................................... 456 

 B.2.1.4 Field Exploration Methods ............................................................... 457 

 B.2.1.4(a) SPT Method .................................................................. 457 

 B.2.1.4(b) CPT Method.................................................................. 457 

 B.2.2 Evaluation of Liquefaction Potential ................................................................ 458 

 B.2.2.1 Simplified Procedure ........................................................................ 459 

 B.2.2.1(a) Cyclic Resistance Ratio, CRR ...................................... 459 

 B.2.2.1(b) Cyclic Stress Ratio, CSR .............................................. 462 

 B.2.2.1(c) Liquefaction Potential................................................... 463 

 B.2.2.2 Numerical Modeling Methods.......................................................... 463 
xv

TABLE OF CONTENTS (Continued)

PAGE 

 B.2.3 Liquefaction Hazards Assessment .................................................................... 464 

 B.2.3.1 Flow Failures .................................................................................... 465 

 B.2.3.2 Lateral Spreading.............................................................................. 467 

 B.2.3.2.1 Youd et al. Emprical Approach .................................... 468 

 B.2.3.2.2 Newmark Time History Analyses................................. 468 

 B.2.3.2.3 Simplified Newmark Charts ......................................... 469 

 B.2.3.2.4 Numerical Modeling ..................................................... 471 

B.3 Evaluation of Soil Settlement Hazard........................................................................... 471 

B.4 Evaluation of Surface Fault Rupture Hazard................................................................ 474 

 B.4.1 Fault Location ................................................................................................... 476 

 B.4.1.1 Interpretation of Aerial Photographs ................................................ 476 

 B.4.1.2 Contacting Knowledgeable Geologists............................................. 476 

 B.4.1.3 Ground Reconnaissance of Site and Vicinity ................................... 476 

 B.4.1.4 Surface Exploration .......................................................................... 476 

 B.4.2 Fault Activity .................................................................................................... 477 

 B.4.2.1 Assess Fault Relationship to Young Deposits/Surfaces ................... 477 

 B.4.2.2 Evaluate Local Seismicity ................................................................ 477 

 B.4.2.3 Evaluate Structural Relationships..................................................... 477 

 B.4.3 Fault Rupture Characteristics............................................................................ 477 

B.5 Evaluation of Flooding Hazard..................................................................................... 479 

APPENDIX C. FRAGILITY CURVE THEORY............................................................... 481 

APPENDIX D. CAPACITY/DEMAND RATIOS FOR BRIDGE MEMBERS AND 

COMPONENTS...................................................................................................................... 485

D.1 General.......................................................................................................................... 485 

D.2 Minimum Bearing or Restrainer Force Demands......................................................... 485 

D.3 Minimum Support Lengths........................................................................................... 486 

D.4 Capacity/Demand Ratios for Expansion Joints and Bearings....................................... 488 

 D.4.1 General.............................................................................................................. 488

 D.4.2 Displacement Capacity/Demand Ratios ........................................................... 490 

 D.4.3 Force Capacity/Demand Ratios ........................................................................ 491 

D.5 Capacity/Demand Ratios for Reinforced Concrete Columns, Walls and Footings...... 491 

 D.5.1 Anchorage of Longitudinal Reinforcement ...................................................... 496 

 D.5.2 Splices in Longitudinal Reinforcement ............................................................ 501 

 D.5.3 Column Shear.................................................................................................... 505 

 D.5.4 Transverse Confinement Reinforcement .......................................................... 511 

 D.5.5 Footing Rotation and/or Yielding ..................................................................... 513 

D.6 Capacity/Demand Ratios for Abutments ...................................................................... 517 

D.7 Capacity/Demand Ratios for Liquefaction Induced Foundation Failure...................... 518 

D.8 Summary....................................................................................................................... 519 
xvi

TABLE OF CONTENTS (Continued)

PAGE 

APPENDIX E. EXAMPLE PROBLEM 5.1: COMPONENT CAPACITY/DEMAND 

EVALUATION OF A 4-SPAN, REINFORCED CONCRETE BOX GIRDER 

BRIDGE ........................................................................................................................... 521

E.1 Introduction................................................................................................................... 521 

E.2 Description of the Example Bridge............................................................................... 521 

E.3 Seismic Rating .............................................................................................................. 521 

 E.3.1 Vulnerability Rating, V..................................................................................... 524 

 E.3.2 Seismic Hazard Rating, E ................................................................................. 526 

 E.3.3 Bridge Rank, R.................................................................................................. 526 

 E.3.4 Priority Index .................................................................................................... 526 

E.4 Detailed Evaluation....................................................................................................... 527 

 E.4.1 Capacity/Demand Ratios – Existing Bridge ..................................................... 527 

 E.4.2 Identification and Assessment of Potential Retrofit Measures......................... 548 

APPENDIX F. EXAMPLE PROBLEM 5.2: COMPONENT CAPACITY/DEMAND 

EVALUATION OF A MULTISPAN SIMPLY-SUPPORTED STEEL GIRDER 

BRIDGE................................................................................................................................... 557

F.1 Introduction................................................................................................................... 557 

F.2 Description of the Example Bridge............................................................................... 557 

F.3 Bridge Evaluation ......................................................................................................... 559 

 F.3.1 Seismic Retrofit Categories .............................................................................. 559 

 F.3.2 Components for Capacity/Demand Ratio Evaluation....................................... 559 

 F.3.3 Analysis Procedure ........................................................................................... 559 

 F.3.4 Uniform Load Method for Case 2..................................................................... 560 

 F.3.4.1 Structural Data – Pier 2....................................................................... 560 

 F.3.4.2 Longitudinal Earthquake Loading ...................................................... 560 

 F.3.4.3 Transverse Earthquake Loading ......................................................... 561 

F.4 Capacity/Demand Ratio for Bearings for Case 1.......................................................... 561 

 F.4.1 Displacement Capacity/Demand Ratio (Case 1)............................................... 561 

 F.4.2 Force Capacity/Demand Ratio (Case 1) ........................................................... 562 

F.5 Potential for Liquefaction in Case 1 ............................................................................. 562 

F.6 Capacity/Demand Ratio for Bearings for Case 2.......................................................... 562 

 F.4.6.1 Displacement Capacity/Demand Ratio (Case 2)............................................... 563 

 F.4.6.2 Force Capacity/Demand Ratio (Case 2) ........................................................... 564 

F.7 Capacity/Demand Ratios for Pier 2 for Case 2............................................................. 564 

F.8 Potential for Liquefaction in Case 2 ............................................................................. 567 

GLOSSARY ............................................................................................................. 577 

REFERENCES........................................................................................................................ 583 
xvii

LIST OF FIGURES 

FIGURE PAGE 

1-1 Overview of the retrofitting process for highway bridges................................................ 2 

1-2 Collapse of the link span at Tower E9 of the San Francisco Oakland Bay 

 bridge due to inadequate seat lengths and anchor bolts.................................................... 8 

1-3 Collapse of the two-level Cypress Viaduct on I-880 in Oakland due to brittle shear 

 failure at the connection between the upper and lower levels of the viaduct ................... 8 

1-4 Collapse of end spans in the Shi Wei bridge in Taichung, due to ground 

 failure and nearby fault rupture......................................................................................... 8 

1-5 Diagonal shear crack in lightly reinforced concrete pier of the Wu Shu 

 bridge in Taichung ............................................................................................................ 8 

1-6 Conceptual relationship between relative effort, increasing hazard and 

 performance criteria implied in this manual ..................................................................... 9 

1-7 Retrofit process for dual level earthquake ground motions............................................ 15 

1-8 Construction of the seismic design response spectrum................................................... 20 

1-9 Determination of seismic retrofit category ..................................................................... 23 

1-10 Seismic retrofitting process for highway bridges subject to upper level ground 

 motion ............................................................................................................................. 27 

1-11 Relative effort to retrofit an (a) ‘standard’ and (b) ‘essential’ bridge with varying 

 service life and hazard level............................................................................................ 29 

1-12 Screening and prioritization process............................................................................... 31 

1-13 Evaluation methods for existing bridges showing relationship between demand 

 analysis and capacity assessment.................................................................................... 41 

1-14 Identification and evaluation of a retrofit strategy.......................................................... 46 

1-15 Selected retrofit measures............................................................................................... 53 

2-1 Variation in short- and long-period spectral acceleration with ground motion 

 return period.................................................................................................................... 61 

2-2 Seismic hazard maps for the conterminous United States.............................................. 65 

2-3 Calculation of short- and long-period spectral accelerations for 100- and 1000-year 

 ground motion return periods.......................................................................................... 73 

2-4 Design response spectrum (five percent damping) construction using two-point 

 procedure......................................................................................................................... 85 

2-5 Comparison of two-point method of five percent damped response spectrum 

 construction with USGS multi-point mapping results for two percent probability 

 of exceedance in 50 years ............................................................................................... 87 

2-6 Time histories and horizontal response spectra (five percent damping) for the 

 fault strike-normal and fault strike-parallel components of ground motion 

 for the Rinaldi recording obtained 4.5 mi (7.5 km) from the fault rupture 

 during the 1994 Northridge, California earthquake........................................................ 90 

2-7 Vertical/horizontal spectral ratios vs. period .................................................................. 92 

2-8 Hazard deaggregation for 0.2 second response spectral acceleration (Ss) for 

 ground motion with three percent probability of exceedance in 75 years 

 (2500-year return period)................................................................................................ 95 
xviii

LIST OF FIGURES (Continued) 

FIGURE PAGE 

4-1 Seismic rating method using indices............................................................................. 112 

4-2 Calculation of bridge vulnerability, V .......................................................................... 114 

4-3 Seismically vulnerable bearings ................................................................................... 115 

4-4 Calculation of vulnerability rating for connections, bearings and seat widths (V1) ..... 119 

4-5 Seismic rating method using expected damage ............................................................ 128 

5-1 Capacity curve .............................................................................................................. 163 

5-2 Idealized capacity.......................................................................................................... 164 

5-3 Demand spectrum ......................................................................................................... 167

5-4 Capacity/demand spectra .............................................................................................. 169 

6-1 Uncoupled elasto-plastic spring model for rigid footings............................................. 183 

6-2 Uncoupled winkler spring model.................................................................................. 183 

6-3 Upper and lower bound approach to define stiffness and capacity .............................. 184 

6-4 Idealized concentration of stress at edge of rigid footings subjected 

 to overturning moment.................................................................................................. 188 

6-5 Rocking of shear wall on strip footing.......................................................................... 189 

6-6 Method for passive pressure capacity of footing or pile cap ........................................ 190 

6-7 Effect of boundary conditions on pile stiffness ............................................................ 193 

6-8 Lateral pile-head stiffness (fixed-head condition) ........................................................ 194 

6-9 Rotational pile head stiffness........................................................................................ 195 

6-10 Cross-coupling pile-head stiffness................................................................................ 196 

6-11 Lateral pile-head stiffness (free-head condition) .......................................................... 197 

6-12 Recommended coefficient of variation in subgrade modulus (f) with depth of sand... 198 

6-13 Recommended coefficient of variation in subgrade modulus (f) with depth of clay.... 199 

6-14 Lateral embedded pile-head stiffness (fixed-head condition)....................................... 200 

6-15 Embedded pile-head rotational stiffness....................................................................... 201 

6-16 Embedded pile cross-coupling pile-head stiffness........................................................ 202 

6-17 Axial load-displacement graphical solution.................................................................. 204 

6-18 Modeling approaches for deep foundations.................................................................. 206 

6-19 Pile footing configuration for moment-rotation study.................................................. 207 

6-20 Axial load-displacement curve for single pile ............................................................. 208 

6-21 Cyclic moment-rotation and settlement-rotation solutions........................................... 209 

6-22 Abutment types ............................................................................................................. 212 

6-23 Design passive pressure zone........................................................................................ 213 

6-24 Characterization of abutment capacity and stiffness .................................................... 214 

6-25 Trial wedge method for determining critical earthquake-induced active forces .......... 216 

6-26 Sketch of Magsaysay bridge showing earthquake damage........................................... 218 

6-27 Landing Road bridge lateral spread .............................................................................. 219 

6-28 Site and damage characteristics for a precast concrete pile subjected to a 

 lateral spread in the Kobe earthquake........................................................................... 220 
xix

LIST OF FIGURES (Continued)

FIGURE PAGE 

7-1 Evaluation methods for existing bridges showing relationship between demand 

 analysis and capacity assessment.................................................................................. 222 

7-2 Confined strength ratio (K) for reinforced concrete members ..................................... 241 

8-1 Deck to girder connection retrofit................................................................................. 258 

8-2 Steel girder diaphragm retrofit...................................................................................... 260 

8-3 Diaphragm free body diagram ...................................................................................... 261 

8-4 Ductile end diaphragms ................................................................................................ 262 

8-5 Girder bracing retrofit................................................................................................... 264 

8-6 Bearing stiffener retrofit ............................................................................................... 265 

8-7 Methods for accommodating girder misalignment using web splice plates................. 266 

8-8 Girders made continuous for live load.......................................................................... 267 

8-9 Retrofit of concrete box girder for flexural capacity .................................................... 268 

8-10 Steel bearing nomenclature........................................................................................... 269 

8-11 Bearing sole plate to girder connection retrofit ............................................................ 271 

8-12 Bearing sole plate to girder with catcher bar retrofit .................................................... 272 

8-13 Expansion bearing retrofit............................................................................................. 273

8-14 Retrofit to increase bearing transverse capacity ........................................................... 274 

8-15 Retrofit to increase anchor plate capacity..................................................................... 275 

8-16 Bearing retrofitting by encasement in steel shell.......................................................... 277 

8-17 Bearing retrofitting by encasement in concrete ............................................................ 278 

8-18 Bearing replacement retrofit using steel pedestal ......................................................... 279 

8-19 Bearing replacement using concrete pedestal............................................................... 280 

8-20 Lead-filled elastomeric isolation bearing...................................................................... 282 

8-21 Friction-pendulum isolation bearing............................................................................. 282 

8-22 Eradiquake isolation bearing......................................................................................... 283 

8-23 Knock-off device in backwall of seat-type abutment ................................................... 285 

8-24 Freyssinet hinge ............................................................................................................ 286 

8-25 Pipe shear key ............................................................................................................... 287 

8-26 Concrete hinge retrofit .................................................................................................. 288 

8-27 Abutment seat extension............................................................................................... 289 

8-28 Pipe seat extenders........................................................................................................ 291 

8-29 Caltrans cable restrainer unit ........................................................................................ 294 

8-30 Caltrans tests of restrainer cable ................................................................................... 296 

8-31 Multi-cable restrainer assembly.................................................................................... 297 

8-32 Bumper block retrofit.................................................................................................... 298 

8-33 Restrainer anchorage schemes ...................................................................................... 300 

8-34 Undesirable restrainer detail......................................................................................... 301 

8-35 Resistance of concrete wall to punching shear ............................................................. 302 

8-36 Restrainer orientation for transversely rigid supports................................................... 303 

8-37 Restrainer orientation for transversely flexible supports.............................................. 303 
xx

LIST OF FIGURES (Continued)

FIGURE PAGE 

8-38 Restrainer at pier with a positive tie to pier .................................................................. 303 

8-39 Required clearances for coring ..................................................................................... 304 

8-40 Available seat width...................................................................................................... 305 

8-41 SDOF model for end-span fixed bearing at abutment .................................................. 307 

8-42 SDOF model for an end-span fixed bearing at pier ...................................................... 308 

8-43 Force displacement relationship for abutment.............................................................. 308 

8-44 2DOF model for the iterative method........................................................................... 315 

8-45 Transverse shear keys ................................................................................................... 317 

8-46 Bearing keeper bracket retrofit ..................................................................................... 318 

8-47 Steel pipe restrainers..................................................................................................... 318 

8-48 Vertical hold down retrofit............................................................................................ 332

8-49 Energy dissipation devices............................................................................................ 335 

8-50 Friction dampers ........................................................................................................... 336 

9-1 Column replacement on San Francisco viaducts .......................................................... 343 

9-2 Column replacement using existing column reinforcement ......................................... 344 

9-3 Column anchorage using headed reinforcement........................................................... 345 

9-4 Replaceable plastic hinge with fuse bars ...................................................................... 346 

9-5 Supplemental column retrofit ....................................................................................... 347 

9-6 Column strengthening by concrete overlay .................................................................. 349 

9-7 Column strengthening by adding reinforcement within steel shell .............................. 350 

9-8 Column strengthening by anchoring steel shell............................................................ 351 

9-9 Typical steel shell retrofit of round column.................................................................. 354 

9-10 Typical steel shell retrofit of rectangular column......................................................... 355 

9-11 Free body diagram of column retrofitting with a steel shell......................................... 356 

9-12 Geometry of an elliptically shaped jacket..................................................................... 360 

9-13 Free body diagram of column retrofitting with a composite shell................................ 367 

9-14 Anchorages for prestress strand retrofit........................................................................ 371 

9-15 Prestressing retrofit by wedging between column and strand ...................................... 372 

9-16 Semi-circular reinforcing bars with couplers................................................................ 373 

9-17 Free body diagram of column retrofitting with external prestressing........................... 374 

9-18 Infill shear wall in multi-column bents......................................................................... 379 

9-19 Steel shell retrofit for vertical capacity preservation .................................................... 380 

9-20 Sierra Point overhead, US 101, near San Francisco, California ................................... 381 

9-21 Column retrofit by flexural strength reduction ............................................................. 382 

9-22 Retrofit of X-braced steel bent...................................................................................... 384 

9-23 Richmond-San Rafael bridge bent retrofit.................................................................... 385 

9-24 Built-up steel columns stiffened by plates.................................................................... 386 

9-25 Built-up steel column retrofitted with steel shell.......................................................... 387 

9-26 Retrofit for wall pier with starter bars........................................................................... 388 

9-27 Typical cap beam configurations for boxgirder bridges............................................... 389 
xxi

LIST OF FIGURES (Continued)

FIGURE PAGE 

9-28 Pier cap strengthening................................................................................................... 391 

9-29 Flexural and shear retrofit of free standing multi-column bents................................... 392 

9-30 Pier cap retrofit tested at University at Buffalo ............................................................ 393 

9-31 Flexural and shear retrofit of integral cap beams.......................................................... 393 

9-32 Horizontal link beams for pier cap force reduction ...................................................... 394 

9-33 Joint retrofit with external concrete jacket.................................................................... 395 

9-34 Free body diagram of shear forces on a knee joint ....................................................... 396 

9-35 Knee joint retrofit with steel plates............................................................................... 398 

9-36 Retrofit of an integral cap beam joint for longitudinal loading .................................... 399 

9-37 “Supergirders” added between piers to reduce torsion in pier caps.............................. 400 

10-1 Abutment types ............................................................................................................. 402 

10-2 California style settlement slabs ................................................................................... 404 

10-3 New Zealand style settlement slabs .............................................................................. 404 

10-4 “Waffle” slabs............................................................................................................... 406 

10-5 Anchor slabs with CIDH piles ...................................................................................... 407 

10-6 Diaphragm abutment overlay........................................................................................ 409 

10-7 Concrete block abutment shear key .............................................................................. 410 

10-8 Pipe shear key at abutment ........................................................................................... 411 

10-9 Abutment seismic shear walls....................................................................................... 412 

10-10 Wingwall overlay.......................................................................................................... 413 

10-11 Wingwall cross-tie ........................................................................................................ 414 

10-12 CIDH pile shear keys.................................................................................................... 415 

10-13 Gravity or “Deadman” anchors..................................................................................... 416 

10-14 Abutment retrofit using soil anchors............................................................................. 417 

10-15 Modified abutment friction anchor ............................................................................... 417 

10-16 Global footing strengthening ........................................................................................ 419 

10-17 Footing retrofit with concrete overlay .......................................................................... 421 

10-18 Bottom reinforcing splice detail.................................................................................... 422 

10-19 Footing strengthening by horizontal prestressing......................................................... 423 

10-20 Footing shear strength retrofitting ................................................................................ 424 

10-21 Footing joint shear forces.............................................................................................. 425 

10-22 Strut and tie model for joint design............................................................................... 427 

10-23 Yield line analysis of footing overlay........................................................................... 433 

10-24 Footing link beam retrofit ............................................................................................. 434 

10-25 Prestressed tie-down anchor ......................................................................................... 436 

11-1 Elevation view of a representative bridge on a liquefiable stratum.............................. 439 

11-2 Pseudo-static stability analysis ..................................................................................... 440 

11-3 Design flowchart........................................................................................................... 446 

11-4 Corrections for P-∆ effects in piles with plastic hinges................................................ 447 

11-5 Vibro-replacement equipment and process................................................................... 448 
xxii

LIST OF FIGURES (Continued)

FIGURE PAGE 

11-6 Vibro-system (stone column) treatment at toe of approach embankment .................... 449 

11-7 Vibro-system (stone column) treatment around a pile-supported bridge pier .............. 450 

11-8 Compaction grout bulb construction............................................................................. 451 

11-9 Effect of restrainers at bent during liquefaction failure................................................ 452 

B-1 Simplified base curve recommended for determination of CRR from SPT 

 data for magnitude 7.5 along with empirical liquefaction data .................................... 460 

B-2 Magnitude scaling factors derived by various investigators......................................... 461 

B-3 Robertson and Wride chart ........................................................................................... 462 

B-4 Soil flexibility factor (rd) versus depth curves developed by Seed and Idriss 

 (1971) with added mean value lines ........................................................................... 463 

B-5 Relationship between residual strength (Sr) and corrected ‘clean sand’ 

 SPT blowcount (N1)60 from case histories.................................................................... 466 

B-6 Simplified displacement chart for velocity-acceleration ratio of 30............................. 470 

B-7 Simplified displacement chart for velocity-acceleration ratio of 60............................. 470 

B-8 Relationship between cyclic stress ratio, blowcount (N1)60, and 

 volumetric strain for saturated clean sands and magnitude 7.5 .................................... 473 

B-9 Schematic diagram for determination of H1 and H2 ..................................................... 474 

B-10 Types of earthquake faults............................................................................................ 475 

B-11 Relationship between maximum surface fault displacement and 

 earthquake moment magnitude for strike-slip faulting................................................. 479 

C-1 Capacity-demand acceleration-displacement spectra showing 

 randomness and/or uncertainty in structural behavior and ground 

 motion response ............................................................................................................ 481

C-2 A normalized fragility curve based on a lognormal probability 

 distribution with a coefficient of dispersion βc = 0.6.................................................... 482 

D-1 Minimum support length requirements......................................................................... 487 

D-2 Effective seat width....................................................................................................... 489 

D-3 Procedures for determining capacity/demand ratios for columns, piers and footings.. 492 

D-4 Effectiveness of poorly anchored transverse reinforcement as a function of 

 ductility indicator.......................................................................................................... 494 

D-5 Effective anchorage length of longitudinal reinforcement ........................................... 496 

D-6 Procedure for determining capacity/demand ratios for anchorage of longitudinal 

 reinforcement ................................................................................................................ 499 

D-7 Radial stresses developed due to bar anchorage........................................................... 501 

D-8 Procedure for determining capacity/demand ratios for splices in longitudinal 

 reinforcement (mm and kPa) ........................................................................................ 503 

D-9 Procedure for determining capacity/demand ratios for column shear .......................... 506 
xxiii

LIST OF FIGURES (Continued)

FIGURE PAGE 

D-10 Resolution of shear demand and shear capacity ........................................................... 509 

D-11 Modes of failure for spread footings............................................................................. 515 

D-12 Modes of failure for pile footings ................................................................................. 516 

E-1 Example bridge ............................................................................................................. 522 

E-2 Expansion joint detail ................................................................................................... 522 

E-3 Column details .............................................................................................................. 523 

E-4 Positive sign convention for abutment, hinge and joint forces..................................... 528 

E-5 Positive sign convention for column forces and moments ........................................... 528 

E-6 Bent 2 column and footing interaction diagrams.......................................................... 536 

E-7 Bents 3 and 4 column and footing interaction diagrams............................................... 537 

E-8 Development of footing interaction diagram at bent 2 ................................................. 538 

F-1 Plan and elevation of example bridge........................................................................... 557 

F-2 Pier 2 bent details.......................................................................................................... 558 

F-3 Column interaction diagram ......................................................................................... 565 

F-4 Foundation interaction diagram .................................................................................... 565 

F-5 Foundation interaction diagram – transverse direction................................................. 566 
 
xxv

LIST OF TABLES

TABLE PAGE 

1-1 Service life categories..................................................................................................... 12 

1-2 Minimum performance levels for retrofitted bridges...................................................... 14 

1-3 Site classes ...................................................................................................................... 18 

1-4 Site factors Fa and Fv....................................................................................................... 19 

1-5 Seismic hazard level ....................................................................................................... 21 

1-6 Performance-based seismic retrofit categories .............................................................. 22 

1-7 Minimum requirements................................................................................................... 24 

1-8 Sample bridge seismic inventory form ........................................................................... 36 

1-9 Evaluation methods for existing bridges......................................................................... 40 

1-10 Cost of various retrofit strategies as percentage of new construction costs ................... 43 

1-11 Matrix of seismic retrofit approaches and section references to associated 

retrofit measures.............................................................................................................. 56 

2-1 Relationship between probability of exceedance of earthquake ground 

motion and return period................................................................................................. 60 

2-2 Site classes ...................................................................................................................... 79 

2-3 Steps for classifying a site............................................................................................... 80 

2-4 Site factors Fa and Fv....................................................................................................... 82 

2-5 Damping adjustment factors ........................................................................................... 88 

2-6 Procedure for obtaining deaggregated seismic hazard from USGS web site ................. 94 

3-1 Estimated susceptibility of sedimentary deposits to liquefaction during 

strong ground motion ................................................................................................... 103 

4-1 Values for PR................................................................................................................. 122 

4-2 Potential for liquefaction-related damage..................................................................... 125 

4-3 NBI fields (attributes) used in determining bridge fragility ......................................... 134 

4-4 Fragility of bridges constructed before 1975 in California and 1990 

elsewhere (non-seismic bridges)................................................................................... 136 

4-5 Fragility of bridges constructed since 1975 in California and 1990 

elsewhere (seismic bridges) .......................................................................................... 137 

4-6 Modification factors (K3D) used to model 3D effects for multi-span bridges .............. 138 

4-7 Modified repair cost ratio for all bridges ...................................................................... 140 

4-8 Expected losses in two example bridges subject to similar earthquakes...................... 141 

5-1 Evaluation methods for existing bridges....................................................................... 148 

5-2 Components for which capacity/demand ratios are required........................................ 155 

5-3 Restrictions on the application of the uniform load method......................................... 157 

5-4 Effective viscous damping ratios and damping factors, BS and BL .............................. 168 
xxvi

LIST OF TABLES (Continued)

TABLE PAGE 

6-1 Surface stiffnesses for a rigid plate on a semi-infinite homogeneous 

elastic half-space........................................................................................................... 186 

6-2 Stiffness embedment factors for a rigid plate on a semi-infinite 

homogeneous elastic half-space.................................................................................... 187 

7-1 Component rigidities..................................................................................................... 226 

7-2 Dead load multipliers (Cv) for bridges subject to vertical acceleration 

 for rock sites.................................................................................................................. 233 

7-3 Values of plastic curvature corresponding to various limit states in 

reinforced concrete columns and beams ....................................................................... 248 

8-1 Basic characteristics of typical isolation bearings for bridges...................................... 283 

8-2 Basic design properties for typical high strength bar sizes........................................... 297 

9-1 Mechanical properties of fibers used in modern fiber-reinforced plastic composites.. 366 

11-1 Summary of ground improvement methods for liquefaction remediation at existing 

 bridges........................................................................................................................... 441 

D-1 Footing ductility indicators........................................................................................... 513 

D-2 List of capacity/demand ratios...................................................................................... 520 

E-1 Maximum elastic moment demands (kN m)................................................................. 535 

E-2 Column and footing overstrength moments.................................................................. 537 

E-3 Revised column and footing overstrength moments (iteration 1)................................. 539 

E-4 Revised column and footing overstrength moments (iteration 2)................................. 540 

E-5 Ultimate moment capacity/elastic moment demand ratios ........................................... 541 

E-6 Capacity/demand ratios for the as-built bridge............................................................. 548 

F-1 Seismic retrofit categories for upper level earthquake (Fv = 1.0, PL2) ........................ 559 
xxvii

LIST OF ABBREVIATIONS AND SYMBOLS 

ABBREVIATIONS 

AASHTO American Association of State Highway and Transportation Officials 

ASL Anticipated Service Life 

ASTM American Society for Testing and Materials 

ATC Applied Technology Council 

ATC/MCEER Joint Venture of Applied Technology Council and Multidisciplinary 

Center for Earthquake Engineering Research 

AVR Abutment Vulnerability Rating 

BHT Becker Hammer Test 

BSSC Building Seismic Safety Council 

BSTRUCT Software for determining plastic deformation in piles 

C/D Capacity/Demand ratio 

Caltrans California Department of Transportation 

CDMG California Division of Mines and Geology 

CD-ROM Compact Disc-Random Access Memory 

CEUS Central and Eastern United States 

CGS California Geological Survey 

CIDH Cast-in-Drilled Hole 

CPT Cone Penetration Test or Cone Penetrometer Test 

CQC Complete Quadratic Combination 

CRR Cyclic Resistance Ratio 

CSABAC Caltrans Seismic Advisory Board Adhoc Committee 

CSR Cyclic Stress Ratio 

CVR Column Vulnerability Rating 

DL Dead Load 

EBF Eccentrically Braced Frames 

FEMA Federal Emergency Management Agency 

FEX Frequency of Exceedance 
xxviii

FHWA Federal Highway Administration 

FLPIER Software for soil foundation interaction analysis 

FRP Fiber Reinforced Polymer composite material 

FS Factor of Safety 

HAZUS Software for estimation of earthquake losses in built environment, 

developed by FEMA (name derived from “Hazards - U.S.”) 

HITEC Highway Innovative Technology Evaluation Center 

IBC International Building Code 

ICBO International Conference of Building Officials 

LL Lower Level earthquake ground motion 

LL Live Load 

LPILE Software for lateral load analysis of piles 

LRFD Load and Resistance Factor Design 

LVR Liquefaction Vulnerability Rating 

MCE Maximum Considered Earthquake 

MCEER Multidisciplinary Center for Earthquake Engineering Research 

MDOF Multiple Degree of Freedom 

NBI National Bridge Inventory 

NCHRP National Cooperative Highway Research Program 

NDP Nonlinear Dynamic Procedure 

NEHRP National Earthquake Hazard Reduction Program 

NSP Nonlinear Static Procedure 

NZS Standards Association of New Zealand 

PEX Probability of Exceedance 

PGA Peak Ground Acceleration 

PL Performance Level 

PTFE Polytetrafluoroethylene (Teflon) 

RCR Repair Cost Ratio or Damage Ratio 

SA Spectral Acceleration 

SCEC South California Earthquake Center 

SDOF Single Degree of Freedom 

SEISAB Software for the seismic analysis of bridges 
xxix

SHL Seismic Hazard Level 

SPC Seismic Performance Category 

SPS Shear Panel System 

SPT Standard Penetration Test 

SRA Seismic Risk Assessment 

SRC Seismic Retrofit Category 

SRSS Square Root of the Sum of the Squares 

SFSI Soil-Foundation-Structure Interaction 

SSI Soil-Structure Interaction 

STRAHNET Strategic Highway Network, part of the National Highway System 

STU Shock Transmission Unit 

TADAS Triangular-plate Added Damping and Stiffness Device 

UHM Ultra High Modulus 

UL Upper Level earthquake ground motion 

ULM Uniform Load Method 

USGS United States Geological Survey 

WUS Western United States 

SYMBOLS 

CHAPTER 1: SEISMIC RETROFITTING OF HIGHWAY BRIDGES 

F = Seismic demand or force. 

FL = Longitudinal seismic demand or force. 

FT = Transverse seismic demand or force. 

Fa = Site factor in short-period range of design spectrum. 

Fv = Site factor in long-period range of design spectrum. 

N = Average Standard Penetration Test (SPT) blowcount. 

P = Priority index. 

PI = Plasticity index. 

R = Bridge rank. 

S1 = Spectral acceleration at 1.0 sec period for reference site (Site Class B). 

SD1 = Spectral acceleration at 1.0 sec period including site effects. 
xxx

SDS = Spectral acceleration at 0.2 sec period including site effects. 

Ss = Spectral acceleration at 0.2 sec period for reference site (Site Class B). 

us = Average undrained shear strength. 

v = Average shear wave velocity for the upper 30 m (100 ft) of the soil profile. s

W = Weight. 

w = Moisture content. 

CHAPTER 2: SEISMIC GROUND MOTION HAZARD 

ξ = Viscous damping ratio. 

ξeff = Effective viscous damping ratio. 

ν = Annual frequency of occurrence. 

dc = Total thickness of cohesive soil layers in the top 30 m (100 ft). 

di = Thickness of a layer between 0 and 30 m (0 and 100 ft). 

ds = Total thickness of cohesionless soil layers in the top 30 m (100 ft). 

Fa = Site factor in short-period range of design spectrum. 

Fv = Site factor in long-period range of design spectrum. 

i = Soil layer (ranges from 1 to n). 

k = Number of cohesionless soil layers in the top 30 m (100 ft). 

N = Average Standard Penetration Test (SPT) blowcount (blows/0.30m or blows/ft). 

Nch = Blowcount for a cohesionless soil layer. 

Ni = Standard penetration test blowcount of a layer. 

PE = Probability of exceedance. 

PI = Plasticity index. 

S1 = Spectral acceleration at 1.0 sec period for reference site (Site Class B). 

SD1 = Spectral acceleration at 1.0 sec period including site effects. 

SDS = Spectral acceleration at 0.2 sec period including site effects. 

SS = Spectral acceleration at 0.2 sec period for reference site (Site Class B). 

us = Average undrained shear strength. 

sui = Undrained shear strength for a cohesive soil layer. 

T = Period of vibration. 

t = Lifetime of bridge. 
xxxi

vsi = Shear wave velocity of ith layer of soil. 

v = Average shear wave velocity for the upper 30 m (100 ft) of the soil profile. s

w = Moisture content. 

CHAPTER 4: SEISMIC RATING METHODS FOR SCREENING AND PRIORITIZATION 

α = Angle of skew. 

A2 = Modified median fragility curve parameter. 

a2 = Median spectral acceleration (at 1.0 sec period). 

B = Width of deck. 

BLOSS = Direct economic loss due to structural damage to a bridge. 

bmax = Maximum transverse column dimension. 

F = Framing factor. 

H = Height. 

HLOSS = Indirect economic loss due to loss of life, injuries, business disruption, traffic 

congestion, and denied access. 

Kshape = Factor relating to the shape of the design acceleration spectrum. 

Kskew = Fragility modification factor to account for bridge skew. 

L = Length of bridge deck 

L = Available support length for superstructure 

Lc = Effective column length. 

N = Minimum recommended support length. 

n = Number of spans in bridge. 

np = Number of piers. 

P = Priority index. 

PR = Total number of points to be deducted from Q for factors known to reduce 

susceptibility to shear failure. 

Ps = Amount of main reinforcing steel expressed as a percent of the column crosssectional area. 

Q = Factor used to determine column vulnerability rating (equation 4-5b) 

R = Bridge rank. 

RCRi = Repair cost ratio for the ith damage state. 

RCRT = Total repair cost ratio.

S = Site factor. 
xxxii

SD1 = Spectral acceleration at 1.0 sec period including site effects (seismic coefficient). 

TLOSS = Total monetary loss to the economy. 

U = Replacement cost of the bridge. 

V = Bridge vulnerability rating. 

VL = Vulnerability to collapse or loss of access due to longitudinal movement. 

VT = Vulnerability to collapse or loss of access due to transverse movement. 

CHAPTER 5: EVALUATION METHODS FOR EXISTING BRIDGES 

α = Angle of skew. 

∆ = Displacement of superstructure. 

∆max = Maximum displacement. 

∆’max = Displacement limit state. 

∆y = Yield displacement. 

μ = Displacement ductility factor. 

ξeff = Effective viscous damping ratio.

ρt = Total area ratio of longitudinal reinforcement. 

φ = Capacity reduction factor. 

ω = Angular frequency. 

ΣQNSi = Sum of the displacement or force demands on a component from non-seismic 

loads. 

Ag = Gross area of the column section. 

B = Width. 

BL = Damping factor used in the long-period range of the spectrum. 

BS = Damping factor used in the short-period range of the spectrum. 

Cc = Capacity coefficient. 

Cd = Seismic demand coefficeint. 

D = Smallest column dimension or diameter. 

Dp = Pile dimension about the weak axis of bending. 

F = Total horizontal force acting on the bridge.

Fy = Yield force. 

f'c = Compressive strength of concrete. 

g = Acceleration due to gravity. 
xxxiii

H = Height. 

K = Lateral stiffness of the bridge. 

k1 = Elastic stiffness in direction considered, transverse or longitudinal. 

k2 = Equivalent post-yield stiffness in direction considered, transverse or longitudinal. 

L = Total length of bridge 

L = Distance between expansion joints. 

M/V = Shear span length of an equivalent cantilever member where M is the end moment 

and V is the shear force. 

Mn = Nominal column yield moment. 

N = Minimum seat width. 

N0 = Available seat width at an abutment or pier cap. 

P = Axial load. 

Pc = Axial capacity of a steel column or timber pile member in compression. 

Pe = Axial load on the bridge column including both gravity and seismic effects. 

pe = Equivalent uniform static seismic loading per unit length of bridge. 

po = Uniform load. 

Py = Axial capacity at yield of a steel column/pile member. 

QEQ = Displacement or force demand for the earthquake loading under consideration. 

R = Ratio of elastic force demand to lateral capacity of pier. 

RC = Nominal ultimate displacement or force capacity of the structural component 

being evaluated. 

Sa = Spectral acceleration. 

Sd = Spectral displacement. 

T = Period, or wall thickness or the smallest cross-sectional dimension. 

vs,MAX = Maximum value of static displacement vs(x). 

VU = Lateral strength capacity at zero displacement. 

W = Weight. 

W’ = Seismic weight per column. 

w(x) = Unfactored dead load of the bridge superstructure and tributary substructure. 

CHAPTER 6: GEOTECHNICAL MODELING AND CAPACITY ASSESSMENT 

γ = Weight density of soil. 
xxxiv

γt = Total unit weight of soil. 

γw = Unit weight of water. 

δc = Axial displacement of the pile head due to compression of the pile under axial 

load but neglecting the surrounding soil. 

λ = Characteristic length of a pile defined by (EIp/Es)

0.25. 

ν = Poisson’s ratio. 

0 σ' = Effective vertical stress. 

φ = Internal friction angle of a soil. 

AE = Axial rigidity of pile. 

B = Footing width. 

c = Compressive strength of a soil. 

D = Pile diameter. 

Dg = Gap width. 

d = Depth of sample. 

dw = Depth of water level. 

E = Modulus of elasticity. 

EI, EIp = Flexural stiffness of pile. 

Es = Subgrade modulus of the soil. 

e = Stiffness embedment factor (table 6-2). 

Fv = Factor of safety. 

f = Coefficient of variation of the inelastic subgrade modulus. 

Go = Initial or low strain shear modulus during cyclic loading. 

g = Acceleration due to gravity. 

H = Height. 

Ix, Iy = Moments of inertia of a footing about the x and y axis, respectively.

K = Soil stiffness parameter (table 6-1). 

L = Length. 

Mc = Ultimate moment capacity. 

(N1)60 = Normalized corrected blowcount. 

P = Vertical load on footing. 

Pp = Total passive force per unit length of wall. 

pp = Passive pressure behind a wall. 
xxxv

Q = Axial load. 

q = Vertical load on a footing (contact stress). 

qc = Ultimate strength of soil (capacity) per unit area. 

vs = Shear wave velocity at small strains. 

Z = Displacement. 

Zc = Displacement when capacity is reached. 

z = Depth below grade. 

CHAPTER 7: STRUCTURAL MODELING AND CAPACITY ASSESSMENT 

α = Corner-to-corner strut angle. 

α = Ratio of average concrete stress in compression zone to confined concrete 

strength. 

β = Stress block factor. 

γ = Reinforcing steel configuration factor. 

∆e = Elastic component of displacement. 

∆p = Plastic component of displacement. 

∆y = Nominal yield displacement. 

εap = Plastic strain amplitude. 

εb = Buckling strain in the longitudinal reinforcing steel. 

εcu = Ultimate compression strain of the confined core concrete. 

εsu = Strain at the maximum stress of the transverse reinforcement. 

εy = Yield strain. 

θp = Plastic hinge rotation. 

θu = Ultimate (total) drift. 

θy = Elastic drift at yield. 

θ = Angle of the principal crack plane. 

κo = Factor related to the centroid of the stress block. 

Λ = Fixity factor. 

μlapφ = Curvature ductility at the initial breakdown of bond in the lap-splice zone. 

ρs = Volumetric ratio of transverse steel. 

ρt = Volumetric ratio of the longitudinal reinforcement. 

ρv = Volumetric ratio of transverse steel. 
xxxvi

ρx, ρy = Volume ratio of transverse hoops or ties to the core concrete in x- and y- directions, 

respectively. 

φ = Strength reduction factor. 

φe = Expected strength factor. 

φο = Overstrength factor.

φp = Plastic curvature. 

φu = Ultimate (total) curvature. 

φy = Nominal yield curvature. 

Abh = Area of one spiral or hoop bar. 

Acc = Area of confined concrete core. 

Ae = Effective shear area. 

Ag = Gross section area. 

Ajh = Area of a beam-column joint in a horizontal plane. 

Ast = Area of steel. 

Av = Area of transverse shear steel. 

Aw = Shear area of beam or column. 

b'' = Width of column normal to y-direction. 

bw = Center-to-center spacing of transverse shear steel across width of rectangular 

column. 

Cv = Dead load multiplier. 

c = Depth from the extreme compression fiber to the neutral axis. 

D = Overall depth of section. 

D’ = Distance between the outer layers of longitudinal steel in a rectangular section. 

D'' = Diameter of transverse hoop or spiral. 

d = Depth to the outer layer of tension steel from the extreme compression fiber. 

d′ = Distance from the extreme compression fiber to the center of the nearest 

compression reinforcing bars. 

db = Diameter of the longitudinal reinforcing bar. 

Ec = Elastic modulus of concrete. 

EcIeff = Effective flexural rigidity. 

Es = Elastic modulus of steel. 

fν = Average axial stress on the joint. 

'

cc f = Confined concrete strength. 
xxxvii

'

cf = Ultimate compressive strength of concrete. 

'

ce f = Ultimate expected compressive strength of concrete. 

fh = Average horizontal axial stress on the joint. 

'

lf = Confining stress supplied by the transverse reinforcement at yield in a circular 

 column. 

' '

x y f ,f A A = Confining stress in the x- and y- directions of a rectangular column, respectively. 

fsu = Ultimate tensile strength of the longitudinal reinforcement. 

fy = Yield stress. 

fye = Expected yield strength of longitudinal reinforcement. 

fyh = Yield stress of the transverse hoops. 

hb = Depth of the cap beam at the joint. 

Ig = Moment of inertia of the gross uncracked section. 

jd = Internal lever arm of the concrete compression member. 

K = Strength enhancement factor. 

Kshape = Shape factor. 

ke = Confinement effectiveness coefficient for spirals and hoop steel. 

L = Length. 

L/D’ = Slenderness ratio. 

Lgap = Clear gap between edge of jacket and bottom of pier cap or top of footing. 

Lp = Equivalent plastic hinge length. 

ls = Length of the lap splice. 

MD = Design moment. 

Me = Expected flexural strength (moment). 

Mn = Nominal yield moment (theoretical yield strength) of the member. 

Mp = Plastic strength capacity. 

Mpo = Flexural moment overstrength capacity. 

MS = Reduced moment. 

Nf = Effective number of equal-amplitude cycles of loading that lead to fracture. 

P = Axial load on member. 

Pe = Axial load on the section. 

pt = Major principal tension stress. 
xxxviii

S = Center-to-center longitudinal spacing of the transverse hoop steel. 

Sd = Design strength. 

Sn = Nominal strength. 

So = Overstrength. 

s = Spacing of spirals or hoops. 

Tn = Natural period of vibration of the bridge. 

Vcf = Final shear strength carried by the concrete. 

Vci = Contribution provided by diagonal tension field in the concrete. 

Vf = Final shear strength. 

Vi = Initial shear strength. 

Vjf = Final (residual) joint shear strength. 

Vjh = Horizontal shear in the joint. 

Vji = Initial shear strength of the beam-column joint. 

Vm = Shear demand based on flexure. 

Vp = Contribution provided arch (strut) action. 

Vs = Contribution to the shear strength provided by rebar truss action. 

vj = Average joint shear stress acting on the joint. 

CHAPTER 8: RETROFIT MEASURES FOR SUPERSTRUCTURES, BEARINGS AND SEATS 

α = Angle of skew. 

β = Frequency ratio. 

ξ = Equivalent viscous damping ratio. 

μ = Ductility capacity. 

{φi} = Mode shape for mode ‘i’. 

Ar = Area of one restrainer. 

D1 = Maximum displacement of the span. 

D2 = Maximum displacement at the top of the pier. 

Das = Available seat width. 

Deq0 = Unrestrained relative expansion joint displacement. 

Dr = Maximum permissible restrainer displacement. 

Drs = Restrainer slack. 

Dy = Restrainer elongation at yield. 
xxxix

E = Modulus of elasticity. 

Fh = Increase in longitudinal resistance resulting from wedging a rocker bearing. 

fy = Yield stress. 

[K] = Stiffness matrix of a two frame system with restrainers. 

KDED = Stiffness of the ductile end diaphragm.

Kr = Restrainer stiffness. 

KSUB = Stiffness of the substructure. 

Lr = Restrainer length. 

[M] = Mass matrix of a two frame system. 

Nr = Number of restrainers. 

P = Dead load / bearing. 

Pi = Participation factor for mode i. 

R = Response modification factor (R-factor). 

s = Restrainer slack. 

CHAPTER 9: RETROFIT MEASURES FOR SUBSTRUCTURE COMPONENTS 

α = Geometric aspect ratio of a column. 

εcu = Ultimate strain of concrete column jacketed with composites. 

εdu = Ultimate design strain of the jacket material taking into account long-term 

environmental degradation. 

εp = Maximum allowable passive strain. 

εsu = Ultimate strain in the shell steel. 

θ = Shear crack angle. 

ρl = Longitudinal reinforcement ratio (As/Ag). 

ρs = Effective volumetric ratio. 

Ab = Area of spliced longitudinal bar. 

Ad = Area of dowel bar. 

Ap = Wire cross-sectional area. 

c = Cover over the main column reinforcing bars. 

D = Diameter. 

D' = Diameter of the pitch circle of the main column reinforcement; distance between 

the outermost reinforcing bars in the direction of shear. 
xl

db = Diameter of longitudinal reinforcement. 

dp = Diameter of wire. 

Ea = Modulus of elasticity of active wrap. 

Ej = Weighted average of the elastic moduli for the active and passive wraps. 

Ep = Modulus of elasticity of passive wrap. 

Es = Modulus of elasticity for steel. 

fa = Active confining stress in the column. 

f'cc = Ultimate confined concrete stress capacity. 

fdu = Ultimate design strength of the jacket material taking into account long-term 

environmental degradation. 

fi = Tensile stress in wire after losses. 

fA = Required level of confinement stress. 

fpu = Ultimate design stress. 

fs = Stress induced in the jacket. 

fy = Yield stress. 

fys = Jacket yield stress. 

g = Gap between the retrofit measure and critical section. 

L = Length. 

p A = Length of plastic hinges. 

s A = Splice length. 

N = Number of main column reinforcing bars. 

s = Spacing. 

ss = Gap between retrofit measure and critical section.

t = Steel jacket thickness. 

ta = Thickness of active wrap. 

tp = Thickness of passive wrap. 

Vsj = Additional nominal shear capacity provided by a steel jacket. 

vi = Maximum interface shear stress. 

vih = Horizontal interface shear stress. 

viv = Vertical interface shear stress. 
xli

CHAPTER 10: RETROFIT MEASURES FOR ABUTMENTS, FOOTINGS, AND 

FOUNDATIONS 

μ = Coefficient of friction between soil and concrete. 

Aeff = Effective bearing area. 

Bc = Column width. 

beff = Column width plus depth for rectangular columns. 

Cin = Column interior. 

Cex = Column exterior. 

D = Column diameter. 

Dc = Column depth. 

DLs = Dead load of slab and any overlaying fill. 

df = Depth of the retrofitted footing. 

FD = Design force. 

fa = Effective axial compressive stress within the joint in the vertical direction. 

ft = Principle tensile stress in the joint region. 

P = Axial load. 

Sa = Spectral acceleration. 

Vjv = Vertical joint shear stress. 

CHAPTER 11: RETROFIT MEASURES FOR BRIDGES ON HAZARDOUS SITES 

Mp = Plastic moment capacity of pile. 

Mpt = Plastic moment capacity at head of pile. 

Vpass = Passive force developed against end diaphragm. 

Vstrut = Force resisted by superstructure and other piers. 

APPENDIX B: EVALUATION OF GEOTECHNICAL HAZARDS 

τav/σ’vo = Average earthquake induced shearing stress divided by the effective overburden 

stress. 

σvo/σ’vo = Ratio of total overburden stress to effective overburden stress. 

amax = Peak ground acceleration in units of g. 

rd = Soil flexibility factor. 
xlii

APPENDIX C: FRAGILITY CURVE THEORY 

βc = Normalized composite log-normal deviation. 

Φ = Standard log-normal cumulative distribution function. 

Ai = Median spectral acceleration necessary to cause the ith damage state to occur. 

DSi = ith damage state. 

Sa = Spectral acceleration. 

APPENDIX D: CAPACITY/DEMAND RATIOS FOR BRIDGE MEMBERS AND 

COMPONENTS 

∆eq(d) = Maximum calculated relative displacement due to earthquake loading. 

∆i(d) = Maximum possible movement resulting from temperature, shrinkage and creep 

shortening. 

∆s(c) = Available capacity of the expansion joint or bearing for movement. 

μ = Ductility indicator. 

ρ(c) = Volumetric ratio of existing transverse reinforcement. 

ρ(d) = Required volumetric ratio of transverse reinforcement. 

Ab = Area of the spliced bar. 

Ag = Gross area of column. 

AL(c) = Effective peak ground acceleration at which liquefaction failures are likely to 

occur. 

AL(d) = Effective acceleration coefficient for the site. 

Atr(c) = Area of transverse reinforcing normal to potential splitting cracks. 

B = Width of superstructure. 

bc = Width of the column. 

bmin = Minimum width of the column cross section. 

c = Lesser of the clear cover over the bar, or half the clear spacing between adjacent 

bars. 

db = Diameter of longitudinal reinforcement. 

'

cf = Concrete compression strength. 

fy = Yield stress of the longitudinal reinforcement. 

fyt = Yield stress of transverse reinforcement. 

H = Height. 
xliii

k3 = Effectiveness of transverse bar anchorage. 

ks = Constant for reinforcing steel with a yield stress of fy. 

L = Distance between joints. 

La(c) = Effective anchorage length of longitudinal reinforcement. 

La(d) = Required effective anchorage length of longitudinal reinforcement. 

Lc = Height of the column. 

Ls = Splice length. 

N(c) = Actual support length provided. 

N(d) = Minimum seat width. 

Pc = Axial compressive load on the column. 

s = Spacing of transverse reinforcement. 

Vb(c) = Nominal ultimate capacity of the component in the direction under consideration. 

Vb(d) = Seismic force acting on the component. 

Ve(d) = Maximum calculated elastic shear force. 

Vf(c) = Final shear resistance of the damaged column. 

Vi(c) = Initial shear resistance of the undamaged column. 

Vu(d) = Maximum column shear force resulting from plastic hinging at both the top and 

bottom of the column. 
xliv

GREEK ALPHABET 

Α α alpha 

Β β beta 

Γ γ gamma 

∆ δ delta 

Ε ε epsilon 

Ζ ζ zêta 

Η η êta 

Θ θ thêta 

Ι ι iota 

Κ κ kappa 

Λ λ lamda 

Μ μ mu 

Ν ν nu 

Ξ ξ xi 

Ο ο omikron 

Π π pi 

Ρ ρ rho 

Σ σ, ς sigma 

Τ τ tau 

Υ υ upsilon 

Φ φ phi 

Χ χ chi 

Ψ ψ psi 

Ω ω omega 
1

CHAPTER 1: SEISMIC RETROFITTING OF HIGHWAY BRIDGES 

1.1. GENERAL 

Many bridges in the United States are inadequate for seismic loads and could be seriously 

damaged or suffer collapse in a relatively small earthquake. Major structural damage has 

occurred in Alaska, California, Washington and Oregon due to earthquake ground shaking. Some 

of these failures occurred at relatively low levels of ground motion and although the risk of 

bridge collapse is lower in the central and eastern United States, ground motions large enough to 

cause damage have a 10 percent chance of occurring within the next 50 years in 37 of the 50 

states. In some states, this could lead to bridge collapse, and to remedy this situation, retrofitting 

or replacement of deficient structures is necessary. This is done by identifying bridges at risk, 

evaluating their vulnerability for collapse or major seismic damage, and initiating a program to 

reduce this risk. 

Retrofitting is the most common method of mitigating risks; however, its cost may be so 

prohibitive that abandoning the bridge (total or partial closure with restricted access) or replacing 

it altogether with a new structure may be favored. Alternatively, doing nothing and accepting the 

consequences of damage is another possible option. The decision to retrofit, abandon, replace, or 

do-nothing requires that both the importance and degree of vulnerability of the structure be 

carefully evaluated. Limited resources will generally require that deficient bridges be prioritized, 

with important bridges in high risk areas being given the first priority for retrofitting. 

This manual contains procedures for evaluating and upgrading the seismic resistance of existing 

highway bridges. Specifically, it contains: 

• A screening process to identify and prioritize bridges that need to be evaluated for seismic 

retrofitting. 

• A methodology for quantitatively evaluating the seismic capacity of a bridge and determining 

the overall effectiveness of alternative retrofitting measures, including cost and ease of 

installation. 

• Retrofit approaches and corresponding techniques for increasing the seismic resistance of 

existing bridges. 

This process is illustrated in figure 1-1. A bridge may be exempt from retrofitting if it is located 

in the lowest seismic zone, or has limited remaining useful life. Temporary bridges and those 

closed to traffic, may also be exempt. Details are provided in section 1.4.6. 

This manual does not prescribe rigid requirements as to when and how bridges are to be 

retrofitted. The decision to retrofit a bridge depends on a number of factors, several of which are 

outside the realm of engineering. These include, but are not limited to, the availability of funding 
2

 

BRIDGE 

INVENTORY 

SEISMIC HAZARD 

• Ground Motions 

• Site effects 

ANTICIPATED 

SERVICE LIFE 

PERFORMANCE 

OBJECTIVES 

SCREEN / PRIORITIZE 

EVALUATE 

RETROFIT 

Is bridge 

exempt?

NEXT 

BRIDGE 

Pass

Fail 

NO 

YES 

Pass 

Fail 

Review 

Figure 1-1. Overview of the retrofitting process for highway bridges. 
3

and a number of political, social, and economic issues. This manual focuses on the engineering 

factors. 

This manual is intended for use on conventional steel and concrete highway bridges with spans 

not exceeding 150 m (500 ft). Suspension bridges, cable-stayed bridges, arches, long-span 

trusses, and movable bridges are not covered. However, many of the procedures and techniques 

presented here can be applied to these types of structures, if appropriate judgment is used. This is 

particularly true for the substructures of truss bridges, for example. Although specifically 

developed for highway bridges, this manual may also be applicable to other types of bridges. 

1.2. BACKGROUND 

Within the United States, the first attempts to seismically retrofit bridges took place in the 

aftermath of the 1971 San Fernando earthquake in southern California. This earthquake has 

often been cited as a watershed event in bridge engineering since it demonstrated quite 

dramatically that the bridge design practices of the time did not guarantee that bridges would 

perform well during an earthquake, even if the earthquake was of moderate intensity. 

Although bridge failures during this earthquake could be attributed to deficiencies in several 

types of structural components, initial retrofit measures focused on the potential for loss of 

support at bridge bearing seats. The principal retrofit strategy was to add restrainer cables or 

high strength bars within bridge superstructures in order to limit relative movements at 

expansion joints, and to tie individual spans to the bridge piers. A program was initiated in 

California to retrofit all bridges that had vulnerable expansion joints. Some believed at the time 

that this could prevent most, if not all, bridge failures at a minimum cost; however, this was not 

the case. 

During the 1987 Whittier Narrows earthquake, a bridge that had been retrofitted with expansion 

joint restrainers suffered severe column damage, threatening the overall stability of the bridge 

(Wipf et al., 1997). This poor performance prompted renewed interest in column retrofitting, 

and research efforts were intensified to develop suitable methods for improving the seismic 

performance of existing reinforced concrete columns. Much of this research was conducted at 

the University of California at San Diego and was initially focused on the addition of steel shells 

(or jackets) to enhance confinement of the reinforcing steel within the columns (Chai et al., 

1992). Unfortunately, the 1989 Loma Prieta earthquake occurred before Caltrans could begin a 

state-wide column retrofitting program. The most dramatic failure during this earthquake was 

the collapse of the Cypress Viaduct in Oakland due to column joint failures. Although this twolevel structure had been retrofitted with expansion joint restrainers prior to the earthquake, large 

portions of the upper deck collapsed onto the lower deck, crushing vehicles and resulting in a 

number of deaths (Housner, 1990). 

The Loma Prieta earthquake also demonstrated that the early restrainer designs were inadequate. 

Many of these designs relied on too few restrainers while others caused failures to occur 

elsewhere in the structure, such as in the diaphragms at either side of expansion joints 

(Yashinsky, 1997). Many of these deficiencies had been identified through research and refined 
4

analysis (Selna and Malvar, 1987) and were not entirely unexpected. Still, it emphasized the 

need to review earlier bridge retrofits. 

Since the Loma Prieta earthquake, there has been a major effort to perform comprehensive 

seismic retrofits on a large number of bridges in California. Initially, this effort was focused on 

bridges with single column piers, which were believed to be the most vulnerable to collapse. 

However, many bridges with multi-column piers collapsed or were severely damaged during the 

1994 Northridge earthquake, and bridges of this type were subsequently added to the Caltrans 

retrofit program (Buckle, 1994). 

Other states besides California now have active programs in seismic retrofitting. Washington 

and Nevada, for example, have sponsored research efforts and started seismic retrofit programs. 

Similarly, countries such as Japan and New Zealand have seismic retrofit programs. Due to these 

parallel efforts, significant progress has been made in the state-of-the-art of retrofitting bridge 

superstructures, columns, and foundations. 

The Federal Highway Administration (FHWA) also became a major sponsor of bridge seismic 

research shortly after the 1971 San Fernando earthquake, including research on the retrofitting of 

existing bridges. An early research project dealing with the retrofitting of bridges was conducted 

at the Illinois Institute of Technology (Robinson et al., 1979). The first known guidelines for 

retrofitting highway bridges were published by FHWA in 1981, in a report titled Seismic Design 

Guidelines for Highway Bridges, FHWA-RD-81-081. This was followed in 1983 by the 

publication of the FHWA Retrofitting Guidelines for Highway Bridges, which contained 

recommendations intended for national use (FHWA-RD-83-007, ATC, 1983). In addition to 

providing formal screening and evaluation procedures, these guidelines presented retrofit 

concepts that in many cases had not been used in practice at that time. Concepts were provided 

for several different bridge components besides expansion joints including bridge columns, 

abutments and footings. Over the years, many of these concepts have been developed and 

refined into methods that are commonly used today. These procedures and techniques are also 

included in the Seismic Design and Retrofit Manual, FHWA-IP-87-6, published by FHWA in 

1987. 

Since 1992, FHWA has sponsored a multi-year research program to advance the state-of-thepractice in seismic retrofitting of bridges. This work was conducted at the National Center for 

Earthquake Engineering Research (NCEER), later to become the Multidisciplinary Center for 

Earthquake Engineering Research (MCEER). The project’s first product was FHWA’s Seismic 

Retrofit Manual for Highway Bridges that was published in 1995 (FHWA-RD-94-052, FHWA, 

1995). This revision reflected advancements in the practice of seismic retrofitting that had 

occurred since 1983. Other products of this research program included improved methods for 

restrainer design (Randall et al., 1999; DesRoches and Fenves, 1998); methods for improving the 

performance of older steel bridge bearings (Mander et al., 1998a); improved methods of analysis 

and retrofit of reinforced concrete columns (Dutta et al., 1999); retrofit methods for multicolumn reinforced concrete piers (Mander et al., 1996a and b); and a better understanding of the 

performance of retrofits that use both conventional and seismic isolation elastomeric bearings 

(Wendichansky et al., 1998). This program culminated in the preparation of this current FHWA 
5

manual, which presents not only the results of FHWA research at MCEER, but also from 

throughout the United States and the world. 

1.3. PHILOSOPHY 

It has been common practice to design new bridges and buildings for a single-level of earthquake 

ground motion. This ground motion, often called the design earthquake, represents the largest 

motion that can be reasonably expected during the life of the bridge. Implied in such a statement 

is the fact that ground motions larger than the design earthquake may occur during the life of the 

bridge, but the likelihood of this happening is small. This likelihood is usually expressed as the 

probability of exceedance, and it may also be described by a return period in years. When setting 

the seismic hazard level, most design specifications that are intended for regions of varying 

seismicity use the same probability of exceedance from one region to another. This ‘uniform 

hazard’ approach is considered to be more rational than using the maximum historical event for 

each region. 

The Standard Specification for Highway Bridges (AASHTO, 2002) in the United States adopted 

a uniform hazard approach following the 1989 Loma Prieta earthquake, and chose a level of 

hazard that had a 10 percent probability of exceedance in a 50-year exposure period (the 

assumed life of a bridge). This corresponded to a ground motion with a return period of about 

500 years. During the development of the AASHTO LRFD Specification in the mid-nineties, the 

life of the average highway bridge was reassessed at 75 years and the exposure period was 

adjusted accordingly (AASHTO, 1998). The probability of exceedance was then raised to 15 

percent to maintain (approximately) the same return period (500 years). 

At the same time as adopting this uniform hazard approach, a corresponding set of performance 

standards was included in the philosophy of the 1992 AASHTO Specifications (AASHTO, 2002). 

These are given in Art. 1.1 of the Specification and summarized below: 

• Small to moderate earthquakes should be resisted within the elastic range, without significant 

damage. 

• Realistic seismic ground motion intensities and forces be used in the design procedures. 

• Exposure to shaking from large earthquakes should not cause collapse of all or part of the 

bridge. Where possible, damage that does occur should be readily detectable and accessible 

for inspection and repair. 

A set of basic concepts for seismic design was derived from this philosophy (Art. 1.3, AASHTO, 

2002), and is summarized below: 

• Hazard to life is minimized. 

• Bridges may suffer damage but should have a low probability of collapse. 

• Function of essential bridges is maintained. 

• Ground motions used in design should have a low probability of being exceeded in the 

normal lifetime of the bridge. 
6

While characterized by a lack of specificity, these criteria were a significant advance over the 

then prevailing requirements for seismic design. 

In like manner, previous retrofit guidelines and manuals have also used a single-level of 

earthquake ground motion (a 500-year event) for representing the earthquake hazard, and 

adopted the same performance criteria as in the then current AASHTO Specifications for bridge 

design. 

The assumption is made in single-level design and retrofit, that if performance under the design 

earthquake is satisfactory, it will be satisfactory at all other levels of ground motion, both 

smaller and larger. Such an assumption is generally not true as seen in recent earthquakes in 

California, Costa Rica, Japan, Turkey and Taiwan (figures 1-2 to 1-5). It would be true for a 

smaller event if elastic performance was required at the design ground motion, and it may also be 

true for a larger event, if it exceeded the design ground motion by only a small margin (i.e., less 

than 50 percent), and there was a sufficient reserve of strength in the bridge to accommodate this 

higher demand. 

However, in many areas of the United States, these larger ground motions can be three or four 

times the design ground motions and may cause instability and collapse. Although such ground 

motions rarely happen, their occurrence should be explicitly considered in the design and retrofit 

process and a ‘multi-level’ rather ‘single-level’ design process should be used. In addition, 

performance requirements should be adjusted for ground motions of different sizes, with higher 

levels of performance being expected for smaller motions and lesser levels of performance for 

larger motions. 

Performance-based design provides a format for addressing these needs in a rational manner. It 

explicitly allows for different performance expectations for bridges of varying importance while 

subject to different levels of seismic hazard. Accordingly, this manual recommends a 

performance-based approach to the seismic retrofitting of highway bridges in the United States. 

This relationship is shown in figure 1-6. Representation of the hazard and performance 

expectations by discrete zones (or levels) is necessary given the current state-of-the-art, and this 

leads to the bar chart shown in this figure. Nevertheless, the trends are the same: high 

performance standards in high hazard zones imply higher costs. 

1.4. SEISMIC PERFORMANCE CRITERIA 

1.4.1. PERFORMANCE LEVELS 

As noted in the previous section, this manual presents a performance-based approach to the 

seismic retrofitting of highway bridges. This means that the expected performance of the 

retrofitted bridge is explicitly recommended for different levels of earthquake ground motion. In 

this manual, performance criteria are defined for four performance levels. These are given as 

follows: 
Performance Level 0 (PL0): No minimum level of performance is recommended. 

Performance Level 1 (PL1): Life safety. Significant damage is sustained during an earthquake 

and service is significantly disrupted, but life safety is assured. The 

bridge may need to be replaced after a large earthquake. 

Performance Level 2 (PL2): Operational. Damage sustained is minimal and full service for 

emergency vehicles should be available after inspection and 

clearance of debris. Bridge should be reparable with or without 

restrictions on traffic flow. 

Performance Level 3 (PL3): Fully Operational. Damage sustained is negligible and full service is 

available for all vehicles after inspection and clearance of debris. 

Any damage is repairable without interruption to traffic. 

 

The terms negligible damage, minimal damage, and significant damage are used in the above 

performance criteria. These terms are explained below: 

• Minimal damage includes minor inelastic response and narrow flexural cracking in concrete. 

Permanent deformations are not apparent and repairs can be made under non-emergency 

conditions with the possible exception of superstructure expansion joints which may need 

removal and temporary replacement. 

• Significant damage includes permanent offsets and cracking, yielded reinforcement, and 

major spalling of concrete, which may require closure to repair. Partial or complete 

replacement of columns may be required. Beams may be unseated from bearings but no span 

should collapse. Similarly, foundations are not damaged except in the event of large lateral 

flows due to liquefaction, in which case inelastic deformation in piles may be evident. 

• Negligible damage includes evidence of movement, and/or minor damage to nonstructural 

components, but no evidence of inelastic response in structural members or permanent 

deformations of any kind. 

Higher levels of performance may be specified by the owner. For example, the following criteria 

might be used for extremely important bridges: 

• No damage is sustained and full service is available to all traffic immediately after the 

earthquake. No repairs are required. 

Generally, the performance criteria vary with level of earthquake ground motion, bridge 

importance and anticipated service life. In this manual, these objectives are defined for two 

ground motion levels (a lower and an upper level), two importance classifications (standard and 

essential), and three service life categories (ASL l, 2 and 3), as discussed below. 

7
8

Figure 1-2 (right). 

Collapse of the link span at Tower E9 of the 

San Francisco Oakland Bay bridge due to 

inadequate seat lengths and anchor bolts. 

Loma Prieta earthquake, 1989 

Figure 1-3 (left). 

Collapse of the two-level Cypress Viaduct on 

I-880 in Oakland due to brittle shear failure at 

the connection between the upper and lower 

levels of the viaduct. 

Loma Prieta earthquake, 1989 

Figure 1-4 (right). 

Collapse of end spans in the 

Shi Wei bridge in Taichung, 

due to ground failure and 

nearby fault rupture. 

Chi Chi earthquake, 1999

Figure 1-5 (left). 

Diagonal shear crack in lightly 

reinforced concrete pier of the Wu 

Shu bridge in Taichung. 

Chi Chi earthquake, 1999
9

HL1 HL2 HL3 HL4PL0

PL1

PL2

PL3

0

1

2

3

4

5

6

7

8

9

10

Relative Effort

Hazard Level

Performance 

Level

PL0 PL1

PL2 PL3

Figure 1-6. Conceptual relationship between relative effort, increasing hazard 

 and performance criteria implied in this manual. 

1.4.2. EARTHQUAKE GROUND MOTION LEVELS 

The lower level (LL) earthquake ground motion is one that has a reasonable likelihood of 

occurrence within the life of the bridge (assumed to be 75 years), i.e., it represents a relatively 

small but likely ground motion1

. It is common practice to use a probability of exceedance to 

characterize the motion, as noted in section 1.3. Accordingly, the lower level motion has a 

relatively high probability of exceedance within the life of a bridge, and a figure of 50 percent is 

recommended for retrofit design. A 50 percent probability of exceedance in 75 years corresponds 

to a return period of about 100 years (table 2-1). 

By contrast, the upper level (UL) earthquake ground motion has a finite, but remote, probability 

of occurrence within the life of the bridge; i.e., it represents a large but unlikely ground motion2

. 

Just as for the lower level motion, it is common practice to use a probability of exceedance to 

characterize this motion. Thus the upper level earthquake ground motion has a relatively low 

 

1

 This ground motion is sometimes called the frequent earthquake. In the NCHRP 12-49 Recommended LRFD 

Guidelines for Seismic Design (ATC/MCEER, 2003) it is called the expected earthquake, and in the Caltrans 

Seismic Design Methodology (Caltrans, 1999) it is called the functional evaluation earthquake (FEE). 2

 This ground motion is sometimes called the rare earthquake. In the NCHRP 12-49 Recommended LRFD 

Guidelines for Seismic Design (ATC/MCEER, 2003) it is the maximum considered earthquake (MCE), and in the 

Caltrans Seismic Design Methodology (Caltrans, 1999) it is the safety evaluation earthquake (SEE).
probability of exceedance within the life of a bridge. In this manual, the upper level motion has a 

7 percent probability of exceedance in 75 years, which corresponds to a return period of about 

1,000 years (table 2-1). 

Spectral ordinates and peak ground accelerations for both the lower and upper level ground 

motions may be found using the CD-ROM included in this manual (inside back cover)3. Bridge 

sites may be identified by zip code or, more accurately, by latitude and longitude. However, 

values given on this CD are in terms of an exposure period of 50 years rather than the 75-year 

bridge life assumed above. Therefore, equivalent exceedance probabilities for a 50-year life are 

required to use this CD, as follows: 

• The return period for an earthquake ground motion with a 50 percent probability of 

exceedance in 75 years is 108 years. However, this return period is only 72 years for a 

ground motion with a 50 percent probability of exceedance in 50 years. For the purpose of 

this manual, this lesser return period is considered to be close enough to the specified value 

of 100 years (table 1-2), and that values for the 50-year life may be used. Therefore, for the 

lower level ground motions, use data from the CD for 50 percent probability of exceedance 

in 50 years. 

• The return period for an earthquake ground motion with a seven percent probability of 

exceedance in 75 years is approximately the same as that for a ground motion with five 

percent probability of exceedance in 50 years (both are about 1,000 years). Therefore, for the 

upper level ground motions, use data from the CD for five percent probability of exceedance 

in 50 years. 

Alternatively, spectral ordinates and peak ground accelerations may be obtained for the upper 

level ground motion from the following web site maintained by the U.S. Geological Survey: 

http://eqhazmaps.usgs.gov. Mapped values are given for regions within the United States, and 

numerical values are given for specific locations according to zip code, or longitude and latitude. 

As with the CD-ROM, these values are expressed in terms of a 50-year bridge life and equivalent 

exceedance probabilities must be found to use this site, as described for the CD. At this time, this 

site does not give spectral ordinates and accelerations for the lower level ground motions (100 

years) and the only known, readily available, source of this data is the CD-ROM included with 

this manual. 

Some performance-based specifications for bridges and buildings have recommended a three 

percent probability of exceedance in 75 years for the upper level ground motions; but these 

specifications are for new construction and not necessarily appropriate for the retrofit of existing 

structures. Seismic resistance is much easier to provide in new structures than in existing ones. 

The selection of the reduced upper level motions for retrofitting is a compromise between the 

need to provide life safety and adequate performance for these less frequent motions and the 

limited resources of the owner. Keep in mind that these performance criteria are general 

recommendations and subject to change by the owner (or engineer) when specific circumstances 

of a particular bridge make it necessary. 

 

3

 This CD contains the seismic hazard maps and curves published by USGS in 1996. Updated maps were published 

in 2002 but not for the 1000-year return period. See section 2.3. 

10
11

 

1.4.3. BRIDGE IMPORTANCE 

Classification of bridge importance based on traffic counts and detour lengths has been proposed 

in the past and importance indices developed. But such quantitative methods do not usually 

include many non-technical issues that directly affect importance and are loosely called socioeconomic factors. Instead, a broad classification based on engineering judgment is preferred, and 

in this manual two such classes are recommended: essential and standard. Essential bridges are 

those that are expected to function after an earthquake or which cross routes that are expected to 

remain open immediately following an earthquake. All other bridges are classified as standard.

The determination of importance is therefore subjective and consideration should be given to 

societal/survival and security/defense requirements when making this judgment. 

An essential bridge is, therefore, one that satisfies one or more of the following conditions: 

• A bridge that is required to provide secondary life safety; e.g., one that provides access to 

local emergency services such as hospitals. This category also includes those bridges that 

cross routes that provide secondary life safety, and bridges that carry lifelines such as electric 

power and water supply pipelines. 

• A bridge whose loss would create a major economic impact; e.g., one that serves as a major 

link in a transportation system, or one that is essential for the economic recovery of the 

affected region. 

• A bridge that is formally defined by a local emergency plan as critical; e.g., one that enables 

civil defense, fire departments, and public health agencies to respond immediately to disaster 

situations. This category also includes those bridges that cross routes that are defined as 

critical in a local emergency response plan and those that are located on identified evacuation 

routes. 

• A bridge that serves as a critical link in the security and/or defense roadway network. 

Security and defense requirements may be evaluated using the 1973 Federal-aid Highway 

Act, which required that a plan for defense highways be developed by each state. Now called 

STRAHNET, this defense highway network provides connecting routes to military 

installations, industries, and resources and is part of the National Highway System. 

1.4.4. ANTICIPATED SERVICE LIFE 

An important factor in deciding the extent to which a bridge should be retrofitted is the 

anticipated service life (ASL). Retrofitting a bridge with a short service life is difficult to justify 

for two reasons: it is not economical and the design earthquake is unlikely to occur during the 

remaining life of the structure. On the other hand, a bridge that is almost new or being 

rehabilitated to extend its service life, should be retrofitted for the longer remaining service life.

Estimating remaining life is not an exact science and depends on many factors such as age, 

structural condition, specification used for design, and capacity to handle current and future 
traffic. Nevertheless, estimates can be made, at least within broad ranges, for the purpose of 

determining a bridge’s remaining service life and, subsequently, a retrofit category. Three such 

categories are used in this manual, as defined in table 1-1. When setting these categories, it was 

noted that new bridges are assumed to have a service life of 75 years in the AASHTO LRFD 

Specification (AASHTO, 1998), and this life span was then divided into three categories for the 

purpose of assigning retrofit levels (retrofit categories) according to age and remaining life. It is 

recognized that many long-span bridges have service lives far greater than 75 years, but these are 

outside the scope of this manual. Bridges in benign climates and those located on low-density 

routes may also have service lives in excess of 75 years. 

Bridges in category ASL 1 are considered to be near the end of their service life and retrofitting 

may not be economically justified. Thus, these bridges need not be retrofitted and are assigned to 

the lowest seismic retrofit category, i.e., category A (see section 1.6). Bridges in category ASL 3 

are almost new, and retrofitting to the standard of a new design may be justified. Those in 

category ASL 2 fall between these two extremes and a lesser standard is acceptable. However, 

the owner may always choose to retrofit to a higher standard as circumstances permit. 

Table 1-1. Service life categories. 

SERVICE LIFE 

CATEGORY 

ANTICIPATED 

SERVICE LIFE 

ASL 1 0 - 15 yrs 

ASL 2 16 - 50 yrs 

ASL 3 > 50 yrs 

Bridges are often rehabilitated toward the end of their service life to address deficiencies that 

have accumulated over time (e.g., deteriorated deck slabs, frozen bearings and damaged 

expansion joints), improve safety, and to accommodate increased traffic volume. As a 

consequence, a bridge with 15 years, or less, of life may, after rehabilitation, have a new service 

life of 35 years, and in so doing, the service life category (ASL) for the bridge has been lifted 

from ASL 1 to ASL 2 (table 1-1). The bridge should now be reevaluated for seismic 

performance, which should be done at the same time as planning the other rehabilitation. In this 

way, retrofit measures (if needed) can be implemented at the same time. By taking advantage of 

the contractor being on site, the cost of the seismic retrofit may be significantly reduced. 

12
1.4.5. SELECTION OF PERFORMANCE LEVEL 

Recommended minimum performance levels are given in table 1-2 according to the level of 

earthquake ground motion, bridge importance and service life category, as defined above. If 

retrofitting to these levels cannot be justified economically, the owner may choose a lower level. 

On the other hand, for certain classes of bridges, the owner may choose a higher level than that 

recommended here. An example of such a case is the bridges on STRAHNET, which are 

critically important to the operation of national or regional transportation routes. It is likely that 

some of these bridges are of sufficient importance to justify site-specific and structure-specific 

performance criteria. Such bridges may fall outside the scope of this manual. 

1.4.6. RETROFITTING PROCESS FOR DUAL LEVEL GROUND MOTIONS 

As noted in section 1.3, retrofitting is only one of several courses of action when faced with a 

bridge that is seismically vulnerable. Others include bridge closure, bridge replacement, and 

acceptance of the damage and its consequences. Bridge closure or replacement is usually not 

justified by seismic deficiency alone and will generally only be an option when other 

deficiencies exist. Therefore, for all practical purposes, a choice is made between strengthening 

and accepting the risk. This decision often depends on the importance of the bridge and on the 

cost and effectiveness of the proposed retrofit. 

Budget constraints and limited resources prevent the simultaneous retrofit of all of the deficient 

bridges on the highway system, and the most critical bridges should be upgraded first. The 

selection and prioritizing of bridges for retrofitting requires an appreciation of not just the 

engineering issues but also the economic, social, and practical aspects of the situation. 

Since it is recommended above that the seismic performance of a bridge be checked for two 

levels of earthquake ground motion (lower level and upper level) the overall retrofitting process 

has two distinct stages: 

• Stage 1. Screening, evaluation and retrofitting for the lower level earthquake ground motion, 

and 

• Stage 2. Screening, evaluation and retrofitting for the upper level earthquake ground motion. 

It is not possible to combine these two stages into one, since the performance criteria for each is 

very different. For example, the criteria for the lower level ground motion includes no structural 

damage and no repair (i.e., elastic behavior is expected) whereas for the upper level ground 

motion, damage is acceptable provided collapse does not occur and, for some bridges, access for 

emergency vehicles is available (i.e., inelastic behavior is expected). 

Each stage comprises the three steps discussed in section 1.1 and is shown schematically in 

figure 1-1, i.e., screening, evaluation, and retrofitting for the relevant ground motion. The 

breakdown of each stage into these steps is illustrated in figure 1-7, and discussed in sections 1.7 

and 1.8. 

13
Table 1-2. Minimum performance levels for retrofitted bridges. 

BRIDGE IMPORTANCE 

and 

SERVICE LIFE CATEGORY 

Standard Essential 

EARTHQUAKE 

GROUND MOTION 

ASL 1 ASL 2 ASL 3 ASL 1 ASL 2 ASL 3 

Lower Level 

Ground Motion 

50 percent probability 

of exceedance in 75 

years; 

return period is about 

100 years. 

PL04 PL3 PL3 PL04 PL3 PL3 

Upper Level 

Ground Motion 

7 percent probability of 

exceedance in 75 

years; return period is 

about 1,000 years. 

PL04 PL1 PL1 PL04 PL1 PL2 

Notes: 

1. Anticipated Service Life categories are: 

• ASL 1: 0 – 15 years 

• ASL 2: 16 – 50 years 

• ASL 3: > 50 years 

2. Performance Levels are: 

• PL0: No minimum level of performance is recommended. 

• PL1: Life safety. Significant damage is sustained and service is significantly 

disrupted, but life safety is preserved. The bridge may need to be replaced after a 

large earthquake. 

• PL2: Operational. Damage sustained is minimal and service for emergency 

vehicles should be available after inspection and clearance of debris. Bridge should 

be reparable with or without restrictions on traffic flow. 

• PL3: Fully Operational. Damage sustained is negligible and full service is 

available for all vehicles after inspection and clearance of debris. Damage is 

repairable without interruption to traffic. 

3. Spectral ordinates and peak ground accelerations may be found for the Upper Level 

earthquake ground motion from http://eqhazmaps.usgs.gov. Ordinates and ground 

accelerations may be found for both the Upper and Lower Level ground motions from 

the CD-ROM: Seismic Hazard Curves and Uniform Hazard Response Spectra for the 

United States, (Frankel and Leyendecker, 2001; see section 2.3). A copy of this CD is 

included with this manual (see inside back cover). 

4. Bridges assigned a Performance Level of PL0 have 15 years, or less, anticipated 

service life (ASL) and are candidates for replacement or rehabilitation. 

14
15

Is Bridge Exempt?

Screen

Evaluate

Screen

Evaluate

Retrofit

Retrofit

Next

Bridge

Lower Level Earthquake

Ground Motion

(Section 1.6)

Yes

No

Pass

Pass

Fail

Fail

Review

Pass

Fail

Pass

Fail Review

Upper Level Earthquake

Ground Motion

(Section 1.7)

Figure 1-7. Retrofit process for dual level earthquake ground motions. 
16

1.4.7. EXEMPT BRIDGES 

A bridge is exempt from retrofitting for both levels of ground motion if it satisfies any one of the 

following criteria: 

• The bridge has 15 years or less of anticipated service life (section 1.4.4). 

• The bridge is ‘temporary’ (i.e., anticipated service life is 15 years or less). 

• The bridge is closed to traffic and does not cross an active highway, rail or waterway. 

A bridge is exempt from retrofitting for the upper level ground motion if it meets the criteria for 

Seismic Retrofit Category A (section 1.6) for this level of motion. 

1.5. SEISMIC HAZARD LEVEL 

The Seismic Hazard Level (SHL) at a bridge site is determined by the intensity of ground 

shaking in the rock below the site and the amplification of this motion by the overlying soils. 

Motions at the surface may be considerably greater than in the rock below. These two factors are 

treated separately below and then combined to define the SHL in section 1.5.4. Other 

geotechnical factors, such as liquefaction and fault rupture, can influence the hazard at a bridge 

site and these are also briefly described in this section. 

1.5.1. GROUND MOTION REPRESENTATION 

Earthquake ground motion may be characterized by either: 

a. The peak ground acceleration expected at a site in a given period of time, from which the 

design spectrum may be drawn based on several simplifying assumptions, or 

b. By two points on the design spectrum from which the remainder of the spectrum may be 

drawn using fewer, but more realistic, assumptions than in (a) above. 

The earlier edition of this manual (FHWA, 1995) and the 1997 AASHTO LRFD Bridge Design 

Specifications (AASHTO, 1998) use the former method, whereas the NCHRP 12-49 provisions 

(ATC/MCEER, 2003) use the latter method, to take advantage of the improved accuracy that it 

offers. This edition of the retrofit manual also uses the latter two-point method. 

The two points used to define the spectrum are spectral ordinates (peak structural accelerations) 

in bridges with periods of 0.2 and 1.0 second. For bridge sites on rock (site class B), these 

ordinates are identified as Ss and S1, respectively. The variation of these ordinates throughout 

the United States has been calculated for various return periods by the US Geological Survey 

(USGS), and the results are available (in both tabular and mapped formats) on the CD-ROM 

provided with this manual (inside back cover). Alternately, results for a selected number of 

return periods may be downloaded from the following web site: http://eqhazmaps.usgs.gov. 

Chapter 2 summarizes the development of these data and the method for constructing a design 
17

spectrum from the given spectral ordinates. In lieu of using national ground motion maps, the 

spectral accelerations Ss and S1 may be found from approved state ground motion maps. 

1.5.2. GEOTECHNICAL FACTORS 

Earthquake ground motions at a bridge site are not only dependent on the intensity of ground 

shaking in the rock below the site but also on the amplification of this motion by the overlying 

soils. The presence of one or more geotechnical hazards at the site, such as liquefaction and 

fault rupture, can also have a significant effect on the motion experienced by a bridge. These 

factors are discussed in this section. 

1.5.2.1 Soil Amplification of Ground Motion 

The behavior of a bridge during an earthquake is strongly related to the soil conditions at the 

site. Soils can amplify ground motions in the underlying rock, sometimes by factors of two or 

more. The extent of this amplification is dependent on the profile of soil types at the site and the 

intensity of shaking in the rock below. Sites are classified by type and profile for the purpose of 

defining the overall seismic hazard, which is quantified as the product of the soil amplification 

and the intensity of shaking in the underlying rock. 

Site class definitions are given in table 1-3, where it is seen that sites are classified by their 

stiffness as determined by the shear wave velocity in the upper 30 m (100 ft). Standard 

Penetration Test (SPT) blowcounts and undrained shear strengths of soil samples from soil 

borings can also be used to classify sites as indicated in table 1-3. See chapter 2 for further 

details, including steps for classifying a site. 

Site factors, for the site classes in table 1-3, are given in table 1-4. Site class B (soft rock) is 

taken to be the reference site category for the USGS and NEHRP MCE ground shaking maps. 

Site class B rock is therefore the site condition for which the site factor is 1.0. Site classes A, C, 

D, and E have separate sets of site factors for the short-period range (site factor Fa) and longperiod range (site factor Fv), as indicated in table 1-4. These site factors generally increase as the 

soil profile becomes softer (in going from site class A to E). Except for site class A (hard rock), 

the factors also decrease as the ground motion level increases, due to the strongly nonlinear 

behavior of the soil. For a given site class, C, D, or E, these nonlinear site factors increase the 

ground motion more in areas having lower rock ground motions than in areas having higher rock 

ground motions. The levels of ground motion for use with table 1-4 are characterized by shortperiod (0.2 second) response spectral acceleration and long-period (1.0 second) response spectral 

acceleration on rock as mapped on USGS and NEHRP MCE national ground motion maps. 

Special considerations for site class F soils are discussed in chapter 2. 

1.5.2.2 Geotechnical Hazards 

Geotechnical hazards at bridge sites that can be triggered by earthquakes include soil 

liquefaction, soil settlement, slope failure (landslides and rock falls), surface fault rupture and 

flooding. In general, these hazards are examined independently of the rock ground motion and 

soil amplification effects described in the previous section. 
18

Table 1-3. Site classes. 

Site 

Class Description 

A Hard rock with measured shear wave velocity, vs > 1500 m/sec (5,000 ft/sec) 

B Rock with 760 m/sec < s v < 1500 m/sec (2,500 ft/sec < s v < 5,000 ft/sec) 

C 

Very dense soil and soil rock with 360 m/sec < s v < 760 m/sec (1,200 ft/sec < s v < 2,500 ft/sec) 

or with either N > 50 blows/0.30m (50 blows/ft) or us > 100 kPa (2,000 psf) 

D 

Stiff soil with 180 m/sec < s v < 360 m/sec (600 ft/sec < s v < 1,200 ft/sec) 

or with either 15 < N < 50 blows/0.30m (15 < N < 50 blows/ft) or 50 kPa < us < 100 kPa 

(1,000 < us < 2,000 psf) 

E 

Soil profile with s v < 180 m/sec (600 ft/sec) 

or with either N < 15 blows/0.30m (N < 15 blows/ft) or us < 150 kPa (1000 psf), 

or any profile with more than 3 m (10 ft) of soft clay defined as soil with PI > 20, w > 40 percent 

and us < 25 kPa (500 psf) 

F 

Soils requiring site-specific evaluations 

1. Peats or highly organic clays (H > 3 m [10 ft] of peat or highly organic clay where H = 

thickness of soil) 

2. Very high plasticity clays (H > 8 m [25 ft] with PI > 75) 

3. Very thick soft/medium stiff clays (H > 36 m [120 ft]) 

Exception: When the soil properties are not known in sufficient detail to determine the site class, site class 

D may be used. Site classes E or F need not be assumed unless the authority having jurisdiction 

determines that site classes E or F could be present at the site or in the event that site classes E or F are 

established by geotechnical data. 

Notes: 

1. s v is average shear wave velocity for the upper 30 m (100 ft) of the soil profile 

N is the average Standard Penetration Test (SPT) blowcount (blows/0.30m or blows/ft) (ASTM D1586) 

 for the upper 30 m (100 ft) of the soil profile 

us is the average undrained shear strength in kPa (psf) (ASTM D2166 or D2850) for the upper 30 m (100 ft) 

 of the soil profile 

 PI is plasticity index (ASTM D4218) 

 w is moisture content (ASTM D2216) 

2. The shear wave velocity for rock, Site Class B, shall be either measured on site or estimated for competent rock 

with moderate fracturing and weathering. Softer and more highly fractured and weathered rock shall either be 

measured on site for shear wave velocity or classified as Site Class C. 

3. The hard rock, Site Class A, category shall be supported by shear wave velocity measurements either on site or on 

profiles of the same rock type in the same formation with an equal or greater degree of weathering and fracturing. 

Where hard rock conditions are known to be continuous to a depth of 30 m (100 ft), surficial shear wave velocity 

measurements may be extrapolated to assess s v . 

4. Site classes A and B should not be used when there is more than 3 m (10 ft) of soil between the rock surface and 

the bottom of a spread footing. 
19

Table 1-4. Site factors Fa and Fv. 

(a) Values of Fa as a function of site class and short-period (0.2-sec) spectral acceleration, Ss 

Spectral Acceleration at Short-Period (0.2 sec), SS

1 

Site 

Class SS < 0.25 SS = 0.50 SS = 0.75 SS = 1.00 SS > 1.25 

A 0.8 0.8 0.8 0.8 0.8 

B 1.0 1.0 1.0 1.0 1.0 

C 1.2 1.2 1.1 1.0 1.0 

D 1.6 1.4 1.2 1.1 1.0 

E 2.5 1.7 1.2 0.9 0.9 

F2 

Notes: 

1. Use straight-line interpolation for intermediate values of SS. 

2. Site-specific geotechnical investigation and dynamic site response analysis should be performed for 

 class F soils.

(b) Values of Fv as a function of site class and long-period (1.0-sec) spectral acceleration, S1

Spectral Acceleration at Long-Period (1.0 sec), S1

1 

Site 

Class S1 < 0.1 S1 = 0.2 S1 = 0.3 S1 = 0.4 S1 > 0.5 

A 0.8 0.8 0.8 0.8 0.8 

B 1.0 1.0 1.0 1.0 1.0 

C 1.7 1.6 1.5 1.4 1.3 

D 2.4 2.0 1.8 1.6 1.5 

E 3.5 3.2 2.8 2.4 2.4 

F2 

Notes: 

1. Use straight-line interpolation for intermediate values of S1. 

2. Site-specific geotechnical investigation and dynamic site response analysis should be performed for class F 

 soils.
20

Procedures for evaluating these hazards are presented in chapter 3. The consequences of these 

hazards in the free-field (e.g. ground settlement and fault displacements) are also addressed in 

that chapter. Methods for evaluating the effects of these hazards on the capacity and 

deformations of the bridge-foundation system are presented in chapter 6, and measures for 

mitigating them are described in chapter 11. 

Assessing geotechnical hazards is a two-part procedure. In the first part, a quick screening 

evaluation is conducted. Generally, this can be accomplished using available information and 

field reconnaissance. If the criteria are satisfied, the risk is considered to be low and further 

evaluations of the hazard are not required. If a hazard cannot be screened out, more detailed 

evaluations are conducted in the second part of this procedure. This usually requires obtaining 

additional data to more rigorously assess the hazard and its consequences. Chapter 3 gives 

guidelines for initial hazard screening, together with a brief overview of methods for detailed 

hazard evaluation. Detailed methods for hazard evaluation are described in appendix B. Hazard 

screening and hazard evaluation should be carried out by geotechnical engineers, geologists, and 

seismologists, who have expert knowledge of these hazards and experience with their evaluation. 

 

1.5.3. DETERMINATION OF DESIGN RESPONSE SPECTRUM 

 

As noted in section 1.5.1, a two-point method is used to define the design response spectrum 

from which earthquake forces are calculated to determine seismic demand on a bridge due to a 

particular earthquake ground motion. Figure 1-8 shows how this spectrum is drawn from the two 

given points. It indicates that the Ss and S1 ordinates are first scaled by the soil factors Fa and Fv

and the resulting products SDS and SD1 are used to plot the spectrum where: 

 SDS = Fa . Ss and SD1 = Fv . S1 (1-1) 

Figure 1-8. Construction of the seismic design response spectrum. 
21

1.5.4. DETERMINATION OF SEISMIC HAZARD LEVEL 

Each bridge is assigned a Seismic Hazard Level (SHL) as indicated in table 1-5, based on SDS

and SD1 , which are defined in section 1.5.3 to be the product of the spectral ordinates Ss and S1

(section 1.5.1) and the soil factors Fa and Fv (section 1.5.2). In the event that two different hazard 

levels are indicated by table 1-5 for the same site, the higher level should be used. 

The two footnotes to table 1-5 effectively limit boundaries for site classes E and F in Hazard 

Levels I and II to those of site class D. This is because of the greater uncertainty in the values of 

Fv and Fa for class E and F soils when ground shaking is relatively low (S1 < 0.10 and Ss < 0.25). 

Table 1-5. Seismic hazard level. 

HAZARD LEVEL Using SD1 = FvS1 Using SDS = FaSs

I SD1 ≤ 0.15 SDS ≤ 0.15 

II 0.15 < S D1 ≤ 0.25 0.15 < SDS ≤ 0.35 

III 0.25 < S D1 ≤ 0.40 0.35 < SDS ≤ 0.60 

IV 0.40 < S D1 0.60 < SDS

Notes: 

1. For the purposes of determining the Seismic Hazard Level for Site Class E soils, the 

value of Fv and Fa need not be taken larger than 2.4 and 1.6 respectively, when S1 is less 

than or equal to 0.10 and Ss is less than 0.25. 

2. For the purposes of determining the Seismic Hazard Level for Site Class F soils, Fv and 

Fa values for Site Class E soils may be used with the adjustment described in Note 1 

above. 

1.6. PERFORMANCE-BASED SEISMIC RETROFIT CATEGORIES 

Seismic retrofit categories (SRC) are used to identify minimum screening requirements, 

evaluation methods and retrofitting measures for deficient bridges. They are determined by the 

anticipated service life, importance, and the seismic hazard exposure of the bridge. There are 

four categories, A through D, in increasing order of rigor and complexity. SRC A is a default 

category, which means that bridges in this category do not need to be screened, evaluated or 

retrofitted. SRC D is the highest category requiring the most rigorous screening, evaluation, and 

retrofitting measures. SRC’s perform a similar function to the SPC’s (seismic performance 

categories) in the previous retrofit manual (FHWA, 1995). (The name has been changed to 

distinguish them from the SPC’s used in the seismic design provisions of the AASHTO Standard 

Specifications (AASHTO, 2002), which are widely used for the seismic design of new bridges.) 

Seismic retrofit categories are given in table 1-6 where they are determined by the performance 

level (PL) required, and the seismic hazard level (SHL) at the site. 
22

Table 1-6. Performance-based seismic retrofit categories. 

PERFORMANCE LEVEL 

During Upper Level Earthquake During Lower Level 

Earthquake HAZARD 

LEVEL PL0: No 

Minimum 

Level 

PL1: Life 

Safety PL2: 

Operational 

PL0: No 

Minimum 

Level 

PL3: Fully 

Operational 

I A A B A C 

II A B B A C 

III A B C A C 

IV A C D A D 

The steps required to determine an SRC are shown in figure 1-9 and are as follows: 

Step 1. From the bridge records and other sources determine the: 

• Importance of the bridge: either standard or essential,

• Anticipated service life of the bridge, and assign a service category: ASL 1 through 

ASL 3 (table 1-1), and 

• Site class based on soil type and profile (table 1-3). 

Step 2. Determine the performance level for the bridge (PL0 through PL3), based on the 

anticipated service life and bridge importance (table 1-2). 

Step 3. Obtain the spectral accelerations Ss and S1, the soil factors Fa and Fv from table 1-4, and 

determine the hazard level using table 1-5. Determine the seismic retrofit category 

required to satisfy the performance objective for this event, using table 1-6. These steps 

are summarized as follows: 

Step 3.1. Obtain the spectral ordinates, Ss and S1, from the CD-ROM in this manual or 

from the USGS web site: http://eqhazmaps.usgs.gov. 

Step 3.2. Obtain the soil factors, Fa and Fv, for the site (table 1-4). 

Step 3.3. Determine the seismic hazard level (SHL) based on FaSs and FvS1 (table 1-5). 

Step 3.4. Obtain the SRC for the required performance level from table 1-6. 

This process is illustrated in example 1.1 in this chapter, and examples 4.1 and 4.2 in chapter 4. 
23

Bridge Importance Anticipated Service

Life, ASL

Spectral Ordinates,

Ss and S1

Spectral Accelerations,

Ss

 and S1

Soil Factors,

Fa and Fv

SEISMIC HAZARD

LEVEL, SHL

PERFORMANCE

LEVEL, PL

SEISMIC RETROFIT

CATEGORY, SRC

Figure 1-9. Determination of seismic retrofit category. 

1.7. RETROFITTING PROCESS FOR THE LOWER LEVEL GROUND MOTION 

 

The lower level earthquake ground motion has a return period of about 100 years and is therefore 

a relatively small motion. The performance required of any bridge for these motions (table 1-2) 

is elastic behavior, which is similar to the requirements for wind and braking loads. Indeed, in 

many parts of the United States, wind and braking loads may be larger than the earthquake loads 

for this return period and may govern for this stage. 

The process of retrofitting bridges for the lower level earthquake ground motion is divided into 

three parts. These are: 

• Preliminary screening of bridge inventory. 

• Detailed evaluation of an existing bridge. 

• Selection of retrofit strategy and design of retrofit measures. 

Note that decisions are made at each step, based on the results obtained thus far, that determine 

the next step in the process. Bridges are passed through a series of checkpoints to assure that 

only those structures actually in need of retrofit will be strengthened. 

As described in section 1.6, seismic retrofit categories (SRC) are used to recommend minimum 

requirements for evaluation and retrofitting. These requirements for the lower level ground 

motion are given in table 1-7. 
Table 1-7. Minimum requirements. 

SEISMIC RETROFIT CATEGORY FOR THE 

LOWER LEVEL EARTHQUAKE 

SEISMIC RETROFIT CATEGORY FOR THE 

UPPER LEVEL EARTHQUAKE ACTION 

A B1 C D A B C D 

Screening Components to be screened: 

NR2 - seat widths, 

connections, columns, walls, footings and liquefaction 

seat widths, connections, columns, walls, footings, abutments and liquefaction 

NR seat widths, 

connections and liquefaction 

seat widths, connections, columns, walls, footings and liquefaction 

seat widths, connections, columns, walls, footings, abutments and liquefaction 

EvaluationEvaluation methods to be used3,4 . See table 1-9. NR - C C NR A1/A2 B/C/D1/D2 C/D1/D2/E 

Seats and Connections NR - Yes Yes NR Yes Yes Yes Columns, walls, footings

NR - Yes Yes NR NR Yes Yes 

Abutments NR - NR Yes NR NR NR Yes 

Retrofitting Components to be retrofitted, if deficient:

Liquefaction NR - Yes Yes NR Yes Yes Yes 

Notes: 

1. Seismic Retrofit Category B is not used when evaluating and/or retrofitting bridges for lower level ground motions. 

2. NR = Not Required. 3. A1/A2 = No analysis; minimum capacity checks 

B = No analysis; component capacity checks C = Component C/D method using elastic dynamic analysis methods D1 = Capacity spectrum method. D2 = Structure C/D method; also called Nonlinear Static Procedure, or Displacement Capacity Evaluation E = Nonlinear dynamic method using inelastic time history analysis 

4. Selection of evaluation method also depends on bridge geometry: for irregular (complex) bridges, use Methods A, C, D2 and E as 

appropriate. 

24
25

The retrofitting process for this level of ground motion is described in the following sections 

where it will be seen to be a force-based approach. Displacements are not explicitly considered 

since they are assumed to be small and within the default capacity of the structure. This is a 

reasonable expectation provided yielding does not occur, bearing restraints do not fail, and soils 

do not soften at this level of ground motion. 

1.7.1. SCREENING AND PRIORITIZATION FOR THE LOWER LEVEL GROUND 

MOTION 

All bridges in the inventory should be checked for their lateral strength in both the transverse and 

longitudinal directions. This strength should be based on the elastic capacities of the members 

and foundations. Previous design calculations for wind and braking loads may be useful here. 

This capacity should then be checked against the seismic force (F) on the bridge, which can be 

conservatively estimated as: 

 F = SDS W (1-2) 

where SDS is defined in equation (1-1) and W is the weight of the superstructure. If the capacity 

is greater than the demand (F), the bridge passes the screen and the process moves to the second 

stage (section 1.8). If the capacity is less than the demand, the bridge should be considered for 

retrofitting at this earthquake level. 

As noted above, a quick estimate of the elastic capacity of a bridge in the transverse direction is 

obtained by applying the factored wind load used in the design of the bridge. In the longitudinal 

direction, the factored braking load may be used for this purpose. If the seismic demand (F) is 

greater than either of these two service loads, a more detailed evaluation is necessary, as 

described in section 1.7.2. 

1.7.2. EVALUATION FOR THE LOWER LEVEL GROUND MOTION 

The seismic demand given by equation 1-2 is very conservative since it makes two simplifying 

assumptions. First, the period of the bridge is assumed to be very short (less than Ts, figure 1-8), 

in both the longitudinal and transverse directions, and second, the entire weight of the 

superstructure is assumed to be mobilized during an earthquake in both the longitudinal and 

transverse directions. 

Therefore, a two-step process is recommended for the detailed evaluation of a bridge for the 

lower level ground motion. These steps are: 

Step 1. Revise the estimate of the seismic demand using improved values for the period of 

vibration. 

Step 1.1. Calculate the period of the bridge in both the longitudinal and transverse 

directions using any of the elastic methods described in Method C (section 1.11). These 

include the Uniform Load Method and the Multi-mode Spectral Method. 
Step 1.2. Obtain the response spectral accelerations in the longitudinal and transverse 

directions (SaL and SaT, respectively), using the periods found in step 1.1 and figure 1-8. 

Step 1.3. Calculate the longitudinal and transverse seismic demands (FL and FT) from: 

 FL= SaL W (1-3a) 

 FT=SaT W (1-3b) 

Step 1.4. Compare FL and FT against total longitudinal braking load, and total transverse wind 

load respectively, and if either one is greater than the corresponding service load, a more 

rigorous evaluation is required. Go to step 2. If, however, both FL and FT are less than the 

corresponding service loads, no further evaluation is necessary and proceed to section 1.8. 

Step 2. Revise the estimate of bridge capacity using explicit member strengths. 

Use Method C (section 1.11 and section 5.4) to calculate capacity/demand ratios for each 

member and component in the lateral load path. Members and components with capacity/demand 

ratios less than unity are flagged for further consideration under section 1.7.3. 

1.7.3. RETROFITTING FOR THE LOWER LEVEL GROUND MOTION 

Once a bridge has been found to be vulnerable to the lower level ground motion, the next step is to 

decide what, if anything, should be done to correct the deficiencies. Decision-making may be 

formalized by exploring retrofit options and associated cost implications using the same process 

described for the upper level motion in section 1.12. In most instances, these deficiencies will be a 

lack of sufficient elastic strength, in which case component strengthening will be the most practical 

approach. The first column of table 1-11 lists a number of strengthening approaches that may be 

applicable. These approaches are described in chapters 8, 9 and 10. 

However, before significant effort is devoted to retrofitting for the lower level earthquake, evaluate 

the structure for the upper level event. It is possible that the bridge is also inadequate for the upper 

level motion. Thus, while addressing the needs for the upper level earthquake, the deficiencies at the 

lower level may be eliminated, and retrofitting specifically for the lower level would no longer be 

necessary. 

1.8. RETROFITTING PROCESS FOR UPPER LEVEL GROUND MOTION 

The process of retrofitting bridges for the upper level earthquake ground motion involves an 

assessment of many variables and requires the use of considerable judgment. Just as for the 

lower level ground motion, it helps to divide the process into three parts. These are: 

• Preliminary screening of bridge inventory. 

• Detailed evaluation of an existing bridge. 

• Selection of retrofit strategy and design of retrofit measures. 

Figure 1-1 illustrates this three-step process and figure 1-10 shows the process in greater detail 

for the upper level ground motion. Note that decisions are made at each step, based on the results 

obtained thus far, that determine the next step in the process. Bridges are passed through a series 

26
27

 Seismic Retrofit Category

(SRC)

(Section 1.6)

Seismic Retrofit

Categories B, C, D

Seismic Retrofit

Category A

Screening &

Prioritization

(Section 1.10)

DESIGN RETROFIT

MEASURES

(Section 1.13)

RETROFITTING NOT

REQUIRED

FAIL

PASS

NO

YES

Detailed

Evaluation

(Section 1.11)

FAIL

Strategize Retrofit

(Section 1.12)

Is retrofitting

warranted?

PASS

RATING METHODS

 1. Indices Method

 2. Expected Damage Method

 3. Seismic Risk Assessment Method

EVALUATION METHODS

(Table 1-9, Figure 1-13)

 A, B. No Analysis Methods

 C. Component C:D Method

 D1. Structure C:D Method (basic)

 D2. Structure C:D Method

 (advanced)

 E. Nonlinear Dynamic Method

CAPACITY ASSESSMENT

(Table 1-9, Figure 1-13)

 1. Member Strength Capacity

 2. Member Deformation Capacity

 3. Foundation Strength and

 Deformation Capacity

RETROFIT MEASURES

(Table 1-11)

 1. Extended seat widths

 2. Expansion joint restrainers

 3. Column jackets

 4. Footing Overlays

 5. Supplemental Piles

 6. Response Modification Devices

 7. Soil Remediation

RETROFIT STRATEGIES

(Table 1-11, Figure 1-14)

 1. Do nothing

 2. Partial Retrofit

 3. Full Retrofit

PERFORMANCE-BASED SEISMIC

RETROFIT CATEGORIES (SRCs)

(Figure 1-9)

 Anticipated Service Life

Bridge Importance

Site Class (Soil Factors)

Performance Criteria

Seismic Hazard Level for

Upper Level Ground Motion

SRC

RE-EVALUATE

Figure 1-10. Seismic retrofitting process for highway bridges subject to 
28

of checkpoints to assure that only those structures actually in need of retrofit will be 

strengthened. 

As described in greater detail in sections 1.6 and 1.9, seismic retrofit categories (SRC) are used 

to recommend minimum requirements for evaluation and retrofitting. The SRC assigned to a 

bridge dictates the level of effort required to screen, analyze and retrofit a bridge, should 

strengthening be necessary. The SRC is determined by the importance of the bridge, its 

anticipated service life, and the seismic hazard level. Using these three factors, it is possible to 

schematically illustrate the relative effort required to retrofit bridges of different importance, in 

different hazard zones, and with different service lives. Figure 1-11 shows this relationship for 

both ‘standard’ and ‘essential’ bridges. 

1.9. MINIMUM REQUIREMENTS FOR UPPER LEVEL GROUND MOTION 

Minimum requirements for screening, evaluation and retrofitting are determined by the Seismic 

Retrofit Category (SRC), as recommended in table 1-7. 

As noted above, bridges in SRC A need not be retrofitted due to their short anticipated service 

life, or the fact that they are located in the lowest seismic zone with minimum performance 

objectives (life safety only). Minimum screening, evaluation and retrofitting requirements are 

therefore given only for SRCs B, C and D. In general, the higher the category, the more rigorous 

the requirements, as seen in table 1-7 

The screening requirements in table 1-7 list those components of a bridge that must be examined 

when setting priorities for retrofitting. Since insufficient seat width and inadequate connections 

are common reasons for bridge failures, the minimum screening requirements (SRC B) start with 

seats and connections. Other components are added in the higher categories. 

The evaluation requirements in table 1-7 give permissible methods of evaluation depending on 

the SRC. They range from the ‘no analysis’ option and simplified elastic methods for SRC B, up 

to the rigorous, nonlinear dynamic analysis methods for bridges in SRC D. 

Table 1-7 lists those components of a bridge that should be retrofitted if found to be deficient. 

Inadequate seats and connections are common reasons for bridge failures and minimum 

requirements (SRC B) start with restrainers, seat width extensions and connection strengthening. 

More extensive retrofit measures such as column jacketing, footing overlays, cap beam 

prestressing, ground remediation and the like, are added in the higher categories. 

Finally, it is noted that the requirements in table 1-7 are minima, and the owner may at any time 

impose more rigorous requirements. 

 
29

HL1 HL2 HL3 HL4

ASL1

ASL2

ASL3

0

1

2

3

4

5

6

7

8

9

10

Relative Effort

Hazard Level

Service Life 

Category

ASL1 ASL2 ASL3

(a)‘Standard’ Bridge 

HL1

HL2

HL3

HL4

ASL1

ASL2

ASL3

0

1

2

3

4

5

6

7

8

9

10 Relative Effort

Hazard Level

Service Life 

Category

ASL1 ASL2 ASL3

(b) ‘Essential’ Bridge 

Figure 1-11. Relative effort to retrofit an (a) ‘standard’ and (b) ‘essential’ bridge 

with varying service life and hazard level. 
30

EXAMPLE 1.1: DETERMINATION OF SEISMIC RETROFIT CATEGORIES FOR BOTH 

UPPER AND LOWER LEVELS OF GROUND MOTION 

 

____________________________________________________________________________ 

An essential bridge in Salt Lake City, UT (zip code 84112) has an anticipated service life of 30 years, and 

is founded on very dense soils with an average shear wave velocity, in the upper 100 ft, of 1,350 ft/sec. 

Determine the seismic retrofit categories (SRC) for this bridge for both the upper and lower levels of 

ground motion. 

Step 1. Importance, Anticipated Service Life, and Site Class 

1.1 Bridge is stated to be essential.

1.2 For an anticipated service life of 30 years, the service life category is ASL 2 (table 1-1) 

1.3 For the given soil description and shear wave velocity, the site class is C (table 1-3) 

Step 2. Performance Criteria 

2.1 From table 1-2, for the lower level ground motion, the performance level is PL3. 

2.2 From table 1-2, for the upper level ground motion, the performance level is PL1. 

Step 3. Seismic Retrofit Category for Upper Level Ground Motion 

3.1 For zip code = 84112, the USGS CD-ROM gives SS = 1.11 and S1 = 0.39 (example 2.1) 

3.2 For site class C and the values obtained for SS and S1: Fa = 1.0 and Fv = 1.4 (table 1-4) 

3.3 FaSS = 1.0 (1.11) = 1.11 

 FvS1 = 1.4 (0.39) = 0.55 

 Seismic Hazard Level (SHL) = IV (table 1-5) 

3.4 Seismic retrofit category for PL1 and SHL = IV, is C (table 1-6, upper level motion). 

 This is the SRC for the upper level ground motion. 

Step 4. Seismic Retrofit Category for Lower Level Ground Motion 

4.1 For zip code = 84112, the USGS CD-ROM gives SS = 0.18 and S1 = 0.05 (example 2.1) 

4.2 For site class C and the values obtained for SS and S1: Fa = 1.2 and Fv = 1.7 (table 1-4) 

4.3 FaSS = 1.2 (0.18) = 0.22 

 FvS1 = 1.7 (0.05) = 0.09 

 Seismic Hazard Level (SHL) = II (table 1-5) 

4.4 Seismic retrofit category for PL3 and SHL = II, is C (table 1-6, lower level motion). 

 This is the SRC for the lower level ground motion. 
31

1.10. SCREENING AND PRIORITIZATION FOR UPPER LEVEL GROUND MOTION 

1.10.1. GENERAL 

Bridge inventories are screened to identify structures that are seismically deficient and to 

prioritize them in order of need for retrofitting. Such a process is intended to be rapid, easy to 

apply and conservative. Bridges found to be deficient at this stage are subject to detailed 

evaluation at the next step (section 1.11 and chapter 5), and any that are later found to be 

satisfactory are excluded from further study at that time. An overview of the recommended 

process for screening and prioritizing an inventory of bridges for retrofitting is illustrated in 

figure 1-12. Details are given in chapter 4. 

Figure 1-12. Screening and prioritization process. 

Seismic Retrofit Category (SRC)

Seismic Retrofit

Category A

Seismic Retrofit Categories:

B, C, D

Retrofitting Not

Required

Compile Bridge and Hazard

Data.

Calculate bridge rank, R,

based on vulnerability and

seismic hazard.

Compile Other Issues, O.

e.g., Bridge Importance,

Network Redundancy,

Nonseismic Rehabilitation

Needs

Calculate Priority Index,

P = f(R,O)
32

Three screening and prioritization methods are described in this manual, two are presented in 

depth, and the third is only briefly noted. Screening programs for bridges in Seismic Retrofit 

Category A need not be conducted beyond the initial determination of SRC, since bridges in this 

category are not required to be retrofitted, regardless of their structural deficiencies or level of 

hazard. 

Many screening and prioritization methods have been proposed in the past (Buckle, 1991). Most 

develop a Seismic Rating System first, and then use the results of this rating exercise to prioritize 

the inventory. Factors considered in the rating exercise usually include structural vulnerabilities, 

and prevailing seismic and geotechnical hazards. Some also include bridge importance and 

network redundancy at this stage, but others use these factors only when prioritizing the list of 

deficient structures. Another important factor to be considered is the political, social, and 

economic context in which the retrofit program is being conducted. Regardless of the process 

used to develop the final prioritized list, all bridges should be subject to detailed evaluation 

before actual retrofitting is undertaken, to confirm the identified structural deficiencies and 

determine the cost-benefits of retrofitting. 

1.10.2. FACTORS TO BE CONSIDERED 

The objective of a screening and prioritization program is to determine which bridge (or set of 

bridges) should be retrofitted first. Factors affecting such a program are the structural and soil 

vulnerabilities, the level of the seismic hazard and several other factors which include: 

• Bridge importance. This has been discussed in section 1.4.3. 

• Network redundancy. This is a measure of route availability and is used to calculate bridge 

importance. Whereas redundancy implies resiliency in the network, and should lessen the 

need for retrofitting, setting priorities based on redundancy alone is not so straightforward. 

For example, the likelihood that alternative routes will be damaged in the same earthquake 

must be considered. Suppose a freeway overpass is highly vulnerable but it can be bypassed 

using adjacent access ramps. Since a convenient detour is nearby, a lower seismic rating 

might be assigned. But this assumes that the ramps remain operational, which may not be the 

case if there is strong ground motion in the area. If, on the other hand, the structure is a 

vulnerable river crossing, and the nearest detour several miles away, the redundancy of the 

network will be low. However, the possibility of the alternate crossing being damaged will be 

low and an alternate route may be available. The length of the detour then becomes the issue 

when deciding priority for retrofitting. Other examples where network redundancy leads to 

unexpected results are given in example 1.2. 

• Age and physical condition. It is generally not wise to spend a large sum retrofitting a bridge 

with relatively few years of service life remaining. This is one reason why bridges in SRC A 

do not need to be retrofitted. It is also true that an unusually high seismic vulnerability may 

be justification to accelerate closure or replacement of such a bridge. Also, a bridge in poor 

physical condition, or one that is already scheduled for structural or functional rehabilitation, 

may be given a higher priority for seismic retrofitting, since cost savings can be achieved by 

performing the non-seismic and seismic work simultaneously. 
33

The above factors are not an exhaustive list, but illustrate some of the principles involved in 

assigning priorities. In most cases, seismic ratings are used to guide decision-making but are not 

the final word. Common sense and engineering judgment are necessary when weighing the 

actual costs and benefits of retrofitting against the risks of doing nothing. 

1.10.3. SEISMIC VULNERABILITY RATING METHODS 

Most vulnerability rating methods assign a structure vulnerability index (from 1-10) and a hazard 

index (1-10), and combine these in various ways to obtain an overall seismic rating. Some 

methods also develop an importance index (1-10) to address daily traffic flow, redundancy and 

the socioeconomic climate. Others use qualitative measures to include these factors. This last 

approach was recommended in the 1995 FHWA Retrofitting Manual (FHWA, 1995) and is also 

included in the current edition (section 1.10.4). Recent progress in seismic risk assessment 

methods has led to the development of fragility functions for specific classes of bridges. These in 

turn have led to loss estimation methodologies for highway systems. These methodologies have 

many potential applications in highway design and retrofitting, including planning for emergency 

response and recovery. They may be used for screening bridge inventories with more rigorous 

results than possible with the above methods, since they quantify the uncertainties surrounding 

bridge and site vulnerability, network redundancy and importance factors. 

The three methods summarized below have increasing complexity, but decreasing conservatism. 

Two of these are further described in later sections of this chapter and in greater detail in chapter 

4. These methods are differentiated by the manner in which structure vulnerability, seismic and 

geotechnical hazards, importance, redundancy and various socioeconomic factors are treated. 

The methods are: 

1. Indices Method (FHWA, 1995) 

Indices are used to characterize the structure vulnerability and hazard level and are then 

combined to give a single rating for each bridge. Indices range from 0 to 10 and are based on 

conservative, semi-empirical rules. Prioritization is determined by this rating together with a 

qualitative assessment of importance, redundancy, non-seismic issues, and socioeconomic 

factors. This is the simplest of the three methods, but it is also the most conservative, since it 

uses arbitrary rules to allow for inherent uncertainties. 

2. Expected Damage Method 

This method compares the severity of expected damage for each bridge in the inventory, for 

the same earthquake. Severity of damage is measured either by sustained damage state(s) or 

by estimating direct economic losses. Bridges with the highest expected damage (and/or loss) 

are given the highest priority for retrofitting. Uncertainty in ground motions, and randomness 

in soil and structure properties, are explicitly addressed by using fragility functions to 

estimate damage-state probabilities (see chapter 4). A qualitative assessment of indirect 

losses, network redundancy, and non-seismic issues is required and the ranking, based on 

fragility, is modified accordingly. 

3. Seismic Risk Assessment Method 

Explicit analysis of the highway network is performed for a given hazard level and the 

resulting damage states used to estimate the effect on system performance as measured by 
34

traffic flow (e.g., increased travel times). Sensitivity of this performance to bridge condition 

is subsequently used to determine bridge retrofit needs and priorities. Independent qualitative 

assessment of non-seismic issues and socioeconomic factors is required. This is the most 

complex of the three methods, but it is also the most rigorous with the least conservatism in 

the final result. 

Methods 1 and 2 are summarized in sections 1.10.4 and 1.10.5 below and described in detail in 

chapter 4. Method 3 is considered outside the scope of this manual and is not discussed further. 

A full description may be found in Werner et al., 2000. 

1.10.3.1. Minimum Screening Requirements 

Minimum requirements for screening bridge inventories, based on Seismic Retrofit Categories, 

are recommended in table 1-7. 

1.10.3.2. Seismic Inventory of Bridges 

The first step in implementing any of the above methods is to compile a bridge inventory with 

the objective of obtaining the following basic information: 

• The structural characteristics of each bridge to determine either the vulnerability rating or to 

select the fragility function as described in chapter 4, and 

• The seismicity and soil conditions at each bridge site to determine the seismic hazard rating 

or select the fragility function as described in chapter 4. 

This information may be obtained from the bridge owner's records, the National Bridge 

Inventory (http://www.fhwa.dot.gov/bridge/nbi.htm), “as-built” plans, maintenance records, the 

regional disaster plan, on-site bridge inspection records, and other sources. A form, such as the 

sample shown in table 1-8, should be used for recording this information and filed with the 

bridge records. 

Much of the information used to assign the required performance level for the retrofitted bridge 

in section 1.4 will also be necessary when making assessments of importance, redundancy, and 

socioeconomic issues in the sections that follow. 

1.10.4. INDICES METHOD 

In this method, the seismic rating of a bridge is determined by its structural vulnerability, the 

seismic and geotechnical hazards at the site, and the socioeconomic factors affecting the 

importance of the structure. Ratings of each bridge are first found in terms of vulnerability and 

hazard, and then modified by importance (societal and economic issues) and other issues 

(redundancy and non-seismic structural issues) as necessary to obtain a final, ordered 

determination of retrofitting priority (Buckle, 1991; FHWA, 1995). 
35

EXAMPLE 1.2: IMPACT OF NETWORK REDUNDANCY ON BRIDGE PRIORITIZATION 

____________________________________________________________________________ 

Example 1.2(a). In the figure at 

right, assume that Bridge A is a 

seismically vulnerable bridge 

and has a high seismic 

vulnerability rating. It is located 

on a major route in series with 

lower-rated Bridges B and C, 

which are also vulnerable to 

earthquakes, but to a lesser 

degree than Bridge A. Assume 

that no convenient detour to this 

route exists and that each 

bridge can be economically 

retrofitted. What priority should 

be given to Bridges B and C?

Answer: Since retrofitting Bridge A alone would improve only one point on the route and do nothing to 

prevent the failure of Bridges B or C, and because construction and administration savings can be 

realized by retrofitting more than one bridge in the same geographical area at a time, Bridges B and C, 

although lower rated than A, should also be considered for retrofitting. 

Example 1.2(b). Suppose in the Example above, Bridge B has a high vulnerability rating but cannot be 

economically retrofitted. What priority should now be given to Bridges A and C? 

Answer: Because Bridge B is in series with Bridges A and C, the route would be closed if Bridge B were 

to collapse. Therefore, unless Bridge B can be replaced at the same time, it may be advisable to give 

Bridges A and C a lower retrofit priority because strengthening of these two bridges alone may not 

prevent closure of the route. 

Example 1.2(c). Consider two bridges 

that have parallel functions, such as 

Bridges D and E in the figure at right. If 

Bridge D has a lower vulnerability 

rating than Bridge E, which bridge 

should be retrofitted first? 

Answer: Since Bridge D has the lower 

rating, it is possible that it could be 

more economically retrofitted than 

Bridge E, since less strengthening is 

required. If this is true, and the 

collapse of one bridge is preferred 

over the loss of both bridges, then it 

might be more logical to retrofit Bridge 

D before Bridge E, even though Bridge 

E had the higher vulnerability rating. 
36

Table 1-8. Sample bridge seismic inventory form. 

BRIDGE SEISMIC INVENTORY DATA FORM 

GENERAL 

Bridge Name ____________________________________________ BIN Number________________________ 

Location ______________________________________________________________________________________ 

Year Built ________________ ADT____________________________ Detour Length_________________ 

Total Length ______________ Feature Carried____________________________________________________ 

Overall Width______________ Feature Crossed___________________________________________________ 

Importance: essential / standard Alignment: straight / skewed/ curved Geometry: regular / irregular 

Seismic Hazard (100-year event): Ss =________ g S1 = _______ g Soil Site Class: A / B / C / D / E _____ 

 (1000-year event): Ss =________ g S1 = _______ g Soil Site Class: A / B / C / D / E _____ 

SUPERSTRUCTURE 

Material and Type ______________________________________________________________________________ 

Number of spans __________ Continuous: yes / no Number of expansion joints________ 

BEARINGS 

Type ____________________ Condition: functioning / not functioning ____________________________________ 

Type of restraint: Longitudinal: ____________________________ Transverse:______________________________ 

Actual support length____________________________________Minimum required length____________________ 

COLUMNS AND PIERS 

Material and Type ______________________________________________________________________________ 

Cross-section: Min. transverse dimension__________________ Min. longitudinal dimension ___________________ 

Height range (low – high): ____________________________ Fixity: Top _____________ Bottom_______________ 

Longitudinal reinforcement (%)_______________ __________ Splices in end zones ? yes / no_________________ 

Transverse confinement steel_____________________________________________________________________ 

FOUNDATIONS AND ABUTMENTS 

Pier foundation type: spread footings / pile footings / pile bent / single shaft / other _________________________ 

Abutment type: seat / integral / other ____________________ On Piles: yes / no 

other_________________________ 

Abutment height _______________ Approach slabs: yes / no Slab length _________________________________ 

Location: cut / fill Wingwalls: yes / no Liquefaction: susceptibility low / moderate / high _______________ 

REMARKS ___________________________________________________________________________________ 
37

This rating system has two parts: quantitative and qualitative. The quantitative part produces a 

seismic rating (‘bridge rank’) based on structural vulnerability and site hazard. The qualitative 

part modifies the rank in a subjective way that accounts for importance, network redundancy, 

non-seismic deficiencies, remaining useful life, and similar issues to arrive at an overall priority 

index. Engineering and societal judgment are key to the second stage of the screening process. 

This leads to a priority index, P, which is a function of rank, importance, and other issues: 

 

 P = f (R, importance, non-seismic, and other issues) (1-4a) 

where P is the priority index, and R is the rank based on structural vulnerability and seismicity. 

In summary, bridge rank is based on structural vulnerability and seismic hazard, whereas retrofit 

priority is based on bridge rank, importance, non-seismic deficiencies, and other factors such as 

network redundancy. A methodology for calculating bridge rank (R) and the assignment of the 

priority index (P) is given in chapter 4 (section 4.2). 

1.10.5. EXPECTED DAMAGE METHOD 

The Expected Damage Method compares the severity of expected damage for each bridge in the 

inventory for the same earthquake, and ranks each bridge accordingly. Severity of damage is 

measured either by the sustained damage state(s) or the estimated direct economic loss. Bridges 

with the highest expected damage (and/or loss) are given the highest priority for retrofitting. 

In its present form, the method does not include indirect economic losses due to loss of life, 

injuries, business disruption, traffic congestion, denial of access to emergency responders and the 

like. These losses will probably exceed the direct losses (i.e., repair costs) but the state-of-the-art 

of loss estimation is currently unable to quantify these costs with any certainty. 

As in the Indices Method, this method has two parts: quantitative and qualitative. The 

quantitative part is based on expected damage and direct economic losses, and is used to obtain a 

bridge rank, R. The qualitative part modifies the rank in a subjective way that takes into account 

such factors as indirect losses, network redundancy, non-seismic deficiencies, remaining useful 

life, and other issues, to obtain an overall priority index. As in the previous method, engineering 

and societal judgment are the key to the second stage of the process. This leads to a priority 

index, P, which is a function of rank, indirect losses, redundancy, and other issues, as follows: 

P = f (R, indirect losses, redundancy and various non-seismic issues) (1-4b) 

where P is the priority index, and R is the rank based on expected damage and direct losses. 

It is seen that bridge rank is based on expected damage and direct losses for a given earthquake, 

whereas retrofit priority is based not only on bridge rank, but also on expected indirect losses, 

network redundancy and non-seismic factors, estimated in a subjective way. Although equation 

1-4b has the same form as equation 1-4a, the terms are calculated in different ways. A particular 

advantage of this method is that it provides a template by which indirect losses may be rationally 

included, as the state-of the art improves. 
38

The estimation of expected damage is a critical step in this method and due to uncertainty in 

earthquake ground motions, and randomness in soil and structure properties, this step is a 

probabilistic one. Fragility functions are used to estimate the probability of a bridge being in one 

or more specified damage states, after a given earthquake. Appendix C summarizes the theory of 

fragility functions and briefly explains how they are obtained. It also describes the six damage 

states most commonly used to characterize expected damage. 

As noted in section 1.10.3, fragility functions are also essential elements in the Seismic Risk 

Assessment Method for screening and prioritizing bridges. These functions are also used in most 

loss-estimation methodologies, including HAZUS, which is under development by the National 

Institute of Building Sciences for the Federal Emergency Management Agency (HAZUS, 1997). 

In order to simplify the method as much as possible, input data requirements are kept to a 

minimum. In particular, all structure attributes may be found in the NBI. Ground motions and 

soils data are based on spectral accelerations and soil types as described in section 1.5. 

A methodology for calculating bridge rank (R) and the assignment of the priority index (P), 

based on expected damage, is given in chapter 4 (section 4.3). 

1.11. METHODS OF EVALUATION FOR UPPER LEVEL GROUND MOTION 

Bridges found to be deficient during screening and prioritization (section 1.10) are subject to 

detailed evaluation using one or more of the methods described in this section. Since the 

screening procedures described above are necessarily conservative, it is likely that a bridge 

identified as deficient during screening will be found satisfactory upon a more detailed 

evaluation. 

The seismic evaluation of a bridge is explicitly or implicitly a two-part process. A demand 

analysis is first required to determine the forces and displacements imposed on the bridge by an 

earthquake; this is then followed by an assessment of capacity to withstand this demand. Most 

evaluation methods express their results as capacity/demand ratios calculated on a componentby-component basis, or for the bridge as a whole (i.e., as a single structural system). 

Six evaluation methods are described in this manual, which are all based, to varying degrees, on 

capacity-demand principles. They are listed below in order of increasing sophistication and rigor, 

and are summarized in table 1-9. All six methods are described in detail in chapter 5. These 

methods emphasize the calculation of demand on a member or component of a bridge. Methods 

for calculating the capacity of a member or component are described in detail in chapters 6 

and 7. The relationships between these methods for demand analysis and capacity assessment are 

shown schematically in figure 1-13. 

Method A1/A2: Connection forces and seat width checks. Seismic demand analysis is not 

required but the capacity of connections and seat width adequacy is checked against minimum 

values (section 5.2). The method is suitable for all single span bridges and others in low hazard 

zones. The method is divided into two categories, A1 and A2. If the short-period spectral 

ordinate Ss < 0.10, Method A1 may be used. Otherwise, Method A2 must be used, which 

requires higher minimum connection capacities than Method A1. 
39

Method B: Component capacity checks. Seismic demand analysis is not required, but the 

relative strength of the members and the adequacy of certain key details are checked against 

recommended minima. This method is suitable for regular bridges in SRC C, subject to 

restrictions on FVS1 (section 5.3). 

Method C: Component capacity/demand method. Seismic demands are determined by an elastic 

analysis such as the uniform load method, multi-mode response spectrum method, or an elastic 

time history method. The uniform load method is adequate for bridges with regular 

configurations; otherwise, the multi-mode method is used as a minimum. Capacity/demand ratios 

are calculated for all relevant components. This method is suitable for all bridges in SRC C and 

D, but gives best results for bridges that behave elastically or nearly so (section 5.4). 

Method D1: Capacity spectrum method. Seismic demands are determined by simple models 

such as the uniform load method, and capacity assessment is based on a simplified bilinear 

lateral strength curve. A capacity spectrum is used to calculate the capacity/demand ratio for the 

bridge, for each limit state. This method is suitable for regular bridges in SRC C and D (section 

5.5). 

Method D2: Structure capacity/demand method. Seismic demands are determined by elastic 

methods such as the multi-mode response spectrum method, or an elastic time history method. 

Capacity assessment is based on the displacement capacity of individual piers as determined by a 

‘pushover’ analysis, which includes the nonlinear behavior of the inelastic components. A 

capacity spectrum is used to calculate the capacity/demand ratio for each pier, bearing, and 

foundation of the bridge, for each limit state. This method is suitable for all bridges in SRC C 

and D. It is also known as the Pushover Method or alternatively the Nonlinear Static Procedure

(section 5.6). 

Method E: Nonlinear dynamic procedure (time history analysis). Seismic demands are 

determined by a nonlinear dynamic analysis using earthquake ground motion records to evaluate 

the displacement and force demands. Capacities of individual components are explicitly modeled 

in the demand analysis. This method is suitable for irregular complex bridges, or when site 

specific ground motions are to be used for a bridge of major importance (section 5.7). 

In summary, Methods A and B are based on default minima with no demand analysis required. 

Methods C and D are capacity-demand methods of varying rigor, and Method E is the most 

rigorous of all the methods and is based on inelastic time history analysis. 

The choice of method to be used for an evaluation is determined by the minimum requirements 

in table 1-7 and the regular/irregular requirements summarized above and in table 1-9. The 

minima in table 1-7 are related to the Seismic Retrofit Category of the bridge under 

consideration, and are based on two principles. First, as the seismic hazard increases, improved 

modeling and analysis for the seismic demands is necessary, because bridge response is sensitive 

to increasing demand. Second, as the complexity of the bridge increases, more sophisticated 

models are required to capture both the demand and capacity with certainty. Note that a higher 

level of analysis may always be used in place of a lower-level method. 
Table 1-9. Evaluation methods for existing bridges. 

APPLICABILITY 

METHOD CAPACITY ASSESSMENT 

DEMAND 

ANALYSIS SRCUL1 Bridge Type COMMENTS 

A – D All single span bridges. A1/

A2 

Connection and Seat Width Checks 

Uses default capacity due to non-seismic loads for connections and seat widths. 

Not required 

B Bridges in low hazard zones. 

Hand method, spreadsheet useful. 

B Component Capacity Checks 

Uses default capacity due to non-seismic loads for connections, seats, columns and foundations. 

Not required C Regular bridges, but subject to 

limitations on F Sv 1. 

Hand method, spreadsheet useful. 

C 

Component Capacity/Demand Method 

Uses component capacities for connections, seat widths, column details, footings, and liquefaction susceptibility (11 items). 

Elastic Methods2: 

• ULM • MM • TH 

C & D Regular and irregular bridges that 

respond almost elastically, such as those in low-to-moderate seismic zones and those with stringent performance criteria. 

Calculates C/D ratios for individual components. This is the C:D Method of previous FHWA highway bridge retrofitting manuals. Software required for demand analysis. 

D1 Capacity Spectrum Method 

Uses bilinear representation of structure capacity for lateral load, subject to restrictions on bridge regularity. 

Elastic Methods2: 

• ULM 

C & D Regular bridges that behave as 

single-degree-of-freedom systems and have ‘rigid’ in-plane superstructures. 

Calculates C/D ratios for complete bridge, for specified limit states. Spreadsheet useful. 

D2 

Structure Capacity/Demand Method 

Uses pushover curve from detailed analysis of superstructure, individual piers and foundation limit states. 

Elastic Methods2: 

• ULM • MM • TH 

C & D Regular and irregular bridges. Calculates C/D ratios for bridge 

superstructure, individual piers, and foundations. Also known as Nonlinear Static Procedure or Displacement Capacity Evaluation Method. Software required for demand and capacity analysis. 

E Nonlinear Dynamic Method 

Uses component capacities for connections, seat widths, columns and footings. 

Inelastic Methods2

• TH 

D Irregular complex bridges, or when 

site specific ground motions are to be used such as for bridges of major importance. 

Most rigorous method, expert skill required. Software essential. 

Notes:1. SRC = Seismic Retrofit Category for upper level earthquake; for the lower level earthquake, the recommended method of evaluation is Method C. 

 2. ULM = Uniform Load Method; MM = Multi-Mode Spectral Method; TH = Time History Method

40
41

METHOD A

sec 5.2

No analysis req'd

Min. seat and force

requirements are

specified

No modelling req'd

METHOD B

sec 5.3

No analysis req'd

Min. seat and force

requirements are

specified

No modelling req'd

Demand on pier

caps and footings

calculated from

column

overstrength,

sec 7.6

METHOD C

sec 5.4

Elastic Analysis

(Uniform Load (ULM)

Multimode (MM)

Time History (TH)

Methods)

Modeling as per

sec 7.3

Seat widths by

inspection

Connection

capacity by

calculation

Seat widths by

inspection

Connection

capacity by

calculation

Columnns must

satisfy min. shear

and confining steel

requirements

Column

overstrength,

sec 7.7

Component capacities

using Appendix D

Capacity/Demand

ratios for:

y Abutment

displacement

y Anchorage

length

y Bearing

connection force

y Column moment

y Column shear

y Confinement

steel

y Footing moment

y Footing rotation

y Liquefaction

potential

y Seat width

y Splice length

METHOD D1/D2

sec 5.5 / 5.6

Elastic Analysis

(Response

spectrum methods

ULM, SM, MM)

Modeling as

per sec 7.3

Either bridge

capacity curve (D1)

or pier capacity

curve (D2)

Forces and

moments in piers

due to overstrength,

sec 7.6

Strength

capacity

of bridge members,

sec 7.7

Deformation

capacity

of bridge members,

sec 7.8

Combination of

seismic force effects,

sec 7.4

Combination of

seismic force

effects,

sec 7.4

METHOD E

sec 5.7

Nonlinear Analysis

 (3 Dimensional,

Time History

Method)

Modeling as per

sec 7.3

Detailed bridge

model includes

member capacities

Forces and

moments in piers

due to overstrength,

sec 7.6

Strength

capacity

of bridge members,

sec 7.7

Deformation

capacity

of bridge members,

sec 7.8

SEISMIC RETROFIT CATEGORY and BRIDGE TYPE

DEMAND ANALYSIS

CAPACITY ASSESSMENT

Figure 1-13. Evaluation methods for existing bridges 

showing relationship between demand analysis and capacity assessment. 
42

1.12. RETROFIT STRATEGIES FOR UPPER LEVEL GROUND MOTION 

1.12.1. GENERAL

Once a bridge is found to be seismically deficient, the next step is to decide what, if anything, 

should be done to correct the deficiencies. Decision-making may be formalized by exploring 

retrofit options and the associated cost implications. This section revisits the objectives of 

retrofitting and discusses the selection of a retrofit strategy (including cost considerations), 

developing a retrofit approach, and identifying retrofit measures. 

A Retrofit Strategy is the overall plan for the seismic retrofit of a bridge. This plan can employ 

more than one retrofit approach and thus several different retrofit measures. Retrofit strategies 

are discussed in section 1.12.2. 

A Retrofit Approach is the philosophy of seismic enhancement adopted for a bridge. 

Strengthening is an example of a retrofit approach. One or more retrofit approaches can be 

employed in the seismic retrofit of a bridge. Various approaches are discussed in section 1.12.3. 

A Retrofit Measure is the physical modification of a component in a bridge for the purpose of 

upgrading overall seismic performance. For example, the addition of a steel shell to a reinforced 

concrete column is a retrofit measure. Retrofit measures for superstructure, substructure, and 

foundation components are briefly summarized in section 1.13, and described in detail in 

chapters 8, 9, and 10, respectively. 

1.12.2. SELECTING A RETROFIT STRATEGY 

1.12.2.1. Objective of Retrofitting and Acceptable Damage 

The objective of retrofitting a bridge is to ensure that it will perform satisfactorily when 

subjected to the design earthquake. Specifically, bridges should be retrofitted to meet the 

performance criteria given in section 1.4 and table 1-2, which are determined by the importance 

of a bridge and its anticipated service life. 

Performance criteria for both new and existing bridges permit considerable structural damage as 

long as collapse is prevented, and the amount of acceptable damage in existing bridges may be 

greater than for new designs. 

There are at least two reasons why accepting some level of damage is an essential ingredient of 

almost all retrofit strategies. These are: 

• Retrofitting is usually more complex and more expensive than providing adequate resistance 

in a new bridge. In some cases, the added cost of preventing damage could be as much, or 

more, than the cost of repairing the damage in the unretrofitted bridge (neglecting indirect 

losses). 
43

• Existing bridges may have a reduced useful or remaining life due to normal ‘wear and tear,’ 

and increasing traffic volume and/or load capacity requirements. This shortened life reduces 

the likelihood that such a structure will experience a damaging earthquake, and increases the 

annualized cost of every dollar spent to prevent earthquake damage. 

There are exceptions to this rule. These include very important bridges, and where the societal 

cost of any damage that disrupts serviceability, or reduces traffic capacity, is unacceptable. 

1.12.2.2. Cost Considerations of Seismic Retrofit 

Cost is a major consideration in seismic retrofitting. As noted above, the average cost of seismic 

retrofitting is much higher than of incorporating seismic resistance into the design of a new 

bridge. In fact, it has been estimated that in extreme cases, seismic retrofitting may be two to 

three times more expensive. 

Data giving the actual cost of retrofitting is hard to find because very few States have completed 

extensive retrofit programs from which cost databases may be compiled. However, table 1-10 

gives some data based on Caltrans’ experience in retrofitting 165 bridges during 1993 and 1994. 

In this table, retrofit costs are expressed as a fraction of the cost of new construction and the 

average cost is seen to be about 15 percent of the cost of building a new bridge in the same time 

period. Also, as might be expected, retrofit costs strongly depend on the strategy employed. 

When only the superstructure is retrofitted (cable restrainers and seat width extenders), the 

average construction cost is about 3.1 percent of new construction. When the substructure is also 

retrofitted, but not the foundation, the average cost rises to 15.4 percent. When the foundation is 

included, the average cost rises further to 28.8 percent. Table 1-10 also shows a significant 

variation in cost when the same strategy is applied to different bridges, due to differences in the 

extent of retrofitting required. Although this data reflects prevailing conditions in the California 

construction market at the time of these retrofits, the trends are believed applicable elsewhere in 

the country. 

Table 1-10. Cost of various retrofit strategies as percentage of new construction costs 1,2. 

RETROFIT STRATEGY 

RANGE Superstructure 

Only3

Superstructure 

and Substructure

Superstructure, 

Substructure and 

Foundations 

TOTALS 

(weighted sum 

all retrofits in 

California, 1993 

and 1994) 

Low 1.3 0.7 2.3 0.7 

Average 3.1 15.4 28.8 15.1 

High 13.2 64.8 232.9 232.9 

Notes: 1. Caltrans data for 165 bridges retrofitted in 1993 and 1994. 

 2. Costs expressed as percentage of new construction for same time frame. 

 3. Superstructure includes restrainers and seat width extensions. 

 
44

Engineering costs for retrofit design are also generally higher than for new construction. It is not 

unrealistic to expect that these costs will be twice the cost of the engineering required for a new 

bridge of similar value. This is because many bridges are unique and often require customized 

retrofit strategies. Standardization of design and retrofit details is therefore difficult to achieve. 

In addition, the detailed seismic evaluation of a bridge and identification of the most appropriate 

retrofit strategy is a time-consuming process that may involve a detailed dynamic analysis and 

potentially, many trial designs investigating possible strategies. 

In addition to higher initial costs, the fact that the life of a retrofit should not exceed the 

remaining service life of the bridge means that the annualized cost of retrofitting is further 

increased over the cost of seismic resistance in new construction. This requires that the benefits 

of retrofitting, particularly for damage prevention, be weighed against these higher costs when 

selecting a retrofit strategy for the bridge. 

1.12.2.3. Other Considerations and Non-Seismic Issues 

Many existing bridges within the United States are either structurally deficient or functionally 

obsolete, in addition to being seismically vulnerable. Either one of these two conditions could 

result in a bridge being rehabilitated, which might present an opportunity for improving the 

seismic resistance of a bridge at the same time. If retrofitting is required immediately, it may be 

advisable for the designer to consider the additional demands that will be placed on the structure 

when the future widening is finally accomplished. A designer should therefore consider both the 

present condition and possible future service that will be required of the bridge. 

Bridge inspection and maintenance needs should also be considered when designing a seismic 

retrofit. This includes access for inspection as well as maintenance activities. The retrofit 

measure itself should not become a maintenance problem. The designer should be aware of the 

needs of maintenance personnel and include them in the selection process of the retrofit strategy. 

Retrofit measures can dramatically alter the appearance of the bridge. The designer should 

therefore be sensitive to the aesthetics of the retrofit design. This could be particularly important 

if the original design has notable aesthetic value or if it is desirable or required to preserve the 

appearance of the bridge for other reasons (e.g., if the bridge is a historic structure). There may 

be alternative retrofit measures that are just as effective, but more aesthetically acceptable. 

It is often not possible to close a bridge to traffic while retrofitting is performed, but some retrofit 

measures require access to portions of the bridge that will disrupt traffic. Rerouting of traffic and 

staged construction will then be necessary, and when this is difficult to achieve, a strategy that 

limits disruption to traffic must be found. Similar constraints may exist with respect to utilities 

on or near the bridge. If they cannot be easily relocated, it may be necessary to redesign the 

retrofit to avoid damage or disruption to these utilities. 

Constructability is always an issue for the bridge designer, but potentially more so in the case of 

seismic retrofitting. The main problem is access to the work area on and around the existing 

bridge and surrounding facilities. Some examples include driving or drilling piles under an 

existing structure with limited headroom, placing concrete under an existing horizontal surface, 
45

and excavating near other adjacent facilities. The designer must carefully think through the steps 

required for construction, and verify that the proposed retrofit can indeed be built. 

Political and environmental constraints often arise during retrofitting and should be identified as 

far ahead as possible during selection of the preferred strategy. For example, the bridge may be 

listed on an historical register, which could limit the types of modifications that can be 

performed. The bridge may be located in an environmentally sensitive area that limits access to 

certain critical elements such as columns or foundations. The bridge may be located in a highly 

urban or residential environment, which may limit certain types of construction activities such as 

pile driving because of vibration and noise. Hazardous wastes are likely to be present around 

bridges in urban environments and, if disturbed, will require remediation. Political opposition to 

certain solutions such as bridge replacement, may also limit the options of the designer. It is 

important that the designer be sensitive to these and other issues early in the retrofit process with 

a view to finding the best solution given the constraints. 

 

1.12.2.4. Identifying and Evaluating a Retrofit Strategy 

Selecting the preferred retrofit strategy can be complicated. Not only is it often a challenge to 

find the right technical solution, it is also a challenge to satisfy a multitude of socio-economic 

constraints. A systematic process should therefore be followed to assure an appropriate strategy 

is selected. A flowchart for such a process is shown in figure 1-14. The various steps are 

discussed below. 

Step 1. Conduct a detailed as-built evaluation. The first step in seismic retrofitting is to perform a 

detailed analytical evaluation of the existing bridge, as it exists in the field. Procedures 

for this step are discussed in section 1.11 and chapters 5 through 7. The goal of this step 

is to assess the response of the bridge to the design earthquake and to identify weaknesses 

that can be addressed by retrofitting. A formal field review of the bridge should be 

performed as part of this step. This is needed to verify the as-built condition of the bridge 

and to identify any constraints on retrofitting. Structural and geotechnical specialists and 

the owner should be involved in this effort. 

Step 2. Identify alternative retrofit strategies. Frequently, there is more than one way to improve 

the performance of an existing bridge, and it is important to identify as many options as 

possible. Many will be quickly eliminated because of excessive cost, constructability or 

other problems as noted in section 1.12.2.3. Solutions that appear viable should be further 

considered in Step 3. Table 1-11 (see section 1.13.2) identifies alternative retrofit 

approaches that might be used to address common deficiencies and directs the designer to 

possible retrofit measures for each approach. This table may therefore be helpful when 

looking for potential retrofit strategies. 

Step 3. Evaluate alternative retrofit strategies. Detailed analytical evaluations of each viable 

retrofit strategy should be performed using the methods described in section 1.11 and 

chapters 5 through 7. This step should also include the preliminary design of the 

elements of the proposed retrofit so that a preliminary cost estimate may be prepared for 

each alternative. 
46

Detailed evaluation of 'as-built' bridge

Is retrofitting

warranted?

Identify alternative strategies

Evaluate retrofitting alternatives

Does the strategy

work?

Present strategy to Strategy Meeting

Is strategy

accepted?

Document strategy selection

'Do nothing'

alternative

YES

NO

NO

YES

NO

YES

Figure 1-14. Identification and evaluation of a retrofit strategy. 
47

Step 4. Conduct a strategy meeting. Since seismic retrofitting involves many complex issues, 

consensus must be achieved on the most appropriate strategy. This may be accomplished 

through a strategy meeting where the designer presents recommendations and cost 

estimates for retrofitting. Representatives of all agencies that have an interest in the 

project should attend this meeting, including the bridge owner, utility companies, 

Federal, state and local government agencies, structural and geotechnical engineering 

specialists, and environmental and citizens groups. 

Step 5. Document the strategy selection. The retrofit decision should be documented in a strategy 

report that becomes part of the permanent record for the project. This report should 

include all the calculations of the as-built and as-retrofitted evaluations, preliminary plans 

and sketches showing the proposed retrofit, a summary of conclusions and 

recommendations, preliminary cost estimates, and a summary of the discussions from the 

strategy meeting. 

1.12.2.5. Do-Nothing and Full-Replacement Options 

When retrofitting a seismically deficient bridge, two possible solutions, at opposite ends of the 

spectrum, should be kept in mind: the ‘do-nothing’ and ‘full-replacement’ options. 

The ‘do-nothing’ option requires the acceptance of damage during a future earthquake. This will 

be a relatively straightforward decision if life safety is the only performance requirement, and the 

expected damage is not a threat to life safety. The most likely cause of loss of life is total 

collapse of a span, but this is a relatively rare event. For example, the toppling or failure of 

individual bearings will not necessarily lead to collapse if the bearing seats are wide enough to 

catch the superstructure. Similarly, foundation failures are unlikely to cause collapse, unless the 

ground deformations are extremely large due to widespread liquefaction or massive ground 

failure such as fault rupture. Fortunately, these occurrences are rare. Nevertheless, judgment 

should be used when assessing collapse potential and, to the extent possible, this potential should 

be carefully evaluated using a detailed analysis. 

The ‘full-replacement’ option may also be an attractive option, particularly when the cost of 

retrofit is on the same order of magnitude as the replacement cost of the bridge. Full replacement 

is generally considered whenever the retrofit costs approach 60 to 70 percent of a new bridge and 

may become even more attractive if the structure has non-seismic structural deficiencies and is 

functionally obsolete. However, in making this recommendation, the designer should also 

consider the cost of demolition and any costs associated with control and rerouting of traffic as 

part of the cost of the replacement alternative. These costs can be significant and may tip the 

scales back toward the retrofit alternative. 

1.12.3. SEISMIC RETROFIT APPROACHES 

Seismic retrofit approaches may be used alone or in combination to develop a seismic retrofit 

strategy for a given bridge. Some of the more common approaches are listed below and 

discussed in sections 1.12.3.1 through 1.12.3.7. 
48

• Strengthening. 

• Improvement of Displacement Capacity. 

• Force Limitation. 

• Response Modification. 

• Site Remediation by Ground Improvement. 

• Acceptance or Control of Damage to Specific Components. 

• Partial Replacement. 

1.12.3.1. Strengthening 

Strengthening is intended to increase the force or moment capacity of one or more deficient 

elements of a bridge. This approach should also take into account any significant increase in 

stiffness due to the strengthening, and its likely effect on structural response. 

It is generally not feasible to economically retrofit a bridge for a major earthquake within the 

elastic range of the members, and therefore, yield is to be expected in many locations. In cases 

where the force or moment demands on a structural element are limited by yielding elsewhere in 

the structure, the strength of the element should be sufficient to resist the demands placed on it 

without damage. Such is usually the case with superstructure and foundation elements where 

forces are controlled by column yielding. When these elements are not strong enough to resist 

the forces or moments generated by the existing or retrofitted column, they must then be 

strengthened. 

Strengthening of a ductile component can reduce the ductility demand on the component and 

thus improve seismic performance. However, it will usually increase the forces or moments in 

adjacent components, which will then also need to be strengthened. 

The addition of restrainer cables or high strength bars at hinges and seats is also considered to be 

a form of strengthening. The added strength and stiffness of restrainers will limit the relative 

movement between superstructure spans or frames, with the goal of preventing failure due to loss 

of support. 

1.12.3.2. Improvement of Displacement Capacity 

Displacement capacity is generally a better indicator of structural performance than member 

strength. A common retrofit approach, therefore, focuses on improving displacement capacity. 

This can be achieved in one of two ways. 

The first technique extends the length of bearing seats to permit greater relative movement at 

bearing locations without loss of support. This is an alternative method to the use of restrainer 

cables or bars, which is a strengthening approach intended to reduce the displacement demand on 

the bearing seat. In many cases, both approaches are used in combination, i.e., both restrainers 

and seat extenders are used to prevent loss of support at the same location. 

The second technique increases the ductility capacity of columns and piers. Large inelastic 

deformations may be required of columns and piers during a major earthquake and ductility 
49

capacity is a measure of their ability to sustain this deformation without collapse or fracture. A 

ductile column or pier can therefore accommodate large imposed deformations, and any 

technique which increases this capacity is considered to be a form of displacement capacity 

enhancement. 

1.12.3.3. Force Limitation 

Forces in critical structural components can be limited by using yielding elements as a structural 

‘fuse’, i.e., a sacrificial element. If one member in a bridge is deliberately designed to yield and 

thus limit the forces that can be transmitted to an adjoining member, the second member is 

‘capacity-protected’ by the first member. If, for example, a column is intended to yield and 

develop plastic hinges, the maximum moment that can be transmitted from the column to the 

foundation is limited by the yield moment of the hinges. In such a case, the foundation is 

‘capacity-protected’ by the column. 

Although force limitation occurs naturally in many structures, several force-limiting retrofit 

methods have been developed to reduce the cost of strengthening of components, particularly 

those that are structurally difficult to implement. 

One method of force limitation uses seismic isolation bearings. Although these bearings can be 

used to modify the dynamic response of a bridge, as discussed in section 1.12.3.4, they can also 

be used as ‘fuses’ that limit the amount of shear force that can be transmitted to a substructure or 

foundation. 

Another example of force limitation is the link-beam concept that has been used for retrofitting 

multi-column piers. With this technique, shear forces are limited by plastic hinging in the 

beams, which in turn limits the magnitude of the moments that can be transmitted to bent caps 

and footings. 

The ductile diaphragm method described in chapter 8 also utilizes the force limitation approach. 

1.12.3.4. Response Modification 

The dynamic response of a bridge to earthquake ground motion determines the force and 

displacement demands that will be placed on various components. It is often possible to retrofit 

the bridge in such a way that the dynamic response will be significantly altered. This could 

reduce force and displacement demands and thus eliminate or reduce the need for retrofitting by 

strengthening or enhancing displacement capacity. An example of this approach is seismic 

isolation, in which the fundamental period of the structure is deliberately increased to reduce the 

force demands. In this case, however, the displacement demands in the isolators may be very 

large and additional damping is usually provided to reduce these demands. Energy dissipators 

and dampers also modify dynamic response and may be used to reduce the need for 

strengthening or improving displacement capacity. 

Another method of response modification is to modify the load path for horizontal inertia forces. 

Retrofitting to strengthen or stiffen an alternative load path may be used to attract forces away 
50

from vulnerable components, and thus reduce or eliminate the need to retrofit them. An example 

is a continuous bridge in which the abutment stiffness and strength are increased. Inertial forces 

are then redistributed to the abutments, which relieve or reduce the forces on the interior columns 

or piers. This approach can be very effective for simply supported spans if the retrofit also 

includes provision for making the deck continuous for live load. Shock transmission units can 

also be used to modify a load path. 

Response modification is a powerful retrofit approach but it requires careful consideration of 

possible side effects, such as changes in the way a bridge responds to service loads. 

1.12.3.5. Site Remediation by Ground Improvement 

Bridges can suffer significant damage when large permanent ground movements occur during an 

earthquake. Liquefaction, lateral spreading, landslides, and fault rupture are some of the more 

common causes of this damage. There are two possible approaches to this problem. 

The first approach is to give the structure the capacity to resist the loading and/or accommodate 

the displacements created by the moving soil without collapse. A critical step in this approach is 

the quantification of the loads and/or the expected ground movement or fault rupture, which is a 

difficult problem to solve. 

The second approach is to modify the soil using ground improvement techniques, such as those 

discussed in chapter 11. Site remediation often involves treatment of large areas of soil and can 

be expensive. In the case of river or stream crossings, improvements may be subject to flooding 

and require protection from scour. Nevertheless, there will be cases where this approach is a 

viable option and should be considered. It is not, however, applicable to sites where fault rupture 

is expected. 

1.12.3.6. Acceptance or Control of Damage to Specific Components 

When collapse prevention is the main goal of retrofitting, it is often acceptable to allow 

component damage as long as this does not jeopardize the overall stability of the structure. 

Often this will mean that no retrofit is required for the members in question. In some cases, 

however, damage must be controlled to prevent collapse. An example of this is the “P” column 

retrofit commonly used by Caltrans (Caltrans, 1996). This retrofit method is intended to 

preserve the vertical load carrying capacity of the column while the column sustains considerable 

damage. It is common to ignore the lateral load carrying capacity of such a retrofit when 

conducting an analysis of the structural response. 

1.12.3.7. Partial Replacement 

There are cases where the required retrofitting of a bridge component may be so extensive that 

partial replacement is the most economical solution. Partial replacement may have several goals, 

such as increasing strength and ductility when a column is replaced. In some cases, the type of 

retrofit performed on an adjacent element may mandate partial replacement. When load-bearing 

elements are replaced, temporary shoring will be required. 
51

1.13. RETROFIT MEASURES FOR UPPER LEVEL GROUND MOTION 

1.13.1. GENERAL 

In the last decade, the range of available retrofit measures has increased markedly. Measures 

have now been developed for deficient superstructures, bearings, beam seats, piers and columns, 

including weak cap beams and column-to-cap beam joints. In addition, techniques for improving 

the behavior of abutments and foundations have been developed, including measures for bridges 

on hazardous sites. This progress is the result of an aggressive research program in California 

and elsewhere and field experience, mainly in California. 

A partial list of these measures is as follows: 

• Diaphragm strengthening. 

• Energy dissipating ductile diaphragms. 

• Provision of longitudinal continuity in simply supported spans. 

• Replacement of bearings. 

• Seismic isolation bearings. 

• Energy dissipators. 

• Seat width extensions and catcher blocks at girder supports and intermediate hinges. 

• Restrainers at girder supports and intermediate hinges. 

• Column replacement. 

• Concrete shells, steel and fiber-composite jackets for columns. 

• Infill shear walls in bents. 

• Cap beam strengthening using prestressing. 

• Supergirders. 

• Anchor slabs behind abutments. 

• Soil and gravity anchors. 

• Abutment shear keys. 

• Footing replacement. 

• Footing overlays. 

• Pile tie-down enhancement. 

• Supplemental piles. 

• Articulation for fault crossings. 

• Site remediation for unstable slopes and liquefaction. 

• Vibro-replacement of soils and stone columns. 

Figure 1-15 illustrates some of the above measures; the details are discussed in chapters 8 

through 11. 
52

1.13.2. SEISMIC RETROFIT MATRIX 

The matrix presented in table 1-11 is a roadmap for the above list of retrofit measures. It shows 

the relationship between the seven retrofit approaches in section 1.12.3 and the retrofit measures 

necessary to make them work. 

For example, if the strengthening approach is chosen to address insufficient seat length, then the 

following measures are recommended in table 1-11 for consideration: 

• Section 8.2.2.1: Providing Longitudinal Continuity, Web and Flange Plates. 

• Section 8.4.2.1: Longitudinal Joint Restrainers. 

Although table 1-11 presents a comprehensive list of approaches and measures, it is not intended 

to preclude innovative retrofit approaches or measures that are not specifically mentioned in this 

manual. 
53

(a) Restrainers 

(b) New Bearings 

(continued)

Figure 1-15. Selected retrofit measures. 
54

(c) Seismic Isolators 

(d) Column Jacketing and Footing Overlay (Schematic) 

(continued)

Figure 1-15 (continued). Selected retrofit measures. 
55

(e) Footing Overlay (Detail) 

(f) Vibro-Replacement Site Remediation 

Figure 1-15 (continued). Selected retrofit measures. 
Table 1-11. Matrix of seismic retrofit approaches and section references to associated retrofit measures. 

RETROFIT APPROACH 

SEISMIC 

DEFICIENCY 

Strengthening 

Displacement 

Capacity 

Enhancement 

Force 

Limitation 

Response 

Modification Site Remediation 

Damage 

Acceptance 

or Control 

Partial 

Replacement 

Superstructure 

deficiencies 

8.2.1.1 Strengthening of Deck to Girder Connection 8.2.1.4 Girder Strengthening 8.2.4 Strengthening of Continuous Superstructures 

Structurally 

deficient 

diaphragms 

8.2.1.2 Diaphragm Strengthening or Stiffening 

8.2.1.3 Energy Dissipating Ductile Diaphragms 

Structurally 

deficient bearings/ 

connections 

8.3.1 Strengthening of Existing Bearings 8.3.3 Strengthening of Superstructure to Substructure Connections 8.4.2.2 Transverse Restrainers 8.4.2.3 Vertical Motion Restrainers 

8.3.2.2 Replacement with Seismic Isolation Bearings 8.4.3 Energy Dissipation Devices 8.4.4 Shock Transmission Units 

8.3.2.1 Conventional Bearings 

Insufficient seat 

length 

8.2.2.1 Web and Flange Plates 8.4.2.1 Longitudinal Joint Restrainers 

8.4.1.1 Concrete Seat Extensions and Catcher Blocks 8.4.1.2 Pipe Extenders 

8.2.2.2 Superstructure Joint Strengthening 8.2.3 Reduction of Dead Load 

Flexurally deficient 

columns or piers 

9.2.1.2 Column Flexural Strengthening 9.2.1.4 Supplemental Column Shear Walls 9.2.2.1 Braced Frames 9.2.2.2 Built-up Compression Members 9.2.3 Concrete Wall Piers 

9.2.1.3 Column Ductility Improvement and Shear Strengthening 9.2.2.2 Built-up Compression Members 

9.2.1.6 Limitation of Column Forces 9.2.2.1 Braced Frames 

9.2.1.5 Preservation of the Vertical Load Capacity of Columns 

9.2.1.1 Column Replacement 

Shear deficient 

columns or piers 

9.2.1.3 Column Ductility Improvement and Shear Strengthening 9.2.1.4 Supplemental Column Shear Walls 

56
RETROFIT APPROACH 

SEISMIC 

DEFICIENCY 

Strengthening 

Displacement 

Capacity 

Enhancement 

Force 

Limitation 

Response 

Modification Site Remediation 

Damage 

Acceptance 

or Control 

Partial 

Replacement 

Structurally 

deficient 

pier caps 

9.3.2 Pier Cap Strengthening 

9.3.3 Reduction of Pier Cap Forces 

9.3.5 Supergirders 

9.3.1.1 Partial Replacement at a Joint 9.3.1.2 Total Replacement 

Structurally 

deficient 

column-to-cap 

joints 

9.3.4 Strengthening of Column and Beam Joints 9.3.5 Supergirders 

9.3.1.1 Partial Replacement at a Joint 

Structurally 

deficient 

column-footing 

joints 

10.3.2 Strengthening of Footings 

10.3.4 Limiting Forces Transmitted to the Footings 

10.3.1 Footing Replacement 

Unstable footings 

10.3.2 Strengthening of Footings 10.4.2 Pile Tie-downs 

10.3.4 Limiting Forces Transmitted to the Footings 

11.4.2 Site Remediation Using Ground Improvement 

10.3.1 Footing Replacement 

Structurally 

deficient footings 

10.3.2 Strengthening of Footings 

10.3.4 Limiting Forces Transmitted to the Footings 

Abutment fill 

settlement 10.2.1 Approach Slabs 

11.4.2 Site Remediation Using Ground Improvement 

Unstable 

abutments 

10.2.2 Anchor Slabs 10.2.5 Transverse Abutment Anchors 10.2.6 Soil and Gravity Anchors 

 10.2.2 Anchor Slabs 

Structurally 

deficient abutments 

10.2.3 Diaphragm Walls 10.2.4 Transverse Abutment Shear Keys 

Excessive ground 

movement 

8.2.2.1 Web and Flange Plates 8.4.2.1 Longitudinal Joint Restrainers 9.2.3 Concrete Wall Piers 10.3.2 Strengthening of Footings 

8.4.1.1 Concrete Seat Extensions and Catcher Blocks 8.4.1.2 Pipe Extenders 9.2.1.3 Column Ductility Improvement and Shear Strengthening 9.2.2.2 Built-up Compression Members 

9.2.1.6 Limitation of Column Force 

11.4.2 Site Remediation Using Ground Improvement 

9.2.1.5 Preservation of the Vertical Load Capacity of Columns 

9.2.1.1 Column Replacement 

57

59

CHAPTER 2: SEISMIC GROUND MOTION HAZARD 

2.1. GENERAL 

One of the first steps in retrofitting a bridge is to characterize the earthquake hazard at the bridge 

site. This is done by developing peak ground accelerations and response spectra for the site and, 

for some bridges, acceleration time histories with and without the effect of spatial variation. The 

following topics are covered in this chapter: 

1. Developing horizontal ground motion response spectra using either: 

• A general procedure based on national ground motion maps and site factors, or 

• A site-specific procedure. 

2. Developing vertical ground motion response spectra. 

3. Developing acceleration time histories. 

4. Assessing the spatial variation in the ground motion. 

The estimation of ground motion at a site involves considerable judgment due to uncertainties in 

the location of seismic sources, the frequency of occurrence of earthquakes of different 

magnitudes, the maximum magnitude of an earthquake that could occur at a particular site, and 

the resulting site ground motions for any given earthquake. Since it is believed that these 

uncertainties are best addressed using a probabilistic approach, this methodology is typically 

used to obtain design ground motions for bridges and buildings. 

As described in chapter 1, two levels of earthquake ground motion are used to determine the 

performance of an existing bridge and appropriate retrofit measures. They consist of a lower 

level ground motion and an upper level ground motion. The lower level motion has a probability 

of exceedance of 50 percent during the service life of the bridge (assumed to be 75 years), 

whereas the upper level motion has a probability of exceedance of seven percent in 75 years. As 

discussed below and shown in table 2-1, these two exceedance probabilities correspond to return 

periods of approximately 100 and 1000 years for the lower and upper level ground motions, 

respectively. 

Using a Poisson probability model1

, the probability of exceedance (PE) of a given amplitude of 

ground motion in the lifetime of the bridge (t) is related to the annual frequency of exceedance 

(ν) of that ground motion amplitude, by the following equation: 

 

1

 In a Poisson model, earthquakes of given magnitudes occur randomly in time at an average rate. The occurrence 

of an earthquake is assumed to have no influence on the timing or probability of a future earthquake. 
60

- t P 1e Eν = − (2-1) 

Rewritten, equation 2-1 gives ν as a function of PE and t as follows: 

E ln(1 P )

t

− − ν = (2-2) 

Return periods2

corresponding to various probabilities of exceedance can be calculated using 

equation 2-2, as shown in table 2-1. For this table, the lifetime of a bridge (t) is taken as 75 years. 

Table 2-1. Relationship between probability of exceedance of earthquake 

ground motion and return period. 

Return Period (years) Probability of 

Exceedance (PE) in 

Bridge Life of 75 

Years (%) 

Annual Frequency 

of Exceedance, ν

Actual Rounded 

50 0.009242 108 100 

15 0.002167 461 500 

7 0.000968 1033 1000 

5 0.000684 1462 1500 

3 0.000406 2462 2500 

 

As described in section 1.5, two spectral accelerations (SDS and SD1) are used to determine the 

seismic loads for bridge screening and evaluation. These accelerations are obtained by 

modifying the spectral accelerations assuming a rock site (SS and S1), to account for the actual 

soil conditions at the site. Figure 2-1 shows the variation in these two accelerations with return 

period for different locations in the United States. Figure 2-1(a) gives the short-period (Ss) 

accelerations and figure 2-1(b) gives the long-period (S1) values. These plots show that for 

bridge sites in the eastern United States, spectral accelerations for earthquake ground motions 

with very long return periods, such as 2500 years, may be three to four times higher than for the 

500-year ground motions at the same site. This ratio varies considerably from state to state across 

the U.S., and is one of the primary reasons for adopting two different levels of earthquake 

ground motion for seismic retrofitting. 

 

2

 The reciprocal of the annual frequency of exceedance is the return period of exceedance. In this manual, the 

reciprocal is simply called the 'return period'. 
61

(a) Ratios of short-period (Ss) spectral acceleration at various return periods 

to short-period spectral acceleration at 475-year return period 

ATC/MCEER, 2003 

(continued)

Figure 2-1(a). Variation in short- and long-period spectral accelerations 

with ground motion return period. 
62

(b) Ratios of long-period (S1) spectral acceleration at various return periods 

to long-period spectral acceleration at 475-year return period 

ATC/MCEER, 2003 

Figure 2-1 (continued). Variation in short- and long-period spectral accelerations 

with ground motion return period. 
63

2.2. BASIC PROCEDURES FOR CHARACTERIZING HORIZONTAL SEISMIC 

GROUND MOTION 

The two basic procedures for developing response spectra and peak accelerations of horizontal 

ground motions are: 

1. A general procedure using national ground motion maps and site factors as described in 

sections 2.3, 2.4 and 2.5. 

2. A site-specific procedure as described in section 2.6. 

The site specific procedure should be used if any of the following apply: 

• Soils at the site require site-specific evaluation (i.e., site class F soils, as defined in section 

2.4) unless a determination is made that the presence of such soils would not result in a 

significantly higher response of the bridge. 

• The bridge is considered to be a major or very important structure by the owner. 

• The site is located within 10 km (6 mi) of an active fault and its response would be 

significantly and adversely influenced by near-fault ground motions. 

Examples of conditions for which site-specific evaluations would not be required for Type F 

soils are given in section 2.4. Section 2.6 describes the characteristics of near-fault ground 

motions that could lead to increased bridge response, but these effects might be considered for 

major bridges only. 

2.3. DETERMINATION OF SPECTRAL ACCELERATIONS, Ss and S1

Both the short-period (Ss) and long-period (S1) spectral accelerations have been mapped by the U. 

S. Geological Survey (USGS) for the United States for various exceedance probabilities (return 

periods). The maps used with the general procedure are the 1996 edition of the probabilistic 

ground motion maps published by the USGS 3,

4

. In California, these maps are produced jointly 

by the California Geological Survey (CGS), and the USGS. These maps are for a reference rock 

site condition (site class B). Procedures for adjusting the ground motions to other site conditions 

are described in section 2.4. However, maps are not available for all possible return periods and 

the USGS has provided ground motion hazard curves for selected longitudes and latitudes to 

provide for those cases not covered by the maps. These curves may be used to develop the 

relationship between the amplitude of a ground motion parameter and its annual frequency of 

exceedance, which allows spectral values to be found for any return period specified by the user. 

These curves are available on a CD-ROM published by the USGS; a copy of this CD is included 

in the manual (see inside back cover). 

 

3

 Updated maps were published by the USGS in 2002, but not for 1000-year return period. See sidebar. A new CD is 

expected in 2005 which will allow data to be automatically generated for any return period. 

4

 Frankel et al., 1996, 1997a, 1997b, 1997c, 2000; Klein et al., 1999; Peterson et al., 1996; Wesson et al., 1999a; 

1999b 
64

2.3.1. USGS PROBABILISTIC GROUND MOTION MAPS 

The USGS probabilistic ground motion maps should be used to obtain horizontal response 

spectral accelerations and peak ground accelerations in rock. Large-scale paper maps are 

available for return periods of 500 (475), 1000 (975) and 2500 (2475) years, and give spectral 

accelerations at 0.0, 0.2, 0.3, and 1.0-second periods of vibration. The 0.0-second values are 

equivalent to peak ground accelerations. All mapped values are for five percent damping. 

Maps can also be downloaded from the USGS web site at http://eqhazmaps.usgs.gov, but only 

for the return periods noted above. In addition to maps, ground motion values are also available 

from this site by entering the longitude and latitude coordinates of the site. If these coordinates 

are not available, the zip code for the region containing the site may be used. The same 

information is also available on a CD-ROM published by USGS (Frankel and Leyendecker, 

2001). A copy of the CD is included in this manual (see inside back cover). 

Note that the above sources of mapped and tabulated data do not include earthquakes with a 100-

year return period. A methodology to obtain 100-year data is described in the next section. 

 

Figure 2-2(a) is a map of the peak ground acceleration for the conterminous U.S. for the upper 

level earthquake (1000-year return period). This map was obtained from the CD-ROM noted 

above. Maps for Alaska, Hawaii, Guam and Puerto Rico are also available from this source. 

Figures 2-2 (b) and (c) are maps of the short-period (Ss) and long period (S1) spectral 

accelerations, respectively, for the upper level earthquake (1000 years). These maps are from the 

same CD and are for the conterminous U.S. Maps for other states and regions are also available. 

Available U.S. Geological Survey Seismic Hazard Maps and CDs 

Maps of peak ground acceleration, and the short- and long-period spectral accelerations (SS and S1) for 

selected return periods are available from the US Geological Survey in either paper form or on CD as 

listed below. Seismic hazard curves are also available on CD from which accelerations and spectral 

ordinates may be found for any return period. Some of this data is also available on the USGS web site 

http://eqhazmaps.usgs.gov. 

 

Year Published Data 

1996 Maps for 48 conterminous states, for return periods 500, 1000, and 2500 years. 

1998 Maps for Alaska and Hawaii, for 500 and 2500 years. 

2001 CD with maps and seismic hazard curves for 48 conterminous states, Alaska, and Hawaii 

based on 1996 data.This CD is provided inside back cover of this report. 

2002 Maps for 48 conterminous states, for return periods 500 and 2500 years. 

 Major difference between 1996 and 2002 Maps is in the hazard models used for 

New Madrid and Charleston regions. 

2003 Maps for Puerto Rico and Virgin Islands, for return periods 500 and 2500 years. 

2005 est. Updated CD with 1996 and 2002 data; also automated means of generating data 

for any return period instead of manual method used in this report (section 2.3.2). 

.
(a) Peak ground accelerations for the upper level ground motion (1000-year return period) 

(continued)

Figure 2-2. Seismic hazard maps for the conterminous United States. 

65
(b). Short-period (Ss) spectral accelerations for the upper level ground motion (1000-year return period) 

(continued)

Figure 2-2 (continued). Seismic hazard maps for the conterminous United States. 

66
(c). Long-period (S1) spectral accelerations for the upper level ground motion (1000-year return period) 

Figure 2-2 (continued). Seismic hazard maps for the conterminous United States. 

67
68

2.3.2. USGS PROBABILISTIC GROUND MOTION DATA FILES FROM CD-ROM 

Instead of using the USGS maps, the spectral accelerations Ss and S1 may be obtained from the 

ground motion data files used to produce the maps which are available on the CD-ROM (Frankel 

and Leyendecker, 2001) included with this manual. 

Spectral accelerations and peak ground motions may be found from this CD for both the lower 

and upper level earthquakes (100- and 1000-year return periods) for any site identified by either 

zip code or longitude and latitude coordinates. The procedure for these calculations is described 

in the following steps. 

Step 1. Load CD titled Seismic Hazard Curves and Uniform Hazard Response Spectra, and using 

either Windows Explorer or Setup from the Start Menu, run “Probabilistic Hazard 3.10.’ 

Display the window shown in figure 2-3(a). 

Step 2. Click box labeled ‘Hazard Curves and Response Spectra for the United States’ and 

display the window shown in figure 2-3(b). 

Step 3. Initialize: 

• Check name box and enter identifier for results to be generated. 

• Check box to include date and time on output file. 

• Click box labeled ‘Open File Selection Menu’ and display window shown in figure 

2-3(c). 

Step 4. Setup files: 

• Under heading ‘Select Directory for Data Files,’ select the drive letter for the CDROM and then choose ‘GM96-Data.’ 

• Under heading ‘Select Directory for Map Files,’ select the drive letter for the CDROM and the choose ‘GM96-Maps.’ 

• Click ‘OK’ if satisfied and display the window shown in figure 2-3(c); otherwise 

click ‘Cancel Selections’ and return to beginning of Step 3. 

Step 5. Choose either ‘Latitude and Longitude’ or ‘5-Digit Zip Code’ to identify the bridge site 

as shown in figure 2-3(d)5

. 

 

5 Latitude and longitude coordinates are the most accurate way to locate a bridge but in the absence of these 

coordinates, a zip code may be used. In the latter case, the seismic hazard at the geographical center of the region 

covered by the zip code is presumed to be representative of that at the bridge site. 
69

Step 6. To obtain values for SS and S1 for the upper level ground motion, go to the box labeled 

‘Select Parameter,’ and select the line ‘U.H.S. for 5 percent PE in 50 years,’ then click 

‘Calculate Spectrum’ (figure 2-3(d))6

. 

Step 7. Values for SS and S1 for the site specified in Step 5 and the return period selected in 

Step 6 are displayed in the box to the right of the screen under the heading ‘Output for 

All Calculations’ (figure 2-3(d)). 

• Read SS from the row with Period = 0.2 sec under column headed ‘PGA or Sa.’ 

• Read S1 from the row with Period = 1.0 sec under column headed ‘PGA or Sa.’ 

Click box ‘Clear Selections’ and display screen shown in figure 2-3(e). 

Step 8. To obtain values for SS for the lower level ground motion, go to the box labeled ‘Select 

Parameter’ and select the line ‘Hazard Curve for 0.2 sec,’ then click ‘Calculate Hazard 

Curve’ (figure 2-3(e))7

. 

Step 9. Values for SS for the site specified in Step 5 and for a range of frequencies of exceedance 

are displayed in the box to the right of the screen under the heading ‘Output for All 

Calculations’ (figure 2-3(e)). The value of SS for a frequency of 0.01 (return period of 

100 years) is obtained by linear interpolation from this table8

. Alternatively, values may 

be read directly from a plot of frequency against acceleration, i.e., from the Hazard Curve, 

as described in Step 10. 

Step 10. Click ‘View Hazard Curve’ to display plot shown in figure 2-3(f). Read SS for annual 

frequencies of 0.01. Note that the value for SS for the 1000-year return period may also 

be read from this same graph (FEX = 0.001 = 1.0E-03) and compared with the value 

obtained in Step 7. Return to the previous screen by clicking ‘Exit Viewer,’ and then 

click ‘Clear Selections’ to display the screen shown in figure 2-3(g). 

Step 11. To obtain values for S1 for the lower level ground motion, go to the box labeled ‘Select 

Parameter’ and select the line ‘Hazard Curve for 1.0 sec,’ then click ‘Calculate Hazard 

Curve’ (figure 2-3(g))7,8. 

Step 12. Values for S1 for the site specified in Step 5 and for a range of frequencies of 

exceedance, are displayed in the box to the right of the screen under the heading ‘Output 

 

6 The upper level ground motion is defined in chapter 1 (table 1-2) as having a seven percent probability of 

exceedance (PE) in 75 years, which is equivalent to a five percent PE in 50 years. Both probabilities of exceedance 

describe an earthquake with a return period of about 1,000 years. 

7 The lower level event is defined in chapter 1 (table 1-2) as having a 50 percent probability of exceedance (PE) in 

75 years, which is equivalent to a return period of about 100 years or an Annual Frequency of Exceedance (FEX) of 

about 0.01. 

8

 This table uses scientific notation to express frequencies of exceedance, and a frequency of 0.01 is therefore 

written as 1.0E-02. 
70

for All Calculations’ (figure 2-3(g)). The value of S1 for a frequency of 0.01 (return 

period of 100 years) is obtained by linear interpolation from this table. Alternatively, 

values may be read directly from a plot of frequency against acceleration, i.e., from the 

Hazard Curve, as described in Step 13. 

Step 13. Click ‘View Hazard Curve’ to display plot shown in figure 2-3(h). Read S1 for annual 

frequencies of 0.01. Note that the value for S1 for the 1000-year return period may also be 

read from this same graph (FEX = 0.001 = 1.0E-03) and compared with the value 

obtained in Step 7. Return to the previous screen by clicking ‘Exit Viewer.’ 

Step 14. Click ‘Exit Program’ to end calculations. 

Example 2.1 illustrates the above process for finding Ss and S1 for both the lower and upper 

earthquake ground motions for a site identified by a zip code. 
71

EXAMPLE 2.1: DETERMINATION OF SS AND S1 VALUES FROM CD-ROM 

Calculate Ss and S1 for a bridge site in Salt Lake City (zip code 84112) for both the lower and upper 

earthquake ground motions (100- and 1000-year return periods). 

Steps 1, 2, 3, and 4. Initialize: 

• Load CD and start ‘Probabilistic Hazard 3.10’ 

• Click ‘Hazard Curves and Response Spectra for the United States’ (figure 2-3(a)) 

• Enter name ‘Demonstration: SLC’ (figure 2-3(b)) 

• Select ‘Include Date and Time on Output’ (figure 2-3(b)) 

• Select ‘Open File Selection Menu on figure 2-3(b) and choose CD-ROM drive d: for both the Data 

Files and Map Files (figure 2-3(c)) 

• Select ‘GM96-Data’ and ‘GM96-Maps’ respectively (figure 2-3(c)), and 

• Click ‘OK’. 

Step 5. Identify bridge site: 

• Select ‘5-digit Zip Code’ and enter ‘84112’ (figure 2-3(d)). 

Steps 6 and 7. Find Ss and S1 for the Upper Level Ground Motion (1000-year return period): 

• Go to ‘Select Parameter’ box and choose VHS ‘5 percent PE in 50 years’ (figure 2-3(d)) 

• Click ‘Calculate Spectrum’ (figure 2-3(d)) 

• In panel at right read Sa = 110.9 percent g = 1.109g at period = 0.2 sec (figure 2-3(d)) 

• In panel at right read Sa = 38.6 percent g = 0.386g at period =1.0 sec (figure 2-3(d)), and 

• Click ‘Clear Selections’ (figure 2-3(d)). 

Steps 8, 9, and 10. Find Ss for the Lower Level Ground Motion (100-year return period): 

• Go to ‘Select Parameter’ box and choose ‘Hazard Curve for 0.2 sec’ (figure 2-3(e)) 

• Click ‘Calculate Hazard Curve’ (figure 2-3(e)) 

• Obtain Ss for frequency of exceedance (FEX) = 0.01 by linear interpolation as follows: 

From the box at right of figure 2-3(e) labeled ‘Output for All Calculations:’ 

Therefore, Ss at FEX =0.01 is given by: 

 Ss = 0.128 + (0.192 – 0.128) (0.01448 – 0.01000) / (0.01448 – 0.00855) 

 = 0.176 g 

Alternatively, Ss may be read directly from the Hazard Curve in figure 2-3(f) by selecting ‘View 

Hazard Curve.’ Here it can be seen that for a frequency of 0.01, the acceleration (Ss) is 

approximately 0.17g. Click ‘Exit Viewer.’ 

• Click ‘Clear Selections.’ 

Frequency of exceedance (FEX) Ss (g) 

1.448E-02 0.01448 0.128 

8.551E-03 0.00855 0.192 
72

Steps 11, 12, and 13. Find S1 for the Lower Level Ground Motion (100-year return period): 

• Go to ‘Select Parameter’ box and choose ‘Hazard Curve for 1.0 sec’ (figure 2-3(g)) 

• Click ‘Calculate Hazard Curve’ (figure 2-3(g)) 

• Obtain S1 for frequency of exceedance (FEX) = 0.01 by linear interpolation as follows: 

From the box at right of figure 2-3(g) labeled ‘Output for All Calculations:’ 

Therefore, S1 at FEX =0.01 is given by: 

 S1 = 0.0427 + (0.0641 – 0.0427) (0.01173 – 0.01000) / (0.01173 – 0.00715) 

 = 0.0508 g 

Alternatively, S1 may be read directly from the Hazard Curve in figure 2-3(h) by selecting ‘View 

Hazard Curve.’ Here it can be seen that for a frequency of 0.01, the acceleration (S1) is 

approximately 0.05g. Click ‘Exit Viewer.’ 

Step 14. Click ‘Exit Program.’ 

A summary of results obtained above is given in the table below. 

Values of Ss and S1 for Bridge Site in Zip Code 84112. 

Ground Motion Ss (g) S1 (g) 

Lower Level: 100-year 0.176 0.051 

Upper Level: 1000-year 1.109 0.386 

Frequency of exceedance (FEX) S1 (g) 

1.173E-02 0.01173 0.0427 

7.146E-03 0.00715 0.0641 
73

(a) Opening window for ‘Probabilistic Hazard 3.10’ 

(b) Initialization window 

(continued)

Figure 2-3. Calculation of short- and long-period spectral accelerations for 100- and 

1000-year ground motion return periods. 
74

 (c) Setup window for data and map files 

(d) Selection of 1000-year uniform hazard spectrum and result for given site

(continued)

Figure 2-3 (continued). Calculation of short- and long-period spectral accelerations for 

100- and 1000-year ground motion return periods. 
75

(e) Selection of Ss (0.2 sec) hazard curve and result for given site 

(f) Example of Ss (0.2 sec) hazard curve for given site 

(continued)

Figure 2-3 (continued). Calculation of short- and long-period spectral accelerations for 

100- and 1000-year ground motion return periods. 
76

(g) Selection of S1 (1.0 sec) hazard curve and result for given site 

(h) Example of S1 (1.0 sec) hazard curve for given site 

Figure 2-3 (continued). Calculation of short- and long-period spectral accelerations for 

100- and 1000-year ground motion return periods. 
77

2.4. SITE CLASSES AND SITE FACTORS

2.4.1. DESCRIPTION OF SITE CLASSES AND SITE FACTORS 

When using the general procedure, sites should be classified and site factors determined in 

accordance with this section. The development of response spectra using national ground motion 

maps (section 2.3) and site factors is described in section 2.5. 

Recommended site classes and site factors are based on studies carried out following the 1989 

Loma Prieta earthquake in California, which culminated in recommendations that have been 

adopted by the NEHRP Building Provisions (BSSC, 1994, 1998), the Uniform Building Code 

(ICBO, 1997), and the International Building Code (IBC) (ICC, 2000)9

. Earthquake data 

obtained during the Loma Prieta earthquake confirmed the validity of these site factors and the 

improvement in accuracy compared to site factors in earlier bridge and building codes. 

Site class definitions are given in table 2-2. Sites are classified according to their stiffness as 

determined by their shear wave velocity in the upper 30 meters (100 feet). Standard Penetration 

Test (SPT) blowcounts and undrained shear strengths of soil samples from soil borings can also 

be used to classify sites as indicated in table 2-2. Steps for classifying a site are given in table 

2-3. Alternatively, SPT blowcounts and undrained shear strengths can be converted into 

estimated shear wave velocities and used for site classification. Procedures given in Kramer 

(1996) can be used for these conversions. 

Shear wave velocities were used when conducting the original studies that defined the site 

classes and may be considered as the fundamental soil property. If the site profile consists of 

layered sands and clays or soil over rock within the upper 30 meters (100 feet), then it is 

particularly desirable to convert the profile to estimated shear wave velocities. If the resulting 

site class does not appear reasonable, or if the project involves special design issues, shear wave 

velocity measurements should be made at the site. 

Site factors for the site classes in table 2-2 are given in table 2-4. Site class B rock (soft rock) is 

taken to be the site category for the USGS ground shaking maps. Type B rock is therefore the 

reference site condition for which the site factor is taken to be 1.0. Site classes A, C, D, and E 

have separate sets of site factors for the short-period range (site factor Fa) and long-period range 

(site factor Fv), as indicated in table 2-3. These site factors generally increase as the site profile 

becomes softer (in going from site class A to E). Except for site class A (hard rock), the site 

factors also decrease as the ground motion level increases, reflecting the nonlinear behavior of 

most soils. Therefore, for a given soil site class C, D, or E, these nonlinear site factors increase 

the ground motion more in areas having lower ground motions than in areas having higher 

ground motions. The levels of ground motion for use with table 2-4 are characterized by shortperiod (0.2-second) response spectral acceleration and long-period (1.0-second) response 

spectral acceleration on rock as shown on USGS national ground motion maps. 

 

9

 Martin and Dobry, 1994; Rinne, 1994; Dobry et al., 2000 
78

2.4.2. CONSIDERATIONS FOR SITE CLASS F SOILS 

As indicated in tables 2-2 and 2-4, a site class F is defined for types of soil for which site factors 

are not given and site-specific investigations are recommended to determine site response effects. 

Site class F is intended for soils for which site response effects may be very strong and not 

reliably quantified except by site-specific studies. Guidelines for determining the response of 

site class F soils by site-specific studies are given in appendix A. 

Also per section 2.2, a site-specific investigation to determine site response for site class F soils 

is not required if a determination is made that the presence of such soils will not result in a 

significantly higher response of a bridge. Such a determination should be made jointly by the 

bridge engineer and the geotechnical engineer. 

Examples of conditions that could lead to such a conclusion are the extent and depth of site class 

F soils. With regard to the extent of these soils, for short bridges with a limited number of spans 

and having earth approach fills, ground motions at the abutments will generally determine the 

response of the bridge. If site class F soils are found only at the piers and are not present at the 

abutments, it might be concluded that the response of the piers will not significantly affect 

overall bridge response. With regard to the depth of these soils, there may be cases where the 

effective depth of ground motion is in stiff soil below a soft surficial layer. If the surficial layer 

is site class F and the underlying soil profile is site class E or stiffer, the surficial soils are 

unlikely to significantly increase bridge response. 

Note that in table 2-2, there is one less category of site class F soils than the four originally 

adopted in the 1994 NEHRP Provisions (BSSC, 1994). This category consists of soils 

vulnerable to potential failure or collapse under seismic loading, such as liquefiable soils, quick 

and highly sensitive clays, and collapsible weakly cemented soils. Such soils should be 

classified on the basis of tables 2-2 and 2-3 assuming that soil failure or collapse will not occur. 

Special analyses to define the amplification of site ground motion in these soils is too severe a 

requirement for ordinary bridge design because such analyses require utilization of either 

effective stress or strength-degrading, nonlinear analytical techniques that are difficult to apply 

even by experts. Also, limited case history data and analytical results indicate that liquefaction 

reduces spectral response rather than increases it, except in some cases at long periods. 

However, in accordance with section 2.2, site-specific analyses should be considered for major 

or very important bridges and advanced analytical techniques, such as effective stress analysis, 

should be used if appropriate. 

Since liquefaction generally reduces response spectral amplitudes, the engineer may wish to 

conduct special analyses of site response to avoid excessive over-estimation of bridge inertia 

loads when liquefaction occurs. 

The deletion of ground-failure-susceptible soils from site class F only affects the requirement to 

conduct site-specific analyses for determining ground motion amplification by these soils. It is 

still necessary to evaluate their potential for failure and their effect on bridge performance as 

indicated in chapter 3. 
79

Table 2-2. Site classes. 

Site 

Class Description 

A Hard rock with measured shear wave velocity, vs > 1500 m/sec (5,000 ft/sec) 

B Rock with 760 m/sec < s v < 1500 m/sec (2,500 ft/sec < s v < 5,000 ft/sec) 

C 

Very dense soil and soil rock with 360 m/sec < s v < 760 m/sec (1,200 ft/sec < s v < 2,500 ft/sec) 

or with either N > 50 blows/0.30m (50 blows/ft) or us > 100 kPa (2,000 psf) 

D 

Stiff soil with 180 m/sec < s v < 360 m/sec (600 ft/sec < s v < 1,200 ft/sec) 

or with either 15 < N < 50 blows/0.30m (15 < N < 50 blows/ft) or 50 kPa < us < 100 kPa 

(1,000 < us < 2,000 psf) 

E 

Soil profile with s v < 180 m/sec (600 ft/sec) 

or with either N < 15 blows/0.30m (N < 15 blows/ft) or us < 150 kPa (1000 psf), 

or any profile with more than 3 m (10 ft) of soft clay defined as soil with PI > 20, w > 40 percent 

and us < 25 kPa (500 psf) 

F 

Soils requiring site-specific evaluations 

1. Peats or highly organic clays (H > 3 m [10 ft] of peat or highly organic clay where H = 

thickness of soil) 

2. Very high plasticity clays (H > 8 m [25 ft] with PI > 75) 

3. Very thick soft/medium stiff clays (H > 36 m [120 ft]) 

Exception: When the soil properties are not known in sufficient detail to determine the site class, site class 

D may be used. Site classes E or F need not be assumed unless the authority having jurisdiction 

determines that site classes E or F could be present at the site or in the event that site classes E or F are 

established by geotechnical data. 

Notes: 

1. s v is average shear wave velocity for the upper 30 m (100 ft) of the soil profile 

N is the average Standard Penetration Test (SPT) blowcount (blows/0.30m or blows/ft) (ASTM D1586) 

 for the upper 30 m (100 ft) of the soil profile 

us is the average undrained shear strength in kPa (psf) (ASTM D2166 or D2850) 

 for the upper 30 m (100 ft) of the soil profile 

 PI is plasticity index (ASTM D4218) 

 w is moisture content (ASTM D2216) 

2. The shear wave velocity for rock, site class B, shall be either measured on site or estimated for competent rock 

with moderate fracturing and weathering. Softer and more highly fractured and weathered rock shall either be 

measured on site for shear wave velocity or classified as site class C. 

3. The hard rock, site class A, category shall be supported by shear wave velocity measurements either on site or on 

profiles of the same rock type in the same formation with an equal or greater degree of weathering and fracturing. 

Where hard rock conditions are known to be continuous to a depth of 30 m (100 ft), surficial shear wave velocity 

measurements may be extrapolated to assess s v . 

4. Site classes A and B should not be used when there is more than 3 m (10 ft) of soil between the rock surface and 

the bottom of a spread footing. 
80

Table 2-3. Steps for classifying a site. 

Step Description 

1 

Check for the three categories of site class F in table 2-2 requiring site-specific evaluation. If 

the site corresponds to any of these categories, classify the site as site class F and conduct a 

site-specific evaluation.

2 

Check for existence of a soft layer with total thickness > 3m (10 ft), where soft layer is defined 

by su < 25 kPa (500 psf), w > 40 percent, and PI >20. If these criteria are met, classify site as 

site class E. 

3 Categorize the site into one of the site classes in table 2-2 using one of the following three 

methods to calculate: 

• s v for the top 30 m (100 ft)) ( s v method) 

• N for the top 30 m (100 ft) (N method) 

• Nch for cohesionless soil layers (PI < 20) in the top 30 m (100 ft) and 

 us for cohesive soil layers (PI > 20) in the top 30 m (100 ft) ( us method) 

To make these calculations, the soil profile is subdivided into n distinct soil and rock layers, 

and in the methods below the symbol i refers to any one of these layers from 1 to n. 

Method A: s v method 

The average s v for the top 30 m (100 ft) is calculated as follows: 

1

1

=

=

= ∑

∑

n

d

i

i

n d

i

vsi i

vs

where: 

=1

∑

n

d

i

i

 is equal to 30 m (100 ft), vsi is the shear wave velocity in m/sec (ft/sec) of a 

layer, and di

 is the thickness of a layer between 0 and 30 m (0 and 100 ft). 

Method B: N method 

The average N for the top 30 m (100 ft) is calculated as follows: 

=

=

=

∑

∑

n

d

i

i 1

n d

i

Ni i 1

N

where: Ni is the Standard Penetration Test blowcount of a layer (not to exceed 100 

blows/0.30m [100 blows/ft] in the above expression). Note that when using 

Method b, N values are for cohesionless soils and cohesive soil and rock layers 

within the upper 30 m (100 ft). Where refusal is met for a rock layer, Ni should be 

taken as 100 blows/0.30m [100 blows/ft] 

(continued) 
81

Table 2-3. Steps for classifying a site (continued). 

Step Description 

3 

continued 

Method C: us method 

The average Nch for cohesionless soil layers in the top 30 m (100 ft) is calculated as follows: 

=

=

∑

s

ch m di

Nchi i 1

d N

where: 

=

∑ =

m

d d i s

i 1

, m is the number of cohesionless soil layers in the top 30 m (100 ft), Nchi

is the blowcount for a cohesionless soil layer (not to exceed 100 blows/0.30m [100 

blows/ft] in the above expression), and ds is the total thickness of cohesionless soil 

layers in the top 30 m (100 ft). 

The average us for cohesive soil layers in the top 30 m (100 ft) is calculated as follows: 

=

=

∑

dc

u k d

i

s

ui i 1

s

where: 

=

∑ = k

i c

i 1

d d , k is the number of cohesive soil layers in the top 30 m (100 ft), sui is the 

undrained shear strength for a cohesive soil layer (not to exceed 250 kPa [5,000 psf] in the 

above expression), and dc is the total thickness of cohesive soil layers in the top 30 m (100 

ft).

Note: When using Method C, if the site class resulting from Nch and us differ, select the site class that gives the 

highest site factors and design spectral response in the period range of interest. For example, if Nch was 

equal to 20 blows/0.30m (20 blows/ft) and us was equal to 38 kPa (800 psf), the site would classify as D or 

E in accordance with Method C and the site class definitions of table 2-2. In this example, for relatively low 

response spectral acceleration and for long-period motions, table 2-4 indicates that the site factors are 

highest for site class E. However, for relatively high short-period spectral acceleration (Ss > 0.75g), short 

period site factors, Fa, are higher for site class D. 
82

Table 2-4. Site factors Fa and Fv

(a) Values of Fa as a function of site class and short-period (0.2-sec) spectral acceleration, SS 

Spectral Acceleration at Short-Period (0.2 sec), SS

1 

Site 

Class SS < 0.25 SS = 0.50 SS = 0.75 SS = 1.00 SS > 1.25 

A 0.8 0.8 0.8 0.8 0.8 

B 1.0 1.0 1.0 1.0 1.0 

C 1.2 1.2 1.1 1.0 1.0 

D 1.6 1.4 1.2 1.1 1.0 

E 2.5 1.7 1.2 0.9 0.9 

F2 

Notes: 

1. Use straight-line interpolation for intermediate values of SS. 

2. Site-specific geotechnical investigation and dynamic site response analysis should be performed for 

 class F soils.

(b) Values of Fv as a function of site class and long-period (1.0-sec) spectral acceleration, S1

Spectral Acceleration at Long-Period (1.0 sec), S1

1 

Site 

Class S1 < 0.1 S1 = 0.2 S1 = 0.3 S1 = 0.4 S1 > 0.5 

A 0.8 0.8 0.8 0.8 0.8 

B 1.0 1.0 1.0 1.0 1.0 

C 1.7 1.6 1.5 1.4 1.3 

D 2.4 2.0 1.8 1.6 1.5 

E 3.5 3.2 2.8 2.4 2.4 

F2 

Notes: 

1. Use straight-line interpolation for intermediate values of S1. 

2. Site-specific geotechnical investigation and dynamic site response analysis should be performed for class F 

 soils.
83

 2.4.3. EFFECT OF SITE CLASS VARIATION ALONG A BRIDGE 

The procedures described above for assessing site effects on ground motions were originally 

developed for computing ground motions at the surface of relatively uniform soil conditions. 

However, soil conditions may differ significantly from one end of a bridge to the other; an 

example is where one abutment is on firm ground or rock and the other is on fill over soft soil. 

These variations are not handled well by simplified procedures and significant errors may result. 

As a general rule, when geotechnical conditions vary significantly along the length of a bridge, 

the site factors described above may be used to construct response spectra at key locations along 

the length. These spectra may then be used to construct an envelope of response spectra for 

design. For major or very important bridges, it may be necessary to use more rigorous numerical 

modeling to represent these conditions, as discussed in section 2.9. 

2.4.4. EFFECT OF DEPTH-OF-MOTION ON SITE CLASSIFICATION AND SITE 

FACTORS 

For short bridges with a limited number of spans, the motion at the abutment will generally be 

the primary mechanism by which energy is transferred from the ground to the bridge 

superstructure. If the abutment involves an earth fill, the site classification should be determined 

on the basis of the soil profile below the fill. The effects of the fill overburden pressure on the 

soil properties (e.g., change in shear wave velocity) should be included in the determination of 

site classification. 

For some bridges, it will be necessary to determine the site classification at the piers. If a pier is 

supported on spread footings, then it is appropriate to define the site class for conditions 

extending below the ground surface. However, if deep foundations (e.g., piles or drilled shafts) 

are used to support the pier, then the input motion may effectively be at some depth below the 

surface, depending on the horizontal stiffness of the soil-pile-footing system relative to the 

horizontal stiffness of the soil-pile system. If the soil-pile-footing is the stiffer of the two, then 

the motion should be defined at the footing. If the soil-pile-footing provides little horizontal 

stiffness or if there is no footing (e.g., pile bents), then the effective input motion to the bridge 

will likely be applied at some depth below the surface. The determination of this depth and the 

effective motion requires considerable judgment and should be evaluated jointly by the 

geotechnical engineer and bridge engineer. If the effective input motion is defined at some depth, 

then it may be overly conservative to define the site class and site factors on the basis of the soil 

profile immediately below the ground surface. Instead, it may be more appropriate to define the 

site class for a soil profile extending 30 m (100 feet) below the depth at which the input motion 

is defined. 

2.5. DEVELOPING HORIZONTAL GROUND MOTION RESPONSE SPECTRA USING 

NATIONAL GROUND MOTION MAPS AND SITE FACTORS 

2.5.1. TWO-POINT PROCEDURE FOR CONSTRUCTING RESPONSE SPECTRA 

The two-point procedure described in this section may be used to construct horizontal response 

spectra. Because the national ground shaking maps (section 2.3) give five percent damped 
84

spectral accelerations, the spectra constructed from these maps are for five percent damping. In 

section 2.5.3, factors are provided for obtaining response spectra for other damping ratios. 

The two-point response spectrum construction procedure uses response spectral accelerations of 

rock ground motion from national ground motion maps (section 2.3) for a short period of 

vibration (0.2 second) and a long period of vibration (1.0 second) along with site factors 

applicable to the short-period and long-period ranges (section 2.4). The following equations and 

figure 2-4 describe the response spectrum construction. 

S FS DS a s = (2-3) 

 

and 

S FS D1 v 1 = (2-4) 

Where SDS is the short-period (0.2 sec) design spectral acceleration, SD1 is the long-period (1.0 

sec) design spectral acceleration, SS is the short-period (0.2 sec) spectral acceleration on rock 

(site class B) from national ground motion maps or CD-ROM, S1 is the long-period (1.0 sec) 

spectral acceleration on rock (site class B) from national ground motion maps or CD-ROM, Fa is 

the short-period site factor interpolated for a given value of SS using table 2-3, and Fv is the longperiod site factor interpolated for a given value of S1 using table 2-3. 

The design spectral acceleration, Sa, of the plateau of the response spectrum is defined by: 

S S a DS = (2-5) 

The design spectral acceleration of the long-period declining branch to the response spectrum is 

defined by: 

D1

a

S S

T = (2-6) 

where T is the period of vibration. 

The period of vibration, TS, of the intersection of the plateau and the long-period branch of the 

spectrum is defined by: 

D1

S

DS

S T

S = (2-7) 

At periods of vibration less than or equal to To, the design spectral acceleration is defined by: 

DS

a DS

O

S S 0.60 T 0.40S

T

= + (2-8) 

where To = 0.2TS. 
85

Note that for T = 0 seconds, the resulting value of Sa is equal to the peak ground acceleration 

(assumed equal to 0.40 SDS based on equation 2-8). 

For long periods of vibration (T > 3 seconds), the spectral acceleration Sa may decrease more 

rapidly with increasing period than implied by equation 2-6. However, there is considerable 

uncertainty as to the influence of very long-period motions and the effect of other factors such as 

earthquake magnitude on the rate of decrease. Therefore, the 1/T rate should be used unless a 

faster rate can be justified based on a site-specific study. 

Figure 2-4. Design response spectrum (five percent damping) construction using twopoint procedure. 

2.5.2. MULTI-POINT PROCEDURE OF RESPONSE SPECTRUM CONSTRUCTION 

Because the 1996 USGS national ground motion mapping provides spectral accelerations at 

several periods of vibration (section 2.3.1) in addition to the periods of 0.2 second and 1.0 

second used for the two-point procedure described above, it is possible to use all of the spectral 

data directly rather than just the 0.2-second and 1.0-second values. This method of response 

spectrum construction is termed the multi-point method and can be used as an alternative to the 

two-point method. However, the multi-point method can lead to ambiguous results for sites 

other than rock (Power and Chiou, 2000). This is because only a short-period site factor and a 

long-period site factor are available (section 2.4) and it is not clear how the site factors vary 

when applied to spectral values at multiple periods. Therefore, judgment is required when 

applying the multi-point method for soil sites. Guidelines for applying site factors for soil sites 

using the multi-point method are: 
86

1. Site factor Fa should be used for periods less than or equal to 0.2 second. 

2. Site factor Fv should be used for periods greater than or equal to 1.0 second. 

3. Between the 0.2 and 1.0 second periods, the resulting soil spectrum should be smooth and not 

contain spectral peaks higher than the larger of the spectral accelerations defined at 0.2 

second and 1.0 second period. 

Analysis of 1996 USGS mapping results indicates that the two-point method can be 

unconservative in comparison to multi-point data for periods longer than 1.0 second at some sites 

in the central and eastern United States (CEUS) having very low seismic ground motion hazard 

(Power et al., 1997, 1998). Therefore, for CEUS sites, it is desirable to check the rock spectrum 

constructed using the two-point method by comparing with map data for periods longer than 1.0 

second. 

Figure 2-5 illustrates differences between the two-point and multi-point method for a soft rock 

site (site class B) for two cities in the CEUS (New York City and Memphis) and a city in the 

western United States (WUS) (Seattle). As illustrated by the multi-period data in the figure, a 

typical difference in rock motion characteristics between the CEUS and WUS is the peaking of 

the CEUS spectrum at a shorter period (0.1 second for the CEUS spectrum and 0.2 second for 

the WUS spectrum). However, the moderate exceedance of the two-point plateau acceleration 

by the very short-period (0.1 second) spectral acceleration is probably not significant for most 

bridges. 

In summary, use of the two-point method of response spectrum construction is a reasonable and 

acceptable approach for response spectrum construction. However, the following checks of the 

two-point-method spectrum at CEUS sites (east of 110°

 west longitude) are desirable: 

1. If periods of vibration greater than 1.0 second are important to structural response, check the 

2.0-second spectral acceleration to assure that the long-period response is not significantly 

underestimated. 

2. If periods of vibration less than 0.2 second are important to structural response, check the 

0.1-second spectral acceleration to assure that the short-period response is not significantly 

underestimated. 

2.5.3. MODIFICATION OF ELASTIC SPECTRAL DEMAND FOR HIGHER OR LOWER 

DAMPING 

Response spectra for structural damping ratios different than five percent can be obtained by 

multiplying the five percent damped spectra constructed using procedures in section 2.5.1 and 

2.5.2 by the period-dependent factors in table 2-5. For spectra constructed using the two-point 

method, the factors for 0.2 second and 1.0 second can be used. The factors in table 2-5 are based 

on empirical studies of the variation of elastic response spectral amplitudes with damping ratio10. 

 

10 Newmark and Hall, 1982; Abrahamson, 1993; Idriss, 1993 
87

Figure 2-5. Comparison of two-point method of five percent damped response spectrum 

construction with USGS multi-point mapping results for two percent probability of 

exceedence in 50 years. 
88

Table 2-5. Damping adjustment factors. 

Ratio of Response Spectral Acceleration for Damping 

Period Ratio ξ to Response Spectral Acceleration for ξeff = 5 percent (seconds) ξeff = 2 percent ξeff = 7 percent ξeff = 10 percent 

0.02 1.00 1.00 1.00 

0.10 1.26 0.91 0.82 

0.20 1.32 0.89 0.78 

0.30 1.32 0.89 0.78 

0.50 1.32 0.89 0.78 

0.70 1.30 0.90 0.79 

1.0 1.27 0.90 0.80 

2.0 1.23 0.91 0.82 

4.0 1.18 0.93 0.86 

2.5.4. OBTAINING PEAK GROUND ACCELERATION FOR GROUND FAILURE 

EVALUATIONS 

As noted in section 2.5.1, the peak ground acceleration is equal to the zero-period acceleration of 

a response spectrum. When peak ground acceleration is needed for an evaluation of the potential 

for soil failure, such as liquefaction or landsliding, it can be obtained as the zero-period 

acceleration from equation 2-8. Alternately, it can be directly obtained for rock site conditions 

from the 1996 USGS maps for peak ground acceleration for a selected probability of exceedance 

or return period. If obtained directly from the 1996 USGS maps, the corresponding soil-site 

acceleration for a given site class can reasonably be assumed to equal the mapped rock site 

acceleration multiplied by the short-period site factor (Fa) in table 2-4(a). In interpolating site 

factors as a function of peak rock acceleration level in table 2-4(a), it can be assumed that peak 

ground (rock) acceleration is equal to 0.4 times the short-period spectral acceleration on rock, Ss. 

2.6. DEVELOPING SITE-SPECIFIC RESPONSE SPECTRA OF HORIZONTAL 

GROUND MOTIONS 

In cases where a site-specific approach to developing response spectra is used, the overall 

objective is to develop ground motions that are more accurate for the local seismic and site 

conditions than can be determined from the general procedure using national ground motion 

maps and site factors (sections 2.3 through 2.5). Accordingly, site-specific studies must be 

comprehensive and incorporate current scientific interpretations for seismic sources and ground 

motion attenuation. It is also important to incorporate uncertainties in modeling in a site-specific 

probabilistic analysis. Examples of these uncertainties include seismic source location, extent, 

and geometry, maximum earthquake magnitude, earthquake recurrence rate, and ground motion 
89

attenuation relationships. A site-specific analysis should be documented in detail and be subject 

to detailed peer review. When response spectra are determined from a site-specific study, the 

design spectra should not be lower than two-thirds of the spectra determined using the general 

procedure described in sections 2.3 through 2.5. 

Where analyses to determine soil amplification are required by table 2-3 for site class F soils (see 

also section 2.4.2), the influence of the local soil conditions should be determined based on sitespecific geotechnical investigations and dynamic site response analyses. Guidelines are 

presented in appendix A for conducting these investigations and analyses. These guidelines are 

applicable for site-specific determination of site response for any site class when the site 

response is determined on the basis of a dynamic site response analysis. 

For sites located within 10 km (6 mi) of an active fault as defined in chapter 3, studies to 

quantify near-fault effects on ground motions should be conducted to determine if these could 

significantly influence bridge response. The USGS and state geological agencies may be 

contacted to determine the locations of known active faults. 

Near-fault effects on horizontal response spectra include: 

1. Higher ground motions due to the proximity of the active fault, 

2. Directivity effects that increase ground motions for periods greater than 0.5 second if the 

fault rupture propagates toward the site, and 

3. Directionality effects that increase ground motions for periods greater than 0.5 second in the 

direction normal (perpendicular) to the strike of the fault. 

If the active fault was included in the development of the national ground motion maps, then 

effect (1) is already included in the ground motions defined by national ground motion maps. 

Effects (2) and (3) are not included in the national maps. These effects are considered to be 

significant only for periods longer than 0.5 second. Normally these effects would be evaluated 

only for major or very important bridges having natural periods of vibration longer than 0.5 

second (Somerville, 1997, and Somerville et al., 1997). 

Figure 2-6 illustrates near-fault effects on both response spectra and time histories of horizontal 

ground motion. As seen in the time history plots, there is a significant long-period ground 

motion pulse that occurs in the direction perpendicular to the fault strike (i.e., fault-normal 

direction). As a result, the response spectrum at long periods is significantly increased in the 

fault-normal direction compared to the fault-parallel direction. 

For purposes of developing deterministic spectra, subsurface faults as well as surface faults 

should be considered if (1) their location is known or can reasonably be inferred, and (2) the fault 

is an active fault as defined in chapter 3. The interfaces between tectonic plates in subduction 

zone regions are considered to be active faults. 
90

Somerville, 1997 

Figure 2-6. Time histories and horizontal response spectra (five percent damping) for the 

fault strike-normal and fault strike-parallel components of ground motion for the Rinaldi 

recording obtained 4.5 mi (7.5 km) from the fault rupture during the 1994 Northridge, 

California earthquake. 
91

2.7. DEVELOPING VERTICAL GROUND MOTION RESPONSE SPECTRA 

Recent studies11 have shown that the ratio of the vertical response spectrum to the horizontal 

response spectrum of ground motions can differ substantially from the nominal two-thirds ratio 

commonly assumed in engineering practice. These studies show that the ratios of vertical to 

horizontal response spectral values are functions of the tectonic environment, subsurface soil or 

rock conditions, earthquake magnitude, earthquake source-to-site distance, and period of 

vibration. Whereas the two-thirds ratio may be conservative for longer periods of vibration 

(greater than about 0.2 second), at shorter periods the ratio of vertical to horizontal response 

spectra may exceed two-thirds and even substantially exceed 1.0 for close earthquake source-tosite distances and periods less than about 0.1 second. 

Figure 2-7 illustrates vertical-to-horizontal response spectral ratios obtained in four studies for a 

moment magnitude 6.5 earthquake, source-to-site distance of 10 km, and for rock and soil site 

conditions. These results are from analysis of ground motion data from the western United 

States (WUS) and other regions having a similar shallow crustal faulting tectonic environment. 

Ground motion modeling analyses by Chiou et al. (2002) suggest that the ratios for the central 

and eastern United States (CEUS) are not greatly different from the ratios for the WUS. 

At present, detailed procedures have not been developed for constructing vertical spectra having 

an appropriate relationship to the horizontal spectra constructed using the general procedure 

described in section 2.5. When developed, these procedures could be used in conjunction with 

deaggregation information on dominant earthquake source-to-site distance and earthquake 

magnitude (section 2.8.1), to construct vertical spectra at any location. For the present, when 

vertical response spectra are required and the horizontal response spectra are constructed using 

the general procedure of section 2.5, use of a vertical-to-horizontal spectral ratio of two-thirds 

throughout the period range is recommended unless the important vertical natural periods of 

vibration of the bridge are less than 0.2 second. In the latter case, analyses should be made to 

determine whether the short-period part of the vertical spectrum should be enriched. Such 

analyses should be made using results of current studies of vertical-to-horizontal spectral ratios, 

such as illustrated in figure 2-7. 

In cases where site-specific horizontal spectra are developed (section 2.6) but vertical spectra are 

also required, results of current studies of vertical ground motions, such as those illustrated in 

figure 2-7, should be used. Vertical-to-horizontal spectral ratios less than two-thirds may be 

used if validated by site-specific vertical spectra but values less than 0.5 are not recommended. 

 

 

11 Abrahamson and Silva, 1997; Silva, 1997; Sadigh et al., 1993; Campbell, 1997; Bozorgnia et al., 1999; Campbell 

and Bozorgnia, 2000a, b; Chiou et al., 2002 
92

Figure 2-7. Vertical/horizontal spectral ratios vs. period 

2.8. DEVELOPING ACCELERATION TIME HISTORIES

2.8.1. GENERAL REQUIREMENTS FOR TIME HISTORIES 

When a time history analysis of a bridge is to be performed, the time histories of ground motions 

to be used in the analysis should represent the seismic environment of the site and local site 

conditions. Site characteristics to be considered include: 

1. The tectonic environment, e.g., subduction zone, shallow crustal faults in western United 

States, or crustal environment in eastern United States. 

2. Earthquake magnitude. 

3. Type of faulting (e.g., strike-slip; reverse; normal). 
93

4. Seismic source-to-site distance. 

5. Local site conditions. 

6. Design or expected ground motion characteristics (e.g., design response spectrum, duration 

of strong shaking, special ground motion characteristics such as near-fault characteristics). 

Recorded time histories from sites with similar characteristics are preferred, but compromises are 

usually required because of the many characteristics defining the environment and the limited 

database of recorded time histories. Use of time histories having similar earthquake magnitudes 

and distances, within reasonable ranges, are especially important parameters because they have a 

strong influence on response spectral content, response spectral shape, duration of strong shaking, 

and near-source ground motion characteristics. The motions selected should be somewhat 

similar in overall ground motion level and spectral shape to the design spectrum to avoid using 

very large scaling factors for the recorded motions and very large changes in spectral content in 

the spectrum-matching approach discussed in section 2.8.3 below. 

If the dominant earthquake is located within about 10 km (6 mi) of the site and has a magnitude 

equal to or greater than 6, then intermediate-to-long period ground motion pulses should be 

included, since these characteristics in the ground motion could significantly influence bridge 

response. Somerville et al. (2000, 2003) provides guidance on the time-domain characteristics of 

near-source ground motion pulses. The response spectral characteristics of near-source ground 

motions are described in section 2.6, and are illustrated in figure 2-6. Similarly, the high shortperiod spectral content of near-source vertical ground motions should be considered (section 2.7). 

As indicated in figure 2-6, the amplitude of near-source horizontal ground motion pulses is 

dependent on the direction of the ground motion relative to the fault, being highest in the fault 

strike-normal (perpendicular) direction and lowest in the fault strike-parallel direction. Selected 

recorded near-source time histories should be transformed to fault-normal and fault-parallel 

directions. When used for a specific bridge, the time histories should be transformed to the 

principal bridge axes, depending on the orientation of the bridge relative to the fault strike. 

When the design response spectrum is defined for a particular probability of exceedance or 

return period, a range of earthquake magnitudes and distances contribute to the spectrum. The 

ground motion hazard from a probabilistic ground motion analysis should be deaggregated to 

determine the predominant magnitude and distance contributions to the hazard to guide the 

selection and development of appropriate time histories. Hazard deaggregation of response 

spectral accelerations constructed using 1996 USGS ground motion maps can be obtained from 

the USGS website [http://eqhazmaps.usgs.gov]. Table 2-6 outlines the steps necessary to obtain 

these data from the web site. 

Two examples of hazard deaggregation are shown in figure 2-8 for New York City and Seattle. 

The example is for 0.2-second spectral acceleration for three percent probability of exceedance 

in 75 years (2500-year return period). For this example, the hazard in New York City is 

dominated by small-to-moderate magnitude earthquakes occurring at close to moderate distances, 
94

and the hazard at Seattle is dominated by moderate-to-large magnitude earthquakes occurring at 

close distances. The weighted average magnitude contribution to the hazard for each city is also 

shown. It should be noted that the relative magnitude and distance contributions for a particular 

location may vary with the ground motion parameters (e.g., relative contributions may differ for 

short-period and long-period motions) and with probability of exceedance. Also, at some sites, 

contributions may be more complex, such as a bi-modal contribution from nearby small-tomoderate magnitude earthquakes and distant large-magnitude earthquakes. 

Table 2-6. Procedure for obtaining deaggregated seismic hazard from USGS web site. 

Step Activity 

1 Access http://eqhazmaps.usgs.gov with any recent browser and open home page. 

2 Select ‘Interactive Deaggregations, 1996’

3 On next screen: 

Enter site name and latitude and longitude coordinates for site. 

Select Return time from menu. 

For example: 5 percent PE 50 yrs for 1000-year return period. 

Select SA frequency from menu. 

For example: 1.0 Hz for S1 (long-period spectral acceleration) or 5.0 Hz for SS (shortperiod spectral acceleration). 

Choose ‘Geographic Deaggregation.’ 

Choose ‘Mean’ Seismograms. 

4 Click ‘Generate Plot(s) and Data.’ 

5 Select ‘Hazard Matrices’ to print tabulated data. 

Select ‘Deaggregated Seismic Hazard Graph,’ to print earthquake contribution to hazard by 

magnitude and distance. Choose desired format: gif, .pdf, .ps. 

Read mean (and modal) earthquake distance and magnitude. 

Select ‘Geographic Deaggregated Seismic Hazard Map’ and ‘Seismograms for Modal or 

Mean Event’ if required. Choose desired formats. 

Note: Deaggregated seismic hazard data is not available on the CD-ROM provided with this manual. 

 Use the website in step 1 to obtain this data. 

2.8.2. METHODS OF SELECTING AND DEVELOPING TIME HISTORIES 

Time histories may be either recorded time histories that are scaled for the bridge location, or 

spectrum-matched time histories. For scaled time histories, the amplitudes of recorded ground 

motions are scaled by a constant factor so that the response spectrum of the time history (with its 

spectral peaks and valleys) is, on average, approximately at the level of the design response 

spectrum in the range of periods of structural significance. For spectrum-matched time histories, 

after simple scaling of recorded time histories as in the previous approach, a spectrum-matching 

procedure is then employed so that the frequency content of the scaled recorded time history is 

adjusted and the spectrum of the modified time history is a close fit or ‘match’ to the design 

response spectrum in the range of periods of structural significance. 
95

(a) New York City, NY 

(b) Seattle, WA 

Figure 2-8. Hazard deaggregation for 0.2 second response spectral acceleration (Ss) for 

ground motion with three percent probability of exceedance in 75 years 

(2500-year return period). 
96

If sufficient recorded motions are not available, simulations of recorded time histories may be 

developed using theoretical ground motion modeling methods that simulate the earthquake 

rupture process and wave propagation from source-to-site through the earth’s crust. Analyses of 

soil column effects on time histories can be made using the methods of site response analysis 

described in appendix A. 

If spectrum-matched time histories are developed, the initial time histories should be 

representative of recorded or simulated motions. The analytical techniques used for spectrum 

matching should be capable of achieving seismologically realistic time series that are similar to 

the time series of the initial time histories selected for spectrum matching. Response spectrummatching approaches include methods in which time series adjustments are made in the time 

domain12 and those in which the adjustments are made in the frequency domain13. Both of these 

approaches can modify existing time histories to achieve a close match to the design response 

spectrum while maintaining, fairly well, the basic time domain character of the recorded or 

simulated time histories. To minimize changes to the time domain characteristics, it is desirable 

that the overall shape of the spectrum of the recorded or simulated time history be similar to the 

shape of the design response spectrum and that the time history initially be scaled so that its 

spectrum is at the approximate level of the design spectrum before spectral matching. 

If hazard deaggregation (section 2.8.1) indicates that different seismic sources dominate 

contributions to different period ranges of a design response spectrum (such as nearby small-tomoderate magnitude earthquakes dominating the short-period range and distant large-magnitude 

earthquakes dominating the long-period range), then it may be appropriate to develop two or 

more sets of time histories representing the different earthquakes and period ranges. In this way, 

individual time histories will be more realistic with respect to spectrum shape, duration, and 

energy. 

2.8.3. REQUIREMENTS FOR COMPATIBILITY OF TIME HISTORIES WITH THE 

DESIGN RESPONSE SPECTRUM 

When using recorded or simulated time histories that are not spectrum matched, they should be 

scaled to the approximate level of the design response spectrum in the period range of 

significance. For each component of motion, an aggregate match of the design response 

spectrum should be achieved for the set of acceleration time histories used for each component 

of motion. To evaluate the match, the mean of the spectra of the time histories should be 

calculated period-by-period. Over the range of periods of structural significance, the calculated 

mean spectrum should not be more than 15 percent lower than the design spectrum at any period, 

and the average of the ratios of the mean spectrum to the design spectrum should be equal to or 

greater than 1.0. 

When developing spectrum-matched time histories, before the matching process, they should be 

scaled to the approximate level of the design response spectrum in the period range of 

significance. Thereafter, the set of time histories for each component should be spectrummatched to achieve the same overall fit requirement as stated above for time histories that are not 

 

12 Lilhanand and Tseng, 1988; Abrahamson, 1992 

13 Gasparini and Vanmarcke, 1976; Silva and Lee, 1987; Bolt and Gregor, 1993 
97

spectrum-matched. Ordinarily this is easy to achieve because each spectrum-matched time 

history will have a spectrum close to the design spectrum. 

When developing three-component sets of time histories by simple scaling rather than spectrum 

matching, it is difficult to achieve a comparable match to the design spectrum for each 

component when using the same scaling factor. It is desirable, however, to use the same scaling 

factor to preserve the relationship between the components. Approaches for dealing with this 

scaling issue include: 

1. Increase the scaling factor to meet the minimum aggregate match requirement for one 

component of motion while exceeding it for the other two. 

2. Use a scaling factor to meet the aggregate match for the most critical component of motion 

with the match somewhat deficient for other components. 

3. Compromise on the scaling by using different factors as required for different components of 

a time history set. 

All these approaches are acceptable, but the third approach is less desirable. The second 

approach requires careful examination and interpretation of the results and possibly dual analyses 

for application of the higher horizontal component in each principal horizontal direction. 

Scaled or spectrum-matched time histories should be integrated to obtain corresponding time 

histories of velocity and displacement. All results should be examined for reasonableness. 

2.8.4. NUMBER OF TIME HISTORIES 

At least three time histories of either recorded, simulated-recorded, or spectrum-matched 

motions, should be used for each component of motion when performing nonlinear inelastic 

dynamic analyses. The design forces and displacements should be taken as the maximum 

response calculated for the three ground motions in each principal direction. However, if a 

minimum of seven recorded, simulated, or spectrum-matched time histories are used for each 

component of motion, the design forces and displacements may be taken as the mean of the 

forces and displacements calculated for each principal direction. These recommendations take 

into account the sensitivity of nonlinear response to both the response spectral content of the 

time histories and the time domain character of the time series (duration, pulse shape, pulse 

sequencing). 

If linear elastic time history dynamic analysis is performed, fewer time histories could be used 

depending on the sensitivity of response to the time histories. Normally, for linear analysis, the 

response is not sensitive to the time domain character of the time series, and a sufficient number 

of time histories would be the number required to achieve an aggregate fit to the design spectrum. 

Using a spectrum-matching approach, this may be a single time history. 

Additional guidance on developing acceleration time histories for dynamic analysis may be 

found in publications by the Caltrans Seismic Advisory Board Adhoc Committee on Soil-
98

Foundation-Structure Interaction (CSABAC, 1999) and the U.S. Army Corps of Engineers 

(2003). 

2.9. SPATIAL VARIATION OF GROUND MOTIONS

Spatial variation in earthquake ground motion is principally due to: 

1. Differences in arrival time of ground motions at bridge supports (out-of-phase motions). This 

is the so-called ‘traveling wave’ or ‘wave passage’ effect. Waves that propagate vertically do 

not show this effect.. 

2. Incoherence or dissimilarities in ground motions at bridge supports due to wave-scattering 

effects as seismic waves propagate through the earth’s crust. 

3. Incoherence of ground motions due to variations in soil profiles at bridge supports. 

Limited studies conducted to date indicate that spatial variation in the ground motion generally 

has little effect on bridge response and can be neglected in design. This is generally true for 

bridges less than 300 to 450 meters (1,000 to 1,500 feet) in total length and that have uniform 

soil conditions (Shinozuka et al., 2000). On the other hand, spatial variation can be significant 

for longer bridges or those with differential soil conditions and its effect should be included in 

design14. The effect of these differential soil conditions at bridge supports can be evaluated 

approximately using one-dimensional soil column analyses. Several computer programs that can 

be used for this purpose are identified in appendix A. These methods do not account for out-ofphase motions or incoherence due to crustal wave scattering. Shinozuka et al. (2000) presents a 

methodology for accounting for differential soil response in the development of out-of-phase, 

incoherent time histories. 

Many bridge sites, especially those at river or valley crossings, are characterized by significant 

variations in both surface topography and subsurface soil stratigraphy (Martin, 1998a). In some 

cases, it may be warranted to use a two-dimensional, rather than one-dimensional, site response 

analysis to capture effects of the spatially varying soil and topographic conditions. Computer 

programs that can be considered for such analyses include the nonlinear analysis programs 

FLAC15 and TENSI-MUSC (Martin, 1998a) and the equivalent linear analysis programs FLUSH 

(Lysmer et al., 1975) and QUAD4M (Hudson et al., 1994). These methods capture twodimensional interaction of the topographic and soil variations, but do not account for out-ofphase motions or incoherence due to crustal wave scattering. 

 

14 Hao et al. (1989), Abrahamson (1985; 1992), Abrahamson et al. (1991), Shinozuka et al. (2000), and CSABAC 

(1999) 

15 ITASCA, 1998; Wang and Makdisi, 1999 
99

CHAPTER 3: GEOTECHNICAL HAZARDS 

3.1. GENERAL 

Geotechnical hazards at highway bridge sites that can be triggered by earthquakes include soil 

liquefaction, soil settlement, slope failure (landslides and rock falls), surface fault rupture, and 

flooding. In this chapter, liquefaction, settlement, fault rupture and flooding are described and 

their adverse consequences to highway bridges are discussed. Procedures for evaluating these 

hazards are also presented. Methods for screening and evaluating slope failure potential 

(landslides and rock falls) are given in Part 2 of this manual and are therefore not included in this 

chapter. 

The consequences of these hazards in the free-field (e.g., ground settlement and fault 

displacements) are also addressed in this chapter. Methods for evaluating the effects of these 

hazards on the capacity and deformations of the bridge-foundation system are presented in 

chapter 6, and measures for mitigating them are described in chapter 11. 

Assessing geotechnical hazards is a two-part procedure. In the first part, a quick screening 

evaluation is conducted. Generally, this can be accomplished using available information and 

field reconnaissance. If the criteria are satisfied, the risk is considered to be low and further 

evaluations of the hazard are not required. If a hazard cannot be screened out, more detailed 

evaluations are conducted in the second part of this procedure. This usually requires obtaining 

additional data to more rigorously assess the hazard and its consequences. In this chapter, 

guidelines for initial hazard screening are provided, and a brief overview of methods for detailed 

hazard evaluation is given. Detailed methods for hazard evaluation are described in appendix B. 

Hazard screening and hazard evaluation should be carried out by engineers and scientists (e.g., 

geotechnical engineers, geologists, and seismologists), who have expert knowledge of these 

hazards and experience with their evaluation. 

3.2. SOIL LIQUEFACTION 

3.2.1. LIQUEFACTION HAZARD DESCRIPTION 

Soil liquefaction is a phenomenon in which a cohesionless soil deposit below the groundwater 

table loses a substantial amount of strength due to strong earthquake ground shaking. The reason 

for strength loss is that some types of cohesionless soil tend to compact during earthquake 

shaking and this tendency for compaction causes pore water pressures in the soil to increase. 

This, in turn, causes a reduction in soil strength. Recently deposited (i.e., geologically young) 

and relatively loose natural soils, and uncompacted or poorly compacted fills, are susceptible to 

liquefaction. Loose sands and silty sands are particularly susceptible. Loose silts and gravels 

also have potential for liquefaction. Dense natural soils and well-compacted fills have low 

susceptibility to liquefaction. Clay soils are generally not susceptible, except for highly sensitive 

clays found in some geographic locales. 
100

Liquefaction has been perhaps the single most significant cause of damage to bridges during past 

earthquakes. Most of the damage has been related to the lateral movement of soil at bridge 

abutments. However, cases involving the loss of lateral and vertical bearing support of 

foundations for bridge piers have also occurred. 

The potential consequences of liquefaction can be grouped into the following categories: 

1. Flow slides. Flow failures are the most catastrophic form of ground failure that can occur due 

to liquefaction. These large slides occur when the downslope static (gravity) loads exceed 

the resistance provided by low shear strengths of liquefied soils (i.e., static factor of safety 

drops below 1.0 due to liquefaction). Because earthquake-induced inertial forces following 

liquefaction are not required to cause flow sliding (gravity forces being sufficient), these 

slides can occur even after the ground stops shaking. Flow slides commonly result in tens of 

meters (yards) of displacement and occur at relatively high velocities, up to tens of 

kilometers per hour (miles per hour). 

2. Lateral spreads. Lateral spread is the most common form of landslide-type movements 

accompanying liquefaction. It can occur on very gently sloping ground (less than one 

percent slope in some cases) underlain by liquefied soil due to combined static forces plus 

seismically induced inertia forces in the soil mass. Lateral movements accumulate during the 

earthquake whenever the static plus seismically induced shear stresses exceed the strength of 

liquefied soil. The resulting lateral movements can range in magnitude from centimeters 

(inches) to several meters (yards) and are typically accompanied by severe ground cracking 

with horizontal and vertical offsets. The potential for lateral movements is increased if there 

is a ‘free face,’ such as a river channel or the sloping shoreline of a lake or bay, toward 

which movements can occur. Lateral spreading or ‘embankment penetration’ can occur 

beneath a bridge approach fill or any highway embankment if the underlying soil liquefies. 

Manifestations are settlements, lateral movements, and cracking of the embankment. 

3. Reduction in foundation bearing capacity. The occurrence of liquefaction beneath, and/or 

laterally adjacent to, bridge foundations can greatly reduce foundation vertical and/or lateral 

capacity, resulting in unacceptable foundation settlements and/or lateral movements. 

4. Ground settlement. Even in the absence of flow sliding, lateral spreading, or reduction in 

foundation bearing capacity due to liquefaction, ground settlements due to soil consolidation 

can occur as liquefaction-induced, excess pore water pressures in the soil dissipate. This 

consolidation occurs over time, perhaps for several days after the earthquake, and may result 

in the settlement and/or differential settlement of foundations located above the liquefied 

layer. Pile foundations extending through liquefied strata into stable ground may be subject 

to downdrag as soils overlying liquefied layers settle relative to the piles. In general, 

magnitudes of total and differential ground movements due to liquefaction-induced soil 

consolidation are less than those associated with flow slides, lateral spreading, or reduction 

in foundation bearing capacity. 
101

5. Increased pressures on retaining walls. The occurrence of liquefaction in the backfill behind 

a retaining wall, such as an abutment backwall or wingwall, will increase pressures on the 

wall, potentially leading to wall failure or excessive deformations. 

3.2.2. INITIAL HAZARD SCREENING FOR LIQUEFACTION 

Prior to initiating screening and liquefaction evaluation procedures for soils that are susceptible 

to liquefaction, a check should be made as to whether liquefaction has previously occurred at the 

site (or in the near vicinity of the site in similar geotechnical conditions) during past earthquakes. 

This check may involve review of the earthquake history of an area and review of published 

post-earthquake reconnaissance reports. If there is evidence that liquefaction has previously 

occurred at the site, then it must be given further consideration. 

In general, the requirement to conduct an evaluation of liquefaction potential is a function of the 

Seismic Hazard Level for the bridge, as described in chapter 1. For Seismic Hazard Levels I and 

II, the potential for liquefaction is generally low. In some cases (e.g., Type E and F soils for 

Seismic Hazard Level II), the peak ground acceleration may exceed 0.14 g (FaSs in excess of 

0.35 g). While this level of peak ground acceleration may cause liquefaction, the magnitude of 

the earthquake causing liquefaction at these hazard levels will generally be less than 6.0 and the 

duration of strong shaking will be relatively short. For magnitudes less than 6.0, liquefaction 

develops slowly at most sites, and affects the structure minimally during dynamic shaking. 

Therefore, the effects of liquefaction on dynamic response can be neglected. In addition, little 

potential exists for permanent movement of the ground, again because of the small size and 

limited duration of seismic events at these levels. 

The potential for liquefaction at Seismic Hazard Levels III and IV is higher, and careful attention 

is needed to determine the potential for, and consequences of, liquefaction at sites with this 

classification. If the mean magnitude contributing to the peak ground acceleration is less than 

6.01, then the discussion above with regard to duration is applicable for Seismic Hazard Levels 

III and IV. For magnitudes between 6.0 and 6.4, a liquefaction analysis may or may not be 

required depending on the combination of ground shaking and the blowcount values for the soils. 

For magnitudes above 6.4, a liquefaction analysis is required. 

In summary, an evaluation of the potential for, and consequences of, liquefaction in soils near the 

surface should be made in accordance with the following requirements: 

• Seismic Hazard Levels I and II: Not required. 

• Seismic Hazard Levels III and IV: Required, unless one of the following two conditions is 

met: 

 - Mean magnitude for the design event is less than 6.0. 

 

1 Mean earthquake magnitudes for a given bridge site and return period may be found on the USGS web site 

(http://eqhazmaps.usgs.gov), where results for the deaggregation of the seismic hazard are given (see section 2.8.1, 

table 2-6 and figure 2-8). 
102

 - Mean magnitude for the design event is less than 6.4 and equal to or greater than 

 6.0, and either the normalized Standard Penetration Test (SPT) blowcount 

 [(N1)60] is greater than 20, or (N1)60 is greater than 15 and FaSs is less than 0.35 g. 

 

If the mean magnitude is greater than or equal to 6.4, or if the above requirements are not met for 

magnitudes between 6.0 and 6.4, the potential for liquefaction and associated phenomena such as 

lateral flow, lateral spreading, foundation bearing capacity reduction, and settlement, should be 

evaluated. 

In addition to the above criteria, it can be assumed that a significant liquefaction hazard does not 

exist for Seismic Hazard Levels III and IV if any of the following screening criteria are satisfied: 

• The geologic materials underlying the site are either bedrock or have very low liquefaction 

susceptibility according to the relative susceptibility ratings shown in table 3-1, which are 

based on geologic age and general depositional environment (Youd and Perkins, 1978). 

Table 3-1 should be applied conservatively if there are uncertainties regarding geologic age 

or depositional environment. 

• The soils below the groundwater table at the site are one of the following: 

1. Clayey soils which have a clay content (grain size < 0.005 mm) greater than 15 percent, 

liquid limit greater than 35 percent, or natural water content less than 90 percent of the 

liquid limit (Seed and Idriss, 1982). However, clayey soils that are highly sensitive, 

based on measured soil properties or local experience, should not be screened out. A 

highly sensitive soil possesses all of the following properties: 

a. Liquid limit less than 40 percent. 

b. Water content greater than 0.9 times the liquid limit. 

c. Liquidity index greater than 0.6. 

d. (N1) 60 less than five or normalized cone penetration resistance, qc1, less than 1 MPa 

(20 ksf). 

 Areas of the United States known to have highly sensitive soils include some coastal 

areas of Alaska, along the St. Lawrence River, some eastern and western coastal areas 

with estuarine soil deposits, near or within saline lakes in the Great Basin and other arid 

areas, and soils resulting from weathering of volcanic ash (Youd, 1998). 

2. Sands with a minimum corrected Standard Penetration Test resistance, (N1)60 value of 30 

blows/0.3m (30 blows/foot) or minimum corrected cone penetration test tip resistance, 

qC1N, of 160 with a sufficient number of tests. The parameter (N1)60 is defined in 

Appendix B. 

• The water table is deeper than 15 m (50 feet) below the existing ground surface or proposed 

finished grade at the site, whichever is lower, including considerations for seasonal, historic 

and possible future rises in groundwater level. 
103

Table 3-1. Estimated susceptibility of sedimentary deposits to liquefaction 

during strong ground motion. 

Likelihood that Cohesionless Sediments, When 

Saturated, Will be Susceptible to Liquefaction 

(by Age of Deposit) Type of Deposit 

General 

Distribution of 

Cohesionless 

Sediments in 

Deposits <500 yr 

Modern 

Holocene 

>11 ka 

Pleistocene 

11 ka - 2 Ma 

PrePleistocene 

>2 Ma 

(a) Continental Deposits 

River channel 

Floodplain 

Alluvial fan and plain 

Marine terraces and 

plains 

Delta and fan-delta 

Lacustrine and playa 

Colluvium 

Talus 

Dunes 

Loess 

Glacial till 

Tuff 

Tephra 

Residual soils 

Sebka 

Locally variable 

Locally variable 

Widespread 

Widespread 

Widespread 

Variable 

Variable 

Widespread 

Widespread 

Variable 

Variable 

Rare 

Widespread 

Rare 

Locally variable 

Very high 

High 

Moderate 

 --- 

High 

High 

High 

Low 

High 

High 

Low 

Low 

High 

Low 

High 

High 

Moderate 

Low 

Low 

Moderate 

Moderate 

Moderate 

Low 

Moderate 

High 

Low 

Low 

High 

Low 

Moderate 

Low 

Low 

Low 

Very low 

Low 

Low 

Low 

Very low 

Low 

High 

Very low 

Very low 

? 

Very low 

Low 

Very low 

Very low 

Very low 

Very low 

Very low 

Very low 

Very low 

Very low 

Very low 

Unknown 

Very low 

Very low 

? 

Very low 

Very low 

(b) Coastal Zone 

Delta 

Estuarine 

Beach 

 High wave-energy 

 Low wave-energy 

Lagoonal 

Fore shore 

Widespread 

Locally variable 

Widespread 

Widespread 

Locally variable 

Locally variable 

Very high 

High 

Moderate 

High 

High 

High 

High 

Moderate 

Low 

Moderate 

Moderate 

Moderate 

Low 

Low 

Very low 

Low 

Low 

Low 

Very low 

Very low 

Very low 

Very low 

Very low 

Very low 

(c) Artificial 

Uncompacted fill 

Compacted fill 

Variable 

Variable 

Very high 

Low 

--- 

--- 

--- 

--- 

--- 

--- 

Youd and Perkins, 1978
104

If the above screening criteria are not satisfied, then the detailed evaluations described in 

appendix B should be used to evaluate a liquefaction hazard and its potential consequences. 

These evaluation approaches are briefly outlined in section 3.2.4. 

3.2.3. EARTHQUAKE GROUND MOTIONS FOR LIQUEFACTION ANALYSIS 

To perform analyses of the liquefaction potential, and some types of analyses to assess the 

consequences of liquefaction, an estimate of the peak ground acceleration and associated 

earthquake magnitude are needed. In addition, one type of evaluation for lateral spreading also 

requires earthquake source-to-site distance. These parameters can be obtained either by using the 

‘general’ procedure based on national earthquake ground motion maps and site factors, or a ‘sitespecific’ procedure, as described in sections 2.5 and 2.6, respectively. 

Design ground motions are based on a probabilistic approach, except near highly active faults 

where ground motions are ‘bounded’ deterministically so as not to exceed levels associated with 

the occurrence of maximum magnitude earthquakes on these faults. When the design ground 

motions for a bridge are based on a probabilistic approach (i.e., developed for a certain 

probability of ground motion exceedance or return period), then it is necessary to deaggregate 

the peak ground acceleration hazard to determine ‘dominant’ magnitudes and distances 

contributing to the hazard, as described in section 2.8.1 and appendix A. The process of 

deaggregation is required because different seismic sources, earthquake magnitudes, and 

distances contribute to the probability of exceedance of a given ground motion in a probabilistic 

analysis. 

When the deterministic approach is applied, design ground motions are estimated for a maximum 

earthquake magnitude occurring at a specific seismic source. It is assumed that the earthquake 

occurs on that portion of the source closest to the site. One or more earthquake sources, and 

associated magnitudes and distances, are selected on the basis of likelihood to cause liquefaction 

at the site. 

3.2.4. SUMMARY OF PROCEDURES FOR DETAILED LIQUEFACTION EVALUATIONS 

3.2.4.1. Evaluation of the Potential for Liquefaction Using Simplified Method

The procedure most widely used in engineering practice for the assessment of liquefaction 

potential is the ‘Simplified Procedure’2

. This procedure essentially compares the cyclic 

resistance ratio (CRR, the cyclic stress ratio required to induce liquefaction in a soil stratum at a 

given depth) with the earthquake-induced cyclic stress ratio (CSR) at that depth from the 

specified design earthquake, as defined by a peak ground surface acceleration and the associated 

earthquake moment magnitude. 

The ratio of CRR to CSR should be greater than 1.0 to preclude the development of liquefaction. 

Even when the ratio of CRR to CSR is as high as 1.5, increases in pore water pressure in the soil 

 

2

 Seed and Idriss (1982), Seed et al. (1983) and (1985), Seed and De Alba (1986), Seed and Harder (1990), Youd 

and Idriss (1997), Youd et al. (2001) 
105

can occur. The potential consequences of these pore pressure increases in weakening the soil 

should be considered during design. 

The most common application of the Simplified Procedure uses an empirically-based 

liquefaction criterion chart, or correlation, that describes the CRR as a function of SPT 

blowcounts from soil borings. However, variations of this procedure use other parameters as a 

measure of soil resistance to liquefaction, namely Cone Penetration Test (CPT) resistance, shear 

wave velocity, and Becker hammer penetration resistance. The use of CPT data and related 

correlations is becoming widely used as either a complementary or alternative method to the use 

of SPT data and correlations. The techniques of, and correlations with, shear wave velocity and 

Becker hammer penetration resistance are typically used for gravelly soils because of the 

difficulty of obtaining meaningful SPT or CPT measurements in these soils (Youd and Idriss, 

1997; Youd et al., 2001). 

3.2.4.2. Numerical Modeling Methods and Laboratory Cyclic Testing for Evaluation of 

Potential for Liquefaction 

For critical projects, the use of numerical modeling of site response and/or the conduct of 

laboratory cyclic liquefaction tests may be appropriate to supplement the simplified method. The 

use of one- or two-dimensional effective stress site response analyses for acceleration time 

history inputs will provide increased understanding of the progressive development of pore water 

pressures during seismic shaking. Because of the great difficulty of obtaining and testing 

undisturbed cohesionless soil samples, a laboratory cyclic testing program to determine soil 

liquefaction resistance is seldom justified. However, such a program may be considered for 

certain soil types that are liquefiable but have liquefaction resistances that are less well 

quantified by the empirically-based correlations (e.g., cohesionless silts). 

3.2.4.3. Evaluation of the Potential for Liquefaction-Induced Flow Slides and Lateral 

Spreads 

For evaluation of the potential for flow sliding and magnitude of lateral spreading displacements, 

a critical parameter for analytical methods is the residual strength of the liquefied soil. The 

quantification of this parameter is described in appendix B. 

The analysis of flow slide potential is essentially a static slope stability analysis in which the 

static factor of safety is evaluated considering the residual strength of the portion of the soil mass 

that has liquefied. Flow sliding is predicted to occur if the static factor of safety of the liquefied 

soil mass is less than 1.0. The estimation of displacements associated with flow sliding cannot 

be easily made. Sliding continues until geometric changes or increases in the strength of the 

liquefied soil cause the factor of safety to increase above 1.0. 

For analysis of the potential for, and amount of, lateral spreading, empirically-based and 

analytical approaches may be used. Empirical approaches are based on observations of lateral 

spreading displacements during past earthquakes. The most widely used analytical approach is 

the simplified Newmark sliding block method in which displacements are estimated based on the 

amplitude of the induced accelerations in the soil mass, the amplitude of the ‘yield acceleration’ 
106

required to initiate movement of the soil mass considering the residual strength of the liquefied 

portion, and other parameters. The yield acceleration coefficient, ky, is determined from a 

pseudo-static stability analysis as the acceleration required to cause the factor of safety to drop to 

1.0. Two-dimensional nonlinear effective stress dynamic analysis approaches are now available 

as a complementary tool for evaluating lateral spreading. 

3.2.4.4. Other Evaluation Procedures 

The evaluation of the effects of liquefaction on vertical or lateral bearing capacity and foundation 

behavior and the effects on earth pressures behind retaining walls is addressed in chapter 6. The 

estimation of consolidation settlement due to liquefaction is considered in the following section. 

3.3. SOIL SETTLEMENT 

3.3.1. SETTLEMENT HAZARD DESCRIPTION 

As noted in section 3.2.1, free-field settlements may accompany liquefaction. These settlements 

are associated with post-earthquake dissipation of pore water pressures induced by liquefaction 

and resulting soil consolidation and densification. If pore pressures develop during ground 

shaking, but not to a level high enough to cause liquefaction, their post-earthquake dissipation 

may also result in settlement, but less than that occurring due to liquefaction. 

Settlements can also occur in dry or partially-saturated soils above the groundwater table and 

thus not be associated with pore pressure development and liquefaction. The mechanism in this 

case is densification during the period of ground shaking as the earthquake-induced cyclic shear 

stresses cause the soil grains to adjust into a denser state of packing. In the case of bridge 

approach embankment fills, a component of settlement may also be associated with small 

amounts of lateral yielding of the embankments, even in the absence of noticeable slope 

displacements or lateral spreading through weak foundation soils. 

Loose natural soils, uncompacted and poorly compacted fills, are susceptible to densification. If 

densification does not occur uniformly over an area, the resulting differential settlements can be 

damaging to a bridge. 

3.3.2. INITIAL HAZARD-SCREENING FOR SETTLEMENT 

It can be assumed that a significant hazard due to differential compaction does not exist if both

of the following screening criteria are satisfied. 

1. The geologic materials underlying the foundations and below the groundwater table do not 

liquefy. 

2. The geologic materials underlying the foundations and above the groundwater table are one 

or more of the following: 

• Pleistocene (geologic age older than 11,000 years), 
107

• Stiff clays or clayey silts, or 

• Cohesionless sands, silts, and gravels with a minimum (N1)60 of 20 blows/0.3 m 

(20 blows/ft). 

Note that these criteria are not intended to be applied to embankment fills. The potential for 

settlement of fills is strongly dependent on the degree of compaction of the fill and settlements 

should be estimated as summarized in the next section. 

3.3.3. SUMMARY OF PROCEDURES FOR DETAILED SETTLEMENT EVALUATIONS 

Appendix B describes simplified procedures and empirical correlations for estimating 

seismically induced settlements in both saturated sands (below the groundwater table) and dry 

sands (applied to dry or partially saturated sands above the groundwater table). These procedures 

are based in part on case history data and in part on results of laboratory cyclic testing programs. 

Methods are also described for extending the evaluations for clean sands to silty sands and 

cohesionless silts. These procedures do not address cohesive soils (clays and clayey silts). In 

general, it is expected that stiff natural deposits of cohesive soils will not experience damaging 

settlements. However, cohesive or partially cohesive fills, including bridge approach 

embankment fills, may be susceptible to significant settlements if not adequately compacted. 

3.4. SURFACE FAULT RUPTURE 

3.4.1. SURFACE FAULT RUPTURE HAZARD DESCRIPTION 

Surface fault rupture refers to the shearing displacements that occur along an active fault trace 

when movement on the fault extends to the ground surface, or the depth of the bridge foundation. 

Displacements can range from centimeters (inches) to several meters (yards). Because surface 

fault displacements tend to occur abruptly, often across a narrow zone, fault rupture can be very 

damaging to a bridge, particularly if it occurs directly below the structure. It is also difficult and 

often cost-prohibitive to mitigate. In the central and eastern United States, east of the Rocky 

Mountains, few active faults have been identified and the hazard of surface fault rupture is 

generally low in comparison to the western United States. 

3.4.2. INITIAL HAZARD SCREENING FOR SURFACE FAULT RUPTURE 

Surface fault ruptures generally are expected to occur along existing traces of active faults. 

Therefore, it can be assumed that a significant hazard of surface fault rupture does not exist if it 

can be established that either: 

• There is no evidence of a fault trace traversing the bridge site, or 

• If a fault trace does cross the bridge site, it has been established that the fault is not an active 

fault. Faults are generally considered to be active faults with a significant potential for 

future earthquakes and displacements if they have experienced displacements during the past 

approximately 11,000 years (Holocene time). 
108

Hazard screening should involve, as a minimum, the following steps: 

1. Review of geologic maps as well as discussions with geologists in government agencies who 

are knowledgeable about the geology of the area, and 

2. Site reconnaissance and review of aerial photographs, looking for geomorphic expression of 

faulting. If there is uncertainty in the fault location or its activity, the screening criteria 

should be applied conservatively. 

3.4.3. SUMMARY OF PROCEDURES FOR DETAILED SURFACE FAULT RUPTURE 

EVALUATIONS 

If a surface fault rupture hazard cannot be screened out, the owner may decide to accept the risk 

or evaluate the hazard and consequences in detail. If detailed evaluations are carried out, they 

should be oriented toward: 

1. Establishing the fault or fault zone location relative to the bridge if it is not clearly 

established in the screening stage, 

2. Establishing the activity of the fault if it traverses the bridge, and 

3. Evaluating fault rupture characteristics; i.e., amount of fault displacement, width of zone of 

displacement, and distribution of slip across the zone for horizontal and vertical components 

of displacement. A probabilistic assessment of the likelihood of different magnitudes of fault 

displacement during the life of the bridge may also be useful in decision-making. 

3.5. FLOODING 

3.5.1. FLOODING HAZARD DESCRIPTION 

Phenomena that can result in earthquake-induced flooding include: 

• Tsunami (large ocean waves generated by vertical displacements of the seafloor during 

offshore earthquakes, 

• Seiche (waves induced in bays and lakes when the body of water is excited by ground 

shaking; e.g., sloshing), 

• Waves generated by landslides within, or into, a body of water, 

• Flooding caused by tectonic movements, such as uplift along a thrust fault damming a river, 

and 

• Failure of an upstream dam and reservoir caused by an earthquake. 
109

3.5.2. INITIAL HAZARD SCREENING FOR FLOODING 

 

The hazard of flooding can be screened out for bridges that are not located near a body of water 

or in an area that could be inundated by the hazard. Tsunami potential is considered to be low in 

coastal regions of the eastern and southern United States and moderate to high along portions of 

the west coast of the United States, Alaska, and Hawaii. Seiche waves usually do not exceed 

several feet in height and would generally pose a potential hazard only to facilities within or 

immediately adjacent to a bay or lake. Many states and federal agencies have delineated 

inundation zones for failures of dams under their jurisdiction. The hazard of earthquake-induced 

flooding should also be viewed in the context of hazards of flooding from non-seismic causes. 

3.5.3. SUMMARY OF DETAILED EVALUATIONS FOR FLOODING 

If it appears that flooding could pose an unacceptable risk to the performance of a bridge 

immediately following an earthquake, detailed evaluations should be directed at assessing the 

potential, severity, consequences, and likelihood of the hazard. 
 
111

CHAPTER 4: SEISMIC RATING METHODS FOR 

SCREENING AND PRIORITIZATION 

4.1. GENERAL 

Screening an inventory of bridges is done to identify seismically deficient bridges and prioritize 

them in order of need for retrofitting. Such a process is intended to be rapid, easy to apply and 

conservative. Bridges found to be deficient at this stage are subject to detailed evaluation at the 

next step, and any that are later found to be satisfactory are excluded from further study at that 

time. 

As noted in section 1.10, two rating systems are identified in this manual for the purpose of 

screening and prioritizing bridges for seismic retrofitting. Both methods are described in detail in 

this chapter. The methods are: 

• Seismic rating method using indices. 

• Seismic rating method using expected damage. 

4.2. SEISMIC RATING METHOD USING INDICES

In calculating the seismic rating of a bridge for prioritization, consideration should be given to 

structural vulnerability, seismic and geotechnical hazards, and the socioeconomic factors 

affecting importance. This is accomplished first, by making independent ratings of each bridge in 

the areas of vulnerability and seismic hazard, and second, by considering importance (societal 

and economic issues) and other factors (redundancy and non-seismic structural issues) to obtain 

a final, ordered determination of bridge retrofit priorities1

. 

The rating system is therefore both quantitative and qualitative. The quantitative part produces a 

seismic rating (called the bridge rank, R) based on structural vulnerability and seismic hazard. 

The qualitative part modifies the rank in a subjective way that takes into account bridge 

importance, network redundancy, non-seismic deficiencies, anticipated service life, and similar 

factors for inclusion in an overall priority index. Engineering and societal judgment are thus the 

key to the second stage of the screening process. This leads to a priority index that is a function 

of rank, importance, and other factors as indicated below in equation 4-1. 

P = f (R, importance, non-seismic, and other factors...) (4-1) 

where P is the priority index and R is the rank based on structural vulnerability and seismicity. 

In summary, bridge rank is based on structural vulnerability and seismic hazard, whereas retrofit 

priority is based on bridge rank, importance, non-seismic deficiencies, and other factors such as 

network redundancy. This method is illustrated in figure 4-1. 

 

1

 Buckle, 1991; Buckle and Friedland, 1995 
112

Performance Level, PL Seismic Hazard Level, SHL

Seismic Retrofit Category, SRC

Seismic Retrofit Category A Seismic Retrofit Categories

B, C, D

Compile structural vulnerability

data

Calculate Bridge Vulnerability, V

RETROFITTING NOT

REQUIRED

Calculate Seismic Hazard

Rating, E

Calculate Bridge Rank, R

R = V E

OTHER FACTORS, O

Bridge importance

Network redundancy

Anticipated service life

Non-seismic rehabilitation

needs

CALCULATE PRIORITY INDEX, P= f (R,O)

Figure 4-1. Seismic rating method using indices. 
113

4.2.1. CALCULATION OF BRIDGE RANK 

As noted above, the bridge rank, R, is based on a structural vulnerability rating, V, and a seismic 

hazard rating, E. Each rating lies in the range 0 to 10 and the rank is found by multiplying these 

two ratings together as shown in equation 4-2. 

 R = VE (4-2) 

Therefore, R can vary from 0 to 100, and the higher the score, the greater the need for the bridge 

to be retrofitted (ignoring at this time the other factors). Recommendations for assigning values 

for V and E are described below. 

4.2.1.1. Vulnerability Rating (V) 

Although the performance of a bridge is based on the interaction of all its components, certain 

bridge components are more vulnerable to damage than others. These are: 

• Connections, bearings, and seats (support lengths). 

• Piers, columns and foundations. 

• Abutments. 

• Soils. 

Of these, connections, bearings and inadequate seat length are the most common reasons for 

bridge failure and the least costly to fix. For this reason, the vulnerability of these components is 

calculated separately from the rest of the structure and assigned a rating, V1. The vulnerability of 

the rest of the substructure, V2, is determined from the sum of the ratings for each of the other 

components that are susceptible to failure. The overall rating for the bridge is then given by the 

greater of V1 and V2, as shown in the flowchart in figure 4-2. 

The determination of these vulnerability ratings requires considerable engineering judgment. In 

order to assist in this process, a methodology for determining these ratings is given in sections 

4.2.1.1(a) and 4.2.1.1(b). 

Vulnerability ratings may range from 0 to 10. A rating of 0 means a very low vulnerability to 

unacceptable damage, a value of 5 indicates a moderate vulnerability to collapse or a high 

vulnerability to loss of access, and a value of 10 means a high vulnerability to collapse. This 

should not be interpreted to mean that the vulnerability rating must assume one of these three 

values. 

For bridges classified in Seismic Retrofit Category (SRC) B (section 1.6 and table 1-6), the 

vulnerability ratings for bearings, transverse restraints, and support lengths need to be calculated 

along with a rating for liquefaction effects for bridges on liquefiable soils. Experience has shown 

that most connection, bearing, and seat deficiencies can be corrected economically. 

For bridges classified as SRC C or D, vulnerability ratings are also required for the columns, 

abutments, and foundations. Experience with retrofitting these components is much more limited 

than for bearings. They are generally more difficult to retrofit and doing so may not be as costeffective. 
114

Calculate Column

Vulnerability Rating, CVR

V = Maximum of V1, V2

Calculate Abutment

Vulnerability Rating, AVR

Calculate Liquefaction

Vulnerability Rating, LVR

V2 = CVR + AVR + LVR ≤ 10

Calculate Vulnerability Rating

for Connections,

Bearings and Seat Widths, V1

Figure 4-2. Calculation of bridge vulnerability, V. 

A comparison of the above two vulnerability ratings, V1 and V2, can be used to obtain an 

indication of the type of retrofitting needed. If the vulnerability rating for the bearings is equal to 

or less than the vulnerability rating of other components, simple retrofitting of only the bearings 

may be of little value. Conversely, if the bearing rating is greater, then benefits may be obtained 

by retrofitting only the bearings. A comparison of these two ratings during the preliminary 

screening process may be helpful in planning the type of comprehensive retrofit program needed, 

but should not serve as a substitute for the detailed evaluation of individual bridges as described 

in chapters 5, 6 and 7. 

4.2.1.1(a). Vulnerability Rating for Connections, Bearings, and Seat Widths, V1

Bearings are used to transfer loads from the superstructure to the substructure and between 

superstructure segments at hinge seats within the span. In this ranking system, bearings are 

considered to include shear keys, keeper bars and similar restraints. Bearings may be either fixed 

bearings, which do not provide for translational movement, or expansion bearings, which do 
115

permit such movements, as shown in figure 4-3. An expansion bearing may provide for 

translation in one orthogonal direction, but not in the other. 

There are five basic types of expansion bearings used in bridge construction. These are: 

1. The rocker bearing, which is generally constructed of steel and permits translation and 

rotational movement. It is the most seismically vulnerable of all bridge bearings because it 

usually has a high aspect ratio, is difficult to restrain, and can become unstable and overturn 

after limited movement. Fixed bearings with high aspect ratios can also be very vulnerable to 

earthquakes. 

2. The roller bearing, which is also usually constructed of steel. It is stable during an 

earthquake, except that it can become misaligned and horizontally displaced. 

3. The elastomeric bearing pad is constructed of a natural or synthetic elastomer and is usually 

internally reinforced with steel shims. It relies on the distortion of the elastomer to provide 

for movement. This bearing is generally stable during an earthquake, although it has been 

known to ‘walk out’ under severe shaking due to inadequate fastening. 

4. The sliding bearing, in which one surface slides over another and which may consist of 

almost any material from an asbestos sheet between two concrete surfaces to PTFE (Teflon) 

and stainless-steel plates. 

5. High-load, multi-rotational bearings such as pot, disk, and spherical bearings. These bearings 

usually have adequate strength for earthquake loads, but their connections have failed in past 

earthquakes. 

Figure 4-3. Seismically vulnerable bearings. 
116

Transverse restraint of the superstructure is almost always provided at the bearings. Common 

types of restraint are shear keys, keeper bars, or anchor bolts. Restraints are usually not ductile, 

and are subjected to large seismically induced forces resulting from a redistribution of force from 

ductile components such as columns. In addition, when several individual bearings with keeper 

bars are present at a support, the bars do not resist transverse load equally because of slight 

variations in clearances. Therefore, individual bars may be subjected to very high forces which 

can lead to failure. Large transverse or longitudinal movements may then occur at the bearings, 

leading to loss of support and span collapse. The expected movement at a bearing is dependent 

on many factors and cannot be easily calculated. The AASHTO Specifications require a 

minimum support length at all bearings in new bridges.2 This requirement is also in the NCHRP 

12-49 provisions (ATC/MCEER, 2003). Because it is difficult to predict relative movement, the 

minimum support length, N, as given in the NCHRP 12-49 provisions, is recommended as the 

basis for checking the adequacy of longitudinal support lengths. 

In metric units, this length is given by: 

 

( ) 2

v 1 B 1 1.25F S

N 100 1.7L 7.0H 50 1 2

L cos

⎡ ⎤ ⎛ ⎞ +

=++ + + ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ α ⎣ ⎦

 (4-3a) 

where: N is the recommended support length measured normal to the face of an abutment or 

pier (m), 

L is the length of the bridge deck from the seat to the adjacent expansion joint, or to the 

end of the bridge deck (m). For hinges or expansion joints within a span, L is the sum 

of the distances on either side of the hinge (m); for single-span bridges, L is the 

length of the bridge deck (m). 

H is the height. For abutment seats, H is the average height of piers or columns 

supporting the bridge deck between the abutment and the next expansion joint (m). 

For pier seats, H is the height of the pier (m); for hinge seats within a span, H is the 

average height of the two adjacent piers (m); for single-span bridges, H=0. 

 B is the width of deck (m), 

α is the angle of skew (0° for a right bridge), and 

SD1 = FvS1, is the product of the long-period soil factor (Fv) and the 1.0 second spectral 

acceleration (S1) as in equation 1-1. 

The ratio B/L need not be taken greater than 3/8. 

In U.S. customary units, this length is given by: 

( ) 2

v 1 B 1 1.25F S

N 4.0 0.02L 0.08H 1.1 H 1 2

L cos

⎡ ⎤ ⎛ ⎞ +

=+ + + + ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ α ⎣ ⎦

 (4-3b) 

 

2

 AASHTO, 1998; AASHTO, 2002 
117

where N, L, H, B and α are as defined for equation 4-3a, but N is given in inches and L, H and B 

are expressed in feet. Again the ratio B/L need not be taken greater than 3/8. 

Since skew has a major effect on the performance of bridge bearings, rocker bearings may 

overturn at heavily skewed supports during only moderate seismic shaking. In such cases, it is 

necessary to consider the potential for collapse of the span, which will depend to a large extent 

on the geometry of the bearing seat. Settlement and vertical misalignment of a span due to an 

overturned bearing may be a minor problem, resulting in only a temporary loss of access that can 

be restored, in many cases, by backfilling with asphalt or other similar material. The potential for 

total loss of support should be the primary criteria when rating the vulnerability of the bearings. 

A suggested step-by-step method for determining the vulnerability rating for connections, 

bearings, and seat widths (V1) is detailed in the flowchart of figure 4-4 and is as follows: 

Step 1. Determine if the bridge has satisfactory bearing details. Such bridges include: 

 a. Continuous structures with integral abutments. 

 b. Continuous structures with seat-type abutments where all of the following conditions 

are met: 

1. Either (a) the skew is less than 20° (0.35 rad), or (b) the skew is greater than 20°

(0.35 rad) but less than 40° (0.70 rad) and the length-to-width ratio of the bridge 

deck is greater than 1.5.

2. Rocker bearings are not used. 

3. The abutment’s bearing seat under the end diaphragm is continuous in the 

transverse direction and the bridge has more than three beams. 

4. The support length is equal to, or greater than, the minimum required length (N) 

given in equation 4-3. 

If the bearing details are determined to be satisfactory, a vulnerability rating, V1, of 0 

may be assigned and the remaining steps for bearings omitted. 

Step 2. Determine the vulnerability to structure collapse or loss of access to the bridge due to 

transverse movement, VT. 

Before significant transverse movement can occur, the transverse restraint must fail. In 

the absence of calculations showing otherwise, assume that the bearing keeper bars 

and/or the anchor bolts in bridges in SRC C and D will fail. Also assume that nominally 

reinforced, nonductile concrete shear keys will fail in bridges in SRC D. 

When the transverse restraint is subject to failure, beams are vulnerable to loss of support 

if either of the following conditions exist: 
118

a. Individual beams are supported on individual pedestals or columns. 

b. The exterior beam in a 2- or 3-beam bridge is supported near the edge of a bearing 

seat regardless of whether the bearings are on individual pedestals or not. 

In either of these cases, the vulnerability rating, VT, should be 10. 

Steel rocker bearings have been known to overturn transversely, resulting in a permanent 

superstructure displacement. All bridges in SRC D are vulnerable to this type of failure. 

Bridges in SRC C are vulnerable only when the skew is greater than 40° (0.70 rad). 

When bearings are vulnerable to a toppling failure but structure collapse is unlikely, the 

vulnerability rating, VT, should be 5. Otherwise VT=0. 

Step 3. Determine the vulnerability of the structure to collapse or loss of access due to excessive 

longitudinal movement, VL. 

VL is determined according to the available support length (L) measured in a direction 

perpendicular to the centerline of the support. This is done by comparing L with the 

minimum required length N, (equation 4-3), as follows: 

If L > N then VL = 0 regardless of bearing type. 

If N > L > 0.5N and rocker bearings are not used, then VL = 5. 

If N > L > 0.5N and rocker bearings are used, then VL = 10. 

If 0.5 N > L then VL = 10 regardless of bearing type. 

Step 4. Calculate vulnerability rating for connections, V1, from values VT and VL, with 

 V1 = greater of VT and VL. 

4.2.1.1(b). Vulnerability Rating for Columns, Abutments, and Liquefaction Potential, V2

The vulnerability rating for the other components in the bridge that are susceptible to failure, V2, 

is calculated from the individual component ratings as follows: 

 V2 = CVR + AVR + LVR ≤ 10 (4-4) 

where CVR is the column vulnerability rating, AVR is the abutment vulnerability rating, and 

LVR is the liquefaction vulnerability rating. 
119

STEP 1

Are bearing details satisfactory?

Rocker Bearings?

STEP 2

Check transverse behavior

Restraint fails?

2- or 3-girder bridge with outside girder

on seat edge?

Pedestals?

Overturning of bearings possible?

Bridge collapse likely?

VT = 0 VT = 5 VT = 10

A B

No

No

No

No

No

No

No

Yes

Yes

Yes

Yes

Yes

Yes

Yes

(continued)

Figure 4-4. Calculation of vulnerability rating for connections, bearings and seat widths (V1). 
120

Overturning of bearings possible?

STEP 3

Check longitudinal behavior

N < L

N/2 < L < N

Rocker Bearings?

VL= 0 VL = 5 VL = 10

A B

Yes

Yes

No

Yes

No

Yes

No

No

STEP 4

V1 = Maximum of VT, VL

V1 = 0

Note: L = actual seat width and N = seat width required by equation 4-3a or 4-3b 

Figure 4-4 (continued). Calculation of vulnerability rating for connections, bearings 

and seat widths (V1). 
121

Suggested methods for calculating each of these component ratings are given in the following 

sections. 

A. Column Vulnerability Rating, CVR 

Columns have failed in past earthquakes due to lack of adequate transverse reinforcement and 

poor structural details. Excessive ductility demands have resulted in degradation of column 

strength in shear and flexure. In past earthquakes, columns have failed in shear, resulting in their 

disintegration and substantial vertical settlement. Column failure may also occur due to pullout 

of the longitudinal reinforcing steel, mainly at the footings. Fortunately, most bridge column 

failures occur during earthquakes that generate high ground accelerations of relatively long 

duration3

. 

The following step-by-step procedure may be used to determine the vulnerability of columns, 

piers, and footings. 

Step 1. Assign a column vulnerability rating, CVR, of 0 to bridges classified as SRC B. 

Step 2. Assign a vulnerability rating, CVR, of 0 if keeper bars or anchor bolts can be 

 relied upon to fail (section 4.2.1.1(a), step 2), thereby preventing the transfer of load to 

 the columns, piers, or footings. 

Step 3. If columns and footings have adequate transverse steel as required by the current 

AASHTO Specifications, assign a column vulnerability rating, CVR, of 0. 

Step 4. If none of the above apply (steps 1 through 3), check the column for shear, splice details, 

 and foundation deficiencies, and give CVR the highest value calculated from the 

 following steps: 

Step 4a. Column vulnerability due to shear failure. 

 CVR = Q – PR (4-5a) 

where: 

 Lc = effective column length, 

 Ps = amount of main reinforcing steel expressed as a percent of the column cross- 

 sectional area, 

 F = framing factor: 

 2.0 for multi-column piers fixed top and bottom, 

 1.0 for multi-column piers fixed at one end, 

 

3

 This may not always be the case, as illustrated by the collapse of the Cypress Street Viaduct in Oakland, 

California, during the 1989 Loma Prieta earthquake. This short-duration earthquake generated peak rock 

accelerations of about 0.1g close to the bridge. Soft soils amplified this figure to approximately 0.25g at the surface, 

which is a relatively modest value. Nevertheless, the viaduct, designed before 1971, collapsed due to poor detailing 

and shear failures in the connections between the upper and lower decks. 

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ = −

s max

c

P Fb

L

Q 13 6 (4-5b) 
122

 1.5 for box girder superstructure with a single-column pier, fixed at top and 

 bottom, and 

 1.25 for superstructures other than box girders with a single-column pier, 

 fixed at top and bottom. 

 bmax = maximum transverse column dimension, and 

 PR = the total number of points to be deducted from Q for factors known to reduce 

 susceptibility to shear failure, as listed in table 4-1. 

Table 4-1. Values for PR. 

Factor PR

Seismic coefficient, SD1 < 0.5 3 

Skew ≤ 20° (0.35 rad) 2 

Continuous superstructure, integral abutments 

of equal stiffness and length-to-width ratio < 4 1 

Grade 40 (or below) reinforcement 1 

Values of CVR less than zero or greater than 10, should be assigned values of 0 and 10, 

respectively. 

Equation 4-5b was derived empirically based on observations of column shear failure in 

bridges during the San Fernando earthquake in 1971. The derivation is given in appendix 

B of the 1983 retrofit guidelines (FHWA, 1983). This expression has since been checked 

against column failures in the 1994 Northridge earthquake and was found to be a reliable 

indicator of column damage (Buckle, 1994). However, the columns in this empirical data 

set are short to medium in height and equation 4-5b may not apply to tall and/or slender 

columns. In these cases, special studies should be undertaken to estimate Q, PR, and 

CVR. 

Step 4b. Column vulnerability due to flexural failure at splices. 

To account for flexural failure when the column longitudinal reinforcement is spliced in a 

plastic hinge location, the following CVR should be used for columns supporting 

superstructures longer than 90 m (300 ft), or for superstructures with expansion joints:

 CVR = 7 for SD1 < 0.5 (4-6a) 

 CVR = 10 for SD1 ≥ 0.5 (4-6b) 

where SD1 is the seismic coefficient defined in equation1-1. 

CVR need not be taken greater than seven unless microzoning confirms SD1 is greater 

than or equal to 0.5. 
123

 

Step 4c. Column vulnerability due to foundation deficiencies. 

The following CVR should be used for columns supported on pile footings that are not 

reinforced for uplift, or for poorly confined foundation shafts. This step is only applicable 

if microzoning yields values of SD1 greater than or equal to 0.5: 

 

 CVR = 5 for 0.5 ≤ SD1 ≤ 0.6 (4-7a) 

 CVR = 10 for SD1 > 0.6 (4-7b) 

Step 4d. Assign overall column vulnerability rating, CVR. 

Set the column vulnerability rating, CVR, to the highest value calculated for CVR in 

steps 4a, 4b, and 4c. 

B. Abutment Vulnerability Rating, AVR 

Abutment failures during earthquakes do not usually result in total collapse of the bridge. This is 

especially true for earthquakes of low-to-moderate intensity. Therefore, the abutment 

vulnerability rating should be based on damage that would temporarily prevent access to the 

bridge. 

Fill settlements are generally of the order of three to five percent of fill height during moderatesized earthquakes but may be as large as 10-15 percent4

 in cases of poorly constructed approach 

fills or when spill-through abutments are used. Additional settlement may occur when the 

abutments are structurally damaged due to excessive earth pressures and/or forces transferred 

from the superstructure. 

Certain abutment types, such as spill-through abutments and those without wingwalls, may be 

more vulnerable to this type of damage than others. Except in unusual cases, the maximum 

abutment vulnerability rating, AVR, need not exceed five. The following step-by-step procedure 

for determining the vulnerability rating for abutments is based on engineering judgment and the 

performance of abutments in past earthquakes. 

Step 1. If bridges are classified as SRC B, assign a vulnerability rating, AVR, of 0. 

Step 2. Determine the vulnerability of the structure to abutment fill settlement. The fill settlement 

in normally compacted approach fills may be estimated as follows: 

a. One percent of the fill height when 0.24 < SD1 ≤ 0.39. 

b. Two percent of the fill height when 0.39 < SD1 ≤ 0.49. 

c. Three percent of the fill height when SD1 > 0.49. 

 

4

 Fung et al., 1971; Elms and Martin, 1979 
124

The above settlements should be doubled if the bridge is a water crossing. When fill 

settlements are estimated to be greater than 150 mm (6 in), assign a vulnerability rating, 

AVR, for the abutment of 5. Otherwise assign a value of 0 for AVR. 

Step 3. Bridges classified as SRC D should be assigned a vulnerability rating, AVR, of 5 

 regardless of the estimated fill settlement, if all of the following conditions are present: 

• Cantilever abutments. 

• Skews greater than 40° (0.70 rad). 

• Distance between the abutment seat and the underside of the footing exceeds 3 m 

(10 ft). 

 Otherwise, assign a value of 0 for AVR unless the fill settlement calculated in step 2 is 

greater than 150 mm (6 in), in which case AVR should be 5. 

C. Liquefaction Vulnerability Rating, LVR 

Although there are several possible types of ground failure that can result in bridge damage 

during an earthquake, instability resulting from liquefaction is the most significant. The 

vulnerability rating for foundation soil is therefore based on: 

• A quantitative assessment of liquefaction susceptibility. 

• The magnitude of the acceleration coefficient. 

• An assessment of the susceptibility of the bridge structure itself to damage resulting from 

liquefaction-induced ground movement. 

The vulnerability of different types of bridges to liquefaction has been illustrated by failures 

during past earthquakes, such as the 1964 Alaskan earthquake (Ross et al., 1973) and various 

Japanese earthquakes (Iwasaki et al., 1972). The damage observed has demonstrated that bridges 

with continuous superstructures and integral supports can withstand large translational 

displacements and usually remain serviceable (with minor repairs). However, bridges with 

discontinuous superstructures and/or substructures with little or no ductility, are usually severely 

damaged as a result of liquefaction. These observations have been taken into account in 

developing the vulnerability rating procedure described below. The procedure is based on the 

following steps: 

Step 1. Determine the susceptibility of foundation soils to liquefaction. 

High susceptibility is associated with the following conditions: 

a. Where the foundation soil, that is providing lateral support to piles or vertical support 

to footings, consists generally of saturated loose sands, saturated silty sands, or nonplastic silts. 

b. Where soils similar to a. (above) underlie the abutment fills or are present as 

continuous seams, which could lead to abutment slope failures. 

Moderate susceptibility is associated with foundation soils that are generally medium 

dense soils, e.g., compact sands. 
125

Low susceptibility is associated with foundation soils that are generally dense soils. 

Step 2. Use table 4-2 to determine the potential for liquefaction-related damage where 

susceptible soil conditions exist. 

 For all sites where SD1 > 0.49, engineering judgment should be applied to determine the 

possibility of greater damage. 

Table 4-2. Potential for liquefaction-related damage. 

Soil Seismic Coefficient, SD1

Susceptibility 

To 

Liquefaction SD1 ≤ 0.14 0.14 < SD1 ≤ 0.24 0.24 < SD1 ≤ 0.39 0.39 < SD1 ≤ 0.49 SD1> 0.49

Low 

Moderate 

High 

Low 

Low 

Low 

Low 

Low 

Moderate 

Low 

Moderate 

Major 

Low 

Major 

Severe 

Low 

Severe 

Severe 

Step 3. In general, bridges subject to severe liquefaction-related damage should be assigned a 

vulnerability rating, LVR, of 10. This rating may be reduced to 5 for single-span bridges 

with skews less than 20° (0.35 rad) or for rigid box culverts. 

Step 4. Bridges subject to major liquefaction-related damage should be assigned a vulnerability 

rating, LVR, of 10. This rating may be reduced to between 5 and 9 for single-span 

bridges with skews less than 40° (0.70 rad), and for rigid box culverts and continuous 

bridges with skews less than 20° (0.35 rad), provided one of the following conditions 

exists: 

a. Reinforced concrete columns that are integral with the superstructure and have a CVR 

less than 5 and a height in excess of 8 m (25 ft). 

b. Steel columns (except those constructed of non-ductile material) that are in excess of 

8 m (25 ft) high. 

c. Columns that are not integral with the superstructure, provided that large movements 

of the substructure will not result in instability. 

Step 5. Bridges subjected to moderate liquefaction-related damage should be assigned a 

vulnerability rating, LVR, of 5. This rating should be increased to between 6 and 10 if the 

vulnerability rating for the bearings, V1, is greater than or equal to 5. 

Step 6. Bridges subjected to low liquefaction-related damage shall be assigned a vulnerability 

rating, LVR, of 0. 
126

4.2.1.2. Seismic Hazard Rating (E) 

For the purpose of this rating, seismic hazard includes both the seismicity of the site and the 

geotechnical conditions. The seismic coefficient (SD1) is used for this purpose, which, as defined 

in section 1.5.3, is the spectral acceleration at 1.0 sec modified by the site amplification factor for 

this period. This coefficient is then scaled to fit the range 0 – 10, so as to give the same 

weighting to the hazard as to the structural vulnerability. Hence the seismic hazard rating is 

given by: 

 E = 10 SD1 ≤ 10.0 (4-8) 

where SD1 is the seismic coefficient at 1.0 sec period, and equals Fv S1 (equation 1-1). 

In locations where the soil properties are not known in sufficient detail to determine the soil 

profile type with confidence, or where the soil profile does not fit any of the specified types, the 

site coefficient (Fv) shall be based on engineering judgment. 

4.2.1.3. Examples 

Examples 4.1 and 4.2 are given to illustrate the process of calculating bridge rank, R, for two 

different bridges of the same length and soil conditions, subject to similar sized earthquakes. 

Additional examples are given in appendix E. 

The results show that a greater vulnerability (higher bridge rank) has been calculated for the 

simply supported bridge (R = 43) than for the continuous bridge (R = 33), but this difference is 

not as great as might be expected. The reason is that, although the second bridge has continuous 

spans, they are supported on steel bearings, which in turn are mounted on pedestals. Although 

total span collapse is unlikely to occur, it is very possible that the beams will come off the 

pedestals despite the continuity. 

 

These same bridges are also analyzed by the expected damage method (section 4.3) with similar 

results (section 4.3.5.4). 

 4.2.2. CALCULATION OF PRIORITY INDEX BASED ON INDICES 

Once a rank has been calculated for each bridge based on equation 4-2, the bridges may be listed 

in numerical order of decreasing rank. This order now needs to be modified to obtain the Priority 

Index (equation 4-1) so as to include such factors as bridge importance, network redundancy, 

non-seismic deficiencies, remaining useful life, and other socioeconomic issues. 

Guidance on assigning importance was given in section 1.4.3 and some discussion of network 

redundancy and non-seismic rehabilitation was provided in section 1.10.2. If a bridge is part of a 

highly redundant highway network with alternative bridges or routes, the likelihood that these 

alternative facilities may also be damaged must be considered. In general, it will not be possible 

to develop a single number by which to scale the seismic rank (equation 4-2) to obtain the 
127

priority index (equation 4-1). Instead, reordering the rank by subjective means, using a 

combination of engineering and societal judgment, will be necessary. In this way, an attempt can 

be made to include all of the technical and societal issues that influence the prioritization of 

bridges for seismic retrofitting. 

4.3. SEISMIC RATING METHOD USING EXPECTED DAMAGE

4.3.1. BACKGROUND 

An alternative method for screening and prioritizing bridges is presented in this section. The 

method compares the severity of expected damage for each bridge in the inventory, for the same 

earthquake, and ranks each bridge accordingly. Severity of damage is measured by either the 

damage sustained or the direct economic losses. Bridges with the greatest expected damage 

(and/or loss) are given the highest priority for retrofitting. 

Direct economic losses may be taken as the cost of repair or replacement of a damaged bridge. 

These losses do not include the so-called indirect costs or socioeconomic costs, which can be 

very significant and often exceed the direct costs. Indirect costs include loss of life, injuries, 

business disruption, delay due to traffic congestion, and denial of access. Estimation of these 

costs is a complex exercise and subject to great uncertainty. 

 

As for the previous method, this method has both quantitative and qualitative parts. The 

quantitative part is based on expected damage and repair costs, and is used to obtain a bridge 

rank, R. The qualitative part modifies the rank in a subjective way that takes into account such 

factors as indirect losses, network redundancy, non-seismic deficiencies, remaining useful life, 

and other issues, to obtain an overall priority index. As in the previous method, engineering and 

societal judgment are the key to the second stage of the process. This leads to a priority index, 

which is a function of rank, indirect loss, redundancy, and other factors, including non-seismic 

rehabilitation needs. Thus the priority index is defined as 

P = f (R, indirect loss, redundancy and various non-seismic factors) (4-9) 

where P is the priority index, and R is the rank based on expected damage and repair costs. 

The bridge rank, R, is based on expected damage and direct losses for a given earthquake, 

whereas retrofit priority is based not only on bridge rank, but also on expected indirect losses, 

network redundancy and non-seismic factors, estimated in a subjective way. The method is 

illustrated by the flowchart in figure 4-5. 

Although equation 4-9 has the same form as equation 4-1, the terms are calculated in different 

ways. A particular advantage of this method is that it provides a template by which indirect 

losses may be included, as the state-of the art improves. 
128

Performance Level, PL Seismic Hazard Level, SHL

Seismic Retrofit Category, SRC

Seismic Retrofit Category A

RETROFITTING NOT

REQUIRED

OTHER FACTORS, O

Bridge importance

Network redundancy

Anticipated service life

Non-seismic rehabilitation

needs

Expected indirect losses

CALCULATE PRIORITY INDEX, P=f(R,O)

Seismic Retrofit Categories

B, C, D

Compile NBI data and standard

bridge fragility functions

Calculate bridge-specific

fragility function

Calculate

Total Repair Cost Ratio, RCR

and Expected Repair Cost

T

Assign Bridge Rank, R,

based on RCR or

Expected Repair Cost

T

Figure 4-5. Seismic rating method using expected damage. 
129

EXAMPLE 4.1: PRESTRESSED CONCRETE BRIDGE WITH SIMPLE SPANS – INDICES 

METHOD 

A four-beam, prestressed concrete bridge is located on very dense soil and soft rock (site class C). 

Constructed in 1968, it has three simply-supported spans, each seated on elastomeric pads directly on 

the cap beams (no pedestals); seat widths are 450 mm (17 3⁄4”). The total length of the bridge is 56 m 

(183’- 8 3⁄4”), with an overall width of 10 m (32’- 9 3⁄4”) and a skew of 32°. Abutment fill height is 7.6 m (24’- 

11 1⁄4”). The bridge carries 65,000 ADT and is considered to be ‘essential.’ 

Calculate the bridge rank R, for this bridge using the indices method (equation 4-2). 

Step A. Determine seismic retrofit category, SRC, and minimum screening requirements

The anticipated service life, performance level, and seismic hazard level are required to find the SRC, as 

follows: 

Anticipated service life (ASL): Age of bridge = 2004-1968 = 36 years 

 Assumed service life = 75 years 

 Anticipated service life = 39 years 

 Service life category = ASL 2 (table 1-1) 

Performance level (PL): Bridge importance is ‘essential’ 

Therefore the performance level for the upper level earthquake is PL1 

(table 1-2). 

Seismic hazard level (SHL): Obtain Ss and S1 from USGS CD-ROM (section 2.3) for upper level 

earthquake. Assume Ss and S1 = 1.40 and 0.28. 

Obtain Fa and Fv from table 1-4 for site class C. 

Calculate SDS and SD1 and obtain SHL from table 1-5 (Eq. 1-1). 

 Results are summarized below: 

Earthquake Ss S1 Fa Fv SDS SD1 SHL 

 Upper 1.40 0.28 1.0 1.52 1.4 0.43 IV 

 

Seismic retrofit category (SRC): Obtain SRC from table 1-6. 

 Earthquake PL SHL SRC 

 Upper PL1 IV C 

 

Minimum screening requirements are given in table 1-7: For SRC = C, minimum requirements are seat 

widths, connections, columns, walls, footings and liquefaction. 

Step B. Vulnerability rating, V 

Use the procedure in section 4.2.1.1 to find vulnerability rating, V: 

Step B.1 Bearing and seat width vulnerability, V1: 

Simple spans, elastomeric bearings, no pedestals, bearings are not vulnerable to toppling, 

Four-beam bridge, gives VT = 0 

 Required longitudinal support, N = 545 (211⁄2”) mm (equation 4-3a) 

 Available support, L = 450 mm (17 3⁄4”); hence 0.5 N < L < N and VL = 5. 

 V1 = greater of VT and VL = greater of 0 and 5 = 5. V1 = 5 

 Use procedure in section 4.2.1.1(b). 

Step B.2 Column, abutment, and liquefaction vulnerability, V2 = CVR + AVR + LVR < 10 
130

For the purpose of this example, assume elastomeric bearings do not fail and seismic loads 

are transferred to columns; assume columns are not vulnerable to shear failure but 

longitudinal steel is spliced in potential plastic hinge region. 

Superstructure has expansion joints (three simple spans) and SD1 = 0.43 < 0.5; hence 

CVR = 7 (section 4.2.1.1 (b)). 

Since SD1 – 0.43, fill settlement behind abutment is estimated at two percent x 7600 mm = 

152 mm (6”); hence AVR = 5 (section 4.2.1.1 (b)) 

The susceptibility of soils in site class C to liquefaction is low, and from table 4-2, the 

potential for liquefaction-related damage is ‘low’; hence LVR = 0. 

Therefore V2 = CVR + AVR + LVR < 10 (equation 4-4) = 7 + 5 + 0 = 12 > 10 

Hence V2 = 10 

Step B.3 Overall bridge vulnerability, V 

 Bridge vulnerability calculated as V = greater of V1 and V2 

= greater of 5 and 10. V = 10 

Step C Seismic hazard rating, E

 Calculate E = 10 SD1 < 10.0 (equation 4-8) = 10 x 0.43 = 4.3, for SD1 = 0.43. 

Step D Bridge rank, R 

 Calculate R = VE (equation 4-2) = 10 x 4.3 = 43 Answer 
131

EXAMPLE 4.2: STEEL GIRDER BRIDGE WITH CONTINUOUS SPANS – INDICES METHOD 

A four-stringer, steel bridge is located on very dense soil and soft rock (site class C). Constructed in 1972, 

it has three continuous spans of equal length, each seated on steel bearings on pedestals (not rocker 

bearings); seat widths are 350 mm (13 3/4”). The total length of the bridge is 56 m (183’- 8 3⁄4”), with an 

overall width of 14 m (45’- 11”) and a skew of 18o. Abutment fill height is 6 m (19’- 8 1⁄2”). The bridge 

carries 20,000 ADT and is considered to be ‘standard.’ 

Calculate the bridge rank R, for this bridge using the indices method (equation 4-2). 

Step A. Determine seismic retrofit category, SRC, and minimum screening requirements 

The anticipated service life, performance level, and seismic hazard level are required to find the SRC, as 

follows: 

Anticipated service life (ASL): Age of bridge = 2004-1972 = 32 years 

 Assumed service life = 75 years 

 Anticipated service life = 43 years 

 Service life category = ASL 2 (table 1-1) 

Performance level (PL): Bridge importance is ‘standard’ 

Therefore, the performance level for the upper level earthquake is PL1 

(table 1-2). 

Seismic hazard level (SHL): Obtain Ss and S1 from USGS CD-ROM (section 2.3). Assume 

Ss and S1 = 1.50 and 0.21 for upper level earthquake. 

Obtain Fa and Fv from tables 1-4a and b for site class C. 

Calculate SDS and SD1 using Eq. 1-1 and obtain SHL from table 1-5. 

 Results are summarized below: 

Earthquake Ss S1 Fa Fv SDS SD1 SHL 

 Upper 1.50 0.21 1.0 1.59 1.50 0.33 IV 

 

Seismic retrofit category (SRC): Obtain SRCs from table 1-6. 

 Earthquake PL SHL SRC 

 Upper PL1 IV C 

 

Minimum screening requirements are given in table 1-7: For SRC = C, minimum requirements are seat 

widths, connections, columns, walls, footings and liquefaction. 

Step B. Vulnerability rating, V 

Use the procedure in section 4.2.1.1 to find vulnerability rating, V: 

Step B.1 Bearing and seat width vulnerability, V1: 

Continuous spans, steel bearings, pedestals, bearings are not rocker bearings and therefore 

not vulnerable to toppling, 4-stringer bridge, gives VT = 10 

 Required longitudinal support, N = 437 mm (17 1⁄4”) (equation 4-3a). 

 Available support, L = 350 mm (13 3⁄4”); hence 0.5N < L < N 

 0.5 (437 mm) < 350 mm < 437 mm; VL = 5. 

 V1 = greater of VT and VL = greater of 10 and 5. V1=10 

Step B.2 Column, abutment, and liquefaction vulnerability, V2 = CVR + AVR + LVR < 10 

Assume failure of bearing keeper bars cannot be relied upon and seismic loads are 

transferred to columns; assume columns are not vulnerable to shear failure but longitudinal 

steel is spliced in potential plastic hinge region. 
132

Superstructure has expansion joints at abutments and SD1 = 0.33 < 0.5; hence CVR = 7 

(section 4.2.1.1 (b)). 

Since SD1 = 0.33, use 1% of fill height for settlement. 

Fill settlement behind abutment estimated at 1 percent x 6000 = 60 mm (2 3/8”); hence 

AVR = 0 (section 4.2.1.1 (b)) 

The susceptibility of soils in site class C to liquefaction is low, and from table 4-2, the 

potential for liquefaction-related damage is ‘low’; hence LVR = 0. 

Therefore V2 = CVR+AVR+LVR < 10 (equation 4-4) = 7 + 0 + 0 = 7 

Hence V2 = 7 

Step B.3 Overall bridge vulnerability, V 

Bridge vulnerability calculated as V = greater of V1 and V2 

= greater of 10 and 7. V = 10 

Step C Seismic hazard rating, E 

 Calculate E = 10 SD1 < 10.0 (equation 4-8) = 10 x 0.33 = 3.3, for SD1 = 0.33. 

Step D Bridge rank, R 

 Calculate R = VE (equation 4-2) = 10 x 3.3 = 33 Answer
133

The estimation of expected damage is a critical step in this method. Due to uncertainty in 

earthquake ground motions, and randomness of soil and structure properties, this step is a 

probabilistic one. Fragility functions are used to estimate the probability of a bridge being in one 

or more specified damage states, after a given earthquake. Appendix C summarizes the theory of 

fragility functions and explains briefly how they are obtained. It also describes the six damage 

states most commonly used to characterize expected damage. 

Fragility functions are also essential elements in the Seismic Risk Assessment Method for 

screening and prioritizing bridges as noted in section 1.10.3. These functions are also used in 

most loss estimation methodologies, including HAZUS, developed by the National Institute of 

Building Sciences for the Federal Emergency Management Agency (HAZUS, 1997). 

To simplify the method as much as possible, input data requirements are kept to a minimum. In 

particular, all necessary attributes of the structure may be found in the National Bridge Inventory 

(NBI) or the bridge owner’s bridge history file. Ground motions and soils data are again based 

on spectral accelerations and soil types as described in sections 1.5.1 and 1.5.2. 

4.3.2. DATABASE REQUIREMENTS 

The NBI database contains 116 fields that are used to describe structural and operational 

characteristics of a bridge, but there is still insufficient detail on each bridge to permit a detailed 

derivation of the required fragility curves. Therefore, for the purpose of screening and 

prioritization, bridge-specific fragility curves are obtained by taking the results for a reference 

bridge fragility curve and scaling them using selected data from the NBI. 

A reference bridge is assumed to be a ‘long’ structure with no appreciable three-dimensional 

(3D) effects present. Such a bridge is identified for each major type of bridge, of which there 

may be five or six types (section 4.3.3). For each reference bridge, median spectral accelerations 

(a2, a3, a4, a5), at period T=1.0 sec, have been developed for each of the damage states (DSi) 

described in appendix C. The results for these reference bridges are then modified by factors 

accounting for skew (Kskew) and three-dimensional arching5

 effects within the plane of the 

superstructure (K3D), using the NBI attributes in table 4-3. This table shows which fields of the 

NBI are used and for what purpose. 

4.3.3. FRAGILITY CURVES FOR REFERENCE BRIDGES 

As described in appendix C, fragility curves are constructed for each of five damage states, for 

each reference bridge. These five damage states are defined as follows: 

• DS1 = no damage (pre-yield) 

• DS2 = slight damage 

 

5

 ‘Arching’ refers to the ability of a bridge deck to distribute lateral loads to the abutments by the in-plane arching 

action of the deck spanning from one abutment to the other. If expansion joints occur within the superstructure, 

arching action cannot develop until all of these joints are closed. 
134

Table 4-3. NBI fields (attributes) used in determining bridge fragility. 

NBI Data Item Definition Kskew K3D Other Use 

1 State X To infer specification used in design 

8 Structure number General identification number 

27 Year built X Infers whether seismic or 

conventional design 

34 Skew X 

42 Service type To select highway bridges 

43 Structure type X To infer base fragility curve from 

Tables 4-4 and 4-5 

45 Number of spans in 

main unit X To infer whether single or multiple 

span 

46 Number of approach 

spans (n) To infer whether bridge is a major 

bridge (major bridge if n>6) 

48 Length of maximum 

span (L) To infer whether bridge is a major bridge (major bridge if L>150 m) 

• DS3 = moderate damage 

• DS4 = extensive damage 

• DS5 = collapse 

Data for the construction of these fragility curves are presented in tables 4-4 and 4-5, where 

median fragility parameters are listed for non-seismic and seismic structures, respectively (see 

definitions below). In the development of the parameters in tables 4-4 and 4-5, certain 

assumptions have been made. These are listed in the rightmost column and are considered to be 

typical of bridge construction practice throughout the United States. Additional assumptions are 

noted below: 

• The spectrum modification factors accounting for hysteretic energy dissipation (BS, BL) were 

taken as suggested in table 5-4. 

• For large bridges, defined as those structures whose main span exceeds 150 m (500 ft), the 

above procedures do not apply. The values for such bridges presented in tables 4-4 and 4-5 

are interpolations based on engineering judgment. 
135

• Bridges built after 1990 (1975 in California), and designed in accordance with prevailing 

seismic provisions, are considered to have been seismically designed, i.e., they are ‘seismic’ 

structures. Bridges built prior to these dates are assumed to have not been designed for 

seismic loads and are called ‘non-seismic’ structures. 

• For the sake of simplicity, and largely due to lack of details on regional pier construction 

practice in the NBI database, it is assumed that continuous bridges have weak bearings/strong 

piers. Anecdotally this appears to be the case, as relatively few continuous bridges were 

constructed in the eastern United States prior to the 1970's. In California, it is assumed that 

concrete bridges identified in the NBI as ‘continuous’ do not, in fact, have continuous 

superstructures but instead are reinforced concrete frames with in-span expansion joints. 

• For the purpose of inferring what bearing types are matched with bridge deck types, it is 

assumed that steel and concrete (reinforced and prestressed) beam bridges possess steel and 

neoprene bearings, respectively. 

• Site class B soils (table 1-3) were assumed when deriving the reference bridge fragility 

curves. The site factors described in section 1.5.2 and listed in table 1-4 are used to modify 

these curves for other site classes. 

4.3.4. SCALING REFERENCE BRIDGE FRAGILITY CURVES TO ACCOUNT FOR SKEW 

AND THREE-DIMENSIONAL EFFECTS 

4.3.4.1. Definition of Parameters: Kskew and K3D

In order to convert the reference bridge fragility curves to a bridge-specific curve for a given 

spectral acceleration, the parameters Kskew and K3D are used as scaling relations as described 

below. 

The reference fragility curve is modified for skew by applying the correction factor, Kskew, 

which is calculated as follows: 

Kskew = cos α (4-10) 

where α is the angle of skew measured from a line normal to the bridge centerline to a line 

parallel to the centerline of bearing. 

Likewise, the reference fragility curve is modified for 3D effects using the correction factor K3D

as tabulated in table 4-6. This factor converts the fragility curve for a long reference bridge to a 

specific right (no skew) bridge with a finite number of spans. 

4.3.4.2. Scaling Relations for Damage States 3, 4, and 5 

The modified median fragility curve parameter is given by 

S

a A K K i

i = skew 3D (4-11) 
136

Table 4-4. Fragility of bridges constructed before 1975 in California and 1990 elsewhere. 

(non-seismic bridges) 

Median Spectral 

Acceleration at T=1.0 sec Classification NBI Class Damage 

State NonCalifornia 

California Assumptions2,3

Multi-column 

piers, simply 

supported 

101 - 106 

301 - 306 

501 - 506 

2 

3 

4 

5 

0.26 

0.35 

0.44 

0.65 

0.33 

0.46 

0.56 

0.83 

D = 0.91 m (non-CA) 

D = 1.5 m (CA) 

Single-column 

piers, box 

girders 

205 - 206 

605 - 606 

2 

3 

4 

5 

Not applicable 0.35 

0.42 

0.50 

0.74 

D = 1.5 m 

Continuous 

Concrete 

201 - 206 

601 - 607 

2 

3 

4 

5 

0.601

0.79 

1.05 

1.38 

 Fb = 0.8 breakage 

Fb = 0.7 residual 

Continuous 

Steel 

402 - 410 2 

3 

4 

5 

0.761

0.76 

0.76 

1.04 

 Fb = 1.0 breakage 

Fb = 0.4 residual 

Single-Span All 2 

3 

4 

5 

0.801

0.90 

1.10 

1.60 

 Fb = 1.1 initial breakage 

Fb = 0.65 residual 

Major Bridges 2 

3 

4 

5 

0.40 

0.50 

0.60 

0.80 

 Interpolation from above 

categories 

Notes: 

1. Case I (section 4.3.4.3) applies for these cases, evaluate Kshape using equation 4-13 

2. D = column diameter 

3. Fb = bearing shear capacity as a fraction of dead load 
137

 Table 4-5. Fragility of bridges constructed since 1975 in California and 1990 elsewhere. 

(seismic bridges) 

Classification NBI Class Damage 

State 

Median Spectral 

Acceleration at 

T=1.0 sec 

Assumptions2,3

Multi-column 

piers, simplysupported 

101 - 106 

301 - 306 

501 - 506 

2 

3 

4 

5 

0.45 

0.76 

1.05 

1.53 

D = 0.91 m 

Single-column 

bents, box 

girders 

205 - 206 

605 - 606 

2 

3 

4 

5 

0.54 

0.88 

1.22 

1.45 

D = 1.5 m 

Continuous 

Concrete and 

Steel 

201 - 206 

402 - 410 

601 - 607 

2 

3 

4 

5 

0.911

0.91 

1.05 

1.38 

Fb = 1.2 initial breakaway 

Fb = 0.7 sliding value 

Single-Span 2 

3 

4 

5 

0.81

0.9 

1.1 

1.6 

Fb = 1.1 initial breakaway 

Fb = 0.65 sliding value 

Major bridges 2 

3 

4 

5 

0.6 

0.8 

1.0 

1.6 

Interpolation from above 

categories 

Notes: 

1. Case I (section 4.3.4.3) applies, evaluate Kshape using equation 4-13 

2. D = column diameter 

3. Fb = bearing shear capacity as a fraction of dead load
138

Table 4-6. Modification factors (K3D) used to model 3D effects for multi-span bridges. 

Type NBI Class K3D

Non-seismic Bridges

< 1990 (1975 in CA)1,2

K3D

Seismic Bridges 

> 1990 (1975 in CA)1

Concrete 101 – 106 1 + 0.25 / np 1 + 0.25 / np

Concrete Continuous 201 – 206 1 + 0.33 / np 1 + 0.33 / np

Steel 301 – 310 1 + 0.09 / np; L > 20 m (65ft) 

1 + 0.20 / np; L < 20 m (65ft) 

1 + 0.25 / np

Steel Continuous 402 – 410 1 + 0.05 / np; L > 20 m (65ft) 

1 + 0.10 / np; L < 20 m (65ft) 

1 + 0.33 / np

Prestressed Concrete 501 – 506 1 + 0.25 / np 1 + 0.25 / np

Prestressed Concrete 

Continuous 

601 – 607 1 + 0.33 / np 1 + 0.33 / np

Notes: 1. np = number of piers = n – 1, where n = number of spans in bridge 

 2. L = length of maximum span (NBI Data Item 48, table 4-3) 

where ai is the median spectral acceleration (at T = 1.0 sec) for the ith damage state as listed in 

tables 4-4 and 4-5, and S is the site factor for the long-period range; that is S = Fv, the 1.0 second 

site factor given in section 1.5.2 and table 1-4 (note a rock site was assumed in deriving the 

reference bridge fragility curves, i.e., S = 1 for the reference curves). 

4.3.4.3. Scaling Relations for Damage State 2 (Slight Damage) 

Case I: When ‘short’ periods govern (T < TS, figure 1-8): 

S

a A K 2

2 = shape (4-12) 

where: 

a2 = the median spectral acceleration (at T = 1.0 sec) for damage state 2, given in tables 

4-4 and 4-5 and identified by superscript 1 in those tables, 

S = site factor for the short period range; that is S = Fa, the 0.2 second site factor given in 

section 1.5.2 and table 1-4 (note a rock site was assumed in deriving the reference 

bridge fragility curves, i.e., S = 1 for the reference curves). 

Kshape relates to the shape of the design acceleration spectrum and is defined as follows: 

s

1

shape S

S K = 2.5 (4-13) 

 S1 and Ss are the spectral accelerations defined in section 1.5.1. 
139

In equation (4-13), the factor 2.5 is the ratio between the spectral amplitude at 1.0 second and 0.2 

seconds for the standard code-based spectral shape from which the reference bridge fragility 

curves were derived. This equation is necessary to ensure all fragility curves possess a common 

format (either peak ground acceleration or spectral acceleration at T = 1.0 second). Note that for 

this case to govern, Kshape ≤1. 

Case II: When ‘long’ periods govern (T ≥ TS, figure 1-8): 

S

a A 2

2 = (4-14) 

where: 

a2 = the median spectral acceleration (at T = 1.0 sec) for damage state 2, given in tables 

4-4 and 4-5 and identified without a superscript 1 in those tables, and 

S = site factor for the long-period range; that is S = Fv, the 1.0 second site factor given in 

section 1.5.2 and table 1-4 (note a rock site was assumed in deriving the reference 

bridge fragility curves, i.e., S = 1 for the reference curves). 

Note that in equations (4-12) and (4-14) no modification is assumed for skew and 3D effects. 

The structural displacements that occur for this damage state are assumed to be small enough 

(generally less than 50 mm (2 ins)) so that the deck joints do not close and the 3D arching effect 

does not develop. 

4.3.5. ECONOMIC LOSSES 

4.3.5.1. Total Economic Losses 

Economic losses may result from the combined effects of direct losses due to structural damage 

to the bridge (BLOSS) and indirect losses resulting from a variety of causes such as loss of life, 

injuries, business disruption, traffic congestion, and denied access (HLOSS). The total monetary 

loss to the economy (TLOSS) may therefore be expressed as: 

TLOSS = BLOSS + HLOSS (4-15) 

Quantification of these losses is difficult to do at the present time, mainly because of the 

uncertainty with HLOSS. Nevertheless BLOSS can be found with some degree of confidence using 

the cost to repair or replace a damaged bridge. Such a methodology is described in the next 

section. 

4.3.5.2. Direct Economic Losses 

Direct economic losses are assumed to be due to the repair and replacement of damaged bridges 

alone and do not include other direct costs such temporary detours. Estimates of these losses may 

be obtained using the repair cost ratios listed in table 4-76

. These ratios (also known as damage 

 

6

 Basoz and Mander, 1999; Basoz and Kiremidjian, 1997 
140

ratios) express repair costs as a proportion of bridge replacement costs. The mean repair cost 

ratio for ‘collapse’ (RCR i=5 for damage state 5) may be expressed as a function of the number of 

spans. Since more than one span may collapse in this state, the assumption is made that no more 

than two spans collapse in any one event. Then RCR5 is given as below: 

but not more than1.0

n

2 RCRi=5 = (4-16) 

where n is the number of spans in the main portion of the bridge. 

Table 4-7. Modified repair cost ratio for all bridges. 

Damage State, i Range of 

Repair Cost Ratio 

Mean 

Repair Cost Ratio, RCR 

1: No damage (pre yield) 0 0 

2: Slight damage 0.01 to 0.03 0.02 

3: Moderate damage 0.02 to 0.15 0.08 

4: Extensive damage 0.10 to 0.40 0.25 

5: Collapse 0.1 to 1.0 equation (4-16) 

If the repair cost ratio is multiplied by the replacement cost of the bridge, then the expected 

direct monetary dollar loss (BLOSS) can be assessed, thus 

B U RCR LOSS T = (4-17) 

where: 

U = the replacement cost of the bridge. 

 RCRT = the total repair cost ratio that is the expected proportion of the total 

replacement cost of the entire bridge resulting from earthquake damage, or the 

direct loss probability, which is defined as follows: 

( ) [ ]

5

T i ia

i 2

RCR RCR P DS | S 1.0 == < ∑ (4-18) 

where: 

[ ] i Sa P = probability of being in damage state DS DS | i for a given spectral acceleration 

Sa at a structural period, T = 1.0 sec, using fragility curve data for this 

particular bridge and damage state, DSi, and 

 RCRi = repair cost ratio for the ith damage state. 
141

4.3.5.3. Indirect Economic Losses 

Earthquake related bridge damage may lead to indirect losses that may in many cases exceed the 

direct costs of the structural damage. These indirect costs may arise from any or all of the 

following: deaths and injuries, restricted access for emergency response and recovery, business 

disruption, loss of utility lines, and weakened security. 

Quantifying these costs is extremely difficult and cannot be done without considering each 

bridge in its functional and societal context. Risk assessment models of complete highway 

systems are under development for this and other purposes, and are a promising tool for 

developing insight into the complex relationships that govern indirect costs (Werner et al, 2000). 

4.3.5.4. Examples 

Two examples, 4.3 and 4.4, are given which illustrate the process of calculating the expected 

losses for two different bridges of the same length and soil conditions, subject to similar sized 

earthquakes. The losses are expressed first, as the expected repair cost ratio (fraction of the 

replacement cost), and then as the repair cost assuming an actual replacement cost. 

The results are summarized in table 4-8, where the greater vulnerability of the simply supported 

bridge (example 4.1) is illustrated by the higher loss ratios and repair costs for this bridge. It is 

noted that these two bridges were also evaluated by the indices method in section 4.2.1.3, where 

a similar result was obtained: the bridge rank for the simply supported bridge was found to be 

higher than for the continuous bridge (43 vs 33, respectively). 

Table 4-8. Expected losses in two example bridges subject to similar earthquakes. 

Example Bridge RCRT BLOSS

1. 

3-span, simply supported, 

prestressed, concrete bridge, total 

length: 56 m (180 ft), 

site class C 

0.224 $138,046 

2. 

3-span, continuous steel beam 

bridge, total length: 56 m (180 ft), 

site class C 

0.039 $ 33,258 

However the difference between the two bridges is more marked in this case, possibly because 

the pedestal supports under the steel beams are not explicitly considered in this method, as they 

are in the indices method. Such data is not in the NBI files and its omission is a shortcoming of 

any method based solely on NBI data. As inventories improve over time (for bridge management 

purposes perhaps), this problem will be overcome and the usefulness of the method will likewise 

improve. 
142

4.3.6. CALCULATION OF BRIDGE RANK BASED ON EXPECTED DAMAGE 

Once the expected damage is known for each bridge in the inventory and expressed as a loss 

(RCRT, or BLOSS), the inventory may be ranked in descending order of expected losses and each 

bridge assigned a rank, R. 

4.3.7. CALCULATION OF PRIORITY INDEX BASED ON EXPECTED DAMAGE 

As defined in equation 4-9, the Priority Index is based on R, the bridge rank, calculated in 

section 4.3.6, and a qualitative assessment of the indirect losses, network redundancy and nonseismic factors, such as impending plans for bridge widening or deck replacement. 
143

EXAMPLE 4.3: PRESTRESSED CONCRETE BRIDGE WITH SIMPLE SPANS – EXPECTED 

DAMAGE METHOD 

A four-beam, prestressed concrete bridge is located on very dense soil and soft rock (site class C). 

Constructed in 1968, it has three simply-supported spans, each seated on elastomeric pads directly on 

the cap beams (no pedestals); seat widths are 250 mm (9 3⁄4”). The total length of the bridge is 56 m 

(183’- 8 3⁄4”), with an overall width of 10 m (32’- 9 3⁄4”) and a skew of 32°. Piers are multi-column frames. 

Abutment fill height is 7.6 m (24’-11 1⁄4”). The bridge carries 65,000 ADT and is considered to be 

‘essential.’ The table below lists data available from the NBI for this bridge. The estimated replacement 

cost of this bridge is $616,000. 

NBI field Data Remarks 

27 1968 Year built 

34 32o Angle of skew 

43 501 Prestressed concrete, simple span 

45 3 Number of spans 

48 23 Maximum span length (m) 

49 56 Total bridge length (m) 

52 10 Bridge width (m) 

 65,000 ADT 

Using only NBI data, calculate the expected total repair cost following the upper level earthquake for this 

site. 

For the upper level event, spectral accelerations SS and S1 are taken to be 1.40 g and 0.28 g, 

respectively, for the bridge site. Corresponding values for site class C soil factors, Fa and Fv, are 1.00 and 

1.52, respectively (table 1-4). 

Since the bridge was constructed in 1968 and is located outside of California, the first row of table 4-4 

and the fifth row of table 4-6 apply, respectively. From table 4-4 for type 501: 

a 0.26g; a 0.35g; a 0.44g; a 0.65g 2345 ====

Note that the value for a2 in table 4-4 is not superscripted to indicate Case I governs; thus the ‘long-period’ 

values govern for this bridge type; therefore Kshape = 1. 

From table 4-6: =+ =+ = − − 3D

0.25 0.25 K 1 1 1.125

(n 1) (3 1)

From equation 4-10: K cos cos 32 0.92 skew = α= =

 

From equation 4-14: == = 2

2

a 0.26 A 0.17

S 1.52 g 
144

From equation 4-11: ( ) == = i

i skew 3D i i

a 0.92 (1.125) A K K a 0.681a

S 1.52

Hence: A345 === 0.24g; A 0.30g; A 0.44g

The figure presents the fragility 

curves for the prestressed concrete 

bridge with simply supported 

spans. As shown in the figure, the 

probability of being in a given 

damage state, when Sa (T = 1 sec) 

= 0.28g, is given in the table below. 

This table also presents the repair 

cost ratios from which the expected 

loss ratios are determined in terms 

of the total replacement cost for the 

entire bridge. 

Damage 

State, i 

(1) 

P[D>DSi|Sa] 

(2)

P[DSi|Sa] 

(3) 

RCRi

(4)

Product 

(3) x (4) = (5)

1 1 0.203 0.00 0.00000 

2 0.797 0.196 0.02 0.00392 

3 0.601 0.147 0.08 0.01176 

4 0.454 0.228 0.25 0.05700 

5 0.226 0.226 0.67 0.15142 

Total probabilities: 1.000 RCRT 0.22410 

Notes: 1. Column (2) is probability of being in a damage state (D) that is equal to, or 

 greater than, damage state i (DSi

), and is read from the fragility curve for this 

 bridge and damage state, for spectral acceleration = 0.28 g. 

 2. Column (3) is probability of being in damage state i, and is calculated by 

 subtracting corresponding rows in column (2), e.g., row 1, col (3) = (row 1, 

 col (2)) – (row 2, col (2)) Exception is last row, which is set equal to last row of 

 col (2). 

Column (4) is read from table 4-7. 

The resulting value of total repair cost ratio (RCRT = 0.224) is used to compute the expected repair cost in 

dollars from equation (4-17) as follows: 

 = ( ) B U RCR LOSS T , where U is the replacement cost of the bridge, estimated at $616,000. 

 

 = $616,000 (0.224) 

 = $138,046 Answer
145

EXAMPLE 4.4: STEEL GIRDER BRIDGE WITH CONTINUOUS SPANS – EXPECTED 

DAMAGE METHOD 

A four-beam, steel bridge is located on very dense soil and soft rock (site class C). Constructed in 1972, it 

has three continuous spans, each seated on steel bearings on pedestals (not rocker bearings); seat widths 

are 250 mm (9 3⁄4’). The total length of the bridge is 56 m (183’ 83⁄4”), with an overall width of 14 m (45’ 11”) 

and a skew of 18o. Abutment fill height is 6 m (19’ 81⁄4”). The bridge carries 20,000 ADT and is considered 

to be ‘standard.’ Data available from the NBI for this bridge is given in the table below. The estimated 

replacement cost of this bridge is $862,400. 

NBI field Data Remarks 

27 1972 Year built 

34 18o Angle of skew 

43 402 Steel girder, continuous 

45 3 Number of spans 

48 23 Maximum span length (m) 

49 56 Total bridge length (m) 

52 14 Bridge width (m) 

 20,000 ADT 

Using only NBI data, calculate the expected total repair costs following the upper level earthquake for this 

site. 

For the upper level event, spectral accelerations SS and S1 are taken to be 1.50 g and 0.21 g, respectively, 

for the bridge site. Corresponding values for site class C soil factors, Fa and Fv, are 1.00 and 1.59, 

respectively (table 1-4). 

Since the bridge was constructed in 1972 and is located outside of California, a non-seismic structure is 

assumed. Therefore, the fourth row of both table 4-4 and table 4-6 applies. From table 4-4 for type 402: 

a 0.76 g; a 0.76 g; a 0.76 g; a 1.04 g 2345 ====

Note that the value for a2 in table 4-5 is superscripted, implying that short period values govern for small 

displacements for this bridge type; therefore Kshape applies. 

From equation 4-13: = = =≤

1

shape

S 0.21 K 2.5 2.5 0.35 1

Ss 1.50

From equation table 4-6: =+ =+ = − 3D

0.05 0.05 K 1 1 1.025

(n 1) 2

From equation 4-10: K cos cos18 0.98 skew = α= =

From equation 4-12: === 2

2 shape

a 0.76 A K 0.35 0.27

S 1.00 g 
146

From equation 4-11: == = 3

3 skew 3D

a (1.025)(0.98)(0.76) A K K 0.48

S 1.59 g 

 == = 4

4 skew 3D

a (1.025)(0.98)(0.76) A K K 0.48

S 1.59 g 

 == = 5

5 skew 3D

a (1.025)(0.98)(1.04) A K K 0.66

S 1.59 g 

The figure presents fragility curves for the 

continuous steel girder bridge. It shows the 

probability of being in a given damage state 

as Sa (T = 1 sec) = 0.21 g for different 

damage states. These probabilities are 

used in the table to compute the expected 

loss ratio of the total replacement cost. 

Damage 

State, i 

(1) 

P[D>DSi|Sa] 

(2)

P[DSi|Sa] 

(3) 

RCRi

(4)

Product 

(3) x (4) = (5)

1 1 0.662 0.00 0.00000 

2 0.338 0.254 0.02 0.00508 

3 0.084 0.000 0.08 0.00000 

4 0.084 0.054 0.25 0.01350 

5 0.030 0.030 0.67 0.02010 

Total probabilities: 1.000 RCRT 0.03868 

Notes: 1. Column (2) is probability of being in a damage state (D) that is equal to, or greater 

 than, damage state i (DSi

), and is read from the fragility curve for this bridge and 

 damage state, for spectral acceleration = 0.21 g. 

2. Column (3) is probability of being in damage state i, and is calculated by subtracting 

corresponding rows in column (2); e.g., row 1, col (3) = (row 1, col (2)) - (row 2, col (2)) 

Exception is last row, which is set equal to last row of col (2). 

3. Column (4) is read from table 4-7.

The resulting value of total loss ratio (RCRT = 0.03868) is used to compute the expected direct loss in 

dollars from equation (4-17) as follows: 

B U RCR LOSS T = × , where U is the replacement cost of the bridge estimated at $862,400 

B $862,400 0.03868 LOSS = ×

 

 = $33,358 Answer
147 

CHAPTER 5: EVALUATION METHODS FOR EXISTING BRIDGES 

5.1. GENERAL 

5.1.1. SUMMARY OF EVALUATION METHODS 

Six evaluation methods are described in this chapter. They are listed below in increasing order of 

sophistication and rigor. Table 1-9 summarizes the key features of these methods and is repeated 

here as table 5-1 for convenience. 

Method A1/A2: Connection forces and seat width checks. Seismic demand analysis is not 

required but the capacity of connections details and seat width adequacy is checked against 

minimum values (section 5.2). The method is suitable for all single-span bridges and others in 

low hazard zones. The method is divided into two categories, A1 and A2. 

Method A1: Connection forces and seat width checks (SS < 0.10). Seismic demand 

analysis is not required but the capacity of the connections must exceed 10 percent of the 

vertical reactions at each connection, and seat width requirements are checked against 

minimum requirements. Suitable for all single-span bridges and others in low hazard 

zones. (section 5.2.1). 

Method A2: Connection forces and seat width checks (SS ≥ 0.10). Seismic demand 

analysis is not required but the capacity of the connections must exceed 25 percent of the 

vertical reactions at each connection, and seat width requirements are checked against 

minimum requirements. Pile reinforcement must also meet minimum requirements. 

Suitable for all single-span bridges and other bridges in low hazard zones. (section 5.2.2) 

Method B: Component capacity checks. Seismic demand analysis is not required, but the 

relative strength of the members and the adequacy of certain key details (including connection 

forces and seat widths) are checked against specified minima. Suitable for regular bridges in 

Seismic Retrofit Category (SRC) C, subject to restrictions on FVS1. (section 5.3). 

Method C: Component capacity/demand method. Seismic demands are determined by an elastic 

analysis such as the uniform load method, multi-mode response spectrum method, or an elastic 

time history method. The uniform load method is adequate for bridges with regular 

configurations; otherwise, the multi-mode method is used as a minimum. Capacity/demand ratios 

are calculated for all relevant components. Suitable for all bridges in Seismic Retrofit Categories 

C and D, but gives best results for bridges that behave elastically or nearly so. (section 5.4) 

Method D1: Capacity spectrum method. Seismic demands are determined by simple models 

such as the uniform load method, and capacity assessment is based on a simplified bilinear 

lateral strength curve for the complete bridge. A capacity spectrum is used to calculate the 

capacity/demand ratio for the bridge, for each limit state. Suitable for regular bridges in SRC C 

and D. (section 5.5) 
Table 5-1. Evaluation methods for existing bridges. 

APPLICABILITY 

METHOD CAPACITY ASSESSMENT 

DEMAND 

ANALYSIS SRC 

UL1 Bridge Type 

COMMENTS 

A – D All single-span bridges. A1/

A2 

Connection and Seat Width Checks 

Uses default capacity due to non-seismic loads for connections and seat widths. 

Not required 

B Bridges in low hazard zones. 

Hand method, spreadsheet useful. 

B Component Capacity Checks 

Uses default capacity due to non-seismic loads for connections, seats, columns and foundations. 

Not required C Regular bridges, but subject to 

limitations on F Sv 1. 

Hand method, spreadsheet useful. 

C 

Component Capacity/Demand Method 

Uses component capacities for connections, seat widths, column details, footings, and liquefaction susceptibility (11 items). 

Elastic Methods2: 

• ULM • MM • TH 

C & D Regular and irregular bridges that 

respond almost elastically, such as those in low-to-moderate seismic zones and those with stringent performance criteria. 

Calculates C/D ratios for individual components. This is the C:D Method of previous FHWA Highway Bridge Retrofitting Manuals. Software required for demand analysis. 

D1 Capacity Spectrum Method 

Uses bilinear representation of structure capacity for lateral load, subject to restrictions on bridge regularity. 

Elastic Methods2: 

• ULM 

C & D Regular bridges that behave as 

single-degree-of-freedom systems and have ‘rigid’ in-plane superstructures. 

Calculates C/D ratios for complete bridge, for specified limit states. Spreadsheet useful. 

D2 

Structure Capacity/Demand Method 

Uses pushover curve from detailed analysis of superstructure, individual piers and foundation limit states. 

Elastic Methods2: 

• ULM • MM • TH 

C & D Regular and irregular bridges. Calculates C/D ratios for bridge 

superstructure, individual piers, and foundations. Also known as Nonlinear Static Procedure, Pushover Method or Displacement 

Capacity Evaluation Method. Software required for demand and capacity analysis. 

E Nonlinear Dynamic Method 

Uses component capacities for connections, seat widths, columns and footings. 

Inelastic Methods2

• TH 

D Irregular complex bridges, or when 

site specific ground motions are to be used such as for bridges of major importance. 

Most rigorous method, expert skill required. Software essential. 

Notes:1. SRC = Seismic Retrofit Category for upper level earthquake; for the lower level earthquake, the recommended method of evaluation is Method C. 

 2. ULM = Uniform Load Method; MM = Multi-Mode Spectral Method; TH = Time History Method

148
149 

Method D2: Structure capacity/demand method. Seismic demands are determined by elastic 

methods, such as the multi-mode response spectrum method, or an elastic time history method. 

Capacity assessment is based on the displacement capacity of individual piers as determined by a 

‘pushover’ analysis which includes the nonlinear behavior of the inelastic components. A 

capacity spectrum is used to calculate the capacity/demand ratio for each pier, bearing and 

foundation of the bridge for each limit state. Suitable for all bridges in SRC C and D. Method is 

also known as the pushover method or alternatively the Nonlinear Static Procedure. (section 5.6) 

Method E: Nonlinear dynamic procedure (time history analysis). Seismic demands are 

determined by a nonlinear dynamic analysis using earthquake ground motion records to evaluate 

the displacement and force demands. Capacities of individual components are explicitly modeled 

in the demand analysis. Suitable for irregular complex bridges, or when site specific ground 

motions are to be used, as in the case of a bridge of major importance. (section 5.7). 

5.1.2. DEMAND ANALYSES 

Methods A and B simply check default capacities against minimum load requirements. No 

explicit demand analysis is required. Methods C, D1 and D2 are capacity/demand methods of 

varying sophistication, and Method E, which is based on inelastic time history analysis, is the 

most rigorous of all of these methods. 

As noted above, no demand analysis is necessary for regular bridges in Method B. This is 

because the design strength for nonseismic loads should be sufficient for the hazard exposure. 

For regular bridges, a single-degree-of-freedom (SDOF) model is sufficiently accurate to 

represent the seismic response, and the capacity spectrum method (Method D1) combines the 

demand and capacity evaluations into one operation. This method is also appropriate for bridges 

with seismic isolation systems. 

For bridges that do not satisfy the requirements of Method D1, an elastic response analysis, using 

either Methods C or D2, must be performed to determine the displacement and force demands in 

various members and components. Two elastic methods are presented in this chapter: the 

uniform load method and the multi-mode response spectrum method. The selection of a method 

is dependent on the configuration of the bridge. 

The uniform load method is suitable for structures with regular configurations which can be 

modeled as SDOF structures. Long bridges, or those with significant skew or horizontal 

curvature, have dynamic characteristics that may prevent modeling in this way, and multi-modal 

methods should be used instead. 

Elastic analyses use linear models which may not adequately represent the inelastic behavior of 

earthquake resisting members during strong ground motion. However, with the proper 

representation and interpretation of inelastic behavior, an elastic analysis can provide a 

reasonable estimate of seismic demand. The model should be based on cracked section 

properties for concrete components and secant stiffness coefficients for the foundations, 

abutments, and seismic isolators, that are consistent with the expected level of deformation. An 

elastic analysis should give superstructure displacements (usually at the center of mass), and the 
150 

forces in earthquake resisting members, such as the top and bottom bending moments in a 

column. 

The nonlinear dynamic analysis method (Method E) is a poweful analytical tool because it is not 

restricted by geometry or nonlinearities in material behavior. The effect of inelastic behavior is 

included explicitly in the demand analysis. Depending on the mathematical model used, the 

deformation capacity of the inelastic members may also be included in the analysis. However, 

the data required to peform such an analysis is extensive and the corresponding mathematical 

model for the bridge is complex and time consuming to develop. Furthermore, a nonlinear 

dynamic analysis requires the selection of a set of time histories of ground motion that represents 

the local hazard and site conditions. For many bridge sites, this will be a significant effort 

requiring the advice of experts in the field. Because of the complexity involved in a nonlinear 

dynamic analysis, results should always be checked for reasonableness against those from other, 

but less rigorous, methods. 

 

Nonlinear dynamic analysis should be used for bridges with earthquake protective systems 

(isolators and/or energy dissipators) that have long periods (> 3 sec) and/or high damping ratios 

(> 30 percent). This is because procedures that use effective stiffness and damping may not 

properly represent the effect of these properties when they reach such high values. In such cases, 

models for the isolators and dissipators should use explicit hysteretic properties and not 

equivalent linear values. 

Regardless of the method of demand analysis used, it is essential that seat widths at abutments, 

piers, and in-span hinges be checked. 

Note that the methods described below should be used in conjunction with the guidelines for 

bridge modelling and analysis given in chapters 6 and 7. 

5.2. METHOD A: CONNECTION DETAILS AND SEAT WIDTH CHECKS ONLY 

In areas of low seismicity, minimum seat widths and connection forces are usually adequate to 

ensure life safety. In such cases, design values are used as minimum requirements in lieu of 

rigorous analysis. For the operational objective, a check on the minimum shear reinforcement in 

concrete piles is also recommended. This same check is also recommended for the life safety 

objective when the Hazard Level is II and higher. To provide for these exceptions, Method A is 

divided into two categories, A1 and A2, as described below. 

5.2.1. METHOD A1 

In Method A1, the horizontal forces used to check the capacity of connections in their restrained 

directions, should not be less than 10 percent of the vertical reaction at that connection due to the 

tributary dead load. These loads are defined as follows: 

• In the longitudinal direction, for each uninterrupted (continuous) segment of a superstructure, 

including simply supported spans, the tributary dead load at the fixed bearings is the total 
151 

dead load of the entire segment.The longitudinal capacity required at the expansion bearings 

in this segment is assumed to be zero. 

• In the transverse direction, for each uninterrupted (continuous) segment of a superstructure, 

including simply supported spans, the tributary dead load is the dead load reaction at each 

bearing. 

The connection detail between an elastomeric bearing and its masonry and sole plates should be 

able to resist the horizontal forces transmitted through the bearing. For all bridges evaluated 

with Method A1, and all single-span bridges, these shear forces should not be less than the 

connection force described above. 

Seat widths should also be checked against the minimum requirements of equation 5-1 below 

(see also figure D-1). 

α

+

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎠

⎞ ⎜

⎝

⎛ = + + + +

cos

(1 1.25F S )

L

B N 100 1.7L 7.0H 50 H 1 2 v 1

2

 (5-1a) 

where N is the minimum seat width (mm), L is the distance between joints (m), H is the tallest 

pier between the joints (m), and B is the width of the superstructure (m). 

Or, in U.S. customary units: 

α

+

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎟

⎠

⎞ ⎜

⎝

⎛ = + + + +

cos

(1 1.25F S )

L

B N 4.0 0.02L 0.08H 1.1 H 1 2 v 1

2

 (5-1b) 

where N is the minimum seat width (in), L is the distance between joints (ft), H is the tallest pier 

between the joints (ft), and B is the width of the superstructure (ft). 

In both equations, α is the angle of skew (zero for a right bridge). The ratio of B/L need not be 

taken greater than 3/8. 

5.2.2. METHOD A2 

In Method A2, horizontal forces used to check the capacity of connections in their restrained 

directions, should not be less than 25 percent of the vertical reaction at that connection due to the 

tributary dead load. These loads are defined in Method A1. 

The connection detail between an elastomeric bearing and its masonry and sole plates should be 

able to resist the horizontal forces transmitted through the bearing. For all bridges evaluated 

with Method A2, and all single-span bridges, these shear forces should not be less than the 

connection force described above. 

Minimum seat widths should be checked against equation 5-1. 
152 

5.3. METHOD B: COMPONENT CAPACITY CHECKS 

A seismic demand analysis is not required for bridges meeting the requirements of Method B, 

but capacity protection principles1

 and minimum detailing requirements must be satisfied. 

Method B permits the rapid evaluation of bridges complying with certain restrictions, without 

the need for full dynamic analysis. Each bridge is evaluated for non-seismic requirements and 

capacity protection requirements, then checked to determine the adequacy of certain details such 

as shear and confining reinforcement. Capacity protection principles are also used to check the 

adequacy of connection details between columns and footings; and between columns, pier caps 

and superstructure. 

There are no evaluation requirements for abutments except that integral abutments need to be 

evaluated for passive pressure. This method will be mainly used in areas of low-to-moderate 

seismic hazard (SHL I and II) where superstructure displacements in the longtitudinal direction 

are expected to be small. As a result, abutments are not expected to contribute to bridge response 

in any significant way, and may be omitted from further evaluation (in these low seismic zones). 

Earthquake loads for foundation design are determined from the column forces using an 

overstrength ratio of 1.0. 

5.3.1. PROCEDURE FOR METHOD B 

Step 1. Check section 5.3.2 for restrictions on structural and site characteristics to determine if 

Method B is applicable. The hazard at the site must not exceed a limitation on FvS1, 

and the structure must meet certain geometric regularity requirements. 

Step 2. Check all connection and seat width requirements as for Method A2. 

 

Step 3. Reinforced concrete columns should be evaluated using non-seismic loading cases and 

checked for minimum longitudinal reinforcement of 0.8 percent. 

Step 4. Reinforced concrete columns should be checked to see whether the reinforcing details 

are adequate for column shear and confinement (check against minimum requirements 

specified in AASHTO, 2002 or AASHTO, 1998). 

Step 5. Steel columns should be evaluated using non-seismic loading cases and should be 

checked for compactness. 

Step 6. Members connecting to columns should be evaluated for their ability to resist the 

moments and shears caused by plastic hinging in the columns, using the principles of 

capacity protection, with an overstrength ratio of 1.4. See sections 7.6 and 7.7. 

 

1

 Capacity protected design means that certain components and/or members of a bridge are protected from excessive 

forces during an earthquake by the yielding of components and/or members elsewhere in the bridge. For example, 

the maximum shear that can be transmitted to a footing by a column is determined by the flexural capacity of the 

column, i.e., the column’s yield strength Thus, the footing is protected by the capacity of the column. 
153 

Step 7. Foundations should be evaluated for their ability to resist the moments and shears 

caused by plastic hinging in the columns, using the principle of capacity protection, 

with an overstrength ratio of 1.0. See sections 7.6 and 7.7. 

5.3.2. RESTRICTIONS ON USE OF METHOD B 

Method B should be used only at sites where: 

 FvS1 < 0.3 cos α (5-2) 

where α is the skew angle of the bridge (zero for a right bridge). 

Additionally, Method B should be used only on structures that comply with the following 

restrictions (notation is defined below): 

• For bridges with concrete column and pile bents: 

Pe < 0.15 f'cAg

ρt > 0.008 

D > 300 mm (12 in) 

2 < M/VD < 7 

• For bridges with wall piers with low percentages of longitudinal steel: 

Pe < 0.07 f'cAg

ρt > 0.0025 

M/VT < 7 

T > 300 mm (12 in) 

• For bridges with steel pile bents framing into reinforced concrete caps: 

Pe < 0.15 Py

Dp > 250 mm (10 in) 

M/VB < 7 

• For bridges with timber piles framing into reinforced concrete caps: 

Pe < 0.1 Pc

Dp > 250 mm (10 in) 

Notation used above is defined as follows: 

Pe = axial load on the bridge column including both gravity and seismic effects, 

Pc = axial capacity of a steel column or timber pile member in compression, 

Py = axial capacity at yield of a steel column/pile member, 

ρt = total area ratio of longitudinal reinforcement, 

f'c = compressive strength of concrete, 
154 

Ag = gross area of the column section, 

D = smallest column dimension or diameter, 

Dp = pile dimension about the weak axis of bending, 

T = wall thickness or smallest cross-sectional dimension, 

B = flange width of a steel H-pile, and 

M/V = shear span length of an equivalent cantilever member (M is the end moment and V is 

the shear force). 

Note that structures with lower axial loads or stronger columns (i.e., more reinforcement and larger 

column or pile sizes) have greater intrinsic strength. Thus, they are able to resist ground motions 

with less damage. However, the ductility of the details still needs to be checked. 

Other restrictions on the use of Method B include the following: 

• Stiffness of individual piers should not vary by more than a factor of two with respect to the 

average stiffness of all piers in the bridge. 

• Maximum span length should not exceed 80 m (260 ft). 

• Longest individual span should not exceed 150 percent of the average span length. 

• Maximum skew angle should not exceed 30 degrees. 

• For horizontally curved bridges, the subtended angle should not exceed 30 degrees. 

• Columns must resist at least 80 percent of the horizontal load generated in the longitudinal 

direction by the tributary area of each column. 

• Method B should not be used if the bridge site has a potential for liquefaction and the piers 

are seated on spread footings. 

• Method B should not be used if the bridge site has a potential for liquefaction and the piers 

are on pile foundations, unless the piles possess ductile details over the length passing 

through the liquefiable soil layer plus an additional length of three pile diameters or 3 m (10 

ft), whichever is larger, above and below this layer. 

5.4. METHOD C: COMPONENT CAPACITY/DEMAND METHOD 

5.4.1. APPROACH 

Method C calculates capacity/demand ratios for bridge components that may be damaged during 

an earthquake. Ratios greater than one indicate sufficient capacity to resist the earthquake 

demand; ratios less than one indicate components in need of attention and possible retrofitting. 

Capacity/demand ratios are therefore used to indicate the need for retrofitting and may also be 

used to assess the effectiveness of various retrofit strategies. 
155 

One feature that distinguishes this method from Method D1, is that the demand is based on the 

elastic response of the structure calculated by either the uniform load method or a spectral modal 

analysis. Another difference is that this method focuses on individual component behavior rather 

than the response of a bridge as a complete structure. In this way, it gives a detailed view of the 

potential deficiencies of a bridge, but may overestimate the overall vulnerability of a bridge and 

imply a greater need for retrofitting than is actually necessary. This is because the method 

ignores ‘system’ response and the ability of a bridge, acting as a system, to redistribute loads 

from one member to another. The error here is small if the bridge responds elastically or nearly 

so. Method C gives conservative results and the degree of conservatism generally increases with 

the extent of plastic hinging in the bridge. If the indicated retrofit needs are high, it may be wise 

to use one of the more refined methods (D or E) to reassess the situation before committing to 

design and construction. 

Components that should be evaluated will vary with the Seismic Retrofit Category of the bridge. 

Table 5-2 indicates components and failure modes that should be checked. For some bridge 

types, failure of certain components will not result in unacceptable damage, and capacity/demand 

(C/D) ratios for these components need not be calculated. For other bridge types, components, 

other than those listed, should be examined if their failure will result in unacceptable 

performance. 

Table 5-2. Components for which capacity/demand ratios are required. 

Seismic Retrofit Category 

Component 

B C and D 

EXPANSION JOINTS AND BEARINGS 

Support Length x x 

Connection Forces x x 

REINFORCED CONCRETE COLUMNS WALLS AND FOOTINGS 

Anchorage x 

Splices x 

Shear x 

Confinement x 

Footing Rotation x 

ABUTMENTS 

Displacement x 

LIQUEFACTION

Lateral Spread x x 
156 

In addition to calculating elastic demands by spectral methods, some minimum requirements can 

be treated as demands. For example, minimum bearing connection forces and minimum support 

length requirements are useful indicators of demand when calculating C/D ratios for bearing 

forces and superstructure displacements 

Seismic capacities are calculated at their nominal ultimate values without the use of capacity 

reduction factors, φ. In cases such as well-detailed reinforced concrete columns, where postelastic behavior is acceptable, C/D ratios are modified by ductility indicators (see appendix D) to 

reflect the capacity of the column to withstand plastic deformation. 

In general, the ability of a bridge to meet seismic demands will be determined by one of the 

following: 

• Displacements at supports or intermediate hinges that result in a loss of support and collapse 

of one or more spans. 

• Ultimate strength of fixed bearings and their anchorages. 

• Ductile capacity of columns, piers, and foundations beyond which an unacceptable 

degradation in strength occurs. 

• Abutment displacements which make the bridge inaccessible after an earthquake. 

• Foundation movements which are excessive and will result in a collapse of the structure or 

loss of access to the bridge. 

The basic equation for determining the C/D ratio, r, for a particular component is: 

(5-3) 

where: 

RC = nominal ultimate displacement or force capacity of the structural component 

being evaluated, 

ΣQNSi = sum of the displacement or force demands on a component from nonseismic 

loads, which are included in the group loading defined by equations 6-1, 6-2, 

7-1 and 7-2 of the AASHTO Standard Specifications (AASHTO, 2002), or 

table 3.4.1-1 of the AASHTO LRFD Specifications (AASHTO, 1998), and 

QEQ = displacement or force demand for the earthquake loading under consideration. 

C/D ratios should be calculated at the nominal ultimate capacity without the use of capacity 

reduction factors, φ, to account for possible understrength and/or undersize members. This is 

done because the objective of a C/D ratio is to determine the most likely level of failure. 

EQ

NSi C

Q

R Q r −Σ =
157 

Since C/D ratios reflect only component failures and not necessarily the state of the bridge as a 

whole, the global effect of one or more component failures must be assessed in a qualitative way 

using engineering judgment. 

A methodology for calculating component C/D ratios is given in appendix D. 

5.4.2. SELECTION OF ELASTIC ANALYSIS METHOD 

The uniform load method may be used for structures satisfying geometric ‘regularity’ 

requirements of table 5-3. For structures not satisfying these requirements, the multi-mode 

method of dynamic analysis should be used, or alternatively an elastic time history method. 

Table 5-3. Restrictions on the application of the uniform load method. 

Parameter Value 

Number of spans 2 3 4 5 6 

Maximum subtended angle for a curved bridge 20° 20° 30° 30° 30°

Maximum span length ratio from span to span 3 2 2 1.5 1.5 

Maximum pier stiffness ratio from span-to-span, 

excluding abutments –– 4 4 3 2 

5.4.2.1. Uniform Load Method 

The uniform load method is described in Article 4.3 of the Standard Specifications (AASHTO, 

2002) and Article 4.7.4.3.2c of the LRFD Specifications (AASHTO, 1998). It is based on the 

fundamental mode of vibration in the longitudinal or transverse direction, assuming an 

equivalent single mass-spring oscillator. The stiffness of this equivalent spring is calculated 

using the maximum displacement that occurs when an arbitrary uniform lateral load is applied to 

the bridge. The spectral acceleration, at the modal period T, is found from figure 1-8 and used to 

calculate the equivalent, uniform load from which design forces are determined. 

This method may be used for both transverse and longitudinal earthquake motions. It is 

essentially an equivalent static method of analysis that uses a uniform lateral load to approximate 

the effect of seismic loads. The method is suitable for regular bridges that respond principally in 

their fundamental mode of vibration. 

While all displacements and most member forces are calculated with satisfactory accuracy, the 

method is known to overestimate the transverse shears at the abutments by up to 100 percent. If 

such conservatism is undesirable, but a single mode representation is appropriate, then the single 

mode spectral analysis method (Article 4.4 of the Standard Specifications (AASHTO, 2002) and 

Article 4.7.4.3.2b of the LRFD Specifications (AASHTO, 1998)) is recommended. This method 

is a subset of the multi-mode spectral analysis method described in the next section. 
158 

The steps in the uniform load method are as follows: 

Step 1. Calculate the static displacements vs(x) due to an assumed uniform load, po. The 

uniform loading po is applied over the length of the bridge; it has the dimensions of 

force/unit length, and may be arbitrarily set equal to 1.0. The static displacement vs(x) 

has the dimension of length. 

Step 2. Calculate the lateral stiffness of the bridge, K, and total weight, W, from the following 

expressions: 

s , MAX

0

V

p L K = (5-4) 

∫ =

L

0

W w(x)dx (5-5) 

where L is the total length of the bridge, Vs,MAX is the maximum value of vs(x), and 

w(x) is the unfactored dead load of the bridge superstructure and tributary substructure. 

The weight should take into account structural elements and other relevant components 

including, but not limited to, pier caps, abutments, columns, and footings. Other loads, 

such as live loads, may also be included. 

Step 3. Calculate the period of the bridge, Tm, using the expression: 

gK

W T 2 m = π (5-6) 

where g is the acceleration due to gravity. 

Step 4. Calculate the equivalent static earthquake loading, pe, from the expression: 

s e

C W

p L = (5-7) 

where Cs = Sa/g; g is the acceleration due to gravity; Sa is defined in figure 1-8 and 

equations 2-5, 2-6 and 2-8; and pe is the equivalent uniform static seismic loading per 

unit length of bridge applied to represent the primary mode of vibration. 

Step 5. Calculate the displacements and member forces for use in evaluation either by applying 

pe to the structure and performing a second static analysis, or by scaling the results of 

Step 1 by the ratio pe/po. 
159 

5.4.2.2. Multi-Mode Spectral Analysis Method 

The elastic multi-mode spectral analysis method should be used for bridges in which coupling 

occurs in more than one of the three coordinate directions within each mode of vibration. As a 

minimum, linear dynamic analysis, using a three-dimensional model to represent the bridge, 

should be used. This method is described in Article 4.5 of the Standard Specifications

(AASHTO, 2002) and Article 4.7.4.3.3 of the LRFD Specifications (AASHTO, 1998). 

 

Ideally, all of the modes of vibration of a bridge will be included in a modal analysis but this 

may lead to excessive, and often unnecessary, computational effort. Instead, a reduced number is 

frequently used to reduce the effort while still maintaining accuracy. This number is usually 

determined by the relative size of the modal participation factors, which most dynamic analysis 

programs compute as contributions to the total base shear and express them as percentages of 

total bridge mass. For regular bridges, the total number of modes should be such as to include at 

least 90 percent of the modal mass. For irregular bridges, or large multi-segment bridges, it may 

be necessary to raise this figure to 95 percent, to ensure accurate results in all of the critical 

members and components. 

The elastic response spectrum shown in figure 1-8 should be used for each mode, and should be 

scaled for damping ratios other than five percent. To scale the five percent spectrum for a 

damping ratio of ξ percent, multiply the spectral ordinates by (0.2ξ)

0.3 for periods greater than 

TS, and by (0.2ξ)

0.5 for periods less than or equal to TS, where TS is defined as D1

DS

S

S in figure 

1-8. ξ should not be taken as greater than 30 percent. If a bridge is to be retrofitted using seismic 

isolation, scaling of the spectrum should only be done for periods greater than 80 percent of the 

effective isolated period. 

Member forces and displacements may be estimated by combining the respective response 

quantities (moment, force, displacement, or relative displacement) from the individual modes by 

the Complete Quadratic Combination (CQC) method2

. Forces and displacements obtained using 

the CQC combination method are adequate for most bridge systems, especially if there is only 

one component to be considered in the ground motion. If the CQC method is not used, the 

square-root-of-the-sum-of-the-squares method (SRSS) may be used when the modal periods are 

well-separated. For modes that have closely spaced periods, the absolute sum of the modal 

responses may be used as an alternative. 

Member forces and displacements due to two or three simultaneous components of ground 

motion should be estimated by the SRSS method. This method assumes these components are 

independent of each other (i.e., they are uncorrelated), which is an adequate assumption when 

evaluating a bridge because the spectrum in figure 1-8 is intended to represent the principal 

directions of ground motion. This assumption may not be valid for near-fault ground motions, 

which can exhibit strong correlation between the horizontal components. 

 

2

 Newmark and Rosenblueth (1979); Der Kiureghan (1981) 
160 

5.4.2.3. Elastic Time History Method 

Elastic time history methods provide displacements and member actions as a function of time, 

assuming all members remain elastic and no displacement limit is exceeded. These methods are 

more rigorous than the uniform load and multi-mode spectral methods described above, and may 

be used for irregular bridges with complex geometries and/or on poor foundations. They do, 

however, require the development of at least three sets of acceleration time histories for the 

bridge site and a minimum of three analyses is therefore performed, one for each set. Each set 

includes three components of the ground motion (two horizontal and one vertical). The 

maximum response for any single quantity (such as the transverse bending moment at the top of 

a particular column) from these three analyses, should be used for design. If more than seven sets 

of ground motions are used, the mean of the responses may be taken for evaluating the demand 

on the bridge. 

In the absence of site-specific ground motions, time histories may be synthetically generated 

using the response spectrum for the site (figure 1-8). Procedures for developing these so-called 

spectrum-compatible time histories are discussed in section 2.8. 

5.4.3. PROCEDURE FOR METHOD C 

Step 1. Decide whether the restrictions on the use of this method given in section 5.4.4 are 

satisfied. If so, an elastic force-based response spectrum analysis may be used. Based 

on the restrictions described below, determine the type of modal analysis to be 

undertaken. 

Step 2. Determine the capacity, Qci , for each of the relevant members in the structure. 

Step 3. Determine the sum of the non-seismic force and displacement demands, ΣQNSi, for each 

of the members in the structure, for each load combination in equations 6-1, 6-2, 7-1 

and 7-2 of the Standard Specifications (AASHTO, 2002), or table 3.4.1-1 of the LRFD 

Specifications (AASHTO, 1998). 

Step 4. Determine the response spectrum parameters Fa, Ss, Fv and S1 (section 2.5). Perform an 

elastic dynamic analysis (section 5.4.2) to determine the seismic demand, QEQi, on each 

of the members. The analysis should reflect the anticipated condition of the structure 

and the foundation during this earthquake. 

Step 5. For each member or component (i), determine the capacity/demand ratio from: 

EQi

ci NSi

i Q

Q Q r − ∑ = (5-8)

If ri > 1.0, the corresponding member has adequate capacity for the level of demand. 

Otherwise, devise retrofit measures that increase the displacement, strength, or ductility 

capacity, of the specific member or component. 
161 

Alternatively, reanalyze the bridge using a more rigorous approach where system 

behavior is accounted for explicitly and the benefical effect of force redistribution due 

to inelastic action is included in the results. See Methods D1, D2 and E. 

5.4.4. RESTRICTIONS ON USE OF METHOD C 

This method is the same as the capacity/demand method described in the previous FHWA bridge 

retrofit manual (FHWA, 1995). As before, its use should be restricted to bridges that will behave 

in a ‘mostly elastic’ fashion, to ensure that the results are not overly conservative. This means 

that the method is mainly applicable to bridges in regions of low-to-moderate seismicity. It is 

also applicable to major structures in highly seismic zones, where a high level of operational 

performance is required which can only be satisfied by an ‘almost elastic’ design strategy. In 

such cases, the method of elastic analysis used for computing the C/D ratios should be the time 

history method described in section 5.4.2.3. 

5.4.5. EXAMPLES 

Two examples of the use of Method C are given in appendices E and F. 

The first example is a four-span reinforced concrete, box-girder bridge similar to those used in 

California at the time of the San Fernando earthquake (1971). It has seat width deficiencies, and 

inadequate reinforcement details in the columns. This example is given in appendix E. 

The second example is a multispan steel beam bridge with simple supports, constructed in 

Pennsylvania in 1968. It has vulnerable steel rocker bearings and concrete columns with 

potentially inadequate reinforcement details. This example is given in appendix F. 

5.5. METHOD D1: CAPACITY SPECTRUM METHOD 

5.5.1. APPROACH 

 

This method of evaluation explicitly includes the inelastic behavior of members and other limit 

states due to bearing failure and unseated beams. It is a powerful technique that can be used to 

make a quick estimate of either the capacity of an existing bridge or the performance of a bridge 

during a given earthquake. The method may be used for either the design of a new bridge or the 

retrofit of an existing structure. 

The method is restricted to bridges that vibrate as single-degree-of-freedom (SDOF) systems, 

i.e., to bridges with regular geometry and uniform distribution of weight and stiffness. It is also 

limited to bridges where the displacements at the tops of all the piers are the same, or nearly so, 

in both the longitudinal and transverse directions. This does not include piers with expansion 

bearings. 

These limitations are described further in section 5.5.5. 
162 

A bridge that cannot be modeled as an SDOF structure or does not satisfy the equal 

displacement requirement, should be evaluated using a multi-modal method and a more 

advanced capacity assessment technique such as that described in Method D2. 

5.5.2. BRIDGE CAPACITY 

5.5.2.1. General 

The capacity of a bridge to resist lateral loads may be expressed by a so-called pushover curve. 

Such curves show the total lateral load acting on the bridge plotted against the deflection of the 

center of mass of the bridge. This point is chosen because the center of mass is where the 

earthquake’s inertial loads are assumed to act (in SDOF structures). In most highway bridges, the 

center of mass will be located within the superstructure. This curve is also a measure of the 

capacity of the bridge for lateral loads of increasing size and may be used to express force and 

displacement capacity for a number of potential limit states. 

Figure 5-1(a) shows an idealized pushover curve for a flexurally ductile structure. It identifies 

several limit states that are important in characterizing the behavior of the bridge under 

increasing load or deformation. These are: 

1. Pseudo-yield point, marking the end of essentially elastic behavior. 

2. Point of maximum plastic deformation before softening (degradation) begins. 

3. Onset of collapse (e.g., due to column rebar rupture in low cycle fatigue or P-∆ effects). 

4. Collapse (e.g., due to the failure of a plastic hinge). 

These limit states are characteristic of bridges with continuous superstructures on ductile 

columns that are capable of large inelastic deformation in well-detailed plastic hinges. 

Bridges with simple spans may exhibit a different set of limit states and these are illustrated in 

figure 5-1(b). They are: 

1. Pseudo-yield point, marking the end of essentially elastic behavior. 

2. Expansion joint closure, followed by span lock-up, and limited plastic deformation. 

3. Bearing failure (e.g., due to weld failures in keeper bars). 

4. Collapse (e.g., due to the unseating of beams). 

These limit states essentially describe brittle behavior and are likely to be found in older bridges 

with simple spans despite adequate column reinforcement details, particularly in regions of lowto-moderate seismicity. 
163 

(a) “Ductile” structure (b) “Brittle” structure 

Figure 5-1. Capacity curve. 

5.5.2.2. Calculation of Bridge Capacity Curve 

In this method, the capacity curves shown in figure 5-1 are approximated by bilinear curves as in 

figure 5-2. Using the notation in these figures, a seismic capacity coefficient Cc is calculated at 

displacement ∆, as follows: 

c

F C

W= (5-9) 

where: 

 F = total horizontal force acting on the bridge 

 = Fy + k2 (∆ – ∆y) for ∆ > ∆y, and 

 = k1 ∆ for ∆ < ∆y. 

 W = weight of seismic mass, usually taken as the weight of the superstructure, 

 Fy = yield force (see note 1 below), 

∆y = yield displacement corresponding to Fy and equals Fy / k1,

 k1 = elastic stiffness in direction considered, transverse or longitudinal (see note 2 

 below), and 

 k2 = equivalent post-yield stiffness in direction considered, transverse or longitudinal 

 (see note 3 below). 

The maximum displacement ∆max, shown in figure 5-1, is set to the lesser of the following three 

displacement limit states, as appropriate: 

1. Plastic hinge rotation: ∆max < θp H 

 where θp = 0.035 for reinforced concrete columns, and H is the clear height of the column. 
164 

(a) “Ductile” structure (b) “Brittle” structure 

Figure 5-2. Idealized capacity. 

2. P-delta (P-∆): ∆max 0.25 Cc W’ (H/P) 

 where W’ is the seismic weight per column, and P is the axial load on the column due to 

gravity loads. See explanation in note 4 below and also a refinement for ∆max for short period 

bridges. 

3. Seat length: ∆max < N0 

 where N0 is the existing seat width at an abutment or pier cap. See note 5 below. 

Note 1. Fy is calculated from the sum of the individual column lateral strengths (Vui) in the 

direction under consideration, 

i.e., Fy = Σ Vui = Σ n

i

M

H

⎛ ⎞ ⎜ ⎟ ⎝ ⎠ (5-10) 

where Mn is the column’s nominal3

 yield moment calculated from a moment interaction curve 

for column i, using column axial loads, dimensions and reinforcement details, and H is the clear 

height of column i. 

This summation is made over all of the columns supporting the superstructure; abutments are 

excluded. Columns with expansion bearings in the direction being analyzed are also excluded 

from this calculation. Since the axial loads are not known at this stage, these loads may be taken 

equal to the gravity load values when calculating Mn. 

Note 2. k1 is the elastic stiffness of the bridge in a lateral direction and will generally be different 

in different directions (e.g., longitudinal and transverse) unless symmetrical, single-column piers 

are used. Cracked sections should be assumed for reinforced concrete columns and a moment of 

inertia equal to 50 percent of the uncracked moment of inertia is recommended. The uniform 

load method may be used to calculate this stiffness (equation 5-4, section 5.4.2.1). 

 

3

 See definition of nominal strength in section 7.5 
165 

Note 3. k2 is the linearized post-yield stiffness used to approximate actual behavior in this part of 

the capacity curve. In the absence of rigorous analysis, k2 may be taken equal to five percent of 

the elastic stiffness, i.e., k2 = 0.05 k1. If elastic-perfectly plastic behavior is assumed to occur in 

the column hinge, k2 = 0. This assumption underestimates the capacity of the column once 

yielding begins and could be on the conservative side. But it will also underestimate the forces 

that the column can transmit to adjacent components (e.g., bearings above and footings below) 

which may lead to their unexpected failure. Setting k2 = 0 is therefore not recommended. See the 

discussion of capacity-protected design in chapter 1. 

Note 4. Bridges supported on slender piers that carry high axial loads are susceptible to 

instability due to so-called P-∆ effects. Inadequate strength can cause progessive ratcheting of 

a bridge sideways, eventually leading to collapse. Pier equilibrium equations show that P-∆ 

effects reduce the lateral stiffness of columns and may even cause this stiffness to become 

negative once yield occurs. Under these conditions, the post-yield stiffnesss is given by 

 '

2 2

P k k

H = − (5-11) 

 2

P if k 0

H

− = =

where k2’ is the post yield stiffness including P-∆ effects, k2 is the post-yield stiffness excluding 

P-∆ effects (see Note 3 above), P is the axial load due to non-seismic sources, and H is the clear 

height of pier from point of fixity of piles, if any. 

This decrease in stiffness leads to a reduction in strength with increasing displacement. The 

above displacement limit state is therefore chosen to limit this reduction to 25 percent of the 

capacity at zero displacement (VU) , 

 i.e., 

 max U

P 0.25V

H

∆ ≤ (5-12) 

Since the lateral strength (VU) can be expressed as the product of the seismic coefficient (Cc ) 

and the effective seismic weight on the pier (W’), the above equation can be rearranged to give: 

'

max C

W 0.25C H

P

⎛ ⎞

∆ ≤ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

 (5-13) 

Note that the ratio W’/P should not be taken as greater than 2.0 for two-span bridges, or greater 

than 1.0 for other bridges. Note also that for bridges with periods less than 1.25 TS (TS is defined 

in figure 1-8), ∆max is reduced by the factor Rd to account for the possible underestimation of 

displacements by the equal-displacement theory of nonlinear response. For such bridges, the 

displacement limit state (∆’max) is given by: 
166 

' max max Rd

∆ ∆ = (5-14) 

where: 

R

1

T

T

R

1 R 1

*

d ⎟ + ⎠

⎞ ⎜

⎝

⎛ = − for R ≥ 1 and T < T*, 

 Rd = 1.0 for R > 1 and T > T*, 

 Rd = 1.0 for R < 1 and any value of T, 

 T* = 1.25 TS, 

 TS = FvS1/FaSS, as defined in figure 1-8, 

 T = period of vibration of bridge, and 

 R = ratio of elastic force on pier (Fel) to the lateral capacity of pier(VU) 

 = Fel / VU. 

Note 5. Ideally, the existing seat width (N0) will be the greater of: 

• 1.5 ∆'max where ∆'max is the displacement of the superstructure at the seat, given by equation 

(5-14), or 

• The minimum seat width given by equations 5-1a or 5-1b. 

In retrofitting some bridges, satisfying this requirement may be very costly. 

5.5.3. EARTHQUAKE DEMAND 

The earthquake demand on a bridge may be represented by a response spectrum. Both 

acceleration and displacement spectra are used, but by far the most common is the acceleration 

spectrum. These spectra, when scaled by seismic mass, give the seismic forces acting through the 

center of mass of the bridge. Figure 1-8 shows the recommended acceleration spectrum for both 

seismic design and retrofit of highway bridges in the United States. This spectrum assumes five 

percent viscous damping in the bridge and should be modified for other damping values, as 

shown in figure 5-3. A value of five percent is appropriate for essentially elastic behavior but 

once yielding occurs, and other forms of damage begin to occur, the damping level increases. 

Two damping factors are introduced for this purpose, BS and BL, for use in the short and long 

period ranges of the spectrum respectively, as shown in figure 5-3. A procedure for calculating 

BS and BL is given in table 5-4, which shows that both factors depend on the displacement 

ductility factor μ, defined as follows:

y

∆

μ = ∆ (5−15) 

where ∆ is the displacement at which ductlity is being calculated, and ∆y is the yield 

displacement. 
167 

Figure 5-3. Demand spectrum. 

A seismic demand coefficient, Cd, is defined as follows: 

a

d

S C

g = (5-16) 

where: 

 Sa = spectral acceleration defined in figure 1-8, 

 = g [FvS1/BLT ] for long period bridges, T > TS, and (5-17a) 

 = g [FaSS/BS ] for short period bridges, T < TS. (5-17b) 

This is the traditional form of a demand spectrum (Sa vs period, T), but it is also convenient to 

express Cd in terms of Sd (spectral displacement) rather than period. To do so, Sd is first written 

as follows: 

 Sd = spectral displacement 

 = Sa / ω2

 

 = Sa [T2

/4π2]

 

= [FvS1/BL] Tg/4π2 (using equation 5-17a) (5-18) 

where ω is the angular frequency (rad/sec) and equals 2π/T. 

Combining equations 5-16, 5-17a and 5-18, to eliminate the period T, gives Cd for long period 

bridges as: 

2

v 1

d S

d L

g F S C for T T

S 2B

⎡ ⎤⎡ ⎤ = ≥ ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦ π

 (5-19a) 

Figure 5-3 shows Cd plotted against Sd for a particular value of damping factor BL. This spectrum 

is seen to have a shape similar to that of the traditional, period-based spectrum and that Cd

decreases as Sd increases. 
168 

Further, by combining equations 5-16 and 5-17b, Cd for short period bridges can be shown to be 

given by: 

a S

d S

S

F S C for T T

B

= < (5-19b) 

Table 5-4. Effective viscous damping ratios and damping factors, BS and BL. 

Substructure Type Effective Viscous 

Damping Ratio, ξeff 

Damping 

Factor, BS

Damping 

Factor, BL

Nonductile, conventionally-designed 

columns 0.05 + 0.16(1-1/μ)

1

Ductile, sesimically-designed columns 0.05 + 0.24(1-1/μ)

1

Sliding bearings 0.20 

[ξeff/0.05]0.5 [ξeff/0.05]0.3

Note: 1. μ = displacement ductility factor

5.5.4. CAPACITY/DEMAND SPECTRUM 

It is possible to combine the capacity curve represented by equation 5-9 (and figure 5-2) and the 

demand spectrum represented by equation 5-19 (and figure 5-3) in a single plot. The result is 

known as a capacity/demand spectrum. There are many possible uses for such a plot, one of 

which provides capacity/demand ratios for a complete bridge subject to a given earthquake, and 

the other calculates bridge response (F, ∆) to a given earthquake. Both applications are described 

below. A step-by-step procedure is given in a subsequent section for the combination of both 

applications. This is Method D1. 

5.5.4.1. Calculation of Bridge Capacity/Demand (C/D) Ratios 

 

Figure 5-4 shows a capacity/demand spectrum in which three limit states are identified on the 

capacity curve. The displacements corresponding to these limit states are known and for the 

purpose of illustration, might be as follows: 

• ∆LS1 = ∆y (yield displacement). 

• ∆LS2 = ∆θp (displacement corresponding to a given plastic hinge rotation = θp H = 0.02H for 

θp = 0.02; values of θp are given in sections 7.8.1 and 7.8.2 for a range of different column 

limit states). 

• ∆LS3 = N0 (available seat width at abutment or pier cap). 

Corresponding to each of these limit states there is capacity coefficient CcLS. 
169 

Figure 5-4. Capacity/demand spectra. 

Also shown in the figure is a demand curve passing through each of the limit states. Equating Cd

to CcLS (and Sd to ∆LS) at each limit state, and using equation 5-19a results in: 

 Cd = CcLS (5-20a) 

2

v 1

cLS

LS L

g F S C

2 B

⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ =

⎣ ⎦ ∆ π ⎣ ⎦

 (5-20b) 

from which ( ) LS

v 1 L cLS LS FS 2 B C

g

∆

= π (5-21a) 

The above result is applicable to bridges with long periods (T > TS). By a similar process, it is 

possible to show that for short period bridges (T < TS), 

 

 (FaSS)LS = BS CcLS (5-21b) 

Note that BL and BS vary from one limit state to another since they are displacement dependent 

(table 5-4), but may be easily calculated because ∆LS is known for each limit state. 

The left hand side of equation 5-21a or 5-21b is a measure of the size of earthquake that would 

cause limit state LS to be reached. In other words, it is an indicator of the capacity of the bridge 

expressed in terms of the size of earthquake required to reach that capacity. This measure may be 

compared with the actual demand on the bridge using the same quantity, (FvS1) or (FaSS), 

expressed as a demand measure i.e., (FvS1)d or (FaSS)d. 

Accordingly, a capacity/demand ratio (rLSi) may be defined for each limit state (i) as follows: 
170 

 ( )

( )

v 1 LSi

LSi S

v 1 d

F S

r for T T

F S

= ≥ (5-22a) 

 ( )

( )

a S LSi

LSi S

a S d

F S

r for T T

F S

= < (5-22b) 

If rLSi > 1.5, the limit state is not likely to be reached and no remedial action is required. 

If 1.0 < rLSi < 1.5, the limit state may be reached and some remedial action may be required. 

If rLSi < 1.0, the limit state is likely to be reached and retrofit measures which increase 

deformability or ductility capacity of the bridge should be considered. These measures might 

include extending the seat widths at pier caps and/or abutments (to improve N0), adding 

restrainers (to increase k1and Fy), and jacketing columns (to improve Mn and θp). 

5.5.4.2. Calculation of Bridge Response 

Capacity/demand spectra may be used to determine the response of a bridge with a known 

capacity spectrum (Cc vs ∆) during a given earthquake with a known demand spectrum 

(Cd vs Sd). The difficulty, however, is that the final displacement is unknown and both the 

capacity coefficient (Cc) and the damping factor (BL) cannot be calculated in advance. Iteration is 

therefore used, starting with an initial estimate for displacement and iterating until the assumed 

value and the calculated value are in agreement. Basic steps in the method are listed below. 

 

Step 1. Determine if the bridge has a long period of vibration by comparing the elastic period T 

with TS (figure 1-8). If not, go to Step 8 (procedure for short period bridges). 

Procedure for long period bridges: 

Step 2. Start iteration by setting ∆ equal to the displacement of the bridge assuming elastic 

behavior, and calculate ductility factor μ (equation 5−15). 

Step 3. Calculate the damping factor BL using table 5-4. 

Step 4. Calculate the capacity coefficient Cc using equation 5-9. 

Step 5. Set Cd = Cc and solve for Sd in equation 5-19a. 

i.e., 

2

v 1

d

c L

g F S S

C 2B

⎡ ⎤ ⎡ ⎤ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ π ⎦

 (5-23a) 

Step 6. Compare Sd with value for ∆ (see Step 2 or previous Step 6) and if in agreement, go to 

Step 7. Otherwise set ∆ = Sd , recalculate μ, and go to Step 3. 
171 

Step 7. Calculate forces in individual piers, bearings and foundations using ∆, and compare 

sum with total lateral force on bridge (base shear) F, using F = CcW. 

Procedure for short period bridges: 

Step 8. Start iteration by setting ∆ equal to the displacement of the bridge assuming elastic 

behavior, and calculate ductility factor μ (equation 5−15). 

Step 9. Calculate the damping factor BS using table 5-4. 

Step 10. Calculate the capacity coefficient Cc using equation 5-9. 

Step 11. Calculate effective stiffness from Keff = Cc W / ∆. 

Step 12. From equation 5-19b, calculate Cd = FaSS / BS. 

Step 13. Calculate Sd from Sd = Cd W / Keff and by substituting results from Steps 11 and 12 

obtain: 

 a S

d

c S

F S S

C B

⎡ ∆ ⎤⎡ ⎤ = ⎢ ⎥⎢ ⎥ ⎣ ⎦⎣ ⎦

 (5-23b) 

Step 14. Compare Sd with value for ∆ (see Step 8 or value from previous Step 14) and if in 

agreement go to Step 15, otherwise set ∆ = Sd , recalculate μ, and go to Step 9. 

Step 15. Calculate forces in individual piers, bearings and foundations using ∆, and compare 

sum with total lateral force on bridge (base shear) F, using F = CcW. 

5.5.5. PROCEDURE FOR METHOD D1 

The two above applications (sections 5.5.4.1 and 5.5.4.2) may be combined into a single 

procedure as described in this section. Several checks are also introduced to assure that 

assumptions made above (both explicit and implicit) are satisfied. This is Method D1. 

The procedure has three parts: 

Part A : Initialization and calculation of bridge capacity. 

Part B : Calculation of C/D ratios. 

Part C : Calculation of bridge response (F,∆). 

Each part is described below. 
172 

PART A: Initialization and Calculation of Bridge Capacity 

Step A1. Obtain spectral ordinates (SS and S1) and site factors (Fa and Fv) for the earthquake 

under consideration from figure 2-2 and table 1-4. Calculate products FaSS and F1Sv. 

Step A2. Calculate transition period TS (figure 1-8) between short period and long period 

portions of the demand spectrum (see also equation 5-14). 

Step A3. Calculate weight of superstructure (W) and elastic stiffness (k1) in both the longitudinal 

and transverse directions of the bridge. Estimate post-yield stiffness (k2). 

Step A4. Calculate elastic period of structure (T) in each direction (longitudinal and transverse) 

using, for example, the uniform load method (section 5.4.2.1). Compare with TS (Step 

A2) and determine whether bridge falls in the short or long period portion of the 

spectrum, in both directions.

Step A5. If the bridge has a short period of vibration (Step A4), calculate elastic response from 

Fel = FaSS W and ∆el = Fel/k1. If the bridge has a long period (Step A4), calculate elastic 

response from Fel = FvS1 W / T and ∆el = Fel/k1. 

Step A6. Calculate Fy in each direction (longitudinal and transverse) from Fy = Σ Vui where Vui is 

calculated from (Mn/H)i for column i, and Mn is nominal moment capacity of the 

column under axial gravity loads only. (See note 1, section 5.5.2.2.) This assumption 

(about axial column loads) may be refined using the procedure in section 7.6.2, once 

the overturning moments, due to earthquake loads acting through the superstructure, are 

known. 

Step A7. Calculate ∆y from ∆y = Fy/k1 

Step A8. Compare ∆el (Step A5) with ∆y. If ∆el > ∆y, the bridge will yield. Proceed to Step B1. If 

∆el < ∆y bridge remains elastic for earthquake described in Step A1. All C/D ratios are 

greater than 1.0, and bridge response (Part C) is the same as that calculated in Step A5. 

Set ∆ = ∆el , F = Fel and go to Step C6. 

PART B: Capacity/Demand Ratio Checks (r) 

Step B1. Determine the number of limit states to be considered and estimate or calculate the 

corresponding displacement(s) ∆LS, for each state. 

Step B2. Calculate the capacity coefficient Cc, at each limit state 

Step B3. If bridge has a short period of vibration (Step A4), calculate the damping factor BS, for 

each limit state using table 5-4. If bridge has a long period (Step A4), calculate the 

damping factor BL, for each limit state using table 5-4. 
173 

Step B4. Calculate the size of the earthquake corresponding to each limit state, using equations 

5-21a and 5-21b as follows: 

 ( ) LS

v 1 L cLS LS FS 2 B C

g

⎡ ∆ ⎤

= π ⎢ ⎥ ⎣ ⎦

 for T > TS (long period response) 

 

 (FaSS)LS = BS CcLS for T < TS (short period response) 

 

Step B5. Calculate the C/D ratio for each limit state using equations 5-22a and 5-22b as follows: 

 ( )

( )

v 1 LS

LS

v 1 d

F S

r

F S = for T > TS (long period response) 

 ( )

( )

a S LS

LS

a S d

F S

r

F S = for T < TS (short period response) 

Step B6. Examine the C/D ratios and for those values equal to, or close to, unity; analytically 

explore potential retrofit measures that might lift these ratios above unity. 

For example, if rLS > 1.5, the limit state is not likely to be reached, 

 if 1.0 < rLS < 1.5, the limit state may be reached, and some remedial action 

 may be required, and 

 if rLS < 1.0,the limit state is likely to be reached and retrofit measures 

 which increase the deformability or ductility capacity of the bridge should 

 be considered. These measures might include extending the seat widths at 

 pier caps and / or abutments (to improve N0), adding restrainers (to 

 increase k1and Fy), and jacketing columns (to improve Mn and θp). 

PART C: Bridge Response (F, ∆) 

Step C1. Start iteration by setting ∆ equal to the displacement of the bridge assuming elastic 

behavior (∆el from Step A5) and calculate ductility factor μ (equation 5-15). 

Step C2. Calculate the appropriate damping factor using table 5-4: 

• BL for T > TS (long period bridges), and 

• BS for T < TS (short period bridges). 

Step C3. Calculate the capacity coefficient Cc using equation 5-9. 

Step C4. Calculate Sd using equations 5-23a and 5-23b: 

• Sd = [g/Cc] [FvS1/2πBL]

2

 for T > TS (long period bridges), and 

• Sd = [∆/Cc] [FaSS/BS ] for T < TS (short period bridges). 
174 

Step C5. Compare Sd with the assumed ∆ (Step C1 or previous Step C5) and if in agreement, go 

 to Step C6, otherwise set ∆ = Sd, recalculate μ, and go to Step C2. 

Step C6. Calculate forces in individual piers, bearings and foundations using ∆, and compare 

sum with total lateral force on bridge (base shear) F, using F = CcW. 

5.5.6. RESTRICTIONS ON USE OF METHOD D1 

The capacity spectrum method is restricted to bridges that: 

1. Behave as single-degree-of-freedom structures, in both the transverse and longitudinal 

directions, and 

2. Have equal displacements (or nearly equal) at the tops of all the piers in both the transverse 

and longitudinal directions, excluding piers with expansion bearings. 

The first of these restrictions requires eligible bridges to have regular geometry and uniform 

distribution of weight and stiffness, and to conform to the requirements of table 5-3. 

The second of these restrictions means that the method is limited to bridges with superstructures 

that either: 

• Behave as nearly-rigid, in-plane diaphragms supported on seat-type abutments with bearings 

that are either elastomeric or have restraints (keeper bars) that can be assumed to fail, or 

• Be of uniform geometry and weight, have sufficient length that the abutments have little 

influence on transverse response, and be supported on substructures of uniform stiffness. 

In general, bridges which satisfy the following criteria will meet the equal-displacement 

requirement: 

• Span length should not exceed 60 m (200 ft). 

• The ratio of the longest to shortest span lengths in frame should not exceed 1.5. 

• The maximum skew angle should not exceed 30o, and the skew of adjacent piers or bents 

should not differ by more than 5o. 

• For horizontally curved bridges, the subtended angle of the frame should not exceed 20o. 

• The ratio of maximum to mimimum pier stiffness should not exceed 2.0, including the effect 

of foundation stiffness. 

• The ratio of maximum to minimum pier lateral strength should not exceed 1.5. 
175 

These requirements are more stringent than the first set and take precedence for the application 

of Method D1. It is possible to check the equal-displacement requirement by an elastic analysis 

using, for example, the uniform load method (section 5.4.2.1). 

5.6. METHOD D2: STRUCTURE CAPACITY/DEMAND (PUSHOVER) METHOD 

5.6.1. APPROACH 

This more advanced procedure for calculating structure capacity/demand ratios is also known as 

the nonlinear static procedure (NSP) or pushover method. As for Method D1, the NSP is a twostep approach. First, it requires a displacement capacity evaluation using a pushover analysis. 

Such an analysis considers each relevant limit state and level of functionality, including P- ∆

effects. Second, it requires a response spectrum analysis to assess the displacement demands on 

the bridge. 

5.6.2. DISPLACEMENT CAPACITY EVALUATION 

The objective of a displacement capacity evaluation is to determine the displacement at which 

the earthquake resisting members of a bridge reach their inelastic deformation capacity. Damage 

states are defined by local deformation limits, such as plastic hinge rotation, footing settlement or 

uplift, and abutment displacement. Displacements may be limited by a loss of capacity such as 

the degradation of strength under large inelastic deformations, or P-∆ effects. 

This evaluation should be applied to individual piers to determine their lateral load-displacement 

behavior. It should be performed independently in both the longitudinal and transverse 

directions, and should identify those components of a pier which are first to reach their inelastic 

capacities. The displacement at which the first component reaches its capacity defines the 

displacement capacity for the pier. The model used for this analysis should include all of the 

components providing resistance, and use realistic force-deformation relationships for these 

components, including abutments and foundations. 

For piers with simple geometries (e.g., a single-column pier), the maximum displacement 

capacity can usually be found by hand calculation, using an assumed plastic hinge mechanism 

and a maximum allowable deformation capacity for the plastic hinges and foundations. If the 

interaction between axial force and moment is significant, iteration will be necessary to 

determine the capacity of the collapse mechanism. 

For more complicated piers or foundations, displacement capacity can be evaluated by a 

nonlinear static analysis, commonly referred to as a pushover analysis. 

As noted above, evaluation of the displacement capacity is conducted on individual piers. 

Although forces may be redistributed from pier-to-pier as the displacement increases and 

yielding begins to occur (particularly so for bridges with piers of different stiffness and strength), 

the objective of this evaluation is to determine the maximum displacement capacity of each pier. 

This capacity is then compared with the results from an elastic demand analysis, which does 
176 

consider the behavior of the bridge as a whole, and includes the effect of piers with different 

stiffnesses. 

The structural model used for the evaluation should be based on the expected capacities of the 

inelastic components. The model for footings and abutments should include the nonlinear force–

deformation behavior, including uplift, gap opening and closing. Stiffness and strength 

degradation of inelastic components, and the effects of loads acting through the lateral 

displacement (P-∆), should be considered. 

The maximum displacement of a pier is achieved when a component reaches maximum 

deformation. Maximum plastic hinge rotations for structural components are given in 

section 7.8.2. The maximum deformation for foundations and abutments are limited by 

geometric constraints on the structure. 

Although this evaluation is based on monotonically increasing displacement, the effects of cyclic 

loading must be considered when selecting an appropriate model and establishing a maximum 

inelastic deformation. This includes strength and stiffness degradation and low-cycle fatigue. 

5.6.3. DEMANDS 

The uniform load method may be used for structures satisfying the ‘regularity’ requirements of 

table 5-3. For structures not satisfying these requirements, the multi-mode spectral analysis 

method of dynamic analysis should be used, or alternatively an elastic time history method. 

These methods are described in section 5.4.2. 

5.6.4. PROCEDURE FOR METHOD D2 

Step 1. Determine the strength and deformation capacity for each pier of the bridge. 

Step 2. For each pier, carry out a nonlinear static pushover analysis until the structural 

displacement reaches the collapse limit state (limit state 5). Note the structural 

displacements, ∆ci, at each of the limit states (i), namely at: 

1. First yield, 

2. Slight damage with cracking, 

3. Moderate damage that is reparable, 

4. Irreparable damage at the limit of life safety, and 

5. Structural collapse. 

Step 3. Determine the sum of the nonseismic displacement demands Σ∆NSdi for each of the load 

combinations given in equations 6-1, 6-2, 7-1 and 7-2 of the Standard Specifications

(AASHTO, 2002), or table 3.4.1-1 of the LRFD Specifications (AASHTO, 1998). 
177 

Step 4. Determine the response spectrum parameters, Ss and S1 and site factors, Fa and Fv 

(figure 1-8 and table 1-4). Perform an elastic dynamic analysis to determine the 

seismic displacement demands, ∆EQdi, on each pier of the bridge. The analysis should 

reflect the anticipated condition of the structure and the foundation during this 

earthquake. 

Step 5. Determine the capacity/demand ratio (rLSi)for each limit state (i) from the following: 

( ) ci NSdi

LSi

EQdi

r

∆ − ∑ ∆ = ∆ (5-24) 

If rLSi > 1.5, the limit state is not likely to be reached and no remedial action is 

 required. 

If 1.0 < rLSi < 1.5, the limit state may be reached and some remedial action may be 

 required. 

If rLSi < 1.0, the limit state is likely to be reached and retrofit measures which increase 

 deformability or ductility capacity of the bridge should be considered. These 

 measures might include extending the seat widths at pier caps and/or 

 abutments, adding restrainers, jacketing columns, and strengthening joints 

 and foundations as needed. 

5.6.5. RESTRICTIONS ON METHOD D2 

This method is a general approach and has few restrictions. However, the capacity analysis is 

limited to a pier-by-pier evaluation which does not necessarily capture the capacity of the bridge 

as a whole. On the other hand, comparison of results with those from nonlinear time history 

analyses, shows this limitation is not signifcant for regular highway bridges. However, it would 

be wise to check the performance of a complex bridge with the potential for substantial inelastic 

behavior, or of a bridge of major importance, using a time history method and explict modeling 

of bridge capacity, such as that described in the next section. 

5.7. METHOD E: NONLINEAR DYNAMIC PROCEDURE 

5.7.1. APPROACH 

The nonlinear dynamic procedure (NDP) is also a two-step method. First, it requires an 

assessment of the seismicity at the site along with the development of a suite of earthquake 

ground motions (acceleration time histories) that represent this level of seismicity. The strength 

and displacement capacity of all members expected to function in a nonlinear fashion needs to be 

determined along with appropriate hysteretic rules to describe member behavior. Then, a 

nonlinear dynamic time history analysis is conducted for each ground motion. The hysteretic 

performance is evaluated and the expected limit state of the bridge assessed. 

Nonlinear dynamic analysis provides displacements and member actions (forces and 

deformations) as a function of time for a specified earthquake ground motion. A minimum of 
178 

three ground motions representing the earthquake being considered, should be used in the 

analysis. Each ground motion should include two horizontal components and one vertical. The 

maximum response for the three ground motions should be used for evaluating performance. If 

more than seven ground motions are used, the mean of the responses may be used. 

5.7.2. PROCEDURE FOR METHOD E 

Step 1 Determine the strength and deformation capacity for each member in the structure. 

Step 2 Determine the response spectrum parameters, Ss and S1, and site factors, Fa and Fv

(figure 1-8 and table 1-4). Develop at least three earthquake acceleration time histories 

with spectra that are compatible with the design response spectrum. Site-specific 

ground motions may also be used, and these may be either derived from a site-specific 

spectrum, or developed directly from the seismicity and geotectonics of the site. 

Step 3 Conduct a nonlinear dynamic time history response analysis for each of the ground 

motions. Check the sensitivity of the results obtained to assumptions made in analysis, 

such as the size of the time step used in the analysis. 

Step 4 Compare the seismic demands with the member capacities and determine the degree of 

damage to the structure for both levels of earthquake. If the damage will be 

unacceptable, devise retrofit measures and reanalyze the bridge to assess the 

effectiveness of these improvements. 

5.7.3. RESTRICTIONS ON METHOD E 

This method is a general approach and has few restrictions. However, it requires considerable 

computational effort and a significant level of skill is needed in interpreting the results. The 

method is particularly useful for structures that have irregular geometry or large variations in 

mass and stiffness properties. Preliminary solutions from simpler methods should always be 

obtained before undertaking a nonlinear time history solution, to bound the results and check for 

meaningful results. 
179

CHAPTER 6: GEOTECHNICAL MODELING AND CAPACITY 

ASSESSMENT 

6.1. GENERAL 

The behavior of a bridge during an earthquake is strongly dependent on the stiffness and strength 

of its foundation system, which includes the abutments and piers, footings, and piles. Dynamic 

response is determined by these two parameters, which in turn influence the earthquake demands 

on the bridge and the distribution of these loads to the structural and foundation components. 

Because foundation retrofit work may be more costly than new construction, and often has to be 

conducted under difficult site conditions, there are significant advantages to be gained if the 

conservatism in normal foundation design can be minimized. To do this, a more detailed and 

higher level of analysis is usually required than for a new design. This may involve developing 

both a coupled linear stiffness-model for the foundation system to be used in elastic spectral 

analyses, and a nonlinear load-displacement relationship for time history solutions. Nonlinear 

solutions not only allow more rigorous models for stiffness to be used, but also permit the 

beneficial effect of hysteretic damping and soil yield to be studied. 

Acceptable bridge performance is usually determined by satisfying certain displacement limit 

states (chapters 1 and 5). To verify that performance has been satisfied it may be necessary to use 

the displacement-based procedures described in chapter 5. If so, the nonlinear load-deformation 

characteristics of the foundation and the effects of mobilizing the ultimate capacity of the 

foundation system should be included in the calculations. Such consequences could include 

permanent foundation deformation. While transient foundation yielding will, in many cases, 

reduce structural displacement demands and reduce both the structural and foundation retrofit 

costs, permanent residual deformation needs to be very carefully assessed. To enable an 

integrated model of the complete bridge to be developed for system evaluation, as described in 

chapter 5, both linearized stiffness and foundation capacity models are needed. The nature and 

development of these foundation models are described in section 6.2. 

In addition to the earthquake-induced inertial loads on the bridge, ground displacement demands 

may arise, due to liquefaction-induced lateral spreading or global instability of abutment slopes. 

Such demands have led to bridge failures and significant structural damage in past earthquakes. 

The problem entails both the estimation of the likely ground deformation, and the resulting soilfoundation interaction to assess the displacement demands on the bridge. Procedures for 

calculating such demands are discussed in section 6.3. 

6.2. FOUNDATION MODELING 

6.2.1. SOIL-FOUNDATION-STRUCTURE INTERACTION 

The rigorous analysis of the dynamic response of a soil-foundation-bridge system in a fullycoupled manner is complex and difficult (Pecker and Pender, 2000). If the soil is idealized as an 
180

elastic continuum, then a substructuring approach may be used. In this case, the problem is 

separated into three steps: 

Step 1. Analyze the influence of the stiffness and geometry of a massless foundation system on 

the free-field ground motion, leading to modified structural input motions at the 

foundation level (i.e., kinematic interaction). 

Step 2. Analyze the frequency-dependent impedance characteristics of the foundation system 

under cyclic loading, in the form of a foundation stiffness matrix. 

Step 3. Analyze the inertial response of the structure to the foundation input motions (from 

step 1), using the pile cap stiffness matrix to account for foundation compliance (i.e., 

inertial interaction). 

Except for cases involving deep, and relatively stiff, foundations in soft soils and for large rigid 

shallow foundations, the effect of kinematic interaction is normally neglected in practice, and the 

foundation-input motions are assumed to be the same as the free-field motions. As a result, the 

inertial interaction of a bridge with its foundation is the major focus of the material presented in 

this chapter. 

 

6.2.1.1. Shallow Footings 

Procedures for evaluating soil-footing-bridge interaction have evolved over time from the theory 

of continuum mechanics. Frequency-dependent stiffness and damping parameters for low 

amplitude, machine-foundation vibration problems have been adapted to earthquake soilfoundation-structure interaction (SFSI) problems1

. But for practical modeling purposes, as 

described in section 6.2.2.1, the frequency dependence of the stiffness parameters may be 

ignored, and static or slow cyclic values for stiffness may be used instead. This is a reasonable 

assumption considering the range of frequencies in earthquake ground motions. Similarly, 

frequency-dependent radiation-damping due to wave propagation away from a footing is not 

considered. For most footing sizes, radiation damping values are small and difficult to quantify 

in a nonlinear soil. 

In addition to developing secant stiffness parameters, the mobilization of the moment capacity of 

the footing is of concern in many retrofit applications, and is addressed in section 6.2.2.1. For 

practical modeling purposes, soil capacity may be evaluated assuming pseudo-static inertial 

loading on a footing from the bridge above. This is a reasonable assumption, considering normal 

factors of safety under static loading, the stiff or dense soils associated with spread footings, and 

the difficulties in determining phase relations between soil and foundation motions in a coupled 

problem2

. 

 

1

 Mylonakis et al. (2002); Pecker and Pender (2000) 2

 ibid 
181

6.2.1.2. Piles 

Two distinct approaches to soil-pile-structure interaction under dynamic loading have been 

developed. One approach evolved from slow cyclic (long period) lateral loading tests on piles 

conducted in the 1970's, which were motivated originally by the need to develop pile design 

criteria for offshore structures subjected to wave loading. These studies led to analytical 

methods based on the use of nonlinear Winkler springs to model soil-pile interaction under cyclic 

lateral and axial loading. The second approach evolved from other studies conducted in the 

1970's, originally motivated by machine foundation vibration problems, where the problem was 

driven by the need to develop frequency dependent stiffness and damping (i.e., impedance 

functions), to determine resonant frequency and amplitude characteristics of supporting pile 

foundations. In this approach, the model used was that of a vibrating mass supported by pile 

foundations in an elastic continuum. 

Elastic impedance functions for piles and pile groups have been studied by numerous researchers 

and a large number of closed form analytical solutions are available3

. Early research on this 

subject was stimulated by the need to establish analysis methods for vibrating machine 

foundations, where the higher frequencies of loading lead to radiation damping and foundation 

stiffness is strongly frequency dependent. In addition, amplitudes of vibration are generally 

small, and the assumption of an elastic soil is reasonable. 

However, soil-pile interaction under larger earthquake-induced inertial loading can lead to 

strongly nonlinear soil behavior, particularly in the vicinity of the pile interface, and the use of 

the elastic approach becomes impractical for routine structural design under seismic loading. 

Fortunately, given the relatively low frequency range of earthquake inertial loading and the 

nature of representative pile foundation systems for bridges and buildings, stiffness functions are 

essentially frequency independent and static loading stiffness values are a reasonable 

approximation. In addition, the radiation damping component of energy loss arising from wave 

propagation away from the foundation is considerably reduced at lower frequencies, particularly 

in the presence of nonlinear soil behavior. Whereas fully coupled nonlinear solutions to the 

seismic loading problem, using finite element methods, are theoretically possible and have been 

used to a limited extent, the analytical complexity is daunting and impractical for routine design. 

Given the complexity of nonlinear coupled models, the Winkler model, as represented by a series 

of independent or uncoupled lateral and axial springs (linear or nonlinear) simulating soil-pile 

interaction in the lateral and axial directions, provides the most convenient means of analyzing 

the response of pile foundation systems to earthquake loading. The pile is modeled by beamcolumn elements, supported by linear or nonlinear spring elements, and both kinematic and 

inertial interaction effects may be included (Matlock et al., 1978, 1981). Free-field earthquake 

ground motions determined from one-dimensional site response analyses may be used as 

displacement input motions for the spring elements. The analysis method is embodied in the 

computer program SPASM (Single Pile Analysis with Support Motion), as described by Matlock 

et al. (1978), and applications to bridges have been developed4

 using this modeling concept. 

 

3

 Novak (1991); Pender (1993); Gazetas and Mylonakis (1998); Pecker and Pender (2000) 4

 Lam and Law (2000); Gazetas and Mylonakis (1998) 
182

Whereas the effects of the kinematic interaction can be significant for some pile-soil 

configurations (for example, larger diameter piles in soft soils or for sudden changes in soil 

stiffness with depth), piles may be assumed to deform in a compatible manner within the freefield. For such cases, free-field displacements are generally much less than those induced by 

inertial interaction. Hence, it can be assumed that inertial interaction dominates pile foundation 

response and that stiffness functions may be represented by values under static or slow cyclic 

conditions with foundation input motions assumed to be near surface free-field motions. This 

approach has had widespread application in the analysis of soil-pile-structure interaction 

analyses (Martin and Lam, 1995) and forms the basis for the stiffness and capacity modeling 

approaches described in section 6.2.2.2. 

6.2.2. STIFFNESS AND CAPACITY OF FOUNDATION COMPONENTS 

Load-deformation characteristics of foundations are required when the effects of foundations are 

to be taken into account in Evaluation Methods D and E (Capacity Spectrum and Structure 

Capacity/Demand Methods, and Nonlinear Dynamic Procedure), as discussed in chapter 5. 

Foundation load-deformation parameters, characterized by both stiffness and capacity, can have 

significant effects on both structural response and load distribution among structural elements. 

Soil-structure systems for bridges can be complex in some cases, but for the purpose of 

simplicity, four foundation types are considered below: shallow footing foundations, pile 

foundations, drilled shafts, and abutments. 

In addition to the load–deformation behavior of foundations being nonlinear, there are other 

difficulties in determining soil properties and static foundation loads for existing bridges. To 

account for the likely variability of the soils supporting the foundations, a best-estimate, 

equivalent, elasto-plastic representation of the load-deformation behavior is recommended. 

In general, soils have considerable ductility unless their stiffness and strength are significantly 

degraded under cyclic action or large deformations. Degradable soils include cohesionless soils 

that are expected to liquefy or build up large pore pressures, and sensitive clays that may lose 

considerable strength when subject to large strains. Soils that are not subject to significant 

degradation will continue to sustain load, but with increasing deformations after reaching 

ultimate capacity. Such soils are necessarily assumed in the evaluation methods below. 

If soils are susceptible to significant strength loss, due to either the direct effect of the earthquake 

vibration, or the foundation loading on the soil caused by the earthquake, then improvement of 

the soil should be considered or special analyses carried out to demonstrate that the loss of 

strength does not result in excessive structural deformations. 

Acceptable levels of foundation deformation are determined by the consequences of these 

deformations on the structure, which in turn depend on the desired level of bridge performance. 

However, it should be recognized that foundation yielding may be accompanied by progressive, 

permanent foundation settlement or deformation during continued cyclic loading, but this 

settlement will probably be less than a few inches where the static factor of safety exceeds 2.5 to 
183

3.0 (Martin and Lam, 2000). In general, foundation deformations will be small if the loads 

transmitted to the foundation do not exceed the ultimate capacity of the soil. Recommended 

procedures for determining the stiffness and capacity of various foundation types are given in 

sections 6.2.2.1 through 6.2.2.4. 

6.2.2.1. Shallow Bearing Footing Foundations 

Force-displacement relationships for soil foundations allow the designer to include their 

properties directly into the structural models used to determine earthquake response. Consider 

the spread footing shown in figure 6-1 with an applied vertical load, P, lateral load, H, and 

moment, M. The soil characteristics might be modeled as two translational springs and a 

rotational spring, each characterized by a linear elastic stiffness and a plastic capacity. However 

it is more common to use a Winkler spring model, acting in conjunction with the foundation, to 

eliminate the rotational spring, as shown in figure 6-2 (NEHRP, 1997a, 1997b). Conversion to 

Winkler springs uncouples the lateral action from the vertical and rotational actions. Springs with 

elasto-plastic properties are the simplest to use and give adequate results in most cases. More 

refined nonlinear models may be used if the importance of the project justifies it. A particular 

feature of the Winkler spring model is that it can capture progressive mobilization of plastic 

capacity during rotational rocking behavior relatively easily. 

Figure 6-1. Uncoupled elasto-plastic spring model for rigid footings. 

Figure 6-2. Uncoupled Winkler spring model. 
184

Figure 6-3. Upper and lower bound approach to 

define stiffness and capacity. 

An upper and lower bound approach to defining stiffness and capacity is often suggested (as 

shown in figure 6-3) because of the uncertainties in the soil properties and the static loads on the 

foundations of existing bridges. Such an approach allows the sensitivity of the bridge to these 

properties to be determined. In such a case, the range of uncertainty represented by the upper and 

lower bounds should be jointly determined by the geotechnical and structural engineer. 

Otherwise, a best-estimate value may be chosen. 

6.2.2.1(a). Stiffness Parameters 

Most shallow spread footings are stiff relative to the soil on which they rest, and for analytical 

purposes, an uncoupled spring model as shown in figure 6-1 is sufficient. The three equivalent 

spring constants may be determined using theoretical static solutions for rigid plates embedded 

in an elastic medium. The solutions require knowledge of the average elastic modulus and 

Poisson’s ratio of the soil in the vicinity of the foundation (say to a depth of 1.5 times the width 

of the footing below the base of the footing). The shear modulus, G, for a soil is then given by: 

E G =

2(1+ ) ν

(6-1) 

where E is the modulus of elasticity, and ν is Poisson’s ratio (0.35 for unsaturated soils and 0.5 

for saturated soils). 

The initial or low strain shear modulus during cyclic loading, G0, is given by 

2

0 s

γ G= v

g

 (6-2) 

where vs is the shear wave velocity at small strains, γ is the weight density (unit weight) of soil, 

and g is the acceleration due to gravity. 

The initial shear modulus for granular soils may also be found from the normalized corrected 

blowcount, (N1)60, and the effective vertical stress as follows (Seed et al., 1986): 
185

( ) 3

0 1 60

' G 20,000 N in psf ≅ σo (6-3) 

where: 

 (N1)60 = blowcount normalized for 1.0 ton per square foot confining pressure and 

 60 percent energy efficiency of hammer, 

0 σ' = effective vertical stress in psf, 

 = γtd - γw(d – dw), 

γt = total unit weight of soil, 

γw = unit weight of water, 

d = depth of sample, and 

dw = depth of water level. 

Either equation 6-2 or 6-3 may be used to calculate the modulus G0. 

Most soils are very nonlinear, and the shear modulus decreases with increasing shear strain. The 

large-strain effective shear modulus, G, can be roughly estimated on the basis of the anticipated 

peak ground acceleration (PGA). For regions of low-to-moderate seismicity (FvS1 < 0.3), a value 

of G = 0.5 G0 is recommended, while for regions of moderate-to-high seismicity (FvS1 > 0.5), a 

value of G = 0.25 G0 is suggested. Interpolation should be used for intermediate values of FvS1. 

The uncoupled stiffness parameters may be obtained from theoretical solutions for a rigid plate 

resting on a semi-infinite homogeneous elastic half-space. Tabular solutions are shown in tables 

6-1 and 6-2 (Gazetas, 1991). 

If the horizontal stiffness of the foundation is similar to, or less than, that of the bridge, the 

foundation properties should be included in the demand and capacity analyses. However, many 

foundation systems are relatively stiff and strong in the horizontal direction, due to the passive 

resistance of the soils on the sides of footings and the friction beneath the footings and may be 

several orders of magnitude greater than the bridge. In these cases, the foundations may be 

considered fixed in the horizontal directions, and foundation rocking will have the greater 

influence on bridge response. 

If the foundation capacity is exceeded during an elastic analysis based on initial estimates of 

foundation stiffness, softening will occur due to soil yield and it will be appropriate to reanalyze 

the bridge with an estimate of the degraded equivalent linear values. When nonlinear pushover 

analyses are used, initial estimates of elastic foundation stiffness should be used in conjunction 

with capacity estimates. 

6.2.2.1(b). Capacity Parameters 

Soil yield, rocking, and uplift of rigid footings under earthquake-induced moment loading can 

reduce the ductility demand in a bridge5

. Accordingly, rotational yield could be allowed to occur 

 

5

 Taylor and Williams (1979); Taylor et al., (1981) 
186

Table 6-1. Surface stiffnesses for a rigid plate on a semi-infinite 

homogeneous elastic half-space. 

Stiffness Parameter Rigid Plate Stiffness at Surface, Ki’ 

Vertical Translation, ′ KZ ⎡ ⎤ ⎛ ⎞ ⎢ + ⎥ ⎜ ⎟ − ν ⎢ ⎝ ⎠ ⎥ ⎣ ⎦

0.75 GL B 0.73 1.54

(1 ) L

Horizontal Translation, ′ Ky

(toward long side) 

⎡ ⎤ ⎛ ⎞ ⎢ + ⎥ ⎜ ⎟ − ν ⎢ ⎝ ⎠ ⎥ ⎣ ⎦

0.85 GL B 2 2.5

(2 ) L

Horizontal Translation, ′ Kx

(toward short side) 

⎡ ⎤ ⎛⎞ ⎛ ⎞ ⎡ ⎤ ⎢ ⎥ +− − ⎜⎟ ⎜ ⎟ ⎢ ⎥ −ν −ν ⎢ ⎥ ⎝⎠ ⎝ ⎠ ⎣ ⎦ ⎣ ⎦

0.85 GL B GL B 2 2.5 0.1 1

(2 ) L (0.75 ) L

Rotation, θ ′ K x

(about x axis) 

⎛⎞ ⎛ ⎞ + ⎜⎟ ⎜ ⎟ − ν ⎝⎠ ⎝ ⎠

0.25

0.75

x

GL B I 2.4 0.5

(1 ) B L

Rotation, θ ′ K y

(about y axis) 

⎡ ⎤ ⎛ ⎞ ⎢ ⎥ ⎜ ⎟ − ν ⎢ ⎝ ⎠ ⎥ ⎣ ⎦

0.15

0.75

y

G L I 3

(1 ) B

L

(Length)

y

y

x x B

(Width)

Plan

d

(Thickness)

D

(Depth)

Section

Homogeneous Soil Properties

G,

z

1. Determine the uncoupled total surface 

stiffnesses, Ki

’, of the foundation element by 

assuming it to be a rigid plate bearing at the 

surface of a semi-infinite elastic half-space 

(see above). 

2. Adjust the uncoupled total surface stiffnesses, 

Ki

’, for the effect of the depth of bearing by 

multiplying by embedment factors (see Table 

6-2), ei

, to generate the uncoupled total 

embedded stiffnesses, Ki.

Ki

 = ei Ki’ 

Note: IX, Iy are moments of inertia of the footing about the x- and y-axes, respectively. 

adapted from Gazetas, 1991
187

Table 6-2. Stiffness embedment factors for a rigid plate on a 

semi-infinite homogeneous elastic half-space. 

Stiffness Parameter Embedment Factors, ei

Vertical Translation, eZ ( ) ⎡ ⎤ ⎡ ⎤ ⎛ ⎞ ⎛ ⎞ +

+ ++ ⎢ ⎥ ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎢ ⎥ ⎣ ⎦ ⎝ ⎠ ⎣ ⎦

0.67

D B 2L 2B

1 0.095 1 13 1 0.2 d

B L LB

Horizontal Translation, ey

(toward long side) ( )

⎧ ⎫ ⎡ ⎤ ⎛ ⎞ ⎪ − + ⎪ ⎡ ⎤ ⎢ ⎥ ⎜ ⎟ ⎛ ⎞ ⎪ ⎝ ⎠ ⎪ ⎢ ⎥ + + ⎢ ⎥ ⎜ ⎟ ⎨ ⎬ ⎢ ⎥ ⎝ ⎠ ⎢ ⎥ ⎣ ⎦ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎣ ⎦ ⎩ ⎭

0.4

0.5

2

d D 16 L B d

2D 2 1 0.15 1 0.52

B BL

Horizontal Translation, ex

(toward short side ( )

⎧ ⎫ ⎡ ⎤ ⎛ ⎞ ⎪ − + ⎪ ⎡ ⎤ ⎢ ⎥ ⎜ ⎟ ⎛ ⎞ ⎪ ⎝ ⎠ ⎪ ⎢ ⎥ + + ⎢ ⎥ ⎜ ⎟ ⎨ ⎬ ⎢ ⎥ ⎝ ⎠ ⎢ ⎥ ⎣ ⎦ ⎪ ⎪ ⎢ ⎥ ⎪ ⎪ ⎣ ⎦ ⎩ ⎭

0.4

0.5

2

d D 16 L B d

2D 2 1 0.15 1 0.52

L LB

Rotation, eθX

(about x axis) 

− ⎛ ⎞ ⎛⎞ ⎛⎞ + + ⎜ ⎟ ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ ⎝ ⎠0.20 0.50 d 2d d B 1 2.52 1

B BD L

Rotation, eθy

(about y axis) 

− ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛⎞ + + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜⎟ ⎝ ⎠ ⎝ ⎠ ⎝⎠ ⎝ ⎠0.60 1.9 0.60 2d 2d d 1 0.92 1.5

L LD

adapted from Gazetas, 1991 

under earthquake loading, provided there is no serious loss of vertical load-bearing capacity and 

the vertical settlements induced are small. Footing yield and uplift are permitted in the FEMA 

273 guidelines for the seismic retrofit of buildings (NEHRP 1997a, 1997b) and illustrated in 

figure 6-4. 

In the absence of moment loading, the vertical load capacity of a rectangular footing (Qc) is 

given by: 

 Qc = qc B L (6-4) 

where qc is the ultimate strength of soil (capacity) per unit area, B is the footing width, and L is 

the footing length. 

If moment loads are present in addition to the vertical load, contact stresses become concentrated 

at the edges of the footing, particularly as uplift occurs. The ultimate moment capacity, Mc, is 

dependent on the ratio of the vertical load to the vertical capacity of the soil. Assuming a rigid 

footing and that the contact stresses are proportional to vertical displacement and remain elastic 

until the vertical capacity is reached, a factor of safety, Fv, can be defined as follows: 
188

P

P

P

P

P

P

M

M

M

M

q

q

q

q

q

q

q

q

q

q

c c

c c

c

c

c

c

<

>

0.5

0.5

Figure 6-4. Idealized concentration of stress at edge of rigid footings subjected to 

overturning moment. 

 Fv = c q

q

 (6-5) 

where q is the vertical load on footing (contact stress). 

If Fv is greater than 2, uplift will occur prior to plastic yielding of the soil, but if Fv is less than or 

equal to 2, the soil at the toe yields before uplift begins. 

With these assumptions, the moment capacity of a rectangular footing may be expressed as:

 Mc = 

c

LP q 1

2 q

⎛ ⎞ ⎜ ⎟ −

⎝ ⎠

 = 

v

LP 1 1

2 F

⎛ ⎞ ⎜ ⎟ −

⎝ ⎠

 (6-6) 

where P is the vertical load on footing, q equals P/BL, B is the footing width, and L is the length 

of footing in direction of bending. 

The nonlinear moment-rotation behavior generated by yield and uplift corresponding to this 

model is illustrated graphically in figure 6-5. 

Rocking behavior has several important consequences on the seismic response of a structure. 

First, rocking results in a decrease in stiffness and lengthening of the fundamental period of the 

bridge. This effect is amplitude dependent and very nonlinear. The result is generally a 

reduction in the maximum seismic response as previously noted. Depending on the ratio of 

initial bearing pressure to the ultimate capacity of the soil, significant amounts of energy may be 
189

Infinitely Strong Soil

Moment

M

Stress

Distribution

Rotation

2

1

1

4

3 5

5

5

6

l

P

M

W

Pl

Pl

Pl

2

2

6

P

2q w

2

c

Possible Stress States

Elastic

Elastic

Elastic

Yield

Yield

Prior to Uplift Prior to Uplift

At Uplift

After Uplift

After Uplift

Limit

Inelastic

l l

l l

l l

qw< q wc

qw< q wc

qw< q wc

q wc

q wc

q wc

2

3

4

5

6

after NEHRP, 1997b 

Figure 6-5. Rocking of shear wall on strip footing. 

dissipated by soil yielding. This behavior also can result in decreased ductility demand on bridge 

piers. 

When plastic yielding of the soil under the footings is permitted, the magnitude of accumulated 

settlement, induced by the static vertical load, must be considered. Model tests and analytical 

studies (Martin and Lam, 2000) suggest that, provided the initial static factor of safety, Fv, is 

greater than 2.5, expected settlements will be small (say a few centimeters or inches) and may be 

acceptable for retrofit purposes. 

Column shear forces are resisted by friction at the base and sides of footings and by passive 

resistance at the face of the footing. For simplicity, it may be assumed that the maximum total 

resistance develops at a deformation equal to two percent of the embedded depth of the footing. 

For most cast-in-place concrete foundations, a value of interface friction of 80 percent of the 

friction angle of the soil is appropriate for evaluating base and side friction. Classical earth 
190

pressure theory can be used to calculate the ultimate passive pressure capacity. A simplified 

approach based on log spiral solutions for granular soils is shown in figure 6-6. 

If foundation forces are calculated from an equivalent elastic analysis using the shear stiffness 

parameters in tables 6-1 and 6-2, and they exceed the horizontal shear capacity of the footing, the 

consequential yield may be approximated by reducing the stiffness parameters and reanalyzing 

the bridge. 

Figure 6-6. Method for passive pressure capacity of footing or pile cap. 
191

6.2.2.2. Pile Footing Foundations 

Pile foundations generally comprise pile groups connected to a footing or pile cap, with pile 

diameters (normally driven piles) usually less than 0.6 m (2 ft) in diameter. These types of 

foundations are the most typical and are found on a variety of bridge types, including those with 

single-columns, multiple-columns, and wall piers. Connection details at the pile cap can also 

vary from pinned to fixed conditions. In general, bridges that are stiff under lateral loading will 

be more sensitive to foundation flexibility and the careful characterization of foundation stiffness 

becomes more important in these circumstances. In other words, foundation stiffness should be 

explicitly included when the foundation is flexible compared to the bridge, such as in soft soil 

conditions, or in the transverse direction under walls and single-column piers. 

The overall seismic response of a pile group foundation depends on the response of individual 

piles under both axial and lateral loading and on the lateral response of the pile cap. The 

stiffness and capacity characteristics of the various components are discussed below. 

For the purpose of computing stiffness and capacity for analyses, the footing or pile cap is 

normally uncoupled from the piles, and contributions from the two components are evaluated 

separately. Based on centrifuge tests (Gadre, 1997), this appears to be a reasonable solution. The 

stiffness and capacity of the pile cap in the lateral direction is normally much higher than the 

contributions from the piles, while rotational stiffness and capacity is usually governed by axial 

loading in the piles due to moments in the columns. 

6.2.2.2(a). Pile Cap – Lateral Stiffness and Capacity 

Because of the possibility of soil settlement, and the uncertain interaction between the footing 

and the piles, contributions to soil stiffness and capacity from the footing’s base and side shear 

are neglected. Hence the primary source of lateral resistance is the mobilization of passive 

pressure on the vertical face of the pile cap. Maximum passive pressures can be computed using 

the same procedure described for shallow footing foundations, as illustrated in figure 6-6. As the 

ultimate capacity is reduced at small deflection values (about two percent of the embedment 

depth), an initial elastic stiffness or lateral load-deflection stiffness may be defined using this 

assumption. A softened secant stiffness may be used if displacement demands are greater than 

this deflection value. 

6.2.2.2(b). Pile-Head Stiffness – Lateral Loading 

In practice, the lateral load-deflection characteristics of a pile are determined by a pushover 

analysis, assuming a beam supported on Winkler springs that are characterized by nonlinear p-y 

curves. This type of pseudo-static soil foundation interaction problem can be analyzed using a 

variety of methods, including discrete beam-column and finite-element analyses. Various 

authors have developed empirical deflection-dependent lateral soil resistance curves for sand and 

clays based on small-diameter pile load tests. The most commonly used p-y curves are those by 

Reese et al. (1974) for sands and by Matlock (1970) for clays. These curves should be linearized 

to develop a consistent set of lateral, rotational, and coupled stiffness values, if a full, 6 x 6 

stiffness matrix is used to represent the pile group in an elastic structural demand analysis (Lam 
192

et al., 1998). Programs such as LPILE (Reese et al., 1997) and FLPIER (Hoit and McVay, 1996) 

may be used if a nonlinear load-deflection relationship for the pile-head is required, based on the 

nonlinear p-y curves. However such curves are very sensitive to pile-head fixity conditions 

(figure 6-7) and the value of the cracked section modulus for the piles. 

In choosing an equivalent secant modulus for such curves for demand analyses, compatibility 

with displacement demand, which would normally be less than a few centimeters or inches in a 

strong earthquake, should be provided. However, given the many uncertainties in site 

conditions, a simplified procedure to develop a linear stiffness for retrofit evaluations is 

recommended below. 

Lam and Martin (1986) found that lateral load-deflection characteristics representing the overall 

stiffness of the soil-pile system is only mildly nonlinear because the elastic pile usually 

dominates the nonlinear soil stiffness. Furthermore, the significant soil-pile interaction zone is 

usually confined to a depth equal to the upper 5 to 10 pile diameters. Therefore, simplified 

single-layer pile-head stiffness design charts are appropriate for lateral loading. 

Single-layer linear design charts are presented in figures 6-8 to 6-11. These charts use a discrete 

Winkler spring soil model in which stiffness increases linearly with depth, from zero at grade 

level where the location of the pile-head is assumed to be located. A linear representation of 

subgrade stiffness is a good first approximation to test data for pile-caps in both sand and clay 

soil conditions. Two parameters may then be used to define this soil-pile system: the pile 

bending stiffness, EI, and the coefficient of variation of the inelastic subgrade modulus, f, as 

defined in equation 6-7: 

 f = Es

z

 (6-7) 

where Es is the subgrade modulus (soil stiffness per unit length of pile) at a particular depth, and 

z is the depth below grade. 

Values of the coefficient f, which has units of force/unit volume, have been published for piles 

embedded in sand carrying normal working loads6

, as a function of corrected Standard 

Penetration Test (SPT) blowcount or relative density (see figure 6-12). Values for piles in clay 

soils are given in figure 6-13 (Lam et al., 1991). Average soil conditions for the upper five pile 

diameters should be used when reading values from these charts, which are applicable for piles 

up to 0.6 m (24 in) in diameter. 

Given the coefficient f, and the bending stiffness of the pile, EI, lateral and rotational stiffness 

can be read directly from the charts in figures 6-8 through 6-11 (Lam et al., 1991). Although the 

charts assume no pile embedment, they do provide reasonable estimates of stiffness for shallow 

embedments up to say 1.5 m (5 ft), in which case the soil above the pile top is neglected. An 

additional set of design charts for piles with pinned heads (i.e., zero bending moment) is given in 

figure 6-11. If the coefficient of variation in subgrade modulus, as shown in figures 6-12 and 

 

6

 Terzaghi (1955); Murchison and O’Neill (1984) 
193

6-13, is used to derive the pile-head stiffness matrix from figures 6-8 through 6-11, the resulting 

pile cap stiffness matrix is appropriate for cap deflections between 6 and 50 mm (0.25 and 2 in). 

Deeper embedment increases the stiffness coefficients. Figures 6-14 through 6-16 give pile cap 

stiffness coefficients for pile embedment depths of 1.5 m and 3 m (5 and 10 ft), based on a 

subgrade modulus that increases linearly with depth. It can be seen from these figures that the 

embedment effect on stiffness is larger for slender piles, and tends to diminish for stiffer piles. 

The lateral stiffness is affected the most. For example, for a pile cap embedded in slightly 

compact sand (f = 2.7 MN/m3

 [10 lb/in3]), figure 6-14 shows an increase in embedment depth 

from 0 to 1. 5 m (0 to 5 ft) can increase the lateral pile cap stiffness by 130 percent for rigid piles 

and 230 percent for slender piles. For a depth increase from 0 to 3 m (0 to 10 ft), the stiffness 

increases by 150 percent for rigid piles and 350 percent for slender piles. 

These figures also illustrate that the effect of the embedment depth is greater in dense sands. For 

example, for a 0.3 m by 0.3 m (12 in by 12 in) concrete pile (EI = 43 x 106

 MPa [6.22 x 106 ksi]) 

embedded 1.5 m (5 ft), the lateral stiffness increases by a 160 percent in slightly compact sand 

(f = 2.7 MN/m3

 [10 lb/in3]) and 260 percent in dense sand (f = 270 MN/m3 [100 lb/in3]). 

1200

800

400

0

0 20 40 60 80 100 120

Fixed

Free

Partially Restrained

Pile Diameter = 1.2 m

Pile Modulus, E = 2.4 x 10 kPa

Pile Length, L = 12 m

Soil Friction Angle, = 34

Coef. of Subgrade Modulus, f = 16 MN/m

Reese’s p-y Criteria

Pile Head Displacement (mm)

7

3

o

Lateral Load (kN)

Figure 6-7. Effect of boundary conditions on pile stiffness 
194

Figure 6-8. Lateral pile-head stiffness (fixed-head condition). 
195

f = 200

f = 100

f = 60

f = 10

f = 5

f = 1

f = 0.5

f = 0.1

Coeff. of Variation of Soil Reaction

Modulus with Depth. f (lb/in ) 3

10

10

10

10

10

10 10 10 10 10

11

10

9

8

7

9 10 11 12 13

Bending Stiffness, El (lb-in )

f (lb/in )3

M ,t

P ,t

1

E (lb/in )2 s

Rotational Stiffness, K (in-lb/radian)

P = K + K

M = K + K

K =

T =

1.499 E I

T

E I

f

t

t

3

( )1/5

2

Figure 6-9. Rotational pile-head stiffness. 
196

10

10

10

10

10

10 10 10 10 10

9

8

7

6

5

9 10 11 12 13

Bending Stiffness, El (lb-in ) 2

f = 200

f = 150

f = 100

f = 80

f = 60

f = 40

f = 20

f = 10

f = 5

f = 1

f = 0.5

f = 0.1

Coeff. of Variation of Soil Reaction

Modulus with Depth, f (lb/in ) 3

Cross- Coupling Stiffness, K (lb)

f (lb/in )3

M ,t

P ,t

1

E (lb/in )2 s

P = K + K

M = K + K

K =

T =

0.999 E I

T

E I

f

t

t

( )1/5

2

Figure 6-10. Cross-coupling pile-head stiffness. 
197

f = 200

f = 150

f = 100

f = 80

f = 60

f = 40

f = 20

f = 10

f = 5

f = 1

f = 0.5

f = 0.1

Coeff. of Variation of Soil Reaction

Modulus with Depth. (lb/in ) 3

10

10

10

10

10

10 10 10 10 10

7

6

5

4

3

9 10 11 12 13

Bending Stiffness, El (lb-in )2

Lateral Stiffness for Free-Head Cond. (lb/in)

Free head Pile Stiffness

f (lb/in )3

P ,t

1

E (lb/in )2 s

= K

=

T =

E I

T

E I

f

3

( )1/5

K

K

2

0.41

Figure 6-11. Lateral pile-head stiffness (free-head condition). 
198

100

80

60

40

20

0

0 20 40 60 80 100

28 29 30 38 41 45 oo o o oo

VERY

LOOSE LOOSE MEDIUM DENSE VERY

DENSE

Recommended by Terzaghi, 1955

(After O’Neill and Murchison, 1983)

f (lb/in ) 3

E (lb/in ) s 2

Above Water Table

Below Water Table

Coeff. of Variation in Subgrade Stiffness, f (lb/in ) 3

Relative Density, D (Percent) r

Friction Angle,

1

Figure 6-12. Recommended coefficient of variation in subgrade modulus (f) 

with depth of sand. 
199

Figure 6-13. Recommended coefficient of variation in subgrade modulus (f) 

with depth of clay. 
200

f = 10

f = 1

f = 0.1

Coeff. of Variation of Soil reaction

Modulus with Depth, f (lb/in ) 3

10

10

10

10

10

10 10 10 10 10

7

6

5

4

3

9 10 11 12 13

Embedment

0’

5’

Bending Stiffness, El (lb-in ) 2

Translational Stiffness (Fixed Head), K (lb/in)

f = 100

f (lb/in )3 f (lb/in )3

M ,t

M ,t

P ,t

P ,t 1 1

E (lb/in )2 s E (lb/in )2 s

P = K + K

M = K + K

t

t

Embedment Depth

Pile Head at Grade Level Embedded Pile Head

10’

Es0

Figure 6-14. Lateral embedded pile-head stiffness (fixed-head condition). 
201

10

10

10

10

10

10 10 10 10 10

11

10

9

8

7

9 10 11 12 13

Embedment

f = 100

f = 10

f = 1

f = 0.1

Coeff. of Variation of Soil Reaction

Modulus with Depth, F (lb/in ) 3

Bending Stiffness, El (lb-in ) 2

Rotational Stiffness, K (in-lb/radian)

f (lb/in ) f (lb/in ) 3 3

E (lb/in ) E (lb/in ) 2 2

E

t

t

t

,

,

, 1 1

Embedment Depth

s

s

s

Pile Head at Grade Level Embedded Pile Head

P = K + K

M = K + K

t

t

0

0’

5’

10’

Figure 6-15. Embedded pile-head rotational stiffness. 
202

10

10

10

10

10

10 10 10 10 10

9

8

7

6

5

9 10 11 12 13

Embedment

f = 100

f = 10

f = 1

f = 0.1

Coeff. of Variation of Soil Reaction

Modulus with Depth, f (lb/in ) 3

Bending Stiffness, El (lb-in ) 2

Cross- Coupling Stiffness, K (lb)

f (lb/in )3 f (lb/in )3

M ,t

M ,t

P ,t

P ,t 1 1

E (lb/in )2 s E (lb/in )2 s

P = K + K

M = K + K

t

t

Embedment Depth

Es

Pile Head at Grade Level Embedded Pile Head

0’

5’

10’

o

Figure 6-16. Embedded pile cross-coupling pile-head stiffness. 
203

6.2.2.2(c). Pile-Head Stiffness – Axial Loading 

The axial stiffness of a pile head is generally assumed to be uncoupled from the stiffnesses in the 

lateral and rotational directions. Axial load-displacement characteristics are also assumed to be 

uncoupled from the lateral load-displacement behavior because much of the soil resistance to 

axial loading comes from relatively deep elevations. As a result, these properties are relatively 

insensitive to the characteristics of the shallower, soil-structure-interaction zone that influences 

lateral loading. However, since the overall resistance to axial loading is derived along a 

significant length of the pile (especially in the deeper soil strata), and the pile is under a 

relatively large static dead load prior to the earthquake, a realistic determination of axial 

properties requires an ability to account for heterogeneous soil layering as well as plastic 

slippage in the upper part of the pile. As a result, linear, homogeneous analytical solutions 

should not be used for axial loading calculations. Instead methods that account for both soil 

layering and plastic slippage at the soil-pile interface should be used, such as the one described 

below. 

A graphical procedure that gives the axial stiffness coefficient and includes layering and slippage 

(Lam and Martin, 1986), is illustrated in figure 6-17 for a 21.3-m (70-ft) long 0.3-m (1-ft) 

diameter pipe pile embedded in sand with φ = 30° and consists of the following steps: 

Step 1. Ultimate compressive capacity. Calculate the ultimate pile capacity from a sitespecific pile capacity analysis using conventional procedures for skin-friction and endbearing capacities for the different soil layers. When assigning values to the skinfriction and end-bearing capacities, make allowance for the construction method (e.g., 

pre-drilling, driving), special conditions (e.g., consolidation due to surcharging, 

corrosion), and dynamic soil behavior (e.g., cyclic strength degradation, soil 

liquefaction). 

Step 2. Rigid pile load–displacement curve. Develop the pile load-displacement curve based 

on the estimated ultimate capacity and published skin-friction and end-bearing pile 

displacement relationships. The resulting load-displacement curve in figure 6-17, which 

was obtained from the sum of the skin friction and end-bearing capacities at each axial 

displacement, will result in a pile-head load–displacement relationship for a pile that is 

axially-rigid. This curve is a lower bound on the actual pile displacements. 

Step 3. Flexible pile load-displacement curve. Calculate the axial displacement at the pile 

head due to the compression of the pile under axial load but neglecting the surrounding 

soil. This displacement is given by: 

c

QL

AE

δ = (6-8) 

 

 where Q is the axial load, L is the length, and AE is the axial rigidity of pile. 

 
204

Obtain the flexible pile solution by adding δc from equation 6-8, to the rigid-pile loaddisplacement curve in figure 6-17, at pile loads that correspond to the rigid pile loaddisplacement curve. This curve is an upper bound on the actual pile displacements. 

Step 4. Actual load-displacement curve. As a first approximation, obtain the actual axial 

load-displacement curve by averaging the curves from steps 2 and 3. The actual 

solution is bounded by the rigid and flexible load-displacement solutions derived in 

steps 2 and 3 and depends on the nature of the soil-pile system. Averaging the results is 

a good place to start. For an end-bearing pile, the actual curve may be closer to the 

flexible-pile solution. For a friction pile at small load levels, the upper portion of the 

pile is compressed first, and thus the load-displacement curve will be closer to the rigidpile curve. At higher loads, slippage occurs along most of the pile and the solution 

moves towards the flexible-pile curve. 

after Lam and Martin, 1986 

Figure 6-17. Axial load-displacement graphical solution. 
205

Step 5. Axial pile stiffness. Calculate a value for the pile secant stiffness from the loaddisplacement curve over the range of expected displacements, and use it to find the 

equivalent axial stiffness coefficient for inclusion in the pile stiffness matrix. Expected 

displacements may be found from the expected range of axial loads, which for a regular 

highway bridge foundation would be in the range of 50 to 70 percent of the ultimate 

pile capacity. 

The above procedure may be used to obtain the corresponding uplift load-displacement curve by 

ignoring end-bearing and using only skin-friction to evaluate capacity. 

In an even simpler approximation, values for pile-head stiffness under normal loading (not 

exceeding the capacity) may be expressed as a multiple of AE/L, with the constant, α, depending 

on the relative amounts of skin friction and end bearing resistance that are mobilized (Gohl, 

1993). For example, a value of α = 1 gives a lower bound to the stiffness (i.e. AE/L), and would 

be appropriate for an end-bearing pile on rock with negligible skin friction. Values of α close to 

2.0 would be reasonable for friction piles with negligible end tip resistance. 

6.2.2.2(d). Pile Group Analysis 

A group of piles may be modeled in a dynamic analysis of a bridge by either a series of 

uncoupled springs or a fully coupled foundation stiffness matrix, as shown in figure 6-18. The 

latter model is the most general and rigorous approach, and requires the determination of the 

stiffness coefficients in a generalized 6 x 6 stiffness matrix for each pile. The matrix represents 

the lateral and rotational (rocking) stiffnesses for and about the two horizontal axes, and their 

cross-coupling terms. The vertical and rotational stiffnesses for and about the vertical axis, are 

also included (Lam et al., 1998). 

The buried pile cap represents an additional but very important component. As previously noted, 

the pile cap is uncoupled from the piles to determine the lateral stiffness coefficients, which are 

then added to the pile stiffness matrix. In many cases, the pile cap stiffness will be significantly 

greater than the lateral stiffness of the pile group. 

For a vertical pile group, the stiffness for the translational displacement terms (the two horizontal 

and one vertical displacement terms) and the cross-coupling terms can be obtained by merely 

multiplying the corresponding stiffness components of a single pile by the number of piles. 

However, the rotational stiffness terms (the two rocking and torsional rotations) require 

consideration of an additional stiffness component. In addition to individual pile-head bending 

moments, a unit rotation at the pile cap will introduce displacements and corresponding forces at 

each pile head (e.g., vertical forces for rocking rotation and lateral pile forces for torsional 

rotation). These pile-head forces will work together among the piles and will result in an 

additional moment reaction on the overall pile group. In most cases, the axial stiffness of the 

piles will dominate the rotational or rocking stiffness of the group. 

The above procedure does not account for ‘group effects’, which relate to the interaction 

between closely spaced piles and the properties of the surrounding soil, particularly if nonlinear 

behavior occurs. In general, pile spacings of less than three to five pile diameters are necessary 
206

Figure 6-18. Modeling approaches for deep foundations. 

before the effects of pile interaction become significant in practical terms. Experimental data on 

p-y multipliers for closely spaced piles in sand, indicate about a 20 percent loss for pile spacings 

of three diameters7

. Detailed discussion on group effects is given by Lam et al. (1998). 

6.2.2.2(e). Pile Group Analyses – Moment-Rotation Capacity 

The moment-rotation characteristics and the capacity of a pile footing depend on the following 

factors: 

• Configuration of the pile footing, including the number of piles and spatial dimensions, and 

• Capacity of each pile for both compression and uplift loading. 

This is illustrated in figure 6-19 by way of an example problem involving a typical pile footing 

(Lam, 1994). The load-displacement curves for each individual pile in the pile group are shown 

in figure 6-20. The pile is modeled as an elastic beam-column, and nonlinear axial soil springs 

are distributed along the pile to represent the soil resistance in both compression and uplift. The 

pile cap is assumed to be rigid. It can be seen from the figure that the ultimate soil capacities of 

the pile for compression and tension are 800 and 400 kN (180 and 90 kips) per pile, respectively. 

This assumes that the connection details and the pile member have sufficient strength to ensure 

that the failure takes place in the soil. The pile is assumed to be 15 m (50ft) long, and have a 

diameter of 0.3m (12in). It is a concrete pile, driven into a uniform medium sand, which has a 

design load capacity of 400 kN (45 tons) per pile. The adopted ultimate capacity values (800 kN 

[180 kips] compression and 400 kN [90 kips] uplift) are default values commonly assumed in 

seismic retrofit projects for the 400 kN (45 ton) class pile. In the example, it is assumed that the 

footing has been designed for a static factor of safety of 2.0, and the piles are loaded to 50 

percent of the ultimate compression capacity before the earthquake. 

 

7

 Brown et al. (1988); McVay et al. (1995) 
207

Lam, 1994 

Figure 6-19. Pile footing configuration for moment-rotation study. 

Figure 6-19 presents various capacity criteria for the pile footing. Under conventional practice, 

the moment capacity of the pile footing would be 3,665 kNm (2,700 kip/ft). This capacity is due 

to the assumption of a linear distribution of the pile reaction across the footing. The moment 

capacity of 3,665 kNm (2,700 kip/ft) is limited by the ultimate compressive capacity value of the 

most heavily loaded pile (800 kN [180 kip] per pile) while maintaining vertical equilibrium of 

the overall pile group (i.e., for a static load of 4,800 kN [1,080 kips]). The lowest part of figure 

6-19 presents the moment capacity that can be achieved from a nonlinear moment-rotation 

analysis of the pile footing, in which the moment increases above the conventional capacity. 

Nonlinear load-displacement characteristics of the pile are included to allow additional load be 

distributed to the other less-loaded piles in the pile group. As shown, a maximum ultimate 

capacity of 5,500 kNm (4,050 kip/ft), which is 50 percent above the conventional capacity, is 

estimated by this nonlinear analysis. 
208

200

150

100

50

0

-50

-100

-100 -50 0.00 0.50 1.00 1.50 2.00 2.50 3.00

Displacement (in.)

Load (kip)

Compression

Uplift

Lam, 1994 

Figure 6-20. Axial load-displacement curve for single pile. 

Figure 6-21 presents the cyclic moment-rotation solutions for the example discussed above. The 

dotted line on the moment-rotation plot defines the monotonic loading path of the momentrotation relationship. Solutions for two uniform cyclic moment loads are presented: a lower 

cyclic moment level of 3,665 kNm (2,700 kip/ft) corresponding to the conventional design 

capacity, and a higher cyclic moment load of 5,500 kNm (4,050 kip/ft). As shown in the figure, 

at the lower cyclic moment of 3,665 kNm (2,700 kip/ft), the moment-rotation characteristic is 

almost linear, and both the moment-rotation characteristics and settlement will equilibrate to a 

final value very quickly, i.e. within a few cycles of loading. However, at the higher cyclic 

moment load of 5,500 kNm (4,050 kip/ft), progressive settlement of the footing is occurring and, 

within about four cycles of loading, the footing settles almost 125 mm (5 in). The moment–

rotation relationship also indicates that some level of permanent rotation of the footing will likely 

occur even if the load is symmetric between positive and negative cyclic moments. The potential 

for the permanent rotation is associated with the change in the state of stress in the soil from a 

virgin (unstressed) condition to the equilibrated state after cyclic loading, unloading, and 

reloading. A similar analysis, using a static factor of safety of 3.0 (rather than 2.0) 

corresponding to a dead load of 3,200 kN (720 kips), resulted in an ultimate moment capacity of 

130 percent of the conventional capacity, and a reduced settlement of about 6 mm (0.25 in) under 

loading cycles at the increased ultimate capacity level. 

Considering the conservatism in the estimates of pile capacity (especially for compressive 

loading), most existing pile footings probably have an static factor of safety for dead load greater 

than 3.0. Hence, it can be speculated that, the potential for significant settlement or rotation of a 

pile footing is not high, except on poor soil sites where cyclic degradation of soil strength can be 

significant. Instead, the most likely cause of foundation failure will be some form of permanent 

rotation of the pile group, if the size of the footing and the number of piles are inadequate. 
209

5

4

3

2

1

0

-1

-2

-3

-4

-5

-0.010

-0.010

0.000

0.000

0.010

0.010

0.020

0.020

0.030

0.030

0.040

0.040

0.050

0.050

0.0

-2.5

- 5.0

4 x 3 Pile Footing

Ultimate Pile Capacity

180k Compression

90k Uplift

Piles Loaded to 90k Static

Compressive Load

Ultimate Moment Capacity

from Nonlinear Analysis

= 4,050 ft-kip

=1.5 Conventional Capacity

Cyclic Moment

= 2,700 ft-kip

Conventional

Design Criteria

4,000 ft-kip

Cyclic

Moment

Cyclic Moment = 2,700 ft-kip

4,000 ft-kip

Cyclic Moment

Rotation (radians)

Settlement (in.) Moment (1,000 ft-kip)

Lam, 1994 

Figure 6-21. Cyclic moment-rotation and settlement-rotation solutions. 

6.2.2.3. Drilled Shafts 

Drilled shafts are large diameter piles (normally greater than 600 mm [2.0 ft] in diameter) that 

are used to support high axial loads and overturning moments. They are constructed by drilling, 

or auguring, a hole in the ground, which may then be lined to support the sides of the excavation. 

When drilling is complete, a reinforcement cage is placed and the cavity filled with concrete. 

Because installation is straightforward, with minimum disturbance to the site, and they have high 
210

load carrying capacity, drilled shafts are often advantageous over conventional small diameter 

piles for many bridge configurations. 

The analysis and design of a drilled shaft is, for the most part, similar to that for a conventionaldriven pile. Important differences exist, however, due to installation procedure, larger diameter, 

and a small length-to-diameter ratio compared to typical smaller driven piles. Also, for bridges, 

the structural arrangement of drilled shaft foundations can be different from that of a more 

conventional foundation. Most bridges supported on drilled shafts are pile-extension type 

structures involving a single shaft, rather than a group of shafts, as in the case of driven piles. 

The lateral loads that act on the bridge deck are resisted by the induced shear and bending 

moment on the drilled shafts, and the overall bridge response is relatively insensitive to axial 

effects in contrast to pile footings. Where a drilled shaft foundation system consists of a single 

shaft supporting a column, compressive and uplift loads on these shafts during seismic loading 

will normally be within limits of load factors used for gravity loading. However, checks should 

be performed to confirm that any changes in axial load do not exceed ultimate capacities in uplift 

or compression. In contrast to driven piles in a group, no reserve capacity exists for a single 

shaft; i.e., if ultimate capacity is exceeded, large deformation can occur. For a pile-extension 

type structure, the cross-coupling between the lateral and rotational stiffnesses is significant and, 

therefore, an uncoupled foundation spring model is not appropriate and a cross-coupled matrix 

formulation should be used instead. 

Various studies (Lam et al., 1998) have found that conventional p-y stiffnesses derived for driven 

piles are too low (soft) for drilled shafts. This stiffer response is attributed to a combination of 

higher unit side friction, base shear at the bottom of the shaft, and the rotation of the shaft. The 

rotation effect is often implicitly included in the interpretation of lateral load tests, as most lateral 

load tests are conducted in a free-head condition. A scaling factor to stiffen the p-y curves equal 

to the ratio of shaft diameter to 600 mm (2.0 ft) is generally applicable (Lam et al., 1998). The 

scaling factor is applied to either the linear subgrade modulus or the resistance value in the p-y 

curves. This adjustment is thought to be also dependent on the construction method. 

A drilled shaft may be represented by an equivalent cantilever beam in a dynamic analysis. The 

beam is founded at a point of fixity, which is located by matching the beam stiffness to that of 

the shaft (Lam et al., 1998). If a static pushover analysis is to be used for capacity evaluations, a 

p-y curve analysis approach will provide more reliable estimates of moment and shear 

distribution and the location of the depth to fixity may need to be revised. 

Because the elastic modulus and effective cross-section of a concrete pile change with strain, the 

flexural rigidity changes with applied load level. For large diameter concrete piles, minor 

cracking develops at a relatively small moments and effective section properties should be used 

when evaluating pile stiffness. In the absence of detailed information regarding steel 

reinforcement and applied axial load, equivalent properties for the cracked section can be taken 

at 50 percent of those for the uncracked section. This will be sufficiently accurate for most pile 

analyses. 

The depth of maximum moment for a drilled shaft will often be located 2 to 3 diameters below 

the ground surface for dense and loose soils, respectively, if the shaft diameter and stiffness are 
211

the same as the column (Lam et al., 1998). For a cracked section with flexural rigidity equal to 

50 percent of the uncracked section, the depth to the maximum moment is shallower, being on 

the order of 1.5 to 2 diameters. 

For most piles, the boundary conditions at the pile tip do not affect the pile performance because 

they are sufficiently far from the structure and the piles can be considered as infinitely long piles 

when the pile length is larger than 3 times a characteristic length, λ, defined as (Lam et al., 

1998): 

p 0.25

s

EI

E

⎛ ⎞ λ = ⎜ ⎟ ⎝ ⎠

 

where EIp is the flexural rigidity of the pile, and Es is the subgrade modulus of the soil. 

For large diameter shafts, it is important to check the pile length, especially for soft soil 

conditions, to determine the need to include the boundary conditions at the pile tip in the 

analyses. Lower lengths are acceptable if the drilled shaft provides sufficient lateral stiffness. 

Lateral load analyses using programs such as LPILE or the FHWA program COM624P would be 

needed for such determinations. 

In many instances, concrete pavement or traffic barriers are constructed around drilled shafts at 

the ground level. Unless a gap is maintained around the shaft, the pavement can provide lateral 

support to the pile, which may in turn influence the pile-head stiffness. 

6.2.2.4. Abutments 

Abutment walls and wingwalls can play a very beneficial role in the performance of a bridge 

during an earthquake, because the back fills behind these walls can resist large inertial loads and 

thus reduce ductility demands elsewhere in the bridge. The extent of this reduction depends on 

the structural configuration (e.g. simple or continuous spans), load-transfer mechanism from the 

bridge to the abutment system (e.g. integral or seat type abutments), effective stiffness and force 

capacity of the wall-soil system, and the level of acceptable damage to the abutment that is 

consistent with the performance criteria. The capacity of the abutments for the inertial load is 

determined by the structural design of the abutment wall and the shear keys, if any, as well as the 

resistance of the soil that can be mobilized. 

6.2.2.4(a). Abutment Capacity – Longitudinal Direction 

Under earthquake loading, the earth pressure action on abutment walls changes from a static 

condition to one of two possible conditions, depending on the movement of the abutment walls, 

the bridge superstructure, and the bridge/abutment configuration. 

For seat-type abutments where the expansion joint is sufficiently large to accommodate the 

movement between the abutment wall and the bridge superstructure (i.e., the superstructure does 

not strike the wall), the earthquake-induced earth pressure on the abutment wall will be the active 

pressure condition. However, when the gap at the expansion joint is too small to accommodate 
212

the relative movements, there will be a transfer of forces from the superstructure to the abutment 

wall. As a result, the active earth pressure condition is no longer valid and the earth pressure 

behind the wall approaches a passive pressure condition due to loading from the bridge 

superstructure. 

For integral or diaphragm abutments as shown in figure 6-22, the abutment stiffness and capacity 

under passive pressure must be defined as described below. 

However, for high, seat-type abutment walls, earthquake-induced active earth pressures will 

continue to act below the backwall if the backwall fails. These active pressures need to be 

considered in checking wall stability. In these circumstances, backwall failure may be a desirable 

condition so as to avoid backward tilt of the wall. This technique is commonly deployed for new 

design and for the same reason, the backwalls of existing abutments might be modified to 

encourage their ‘early’ failure. This retrofit measure is strongly recommended for bridges in 

Seismic Hazard Levels III and IV. 

On Piles

Stub

Integral

Cantilever

Strutted Rigid Frame

Seat-type

On Spread Footing

Figure 6-22. Abutment types. 
213

For Seismic Hazard Levels III and IV, the resistance due to passive pressure in the soils behind 

the abutment walls, will usually be mobilized by large longitudinal superstructure displacements. 

For design purposes, static passive pressures may be used without reduction for inertial loading 

effects in the abutment backfill. The passive pressure zone that is mobilized by the displacement 

of the abutment, extends beyond the active pressure zone normally adopted for static serviceload design, as illustrated schematically in figure 6-23. Hence, the soil properties (compressive 

strength, c, and internal friction angle, N) in the zone will control passive pressures. 

Abutment stiffness and passive pressure capacity should be characterized by a bi-linear 

relationship as shown in figure 6-24. For seat-type abutments, knock-off backwall details should 

be used with superstructure diaphragms that are designed to accommodate passive pressures. 

Passive pressures may be assumed uniformly distributed over the height of the backwall or 

diaphragm. Thus, the total passive force per unit length of wall, Pp, is: 

 Pp = pp H (6-9) 

where pp is the passive pressure behind the backwall, and H is the wall height. 

Approach Slab Active Pressure Zone

Tie

Granular

Drainage

Material

H

Passive Pressure Zone

45 60

H

o o

Footing

Figure 6-23. Design passive pressure zone. 
214

ForcePP

Dg

D

D

K K eff2 eff2

ForcePP

Actual Behavior

K = Keff1

eff1

Deflection

Seat Abutments Integral or Diaphragm Abutments

i

i

K

K

eff

eff

Figure 6-24. Characterization of abutment capacity and stiffness. 

6.2.2.4(b). Calculation of Best-Estimate Passive Force, Pp 

If the strength characteristics of compacted or natural soils in the passive pressure zone (total 

stress strength parameters c and N) are known, then the passive force for a given height, H, may 

be calculated using accepted procedures. These procedures should account for the interface 

friction between the wall and the soil. (A conservative value equal to 50 percent of the friction 

angle of the backfill adjacent to the wall is recommended.) The strength properties used shall be 

indicative of the entire passive pressure zone as indicated in figure 6-23. Therefore, backfill 

properties adjacent to the wall in the active pressure zone may not be appropriate. 

If default passive pressures are to be used for design, then the following values might be used: 

1. For cohesionless, non-plastic backfill (fines content less than 30 percent), the passive 

pressure, pp, may be assumed equal to H/10 MPa (H in meters) or 2H/3 ksf (H in feet). 

2. For cohesive backfill (clay fraction > 15 percent), the passive pressure, pp, may be assumed 

equal to 0.25 MPa (5 ksf) provided the estimated unconfined compressive strength is greater 

than 0.20 MPa (4 ksf). 

Alternatively, the approximate approach shown in figure 6-6 may be adopted. 

6.2.2.4(c). Abutment Stiffness – Longitudinal Direction 

For integral or diaphragm abutments, an initial secant stiffness, Keff1 as shown in figure 6-24, 

may be calculated as follows: 

p

effl

P

K

0.02H = (6-10) 
215

where it is assumed that the displacement Deff where Pp is mobilized equals 0.02H. 

If computed abutment forces exceed the capacity, the stiffness should be reduced iteratively 

(Keff1 to Keffn) until abutment displacements are consistent (within 30 percent) with the assumed 

stiffness. For seat-type abutments, the expansion gap should be included in the calculation of 

secant stiffness. Thus: 

( )

p

effl

g

P

K

0.02H D = +

(6-11) 

where Dg is the gap width. 

Where nonlinear or pushover analyses are conducted, values of Pp and the initial estimate of Keff

should be used to define the bilinear load-displacement behavior of the abutment for the 

assessment of capacity. 

For partial- or full-depth seat abutment walls, where knock-off backwalls are activated, the 

design of the remaining portion of the wall, and a stability check under the action of continuing 

earthquake-induced active pressures, should be evaluated. For the ‘no collapse’ performance 

criteria, and assuming conventional cantilever retaining wall construction, horizontal wall 

translation under dynamic active pressure loading is acceptable. However, rotational instability 

may lead to collapse and thus should be prevented. 

The design approach is similar to that of a free-standing retaining wall, except that lateral forces 

from the bridge superstructure should be included in equilibrium evaluations, as the 

superstructure moves away from the wall. Earthquake-induced active earth pressures should be 

computed using horizontal accelerations that are at least equal to 50 percent of the site peak 

ground acceleration, which assumes that limited sliding of the wall has occurred during the 

earthquake. A limiting equilibrium condition should be checked in the horizontal direction. To 

ensure safety against potential overturning about the toe, a restoring moment of at least 50 

percent more than the overturning moment should exist. If necessary, wall design (initially 

based on a static loading condition) should be modified to meet the above condition. 

For the case of seismic active earth pressures, the Mononobe-Okabe equations are often used. 

These are based on Coulomb failure wedge assumptions and a cohesionless backfill. For high 

accelerations and/or for backslopes, these equations lead to excessively high pressures that tend 

to infinity at critical acceleration levels or backslope angles. For the latter conditions, no real 

solution to the equations exist, implying that equilibrium is not possible (Das, 1999). For 

example, in a horizontal backfill of sand with a friction angle of 40 degree, a wall friction of 20 

degrees, and a peak acceleration of 0.4g, the failure surface angle is 20 degrees to the horizontal. 

For a peak acceleration of 0.84g, the active pressure becomes infinite, implying a horizontal 

failure surface. 
216

Figure 6-25. Trial wedge method for determining critical earthquake-induced active 

forces. 

Clearly, for practical situations, cohesionless soil is unlikely to be present for a great distance 

behind an abutment wall and encompass the entire failure wedge under seismic conditions. In 

some cases, free-draining cohesionless soil may only be placed in the static active wedge (say at 

a 60° angle) with the remainder of the soil being cohesive embankment fill or even rock. Under 

these circumstances, the maximum earthquake-induced active pressure should be determined 

using trial wedges as shown in figure 6-25, with the strength on the failure planes determined 

from the strength parameters for the soils through which the failure plane passes. This approach 

will provide more realistic estimates of active pressure. 
217

6.3. GROUND DISPLACEMENT DEMANDS ON FOUNDATIONS 

6.3.1. SOURCES OF DEMAND 

The two principal sources of ground displacement demands on foundations are due to ground 

settlement and lateral spreading during liquefaction. Both are discussed below. 

6.3.1.1. Earthquake-Induced Settlement 

Procedures for calculating the magnitude of earthquake-induced settlement in both saturated and 

unsaturated sands are described in section 3.3 and appendix B, and are based primarily on the 

Tokimatsu and Seed (1987) procedures. A methodology for estimating settlements of highway 

embankment fills is also available in Part 2 of this manual. 

Settlements should be estimated for each foundation component. However, the force demands 

on the bridge due to settlement will be a function of size of the relative settlement between 

adjacent piers and/or abutments, and the flexibility and/or articulation of the bridge. 

Displacement demands will determine collapse potential due to the possible unseating of the 

superstructure. The minimum seat widths defined in equations 5-1a and 5-1b, include an 

allowance for possible ground rotation due to settlement, which may lead to unseating at the top 

of the pier. Computed demands on the seats, due to footing rotation, should be checked against 

these minima. Additional demands due to lateral spreading are discussed in the following 

section. 

Even though collapse may have been prevented due to adequate seat widths, differential 

settlements may adversely affect performance criteria, and these should be assessed against 

recommended tolerable limits.8

6.3.1.2. Liquefaction-Induced Lateral Spreads 

Large rigid-body movements of abutments and piers (both sliding and tilting) can occur when a 

bridge site liquefies and lateral spreading takes place. Such movements may cause catastrophic 

damage to the piers and foundations and/or unseat the superstructure. 

The Magsayay Bridge was damaged in the 1990 Luzon, Philippines earthquake due to 

liquefaction (Hall and Scott, 1995). Figure 6-26 shows the collapse mode of four simply 

supported spans, which was caused by about 2 m (6.5 ft) of lateral spread at the west abutment. 

As the east abutment did not move, it is possible that a continuous deck may have prevented 

collapse due to strut action. 

 

8

 Cooke and Mitchell (1999); Part 2 of this manual 
218

Figure 6-26. Sketch of Magsaysay bridge showing earthquake damage. 

Liquefaction also occurred at the Landing Road Bridge during the 1987 earthquake in 

Edgecumbe, New Zealand (Berrill et al., 1997). In this case, the approach spans over the 

riverbank were supported on concrete wall piers founded on battered prestressed concrete piles, 

as shown in figure 6-27. Liquefaction-induced lateral spreads in the liquefiable sand layer were 

of the order of 2 m (6.5 ft). Back analyses and observations from site excavations indicated that 

the piles successfully resisted the passive pressures mobilized against the piers, although cracks 

in the piles suggested that plastic hinges were on the verge of forming, as shown schematically in 

figure 6-27. 

In the 1995 earthquake in Kobe, Japan, widespread liquefaction occurred and field investigations 

using borehole cameras and slope indicators showed that failures of piles in lateral spread zones 

were concentrated at the interfaces between liquefied and non-liquefied layers, as well as near 

pile heads. Lateral pile analyses using p-y interface springs together with pile deformations 

induced by estimated ground displacement profiles were consistent with observed pile 

performance, as can be seen in figure 6-28 (Tokimatsu and Asaka, 1998). 

When the potential for liquefaction is identified and significant lateral spreading is expected, the 

ability of the bridge to accommodate these movements needs to be assessed. This may require a 

soil-foundation-structure-interaction analysis to be performed. If the bridge is unable to meet the 

required performance criteria, retrofitting will be necessary assuming it is not feasible to relocate 

the bridge to a less vulnerable site. Two retrofit options may be considered: 
219

 Berrill et al., 1997 

Figure 6-27. Landing Road bridge lateral spread. 
220

1. Retrofit the foundation / bridge structural system to accommodate the estimated ground 

movements. This may involve strengthening the foundation system and providing 

articulation in the superstructure to accommodate the movements. The cost of this work may 

be compared with the cost of the ground remediation option below. 

2. Remediate the site to prevent liquefaction or minimize ground displacement demands. Such 

methods are also discussed in chapter 11. 

0

5

10

15

20

25

-0.15 0 0.15 50 0 0 50

10

20

30

40

50

D (cm) cy

Mc

(a) Mu (b) (c) (d) Liquefied Layer

Crack

Crack

Bending Moment (MN m) . f (cm) cy N-value Damage

Depth (m)

after Tokimatsu and Asaka, 1998 

Figure 6-28. Site and damage characteristics for a precast concrete pile subjected to a 

lateral spread in the Kobe earthquake. 

6.3.1.3. Slope Stability 

The potential for earthquake-induced slope failures or landslides next to a bridge (where 

landslide debris could damage the structure) or due to the instability of the bridge site itself 

should be evaluated. Such failures may occur in rock or soil slopes, and may take the form of a 

major landslide or limited slope movement. Methods of stability analysis and retrofit measures 

are briefly summarized in chapter 11. Liquefaction-induced instability is addressed in section

6.3.1.2. 
221

CHAPTER 7: STRUCTURAL MODELING AND CAPACITY 

ASSESSMENT 

7.1. GENERAL 

As noted in section 1.11 and again in chapter 5, bridge evaluation is essentially a two-part 

process. A demand analysis is first required to determine the forces and displacements imposed 

on a bridge by an earthquake. This is followed by an assessment of the capacity to withstand the 

demand. This process is illustrated in figure 7-1. If the bridge has the capacity to withstand the 

demand, retrofitting is not required, but if the capacity is less than the demand, a decision needs 

to be made as to the extent of retrofitting that will be undertaken. Many evaluation methods 

express their results as capacity/demand ratios; a ratio less than one indicates a need for 

retrofitting. 

Five evaluation methods (A - E) are presented in chapter 5, and several demand analysis methods 

are also discussed. This chapter continues this discussion with sections on load path, bridge 

modeling (stiffness, mass and damping), and combination of seismic force effects. 

The remainder of this chapter describes methods for assessing the strength and deformation 

capacities of bridge members (principally columns) for use in the evaluation methods described 

above. The focus is on individual member behavior. Chapter 5 describes how this information is 

used to assess the overall capacity of a bridge (Methods D1, D2 and E). 

In some instances, preset rules are given as a pragmatic way of providing the required member 

properties that are necessary for both capacity and demand analyses for an entire bridge. Where 

simplified rules are given, it is with the intent to be reasonably (but conservatively) accurate and 

simple (and thus economical) to apply. For new or unusual situations not covered by these 

guidelines, member behavior models should be verified by experimental studies. 

7.2. LOAD PATH FOR LATERAL FORCES 

For a bridge to survive a major earthquake without collapse, a clear and straightforward load 

path must exist from the superstructure (where the inertia forces act) to the substructure and 

foundations. Furthermore, all members and components, including connections, in that load path 

must be capable of resisting the imposed forces and deformations. These members and 

components include slab-to-beam connections, beam flanges and webs, diaphragms, crossframes, bearing assemblies, anchor bolts, masonry and sole plates, columns, abutments, footings 

and foundations. 

Members and components in the load path form an earthquake resisting system in which the load 

is attracted to each element, and its performance is determined by the strength and stiffness 

characteristics of other elements in the path. Past earthquakes have shown that, when one of 

these elements responds in a ductile manner, or allows movement, damage is limited. Most 
222

METHOD A

sec 5.2

No analysis req'd

Min. seat and force

requirements are

specified

No modelling req'd

METHOD B

sec 5.3

No analysis req'd

Min. seat and force

requirements are

specified

No modelling req'd

Demand on pier

caps and footings

calculated from

column

overstrength,

sec 7.6

METHOD C

sec 5.4

Elastic Analysis

(Uniform Load (ULM)

Multimode (MM)

Time History (TH)

Methods)

Modeling as per

sec 7.3

Seat widths by

inspection

Connection

capacity by

calculation

Seat widths by

inspection

Connection

capacity by

calculation

Columnns must

satisfy min. shear

and confining steel

requirements

Column

overstrength,

sec 7.7

Component capacities

using Appendix D

Capacity/Demand

ratios for:

y Abutment

displacement

y Anchorage

length

y Bearing

connection force

y Column moment

y Column shear

y Confinement

steel

y Footing moment

y Footing rotation

y Liquefaction

potential

y Seat width

y Splice length

METHOD D1/D2

sec 5.5 / 5.6

Elastic Analysis

(Response

spectrum methods

ULM, SM, MM)

Modeling as

per sec 7.3

Either bridge

capacity curve (D1)

or pier capacity

curve (D2)

Forces and

moments in piers

due to overstrength,

sec 7.6

Strength

capacity

of bridge members,

sec 7.7

Deformation

capacity

of bridge members,

sec 7.8

Combination of

seismic force effects,

sec 7.4

Combination of

seismic force

effects,

sec 7.4

METHOD E

sec 5.7

Nonlinear Analysis

 (3 Dimensional,

Time History

Method)

Modeling as per

sec 7.3

Detailed bridge

model includes

member capacities

Forces and

moments in piers

due to overstrength,

sec 7.6

Strength

capacity

of bridge members,

sec 7.7

Deformation

capacity

of bridge members,

sec 7.8

SEISMIC RETROFIT CATEGORY and BRIDGE TYPE

DEMAND ANALYSIS

CAPACITY ASSESSMENT

Figure 7-1. Evaluation methods for existing bridges showing relationship between demand 

analysis and capacity assessment. 
223

retrofit approaches deliberately introduce ductility into selected substructure members (usually 

columns) for this reason. 

Elastic methods of analysis may be used to calculate the forces in the load path; but if extensive 

yielding occurs, the actual distribution of forces may be quite different from that calculated. A 

more refined distribution may be found by calculating the forces in the columns and other 

substructure elements assuming a sufficient number of hinges have formed to allow a collapse 

mechanism to develop. Superstructure forces are then calculated from the substructure values, 

including an allowance for overstrength (section 7.6). 

However, in lieu of this, the simpler elastic distribution of forces may be used, so long as the 

applied forces are in equilibrium with the inelastic substructure forces. The eccentricity of the 

superstructure inertia force from the center of substructure resistance should be considered in this 

analysis. 

The analysis and seismic evaluation of the diaphragms and cross-frames should consider 

horizontal supports at an appropriate number of bearings. Slenderness and connection 

requirements of bracing members that are part of the lateral force resisting system should comply 

with the recommendations given for main members. 

Elements not in the seismic load path will also deform during an earthquake, and should be 

checked to ensure they have sufficient capacity to maintain their load resistance. 

In lieu of refined analysis, a conservative load path should be assumed. For example, in bridges 

with concrete decks that provide horizontal diaphragm action, or with a horizontal bracing 

system in the plane of the top flange, the lateral loads applied to the deck may be assumed to be 

transmitted directly to the bearings through the end cross-frames. The development and analysis 

of the load path is then similar to that used for wind loading. 

Where the bridge deck cannot provide horizontal diaphragm action and there is no lateral bracing 

in the plane of the top flange, lateral loads applied to the deck should be distributed through the 

intermediate cross-frames to the bottom lateral bracing or the bottom flange, and then to the 

bearings under the end cross-frames, in proportion to their relative rigidity and the respective 

tributary mass of the deck. 

A continuous load path is necessary for the transmission of the superstructure’s inertial forces to 

the substructure. Concrete decks have significant rigidity in their horizontal plane, and in shortto-medium span slab-on-beam bridges, their response approaches rigid body motion. Therefore, 

the lateral loading of the intermediate cross-frames is minimal, consisting primarily of local 

tributary inertia forces from the beams themselves. 

It is unlikely that all bearings will see their peak load simultaneously, particularly in the 

transverse direction. Uneven engagement of side-bar restraints in steel bearings is likely to occur 

and, if so, some bearings will attract high concentrations of transverse load and others will see 

negligible loads. Member loads within the cross-frames will then be significantly different from 

those assumed, and this redistribution should be considered when evaluating cross-frame 
224

members. For the same reason, caution should be exercised when evaluating bearing details 

including transverse restraints and anchor bolts. This problem is less likely to occur with 

elastomeric bearings that are bolted to their masonry and sole plates. 

7.3. MODELING RECOMMENDATIONS FOR BRIDGE STRUCTURES 

Mathematical models used for dynamic analysis should include the strength, stiffness, mass, and 

energy dissipation characteristics of the structural members and components of the bridge. 

Depending on the method of dynamic analysis, different approximations may be used for 

modeling these quantities. One-dimensional, beam-column members are sufficient for the 

earthquake analysis of most regular bridges,1

 and grid or finite element models are usually not 

necessary. Joint size should be included in these models. The advantage of these simple models 

(sometimes referred to as spine models or stick models) is that they permit rapid interpretation of 

results and a quick check on load path and equilibrium. 

Irregular bridges1

 include those with skew and horizontal curvature, and three-dimensional 

models should be used in these cases to more carefully represent the load path, particularly at 

piers and abutments. Short columns or piers may be modeled as a single element, but tall 

columns should have two or more elements. This is particularly true if the piers have significant 

mass as in the case of a concrete bridge, or are framed systems as in the case of a steel tower. 

It is not necessary to model bridges with multiple segments as one complete bridge. Each 

segment (also called a longitudinal frame or simply a frame) should have sufficient capacity to 

resist the inertial loads generated within that segment and may be analyzed as a freebody or 

stand-alone structure. However when the segments have large differences in their modal periods, 

out-of-phase motion may occur and the frames may adversely interact with each other, causing 

impact and/or unseating at the hinges. 

 

To account for these effects, the number of segments that should be included in a model depends 

on the ratio of the fundamental periods. For bridges in which the period ratio of adjacent frames 

is less than 0.70 (shorter period divided by longer period), it is recommended that the model be 

limited to five frames. The first and fifth frames in the model are considered to be boundary 

frames, representing the interaction with the remainder of the structure. The response of the 

three interior frames can then be used for evaluation. For bridges with frames that have period 

ratios between 0.70 and 1.0, less than five frames may be used in the model if desired. For a 

bridge with more than five frames, several different models are required to be analyzed. 

Response of a bridge in two orthogonal horizontal directions should be determined in the seismic 

analysis and the results combined to determine demand forces and deformations (see 

section 7.4). 

If the bridge is located within 10 km (6 mi) of a known active fault, the response in the vertical 

direction should also be determined and combined with the horizontal response. These bridges 

should also be subjected to a site-specific study to determine both the horizontal and vertical 

 

1 Regular and irregular bridges are as defined in AASHTO, 1998 and 2002. 
225

ground motions for use in the above analyses. Explicit analysis for vertical motions need not be 

carried out for regular bridges located more than 10 km (6 mi) from an active fault. For irregular 

bridges, such as those with long flexible spans, C-shaped piers, or with other large eccentricities 

in the load path for vertical loads, an analysis in the vertical direction should be included (see 

section 7.5). 

7.3.1. DISTRIBUTION OF MASS 

Modeling the mass of a bridge should consider the degree of discretization and the expected 

motion during an earthquake. The number and location of the displacement degrees of freedom 

in a bridge model determine the way mass is represented and distributed throughout the 

structure. As most of the mass of a bridge is in the superstructure, four to five elements per span 

are generally sufficient to represent the superstructure. For spine models of superstructures, the 

beam elements have the same neutral axis as the superstructure and rigid links are then used to 

locate the mass centroid relative to the neutral axis of these elements. 

For single column piers, C-shaped piers, or other unusual configurations, the rotational mass 

moment of inertia of the superstructure about the longitudinal axis should be included. 

The inertia of live loads need not be included in the seismic analysis. However, the probability 

of a large live load being on the bridge during an earthquake should be considered when 

evaluating bridges with high live-to-dead load ratios, including those bridges which are located 

in metropolitan areas where traffic congestion is likely. 

7.3.2 DISTRIBUTION OF STIFFNESS AND STRENGTH 

The mathematical model should represent the stiffness of individual structural elements 

considering material properties and section dimensions. When an elastic analysis is used to 

determine the response of an inelastic structure, significant approximations are necessary. The 

most important assumption is that stiffness may be based on an equivalent linearized value. For 

inelastic columns, common practice is to use cracked section properties for concrete members 

and full section properties for steel members. This value for stiffness is sometimes called a 

tangential stiffness. However, for seismic isolators, abutments, and foundation soils, stiffness is 

calculated from the maximum expected deformation. This value is then known as a secant 

stiffness. The distribution of forces from an elastic analysis should be carefully reviewed to 

verify that the results are consistent with the expected nonlinear behavior of the earthquake 

resisting elements. 

For Method D2, the Structure Capacity/Demand Method (Pushover or Nonlinear Static Method)

described in section 5.6, the mathematical model should use strength values based on expected 

material properties. For nonlinear dynamic analysis, the model should use the actual stiffness 

and strength values of the hysteretic elements under seismic loads. 
226

7.3.2.1. Substructures 

The stiffness of structural steel members should be based on elastic properties. For reinforced 

concrete piers, stiffness should be based on cracked section properties, as explained in this 

section. 

Analytical methods for seismic demand generally use stiffness values that are representative of 

deformations close to the deformations at yield. At these levels of deformation, even prior to 

yield, reinforced concrete elements crack. The effect of cracking on stiffness depends on the 

cross-section, longitudinal reinforcement ratio, axial load, and amount of bond slip. The cracked 

flexural stiffness of a reinforced concrete member can be obtained by moment–curvature 

analysis of the cross-section, with modifications for bond-slip. In many cases, it is impractical to 

compute the effective stiffness in this manner, and instead, effective stiffness quantities are 

assumed based on component rigidity values such as those shown in table 7-1. 

Table 7-1. Component rigidities. 

Component Flexural 

Rigidity 

Shear 

Rigidity 

Axial 

Rigidity 

Reinforced concrete columns, beams and caps 

where cracking (but not hinging) is expected 

0.5 Ec Ig 0.4 Ec Aw Ec Ag

Prestressed concrete beams, caps and piles 

where cracking is not expected 

Ec Ig 0.4 Ec Aw Ec Ag

Concrete columns, piles and walls where plastic 

hinging is expected ε

n

y

M D'

2

0.2 Ec Ag 0.5 Ec Ag

Note: Ec = elastic modulus of concrete, 

 Ig = moment of inertia using gross dimensions, 

 Aw = shear area of column or beam, 

 Ag = cross-sectional area of column or beam using gross dimensions, 

 Mn = nominal moment capacity of column or beam, 

 D’ = distance between outer layers of longitudinal reinforcement, and 

εy = yield strain of steel reinforcement. 

From table 7-1 it is noted that for flexurally-dominated regions, where plastic hinging is 

expected to occur at the ends of the member, the effective flexural rigidity, EcIeff , may be 

determined from the ‘theoretical’ yield curvature given by: 

y n y c eff

2 M

 = = 

D E I

ε φ ′ (7-1) 

where: 

εy = yield strain = fy / Es, 

 fy = yield stress (theoretical value, i.e., specified value), 
227

 Es = elastic modulus of steel, 

 D' = center-to-center distance between the outer layers of longitudinal reinforcement in a 

rectangular section normal to axis of bending, or pitch circle diameter of the 

longitudinal reinforcement in a circular section, generally assumed to be 80 percent 

of overall beam depth or overall column diameter, and 

 Mn = nominal yield moment (theoretical yield strength) of the member (see section 7.5). 

Thus the effective flexural rigidity of a severely cracked structural concrete column is: 

n c eff

y

M D E I = 

2

′

ε

 (7-2) 

This rigidity will generally be somewhat less than 0.5 EcIg (Ig is the moment of inertia of the gross, 

uncracked section), which is a commonly assumed value for reinforced concrete members, when 

only a moderate amount of cracking and no plastic hinging is expected. 

Table 7-1 also lists effective values for the shear and axial rigidity of cracked and uncracked 

reinforced concrete beams and columns with and without plastic hinges. In addition, the torsional 

stiffness of a cracked reinforced concrete column may be taken as 20 percent of the uncracked 

value. 

For a displacement capacity assessment (pushover method), the strength of structural steel 

components in the model should be based on their expected plastic capacity. The flexural 

strength of reinforced and prestressed elements should be based on the expected material 

properties of the steel and concrete. The objective here is to determine the displacement at 

which the inelastic components reach their deformation capacity. The deformation capacity is 

usually expressed in terms of a maximum plastic rotation of hinge zones. The maximum 

deformation capacity is the sum of the deformation at yield and the plastic deformation (see 

equation 7-31). 

The stiffness of other elements that are not subjected to inelastic response and damage should be 

based on elastic properties, including the effects of concrete cracking. The stiffness of pier caps 

should be included in the model. Pile caps and joints in reinforced concrete substructures may 

be assumed to be rigid. 

7.3.2.2. Superstructures 

The stiffness of the superstructure should be consistent with the load path (section 7.2) including 

consideration of composite behavior between girders and decks, and effective width of the 

superstructure components that are monolithic with piers. 

For a stick model of the superstructure, the stiffness can be represented by equivalent section 

properties for axial deformation, flexure about two axes, torsion, and possibly shear deformation 

in two directions. The calculation of the section stiffness should represent reasonable 

assumptions about the three-dimensional flow of forces in the superstructure, including 

composite behavior. 
228

The effects of skew can be neglected in the model of the superstructure. However, for large 

skew angles, the geometry of the connection between superstructure and piers must be included. 

For reinforced concrete box girders, the effective stiffness may be based on 75 percent of the 

gross stiffness to account for cracking. For prestressed concrete box girders, the full gross 

stiffness should be used. The torsional stiffness may be based on a rational shear flow without 

reduction due to cracking. 

The flexural stiffness of the superstructure taken about a transverse axis should be reduced near 

piers when there is a moment transfer between the superstructure and pier because of shear lag 

effects. The reduced stiffness should be used in modeling the superstructure. 

 7.3.3. IN-SPAN HINGES 

Two different models are used to represent expansion bearings and in-span hinges. First, a 

compression model is used that assumes the joint at the bearing or hinge is closed and can 

transfer longitudinal forces. Then, a tension model is used that assumes the joint or hinge is 

open and cannot transfer longitudinal forces. The stiffness of restraining devices, such as cable 

restrainers or shock transmission units, is included in the tension model. Two separate analyses 

are run, one for each model, and the larger value for a particular force or deformation is used in 

design. 

The use of compression and tension models is expected to provide a reasonable bound on forces 

(compression model) and displacements (tension model). A compression model need not be 

considered for expansion bearings if calculations show that longitudinal forces cannot be 

transferred through the superstructures at the bearing. 

7.3.4. DAMPING 

Energy dissipation in a bridge, including that developed by the footings and abutments, may be 

represented with sufficient accuracy by viscous damping. The selection of an effective viscous 

damping ratio depends on the type of dynamic analysis and the configuration of the bridge. In an 

elastic response spectrum analysis, this ratio is based on the energy dissipation due to small-tomoderate deformations of the members and soil. 

Damping may be neglected in the calculation of natural frequencies and associated mode shapes. 

The effects of damping should be considered when the dynamic response for seismic loads is 

considered. This is usually done by scaling the earthquake response spectrum for the correct 

amount of damping. 

Suitable damping values may be obtained from field measurements of induced free vibrations or 

from forced vibration tests. In lieu of measurements, the following values may be used for the 

equivalent viscous damping ratio: 
229

• Concrete bridges: five percent. 

• Welded and bolted steel bridges: two percent. 

• Timber bridges: five percent. 

For a single-span or two-span bridge with abutments that are expected to develop significant 

passive pressure in the longitudinal direction, a damping ratio of up to 10 percent may be used 

for the longitudinal vibration modes. 

Equivalent viscous damping may be used to represent the energy dissipated in the cyclic loading 

of members beyond yield, but only when a secant stiffness model is used for the entire bridge. 

For single-degree-of-freedom models, the equivalence can be established within a satisfactory 

degree of accuracy. For bridges with seismic isolation or with fuses and knock-off walls, only an 

approximate equivalence can be established. Equivalent viscous damping should not be used to 

represent inelastic energy dissipation for any other model or method of dynamic analysis. 

7.3.5. PERMANENT GROUND MOVEMENT 

In general, the effect of gross soil movement and liquefaction should be included in the analysis. 

However, the need for sophisticated modeling of foundations and abutments depends on the 

sensitivity of the structure to foundation movements and the degree of conservatism that can be 

tolerated in the calculated forces. When gross soil movement or liquefaction is possible, and an 

analysis is required, the model should represent the change in support conditions and additional 

loads imposed on the substructure due to soil movement. 

When the results of a seismic analysis are expected to be sensitive to foundation properties, these 

properties should be bounded between upper and lower limits of strength and stiffness and 

multiple analyses carried out. Strength and stiffness values should, however, be consistent with 

the expected deformations of the soil and the goals of the analysis. The sensitivity of results to 

the assumptions made about these properties may then be used to direct site investigations for the 

bridge under consideration. 

See also chapter 11, which describes retrofitting measures for bridges on hazardous sites. 

7.4. COMBINATION OF SEISMIC FORCE EFFECTS 

7.4.1. SEISMIC LOADING IN ONE DIRECTION 

Maximum response quantities for use in design, due to seismic loading in one direction, should 

be based on a combination of modal responses in that direction. The preferred method for this 

combination is the Complete Quadratic Combination (CQC) method (section 5.4.2.2), because it 

combines modes with closely-spaced periods more rigorously than other modal combination 

rules. 
230

7.4.2. SEISMIC LOADING IN TWO OR THREE ORTHOGONAL DIRECTIONS 

When combining the responses of two or three orthogonal directions, the design value of any 

quantity of interest (displacement, bending moment, shear or axial force) should be obtained 

either by the ‘square root of the sum of the squares’ (SRSS) rule or the 100-40 percent 

combination rule. 

The SRSS rule is the most appropriate one for combining the contribution of orthogonal and 

uncorrelated ground motion components into a single design force. The method is especially 

recommended if the vertical components of the ground motion are being used in combination 

with the horizontal components (Button et al., 1999). However, the familiar 100-30 percent 

combination rule is also suitable, particularly if the second component is increased from 30 to 40 

percent. This option is, however, more conservative than the SRSS combination rule. 

These two alternative combination rules may be summarized as follows. 

• SRSS Combination Rule – the design value is the SRSS combination of the response quantity 

from each of the orthogonal directions: 

V 2

x

L 2

x

T 2

x x M = (M ) + (M ) + (M ) (7-3) 

 

 where Mx

T

 , Mx

L

 , Mx

V are the x-components of moment calculated from a transverse, 

longitudinal and vertical analysis, respectively. 

• 100–40 percent Combination Rule – the design value is obtained from the largest value given 

by the following three load cases. 

Load Case 1 (LC1) – 100 percent of the absolute value of the response resulting from the 

analysis in one orthogonal direction (transverse) is added to 40 percent of the responses 

resulting from analyses in the other two orthogonal directions (longitudinal and vertical): 

 V

x

L

x

T

x

LC1 Mx = M + 0.4M + 0.4M 

Load Case 2 (LC2) – 100 percent of the absolute value of the response resulting from an 

analysis in one of the other orthogonal directions (longitudinal) added to 40 percent of the 

responses resulting from analyses in the other two orthogonal directions (transverse and 

vertical): 

 V

x

L

x

T

x

LC2 Mx = 0.4M + M + 0.4M 

Load Case 3 (LC3) – 100 percent of the absolute value of the response resulting from an 

analysis in one of the other orthogonal directions (vertical) added to 40 percent of the 

response values resulting from analyses in the other two orthogonal directions (transverse and 

longitudinal): 
231

 V

x

L

x

T

x

LC3 Mx = 0.4M + 0.4M + M 

 The response to be used in design is given by the largest value from these three load 

cases, 

 i.e., Mx = maximum of [ Mx

LC1, MxLC2, MxLC3] (7-4) 

If the seismic loading is applied in only two orthogonal directions (longitudinal and transverse), 

the above rules still apply, but Mx

V is set equal to zero, and Load Case 3 need not be calculated 

i.e., Mx

LC3 = 0. 

7.4.3. COMBINATION OF RESPONSE QUANTITIES FOR BIAXIAL DESIGN 

When the biaxial response of a member is important (e.g., for circular columns), it is necessary 

to find the maximum resultant moment on the column section and the corresponding axial load. 

This is not easy because the maxima of the three components (i.e., axial force, P, and bending 

moments about two local axes, Mx and My) are not likely to occur at the same time. A 

sophisticated approach to finding the critical combination is difficult to justify for most bridges, 

and a simpler approach may be used. Again two choices, based on the SRSS combination rule 

and the 100-40 percent rule, are available. They are as follows. 

For the SRSS combination, the two components to be combined, Mx and My, are first found from 

equation 7-3 and then a 100-40 percent rule is used to find a vector sum. There are two possible 

vector sums and the maximum is taken as the moment resultant MR. This moment is then used 

with the maximum positive or negative axial force to evaluate the column. 

i.e., 

 MR = maximum of ( ) ( ) 2 2 2 2 M 0.4M , or 0.4M M x y xy

⎡ ⎤ + + ⎢ ⎥ ⎣ ⎦ (7-5) 

and MR is used with the maximum axial load of ± P to evaluate the column. 

For the 100-40 percent combination rule, the Mx and My components from each load case are 

combined to obtain the vector sum, and the maximum moment of the three load cases is taken as 

the moment resultant MR. This moment is then used with the maximum positive or negative axial 

load to evaluate the column. 

i.e., 

MR = maximum of ( )( ) ( )( ) ( )( ) 22 2 2 2 2 LC1 LC1 LC2 LC2 LC3 LC3 M M , M M ,or M M xy x y x y

⎡ ⎤

⎢ ⎥ ++ +

⎣ ⎦

 

 (7-6) 

and MR is used with the maximum axial load of ± P to evaluate the column. 
232

7.4.4. VERTICAL ACCELERATION EFFECTS 

A limited study of the effect of vertical accelerations on the behavior of bridges indicates that 

these effects can be significant (Button et al., 1999). This is particularly true for some response 

parameters (such as superstructure moments and shears, and column axial forces) and certain 

bridge types (such as those with long flexible spans, C-shaped piers, or with other large 

eccentricities in the load path for vertical loads). The study was based on vertical response 

spectra developed from recorded ground motions in the western United States (Silva, 1997). 

Response spectra for vertical ground motions are discussed further in section 2.7. 

Until more is known about the characteristics of vertical ground motions in the eastern United 

States and those areas affected by subduction zones, such as in the Pacific Northwest, 

recommendations for design should be primarily cautionary and optional. However, vertical 

acceleration effects may be significant and should be assessed for important bridges. For this 

purpose, the following recommendations may be helpful. 

The impact of vertical ground motion may be ignored if the bridge is greater than 50 km (30 mi) 

from an active fault. If the bridge site is within 10 km (6 mi) of an active fault, then a sitespecific study is required if the response of the bridge could be significantly and adversely 

affected by vertical ground motions. In such cases, response spectra and acceleration time 

histories should be developed for use in the response analysis of the bridge. For vertical 

response forces, the linear analysis should use the CQC modal combination method and the 

SRSS directional combination method. 

If the bridge is more than 10 km (6 mi) but less than 50 km (30 mi) from an active fault, a sitespecific study should be performed to evaluate the effects of vertical ground motion. 

In lieu of a dynamic analysis that explicitly includes the effect of vertical ground motion, the 

following variations in column axial loads and superstructure moments and shears should be 

included in the evaluation of the columns and superstructure to account for the effects of vertical 

ground motion. 

• Column axial loads = (1 + CV) axial forces due to dead load. 

• Superstructure bending moments = (1 + CV) bending moments due to dead load. 

• Superstructure shears = (1 + CV) shear forces due to dead load. 

In the above bullets, CV is a dead-load multiplier and recommended values are given in table 7-2. 

For superstructures, CV is specified at both a mid-span location and a pier. Linear interpolation 

may be used to determine values of CV for points between these two locations. 
Table 7-2. Dead load multipliers (Cv) for bridges subject to vertical acceleration for rock sites. 

Distance from Fault (km), D 

Member Force / Moment 

0≤ D<10 10≤D<20 20≤D<30 30≤D<40 D ≥ 40 

Pier axial force 0.7 0.3 0.2 0.1 0.1 

Superstructure shear force at pier 0.7 0.4 0.2 0.1 0.1 

Superstructure bending moment at pier 0.6 0.3 0.2 0.1 0.1 

Superstructure shear force at mid-span 0.1 0.1 0.1 0.1 0.1 

Superstructure bending moment at 

mid-span 1.4 0.7 0.4 0.3 0.2 

Note: 

The dead load multipliers given above are in addition to the dead load. Thus the actual ‘load factor’ is 

1.0 plus/minus the tabulated values. 

Button et al., 1999

7.5. MEMBER STRENGTHS 

It is difficult to calculate the precise capacity of a bridge because the actual strength of concrete 

and steel may vary considerably from their specified values. Furthermore, deviations from 

specified dimensions due to construction tolerances and simplifying assumptions made in design 

can lead to both under- and over-capacity in the field. However, it is possible to define different 

levels of member strength, which are then used in various types of calculations for capacity 

assessment of a complete bridge. Four strength levels are defined as follows: 

• Nominal strength, Sn

The nominal strength of a member is obtained from failure theory for a section of the 

member and is based on assumed section geometry and specified material strengths. Other 

strength levels are usually related to the nominal strength. 

• Design strength, Sd

The design strength of a member allows for approximations in calculations and variations in 

material strengths, workmanship, and dimensions. Each of these may be within tolerable 

limits, but in combination, they may result in a capacity that is less than the nominal 

strength. A strength reduction factor (φ) is used to relate the design strength to the nominal 

strength, where φ is always less than unity. (The resistance factors in the AASHTO LRFD 

Specifications (1998) or the current LRFD specification may be used here.) 

i.e., Sd = φ Sn (7-7a) 

233
• Expected strength, Se

The expected strength takes into account the fact that the material strengths are generally 

greater than the specified strengths. Expected strengths can be found from routine testing of 

material samples taken from existing bridges, but if these are not available, they are based 

on previous experience with the same materials in other bridges. An expected strength 

factor (φe) is used to relate the expected strength to the nominal strength, where φe is 

always greater than unity. 

i.e., Se = φe Sn (7-7b) 

Values for φe may be obtained by taking material samples in the field and testing for actual 

strength, or by taking values from the mill certificates provided by the supplier at the time 

of construction, if these are still available in the bridge file. In the absence of such data, φe

may be taken as 1.2 and 1.3 for steel and concrete, respectively. 

• Overstrength, So 

Overstrength takes into account all possible factors contributing to a strength increase 

above the nominal value. These include but are not limited to: 

- Steel strength higher than the specified yield strength, 

- Additional steel strength due to strain hardening at large deformations, 

- Concrete strength higher than specified, 

- Section sizes larger than assumed, 

- Axial compression in flexural members due to lateral restraint, 

- Additional reinforcement placed for construction purposes and unaccounted for in design, 

Conservative assumptions made in the derivation of the equations for nominal strength. 

An overstrength factor (φo) is used to relate the overstrength to the nominal strength, where 

φo is always greater than unity. 

i.e., So = φo Sn (7-7c) 

7.6. MEMBER ACTIONS IN PIERS USING CAPACITY PRINCIPLES 

Capacity design principles are used to calculate member actions in bridge piers due to plastic 

hinging. This process is described in this section for both single-column and multi-column piers 

and for extensions to drilled shafts and piled bents are also given. These procedures are the same 

as those given in Article 7.2.2 of the AASHTO Standard Specifications (AASHTO, 2002). A 

worked example may be found in appendix A of the AASHTO Guide Specifications for Seismic 

Design of Highway Bridges (AASHTO, 1983). 

7.6.1 SINGLE COLUMN PIERS

Column shear forces and moments in the superstructure, pier caps, and footings should be 

calculated for the two principal axes of a column and in the weak direction of a pier as follows: 

234
235

Step 1. Determine the column overstrength moment capacities (sections 7.5 and 7.7.1.2). For 

reinforced concrete or structural steel columns, apply an overstrength factor (φo) of 1.4 

to the nominal moment, and use a column axial load equal to the sum of the axial load 

calculated from an elastic analysis and the column dead load. Column overstrength 

moments should be distributed to the connecting structural elements. 

Step 2. Using the column overstrength moments, calculate the corresponding column shear 

force(s) that satisfy static equilibrium. For columns that have flares that are integral 

with the superstructure, the shear should be calculated as the larger of: 

1. The value obtained from the overstrength moments at the top of the flare and 

bottom of the column immediately above the top of the footing, using the clear 

height from top of footing to underside of pier cap, and 

2. The value obtained from using the overstrength moments at the bottom of the flare 

and top of the footing, with the column height reduced to the distance between 

them. 

This method is satisfactory for most columns, but may be unconservative for shear when the 

column contributes a significant portion of the total mass of the bridge. 

If the footing of a column is significantly below ground level, consideration should be given to 

the possibility of a plastic hinge forming below ground level. In such a case, the hinge may not 

form directly above the footing as assumed in Step 2 above, due to restraint provided by the 

overlying soil. If this can occur, the estimated length of the column between plastic hinges 

should be used in Step 2 to calculate the shear force. 

For pile bents or drilled shafts, the length of the column between plastic hinge locations should 

again be used. Assume that hinges below ground level occur no lower than two pile/shaft 

diameters below the mud line. 

In summary, column forces corresponding to plastic hinging in a single column are: 

• Axial forces using unreduced maximum and minimum axial load from elastic analyses, plus 

dead load. 

• Bending moments as calculated in Step 1. 

• Shear forces as calculated in Step 2. 

7.6.2. MULTI-COLUMN PIERS 

The forces in piers with two or more columns should be calculated both in the plane of the pier 

and perpendicular to the plane of the pier. Forces perpendicular to the plane of the pier should 

be calculated as for single columns in section 7.6.1 above. Forces in the plane of the pier should 

be calculated as follows: 
236

Step 1. Determine the column overstrength moment capacities. Use an overstrength factor (φo) 

of 1.4 for both reinforced concrete and structural steel to obtain these moments. Set the 

axial load equal to the dead load. 

Step 2. Using the column overstrength moment capacities, calculate the corresponding column 

shear forces. Sum the column shears in the pier to determine the maximum shear force 

for the pier. Note that if a partial-height wall exists between the columns, the effective 

column height is taken from the top of the wall. For flared columns and foundations 

below ground level, see Step 2 in section 7.6.1. For pile bents, the portion of pile from 

the pile cap to a length of two diameters below the mud line should be used to calculate 

the shear force. 

Step 3. Apply the pier’s shear force to the center of mass of the superstructure above the pier, 

and, using static equilibrium, calculate the axial forces in the columns due to this force 

when the column overstrength capacities are reached. 

Step 4. Combine these column axial forces with the dead load axial forces, and calculate 

revised column overstrength moments. With the revised overstrength moments, 

calculate the column shear forces and the maximum shear force for the pier. If the 

maximum shear force for the pier is not within 10 percent of the value determined in 

Step 2, use this maximum pier shear force and return to Step 3. 

In summary, column forces corresponding to plastic hinging in individual columns in the plane 

of a pier are: 

• Axial forces using the maximum and minimum axial load and dead load, plus or minus the 

axial load determined from the final iteration of Step 3. 

• Column overstrength moment capacities corresponding to the maximum compressive axial 

load specified in Step 1 with an overstrength factor of 1.4 for reinforced concrete and 

structural steel. 

• Shear forces corresponding to the final column overstrength moments in Step 4 above. 

7.7. STRENGTH CAPACITY OF BRIDGE MEMBERS 

Guidelines for determining the capacity of bridge members in terms of strength are given in this 

section for use when assessing the strength capacity of complete bridges or bridge substructures 

as required in Methods D1 and D2 of chapter 5. Guidelines for determining displacement 

capacity are given in the following section (section 7.8). 

In particular, procedures are given for calculating the following: 

• Flexural strength of reinforced concrete columns and beams (section 7.7.1): 

- Expected flexural strength (7.7.1.1), 

- Flexural overstrength capacity (7.7.1.2), and 
237

- Flexural strength of columns with lap-splices in plastic hinge zones (7.7.1.3). 

• Shear strength of reinforced concrete columns and beams (section 7.7.2): 

- Initial shear strength (7.7.2.1), and 

- Final shear strength (7.7.2.2). 

• Shear strength of reinforced concrete beam-column joints (section 7.7.3): 

- Maximum beam-column joint strength (7.7.3.1), and 

- Cracked beam-column joint strength (7.7.3.2). 

7.7.1. FLEXURAL STRENGTH OF REINFORCED CONCRETE COLUMNS AND BEAMS 

Several types of member flexural strength are required for a detailed capacity assessment of a 

bridge. These include the: 

• Expected flexural strength based on material expected strength factors and equation 7-7b. 

• Flexural overstrength based on member overstrength factors and equation 7-7c. 

• Expected flexural strength of columns with lap-splices located in potential plastic hinge 

zones, based on expected strength factors. 

7.7.1.1. Expected Flexural Strength 

The flexural strength of members (either with or without axial loads) is determined based on 

monotonic behavior, where strains are assumed to vary linearly across the section, and the 

stresses are related to strains through material property laws. If a moment-curvature analysis is 

undertaken, then it is possible to define the expected yield strength and determine the maximum 

overstrength capacity and deformation capacity. However, for most situations, this effort is not 

justified and simplified methods, as described below, are used. 

The expected flexural strength (at theoretical yield) should be based on a modified form of the 

nominal strength calculations required in the AASHTO LRFD Specification (AASHTO, 1998). A 

rectangular stress block for unconfined concrete and elasto-plastic behavior for steel is assumed. 

The following assumptions are made: 

• Maximum concrete strain at the outermost concrete fiber is 0.003. 

• Average concrete stress in the effective compression zone is 0.85f'ce, where f'ce is the 

expected concrete strength. 

• Expected yield stress of the longitudinal reinforcement (fye) is 300 MPa (43.5 ksi) and 450 

MPa (65.3 ksi) for Grade 40 and Grade 60 steel respectively, in lieu of material tests on 

samples taken from the bridge. 

• Effective depth of the concrete stress block is a = β1 c, 
238

where c is the neutral axis depth, and β1 = 0.85 for '

ce f ≤ 30 MPa (4.3 ksi) and β1 = 0.65 for 

'

ce f ≥ 60 MPa (8.7 ksi). Linear interpolation is used for β1 if 30 MPa < 'ce f < 60 MPa (4.3 

ksi < '

ce f < 8.7 ksi). 

The expected flexural strength (Me) is calculated using a plastic analysis approach as given 

below (Mander et al., 1998b): 

2

e b

' '

e b ce ce g g

' ce g t b ce g ' '

ce ce g g

P P - M M f f A A = 1 - 

f AD f D A P P - f f A A

⎡ ⎤ ⎛ ⎞ ⎢ ⎥ ⎜ ⎟ ⎛ ⎞ ⎢ ⎥ ⎜ ⎟ ⎢ ⎥ ′ ⎜ ⎟⎜ ⎟ ⎝ ⎠ ⎢ ⎥ ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦

 (7-8) 

where: 

e

'

ce g

P

f A

 = axial stress ratio based on gravity load and seismic actions, 

ye t

t ' '

ce ce g

f P f f A

= − ρ = axial tensile capacity ratio of the column based on the expected material 

properties, 

b

g

' 1

ce

P = 0.425 = 

f A

 β axial load capacity ratio at the maximum nominal (balanced) moment on the 

section, 

b

'

ce g

M

f AD

 = ye b o

shape t ' '

ce ce g

f D' P 1 K

f fA D 2

⎛ ⎞ − κ ⎜ ⎟ ρ +

⎝ ⎠

, 

β1 = stress block factor ( ≤ 0.85), 

D' = pitch circle diameter of the reinforcement in a circular section, or the out-to-out 

dimension of the reinforcement in a rectangular section, generally assumed to be equal 

to 80 percent of overall diameter of column, D, 

Kshape = shape factor: 

= 0.32 for circular sections, 

 = 0.375 for square sections with 25 percent of the reinforcement placed in each face, 

= 0.25 for walls with strong axis bending, and 

= 0.50 for walls with weak axis bending. 

κo = factor related to the centroid of the stress block: 

 = 0.6 for circular sections, and 

 = 0.5 for rectangular sections. 

7.7.1.2. Flexural Overstrength Capacity 

In any method of capacity assessment that is based on a collapse mechanism, it is necessary to know 

the upper bound on flexural capacity, i.e., the flexural moment overstrength capacity, Mpo , of the 
239

columns, piers, and piles that form the primary load resisting mechanism. Three different 

approaches are described below for the calculation of Mpo. 

Method 1. A compatibility section analysis is made to calculate the flexural overstrength 

capacity, Mpo, taking into account the expected strengths of the materials, the properties of 

confined concrete, and the strain hardening effects of the longitudinal reinforcement. 

Method 2. The flexural overstrength capacity, Mpo, is estimated by scaling the expected strength 

using an empirical factor, e.g., Mpo = 1.4 Me, where Me is the expected flexural strength. 

Method 3. The flexural overstrength capacity, Mpo, is estimated from a sectional plastic analysis 

using the properties of confined concrete and the ultimate tensile strength for the stress in the 

longitudinal reinforcement. (Mander et al., 1998a). In this method, Mpo is given by 

2

e bcc

' ' po g g bo c c

' ' to bcc cg c g ' '

c c g g

P P - M M f f A A = 1 - 

fAD f D A P P - f f A A

⎡ ⎤ ⎛ ⎞ ⎢ ⎥ ⎜ ⎟ ⎛ ⎞ ⎢ ⎥ ⎜ ⎟ ⎢ ⎥ ⎜ ⎟⎜ ⎟ ⎝ ⎠ ⎢ ⎥ ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦

 (7-9) 

where: 

e

'

c g

P

f A

= column axial stress ratio based on gravity loads and seismic actions, 

to su

' '

cg c

P f

fA f = − ρ axial tensile capacity ratio of the column, 

bcc cc

'

c g g

P A = 0.5 = 

f A A αβ axial load capacity ratio at the maximum confined (balanced) moment on 

 the section, 

bo su bcc o

shape t ' ''

cg c cg

M f P1 D' K

fAD f fA D 2

⎛ ⎞ − κ

= ρ + ⎜ ⎟

⎝ ⎠

, 

D' = pitch circle diameter of the reinforcement in a circular section, or the out-to-out 

 dimension of the reinforcement in a rectangular section (normal to the axis of 

 bending), generally assumed to be equal to 80 percent of overall diameter of column, 

 D (or overall depth of beam), 

fsu = ultimate tensile strength of the longitudinal reinforcement 

 = 1.5 fye unless determined otherwise by coupon tests, 

α, β = stress block factors for confined concrete defined below, 

Acc = area of confined concrete core, and 

Ag = gross section area. 

Stress block factors α, β are as follows: 

α = ratio of average concrete stress in compression zone to confined concrete strength 
240

 = 0.85 + 0.12(K – 1)0.4, 

K = strength enhancement factor due to the confining action of the transverse 

reinforcement, and is given below for circular and rectangular sections 

 = f'cc/ f'ce, 

'

cc f = confined concrete strength, and 

β = depth of stress block 

 = 0.85 + 0.13(K – 1)0.6. 

For circular sections, the confined strength parameter (K) is given by Mander et al. (1988): 

' '

' '

ce ce

f f K = 2.254 1 + 7.94 2 1.254

f f − − A A 

where: 

' e yh s

1 f = k f 2 A ρ = lateral stress supplied by the transverse reinforcement at yield, 

bh s

4 A

 = 

s D" ρ = volumetric ratio of spirals or circular hoops to the core concrete, 

e

cc

(1 - s / D") k = 

(1 - )

χ

ρ

 = confinement effectiveness coefficient for spirals and hoop steel, 

χ = coefficient with values of 0.5 and 1.0 for spirals and hoops, respectively, 

s = spacing of spirals or hoops, and 

D'' = diameter of transverse hoop or spiral (measured to the centerline of the hoop). 

For rectangular sections, the confined strength parameter (K) is obtained from figure 7-2 which 

uses the x- and y- confining stresses ( '

x fA and '

y fA respectively) to give K (Mander et al., 1988; 

Paulay and Priestley, 1991). Stresses '

x fA and '

y fA are defined as follows: 

'

x e yh x f = A k f ρ is the lateral confining stress in the x-direction, 

'

x e y yh f = k A ρ f is the lateral confining stress in the y-direction, 

sx x

A

 = 

s D" ρ is the volume ratio of transverse hoops or ties to the core concrete in x-direction, 

sy

y

A

 = 

s b" ρ is the volume ratio of transverse hoops or ties to the core concrete in y-direction, and 

ke = confinement effectiveness coefficient for rectangular sections with hoops or ties: 

 = 0.75 for rectangular columns, and 

 = 0.6 for rectangular wall sections. 

fyh = yield stress of the transverse hoops, 

D'' = width of column normal to x-direction (measured to the centerline of hoops or ties), 

 and 

b'' = width of column normal to y-direction (measured to the centerline of hoops or ties). 
241

Paulay and Priestley, 1991 

Figure 7-2. Confined strength ratio (K) for reinforced concrete members. 

7.7.1.3. Flexural Strength of Columns with Lap-Splices in Plastic Hinge Zones 

The flexural strength of a column is reduced when the longitudinal steel is lapped in the plastic 

hinge zone, which is a common occurrence near the base of columns in older bridges. As a first 

approximation, the reduction in strength is assumed to depend solely on the length of the 

splice, llap, compared to the minimum required value. The reduced moment (Ms) is then given by: 

 Ms = Me (llap / ls) but not greater than Me (7-10) 

where: 

 Me = expected flexural strength defined in section 7.5, 

 ls = theoretical lap-splice length and is determined from: 

 b '

ce

ye

s d

f

f

l = 0.4 , (7-11) 

 db = diameter of the longitudinal reinforcing bar in the lap, 

 fye = expected yield strength (MPa) of the longitudinal reinforcement in the lap-splice 

zone, and 

'

ce f = expected strength (MPa) of concrete surrounding the lap-splice zone. 
242

7.7.2. SHEAR STRENGTH OF REINFORCED CONCRETE COLUMNS AND BEAMS 

The shear resistance of cracked structural concrete members is reduced by load reversals and 

increasing plastic hinge rotations. As a consequence, two shear strength states are defined: 

1. Vi the initial shear strength, and 

2. Vf the final shear strength. 

The difference between the two states is due to deterioration in the concrete as the cyclic loading 

progresses with a corresponding reduction in the concrete contribution, Vc, to the shear capacity. 

This contribution diminishes due to the presence of widely-spaced tensile cracks and yielding of 

the longitudinal bars. Expressions for the initial and final shear strengths are given in the 

following sections. 

7.7.2.1. Initial Shear Strength, Vi

Initial shear strength, Vi, is given by: 

 Vi = Vs + Vp + Vci (7-12) 

 

where: 

 Vs = contribution to the shear strength provided by rebar truss action, 

 Vp = contribution provided arch (strut) action, and 

 Vci = contribution provided by diagonal tension field in the concrete. 

Each of these contributions is described below. 

The shear resistance provided by truss action in the reinforcing steel is given by: 

s v yh

D"

V = cot A f

s

θ (7-13) 

where: 

 Av = shear area provided by the transverse hoops: 

 = 2 Abh for rectangular columns, and 

 = π Abh / 2 for circular columns. 

 Abh = area of one spiral or hoop bar, 

 fyh = yield stress of the transverse hoops or spirals, 

 D'' = diameter of transverse hoop or spiral (measured to the centerline of the hoop), 

 s = center-to-center longitudinal spacing of the transverse hoop steel, 

 θ = angle of the principal crack plane (Kim and Mander, 1999) and given by 

tan α

A

1.6 A

tan θ

0.25

t g

v e ≥ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

Λρ

ρ = , (7-14) 

 
243

Λ = fixity factor: 

 = 2 for fixed-fixed column conditions, and 

 = 1 for fixed-pinned conditions. 

 Ae = effective shear area (assumed as 80 percent of the gross area, Ag, for rectangular 

and circular sections), 

ρt = ratio of the total area of longitudinal reinforcement to the gross section area 

 = Ast / Ag, 

ρv = volumetric ratio of transverse steel: 

 = Av / bws for rectangular columns, and 

 = ρs/2 = 2 Abh/sD'' for circular columns. 

 Av = area of transverse shear steel, 

 bw = center-to-center spacing of transverse shear steel across width of rectangular 

column, 

 tan α = corner-to-corner strut angle: 

 = jd/L, 

 jd = internal lever arm of the concrete compression member, and 

 L = length of the column. 

The shear resistance provided by the arch (strut) action is given by: 

α

Λ P tan 2 Vp = (7-15) 

where Λ is the fixity factor defined above, P is the axial load on member (compression only), 

and tan α is the corner-to-corner strut angle defined above. 

The shear resistance provided by the diagonal tension field in the concrete is: 

V = 0.3 f Ae '

ce ci (7-16) 

where Ae is the effective shear area equal to 0.8Ag. 

7.7.2.2. Final Shear Strength, Vf

The final shear strength is given by: 

 Vf = Vs + Vp + Vcf (7-17) 

In this equation, Vs and Vp are as defined above, while Vcf is the final shear strength carried by the 

concrete, which is reduced to allow for plastification, cracking, and cyclic loading effects. This is 

given by: 

V = 0.05 f Ae '

ce cf (7-18) 
244

7.7.3. SHEAR STRENGTH OF BEAM-COLUMN JOINTS 

The shear strength of beam–column joints can be assessed in a manner similar to column 

elements. A principal stress approach is usually adopted, as this is a major determinant in joint 

performance. A beam–column joint will remain essentially elastic and uncracked, providing that 

the principal tensile stress in the joint is less than 0.29 '

ce (f ) MPa. When the principal tensile 

stress exceeds this level, diagonal cracking of the joint can be expected. The joint strength will 

be limited to a maximum principal stress of 0.42 '

ce (f ) MPa, at which point, full diagonal 

cracking of the joint will have developed. If the joint shear demand arising from the flexural 

overstrength capacity exceeds this limit, then the joint is said to be shear-critical and the 

structural performance will be governed by the reduced joint strength. The shear strength of the 

beam-column joint is therefore given by: 

 Vj = νj Ajh (7-19) 

in which Ajh is the area of the beam-column joint in a horizontal plane, and νj is the average joint 

shear stress acting on the joint. The latter can be found from a Mohr’s circle analysis for principal 

stresses in a joint subject to the combined actions of fv, fh and vj as follows: 

( ) t v h v h

2

j t v = p − p f + f + 2f f (7-20) 

where: 

 fν = average axial stress on the joint (compression is negative), 

 fh = average horizontal axial stress on the joint (has a zero value unless the joint has 

 horizontal prestress, or similar, applied), and 

 pt = major principal tension stress (tension is positive). This term dominates 

performance of the joint; recommended values are given in following sections. 

7.7.3.1. Maximum Beam-Column Joint Strength, Vji

The maximum joint shear capacity is achieved when full diagonal cracking develops. This occurs 

when the principal tensile stress in the joint is given by: 

' t ce p = 0.42 (f ) MPa (7-21) 

where f 'ce is the expected concrete strength in MPa. This principal stress is used to calculate Vji

using equation 7-20. 

7.7.3.2. Cracked Beam–Column Joint Strength, Vjf

The residual joint shear capacity is maintained after the strength has deteriorated. This occurs 

when the principal tensile stress in the joint is given by: 
245

t ce p = 0.29 (f' ) MPa (7-22) 

where '

ce f is the expected concrete strength in MPa. This principal stress is used to calculate Vjf

using equation 7-20. 

7.8. DEFORMATION CAPACITY OF BRIDGE MEMBERS 

A displacement capacity evaluation of a bridge, or pushover analysis (Method D2, chapter 5), 

should be able to track the nonlinear relationship between load and deformation for the columns 

and beams as the lateral load is monotonically increased from an initial elastic condition to 

failure. This requires the estimation of the capacity of each of the critical structural members, 

from first yield until collapse, and at intermediate limit states. Therefore, member performance 

needs to be found in terms of force versus deformation, moment versus rotation, or shear force 

versus distortion. 

Cap beams, columns, footings and piles may be modeled as line elements with concentrated 

plastic hinges at their ends. Other models, such as those with distributed plasticity, distributed 

flexibility, or general-purpose nonlinear finite element approaches may also be used, provided it 

can be demonstrated that their theoretical response agrees with experimental results from 

members similar to those being modeled. 

Guidelines for determining the deformation capacity of bridge components are given in this 

section for use in calculating the strength-deformation capacity of bridge members as required in 

the structure capacity/demand methods described in chapter 5 (Methods D1 and D2). 

In particular, procedures are given for calculating the following: 

• Plastic curvatures, rotations and drift angles (section 7.8.1). 

• Deformation-based limit states (section 7.8.2). There are eight of these. 

• Neutral axis depth in columns and beams (section 7.8.3). 

7.8.1 PLASTIC CURVATURES AND HINGE ROTATIONS IN A CANTILEVER BEAM 

7.8.1.1. Deflections and Plastic Curvature, φp

The ultimate displacement, ∆u, of a cantilever column under a lateral (shear) load, F, is given by: 

∆u = ∆e + ∆p (7-23) 

where ∆e is the elastic component of displacement, and ∆p is the plastic component of 

displacement. 

Each of these two components is described below. 
246

For the elastic component, ∆e: 

3

e

c eff

F L

3E I

∆ = (7-24) 

where L is the length from the fixed end to the free tip (or the inflection point of a fixed-fixed 

beam-column element), and EcIeff is the effective flexural rigidity (equation 7-2) that reflects the 

degree of cracking in the member. 

When the plastic strength of the member (Fp, Mp), is reached ∆e = ∆y, and the nominal yield 

displacement is given by: 

3 2

p p 2

y y

c eff eff

FL M L 1 L

3E I 3EI 3 ∆= = = φ (7-25) 

where Mp is the plastic moment capacity and φy is the nominal yield curvature given by 

modifying equation 7-1 such that: 

 

E D'

2 f

D'

2

s

y y

y = ε

φ = (7-26) 

For the plastic component, ∆p: 

∆p = φp Lp (L – 0.5Lp) (7-27) 

where Lp is the equivalent plastic hinge length, and φp is the plastic curvature as defined below. 

The equivalent plastic hinge length is given by the semi-empirical equation: 

 

 Lp = 0.08L + 4400 εy db (7-28) 

where db is the diameter of the longitudinal tension reinforcement, and L is the shear span or 

effective height (i.e., L = M/V). 

If the column has been jacketed as part of the retrofitting process, then the equivalent plastic 

hinge length is modified to give: 

 

 Lp = Lgap + 8800 εy db (7-29) 

where Lgap is the clear gap between the edge of the jacket and the bottom of the pier cap or top of 

the footing. 
247

The plastic curvature capacity depends on the failure mode and is highly dependent on the 

quantity of transverse reinforcement that is present. Values of φp for several deformation-based 

limit states are given in section 7.8.2. φP can be expressed in terms of the ultimate (total) 

curvature φu as follows: 

φp = φu – φy (7-30) 

7.8.1.2. Plastic Hinge Rotation, θp

The analysis in section 7.8.1.1 for deformation is in terms of absolute displacement quantities. It 

is often more convenient to express deformation behavior in relative terms, i.e., drift which is the 

ratio of lateral displacement to column height, and is the same as the angular rotation of the 

member chord. Thus, in terms of drifts (or rotations): 

θu = θy + θp (7-31) 

where the elastic drift at yield is given by: 

D'

L

3

2 L

3

1

L y y

y

y = φ = ε

∆

θ = (7-32) 

and the plastic drift is given by: 

θp = φp Lp (7-33) 

The elastic component of the total drift is related to the member slenderness (L/D’) and the 

plastic drift is given by the plastic hinge rotation, which in turn is related to the plastic curvature 

within the hinge zone. 

7.8.2. CHARACTERIZATION OF DEFORMATION-BASED LIMIT STATES 

The plastic rotational capacity of a member should be based on the governing limit state for that 

member. As shown in the previous section, this rotation may be found using equation 7-33. The 

governing limit state is the state that has the least plastic rotational (or plastic curvature) 

capacity. Plastic curvatures (and therefore plastic rotations) for the following limit states are 

developed in this section and summarized in table 7-3: 

• Compression failure of unconfined concrete. 

• Compression failure of confined concrete. 

• Compression failure due to buckling of the longitudinal reinforcement. 

• Longitudinal tensile fracture of reinforcing bar. 
248

• Low cycle fatigue of the longitudinal reinforcement. 

• Failure in the lap-splice zone. 

• Shear failure of the member that limits ductile behavior. 

• Failure of the joint. 

Table 7-3. Values of plastic curvature corresponding to various limit states 

in reinforced concrete columns and beams. 

Column or Beam Limit State Plastic Curvature, φP

1,2 Equation 

Compression failure, unconfined 

concrete y

cu

p c − φ ε φ = 7-34 

Compression failure, confined concrete ( ) y

cu

p c d − φ − ′′

ε φ = 7-35 

Buckling of longitudinal bars y

b

p (c d )

− φ − ′

ε φ = 7-37 

Fracture of longitudinal reinforcement y

s max

p (d c)

− φ −

ε φ = 7-39 

Low-cycle fatigue of longitudinal 

reinforcement D'

2

(d d )

2 ap ap

p

ε = − ′ε

φ = 7-40 

Lap-splice failure: 

(a) long / confined lap-splices 

(b) short / unconfined lap-splices 

See low cycle fatigue 

φp = (μlapφ + 7)φy Section 7.8.2.6a 

7-46 

Shear failure: 

(a) brittle 

(b) semi-ductile 

φp = 0 

y

i f

m f

p 2

V V

V V 5 φ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

− φ =

7-48 

7-49 

Joint or connection failure: 

(a) weak joint / strong column 

(b) semi-ductile

θP = 0.04 rad

y

ji jf

jh jf

p 2

V V

V V

4 φ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

− φ =

Section 7.8.2.8a 

7-51

Notes: 1. Hinge rotation θp = φp Lp where Lp is length of plastic hinge 

2. Notation is defined in section 7.8.2. 
249

7.8.2.1. Compression Failure of Unconfined Concrete 

The plastic curvature corresponding to compression failure in unconfined concrete is given by: 

y

cu

p c − φ ε φ = (7-34) 

where εcu is the ultimate concrete compression strain for concrete, which should be limited to 

0.005 for unconfined concrete, and c is the depth from the extreme compression fiber to the 

neutral axis. The location of the neutral axis is defined in section 7.8.3. 

7.8.2.2. Compression Failure of Confined Concrete 

For concrete confined by transverse hoops, cross-ties, or spirals, the compression strain is limited 

by first fracture within the confining steel. While this type of failure depends on the cyclic load 

history, a conservative estimate of the plastic curvature can be obtained from: 

( ) cu p y c d"

ε φ = − φ − (7-35) 

where: 

 c = depth from the extreme compression fiber of the cover concrete (which is 

 expected to spall) to the neutral axis, 

 d" = distance from the extreme compression fiber of the cover concrete to the 

 centerline of the perimeter hoop (thus, c – d" is the depth of confined concrete 

 under compression), and 

εcu = ultimate compression strain of the confined core concrete, as given by: 

cc

s yh su

cu f

1.4 f

0.005 ′

ρ ε

ε = + (7-36) 

εsu = strain at the maximum stress of the transverse reinforcement, 

 fyh = yield stress of the transverse steel, 

ρs = volumetric ratio of transverse steel, and 

'

cc f = confined concrete strength. 

For bridge columns that have confined concrete details, it is unlikely that this failure mode will 

govern, as low cycle fatigue is a more likely controlling mechanism. This is because the axial 

load in bridge columns is relatively low, and the transverse reinforcement is primarily required to 

provide restraint against buckling of the longitudinal reinforcement. 
250

7.8.2.3. Buckling of Longitudinal Bars 

If a compression member has inadequate transverse reinforcement and the spacing of the spirals, 

hoops, or cross-ties, s, in potential plastic hinge zones exceeds six longitudinal bar diameters 

(i.e., s > 6db), then local buckling at high compressive strains in the longitudinal reinforcement is 

likely. The plastic curvature of this failure mode can be determined from: 

b

p y (c d ')

ε φ = − φ − (7-37) 

where d’ is the distance from the extreme compression fiber to the center of the nearest 

compression reinforcing bars, and εb is the buckling strain in the longitudinal reinforcing steel. 

If 6db < s < 30db, the buckling strain may be taken as twice the yield strain of the longitudinal 

steel, i.e., 

s

y

b E

2f

ε = (7-38) 

7.8.2.4. Fracture of the Longitudinal Reinforcement 

Tensile fracture occurs when the tensile strain reaches a critical level, as given by εsmax. This 

failure mode is only likely under near-field impulse-type ground motions where there is 

essentially a monotonic (pushover) response. The plastic curvature in this case is given by: 

y

s max

p (d c)

− φ −

ε φ = (7-39) 

where d is the depth to the outer layer of tension steel from the extreme compression fiber, and c 

is the depth to the neutral axis. 

The tensile strain εsmax should be limited to a value less than or equal to 0.10. 

7.8.2.5. Low Cycle Fatigue of Longitudinal Reinforcement 

Since earthquakes induce cyclic loads in bridges, low cycle fatigue failure of the longitudinal 

reinforcement is possible. This is especially so if the column is well confined and other types of 

failure, as described above, are prevented. The plastic curvature that leads to a low cycle fatigue 

failure is given by: 

ap ap

p

2 2

(d d ') D'

ε ε

φ= = − (7-40) 
251

where: 

 D’ = distance between the outer layers of longitudinal steel in a rectangular section 

 (d – d’), or the pitch circle diameter of the longitudinal reinforcement in a circular 

 section, 

εap = plastic strain amplitude, as given by: 

εap = 0.08 (2Nf )

-0.5 (7-41) 

 Nf = effective number of equal-amplitude cycles of loading that lead to fracture, which 

 can be approximated by: 

 Nf = 3.5 (Tn)

-1/3 (7-42) 

 

 provided that: 2 ≤ Nf ≤ 10, and 

 Tn = natural period of vibration of the bridge. 

7.8.2.6. Failure in the Lap-splice Zone 

It is common to have a lap-splice zone at the base of a column where the starter bars from the 

footing or pile cap are lapped with the flexural reinforcement of the column. This is generally 

also the location of the plastic hinge zone. The presence of the lap-splice within the plastic hinge 

may lead to two different behavior modes, which depend on the length of the lap-splice that is 

provided, llap, compared to the required length, given by: 

 

b '

ce

ye

s d

f

f

l = 0.4 (7-43) 

where db is the diameter of the longitudinal reinforcing bar in the lap, fye is the expected yield 

strength of the longitudinal reinforcement in the lap-splice zone, and '

ce f is the expected strength 

of concrete surrounding the lap-splice zone (MPa). 

7.8.2.6(a). ‘Long’ or Confined Lap-splice, llap > ls

Some lap-splice zones in existing bridge columns were designed as tension splices and generous 

lap lengths were provided by some 40 bar diameters or more. It is possible that if the bond 

between the reinforcing steel and column concrete is satisfactory, the effective plastic hinge is 

reduced in length and behavior is then governed by low cycle fatigue of the longitudinal bars. 

From equation 7-28, it follows that the effective plastic hinge is concentrated at the beginning of 

the lap and is given by: 

 Lp = the minimum of yb yb ⎡(8800 d ) or (0.08L + 4400 d ) ε ε ⎤ ⎣ ⎦ (7-44) 

Equation 7-33 may be used to assess the rotational capacity of a long lap-splice. 
252

7.8.2.6(b). ‘Short’ and Unconfined Lap-splice, llap ≤ ls

Many lap-splices in bridge columns were designed as compression splices with a lap of 20 bar 

diameters or less. Initially, such a lap-splice may function quite well and be capable of 

sustaining the flexural strength capacity of the column. However, under the effect of earthquakeinduced cyclic loading, the bond in the lap-splice zone deteriorates and the moment capacity of 

the hinge is reduced. 

For a short and unconfined lap-splice zone where llap ≤ ls, the component has a reduced ductility 

that depends on the curvature ductility of the member. The effective plastic hinge length can be 

taken as: 

 Lp = Llap (7-45) 

The lap-splice-limited plastic curvature capacity is given by: 

φp = (μlapφ + 7) φy (7-46) 

where: 

φy = yield curvature, and 

μlapφ = curvature ductility at the initial breakdown of bond in the lap-splice zone: 

 = 0 when Ms < Me, where Ms is given by equation 7-10; i.e., when the lap-splice 

 strength (Ms) is less than the expected flexural strength (Me), the deterioration in 

 strength commences as soon as the moment reaches Ms and 

 = curvature ductility in the member when it reaches an ultimate strain of 

 0.002, when Me < Ms < Mpo, i.e., when the strength of the lap-splice (Ms) 

 exceeds the expected flexural strength (Me), but is less than the flexural 

 overstrength (Mpo), strength deterioration can still be expected but will be 

 delayed. 

7.8.2.7. Shear Failure 

If the shear strength of the member is less than the shear demand (based on its flexural strength), 

the plastic rotation will be limited. Two limiting cases are: (a) brittle shear, and (b) semi-ductile 

shear. These cases are based on the shear strength relative to the flexural strength. The shear 

demand (Vm), based on flexure, is given by: 

 Vm = Mp / L (7-47) 

where Mp is the expected plastic moment capacity, and L is the distance to the inflection point 

from the point of maximum moment in the member. 
253

7.8.2.7(a). Brittle Shear, Vi ≤ Vm

When the initial shear strength (Vi) is less than or equal to the shear demand (Vm), the member is 

considered to be ‘shear-critical’ and will fail in shear in a brittle manner. Since the member has 

no ductility capacity, 

φp = 0 (7-48) 

7.8.2.7(b). Semi-ductile Shear, Vf < Vm < Vi

If the shear demand, based on flexure, lies between the initial (Vi) and final shear strength (Vf) of 

the member, the member has limited ductility capacity given by: 

y

i f

m f

p 2

V V

V V 5 φ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

− φ = (7-49) 

where φy is the yield curvature, Vi is the initial shear strength (as defined previously) which is 

maintained until a curvature ductility of three is reached after which deterioration commences, 

and Vf is the final (residual) shear strength that is maintained when the curvature ductility 

exceeds eight. 

However, if Vm < Vf, the shear demand is less than the final shear strength of the member and 

the rotational capacity is limited by flexure and not shear. 

7.8.2.8. Joint or Connection Failure 

If the shear induced in a joint between a beam and a column is less than the shear strength of the 

joint, the joint will remain elastic, and the ductility of the beam-column system will be governed 

by the plastic hinges in the beam and column. 

Conversely, if the shear demand exceeds a certain critical level of principal stress in the joint, the 

joint may fail and the ductility of the beam-column system will be limited. 

Although damage (in the form of plastic rotations) will take place in the joint itself, it is difficult 

to model such behavior. Instead, it is more convenient to modify the response of the adjoining 

plastic hinges in the columns. The analysis therefore can proceed in a manner similar to the case 

of a shear-critical column, as described earlier. 

 

The horizontal shear in the joint (Vjh) is caused by flexure in the column and is given by: 

 Vjh = Mp / hb (7-50) 

where Mp is the expected plastic moment flexural strength of the column, and hb is the depth of 

the cap beam at the joint. 
254

7.8.2.8(a). Weak Joint and Strong Column, Vji ≤ Vjh

When the joint shear strength (Vji) is less than the horizontal shear in the joint due to flexure in 

the column (Vjh), joint performance will influence column behavior. The joint strength will 

deteriorate at the onset of full cracking (i.e., when the principal tensile stress in the joint reaches 

or exceeds 0.42 '

c (f ) until a plastic rotation of θp = 0.04 rad occurs (i.e., when the principal 

tensile stress in the joint is 0.29 '

c (f ) . The plastic rotation, which actually occurs in the joint, 

may be assumed to occur within the column’s plastic hinge zone. 

7.8.2.8(b). Semi-ductile Shear, Vjf < Vjh < Vji

When the column causes a shear demand that lies between the initial and final shear strength of 

the joint (i.e., Vjf < Vjh < Vji), the joint has limited ductility capacity, which can be calculated 

from the curvature ductility of the member as follows: 

y

ji jf

jh jf

p 2

V V

V V

4 φ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

−

− φ = (7-51) 

where φy is the yield curvature, Vji is the initial shear strength of the beam-column joint (as 

defined previously) which is sustained until a curvature ductility of three is reached after which 

deterioration commences, and Vjf is the final (residual) joint shear strength that is sustained when 

the curvature ductility exceeds seven. 

However, if Vjf > Vjh, the shear strength is greater than the flexural demand, and the rotational 

capacity is limited by beam or column flexure and not joint shear. 

7.8.3. NEUTRAL AXIS DEPTH IN COLUMNS AND BEAMS 

In several of the above failure modes, it is necessary to find the neutral axis depth for a given 

value of strain, usually in an extreme fiber. This can be done either by a moment-curvature 

analysis, or by a plastic section analysis. The moment-curvature analysis generally requires 

detailed computational modeling of the entire section and is more exact. The plastic section 

analysis is based on equilibrium of the section and several simplifying assumptions, however, its 

accuracy is adequate for purposes of this manual. 

For rectangular sections that are doubly reinforced, the neutral axis depth ratio is given by: 

'

e ty c

'

c g

' ty c

P f /f

f A 1 2d '/ D c

D 2 f /f

1 2d '/ D

⎛ ⎞⎛ ⎞ γρ ⎜ ⎟⎜ ⎟ + − ⎝ ⎠⎝ ⎠ = γ ρ αβ+ −

 (7-52) 
255

For circular sections, the neutral axis depth ratio is given by: 

0.725

e y

t ' '

cg c

P f 1 2c / D 0.5

c 1 fA f 1 2d '/ D

D 1.32

⎡ ⎤ ⎛ ⎞ − ⎢ ⎜ ⎟⎥ + ρ ⎝ ⎠ − ⎢ ⎥ = β α

⎣ ⎦

 (7-53) 

where: 

 c = depth to neutral axis, 

 D = overall depth of section, 

 Pe = axial load on the section, 

'

cf = expected concrete strength, 

 fy = expected yield strength of the longitudinal reinforcement, 

 Ag = gross cross-section area, 

 d′ = depth from the extreme compression fiber to the center of the compression 

 reinforcement, 

ρt = volumetric ratio of the longitudinal reinforcement, 

α, β = concrete stress block parameters as previously defined in section 7.7.1.2, and 

γ = reinforcing steel configuration factor: 

 = 0.5 for square sections with steel placed symmetrically around the perimeter, 

 = 0.0 for rectangular beam sections with steel lumped at the outer (top and bottom) 

 faces, 

 = 0.0 for wall section bending about the weak (out-of-plane) axis, and 

 = 1.0 for wall sections bending about the strong (in-plane) axis. 
 
257

CHAPTER 8: RETROFIT MEASURES FOR 

SUPERSTRUCTURES, BEARINGS, AND SEATS

8.1. GENERAL 

This chapter discusses measures for retrofitting a bridge from the bearing seats up to and 

including the superstructure. The most common and serious seismic deficiencies are often at the 

bearings and bearing seats, and can potentially lead to a loss of support and collapse of the 

bridge. The focus of early seismic retrofitting efforts was on correcting these deficiencies, and 

this continues to be a significant part of current retrofitting practice. Retrofit measures include 

restraining devices, bearing seat extensions, bearing strengthening, and bearing replacement. 

Other weaknesses in the superstructure are also of concern. For example, the superstructure 

must be capable of transferring its inertial load to the substructure without compromising the 

stability of the beams. Control of plastic hinging in the superstructure will also be a concern in 

some cases. In addition, there are several retrofit measures for superstructures that can be used 

to reduce or redistribute load to the substructure. These methods include the use of special 

energy dissipating devices or isolation bearings, reduction of superstructure dead load, and 

provisions for superstructure continuity. 

The following sections discuss some of the most common retrofit measures in greater detail. 

Many of these measures have been used extensively and have evolved to their current state over 

time, while others have had only limited use or are concepts developed as part of recent research 

efforts. The measures presented are not intended to cover all possible situations, and variations 

of these concepts may be possible and, perhaps, necessary for some applications. 

8.2. BRIDGE DECKS AND GIRDERS 

This section discusses retrofitting measures that are applicable to bridge decks and girders. It 

does not include a discussion of cable and bar restrainers, which are covered in section 8.4. 

8.2.1. LATERAL LOAD PATH ENHANCEMENT 

Retrofitting may be required to assure that inertial forces in the superstructure are effectively 

transmitted to the bearings and substructure. These measures are particularly applicable to steel 

girder bridges where a substantial portion of the mass is in the concrete deck slab and the girders 

and cross frames of the superstructure that may be inadequate to transmit the corresponding 

inertial forces to the substructure. In many cases, they are also applicable to pre-cast concrete 

girder bridges, which can experience the same deficiencies. 

8.2.1.1. Strengthening of Deck to Girder Connection 

Many older steel girder bridges are not designed for composite action and therefore do not have a 

positive connection between the deck slab and the girders. This could result in a separation of the 

deck from the girders during a strong earthquake. In many cases, this may be tolerable if it does 
258

not result in the formation of a collapse mechanism, and may improve overall performance if it 

protects other vulnerable structural elements from damage. In other cases, however, it may be 

desirable to strengthen the deck-to-girder connection. This situation could occur when 

unacceptable movement of the deck is expected, when a higher level of seismic performance is 

required, or when a positive deck-to-girder connection is a component of a larger seismic retrofit 

strategy. Figure 8-1 shows one method that has been used in the past to improve the deck-togirder connection. 

CL Girder Existing Concrete Deck

Two rows of threaded

rods at a specified pitch

for entire length of the

girders.

Drill and bond (epoxy

cartridge) into existing

concrete deck. Stagger

as shown below.

CL Girder

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Figure 8-1. Deck to girder connection retrofit. 

8.2.1.2. Diaphragm Strengthening or Stiffening 

Many existing diaphragms in steel or pre-cast concrete beam bridges were not designed for high 

seismic loads and are not strong enough, or sufficiently ductile, to transfer the inertial forces 

from the deck into the bearings. This may protect the bearings from damage, but could also 

jeopardize the transverse stability of the beams if deformation of the diaphragm is large enough. 

If it is not practical to make diaphragms strong enough to resist loads elastically, brittle or nonductile diaphragm failure modes could occur. This would include shear failure of bolted or 

riveted connections or inelastic buckling of compression members within the diaphragm. 

Generally, this will require strengthening or bracing of the diaphragm in such a way as to make 

axial yielding in a diaphragm member the principle mode of failure. 
259

If existing riveted or bolted connections are not strong enough to resist the loads applied to them, 

then retrofitting might include removal of the existing rivets or bolts and replacement with 

stronger connectors (e.g., higher grade material or larger diameter bolts). When only bolts are 

used in a connection, their ultimate strength as bearing connectors may be used. The use of 

welds should also be considered if the weldability of the existing steel is acceptable and the 

direction of applied stress is conducive to a welded connection. Welds may be considered to 

complement the strength of existing rivets and bolts if capacity design principles are used to 

design the connection, and the rivet or bolt design strength is strain-compatible with the welds. 

This could be the case if the design capacity of the existing bolts is limited to that of a frictiontype connection. 

A method of increasing the inelastic buckling strength of a diagonal member is shown in figure 

8-2. This method involves the addition of a relatively stiff member, such as a steel tube, to brace 

the existing diaphragm member against buckling. Slotted bolt holes are used to connect the 

bracing member so that it does not add to the yield strength of the existing diaphragm member. 

This is important, because the diaphragm will be subject to load reversals during the earthquake, 

and some members will act in both tension and compression. Yielding of these members will 

limit forces in other elements of the diaphragm and result in ductile behavior. This phenomenon 

is illustrated in the free body diagram for a typical diaphragm shown in figure 8-3. 

Axial yielding of mild steel members in the diaphragm can be relatively ductile if inelastic 

buckling is prevented. However, it is important to consider the displacement ductility of the 

entire lateral force resisting system (diaphragms, girders, bearings, piers and foundations) in 

determining the adequacy of the critical diaphragm member. This can be accomplished by 

performing a two-dimensional, nonlinear, static ‘pushover’ analysis of the system with a target 

displacement equal to 150 percent of the displacement from an elastic analysis in which the 

stiffness of all elements is considered. With this analysis, it is suggested that the design may be 

considered satisfactory for collapse prevention if no brittle or non-ductile failures are indicated 

(e.g., fracture of the net section at the connections) and axial elongations of diaphragm 

components limit relative transverse displacement at the top of the girder to two percent of the 

girder height. 

8.2.1.3. Energy Dissipating Ductile Diaphragms 

A seismic retrofit strategy that replaces existing end diaphragms in the steel superstructure with 

specially detailed ductile end diaphragms may provide a desirable energy dissipation mechanism 

in the structure in the transverse direction. These ductile diaphragms can be specially designed 

and calibrated to yield before the substructure elements (foundation and bearings). This strategy 

may be adopted when the substructure elements have limited ductility and cannot be relied upon 

to achieve a stable ductile response (e.g., stiff wall piers). Ductile diaphragms should be capable 

of dissipating energy in a stable manner without strength degradation during cyclic deformation. 

Various types of systems capable of stable passive seismic energy dissipation may be adopted for 

this purpose. Among these are eccentrically braced frames (EBF)1

, shear panel systems (SPS)2, 

 

1

 Malley and Popov, 1983; Kasai and Popov, 1986 

2

 Fehling et al., 1992; Nakashima, 1994 
260

and steel triangular-plate added damping and stiffness devices (TADAS)3

. TADAS devices are 

popular in building applications and have been studied for bridge applications4

. Although 

concentrically braced frames can also be ductile, they may not be desirable because they may be 

stronger than calculated, and their hysteretic curves may exhibit pinching and some strength 

degradation. Typical energy dissipating, ductile diaphragms are shown in figure 8-4. 

Figure 8-2. Steel girder diaphragm retrofit. 

 

3

 Tsai et al.,1993 

4

 Zahrai and Bruneau, 1998, 1999 
261

Figure 8-3. Diaphragm free body diagram. 

The ductile diaphragm approach may not be effective when the substructure is significantly more 

flexible than the superstructure. Therefore, bridges with wide piers, wall piers, or other stiff 

substructure elements of similar limited ductility, are candidates for the implementation of this 

system. Ductile diaphragms can also be designed to yield before the bridge piles, thus preventing 

the development of damage below ground where it is difficult to inspect following an 

earthquake. Recent studies have provided methods to account for the effect of substructure 

flexibility5 on the ductility demands of ductile diaphragms. This is reflected in the recommended 

response modification factor (i.e., R-factor) for design of ductile end diaphragms recommended 

by equation 8-1: 

 

5

 Alfawakiri and Bruneau, 2000 
262

R/C Slab R/C Slab

R/C Slab

TADAS

a) Eccentrically Braced Frames b) Shear Panel System (SPS)

c) TADAS

Figure 8-4. Ductile end diaphragms. 

DED

SUB

DED

SUB

K

K R K 1

K

⎛ ⎞ μ + ⎜ ⎟

= ⎜ ⎟

⎜ ⎟ + ⎜ ⎟ ⎝ ⎠

 (8-1) 

where μ is the ductility capacity of the end diaphragm itself, which should not be taken to be 

greater than four, KDED is the stiffness of the ductile end diaphragm, and KSUB is the stiffness of 

the substructure. 

 

Typically, increased substructure flexibility results in higher ductility demands in ductile end 

diaphragm systems. The capacity of the ductile system, or device, should not be exceeded for 

the maximum expected lateral drift of the bridge at the diaphragm location. A drift limit of two 

percent of the girder height is recommended until further experimental evidence is provided to 

demonstrate that higher values may be acceptable. 
263

Inertia forces due to the mass of the pier cap will not be reduced by ductile end diaphragms, and 

must be resisted by the substructure. Refined analyses should consider this condition if the mass 

of the pier cap is a significant portion of the total superstructure mass. 

Ensuring an adequate load path within the superstructure can enhance the effectiveness of ductile 

end diaphragms. In general, a continuous path is necessary for the transmission of the 

superstructure inertia forces to the substructure. Concrete decks have significant rigidity in their 

horizontal plane, and in short-to-medium steel beam spans, their response approaches rigid-body 

motion. Therefore, the lateral loads resisted by the intermediate diaphragms are minimal, and 

the load path for seismic loads is effectively through the deck and the end diaphragms to the 

bearings. 

It is important to note that the contribution of girders to lateral stiffness at the end diaphragms 

can be significant and should not be neglected. For this reason, ductile diaphragms are typically 

more effective in longer span bridges, and may be of limited benefit for very short span bridges. 

The retrofit design should consider the combined stiffness and strength of end diaphragms acting 

in conjunction with the girders (including the effect of the girder’s bearing stiffeners) in 

establishing the diaphragm's strength and therefore the design forces for capacity protected 

elements such as bearings. 

 

Ductile energy dissipating elements in the end diaphragms should be laterally braced at their 

ends to prevent out-of-plane instability. It is recommended that these lateral supports and their 

connections be designed to resist six percent of the nominal strength of the beam flange. To 

prevent lateral torsional buckling of beams in the end diaphragms, the unsupported length of 

these beams should be typically limited to 200 f

y

b

f

, where bf is the width of beam flange in 

meters and fy is the yield strength of steel in MPa. 

The energy dissipating ductile end diaphragm concept is relatively new and has yet to be 

implemented in practice. However, considerable research related to these members has been 

conducted6

, which has demonstrated the effectiveness and validity of the design methods 

described above. Further information is given in the recommended provisions for seismic design 

developed for the NCHRP 12-49 project (ATC/MCEER 2003). 

8.2.1.4. Girder Strengthening 

Girders may be subjected to high transverse bending stresses in the vicinity of the bearings 

during an earthquake. This will be most critical when diaphragms are not provided or when 

diaphragms consist of horizontal beam type elements located some distance above the bearings. 

In these cases, the principal resistance to transverse bending is that provided by web stiffeners at 

the bearings, which may be inadequate for this purpose. If stresses and distortions are too high, 

the ability of the girder web to support dead load may be compromised. Supplemental braces as 

shown in figure 8-5 may be used to prevent transverse distortion of the girders, or bearing 

stiffeners may be strengthened as shown in figure 8-6. 

 

6

 Zahrai and Bruneau, 1998 and 1999; Sarraf and Bruneau, 1998a and 1998b 
264

Existing

Diaphragm

New Structural “T”

Gusset

New Gusset

New Brace

Girder with

Bearing

Stiffeners

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Figure 8-5. Girder bracing retrofit. 

8.2.2. PROVIDING LONGITUDINAL CONTINUITY 

Several retrofit measures other than cable or bar restrainers are available for providing 

superstructure continuity in the longitudinal direction during an earthquake so that seismically 

induced forces can be better shared between the supports. Some of these methods are discussed 

below. 

8.2.2.1. Web and Flange Plates 

It is often possible to tie steel beams together using web plates as shown in figure 8-7. It may be 

necessary to use shims to get the web splice plates to align properly with the beam webs. If the 

plates are located at an expansion joint, bolt holes may be slotted to provide for thermal 

expansion and contraction of the superstructure. This retrofit measure also provides for vertical 

support of one of the girders if it becomes unseated. To prevent bolt failure due to vertical and 

horizontal impact loading and misalignment or slotting of the bolt holes, bolts should be 

designed as bearing 
265

Existing

Diaphragm

Girder with

Bearing

Stiffener

New Stiffener

Knee Brace - Bolt

to Bearing Stiffener

and Diaphragm

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Figure 8-6. Bearing stiffener retrofit. 

connectors with large factors of safety (i.e., two or greater). Also, relative transverse movement 

of the beams should be restrained to prevent tearing of the web. 

8.2.2.2. Superstructure Joint Strengthening 

It is often possible to make a structure continuous for live load across existing joints without 

significantly affecting its ability to accommodate changes in temperature. This is true if the 

bridge is relatively short or if the joint in question involves fixed bearings on both sides of the 

joint. This can be done by removing a portion of the deck on either side of the existing joint, 

connecting the girders together with splice plates at the flanges, and reconstructing the deck so as 

to be continuous. This retrofit method is illustrated in figure 8-8. This method requires 

verification of the ability of the girders to accommodate the resulting negative moments that will 

occur under service load conditions. 
266

Large Offset of

Girder Webs

Section A2 - A2 Girders End Bracing

Splice Plate

C Bent L

C Bent L Small Offset of

Girder Webs

Section A1 - A1

Shim

Elevation of Girders

Splice Plate - Place Bolts 

Through Holes Drilled in Girders - 

Use Slotted Holes to Accommodate 

Thermal Movement

A1

A2

A1

A2

Figure 8-7. Methods for accommodating girder misalignment using web splice plates. 
267

Remove and Replace

Existing RC Deck

Weld New Steel

Splice Plate to

Top Flange

Steel Girder B

Double Angle to

Splice Bottom

Flange

Bearings

(Fixed)

Steel Girder A

Bearing

Stiffeners

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Figure 8-8. Girders made continuous for live load. 

8.2.3. REDUCTION OF DEAD LOAD 

It is sometimes possible to reduce dead load on a bridge superstructure. Although this will often 

decrease the natural period of vibration, and therefore result in a slight increase in the spectral 

acceleration, the net effect will be to reduce the displacement and ductility demands on the 

bridge. This reduction in seismic demand may be enough to improve seismic performance. 

Removal of heavy concrete barrier rails and accumulated overlays of asphalt concrete or other 

materials, or replacement of a heavy concrete deck with light weight concrete or other material, 

are some of the options that may be considered. 

8.2.4. STRENGTHENING OF CONTINUOUS SUPERSTRUCTURES 

It is generally not desirable to allow extensive plastic hinging to occur in a superstructure that is 

integral with the substructure, since the superstructure is usually not designed to behave in a 

ductile manner. To prevent this, the superstructure can be strengthened to restrict plastic hinging 

to the piers. This is preferable since piers are more easily retrofitted for ductility enhancement. 

Figure 8-9 illustrates how this can be done for a concrete box girder structure that lacks 

significant positive moment capacity near a support. Additional retrofitting would be required if 

the negative moment capacity of the superstructure was also inadequate. To prevent excessive 

torsion from developing in the pier cap near the column, retrofitting should be done 

symmetrically about a column within a width equal to the width of the column plus twice the 
268

Figure 8-9. Retrofit of concrete box girder for flexural capacity. 

depth of the superstructure for box girders and solid superstructures, or the width of the column 

plus the depth of the superstructure for stringer-type superstructures. 

8.3. BEARINGS, ANCHORAGES AND PEDESTALS 

When the superstructure can become dislodged from its bearings, the designer has several 

options to consider. If the seat that supports the bearings is wide enough and the superstructure 

can drop only a short distance, it may acceptable to allow the bearings to fail as long as a 

collapse mechanism is avoided and the resulting level of performance is acceptable. If the height 

of the vulnerable bearings is large, it is usually advisable to retrofit the structure. This can be 

done by strengthening the existing bearings, replacing them with new bearings, or providing an 

elevated seat extension (i.e., a catcher block) that will maintain the grade of the superstructure in 

the event of bearing failure. This section deals with bearing strengthening and replacement. 

Catcher blocks are discussed in section 8.4 
269

Bearing types that have fared poorly in past earthquakes are shown in figure 4-3. Although 

different bearings have different configurations, failure will usually occur at the connection 

between the bearing and the girder, at keeper plates, at the connection between the bearing and 

the masonry plate, or at the anchor bolts that connect the masonry plate to the support. In some 

cases, bearings will be mounted on concrete pedestals that must also be considered as a possible 

cause for bearing failure. Steel rocker bearings are particularly vulnerable to damage during an 

earthquake. Figure 8-10 shows the nomenclature for two typical steel bearing types. 

New or strengthened bearings and their restraining components should be capable of resisting the 

longitudinal, transverse, and vertical forces obtained by analysis. 

Figure 8-10. Steel bearing nomenclature. 
270

8.3.1. STRENGTHENING OF EXISTING BEARINGS 

Strengthening is not usually the preferred retrofit measure for steel rocker bearings because of 

their poor maintenance history, and replacement may be preferred instead. But there may be 

cases where strengthening is the best option, in which case the focus should be on those 

components of the bearing that are most likely to fail. Note that if bearings are strengthened, 

failure and/or yielding may shift to other locations within the structure - usually the piers. The 

designer should verify that yielding at this new location is acceptable, and that there is sufficient 

strength and/or ductility to achieve the desired performance of the bridge.

8.3.1.1. Expansion Bearings 

Expansion bearings are intended to allow free movement in the longitudinal direction and 

usually have keeper plates, pintels, anchor bolts, or some other mechanism, to prevent transverse 

movement and resist transverse loads. Retrofitting to increase the strength of these connections is 

frequently necessary to meet minimum force requirements and prevent loss of support in the 

transverse direction. Care should be taken that these measures do not impede the longitudinal 

movement of the bearing and inadvertently restrict longitudinal expansion. 

 

8.3.1.1(a). Sole Plate to Girder 

Sole plates are usually connected to girders with bolts or rivets, which often lack the shear 

capacity required to prevent the sole plate from breaking loose from the girder during an 

earthquake. This deficiency can be overcome by replacing existing bolts and rivets with stronger 

bolts. If this does not provide sufficient strength, it may be possible to use a supplemental 

connector plate that will put the bolts in double shear as shown in figure 8-11. These new bolts 

must be designed to resist both tension and shear simultaneously, and the flange and sole plates 

must be capable of resisting these forces. 

Another retrofit measure is to provide for an extended range of transverse movement of the beam 

on the bearing. The details used for this retrofit will depend on the details of the bearing, but one 

method that was proposed for bearings tested in the laboratory is shown in figure 8-12 (Mander 

et al., 1996a). In this case, keeper plates are allowed to fail. The steel catcher bars that are 

welded to the sole plate would accommodate the resulting sliding of the bearing on the sole 

plate. Transverse movement is limited to half the width of the bearing. In this case, vertical legs 

extending downward from the catcher bars resist extreme movement provided vertical 

accelerations are not excessive. 
271

Bottom

Flange Bearing Stiffener

Bolts to Replace

Existing Rivets

New Plates

Existing

Bearing (Fixed)

Figure 8-11. Bearing sole plate to girder connection retrofit. 

8.3.1.1(b). Bearing to Masonry Plate 

Steel rocker bearings provide very little resistance to movement in the longitudinal direction. 

Recent laboratory testing has shown that it is possible to retrofit this type of bearing to increase 

resistance to displacement beyond a predetermined range (Mander et al., 1996c). This allows 

freedom of movement for thermal expansion and contraction of the superstructure, but resists 

excessive movement due to an earthquake. This is done by welding steel wedges to the bearing 

plate as shown in figure 8-13. The edges of the wedge plate should be sealed to prevent the entry 

of moisture and thus eliminate potential corrosion between the wedges and the bearing plate. 

The resulting increase in longitudinal resistance, Fh, is given as a function of the bearing dead 

load reaction, P, and the angle of the wedges, α, per equation 8-2. 

 Fh = P tan α (8-2) 

 

In the transverse direction, the connection between bearing and bearing plate is sometimes 

provided by pintles that engage holes in the rocker. In other cases, keeper bars or anchor bolts 

provide resistance to transverse force. Any of these transverse force-resisting devices can fail in 

shear and result in unacceptable transverse movement of the superstructure. The type of retrofit 

selected to prevent excess transverse movement will depend on the configuration of the bearing. 

One possible method is shown in figure 8-14. 
272

Figure 8-12. Bearing sole plate to girder with catcher bar retrofit. 
273

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Bearing

Steel Wedge

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Figure 8-13. Expansion bearing retrofit. 
274

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Figure 8-14. Retrofit to increase bearing transverse capacity. 
275

8.3.1.1(c). Masonry Plate to Substructure 

Masonry plates are usually connected to the substructure with anchor bolts cast into concrete 

bearing seats. These bolts must resist both shear and tension when the bearing is subjected to 

transverse forces, and if they should fail, the resulting sliding of the bearing may cause excessive 

displacement of the superstructure. The capacity of the masonry plate to resist these forces can 

be increased by adding more anchor bolts, which may require making the plate larger. This can 

be done by drilling and grouting new anchor bolts into the concrete bearing seat and connecting 

these new anchor bolts to the existing masonry plate. One method for doing this is shown in 

figure 8-15. 

Figure 8-15. Retrofit to increase anchor plate capacity. 
276

8.3.1.2. Fixed Bearings 

In the case of fixed bearings, horizontal forces are resisted in both the longitudinal and transverse 

direction. There are several locations within a bearing assembly where failure can occur. 

Usually, the sole plate-to-beam connection, keeper bars, and masonry plate-to-substructure 

connection are the weakest points. In some cases, bearings are mounted on concrete or masonry 

pedestals that may fail during an earthquake. 

8.3.1.2(a). Sole Plate to Girder 

There is very little difference between retrofitting a sole plate-to-beam connection for a fixed 

bearing than for an expansion bearing. The primary difference is that the connection in a fixed 

bearing must carry large longitudinal as well as transverse forces. See section 8.3.1.1(a) and 

figures 8-11 and 8-12 for suggested retrofit measures. 

8.3.1.2(b). Masonry Plate to Substructure 

Anchor bolts on fixed bearings are usually subjected to simultaneous tension and shear during an 

earthquake. This makes it more difficult to retrofit the masonry plates by simply adding more 

anchor bolts. An alternative method is bearing encasement, which can also be used to increase 

the strength of concrete and masonry pedestals. One approach (Mander et al., 1996c) is to place 

a steel shell around the portion of the bearing to be encased, up to the pivot point of the bearing. 

The concrete within the shell is anchored to the bearing seat by dowels that are drilled and 

bonded into the existing concrete. If insufficient length exists to fully develop the dowels in the 

new concrete encasement, they may be welded to the steel shell. A typical retrofit detail is 

shown in figure 8-16. 

Another method of bearing embedment is shown in figure 8-17. This will prevent shear failure 

and toppling of the bearings. In addition, if the spans are unseated from the bearings, the 

concrete cap will limit the vertical drop. The concrete cap can also act as a shear key and an 

anchorage for vertical motion restrainers. 

8.3.2. BEARING REPLACEMENT 

8.3.2.1. Conventional Bearings 

Replacement is the preferred retrofit measure for bearing types that have performed poorly in 

past earthquakes. This is because seismically vulnerable bearings also have a history of 

maintenance problems. But many of the newer bearings have smaller vertical dimensions than 

the ones they are replacing, and it will be necessary to compensate for the difference in height as 

part of the retrofit. This may be done by using either a steel pedestal (figure 8-18) or a concrete 

pedestal (figure 8-19). In the latter case, the pedestal can be constructed to a higher elevation 

between beams to provide a transverse shear key and vertical motion restrainers may be 

anchored into the pedestal. 
277

Figure 8-16. Bearing retrofitting by encasement in steel shell. 
278

Figure 8-17. Bearing retrofitting by encasement in concrete. 
279

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Figure 8-18. Bearing replacement retrofit using steel pedestal. 
280

New Bolt - Drilled and Grouted

Into Existing Concrete

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Bearing Pad

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Figure 8-19. Bearing replacement using concrete pedestal. 
281

Wherever possible, the replacement bearings at expansion and fixed ends of a girder should be of 

the same type so that the girder end rotations are similar and symmetry is preserved. 

8.3.2.2. Replacement with Seismic Isolation Bearings 

Use of isolation bearings as a seismic retrofit measure may have the dual benefit of replacing a 

vulnerable bearing while also protecting other structural components from damage. More than 

150 bridges in the U.S. have been successfully retrofitted using this method. 

Seismic isolation bearings may be used to reduce the response of a bridge during an earthquake. 

This is done by increasing the fundamental period of vibration of the bridge, which in most cases 

will reduce the accelerations in the superstructure and the inertia forces transmitted to the 

substructure. At the same time, the relative displacements between the superstructure and the 

substructure increase; these are kept to acceptable levels by the energy dissipating characteristics 

of the bearing. An analysis should be performed to determine the magnitude of these forces and 

displacements to assure that the capacity of the bridge will not be exceeded. 

Isolation bearings should be stiff under service conditions, and flexible under seismic loads. 

This is because the isolators should be able to resist service loads with only a minimum amount 

of movement, but soften at higher amplitudes to give the required flexibility that will effectively 

isolate the bridge during an earthquake. This usually means that the force-deflection 

characteristics of these devices will be nonlinear. In most isolators, these nonlinear 

characteristics are also used to provide hysteretic energy dissipation. 

Isolators currently being used for bridges include both rubber- and friction-based systems. The 

majority of bridge applications in the United States are rubber-based and the principal type is the 

lead-filled elastomeric bearing7

. Friction-based systems, principally the friction pendulum 

bearing, have also been used in a number of bridge applications and are gaining in popularity. 

Both rubber- and friction-based isolation bearings have been used extensively in other countries. 

Seismic isolation can be provided by conventional elastomeric or sliding bearings, when used in 

conjunction with separate energy dissipation devices. An example of such an isolator is the 

EradiQuake bearing. Examples of isolation bearings are listed in table 8-1 and illustrated in 

figures 8-20 through 8-22. Energy dissipating devices are discussed further in section 8.4.3. 

A number of these isolation systems have been evaluated in the HITEC8

 program on behalf of 

Caltrans and FHWA (HITEC, 1999a). These include elastomeric, sliding and rolling bearings 

that are capable of resisting the types of loadings experienced in bridges. Summaries of test 

results for eight of these systems are available through HITEC (HITEC, 1998a through h, 

inclusive). 

 

7

 Buckle 2003 

8

 Highway Innovative Technology Evaluation Center, ASCE Civil Engineering Research Foundation, Washington 

DC 
282

Lead

Steel Rubber

Figure 8-20. Lead-filled elastomeric isolation bearing. 

Figure 8-21. Friction-pendulum isolation bearing. 
283

Figure 8-22. Eradiquake isolation bearing. 

Table 8-1. Basic characteristics of typical isolation bearings for bridges. 

Bearing Type Flexible 

Element 

Energy 

Dissipation 

Rigidity for 

Service Loads 

Lead-filled 

elastomeric bearings 

Standard 

elastomeric bearing 

Plastically deformed

lead core Elastic stiffness of lead core 

Eradiquake bearings 

Flat slider with low 

friction coefficient 

and uniaxial 

polyurethane springs 

Friction No slip until friction coefficient 

exceeded 

Friction-pendulum 

bearings 

Spherical slider with 

low friction coefficient Friction No slip until friction coefficient exceeded 

Note: Some of the above hardware is patented or proprietary and is only available through licensed suppliers. 

The selection of isolator type is an important decision, since both short-term and long-term 

performance characteristics are important. In the short-term, resistance to wind and braking 

loads without excessive displacement requires rigidity at small deformations, as noted above. 

However, the same device must also permit thermal expansion to occur in the superstructure 

without overstressing the substructure. In the long-term, reliability of performance is essential. 

It may be decades before the design earthquake occurs, and over this period of time, the isolator 

properties must remain stable. In this regard, the ideal isolator is maintenance free, does not 

require precise field tolerances to operate successfully, and is constructed from materials that are 

chemically inert and resistant to atmospheric pollutants and de-icing salts. 
284

All isolation systems should satisfy rigorous testing requirements and quality control standards. 

The physical properties of many isolators vary with temperature and this can be important in 

areas subject to low temperatures. Guidance on suitable testing requirements, including lowtemperature performance testing, is provided in AASHTO, 1999. Properties likely to exist under 

field conditions should be used in analysis. 

Bridges that already have bearings at every pier and abutment are good candidates for seismic 

isolation because isolator installation is relatively straightforward. In cases where a shallowprofile bearing such as an elastomeric pad is being replaced, there may be insufficient headroom 

to make a direct exchange with an isolator. Extra effort will then be required to create the 

necessary space to install the device. This is unlikely to be the case when replacing steel 

bearings, and in some cases, the reverse situation will be true, requiring the use of a pedestal (see 

section 8.3.2.1). 

Isolation may also be used to retrofit an integral bridge, since the columns may be cut directly 

under the pier cap to create a space for the isolators. 

When a bridge is isolated, it must be free to move in any horizontal direction in order for the 

isolation to be effective. This is not usually a problem in the transverse direction; but in the 

longitudinal direction, special care is necessary at the abutments. This is because the clearance 

at existing expansion joints for most abutments will be insufficient to accommodate the expected 

movements under seismic loads. If the clearance is not increased, impact between the 

superstructure and the backwall will most likely occur. The resulting damage to the backwall is 

unlikely to cause collapse or close the bridge for any period of time since temporary repairs can 

be implemented quickly and access restored. This leads to an approach of not providing the 

required clearance at the time of retrofitting, but waiting until the wall is damaged in an 

earthquake. When the wall is repaired, adequate clearances can be provided for future 

earthquakes. While this approach is not recommended for new bridges, it has merit in retrofit 

situations. 

The backwall can also be modified at the time of retrofit. One way to do this is shown in figure 

8-23. A sacrificial knock-off element is incorporated into the top of the wall, and can be 

replaced if dislodged in an earthquake. Impact is still likely to occur in the longitudinal direction, 

but the consequences are minor. Another option is to provide for a much larger clearance so that 

impact does not occur. In this case, the gap must be bridged with an expansion joint. This may 

be the ideal solution from a structural response point-of-view, but may not be the most 

economical one since road joints that can accommodate large opening and closing movements 

are expensive and difficult to maintain. 
285

Expansion Joint

Bearing

Friction or

Settlement Slab

Backwall Sacrificial

Knock-Off Element

Superstructure

Abutment

Figure 8-23. Knock-off device in backwall of seat-type abutment. 

8.3.3. STRENGTHENING OF SUPERSTRUCTURE TO SUBSTRUCTURE 

CONNECTIONS 

Some integral bridges employ concrete hinges (also called Freyssinet hinges) for the connection 

between the superstructure and the substructure. These hinges allow rotational movement in the 

longitudinal direction, but resist transverse rotation. They also generally resist shear forces in 

both directions. These hinges may be located either at the top or bottom of piers and abutments 

as is shown in figure 8-24. When the steel reinforcing is inadequate to resist the shear forces in 

either direction, or the moments in the transverse direction, retrofitting may be necessary. A 

method that can be used to increase the shear resistance of these hinges is shown in figure 8-25. 

It involves coring a hole through the joint and inserting a steel pin such as a thick-walled pipe. 

Another method of increasing the transverse moment resistance of a hinge is shown in figure 

8-26. This method uses supplemental tie-down piles and a prestressed concrete cap beam to 

strengthen a concrete hinge at the bottom of a single column pier that is deficient in transverse 

overturning capacity. 

Methods used to strengthen these types of hinges will vary based on the configuration of the 

bridge being retrofitted. The design forces should be determined from an analysis of the 

structure. In many cases, these forces are dictated by plastic capacity elsewhere in the bridge, for 

example, the moment capacity at the opposite end of the column to the hinge. The designer must 

also check that the retrofit allows the bridge to perform adequately under service conditions. 
286

Figure 8-24. Freyssinet hinge. 

8.4. EXPANSION JOINT RETROFIT 

This section presents several methods for preventing loss of support at expansion joints during 

earthquakes. Two basic retrofitting approaches are discussed. The first provides additional 

displacement capacity, and the second reduces the displacement demand. These approaches may 

be used alone or in combination to prevent failure due to loss of support. 

8.4.1. BEARING SEAT EXTENSIONS 

Increasing the seat width increases the displacement capacity at an expansion joint, often without 

affecting the dynamic characteristics of the bridge. Two ways of doing this are discussed below. 

8.4.1.1. Concrete Seat Extensions and Catcher Blocks 

Seat extensions are used when too many restrainers would be required to limit movement at an 

expansion joint to 67 percent of the available seat width (see section 8.4.2.1 (f)). By using seat 

extensions to increase the range of movement, a smaller number of restrainers can be used. If 

installed at abutments, these extensions should preferably be supported directly on the footing, as 

shown in figure 8-27. 
287

Cast-in-place Deck and Diaphragm

Core Vertical Hole into

Pier Concrete

(Between Girders)

Insert Concrete

Filled Steel Pipe

A A

Patch Core Hole

after Placing Pipe

Precast Girders

Pipe XX-strong

Polystyrene

Pre-molded Expansion

Joint Filler

Section A-A

Figure 8-25. Pipe shear key. 
288

Figure 8-26. Concrete hinge retrofit 
289

Figure 8-27. Abutment seat extension. 

Extensions that are anchored to the vertical face of a concrete abutment or pier, with dowels or 

anchor bolts, must be designed to carry the large vertical and horizontal forces that will be 

imposed if the bearings fail. These include the impact of the falling superstructure and the 

horizontal forces from the earthquake. When these forces are eccentric to the dowels and anchor 

bolts, the resulting moments must be considered in designing the anchorage to the abutment or 

pier. When feasible, post-tension the seat extensions to the substructure. 

When the bridge is supported on high bearings, the seat extension should be elevated to just 

below the level of the beams. If these so-called ‘catcher blocks’ are used, consideration should 

be given to future inspection and maintenance of the bearings. One approach is to use a 

removable steel assembly for the raised portion of the seat extension. An oversized sole plate 

that is fastened below the bottom flange of the girder has also been used with some older 

bearings. There must be sufficient clearance for this plate to avoid impact or interference with 

the seat extension. 
290

If bearing seats are extended, their width should be increased to the minimum seat width 

recommended in section 5.2.1. These widths reflect the possibility of large relative movements 

at the bearings resulting from the overall inelastic response of the bridge, possible independent 

movement of different parts of the substructure, and out-of-phase rotation of abutments and piers 

resulting from traveling surface wave motion. 

The following load cases should be used to design the seat extensions: 

1. Vertical load equal to twice the dead load reaction. 

2. Vertical load equal to the dead load reaction plus a horizontal load due to the earthquake. 

The first case considers vertical forces only and is intended to account for the large impact forces 

that can result when the superstructure drops from the bearings onto the seat. The second case 

considers both the horizontal and vertical loads that can develop when the superstructure is 

resting on the bearing seat extension and is still being subjected to earthquake ground motions. 

Depending on the type and amount of traffic likely to be present on the bridge at any time, 

consideration should also be given to including a portion of the live load in the vertical load for 

both of these load cases. The horizontal earthquake loading should be equal to the lesser of the 

dead load reaction times, the spectral acceleration coefficient, or the dead load times the 

maximum feasible coefficient of friction between the girder and seat extension. 

8.4.1.2. Pipe Extenders 

For seats at in-span expansion joint hinges, pipe extenders may be used to increase seat width 

capacity. A typical detail is shown in figure 8-28. These extenders may also be used as 

transverse shear keys. 

As for seat extensions, pipe extenders are used when too many restrainers are required at an inspan hinge to limit movement to 67 percent of the available seat width (see section 8.4.2.1 (f)). 

By using pipe extenders to increase the range of movement, a smaller number of restrainers can 

be used. 

Pipe extenders will be subjected to both vertical gravity loading and transverse earthquake 

loading. This will result in both flexural and shear stresses within the pipe as well as complex 

stresses within the concrete diaphragm in which the pipe is anchored. Caltrans has conducted 

field tests of 200 mm (8 in) diameter double extra strong pipe extenders and found that failure 

occurs in the pipe at an ultimate load of approximately 1070 kN (240 kips) and an extension of 

200 mm (8 in). Based on these findings, it is recommended that shear capacity be limited to 

930 kN (210 kips) for resolved vertical and horizontal shear and 800 kN (180 kips) for shear in 

either of the orthogonal directions. These values are for a normal joint, and horizontal shear 

capacity will be reduced in skewed joints. Despite these findings, it is good practice to limit seat 

extender capacity to 445 kN (100 kips) unless space limitations prevent it (Caltrans, 1996). 

Calculated shear demands on multiple pipe extenders should be increased by 130 percent to 

account for potential misalignment of the extenders. Consideration should also be given to the 
291

participation of existing concrete shear keys in resisting transverse shear. For the load case in 

which transverse shears are high and the expansion joint opening is relatively low, existing 

transverse shear keys can be engaged. If the pipe extenders engage before the concrete shear 

keys, then the ductility of the pipe extenders will permit the eventual participation of the shear 

keys during a strong earthquake. If the shear keys engage first, they are likely to experience 

brittle failure before the pipe extenders take up load, and therefore would not contribute to the 

ultimate shear resistance at the expansion joint. 

Figure 8-28. Pipe seat extenders. 

8.4.2. RESTRAINERS 

Many of the bridge failures observed in earthquakes prior to and including the 1971 San 

Fernando earthquake in southern California resulted from loss of support of the superstructure at 

the bearing seats. Such failures usually cannot be repaired and result in the demolition and 

reconstruction of the collapsed spans. Because of the catastrophic consequences of loss of 

support, and the perception that they could be easily prevented at relatively low cost, early 

retrofitting programs focused on preventing such failures. These programs, which were first 

undertaken by Caltrans, involved the addition of longitudinal restrainer cables and bars to limit 

relative movements at expansion joints. 

Subsequent earthquakes showed that longitudinal restrainers may have prevented loss of support 

at expansion joints in some cases, but were unable to prevent serious damage or even collapse in 

many others. Bridges that had been retrofitted with restrainer cables failed in both the 1989 
292

Loma Prieta and 1994 Northridge earthquakes. Many of these failures resulted from flaws in the 

design of the restrainer assemblies themselves. Restrainers ruptured, anchorage plates pulled 

through concrete diaphragms, swaged fittings pulled away from cables, and anchorage nuts had 

apparently worked loose from the ends of cable units. Similar failures of Japanese restraining 

devices were observed during the 1995 Kobe earthquake. 

Restrainer failures drew attention to the need to carefully design restrainer systems. Restrainers 

must not only be stiff enough and strong enough to prevent joints from separating, but the 

remainder of the bridge must be able to resist the forces developed in restrainers (Selna and 

Malvar, 1987). The number of restrainers in a typical Caltrans multi-cable unit was subsequently 

limited and strengthening was required to prevent punching shear failures of concrete 

diaphragms in the expansion joints of continuous box girder bridges. Restraining devices may 

also transmit higher forces to other bridge components such as bearings and columns, and may 

cause their failure if not properly designed. 

On the other hand, dynamic analysis of bridge and restrainer systems has demonstrated that a 

large number of restrainers is often required to limit joint movement to acceptable levels, 

particularly for high seismic loadings. In some of these cases, it is almost physically impossible 

to place the required number of restrainers in the bridge. In other cases, the required number of 

restrainers could severely overstress components elsewhere in the bridge. 

Due to the need to limit restrainer forces to protect other structural elements on the one hand, and 

the need to increase restrainer forces to accomplish their intended purpose on the other, 

restrainers alone may not be the best retrofitting solution in many cases. Alternatives that should 

be considered, either alone or in conjunction with restrainers when a loss of support failure is 

possible, include seat extensions, bearing strengthening, and bearing replacement with 

conventional or isolation bearings. 

8.4.2.1. Longitudinal Joint Restrainers 

Longitudinal joint restrainers are intended to limit the relative displacement at expansion joints 

and thus decrease the chance of a loss of support at these locations. When bearing anchor bolts 

and similar details are inadequate to prevent a loss of support at fixed bearings, longitudinal 

restrainers can also be used to transfer the longitudinal inertia force of the superstructure into the 

substructure. Because most restrainers can carry only tension forces, it will be necessary to 

provide restrainers at both ends of the span in such cases. 

Restrainers should be designed to resist the maximum calculated force within their elastic range. 

Two symmetric restrainers per joint will minimize eccentric movement of the joint. An adequate 

gap should be provided to permit normal in-service movements. For joints located at piers, 

restrainers should provide a direct and positive connection between the superstructure and the 

pier, unless the pier caps are wide enough to prevent a loss of support at the end of the span and

the anticipated maximum movement of the superstructure will not cause excessive damage. 

Steel cables and bars acting in direct tension have been the most frequently used method for 

restraining expansion joints against excessive movements. These devices do not dissipate 
293

significant amounts of energy because they are intended to remain elastic. Cable and bar 

restrainers may allow the ends of girders to be damaged by pounding, but the damage will 

usually be reparable and not extensive enough to allow the spans to lose support. Although 

cables and bars are not ideal restrainers, they are relatively simple to install and are an 

economical means of preventing a catastrophic failure during an earthquake. 

There is no established rule for choosing between cables and bars. Often cables have the 

economic advantage, since shorter lengths are possible for a given amount of joint movement. In 

addition, cables are flexible and able to accommodate transverse and vertical movements. If bars 

are used, transverse and vertical restrainers may be required to prevent shear and flexural 

distortion in the bars. 

8.4.2.1(a). Restrainer Cables 

The single restrainer cable unit is the fundamental component used in most restrainer cables. 

The most commonly used restrainer cable consists of a galvanized 19 mm (0.75 in) diameter 

steel cable that is identical to the type commonly used to anchor guard railing. The cable has 

galvanized, cold-swaged fittings and 25 mm (1 in) diameter ASTM A-449 threaded studs at each 

end. Washers and nuts use a special thread locking system consisting of a cleaner, primer, and 

anaerobic adhesive to prevent the nuts from working loose over time. 

Spring washers and cable yield indicators can be added to the basic restrainer cable unit. The 

washers are designed to keep tension in the cable while allowing movement due to temperature 

effects. The yield indicators provide post-earthquake evidence that the cables have been 

stressed. Once these indicators have yielded, there will be additional restrainer slack, or free 

movement, before they become effective again. Details of a typical Caltrans restrainer cable unit 

are shown in figure 8-29. 

An individual restrainer cable has a cross-sectional area of 143 mm2 (.222 in2

) and a minimum 

breaking strength of 205 kN (46 kips). For load and resistance factor design purposes, cables are 

assumed to have a tensile capacity of 174 kN (i.e., 0.85 x 205 kN) (39.1 kips; i.e., 0.85 x 46 

kips). A cable is more flexible than a steel bar of equivalent length and cross-sectional area. 

When cables are initially tensioned, they undergo a conditioning in which individual wires 

within the cable shift to form a more compact shape. In the case of restrainer cables, this results 

in an effective modulus of elasticity of approximately 69,000 MPa (10,000 ksi). Once this 

conditioning has taken place, a modulus of elasticity of 124,000 MPa (18,000 ksi) can be used. 

Cable length is defined as the distance between the anchored end of a cable unit and the point of 

no movement on the cable. In the case of looped restrainers, this point will be near the center of 

the loop. 

A complete description of these cable units is given in the Caltrans Standard Specifications, 

section 75-1.035 – Bridge Joint Restrainer Units (Caltrans, 1995). 
294

25 mm Diameter Stud

Face of Concrete

Bearing Plate

Thick Steel Washer

Steel Spherical Washer

Disc Springs

Cable Yield

Indicator

Threaded

.

Section B-B

Front of Disc

Reduced Section

Wall Thickness

Washer, Nut & Thread

Locking System, Each End

 . 

.

Disc

45° Bevel Gap

Swaged Fitting & Stud

19 mm Cable

End View Plan

Section A-A As Installed on Stud

Cable Yield Indicator Disc Springs

Figure 8-29. Caltrans cable restrainer unit. 
295

Caltrans has conducted research to study the performance of cable under repeated cycles of 

loading near or beyond the yield stress. The graph shown in figure 8-30(a) was developed by 

loading 2.9 m (9.5 ft.) long specimens to 0.85fult for 15 cycles, and then to failure. On the first 

cycle of loading, the cable underwent a conditioning in which slack in the strands was taken up. 

On subsequent loadings, the specimen elongated approximately 38 mm (1.5 in) before yield. 

Total elongation after the initial conditioning was approximately 115 mm (4.5 in) prior to failure. 

In a second series of tests, specimens were loaded by applying 25 mm (1 in) increments of 

displacement up to failure. Between each displacement increment, the specimen was unloaded 

to zero tension. Typical results from these tests are shown in figure 8-30(b). The wire rope 

failed at 127 mm (5 in) of elongation, which is slightly less than that demonstrated by the first 

series of tests. 

When a large number of cables are required to limit displacements to an acceptable level, it is 

usually necessary to use multi-cable assemblies. These assemblies simplify the anchorage of a 

large number of restrainers and minimize congestion at the joint. Caltrans has used a standard 

multi-cable unit in many retrofitting projects involving concrete box girder bridges. A detail of 

this restrainer assembly is shown in figure 8-31. Installation requires access to the interior cells 

of the box girder on both sides of an in-span expansion joint. It is generally preferable to cut 

access openings through the soffit slab since this will have the least effect on traffic during 

construction. These access holes are typically left in place and covered with a metal plate that 

can be removed if access to the restrainers is necessary at some time in the future. Access 

through the deck slab is possible, but will interfere with traffic. 

Laboratory testing of the early designs of these restrainer assemblies demonstrated that the 

ultimate capacity of multi-cable assemblies was limited by a punching shear failure of the anchor 

plates through the concrete diaphragm. To prevent this type of failure, it is often necessary to 

strengthen the diaphragms using cast-in-place concrete bolsters. 

8.4.2.1(b). High Strength Bar Restrainers 

Galvanized ASTM-A 722 high strength bars have also been used as longitudinal earthquake 

restrainers. These bars are less flexible, but more ductile, than standard restrainer cables. 

Because the goal of restrainer design is to keep restrainers within the elastic range, the added 

ductility is not considered a major advantage, and the reduced flexibility requires the use of 

longer bars or a larger restrainer slack to provide for the same range of movement as cables. If 

this is not done, then a larger restrainer design force is required and this in turn leads to a greater 

number of bars. Therefore, bars are used far less frequently than cables. 
296

Figure 8-30. Caltrans tests of restrainer cable. 
297

Figure 8-31. Multi-cable restrainer assembly. 

Bars should also comply with ASTM A-722 supplementary requirements, which require a 

minimum elongation of seven percent in 10 bar diameters. The modulus of elasticity of high 

strength bars may be assumed to be 200,000 MPa (29,000 ksi). Table 8-2 provides other basic 

design properties for a number of standard bar sizes. 

Table 8-2. Basic design properties for typical high strength bar sizes. 

Diameter 

mm (in) 

Area 

mm2 (in2) 

Ultimate 

Strength 

MPa (ksi) 

Yield Strength

MPa (ksi) 

Yield Force 

kN (kips) 

 25 (1) 

 32 (11⁄4) 

 35 (11⁄2) 

 549 (0.85) 

 807 (1.25) 

 1020 (1.58) 

 1030 (150) 

 1030 (150) 

 1030 (150) 

 827 (120) 

 827 (120) 

 827 (120) 

 703 (102) 

 1030 (150) 

 1310 (190) 

Galvanizing can cause field problems during the installation of high strength bars. Both threaded 

bars and smooth bars with threaded ends have been used in the past. Threaded bars are 

galvanized after being threaded. Therefore, the bar ends must be hot-brushed immediately after 

galvanizing. Even then, placement of end nuts can be difficult. Smooth bars are usually 

threaded after being galvanized. After installation, the ends are coated with zinc-rich paint. 
298

Neither galvanizing nor threading compromise the strength or the anchorage requirement of 

either type of bar. If any damage to the galvanizing occurs, zinc-rich paint must be applied to the 

affected area. 

Another problem is that standard locking devices are not always effective on threaded bars. 

Steps must be taken to prevent lock nuts from vibrating off such bars. Restrainer devices used in 

easily accessible areas should have bolt threads peened after installation, to prevent loss of 

components to vandalism. 

Bars longer than 9 m (30 ft) should be avoided for two reasons: stock lengths are usually 9 m (30 

ft), and it is difficult to galvanize bars longer than this length. 

8.4.2.1(c). Bumper Blocks 

An alternative method for restricting the relative longitudinal movement of the superstructure is 

to use bumper blocks. These devices are usually connected to the bottom flanges of girders and 

project downward to engage the substructure and thus restrict movement. This retrofit method is 

illustrated in figure 8-32. 

Steel Girder B

Bumper Block

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Figure 8-32. Bumper block retrofit. 
299

8.4.2.1(d). Anchorage of Restrainers 

Restrainers must be physically attached to the existing bridge and care should be taken that, in 

doing so, critical components are not weakened or overloaded. Anchorage takes the form of 

steel brackets, anchor plates and looping of the cable through holes drilled or cored into existing 

members. To prevent damage to the cables, they should not be bent to a radius less than 100 mm 

(4 in). A number of restrainer anchorage schemes are shown in figure 8-33. All of these 

connections should be designed for a force equal to 125 percent of the nominal breaking strength 

of the restrainers. In addition, structural members subject to brittle failure should be capable of 

resisting this same force. Restrainers should be placed symmetrically to minimize the 

introduction of eccentricities. The consequences of a premature restrainer failure should also be 

carefully considered. For example, the restrainer detail shown in figure 8-34 may be undesirable 

because, in the event of a premature failure of one of the cables, the resulting eccentric load 

could tear the web out of the girder and cause a serious loss of structural capacity, unless the web 

has been adequately reinforced to prevent such a failure. Both restrainer connections and 

existing structural elements should be capable of resisting the eccentricities caused by variations 

in the restrainer forces of at least 10 percent of the nominal ultimate restrainer capacity. 

Concrete diaphragms or walls that provide anchorage for restrainers should be capable of 

resisting a punching failure. If not, they should be strengthened to resist a force equal to 125 

percent of the nominal breaking strength of the restrainers. As previously mentioned, this type 

of strengthening is often required at expansion joint diaphragms. The size of plates to resist 

punching shear can be selected from the chart in figure 8-35. 

8.4.2.1(e). Design and Construction Issues 

Longitudinal restrainers should be oriented along the direction of expected movement. If piers 

are rigid in the transverse direction, as shown in figure 8-36, the superstructure will move 

primarily along the longitudinal axis of the bridge and restrainers should be placed accordingly. 

However, in a skewed bridge with transversely flexible supports, significant superstructure 

rotation can occur. In this case, restrainers will be more effective if placed normal to the 

expansion joint, as shown in figure 8-37. This arrangement should only be used when 

movements that could shear the restrainers are minimal. 
300

Chip to Provide Curved

Surface at Corner

Cable Restrainers

Minimun Clearance

(Typical)

Bearing Plate

Core Hole

End Diaphragm

CL Bent

Expansion

Countersink Bolt Heads

Into Deck - Holes Drilled

Through Deck

Anchorage in Deck

Restrainer

Cable

Cored Hole

0.076 m Cored Hole

0.044 m Cable (typ)

0.050 m Cored Hole

0.46 m

1.83 m

Joint

Pier

End Diaphragm

Figure 8-33. Restrainer anchorage schemes. 
301

A A

Elevation

Section A-A

Figure 8-34. Undesirable restrainer detail. 

When an expansion joint exists at a pier, restrainers at the expansion joint should provide a 

positive connection to the pier, as shown in figure 8-38. This detail will tend to prevent the 

bearings from becoming unseated. Since each of the restrainers can only resist movement in one 

direction, and because closure of the expansion joint will transfer the inertia forces of one span to 

the adjacent span, each restrainer must resist the inertia forces of both spans in conjunction with 

other load-resisting elements. Depending on the configuration of the restrainers at adjacent 

expansion joints, it is possible that the inertia forces of other spans should also be included. Note 

that the restrainers are connected to the bottom flange in figure 8-38. While this will prevent the 

possibility of tearing the web, it will also reduce vertical clearance under the bridge. 

Consideration should also be given to minimizing access difficulties during construction and 

maintenance. For example, in box girders, the number of bays in which restrainers are placed 

should be kept to a minimum. 

Many retrofit techniques for bearing and expansion joints will require coring of existing 

concrete. When coring is to be used, there are at least two issues that should be considered. The 

first is the clearance required for coring equipment. The minimum distance between the center 

of a cored hole and an adjacent surface should be 75 mm (3 in). For holes larger than 150 mm (6 

in), the edge of the hole may be flush against the adjacent surface. In addition, cored holes 

should be located so that a minimum of 1.2 m (4 ft) of clearance exists on at least one side from 

the centerline of the hole. These clearances are shown in figure 8-39. 
302

Square

Steel

Bearing

Plate

Assume: (Existing Bridge)

Figure 8-35. Resistance of concrete wall to punching shear. 
303

Piers restrain rotation or

transverse movement of

superstructure

Restraint parallel to direction

of movement of superstructure

Piers Restrain Rotation or

Transverse Movement of

Superstructure

Restraint Parallel to Direction

of Movement of Superstructure

Figure 8-36. Restrainer orientation for transversely rigid supports. 

L

Columns permit rotation

of superstructure

Restraint normal to

C hinge or bearings

Columns Permit Rotation

of Superstructure

Restraint Normal to

C Hinge or Bearings L

Figure 8-37. Restrainer orientation for transversely flexible supports. 

Restrainer Cable

Figure 8-38. Restrainer at pier with a positive tie to pier. 
304

Figure 8-39. Required clearances for coring. 

The second issue is the potential for interference with primary reinforcing steel, expansion joint 

hardware, and prestressing tendons. Special care should be taken to avoid cutting structurallycritical reinforcement and prestressing bars, or large multi-wire or multi-strand tendons, in posttensioned bridge components. If the type of prestressing system used cannot be determined from 

the “as built” plans or construction records, bars and large tendons should be assumed. 

Construction personnel should be alerted to the presence of these elements so that appropriate 

precautions can be taken in the field. When possible, drilling should be specified instead of 

coring, since there is less potential for damaging reinforcing or prestressing steel. 

The response of a bridge that has been retrofitted with restrainers is nonlinear, even if the 

columns and foundations remain elastic. This is because restrainers at expansion joints are 

initially slack and intended to engage only after some movement has taken place. The restrainers 

are also essentially tension-only devices and are ineffective while the joints are closing. 

Furthermore, the impact that occurs at joint closure is a highly nonlinear problem that cannot be 

solved rigorously by simplified means. 
305

Because restrainer behavior is very nonlinear, an accurate analysis of restrainer forces can only 

be obtained from a nonlinear dynamic analysis of the entire bridge. Since this is impractical in 

most cases, approximate static analysis procedures are commonly used to evaluate the need for, 

and design of, restrainers. These procedures generally yield better results than those obtained 

from an elastic dynamic analysis. When restrainers are required, the restrainer force capacity 

should be determined according to the procedures described in section 8.4.2.1(g). 

For all single span bridges and bridges in Seismic Retrofit Category B (section 1.10), a dynamic 

analysis is not necessary and the design force is given by Methods A1 and A2 (section 5.2). 

Restrainers should develop this force capacity before movement, equal to 67 percent of the 

available seat width as shown in figure 8-40, is exceeded. In areas of low seismicity, it may be 

desirable to restrain joints having narrow seats by using short, stiff restrainers designed to 

function below their yield capacity. The stiffness of the restrainers will result in small joint 

movements, while restrainer forces will be kept to reasonable levels because of the low 

seismicity. However, for the purpose of designing restrainer anchorages and evaluating the 

effect of restrainers on other structural members, design forces should be assumed to be 125 

percent of the nominal restrainer breaking strength. 

100 mm

Available

Seat Width

Seat

Expansion Joint Gap

Note: The 100 mm dimension shown

provides for nominal “reasonable”

allowable seat width. A larger or

smaller dimension may be required.

The expansion joint gap for new

structures should be the maximum

estimated opening.

Figure 8-40. Available seat width. 

Section 8.4.2.1(f) describes a simplified method for evaluating the need for restrainers in multispan simply supported bridges. Simplified methods for verifying the number, size, and length of 

restrainers is presented in section 8.4.2.1(g). Both iterative and non-iterative methods are 

described. Although these methods are not rigorously correct, they yield acceptable results in 

most cases. The two restrainer design methods given below were developed principally for 

restrainers at in-span expansion joints in continuous superstructures, but can be adapted to 

simply-supported spans by considering every span to be a ‘frame’ (DesRoches and Fenves 

1998). 

Both restrainer design methods are based on limiting joint movement to 67 percent of the 

available seat width, as shown in figure 8-40, which may call for a large number of restrainers in 

some cases. When this occurs, increasing the available seat width and allowable restrainer 
306

displacement will reduce the number of restrainers. Seat extenders may be used to increase the 

available seat width. Increasing the length of restrainers or increasing the restrainer slack will 

increase the allowable restrainer displacement. 

8.4.2.1(f). Evaluation of the Need for Restrainers for Simply Supported Spans 

The following procedure may be used to determine the adequacy of the connection between the 

span and substructure, and to determine the unrestrained displacements of the structure. If the 

connection is adequate and the unrestrained displacements are less than 67 percent of the 

available seat width, no restrainers are required. If the connection is not adequate and the 

unrestrained displacements are more than 67 percent of the available seat width, restrainers 

should be designed using the procedures described in section 8.4.2.1(g). 

In this evaluation, it is necessary to distinguish between fixed and expansion bearings, and to 

consider the location of the bearing. The evaluation procedure varies for bearings located at 

abutments, end-span piers, and the remaining piers. In all cases, an equivalent single degree of 

freedom (SDOF) system is analyzed to determine bearing forces and bearing displacements. The 

following paragraphs define the procedures for determining the adequacy of the connection and 

bearing seat for each case in which one of the spans supported at a pier has fixed bearings while 

the other span at this pier has expansion bearings. These procedures could be adapted to interior 

piers with two sets of expansion bearings or two sets of fixed bearings by modifying the 

participating masses and stiffness of the equivalent SDOF system, as required. 

Fixed Bearings at Abutments – The SDOF system for this case assumes the span is moving away 

from the abutment. Therefore, the stiffness of the unrestrained system is derived from the 

nonlinear behavior of the bearing. The bearing is initially very stiff and, therefore, the equivalent 

maximum elastic force applied to the bearing is simply the weight of the span multiplied by the 

peak ground acceleration, which may be assumed to be given by 0.4 FaSs (see figure 1-8). If the 

equivalent elastic force exceeds the capacity of the bearings, then the bearings are assumed to 

fail and it will be necessary to check the resulting bearing displacement. This is done using an 

iterative procedure in which the bearing is assumed to have a perfectly elastoplastic forcedisplacement relationship. Therefore, the effective bearing stiffness in the SDOF model is 

reduced until the resulting bearing force equals the capacity of the bearing. This stiffness 

represents the secant stiffness on the force displacement diagram of the bearing at maximum 

displacement. No corresponding increase in damping should be assumed. The resulting 

maximum displacement is compared to the available seat width to determine whether or not 

restrainers are required. This case and procedure are illustrated in figure 8-41. 

End-Span Fixed Bearings at Piers – In this case, the SDOF system is derived by assuming the 

unrestrained span is moving toward the abutment. Therefore, in addition to the stiffness of the 

bearing, the stiffness of the column and foundation must be considered. If displacements are 

large enough to close the gap between the span and the abutment, then the abutment stiffness 

will also play a role in the seismic response of the bearing. This case and a model of the 

assumed SDOF system is shown in figure 8-42. 
307

Figure 8-41. SDOF model for end-span fixed bearing at abutment. 

As previously described, the procedure first involves finding the maximum equivalent elastic 

force in the bearing to determine if the bearing will fail. Initially, only the stiffness of the 

column and foundation is considered. If the resulting displacements are smaller than the gap at 

the abutment, bearing forces are compared with bearing capacity to determine if a bearing failure 

will occur. If the displacement at the abutment is larger than the gap, the stiffness of the SDOF 

system must be adjusted to consider the stiffness of the abutment. This is done iteratively by 

first assuming a value for the abutment stiffness and adjusting that stiffness until there is force 

and displacement compatibility at the abutment. The force displacement curve for the abutment 

is assumed to be bilinear once the gap is closed, as shown in figure 8-43. Abutment stiffness in 

the SDOF model is assumed to correspond to a chord from the at-rest point to the point of force 

and displacement compatibility on this curve (Point A). Once the correct abutment stiffness has 

been identified, the bearing force is compared with bearing capacity to determine if yielding of 

the bearing will occur. 
308

Figure 8-42. SDOF model for an end-span fixed bearing at pier. 

Figure 8-43. Force displacement relationship for abutment. 
309

If the bearing yields, it will be necessary to perform a new analysis to assess whether the bearing 

seat is adequate to prevent unseating of the span. In this case, the equivalent bearing stiffness 

will be included in the SDOF system (see figure 8-42). Because of the nonlinear behavior of the 

bearings and the abutment, a trial and error procedure will be required to determine the combined 

secant stiffness of the two elements. Since force and displacement compatibility must be 

achieved in both the bearings and the abutment simultaneously, it will be necessary to adjust the 

combined stiffness of both elements until such compatibility is achieved. To accomplish this, the 

forces and displacements in each element derived from the SDOF analysis must be tracked and 

their combined stiffness adjusted accordingly. The analysis converges on the correct solution 

when these forces correspond to the combined forces from the assumed force and displacement 

curves at the displacement derived from the same analysis. Once this has been achieved, the 

relative displacement at the pier bearing seat may be calculated as the force in the bearing 

divided by the assumed bearing stiffness. This relative displacement must be less than 67 

percent of the available bridge seat to provide an adequate factor of safety against unseating of 

the span. 

Fixed Bearings at Interior Spans – The SDOF model and analysis procedures used for this case 

are similar to those used for an end span fixed bearing at a pier, except that the participation of 

the abutment is not included. It is recommended that the mass include the pier cap mass and 

tributary column mass (may be assumed to be one-third of the column mass), in addition to the 

mass of the span. 

Expansion Bearings at Abutments – The analysis of expansion bearings is limited to determining 

relative displacements. At abutments, this displacement is determined by considering the 

displacement of the end span. Since the critical response occurs when the span moves away 

from the abutment, only the mass of the span and the effective stiffness of the adjacent pier, 

foundation, and fixed bearing are considered. The methods for determining effective stiffness 

are similar to those used in previous cases. Because the abutment in this case is assumed not to 

move with respect to the ground, the movement of the span and the relative movement at the 

abutment are assumed to be equal. 

End Span Expansion Bearings at Piers – For the case of an expansion bearing located at a pier 

that supports an end span, the critical response involves the pier moving out from under the 

bearing. Because the abutment end of the span is assumed to remain fixed relative to the ground, 

the maximum relative displacement at the pier can be determined by finding the maximum 

relative displacement at the top of the pier. In this case, it is assumed that the adjacent span has 

fixed bearings at the pier in question. The displacement at the top of this pier is determined by 

considering the response of a SDOF system consisting of the mass of the adjacent span and the 

equivalent stiffness of the pier, foundation, and fixed bearing system. The relative displacement 

at the top of the pier is determined by dividing the inertia force at the top of the column by the 

stiffness of the column and foundation system. 

Expansion Bearings at Interior Spans – This case differs from the previous case in that the span 

with the expansion bearings can move independently of the pier supporting it. Therefore, in 

addition to calculating the movement at the top of the pier, it is also necessary to determine the 

movement of the span. This is done by modeling the span as a SDOF system with a mass equal 

to the span and an equivalent stiffness equal to the adjacent pier, foundation, and fixed bearing 
310

system. The displacement at the top of the supporting system is determined as in the previous 

case. 

Once these displacements are determined, they are combined using the Complete Quadratic 

Combination (CQC) rule (section 5.4.2.2) to find the maximum relative displacement. The 

application of the CQC rule is given in equation 8-3: 

2 2 D D 2 DD D r 1 12 1 2 2 = +ρ + (8-3a) 

where: 

D1 is the maximum displacement of the span, 

 D2 is the maximum displacement at the top of the pier, 

( )

( ) ( )

3 2 2

12 2 2 2 2

8 1

1 41

ξ +β β ρ =

− β + ξ β + β

 , (8-3b) 

ξ is the equivalent viscous damping ratio, and 

β is the ratio of the vibration frequencies (mode 1 to mode 2). 

Whenever the above analyses indicate the need for restrainers, the restrainers should be designed 

(i.e., number, size, length, and slack) using the methods described in section 8.4.2.1(g). This 

should be done by assuming that the bearings have failed and can no longer resist horizontal 

forces. Only the restrainers should be assumed to contribute to the lateral strength and stiffness 

of the system. In addition, restrainer design forces should be modified by a skew factor of 

1/cosα, where α is the skew angle. 

8.4.2.1(g). Restrainer Design Methods 

The number of 19 mm (0.75 in) diameter restrainer cables required to limit the relative 

displacement of expansion joints is dependent on many factors (DesRoches and Fenves, 1998). 

One of the most critical parameters is the ratio between the longitudinal periods of vibration of 

adjacent frames or spans that are being restrained. Simplified methods are valid only within 

specific ranges of this parameter. 

When the ratio of the periods (smallest period to largest period) is 0.3 or less, a rigorous 

nonlinear dynamic analysis of the bridge must be used to reliably calculate the number and 

length of restrainers that are required. This procedure requires computer software that is not 

readily available, nor is it easy to use. In addition, the number of restrainers required is often so 

large that it is difficult or impossible to fit them into the bridge. Therefore, restrainers are not 

recommended for bridges with small seat widths when the ratio of the periods of adjacent frames 

or spans is less than 0.3. Other retrofit measures should be considered in this case. 

Comparison of simplified methods with nonlinear dynamic analysis results show that many of 

the existing design methods are not reliable at low period ratios. For example, the current 

Caltrans method (Caltrans, 1989) underestimates the required number of restrainers for period 

ratios below approximately 0.6, but grossly overestimates the number of restrainers required for 
311

period ratios above this level. The current AASHTO method estimates a number of restrainers 

that is independent of the period ratio. This is less than the number of restrainers estimated by 

nonlinear dynamic analysis for period ratios under approximately 0.5, and more than the number 

of restrainers estimated for higher ratios of the periods. 

Two new design methods have recently been developed (DesRoches and Fenves, 1998) which 

are more applicable to retrofitted structures than the previous methods. For ratios of periods 

between 0.3 and 0.6, an iterative method is recommended. This procedure may require several 

iterations to converge to the correct solution. For ratios greater than 0.6, a single-step simplified 

method is available that requires less design effort than the first. Both methods are described 

below and illustrated in examples 8-1 and 8-2. 

The Iterative Method

This method utilizes the substitute structure method (Gulkan and Sozen, 1974) to determine the 

required number of restrainers. The span or frame properties that are used in this procedure are 

the frame stiffnesses, K1 and K2, frame masses, m1 and m2, and target displacement ductility of 

the frames, μ. The properties of the restrainers are the length Lr, modulus of elasticity E, yield 

stress fy, and restrainer slack Drs,. All analysis is performed using the applicable design spectra. 

The procedure includes the following steps: 

Step 1. Calculate Maximum Allowable Expansion Joint Displacement 

Step 1a. Calculate the elongation capacity of a restrainer, Dr, as follows: 

Dr = Dy + Drs (8-4) 

where Dr is the maximum permissible restrainer elongation, Dy is the restrainer 

elongation at yield (equation 8-5), and Drs is the slack in the restrainer system. 

The yield elongation is given by: 

Dy = fyLr/E (8-5) 

where fy is the yield stress of restrainer, Lr is the restrainer length, and E is the initial 

modulus of elasticity of restrainer (before initial stretching). 

The restrainer slack, Drs, is the clearance provided to accommodate thermal expansion 

and other non-seismic effects. 

Step 1b. Compare the available expansion joint seat width with the maximum 

permissible restrainer elongation, Dr. 

If the maximum permissible restrainer displacement, Dr, is greater than 67 percent of the 

available seat width Das (see figure 8-40), there is an insufficient safety factor against the 

expansion joint becoming unseated before the restrainer capacity is reached. In this case, 
312

either the seat width must be increased or Dr must be reduced by one or more of the 

following: 

• Shortening the restrainers. 

• Decreasing the restrainer slack. 

• Reducing the stress in the restrainers to a value less than yield. 

Step 2. Compute the Unrestrained Relative Expansion Joint Displacement, Deq0

The unrestrained relative expansion joint displacement (Deq0) can be obtained using the 

CQC rule defined in equation 8-3. The frame displacements, D1 and D2, are determined 

from the following, based on the design spectrum corresponding to the effective damping 

coefficient given by equation 8-7: 

( ) i

2

eff

i v 1 eff eff

T

D FS T , 2

⎛ ⎞ = ⎜ ⎟ ξ

⎝ ⎠ π

 (8-6) 

where i

i

i

eff

eff

m T 2

K

= π = 

i

i

eff

w 2

gK

π and i

i

eff

K K = μ . 

The effective damping coefficient, ξeff , may be estimated as: 

eff

0.95 1 0.05

0.05

− − μ

μ ξ= +

π

 (8-7) 

When design spectra are not available for ξeff, the value of FvS1 for five percent damping, 

[FvS1]0.05, may be modified by a coefficient cd to obtain values at ξeff. 

That is 

 [FvS1]ξeff = cd [FvS1]0.05 (8-8a) 

where: 

d

eff

1.5 c 0.5

40 1

= + ξ + . (8-8b) 

If Deq0 ≤ 0.67 of the available seat width (Das), no restrainers are required. 

If Deq0 > 0.67Das, restrainers must be provided, in accordance with step 3. 

 
313

Step 3. Estimate the Initial Restrainer Stiffness 

The initial estimate of restrainer stiffness required to limit expansion joint displacement is 

obtained from the incremental stiffness expression given in equation 8-9: 

( ) mod 0

0

eff eq r

r

eq

K DD

K

D

− = (8-9) 

where: 

( ) 1 2

mod

1 2

eff eff 1 2

eff

eff eff 1 2

K K K K K

K K KK = = + μ+

 (8-10) 

Step 4. Calculate Relative Expansion Joint Displacement from Modal Analysis 

The maximum relative expansion joint displacement is determined from a two-degree-offreedom (2DOF) modal analysis of the frames using the restrainer stiffness determined in 

step 3. This 2DOF system is shown in figure 8-44. The relative displacement of the 

expansion joint is obtained from the CQC, and is given by: 

1 2 12

2 2 D D D 2 DD eq eq eq 12 eq eq = + +ρ (8-11) 

where: 

 ( )

( ) ( )( )

3 2

12 1 2

12 2 2 2 2 22

12 1 2

8

1 41 4

ξ ξ ξ + βξ β ρ =

−β + ξξβ + β + ξ + ξ β

 (8-12) 

 Deqi = PiSa (Teffi, ξeffi) (8-13) 

{ } [ ]{ }

{ } [ ]{ } ( ) { } { }

T

i T

i i T

i i

M 1

P a participation factor for mode "i"

K

φ = φ=

φ φ

 (8-14) 

β = frequency ratio (mode 1 to mode 2), 

 {φi} = mode shape for mode ‘i’, 

 [M] = mass matrix of a two frame system, 

 [K] = stiffness matrix of a two frame system with restrainers, and 

 {a}T

 = [−1 1], 

 { } ⎭

⎬

⎫

⎩

⎨

⎧ = 1

1

1 . 

Note that in equation 8-12, ρ12 may be calculated using the formula provided in equation 

8-3b if the effective damping coefficient, ξ, is equal for both frames. 
314

Solution of the above equations first requires that the equations of motion for the 2DOF 

system be solved. This can be accomplished by computer analysis or by hand 

calculations. 

If Deq > Dr, go to step 5 and then back to step 4. Otherwise, go to step 6 and calculate the 

required number of restrainers. 

Step 5. Calculate the Incremental Restrainer Stiffness Required to Limit Expansion Joint 

Displacement 

The incremental restrainer stiffness is: 

( ) ( ) j

j 1 j mod j

j

eq r

r r eff r

eq

D D

K KK K + D

−

=+ + (8-15) 

Steps 4 and 5 are repeated until Deq ≤ Dr.

Step 6. Calculate the Number of Restrainers 

 

Once the required restrainer stiffness is calculated, the number of restrainers is 

determined by: 

y r

r r

r f A

K D N = (8-16) 

where Ar is the area of one restrainer, which for typical 19 mm (0.75 in) diameter cables 

is 143 mm2

 (0.22 in2). 

A restrainer design example using the iterative method is given in example 8-1. 

The Single Step (Simplified) Method

The single step method is reliable only when the ratio of the periods is greater than 0.60 and the 

ratio of restrainer displacement capacity to initial unrestrained displacement, (η, where η = 

Dr/Deq0), is between 0.20 and 0.50. If these criteria are not met, the iterative method should be 

used. The following steps are required in the single-step method. 

Step 1. Calculate Maximum Allowable Expansion Joint Displacement 

Same as Step 1 for the iterative method. 

Step 2. Compute Unrestrained Relative Expansion Joint Displacement 

Same as Step 2 for the iterative method. 
315

Figure 8-44. 2DOF model for the iterative method. 
316

Step 3. Calculate Required Restrainer Stiffness 

mod

2

r eff

0.50 K K 0.50 ⎡ − η ⎤

= + ⎢ ⎥ ⎣ η ⎦

 (8-17) 

where mod K is given by equation 8-10 and eff

r

eq0

D

D η = . 

 

Step 4. Calculate Number of Restrainers 

y r

r r

r f A

K D N = (8-18) 

Examples of restrainer design using the single step method are given in examples 8-2 and 8-3. 

8.4.2.2. Transverse Restrainers 

8.4.2.2(a). General 

Transverse restrainers are necessary in many cases to keep the superstructure from sliding off its 

supports should the bearings fail in the transverse direction. Particularly vulnerable conditions 

exist when any of the following conditions exist: 

• High concrete pedestals are used as bearing seats under individual beams. 

• Bearing seats are narrow and highly skewed. 

• Steel rocker bearings are relatively tall. 

• Transverse distance between the edge bearing and the edge of the seat is small. 

Whenever transverse movement might lead to a loss of support, transverse restraint should be 

provided as a retrofit measure. 

8.4.2.2(b). Shear Keys 

Concrete shear keys that are doweled into the bridge seats can be used to resist transverse 

movement at supports. These keys may be placed between beams and outside the edge beams if 

there is sufficient room, and should be designed to carry the expected forces elastically. 

Transverse diaphragms or cross frames at these locations should be strong enough to transmit the 

design force. Shear keys can sometimes be incorporated in new bearing pedestals. An example 

of a transverse concrete shear key is shown in figure 8-45. 
317

Figure 8-45. Transverse shear keys. 

8.4.2.2(c). Keeper Brackets 

Transverse restrainers can sometimes be added as part of a bearing retrofit. These restrainers 

take the form of supplemental keeper brackets that must be designed to carry loads elastically. If 

sufficient deformation of the beams can take place to engage all of the brackets, they may be 

considered to be 100 percent effective. If not, some engineering judgement must be used to 

allow for variable gaps that could cause one bracket to be overloaded with respect to others at the 

same support. An example of a keeper bracket used in a bearing retrofit is shown in figure 8-46. 

8.4.2.2(d). Steel Pipe Restrainers 

Another method that has been used to provide transverse restraint in concrete structures employs 

a double extra-strong steel pipe filled with concrete that passes through the joint (see figure 

8-47). This design assumes that the pipe bears against the walls of the cored hole. The full 

concrete compressive strength may be used in well-reinforced expansion joint diaphragms. 

However, the full strength of acute corners at highly skewed joints should not be relied upon to 

resist the full shear force, because such corners can easily break off. 
318

Figure 8-46. Bearing keeper bracket retrofit. 

Figure 8-47. Steel pipe restrainers. 
319

EXAMPLE 8.1: RESTRAINER DESIGN BY ITERATIVE METHOD 

A six-span bridge divided into two, three-span segments as shown in the figure below. Design restrainers 

for the expansion-joint seat where the two segments meet. Calculate the number and size of the 

restrainers required using the iterative method. 

Assume the following: 

 Seat width, N = 305 mm (12 in) 

 Concrete cover on vertical faces at seat, dc = 50 mm (2 in) 

 Restrainer yield stress, fy = 1,214 MPa (176 ksi) 

 Restrainer modulus of elasticity, E = 69,000 MPa (10,000 ksi) 

 Restrainer length, Lr = 5.4 m (18 ft) 

 Restrainer slack, Drs = 25 mm (1 in) 

 Response spectrum for site: figure 1-8 

 Short period coefficient, FaSs = 1.75 

 Long period coefficient, FvS

 = 0.70 

 Target displacement ductility of the frames, μ = 4 

 Frame stiffnesses, K1 and K2 = 357 and 89.3 kN/mm, respectively 

 (2040 and 510 K/in, respectively) 

 Frame weights, W1 = W2 = 22.3 MN (5000 K) 

STEP 1. CALCULATE ALLOWABLE EXPANSION JOINT DISPLACEMENT 

Dr = 95 + 25 = 120 mm (4.7 in) 

Das = 305 – 25 – 100 = 180 mm (7.1 in) 

2/3Das = Dr OK 
320

STEP 2. COMPUTE EXPANSION JOINT DISPLACEMENT WITHOUT RESTRAINERS 

The effective stiffness of each frame modeled as a substitute structure is: 

eff 1 1 K K 357 4 89.3 kN/mm (510 kips/in) = μ= =

eff 2 2 K K 89.3 4 22.3 kN/mm (128 kips/in) = μ = =

Therefore, the effective natural period of each frame is given by: 

1

1

1

eff

eff

W 22.3 1000 T 2 2 1.0 sec

gK 9800 89.3

⋅ =π =π = ⋅

2

2

2

eff

eff

W 22.3 1000 T 2 2 2.0 sec

gK 9800 22.3

⋅ =π =π = ⋅

The effective damping and design spectrum correction factor is: 

eff

0.95 1 0.05 0.95 1 0.05 2

2 0.05 0.05 0.19

3.142

−− μ − −⋅ μ ξ= + = + =

π

d

eff

1.5 c 0.5 0.67

40 1

= +=

ξ +

Therefore, the frame deflections are calculated as follows: 

( ) 1

1

2 2

eff

1 d a eff

T 1.0 D gc S T ,0.05 9800 0.67 0.7 116 mm (4.6 in) 2 2 3.142

⎛ ⎞ ⎛ ⎞ = = ⋅ ⋅ ⋅= ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ π ⋅ ⎝ ⎠ ⎝ ⎠

( ) 2

2

2 2

eff

2 d a eff

T 2.0 D gc S T ,0.05 9800 0.67 0.35 232 mm (9.2 in) 2 2 3.142

⎛ ⎞ ⎛ ⎞ = = ⋅ ⋅⋅= ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ π ⋅ ⎝ ⎠ ⎝ ⎠

The relative displacement of the two frames can now be calculated using the CQC combination of the two 

frame displacements as given by equation 8-3. In this case the frequency ratio, β, is 2.0. 

( )

( ) ( )

( )( )

( ) () ()

3 3 2 2 2 2

12 2 2 22 2 2 22

8 1 8 0.19 1 2 2

1 4 1 1 2 4 0.19 2 1 2

ξ +β β ⋅ ⋅ + ⋅

ρ= =

−β + ξ β +β − + ⋅ ⋅ ⋅ +
321

Because D1 and D2 are of opposite sign, equation 8-3 becomes 

0

2 22 2 D D 2 D D D 116 2 0.21 116 232 232 237 mm (9.3 in) eq 1 12 1 2 2 = −ρ + = − ⋅ ⋅ ⋅ + =

>2/3Das = 120 mm. 

Therefore, restrainers are needed. 

STEP 3. DETERMINE FIRST ESTIMATE OF RESTRAINER STIFFNESS 

1 2

mod

1

eff eff

eff

eff eff2

K K 89.3 22.3 K 17.9 kN/mm (102 kips/in) K K 89.3 22.3

⋅ === + +

mod 0 ( ) ( ) 0

0

eff eq r

r

eq

K DD 17.9 237 120

K 8.84 kN/mm (53.5 kips/in) D 237

− ⋅ − ===

STEP 4. CALCULATE RELATIVE EXPANSION JOINT DISPLACEMENT FROM MODAL ANALYSIS 

Solve Modal Equations: 

{ } {} 10 0

i

0 20

eff r r 2 1

eff

r eff r 2

KK K m 0

K KK 0 m

⎡ ⎤ + − ⎡ ⎤ ⎢ ⎥ φ = ω ⎢ ⎥ φ

− + ⎣ ⎦ ⎣ ⎦

{ } {} i

2

eff

98.1 8.84 2.28 0

8.84 31.1 0 2.28

⎡ ⎤ ⎡⎤ − φ = ω φ ⎢ ⎥ ⎢⎥ ⎣ ⎦ ⎣⎦ −

1 2

2 2

eff eff 2 2

1 1 13.2 , 43.9

sec sec

ω= ω=

eff eff 1 2 T 2 / 13.2 1.73 sec, T 2 / 43.9 0.95 sec =π = =π =

{ } 1 2 { } 0.13 1.00 , 1.00 0.13

⎧⎫ ⎧ ⎫ φ= φ= ⎨⎬ ⎨ ⎬ ⎩⎭ ⎩ ⎭ −
322

Calculate Participation Factors: 

{ } {}

{} {} ({} { }) { }

{ }

{ }

T

1 T 2

1 1 T

1 1

2.28 0 1 0.13 1.00 M 1 0 2.28 1 0.13 P a 1 1 0.074 sec

K 98.1 8.84 0.13 1.00 0.13 1.00

8.84 31.1 1.00

⎡ ⎤⎧⎫

⎢ ⎥ ⎨ ⎬ φ ⎡ ⎤ ⎣ ⎦ ⎣ ⎦⎩⎭ ⎛ ⎞ ⎧ ⎫ = φ= − = ⎜ ⎟ ⎨ ⎬

φ φ ⎡ ⎤ ⎡ ⎤⎧ ⎫ − ⎝ ⎠ ⎩ ⎭ ⎣ ⎦ ⎢ ⎥ ⎨ ⎬ ⎣ ⎦⎩ ⎭ −

{ } {}

{} {} ({} { }) { }

{ }

{ }

T

2 T 2

2 2 T

2 2

2.28 0 1 1.00 0.13 M 1 0 2.28 1 1.00 P a 1 1 0.022 sec

K 98.1 8.84 1.00 0.13 1.00 0.13

8.84 31.1 0.13

⎡ ⎤⎧⎫ − ⎢ ⎥ ⎨ ⎬ φ ⎡ ⎤ ⎣ ⎦ ⎣ ⎦⎩⎭ ⎛ ⎞ ⎧ ⎫ = φ = − =− ⎜ ⎟ ⎨ ⎬

φ φ ⎡ ⎤ ⎡ ⎤⎧ ⎫ − − ⎝ ⎠ ⎩ ⎭ ⎣ ⎦ − ⎢ ⎥ ⎨ ⎬ ⎣ ⎦⎩ ⎭ − −

Calculate new relative displacement at expansion joint: 

( )

( )

1

2

eq 1 d a

eq 2 d a

D P gc S 1.73,0.05 0.074 9800 0.67 0.405 197 mm (7.7 in)

D P gc S 0.95,0.05 0.022 9800 0.67 0.737 106 mm (4.2 in)

= = ⋅ ⋅⋅ =

= =− ⋅ ⋅ ⋅ =−

β = = 1.73 / 0.95 1.8

2 3/2

12 22 2 2

8(0.19) (1 1.8) 1.8 0.27

(1 1.8 ) 4 (0.19) (1.8) (1 1.8)

⋅+ ⋅

ρ= =

− +⋅ ⋅ ⋅+

2 2 D (197) ( 106) 2(0.27)(197)( 106) 197 mm (7.8 in) eq = +− + − =

Because Deq > Dr, Continue to Step 5 

STEP 5. CALCULATE NEW RESTRAINER STIFFNESS 

( ) ( ) 0 mod 0

eq r

r1 r eff r

eq

D D (196 120) K K K K 8.84 (17.9 8.84) D 196

 19.2 kN/mm (110 kips/in)

− − =+ + = + +

=

STEP 4. SECOND ITERATION 

1 2

2 2

eff eff 2 2

1 1 15.4 , 48.1

sec sec

ω= ω=
323

eff eff 1 2 T 1.60 sec, T 0.91sec = =

{} { } 1 2

0.27 1.00 , 1.00 0.27

⎧⎫ ⎧ ⎫ φ= φ = ⎨⎬ ⎨ ⎬ ⎩⎭ ⎩ ⎭ −

2 2 P 0.054 sec , P 0.017 sec 1 2 = =−

eq1 D 0.054 9800 0.67 0.438 156 mm (6.1in) = ⋅ ⋅⋅ =

eq2 D 0.017 9800 0.67 0.769 88 mm (3.5 in) =− ⋅ ⋅ ⋅ = −

β = 1.60/0.91 = 1.77 

2 3/2

12 22 2 2

8(0.19) (1 1.77) 1.77 0.29

(1 1.77 ) 4 (0.19) (1.77) (1 1.77)

⋅+ ⋅

ρ= =

− +⋅ ⋅ ⋅+

( )( ) 2 2 D 156 88 2 0.29 156 ( 88) 155 mm (6.1in) eq = +− + ⋅ ⋅ ⋅− =

Since Deq > Dr, Continue to Step 5 

STEP 5. CALCULATE REQUIRED RESTRAINER STIFFNESS 

K 19.2 (17.9 19.2)(155 120)/155 27.6 kN/mm (158 kips/in) r2 = ++ − =

STEP 4. THIRD ITERATION 

1 2

2 2

eff eff 2 2

1 1 16.5 , 52.3

sec sec

ω= ω=

eff eff 1 2 T 1.55 sec, T 0.87 sec = =

{} { } 1 2

0.36 1.00 , 1.00 0.36

⎧⎫ ⎧ ⎫ φ= φ = ⎨⎬ ⎨ ⎬ ⎩⎭ ⎩ ⎭ −
324

2 2 P 0.044 sec , P 0.014 sec 1 2 = =−

eq1 D 0.044 9800 0.67 0.452 131mm (5.2 in) = ⋅ ⋅⋅ =

eq2 D 0.014 9800 0.67 0.805 73 mm (2.9 in) =− ⋅ ⋅ ⋅ = −

β = 1.60/0.91 = 1.78 

2 3/2

12 22 2 2

8(0.19) (1 1.78) 1.78 0.29

(1 1.78 ) 4 (0.19) (1.78) (1 1.78)

⋅+ ⋅

ρ= =

− +⋅ ⋅ ⋅+

()( ) 2 2 D 131 73 2 0.29 131 ( 73) 130 mm (5.1in) eq = +− + ⋅ ⋅ ⋅− =

Since Deq > Dr, Continue to Step 5 

STEP 5. CALCULATE REQUIRED RESTRAINER STIFFNESS 

K 27.6 (17.9 27.6)(130 120)/130 31.1kN/mm (178 kips/in) r3 = ++ − =

STEP 4. FOURTH ITERATION 

1 2

2 2

eff eff 2 2

1 1 16.8 , 54.1

sec sec

ω= ω=

eff eff 1 2 T 1.53 sec, T 0.85 sec = =

{} { } 1 2

0.39 1.00 , 1.00 0.39

⎧⎫ ⎧ ⎫ φ= φ = ⎨⎬ ⎨ ⎬ ⎩⎭ ⎩ ⎭ −

2 2 P 0.041sec , P 0.013 sec 1 2 = =−

eq1 D 0.041 9800 0.67 0.458 122 mm (4.8 in) = ⋅ ⋅⋅ =

eq2 D 0.013 9800 0.67 0.824 68 mm (2.7 in) =− ⋅ ⋅ ⋅ = −
325

β = 1.53/0.85 = 1.8 

2 3/2

12 22 2 2

8(0.19) (1 1.8) 1.8 0.28

(1 1.8 ) 4 (0.19) (1.8) (1 1.8)

⋅+ ⋅

ρ= =

− +⋅ ⋅ ⋅+

( )( ) 2 2 D 122 68 2 0.28 122 ( 68) 122 mm (4.8 in) eq = +− + ⋅ ⋅ ⋅− =

Deq ≈ Dr, Say OK - Continue to Step 6 

STEP 6. CALCULATE NUMBER OF RESTRAINERS 

K 31.1 kN/mm (178 kips/in) r =

r r r

y r

K D 31.1 0.120 N 21.5

F A 1214 0.000143

⋅ == = ⋅ Use 22 – 19 mm restrainer cables 
326

EXAMPLE 8.2: RESTRAINER DESIGN BY THE SINGLE-STEP (NON-ITERATIVE) METHOD 

(SIMPLIFIED) 

A six-span bridge has two, three-span segments as shown in the figure below. Restrainers are to be 

designed for the hinge seat where the two segments meet. This is the same bridge as in example 8-1. 

Calculate the number and size of the restrainers required using the single step (simplified) method. 

Assume the following: 

 Seat width, N = 305 mm (12 in) 

 Concrete cover on vertical faces at joint, dc = 50 mm (2 in) 

 Restrainer yield stress, fy = 1,214 MPa (176 ksi) 

 Restrainer modulus of elasticity, E = 69,000 MPa (10,000 ksi) 

 Restrainer length, Lr = 5.4 m (18 ft) 

 Restrainer slack, Drs = 25 mm (1 in) 

 Response spectrum for site: figure 1-8 

 Short period coefficient, FaSs = 1.75 

 Long period coefficient, FvS

 = 0.70 

 Target displacement ductility of the frames, μ = 4 

 Frame stiffnesses, K1 and K2 = 357 and 89.3 kN/mm, respectively 

 (2040 and 510 K/in, respectively) 

 Frame weights, W1 = W2 = 22.3 MN (5000 K) 

 

STEP 1. CALCULATE ALLOWABLE EXPANSION JOINT DISPLACEMENT 

Restrainer elongation at yield. 

Dy = fyLr/E = 1,214 MPa (5,400 mm) / 69,000 MPa = 95 mm (3.7 in) (Eq. 8-5) 

Dr = Dy + Drs = 95 mm + 25 mm = 120 mm (4.7 in) 

Das = N - gap – 2do = N - 25 - 2dc = 305 mm – 25 mm – 2 (50 mm) = 180 mm (7.1 in) 

Therefore Dr = 2/3 Das = (180 mm) = 120 mm 

Dr = 120, 2/3 Das = 120 
327

And if Dr < 2/3 Das ⇒ restrainer design is feasible for given properties. 

r as

2 D 1 D 120

3

=ω =

STEP 2. COMPUTE EXPANSION JOINT DISPLACEMENT WITHOUT RESTRAINERS, Deq0

The effective stiffness of each frame is: 

eff 1 = μ = = 1 K K 357 KN/mm 4 89.3 kN/mm (510 k/in)

eff 2 = μ = = 2 K K 89.3 KN/mm 4 22.3 kN/mm (128 k/in)

Therefore, the effective natural period of each frame is given by: 

=π =π = 1

1

1

eff 2

eff

W 22.3 (1000) KN T 2 2 1.0 sec

gK 9800 mm/ sec (89.3 KN/mm)

 See STEP 2 in the Iterative Method 

=π =π = 2

2

2

eff 2

eff

W 22.3 MN (1000) T 2 2 2.0 sec

gK 9800 mm/ sec (22.3 KN/mm)

The effective damping coefficient is: 

−− μ − −⋅ μ ξ= + = + =

π

eff

0.95 1 0.05 0.95 1 0.05 2

2 0.05 0.05 0.19

3.142 (Eq. 8-7) 

The design spectrum correction factor is: 

d

eff

1.5 c 0.5 0.67

40 1

= +=

ξ + (Eq. 8-8b) 

Therefore, the frame deflections are calculated as follows: 

( ) ⎛ ⎞ ⎛ ⎞ == = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ π ⋅ ⎝ ⎠ ⎝ ⎠

1

1

2 2

eff 2

1 d v 1 eff

T 1.0sec D gc F S T ,0.05 9800 mm/ sec (0.67)(0.7) 119 mm (4.6 in) 2 [2] 3.142 (Eq. 8-6) 

For 2

2

v 1

eff a

eff

F S 0.7 T 2; S .35g T 2 = = ==

( ) ⎛ ⎞ ⎛ ⎞ == = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ π ⎝ ⎠ ⎝ ⎠

2

2

2 2 2 eff 2

2 d a eff

T 2.0sec D gc S T ,0.05 9800 mm/ sec (0.67) 0.35 236 mm (9.3 in) 2 2(3.142)

The relative displacement of the two frames can now be calculated using the CQC combination of the two 

frame displacements, as obtained from equation 8-3a. In this case, the frequency ratio, β, is 2.0. 
328

( )

( ) ( )

( ) ( )

( ) ( ) ( )

3 3 2 2 2

eff

12 2 2 2 2 2 22

eff

8 1 8 0.19 1 2 2

.203

1 4 1 1 2 4 0.19 2 1 2

ξ +β β ⋅ ⋅ + ⋅

ρ= = =

−β + ξ β +β − + ⋅ ⋅ ⋅ + (Eq. 8-3b) 

= +ρ + = + ⋅ + = 0

2 22 2 D D 2 D D D 119 2(0.203) 119(236) 236 285 mm (11.2 in) eq 1 12 1 2 2 (Eq. 8-3a) 

Since 23Das = 120 mm and Deq0 > 2/3 Das =120 mm (4.7 in), restrainers are needed. 

But Teff1/Teff2 = 1.0 / 2.0 = 0.5 is less than 0.6 

η = Dr/Deq0 = 120 / 285 = 0.421 is within range 0.2 and 0.5, 

and therefore the single step (non-iterative) method cannot be used for this bridge, since the ratio of the 

periods exceeds the allowable range. 
329

EXAMPLE 8.3: RESTRAINER DESIGN BY THE SINGLE-STEP (NON-ITERATIVE) METHOD

A six-span bridge has two, three-span segments as shown in the figure below. Restrainers are to be 

designed for the hinge seat where the two segments meet. This is the not same bridge as in example 8-2; 

frame stiffnesses have been changed, the hinge seat is smaller and the restrainer cables are shorter. 

Calculate the number and size of the restrainers required using the single step (non-iterative) method. 

Assume the following: 

 Seat width, N = 250 mm (10 in) 

 Concrete cover on vertical faces at joint, dc = 50 mm (2 in) 

 Restrainer yield stress, fy = 1,214 MPa (176 ksi) 

 Restrainer modulus of elasticity, E = 69,000 MPa (10,000 ksi) 

 Restrainer length, Lr = 2.0 m (9.8 ft) 

 Restrainer slack, Drs = 25 mm (1 in) 

 Response spectrum for site: figure 1-8 

 Short period coefficient, FaSs = 1.75 

 Long period coefficient, FvS

 = 0.70 

 Target displacement ductility of the frames, μ = 4 

 Frame stiffnesses, K1 and K2 = 357 and 150 kN/mm, respectively 

 (2040 and 914 K/in, respectively) 

 Frame weights, W1 = W2 = 22.3 MN (5000 K) 

 

STEP 1. CALCULATE ALLOWABLE EXPANSION JOINT DISPLACEMENT 

Dr = 53 + 25 = 78 mm (3.1 in) 

Das = 250 – 25 – 100 = 125 mm (4.9 in) 

2/3Das > Dr OK 
330

STEP 2. COMPUTE UNRESTRAINED EXPANSION JOINT DISPLACEMENT 

The effective stiffness of each frame modeled as a substitute structure is: 

eff 1 1 K K 357 4 89.3 kN/mm (510 kips/in) = μ= =

eff 2 2 K K 160 4 40.0 kN/mm (228 kips/in) = μ= =

Therefore, the effective natural period of each frame is given by: 

1

1

1

eff

eff

W 22.3 1000 T 2 2 1.0 sec

gK 9800 89.3

⋅ =π =π = ⋅

2

2

2

eff

eff

W 22.3 1000 T 2 2 1.5 sec

gK 9800 40.0

⋅ =π =π = ⋅

The effective damping and design spectrum correction factor is: 

eff

0.95 1 0.05 0.95 1 0.05 2

2 0.05 0.05 0.19

3.142

−− μ − −⋅ μ ξ= + = + =

π

d

eff

1.5 c 0.5 0.67

40 1

= +=

ξ +

Therefore, the frame deflections are calculated as follows: 

( ) 1

1

2 2

eff

1 d a eff

T 1.0 D gc S T ,0.05 9800 0.67 0.7 116 mm (4.6 in) 2 2 3.142

⎛ ⎞ ⎛ ⎞ = = ⋅ ⋅ ⋅= ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ π ⋅ ⎝ ⎠ ⎝ ⎠

( ) 2

2

2 2

eff

2 d a eff

T 1.5 D gc S T ,0.05 9800 0.67 0.47 176 mm (6.9 in) 2 2 3.142

⎛ ⎞ ⎛ ⎞ = = ⋅ ⋅⋅= ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ π ⋅ ⎝ ⎠ ⎝ ⎠

The relative displacement of the two frames can now be calculated using the CQC combination of the two 

frame displacements as given by equation 8-3. In this case the frequency ratio, β, is 1.5. 

( )

( ) ( )

( )( )

( ) () ( )

3 3 2 2 2 2

12 2 2 22 2 2 22

8 1 8 0.19 1 1.5 1.5

0.45

1 4 1 1 1.5 4 0.19 1.5 1 1.5

ξ +β β ⋅ ⋅ + ⋅

ρ= = =

−β + ξ β +β − + ⋅ ⋅ ⋅ +

0

2 22 2 D D 2 D D D 116 2 0.45 116 176 176 161 mm (6.4 in) eq 1 12 1 2 2 = −ρ + = − ⋅ ⋅ ⋅ + =

>2/3Das = 83 mm. Therefore, restrainers are needed. Because Teff1/Teff2 is greater than 0.6 and because 

Dr/Deq0 is between 0.2 and 0.5, the non-iterative method is applicable. 
331

STEP 3. CALCULATE RESTRAINER STIFFNESS 

1 2

mod

1

eff eff

eff

eff eff2

K K 89.3 40.0 K 27.6 kN/mm (158 kips/in) K K 89.3 40.0

⋅ === + +

0

r

eq

D 78 0.484

D 161 η= = =

( ) mod

2

r eff

0.50 0.50 0.235 K K 0.50 27.6 0.50 28.9 kN/mm (165 kips/in) 0.484

⎡ ⎤ − η ⎡ ⎤ − = + =⋅ + = ⎢ ⎥ ⎢ ⎥ η ⎣ ⎦ ⎣ ⎦

STEP 4. CALCULATE NUMBER OF RESTRAINERS 

r r r

y r

K D 28.9 0.078 N 13.0

F A 1214 0.000143

⋅ == = ⋅ Use 14 – 19 mm restrainer cables 
332

8.4.2.2(e). Design Forces 

Transverse bearing restrainers are usually designed to resist load elastically. Studies have shown 

that when columns yield, additional forces will be transferred to elements that are not designed to 

yield. In addition, transverse restrainers will be installed with slightly different construction 

tolerances that will cause them to engage and resist load unevenly. To account for possible 

increased load due to these effects, the elastic forces derived from an analysis should be 

increased by 25 percent for design. 

Although a dynamic analysis is not necessary for any single span bridge or a bridge in Seismic 

Retrofit Category B, transverse restrainers in these bridges are required to resist a minimum 

horizontal load as described in Methods A1 and A2 (section 5.2). 

8.4.2.3. Vertical Motion Restrainers 

Vertical hold-down devices may be used at bearings to prevent uplift that, if free to occur, could 

result in damage or loss of stability. Although uplift by itself is unlikely to result in the collapse 

of a span, vertical hold-down devices should be considered whenever the vertical seismic forces 

exceed the dead load reaction. Vertical motion restrainers are usually not economically justified 

unless some additional bearing retrofit is being performed and the bridge is in Seismic Retrofit 

Category D. An example of a possible hold-down detail is shown in figure 8-48.

Vertical accelerations are often not included in a dynamic response analysis, but if uplift is an 

issue, an analysis which includes the vertical component of the ground motion should be used. 

Figure 8-48. Vertical hold down retrofit. 
333

8.4.3. ENERGY DISSIPATION DEVICES 

Conventional expansion bearings, such as elastomeric bearing pads or sliding bearings, are 

natural isolators. However, these bearings do not generally dissipate much energy, and 

differential displacements at bearing locations may therefore be unacceptably large. In these 

cases, the retrofit designer can obtain functional performance similar to that provided by seismic 

isolation bearings by adding separate energy dissipating devices alongside conventional 

bearings. Energy dissipation devices may also be used at other structural locations where 

relative movement of the structural components is expected during an earthquake. Over the 

years there have been many proposals for energy dissipating devices including steel cantilever or 

beam elements that can deform plastically, lead extrusion devices, viscous dampers, and friction 

dampers. Each of these devices relies on dissipation of energy during movement to keep 

movements to acceptable levels. 

Rigidity under service loads is provided by the initial stiffness of these devices. Some of the 

mechanisms used by these devices, such as lead extrusion, and viscous or friction damping, are 

able to accommodate service load movements if properly designed. However, devices that rely 

on the plastic deformation of steel are likely to experience fatigue failure over time if they are 

allowed to yield under service loads. Many of these devices will also have difficulty 

accommodating orthogonal movements. These issues should be given careful consideration 

during design. 

Energy dissipation devices have been developed and implemented for seismic applications for a 

number of years. Figure 8-49 shows a number of energy dissipation devices that were originally 

developed in Japan and New Zealand. Most of these rely on the plastic deformation of steel 

elements, but one device uses lead extrusion to dissipate energy. Figure 8-50 shows two friction 

dampers. 

A number of energy dissipators have also been developed and tested in the U.S., specifically for 

bridge applications9

. Several of these devices were evaluated in the HITEC program on behalf 

of Caltrans and FHWA (HITEC, 1999a). These included large piston and cylinder type shock 

absorbers that are capable of resisting the types of loadings experienced in bridges. Summaries 

of test results for three damper technologies are available through HITEC (HITEC, 1999b,c,d). 

8.4.4. SHOCK TRANSMISSION UNITS 

Shock transmission units (STUs) are specially fabricated devices that allow very slow movement 

to occur, such as that caused by temperature change, but become rigid under rapid motion, such 

as would occur during an earthquake. This characteristic allows these devices to be placed at the 

expansion joints between superstructure segments, or between the superstructure and 

substructure at expansion bearings. A typical application would be a continuous bridge where 

only one line of bearings is designed to take longitudinal loads at a fixed bearing, and the 

remaining supports are expansion bearings. By placing STUs at the expansion bearings, 

longitudinal load is shared by all supports, thus reducing the load at the fixed bearing. 

 

9

 Whittaker et al., 1989; Richter et al., 1990; Constantinou et al., 1992; Aiken et al., 1993; Bergman and Hanson, 

1993; Tsai et al., 1993; Grigorian et al., 1993 
334

STUs which are typically composed of a piston and cylinder filled with a viscous fluid, are 

usually custom manufactured for each application. The designer should specify the required 

maximum load capacity of the device, maximum range of movement, and maximum rate of 

movement or acceptable drag forces that can result from this movement. It is difficult for most 

manufacturers to achieve a drag force of less then two percent of the rated capacity of the unit. 

The rated capacity may therefore be limited by the fatigue strength of the structural elements of 

the bridge. When unknown, the rate of movement of an STU can be assumed to be 0.0001 L per 

hour, which corresponds to a temperature change of 25 degrees Celsius in one hour. The 

distance, L, is the effective bridge length subject to thermal movement and is limited to 150 m 

(500 ft) for the above rule of thumb to apply. The designer should also specify the temperature 

range in which the STU will operate, since their performance may be temperature dependent. 

STUs are unidirectional devices that must be mounted in such a way as to accommodate out-ofplane movement. One method of doing this is to use ball joints at either end of the unit where it 

is mounted to the structure. Generally, the designer will also specify or design the mounting 

hardware. The structure and brackets must be able to resist drag forces without fatigue. During 

an earthquake, STUs may be assumed to be rigid even though they may experience a small 

movement prior to ‘locking up.’ This small movement can create impact forces, so it is 

recommended that the STU, brackets, and non-ductile structural elements be designed for 125 

percent of the calculated required forces. 

Because these are custom-manufactured devices, it is strongly recommended that they be 

prequalified for use in a retrofit project. This includes satisfying the requirements of a series of 

performance tests such as: 

• Seal wear test. 

• Cyclic load test. 

• Crag force test. 

• Overload test at 150 percent of the rated capacity. 

• Fatigue load test. 

Units should also be proof-tested using the overload test. 

Because of the small number of moving parts, these devices should require relatively low levels 

of maintenance. Nevertheless, it is recommended that they receive periodic inspection. Of 

particular concern are the seals, which could potentially leak. It is recommended that STU 

installations be designed in such a way that they can be easily inspected and replaced if they 

become worn during the life of the bridge. These devices should be protected from a corrosive 

environment by painting or galvanizing exposed components. 
335

Figure 8-49. Energy dissipation devices. 
336

Figure 8-50. Friction dampers. 
337

CHAPTER 9: RETROFIT MEASURES FOR SUBSTRUCTURE 

COMPONENTS 

9.1. GENERAL 

Although retrofit measures for improving the seismic resistance of existing bridge substructures 

(columns, cap beams, and foundations) were proposed in early bridge retrofitting manuals 

(FHWA, 1983), very little actual work of this type was done until the 1990’s. Some bridges that 

had been retrofitted with hinge restrainers alone failed in both the 1987 Whittier Narrows 

earthquake and the 1989 Loma Prieta earthquake, demonstrating the need to also retrofit the 

substructures of many bridges (Housner, 1990). 

A significant portion of the initial column research provided insight into the effectiveness of 

different retrofit measures to improve both flexural and shear strength, and flexural ductility of 

reinforced concrete bridge columns (Chai et al., 1992). As a result, standards were developed for 

evaluating bridge columns and standard techniques were adopted for improving their ductility 

and shear resistance (Caltrans, 1996). This was accomplished by encasing reinforced concrete 

columns in circular or elliptical steel shells or by wrapping them with fiber composite materials. 

These methods were shown in the laboratory to improve flexural ductility and shear strength and 

to prevent the failure of starter bar splices located within potential plastic hinge zones. These 

methods, which have now been implemented on a large number of California bridges, proved to 

be effective in practice by preventing several bridge failures during the 1994 Northridge 

earthquake (Buckle, 1994). 

The marketplace also responded to the need for column retrofitting, through the development of 

other retrofitting systems aimed at improving column ductility. Some of these have been 

successfully tested in the laboratory and have been used in retrofitting demonstration projects1

. 

Caltrans’ research has added to the body of knowledge about substructure retrofitting. In 

addition to addressing other topics related to the retrofit of reinforced concrete columns, this 

research has investigated the retrofitting of wall piers2

, bent caps3, and column footings4. 

Significant research has also been conducted by other State Departments of Transportation, and 

the Federal Highway Administration. 

Concurrent with the relatively large amount of research relating to the seismic retrofitting of 

bridges, there has been a major acceleration of seismic retrofitting programs throughout the 

country. In California alone, this has involved the seismic evaluation and retrofitting of 

approximately 2,200 bridges (Roberts, 1998). Many economical and effective retrofit measures 

 

1

 Fyfe, 1994; Seible et al., 1995; Xiao et al., 1995 

2

 Haroun et al., 1994 

3

 Ingham et al., 1993; Lowes and Moehle, 1994; Thewalt and Stojadinovic, 1995; Zayati et al., 1993 

4

 Xiao et al., 1994 
338

have been developed and implemented in the course of this work, and retrofitting techniques and 

procedures have been refined. 

This chapter summarizes the current state of practice related to the seismic retrofitting of bridge 

substructures. It contains sections on bridge columns, pier caps, and column-to-cap beam joints. 

These components have been found to be particularly vulnerable during recent damaging 

earthquakes. 

The use of an earthquake protective system, such as seismic isolation, might be an acceptable 

alternative, or supplement, to many of the approaches described in this chapter (see the overview 

of different retrofitting approaches in section 1.13). Note that the optimal retrofit approach for a 

given bridge may well be a combination of more than one measure. For example, if isolation 

cannot reduce the seismic demand to a level below the existing column capacity, then column 

retrofitting may appear to be the only option. However, the required amount of conventional 

retrofitting (e.g., column wrapping) may be significantly less if isolation is used, and the optimal 

strategy may be a combination of both measures. 

9.2. RETROFIT MEASURES FOR PIERS 

Reinforced concrete columns constructed before 1971 are commonly deficient in flexural 

ductility and shear strength. Reduced column ductility, which results in a rapid loss of flexural 

strength, is likely to occur when lap splices exist within potential plastic hinge zones at the base 

of columns. Even when starter bars are not used, inadequately confined plastic hinge zones will 

lose flexural strength due to the crushing of concrete and the buckling of longitudinal 

reinforcement. The shear strength of an inadequately reinforced plastic hinge will also degrade, 

leading to the potential for shear failure. If the existing reinforcement within the zone of a 

potential plastic hinge is adequate, or the ductility of that zone is enhanced through retrofitting, 

plastic hinging may still occur, but outside the intended plastic hinge zone. This may be due to 

premature termination of longitudinal bars, the presence of architectural flares, or the unintended 

stiffening effect of median barriers, sidewalks, or pavement. Designers should be aware of this 

possibility. 

In most cases, it will not be necessary (or even desirable) to increase the flexural strength of 

columns, provided the ductility of the column can be assured or improved. Even though smaller 

seismic design loads were used in the past, adequate flexural strength often exists because of the 

conservatism inherent in working stress design widely used in the past. In fact, excess existing 

flexural strength is often undesirable because it leads to higher plastic shear forces within the 

column and larger foundation forces. When it is necessary to increase column flexural strength 

as part of a retrofit, corresponding increases in shear strength and the strength of foundations are 

usually required. 

Wall piers usually perform better than multi-column or single-column piers during an 

earthquake. In most cases, sufficient horizontal and vertical reinforcement is present to 

withstand earthquake loading along the strong axis of the pier, although excessive forces in 

foundation piles may result if the wall does not yield in its strong direction. In the weak 

direction however, walls behave much like columns, except that these walls possess significant 
339

ductility even when the amount of transverse reinforcement is less than that specified for a new 

bridge. This is particularly the case when there are no starter bar splices in the main vertical 

reinforcement5. Therefore, retrofitting of wall piers is less common than it is for conventional 

reinforced concrete columns. 

Methods of retrofitting reinforced concrete columns include: 

• Complete or partial replacement. 

• Addition of supplemental columns. 

• Shear or flexural strengthening. 

• Improvement of column ductility. 

The most popular of these methods is ductility improvement, which is possible using one or 

more of the following techniques: 

• Steel jacketing. 

• Active confinement by prestressing wire. 

• Active or passive confinement by a composite fiber/epoxy jacket. 

• Reinforced concrete jacketing. 

Of these techniques, the steel jacket and composite fiber/epoxy jacket are the most widely used. 

The following sections describe these methods in greater detail. 

9.2.1. REINFORCED CONCRETE COLUMNS 

9.2.1.1. Column Replacement 

Total or partial column replacement will require placement of temporary shoring to carry the 

weight of the bridge while the column is being removed and replaced. This shoring must 

generally be capable of resisting the horizontal loads produced by small earthquakes and, in 

some cases, must support live loads in addition to the weight of the bridge. In general, shoring 

must be capable of carrying at least 50 percent of the shear capacity of the column being 

removed, but not less than the shear given by 50 percent of the spectral value for the lower level 

event for the site (section 1.4.2 and figure 1-8). 

 

5

 Haroun et al., 1994; Abo-Shadi et al., 2000 
340

9.2.1.1(a). Total Replacement 

Total column replacement will sometimes be the most appropriate method for retrofitting the 

bridge substructure. It will usually be necessary to modify or replace both the footing supporting 

the column and the pier cap to which the column is connected. In fact, it is often the need to 

replace the foundation, or strengthen the pier cap, that mandates column replacement in the first 

place. Replacement should also be considered when the column is damaged or deteriorated, 

column flexural capacity is grossly inadequate, or where space limitations or architectural 

considerations preclude other retrofitting alternatives. 

Column replacement was used extensively in San Francisco following the 1989 Loma Prieta 

earthquake (figure 9-1). Some of these columns were damaged during the earthquake, but most 

were replaced as part of a seismic upgrading program for these structures. Because of a lack of 

alternative routes and the fact that many of these structures were two-level viaducts, carefully 

designed shoring systems were required to avoid interference with traffic. 

A key issue in column replacement is the connection of the column to the existing structural 

members. An example of a column replacement illustrating this point is shown in figure 9-2. 

Note that the existing column steel was left in place and allowed to extend into the new column, 

thus forming a moment-resistant connection with sufficient ductility to withstand large 

earthquake loads. Anchoring column reinforcement by drilling and grouting is not desirable 

unless a considerable embedment length is provided. This method should be used with caution 

when the reinforcement will be subjected to several cycles of loading beyond the yield strain of 

the steel. It may be possible to add new ‘headed’ reinforcement in the column by drilling 

through connecting members such as pier caps. In this case, the headed steel is anchored on the 

opposite side of the column, as shown in figure 9-3. In all cases, the structural adequacy of the 

pier cap-to-column joint must be considered. 

When a concrete column is replaced below an existing bridge, it will usually be necessary to 

create a seal between the top of the column form and the soffit of the pier cap so that concrete 

can be placed under pressure, thus assuring full contact between existing and new concrete. It 

may sometimes be necessary to provide cored vent holes in the overlaying member to prevent 

the formation of air pockets. Roughening of the concrete contact surface is also recommended. 

9.2.1.1(b). Partial Column Replacement 

Partial column replacement involves the removal of surface concrete in the region of the 

potential plastic hinge zone. The main vertical column reinforcement is cut and replaced by 

machined ‘fuse bars’ that are, in turn, connected to the existing main steel outside of the plastic 

hinge zone (see figure 9-4). The connection is made by welded splice plates to which threaded 

couplers are welded so that the fuse bars can be replaced in the future. This requires knowledge 

of the weldability of the existing reinforcement so that the correct welding procedures can be 

developed. The connections are designed to assure that yielding occurs in the fuse bars. The 

new hinge is then wrapped with high strength cable to provide the necessary confinement. High 

performance concrete is then used to cover the fuse bars and cable. The procedure is described 

in greater detail in example 9-1. 
341

EXAMPLE 9.1: REPLACEABLE HINGE 

A column plastic hinge zone with lap splices is to be retrofitted with a replaceable hinge constructed with 

high strength fuse bars that can be removed and replaced after an earthquake. 

Connector plates are welded to the existing longitudinal rebar such that each connector plate engages 

two existing rebar. This is done on the starter bars in the footing and on the column bars. Threaded 

studs are welded to the plates, to which threaded couplers are attached. Machined fuse bars are 

connected between the threaded studs by means of the couplers. 

The existing column is 1.22 m (4 ft) in diameter with 50 mm cover and contains 20 #35 (metric) (#11, cus) 

reinforcing bars. The concrete strength is 35 MPa (5000 psi) and the yield stress of the reinforcing steel 

is 276 MPa (40 ksi). Fuse bars are made of steel that has a yield stress of 855 MPa (124 ksi) and an 

ultimate strength of 990 MPa (144 ksi). The structure has a longitudinal period = 0.7 seconds. 

Calculate the fuse bar area and length, size of connector plate, and transverse reinforcement details. 

STEP 1. DETERMINE FUSE BAR AREA 

To assure that the fuse bars yield before the existing main reinforcing bars: 

y

f b

su

2f 2(276) d d 0.035 m 0.035(0.747) 0.026 m (1.0 in) f 990

<= = =

Therefore, the fuse bars will be machined to a diameter of 26 mm (1 in). 

STEP 2. DETERMINE FUSE BAR LENGTH 

The length of the fuse, Lf, is determined so that steel strains remain below levels that will result in low 

cycle fatigue during the design earthquake (Dutta and Mander, 1998). Therefore: 

( ) P 0.5 L D 2N f f 0.16

θ = ′

where 

θp = plastic hinge rotation determined from an analysis; in this case assume θp = 0.03 

D' = distance between extreme tension and compression bars = (Diameter – cover – spiral ties – 1⁄2 db) 

 = 1.22 m– 2 (0.05 + 0.019 + 0.018 m) = 1.046 m 

f 3

f

7 N effective number of cycles imposed by the seismic event where in this case T 0.7 sec. T

N 8

= ≈=

≈

Therefore: 
342

= ( ) [ ]

= ⇒

0.5

f

0.03rad L 1.046 m 2 8

0.16

0.785 m (31in) 0.8 m (31.5 in)

 

STEP 3. DETERMINE SIZE OF CONNECTOR PLATE AND WELD 

For E70XX electrodes and a 4.8 mm (0.19 in) weld size: 

2 2

f su d f 0.026 990 l 0.56 0.56 0.327 m (13 in) w s 217 0.0048 217 w

== =

Based on AISC LRFD, the minimum plate thickness is: 

tplate = size of weld + 1.6 mm = 4.8 mm + 1.6 mm = 6.4 mm (0.25 in) 

STEP 4. DETERMINE TRANSVERSE REINFORCEMENT 

For concrete confinement: 

ρs = effective volumetric ratio 

( )

2 '

c e y g

s t ' ' sf cc cg c

f P f A

0.008 12 1

U A fA f

⎛ ⎞

ρ = + ρ − ⎜ ⎟ ⎝ ⎠ where ρt = = =

2 s

2

g

A (0.000962m ) 20 0.0164

A 1.17m

( ) ( ) ( )

2

s

35 4.45 855 0.008 12 0.00455 0.735 0.0011

110 35 1.17 35 ρ= + =

For antibuckling: 

y

s t

b yh

D s 1.22m 1.22 855 f 0.025 0.025 0.00455 0.0031

s d f 0.026mm 0.026 1489

⎛ ⎞ ⎛⎞

ρ= ρ = ⋅ = ⎜ ⎟ ⎜⎟ ⎝ ⎠ ⎝⎠

For shear resistance: 

ρ ⎡ ⎤ ⎛ ⎞ − φ ρ= Λ − ⎢ ⎥ ⎜ ⎟

φ + ρ ⎣ ⎦ ⎝ ⎠

= ⋅⋅=

'

t su g e cg

s ' yh cc t su c

f A 0.65 P f A

0.75 1

f A 0.65 1.2 f f

0.00455 990 0.75(1.0) 0.735 0.351 0.0007 0.85 1489

Antibuckling therefore controls, and ρs = 0.0031. This requires 13 mm (0.5 in) strand @ 100 mm (4 in) 

spacing. 
343

Figure 9-1. Column replacement on San Francisco viaducts. 
344

Figure 9-2. Column replacement using existing column reinforcement. 
345

Figure 9-3. Column anchorage using headed reinforcement. 

This retrofit method is intended to provide ductile response during an earthquake while 

effectively protecting other structural components. Capacity design principles (see Chapter 5) 

are employed to assure yielding in the fuse bars before damage occurs elsewhere in the column. 

The method has the advantage that the fuse bars are replaceable following an earthquake. An 

alternative design approach, in which these replaceable hinges are constructed as part of new 

precast columns, has also been proposed. 

9.2.1.1(c). Supplemental Columns 

New supplemental outrigger columns to provide ductile resistance to horizontal loads have also 

been used when space is available, or when column replacement, with its required shoring, is an 

undesirable solution. These columns are usually tied into existing or new pier caps. This is 

accomplished by welding or mechanically splicing to existing pier cap reinforcement, or by 

prestressing through concrete overlays tied to the sides of existing pier caps. Because of the 

potential for introducing high torsion forces into these pier caps, supplemental outrigger columns 

are often ‘pinned’ at the top to minimize moment transfer. 

Another design issue is displacement compatibility between the existing columns, which must 

continue to support the weight of the bridge, and the new, more ductile, supplemental columns. 

This often requires that existing columns be retrofitted to preserve a column’s vertical capacity, 

as described in section 9.2.1.5. An example of a retrofit using supplemental columns is shown in 

figure 9-5. 
346

610

Steel

Plate

Section A-A

Starter Bar

Main Bar

Hexagonal Coupler

165

Detail A

Pedestal

(Reinf. Not Shown)

762

3049

2337 1524 914

North-South Elevation East-West Elevation

19 mm Dia. Original

Rebar

11 mm Weld (TYP.)

216x114x13 Steel Plate (TYP.)

Hexagonal Couplers (TYP.)

Lock Nut (TYP.)

Original Rebars

Cut at Midlength

114

406

29 mm Threaded

Stud

152

18

584

Hexagonal

Couplers

Lock Nut

Detail A Fuse Bar Details

A A

Figure 9-4. Replaceable plastic hinge with fuse bars. 
347

Figure 9-5. Supplemental column retrofit. 

9.2.1.2. Column Flexural Strengthening 

In general, it is not necessary or even desirable to increase the flexural strength of a reinforced 

concrete bridge column. Increased column strength results in increased shear capacity, which in 

turn attracts increased forces to foundations and pier caps. However, it will occasionally be 

necessary to increase the flexural strength of a column. For example, increasing column strength 

at a lap splice in the reinforcement will force the plastic hinge to form in a region of the column 

away from the splices. 

Three methods for increasing column strength are discussed below. 

 

9.2.1.2(a). Concrete Overlays 

Applying full or partial height concrete overlays to the face of an existing column can increase a 

column’s flexural strength. A sufficient number of dowels must be provided for shear transfer 

between the overlay and the existing column and a roughened contact surface can also help in 

this regard. 

The extra concrete itself will add some flexural strength because of the increased moment arm 

between the vertical column reinforcement and the compression block in the new concrete. To 

provide additional flexural strength, it will be necessary to add vertical reinforcement in the 

overlays. The additional reinforcing steel must be anchored into existing structural members, 

which presents some of the same problems encountered with column replacement. It may be 

possible to anchor this reinforcing into newly constructed portions of the pier cap or footing, or 

use the headed reinforcement approach discussed earlier and shown in figure 9-3. 

The flexural ductility of concrete overlays is also an issue. This is most easily and effectively 

done when the existing column is circular or nearly circular in cross-section. Providing 
348

continuous transverse hoops that are connected with mechanical couplers will generally provide 

sufficient confinement for ductility. When the column cross-section is rectangular, closed ties 

will effectively support only the corner bars and it will be necessary to provide lateral support for 

other bars by drilling and grouting adequate transverse reinforcement through the existing 

column. In designing this transverse reinforcement, due consideration should be given to its 

anchorage and its effectiveness in providing confinement. An example of column flexural 

strengthening by concrete overlays is shown in figure 9-6. This retrofit method is discussed 

further in section 10.3.2 

9.2.1.2(b). Added Reinforcement in Conjunction with Steel Shell 

The strength of an existing column can also be increased by adding extra longitudinal 

reinforcement in the grout space between a steel shell and the column. This will require 

adequate shear transfer between the grout and the existing column, which can generally be 

assured by roughening the existing concrete surface or providing drilled and grouted dowels on 

the surface of the existing column. A detail of an application of this retrofit method is shown in 

figure 9-7. In this case, a pinned column base was made fixed through this technique. Most of 

the additional column steel was anchored into a newly cast concrete footing, but some was 

anchored by drilling and grouting into the existing footing concrete. If the new steel is anchored 

into existing concrete, it is necessary to provide sufficient embedment length to assure that the 

reinforcement can achieve high levels of plastic strain. 

9.2.1.2(c). Composite Steel Shell 

A steel shell can also contribute to flexural strength if sufficient shear transfer is provided 

between the shell and the grout. This is done by coating the inside of the steel shell with an 

epoxy adhesive impregnated with grit. Steel shear rings, on the inside of the shell, consisting of 

welded reinforcing bar hoops, steel bars, or weld beads can also be used for this purpose. High 

strength adhesives can be used to bond steel plates to the surface of the existing column. 

Because it is difficult to develop full flexural capacity at the ends of these shells or overlays, this 

retrofit method is best used to force yielding of the rebar to occur away from a vulnerable 

column region, such as at a splice location or where reinforcing steel has been prematurely 

terminated. 

Some development of strength at the ends of steel shells is also possible. Anchor bolts may be 

used to connect the steel shell at the base of a column to the footing. Tests have demonstrated the 

effectiveness of this retrofitting technique for increasing both flexural strength and ductility, and 

have shown that flexural yielding of the anchor plate was an efficient means of dissipating 

energy, provided that the anchor bolts themselves could be capacity-protected. A detail of this 

retrofit method is shown in figure 9-8.
349

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Existing Concrete Column

Epoxy

Bonding

New

Reinforcing

New Concrete

Drilled and

Grouted Dowels

Section A - A

Elevation

A A

Figure 9-6. Column strengthening by concrete overlay. 
350

Figure 9-7. Column strengthening by adding reinforcement within steel shell. 
351

Figure 9-8. Column strengthening by anchoring steel shell. 
352

9.2.1.3. Column Ductility Improvement and Shear Strengthening 

One of the principal reasons for the low ductility of existing reinforced concrete columns was the 

practice of using starter bars at the base of columns where plastic hinges are likely to form. 

Failure in the splice between the starter bars and the longitudinal reinforcement generally occurs 

when the concrete surrounding the reinforcing bars splits due to radial stresses that are produced 

as the deformed bar tries to pull out of the concrete socket into which it is cast. Once split, the 

concrete socket dilates and allows the bar to pull free. Tests have indicated that the critical radial 

dilation strain, εd, is on the order of 0.001 for splice lengths of 20 db and reinforcement yield 

strengths of 276 MPa (40 ksi). If the column has enough confinement at a radial strain of 0.001, 

then splice failures can be prevented. Because existing columns usually have either insufficient 

or poorly detailed transverse reinforcement that cannot reliably provide this level of 

confinement, it is necessary that a retrofit method be employed to do so. The required level of 

confinement stress, fl, is given by Priestley and Seible, 1991, as: 

( )

b y

b s

A f

f

D 2d c

2n

≥ ⎛ ⎞ π ′ ⎜ ⎟ + + ⎝ ⎠

A

A

 (9-1) 

where db is the diameter of spliced longitudinal bar, Ab is the area of spliced longitudinal bar, 

fy is the yield stress of splice longitudinal bar, D' is the diameter of the pitch circle of the main 

column reinforcement, N is the number of main column reinforcing bars, c is the cover over 

these bars, and ls is the splice length. 

Equation 9-1 assumes that no slippage of the splice will occur when the stress in the main 

reinforcing steel is less than 1.4 fy, which is typically the ultimate stress for this steel. The splice 

length must be at least long enough to prevent the shearing of concrete between bar 

deformations, which can also cause a splice bar to pull out. Extra confinement cannot prevent 

this type of failure. The minimum splice length required is: 

by by

s ' '

c c

0.25d f 0.021d f

(MPa units) (psi units)

f f

A ≥ = (9-2) 

Flexural retrofit measures should extend from the critical section to the location where the 

moment has decreased to 75 percent of the maximum moment, but not less than a distance equal 

to the column diameter. The higher level of confinement required for lap splices needs to be 

provided only over the length of the lap splice. 

A relaxation of the requirements of equation 9-1 is allowed if slip is permitted at moderate 

displacement ductilities. In this case, the maximum tensile stress in the reinforcement could be 

assumed to be 1.0 fy, which assumes no strain hardening of the longitudinal reinforcement. Bond 

slip will occur at moderate ductilities (typically in the range of three to five), but the constant 

confining stress will provide a rather ductile response with only gradual degradation of 

performance as a result of dependable friction across the displacing surfaces of the fracture 
353

plane. The required confining stress would then be 28.5 percent less than that given by equation 

9-1, where the value 0.285 is calculated from (1.4 – 1.0)/1.4. 

The ductility of existing columns is less than that of new columns even if there are no starter bar 

splices near the footing. Excessive flexural deformations in a plastic hinge will subject concrete 

and steel to excessive levels of strain. When concrete strains reach a level of about 0.005, cover 

concrete begins to spall. When this occurs, inadequate or poorly detailed transverse column 

reinforcement will be subjected to increased stresses, which may cause ties, hoops, and 

discontinuous spirals to stretch, slip, or pull free from the column, thus becoming ineffective in 

providing passive confinement of the column core. This in turn will cause the column core 

concrete to crush and vertical reinforcing steel to buckle. Eventually, the core concrete will 

fracture badly and lose its capacity to resist axial, flexural, and shear forces. When this occurs, 

the column will disintegrate and collapse of the bridge may occur. 

Retrofitting will probably affect a column’s elastic (cracked-section) stiffness, and this should be 

taken into consideration when analyzing the bridge for design forces. For a steel jacket retrofit, 

average increases in stiffness between 10 and 15 percent for a partial height retrofit and 30 

percent for a full height retrofit may be expected (Priestley and Seible, 1991). External 

prestressed confinement steel and fiber composite jacket retrofits have a negligible influence on 

the column stiffness (Wipf et al., 1997). 

9.2.1.3(a). Steel Jacketing 

This technique was originally developed for circular columns, as shown in figure 9-9 and is 

currently the preferred method used by Caltrans for the seismic retrofits of bridge columns 

(Caltrans, 1996). Two steel plate half-shells, which have been rolled to a radius equal to the 

column radius plus 13 to 25 mm (0.5 to 1 in) for clearance, are positioned over the portion of the 

column to be retrofitted, and the vertical seams are then welded. The gap between the jacket and 

the column is grouted with a pure cement grout, after flushing with water. A vertical space of 

about 50 mm (2 in) is typically provided between the end of the jacket and any supporting 

member (i.e., the footing or cap beam) to avoid the possibility of the jacket acting as 

compression reinforcement by bearing against the supporting member at large drift angles. 

Significant increases in flexural strength are possible from this source, which may then result in 

undesirable overload of the adjacent members. 

The construction and fabrication procedures typically used for steel jackets place constraints on 

their design. Limitations on handling stresses require that the shells have a minimum thickness 

of 10 mm (0.375 in), and restrictions on bending thick plates require a maximum thickness of 25 

mm (1 in) (Caltrans, 1996). 

A steel jacket is effective as passive confinement, but confinement is not provided until the radial 

expansion of the concrete column induces circumferential stresses in the steel shell. This radial 

expansion occurs as a result of bulging caused by high axial compression strains in the concrete 

and dilation around longitudinal reinforcement caused by vertical cracking near the bar splices. 

Similar radial dilation occurs with the development of diagonal shear cracks in the concrete of 

the column. 
354

Figure 9-9. Typical steel shell retrofit of round column. 

For rectangular columns, the recommended procedure is to use an oval jacket, which provides 

continuous confining action similar to that for a circular column. Because of increased space 

between the shell and the existing column, small sized aggregate is added to the grout that is 

placed in the gap between the column and the shell, as shown in figure 9-10. Rectangular 

columns retrofitted in this manner perform very well in flexure and shear. Attempts to retrofit 

rectangular columns with rectangular jackets have been less successful, even when the jackets 

have been extensively stiffened. This is because the confining action of the rectangular jackets 

can only be developed as a consequence of lateral bending of the jacket sides, which is a very 

flexible action, compared to the membrane action developed in an oval or circular jacket. 

However, if the radius of a portion of the oval is very large, it may be necessary to stiffen it with 

braces or to support it at interim locations with bolts drilled through the column, which will help 

achieve the required confinement. 

Rectangular steel jackets are effective in enhancing the performance of ‘shear critical’ columns. 

These jackets can improve column ductility by eliminating the brittle shear mode of failure, but 

the failure mode may then shift to a flexural one for which the rectangular jacket can provide 

only limited assistance, for the reasons described above. 
355

Figure 9-10. Typical steel shell retrofit of rectangular column. 

The designer should also consider the potential distortion of a steel shell due to the hydrostatic 

pressures created as concrete is poured into the space between the shell and the column. This is 

particularly important for large oval-shaped shells placed around rectangular columns. Because 

the clearances between the shell and existing concrete at the corners of the column are small and 

may prevent the flow of the concrete, it may be necessary to provide grouting ports at all four 

lobes of the ellipse and to require that the concrete be placed in nearly equal lifts. 

Performance of Lap Splices – For a circular column, the required confinement stress, fl, can be 

related to the characteristics of this retrofit concept by reference to figure 9-11, which shows a 

free body diagram of a half-column section. Equilibrium requires that: 

 2 tfs = fl D (9-3) 

where t is the steel jacket thickness, fs is the stress induced in the jacket, and D is the diameter of 

the column. 

For a steel modulus of elasticity Es, equal to 200 GPa (29,000 ksi), fs will be 200 MPa (29 ksi) at 

a strain of 0.001. Substituting this result into eq. 9-3 and rearranging terms gives the following: 

s

fD fD fD t (mm) (in) 2f 400 58

≥= = AA A (9-4) 

where the confinement stress, fl, is expressed in MPa or ksi, as appropriate. 
356

D

Jacket

t

f f’ f s l s

Figure 9-11. Free body diagram of column retrofitting with a steel shell. 

Equation 9-4 may be simplified by assuming that fl is approximately 2.07 MPa (300 psi), which 

is the value used in the development of this method (Chai et al., 1991). This assumption will 

provide satisfactory performance where volumetric ratios of longitudinal reinforcement are less 

than 2.5 percent and axial loads, P, are less than 0.15f′cAg. 

Columns satisfying the above requirements will sustain drift angles of four percent with an 

adequate reserve of displacement capacity. Such retrofitting generally provides sufficient 

ductility to allow the retrofitted column to be designed using a response modification (R-factor) 

method. In this case, R-factors of four for single column piers, and six for multi-column piers, 

may be used. Higher R-factors can be justified in some cases (Caltrans, 1996). 

Flexural Confinement – Increasing flexural confinement by retrofitting is often done by 

simplified means that yield conservative results. For example, only 83.3 percent of the steel 

shell thickness given by equation 9-4 is needed when lap splices are not present. This 

corresponds to a passive confinement of 1.72 MPa (250 psi) at radial strains of 0.001. The 

R-factor approach described above is then used for columns retrofitted in this manner. 

A more rigorous and less conservative design approach relies on determining material strains 

that will occur under seismic loading, and then designing the steel shell to provide the 

confinement necessary to achieve these strain levels. By equating the strain energy in confined 

concrete to the strain energy in the confining steel, it is possible to determine ultimate strains in 

the concrete based on the ultimate achievable strains in the confining steel (Mander et al., 1988). 

Applying this principle to a circular steel shell of constant thickness, t, leads to the following 

equation for ultimate concrete strain capacity: 

ys su

cu '

cc

5.6tf

0.004

Df

ε

ε= + (9-5) 
357

where fys is the yield stress in the shell steel, εsu is the ultimate strain in the shell steel (a value of 

0.10 is recommended for A36 steel), D is the diameter of the steel shell, and f'cc is the ultimate 

confined concrete stress capacity. 

The value of f'cc can be related to the lateral confinement stress, which equation 9-3 shows is 

related to the shell thickness, t. 

Therefore: ' ' s s cc c ' '

c c

15.88tf 4tf f f 1.254 2.254 1

Df Df

⎛ ⎞

=− + + − ⎜ ⎟

⎝ ⎠

 (9-6) 

By reorganizing equation 9-5 and solving for the shell thickness, t: 

( ) ' cu cc

ys su

.004 Df

t

5.6f

ε −

≥

ε

 (9-7) 

Equation 9-7 is solved for t for known values of εcu and '

cc f . An initial, but conservative, value 

for t may be obtained by assuming concrete strain demands of 0.02 and confined concrete stress 

demands of 1.7 '

cf . 

Once an initial value for t is determined, a more accurate value of εcu for a given earthquake 

response can be determined using a computer program to perform moment-curvature analysis of 

the retrofitted section. This analysis relates concrete and reinforcing steel strains to various 

degrees of plastic curvature of the column. Several computer programs are commercially 

available for this purpose. This analysis also allows the development of a simplified inelastic 

moment-curvature relationship for use in an inelastic static analysis. In this analysis, the frame 

containing the subject column(s) is incrementally deformed (or ‘pushed over’) to a target 

displacement equal to a multiple of the displacement obtained from an elastic analysis (a factor 

of 1.5 is suggested to provide an adequate factor of safety). At each increment of deformation, 

the plastic hinges, which are assumed to have a length lp, are allowed to deform according to the 

assumed moment-curvature relationship, which will vary with the column’s axial load. At the 

target displacement, it will be possible to identify the plastic hinges and determine the 

corresponding plastic rotation at each location. The plastic curvature at a section is determined 

by dividing the plastic rotation by the effective plastic hinge length, lp. The revised target 

concrete strain, εcu, is then determined from the moment-curvature analysis. If necessary, the 

shell thickness can be adjusted and the procedure repeated until the solution converges to an 

acceptable value for shell thickness. An example of a steel shell design using both the simplified 

procedure and this iterative procedure is given in example 9-2. 

The plastic hinge length will be shortened due to the clamping action of the retrofit measure, 

particularly if the plastic hinge contains lap-spliced longitudinal reinforcement. For example, the 

hinge length for a column with a steel shell retrofit is given by 

lp = sg + 2χdb (9-8) 
358

where sg is the gap between the retrofit measure and critical section (typically about 50 mm 

(2 in)), db is the diameter of longitudinal reinforcement, and χ equals 6 for grade 40 reinforcing 

bars (i.e., fy = 275 MPa (40 ksi)) or 9 for grade 60 reinforcing bars (i.e., fy = 414 MPa (60 ksi)). 

Equation 9-8 has been developed for steel-jacketed columns, both with and without lap splices. It 

is expected to give conservative estimates of hinge length for columns that have externally 

prestressed confinement systems or fiber composite jacket retrofits where there are no lapspliced reinforcing bars in the plastic hinge region. 

Shear Strength Enhancement – The shear resistance of a passive circular steel jacket may be 

found by analogy to hoop or spiral reinforcement. The jacket is considered to act like a spiral 

bar of area Av at spacing s, equal to Av/t. The additional nominal shear capacity Vsj, provided by 

the jacket, is given by: 

V tf Dcot sj ys 2

π = θ (9-9) 

where fys is the jacket yield stress, and θ is the shear crack angle measured from the vertical, and 

is given by the equations 9-10 or 9-11 below, provided that the longitudinal reinforcement is not 

terminated in the length of column encompassed by the failure plane defined by the angle. 

1

4 v

v

1

v

n 1.26

tan

1 n

−

⎛ ⎞ ρ ρ + ⎜ ⎟ ρ θ= >α ⎜ ⎟

⎜ ⎟ + ρ

⎝ ⎠

A for fixed-pinned columns (9-10) 

1

4 v

v

1

v

n 0.46

tan

1 n

−

⎛ ⎞ ρ ρ + ⎜ ⎟ ρ θ= >α ⎜ ⎟

⎜ ⎟ + ρ

⎝ ⎠

A for fixed-fixed columns (9-11) 

where: 

ρv = 2t/r, 

ρl = As/Ag (longitudinal reinforcement ratio), 

 n = Es/Ec, 

α = D'/L (geometric aspect ratio of the column), 

 D' = distance between the outermost reinforcing bars in the direction of shear, and 

L = twice the distance from the point of fixity to the point of zero moment within the 

 column. 

Retrofit Design Criteria for Rectangular Columns – The principles developed above can be 

extended to rectangular columns. At this time, only the elliptical-steel jacket method of retrofit, 

as shown in figure 9-10, has been demonstrated in the laboratory to have adequate confinement 

due to hoop action in the shell (Sun et al., 1993). Stiffened flat steel plates have also been used 
359

in the field and tested in the laboratory (Priestley et al., 1992) but their performance under cyclic 

loads is inferior to the elliptical jacket. In some cases, flat plates may be desirable on wide (or 

architecturally flared) columns for aesthetic reasons. In these cases, the required confining 

pressure will need to be provided by plate bending. A thicker or a stiffened plate, in conjunction 

with bolts placed in holes drilled through the column, may be required to accomplish this. 

Rectangular columns that are not square generally have greater stiffness, and thus greater 

ductility demand, in the direction of the longer side (usually the transverse direction). Square 

columns or those that are almost square, can be retrofitted with a circular shell, but those that 

have significant plan aspect ratios should be fitted with elliptical jackets. As a consequence, the 

curvature of the shell surface for a rectangular column varies continuously. 

The equation of an ellipse may be expressed as: 

2 2

2 2

x y

x y 1

B B

+ = (9-12) 

where Bx and By are defined in figure 9-12. The extreme radii of an elliptical jacket in the two 

principal directions are: 

2 2

y x

1 3

x y

B B r and r

B B = = (9-13) 

The jacket radius at the corner of the column section, r2, may be taken as the average of r1 and r3. 

In practice, it simplifies shell fabrication if two constant radius segments are joined to 

approximate an ellipse. Standard shapes have been developed for various cross-section 

dimensions (Caltrans, 1996). 

The design equations for flexural integrity and ductility capacity may be adapted from design 

equations for circular columns using an average radius of the ellipse over the extent of the 

compression zone. A reasonable approximation to this could be obtained by taking the average 

of the jacket radius at the column section corner, r2, and at the principal axis under consideration. 

With reference to figure 9-12 for a rectangular column with cross-sectional dimensions of 2a in 

the x-direction, and 2b in the y-direction, the average radius for the shell in the x direction is: 

 rx = (r1 + r2) / 2 (9-14) 

and in the y-direction is: 

 ry = (r3 + r2) / 2 (9-15) 

The appropriate confinement equations can now be used to determine the required shell 

thickness by substituting D = 2rx or D = 2ry. 
360

Figure 9-12. Geometry of an elliptically shaped jacket. 
361

EXAMPLE 9.2: STEEL SHELL RETROFIT 

Given: a 9.15 m (30 ft) high single column pier that is 1.22 m (4.0 ft) in diameter. It is assumed to be 

rigidly connected to a footing and unrestrained at the top. The dead load of the superstructure is 4.45 MN 

(1000 kips). The structure is located at a site where FvS1 = 0.6. 

The main reinforcement in the column is 20 #35 (metric) (#11 cus) grade 40 reinforcing bars that are not 

lap-spliced and are confined by #13 (#4 cus) hoops at 305 mm (12 in) spacing. There is 50 mm (2 in) 

cover over the #13 (#4 cus) hoops. The concrete strength is assumed to be approximately 35 MPa (5000 

psi) and the effective moment of inertia of the section is 0.0544 m4 (6.3 ft4

). 

Moment capacity M 4.5 MNm n = (3317 k/ft) 

Shear capacity V 0.492 MN n = (110.6k) 

The column details are non-ductile and it is proposed to use a circular steel shell to increase the ductile 

capacity of the column. Calculate the thickness of the shell required for flexural confinement and check 

shear strength requirements. 

STEP 1. SIMPLIFIED DYNAMIC ANALYSIS 

Stiffness for single curvature. 

Stiffness ( )( ) 4

eff c 3 33

3EI 3 27800MPa 0.0544m

k 5.93 MN/m

L 9.15 m

⋅ == = (406 k/ft) 

Mass == = 2

2

W 4.45MN m 0.454 MNsec /m

g 9.8m/ sec (31.0 kּ sec2

 / ft) 

Period 

2

c

m 0.454 MNsec T 2 2 1.74 sec

k 5.93 MN/m

=π =π =

20 -#35 (#11)

Radius =

0.610 m (2.0 ft)

#13 @ 305 mm Hoops

(#4 @ 12”)

A A Steel Shell

Retrofit

9.15 m

(30 ft)

Single Column Pier 
362

v 1 a

F S 0.6 S 0.345

T 1.74sec == =

V S M 0.345 (0.454x9.8) 1.54 MN EQ a g == = (346 k) 

M V L 1.54 MN (9.15m) 14.05 MN/m EQ EQ == = (10,360 k/ft) 

EQ

EQ

c

V 1.54 MN 0.26 m (0.85 ft) k 5.93 MN/m

∆= = =

STEP 2. SIMPLIFIED SHELL DESIGN 

Plate thickness: J ≥= = A ⇒

f D 2.07 MPa (1220) mm t 6.3 mm (0.25 in) say 6 mm (0.25 in) 400 400 (Eq 9-4) 

Fs was taken as 100 MPa. 

Ductility factor: 

R-factor: eff == =< EQ ⇒

n

M 14.05 MNm R 3.12 4 OK

M 4.50 MNm (for Mn derived from section analysis) 

Using t = 10 mm (0.4 in) as the minimum thickness required for field handling: 

Length of plastic hinge zone: 

Ap == = 0.25L 0.25 (9.15 m) 2.29 m (7.5 ft) use 2.5 m (8.2 ft) ⇒

STEP 3. RIGOROUS DESIGN CHECK 

The first step in a rigorous design check is to perform a computer based moment-curvature analysis that 

considers the concrete stress and strain enhancement resulting from shell confinement and the strain 

hardening of the steel. Such programs may model the column section as a series of concrete and steel 

elements with these enhanced stress-strain properties. The shell is only considered effective as confining 

reinforcement. Plane sections are assumed to remain plane as the section is rotated incrementally while 

maintaining force equilibrium. The results can be presented as force-displacement curves at the top of 

the column. The following plot, obtained from a Caltrans in-house program, reflects the forcedisplacement curve for this column retrofitted with a 10 mm (0.4 in) thick steel shell. 
363

0.700

0.600

0.500

0.400

0.300

0.200

0.100

0.000

0.000 0.400 0.100

Horizontal Force vs. Displacement

Displacement - m

Force - MN

0.200 0.300 0.500 0.600 0.700

The following values were obtained from the moment-curvature analysis: 

θ = P 0.0591 radians (plastic rotation capacity) 

ε = cu 0.03211

∆ = Y 0.10 m (4 in) (idealized yield displacement – see plot) 

The next step is to perform a pushover analysis of the bent to the following target displacement, and to 

calculate the plastic rotation demand at the critical section of the column. In this case, the pushover 

analysis can be calculated by hand, but in a more complicated structure, a nonlinear computer program 

must be used. 

∆= ∆ = = T EQ 1.5 1.5 (0.26) m 0.39 m (1.3 ft) (target displacement derived from elastic analysis) 

∆ =∆ −∆ = − = P TY 0.39 m 0.10 m 0.29 m (0.95 ft) 

P

P

0.29 m 0.0317 radians

L 9.15 m

∆ θ= = = (plastic rotation demand from “pushover”) 

θP < 0.0591 radians ⇒ OK 

Notice that, based on rigorous analysis, the retrofitted column can accommodate nearly twice the plastic 

rotation demand that would be imposed on it during the design earthquake. There would be a case for 

reducing the shell thickness if constructability were not an issue. Out of curiosity, the above process was 

repeated until the shell thickness converged on a value that was just sufficient to resist the plastic 

rotational demands. The value obtained was approximately 4 mm (0.16 in), even less than obtained by 

the simplified approach. 

STEP 4. SHEAR DESIGN CHECK 

V 1.5V 1.5 (0.492) MN 0.74 MN P N == = (166 k) 

= φ

= => ⇒

'

cg c

2

V 0.8A (0.083) 2 f

0.85(0.8) 1.17m (0.083) 2 35MPa 0.78 MN (175.3 k) 0.74 MN (166 k) OK

Shear does not control outside the plastic hinge region. 
364

The effective confinement will thus be less in the direction of the shorter side (usually the 

longitudinal direction), which will therefore be the weak direction of the column and may govern 

design. In piers with large plan aspect ratios, it may be impossible to obtain adequate 

confinement in the direction of the shorter side. In these cases, it will be necessary to add 

stiffeners or tie rods to assist the shell in providing the required confinement. However, it will 

frequently be found that a realistic assessment of displacement capacity in the weak 

(longitudinal) direction of the bridge will indicate that no retrofit is necessary, and the design of 

the jacket will be governed by the requirements in the strong (transverse) direction. 

9.2.1.3(b). Fiber Composite Jacketing 

There are a number of proprietary techniques for jacketing or wrapping deficient concrete 

columns that use advanced fiber composites to increase the flexural ductility and shear strength, 

and to correct lap splice length deficiencies at ends of columns. These composites are usually 

high strength glass (E-glass), carbon, or aramid fibers oriented primarily in the circumferential 

direction of the column, and bound in a polyester, vinyl ester, or epoxy resin matrix. The 

resulting material has anisotropic properties, which allows the flexural ductility, splice strength 

and shear strength of a column to be improved without significantly affecting flexural strength 

and stiffness. Since the properties of a cured composite laminate depend on the particular fiber 

and resin combination, it is advisable to use only components that have been developed for use 

as part of a system and thoroughly tested. Detailed guidance on the use of composites for 

column wraps is available in ACI’s Design and Construction of Externally Bonded FRP Systems 

for Strengthening Concrete Structures (ACI, 2002). 

Current design philosophy is to install column confinement as a passive system that is engaged 

only when the concrete column is loaded and dilates. In the past, there has been some 

experimentation with active systems where the fibers are loaded or prestressed during 

installation. However, extreme caution must be taken with an active system because of the 

natural tendency of composite materials to creep under sustained loading, which can lead to 

rupture of the wrap. This can occur even at relatively low stresses and is exacerbated by the 

difficulty in obtaining uniform stresses in the wrap. Exterior steel prestressed cables, used in a 

similar fashion to steel hoops, have been used successfully by Illinois DOT to retrofit inadequate 

length lap splices. 

One of the first fiber composite wrapping systems to be developed consisted of glass and aramid 

fibers in an epoxy matrix (Fyfe, 1994). The fibers were woven into a fabric, which was saturated 

in an epoxy resin in the field and then hand wrapped around an existing column. Additional 

confinement may be provided in critical regions, such as the bottom of columns. This approach 

has been successful in enhancing the flexural ductility and shear strength of circular columns in 

the laboratory (Priestley et al., 1994). 

In the past, another form of passive confinement involved the use of a special wrapping machine 

that placed epoxy impregnated carbon fiber ‘tows’ (a tow is a series of straight-laid fibers) 

directly onto the column (Seible et al., 1995). The tows were wound up the column at 6 rpm and 

heat cured for at least two hours at temperatures up to 121 degrees centigrade. After curing, the 

jacket was protected with an acrylic emulsion spray coating. This proprietary system is no 
365

longer in use. However, carbon systems can be effective when applied in the same manner as 

E-glass systems. 

Rectangular columns have also been tested with fiber composite shell retrofits (Seible and 

Priestley, 1993; Seible et al., 1995). One approach, which utilizes an elliptical jacket placed over 

a section of the column that has been built up into an elliptical shape, has been shown to increase 

both the ductility and shear strength of the column. Another approach allows jacketing to be 

placed directly on the surface of a rectangular column. This method will increase the shear 

resistance of the column and force a flexural failure with increased, but limited, ductility 

provided that the cross-sectional dimensions of the column are not too large. 

At least two systems now available use prefabricated fiber composite shells (Xiao et al., 1995; 

Steckel et al., 1998). One of these systems uses a series of shells comprised of high strength 

fibers in a polymer matrix similar to field-installed systems, except that they are manufactured 

under factory controlled conditions. Field installation of this passive confinement system 

involves bonding an initial prefabricated shell to the existing column. Subsequent shells are 

placed over the first shell, with the shell seams rotated around the column so that they are 

overlapped, until the desired jacket thickness is achieved. These shells are clamped to the 

existing column using straps, until the bond has fully cured. This system can be quickly installed 

without the need for heavy equipment or skilled labor. 

Fiber composite column jacketing systems have been developed and tested (Matsuda et al., 1990; 

Ogata et al., 1993). In addition to using fiber composites to increase flexural ductility and shear 

strength, some of the fibers may be placed along the axis of the column to increase flexural 

strength. 

Designers should be aware of potential freeze-thaw problems when considering column 

jacketing in cold regions. A potential problem exists when retrofitting using continuous fiber 

composite jackets. Moisture can collect between the jacket and the concrete column and freeze, 

resulting in expansive pressures that can cause a rapid deterioration of the concrete or damage to 

the wrap. The potential for this deterioration can be reduced by applying the jacket in relatively 

narrow bands (e.g., 150 mm (6 in)) with an epoxy-free gap between bands equal to or greater 

than the width of the bands, but not greater than six times the main reinforcing bar diameter plus 

the clear cover over the bar (Aquino et al., 2004). Tests (Jin et al., 1994) have also shown that 

discrete fiber composite straps placed around an existing column could increase its flexural 

ductility and shear strength. If a continuous wrap is used, precaution should be taken to prevent 

water ingress from the top. Repair of leaking joints at piers and/or waterproofing at of the top of 

the column is very important. 

Another drawback of fiber composite systems is their susceptibility to absorbed moisture 

(Steckel et al., 1998). Glass fibers can lose significant strength due to moisture absorption, 

which can be accelerated by temperature variation. The polyester epoxy matrix used in glass 

fiber composites can provide a measure of protection against this type of damage, but is very 

sensitive to the type of material used and the manner in which the composite is fabricated and 

installed. 
366

Carbon fibers themselves are not affected by moisture, but the quality of the polyester epoxy 

matrix is critical to the performance of the composite. Carbon fiber systems have shown loss of 

tensile strength at high temperatures due to moisture absorption. This may require redesign of 

the epoxy matrix component in order to satisfy design and performance requirements. 

Collaboration between the designer and composite material supplier is encouraged. 

These types of problems can be addressed by careful testing of prospective systems under 

extreme or accelerated service conditions, good quality control during construction and 

application in the field, and design with sufficient safety factors to account for prospective 

strength loss. It is important to note that although the strength of fiber composite systems is 

susceptible to environmental factors, the stiffness of these systems remains constant. This is 

encouraging since their effectiveness is stiffness sensitive. 

Table 9-1 shows mechanical properties for a number of different fiber types. This is provided 

for reference only and it should be noted that the properties of the cured laminate may differ 

from the fiber properties. 

Table 9-1. Mechanical properties of fibers used in modern fiber-reinforced plastic composites. 

Type of Fiber 

Ultimate 

Strength, 

MPa (ksi) 

Ultimate 

Strain 

Modulus of Elasticity, 

GPa (ksi) 

E-Glass 2410 (350) 0.020 41 (6,000) 

S-Glass 3450 (500) 0.030 41 (6,000) 

CF-Pan 4140 (600) 0.020 228 – 345 (33,000 – 50,000) 

C-Pitch-GP 1380 (200) 0.003 41 (6,000) 

Pitch UHM 2760 (400) 0.005 483 – 827 (70,000 – 120,000) 

Aramid 3450 (500) 0.020 69 – 138 (10,000 – 20,000) 

Ceramic 690 (100) 0.020 69 – 276 (10,000 – 40,000) 

Nylon 345 (50) 0.050 – 0.500 3.5 (500) 

A procedure for the design of fiber-composite column jackets is described below. 

Performance of Lap Splices – In the active/passive confinement system of figure 9-13, an active 

wrap is stressed to produce a reliable, after-creep, active confining stress in the column, and an 

additional passive wrap is also provided. Both layers develop (additional) hoop stress as the 

jacket expands to a strain of 0.001. Wrap thicknesses can be found from equilibrium as follows: 

 2 (taEa + tpEp) (0.001) = D (fl – fa) (9-16) 

where ta is the thickness of active wrap, Ea is the modulus of elasticity of active wrap, fa is the 

active confining stress in the column, tp is the thickness of passive wrap, and Ep is the modulus of 

elasticity of passive wrap. 
367

Rearranging equation 9-16 gives: 

 taEa + tpEp ≥ 500 D (fl - fa) (9-17) 

In a passive system, the active layer is eliminated and equation 9-17 becomes: 

 tpEp ≥ 500 D fl (9-18) 

As with steel jackets, the required confinement stress, fl, can be determined from equation 9-1, 

or a simplified approach can be used in which fl is set equal to 2.07 MPa (300 psi) for a typical 

column with Grade 40 starter bars and splice lengths of 20 db. 

It is likely that columns with bars that have inadequate lengths of lap-splices can be safely 

retrofitted with thinner jackets than those determined by the above design procedure, since there 

is good experimental evidence that columns can tolerate jacket dilation strains up to 0.003 

without a loss in strength (Hawkins, 2000). The Illinois DOT uses this higher value for column 

composite jackets, but requires that any circumferential crack in the lap-splice zone be limited to 

0.8 mm (0.03 in) in width. This width may be exceeded if the shear capacities along the crack 

are calculated and found to be otherwise acceptable. The advantage of a thinner jacket is that it 

will allow the development of a greater plastic hinge length, and thus there will be less chance of 

bar fracture than with a thicker jacket subject to the same plastic rotation. 

after Priestley et al., 1992 

Figure 9-13. Free body diagram of column retrofitting with a composite shell.

Flexural Confinement – The jacket thickness requirements for fiber composites can be 

determined using moment-curvature analysis of the retrofitted column in conjunction with a 

nonlinear static (‘pushover’) analysis. This approach, which is described in section 9.2.1.3(a) for 

steel jacketing, is also applicable for fiber composites. Because the stress-strain curves for high 

strength fiber composites used for column retrofitting are essentially linear up to failure, the 

ultimate strain in concrete (εcu) confined by these jackets is given in equation 9-19: 

s du du cu '

cc

2.5 f 0.004

f

ρ ε

ε= + (9-19) 
368

where fdu is the ultimate design strength of the jacket material taking into account long-term 

environmental degradation, εdu is the ultimate design strain of the jacket material taking into 

account long-term environmental degradation, and ρs is the effective volumetric ratio. 

 ( ) a p

s

4t t

D

+

ρ = (9-20) 

Equations 9-19 and 9-20 may be used to solve for the required combined thickness of both active 

(ta ) and passive (tp) layers of the composite jacket, as shown in equation 9-21: 

( ) '

cc a p cu

du du

Df t t 0.1 0.004) f

+ = ε−

ε

 (9-21) 

If the properties of active and passive wraps are very different, a weighted average for fdu and εdu

should be used. 

A simplified design approach may be used in which a confinement stress of 1.72 MPa (250 psi) 

is provided at a dilation strain of 0.004. This results in the following jacket thickness 

requirement to assure adequate displacement ductility to allow the retrofit to be designed using 

an R-factor of four in most typical columns: 

a p

j

215D

t t

E

+ ≥ (mm) (9-22a) 

a p

j

31D

t t

E

+ ≥ (in) (9-22b) 

where Ej is the weighted average of the elastic moduli for the active and passive wraps (MPa or 

ksi units). 

Shear Strength Enhancement – Following an approach similar to that used for steel jackets, the 

enhancement in column shear strength for fiber composite jackets is: 

V t E t E T Dcot sj a a p p p a ( ) ( ) 2

π = + ε+ θ (9-23) 

where Ta is the active tensile force per unit height of the jacket due to active pressure, Ta = fa

D/2, and θ is the shear crack angle given by substituting Ej for Es in equations 9-10 and 9-11. 

Tests have shown that column retrofits designed assuming a maximum allowable passive strain 

due to shear (εp) = 0.006 and θ = 30 degrees, have ductile flexural response. A design example 

for a composite shell retrofit is given in example 9.3. 
369

EXAMPLE 9.3: FIBER COMPOSITE SHELL RETROFIT 

Given: a 6.1 m (20 ft) high single column pier that is 1.68 m (5.5 ft) in diameter. It is assumed to be rigidly 

connected to the footing. The dead load from the superstructure is 9.9 MN (2250 kips). The structure is 

located in a site where FvS1 = 0.5. 

The main reinforcement in the column is 28 #42 (metric) (#14 cus) grade 40 reinforcing bars that are not 

lap spliced and are confined by #13 (#4 cus) hoops at 305 mm (12 in) spacing. Concrete cover is 50 mm 

(2 in) over the #13 (#4 cus) hoops. The concrete strength is assumed to be approximately 35 MPa (5000 

psi) and the effective moment of inertia of the section is 0.194 m4

 (22.5 ft4). Moment capacity, 

Mn = 13.57 MNm. 

The column details are non-ductile and it is proposed to use a passive composite shell to retrofit the 

column. Calculate the thickness of the shell required for flexural confinement and check shear strength 

requirements. Assume Ej

 = 42000 MPa. 

STEP 1. SIMPLIFIED DYNAMIC ANALYSIS 

Stiffness 

4

eff c 3 33

3EI 3(27800 MPa) 0.194m k 71.3 MN/m

L 6.10 m

⋅ == = (4940 k/ft) 

Mass == = 2

2

W 9.90 MN m 1.01MNsec /m

g 9.8 m/ sec (69.2 · k-sec2

/ft) 

Period 

2

c

m 1.010 MN sec /m T 2 2 0.748 sec

k 71.3 MN/m

− =π =π =

28 -#42 (#14)

Radius =

0.840 m (2.75 ft)

#13 @ 305 mm Hoops

(#4 @ 12”) A A

6.10 m

(20 ft)

Single Column Pier 

v 1 a

F S 0.5 S 0.668 g T .748 == =

V S W 0.668 (9.90MN) 6.61 MN EQ a == ⋅ = (1486 k) 

M V L 6.61MN (6.10 m) 40.34 MNm (29,745 k/ft) EQ EQ == =
370

EQ

EQ

c

V 6.61MN 0.093 m

k 71.3 MN/m

∆= = = (0.3 ft) 

STEP 2. SIMPLIFIED PROCEDURE FOR FLEXURAL CONFINEMENT 

p

j

215D 215 (1.68 m) t 0.0086 m

E 42000 MPa == = (0.34 in) ⇒ minimum shell thickness (Eq. 9-22a) 

eff == =< EQ ⇒

n

M 40.34 R 2.97 4 OK

M 13.57

where Mn is derived from section analysis. 

STEP 3. SHEAR STRENGTH ENHANCEMENT 

n

P

1.4 M 1.4 (13.57 MN m) V 3.11 MN

L 6.10 m

− == = (699 k) 

' 2 V 0.8A (0.083) 2 f 0.85(0.8) 2.21m (0.083) 2 35 MPa 1.48 MN cg c =φ ⋅ = ⋅ ⋅ = (333 k); i.e., concrete 

confined within the shell is 100 percent effective in resisting shear. 

P c

sj

V V 3.11MN 1.48 MN V 1.922 MN

0.85

− − == = φ (432 k) 

Solving for ta from formula 9-23: 

sj

a

i P

2V 2(1.36 MN) t

E Dcot 3.142(42000 MPa)(0.006) 1.68 m(1.0)

0.003 m (0.08 in) 0.0086 m (0.34 in) OK

≥ =

πε θ ⋅

= < ⇒

No additional shell thickness is required for shear strength enhancement, but the shell should be full 

height. The portion of shell above the plastic hinge zone may be reduced in thickness to reflect only 

shear requirements.
371

9.2.1.3(c). External Prestressing Steel 

An improved form of confinement may be achieved by wrapping prestressing wire under tension 

around a column. Active pressure, rather than passive pressure, provides the lateral confining 

stresses needed to increase flexural ductility, although passive pressure will also add to the 

confinement. This procedure has successfully increased the flexural ductility of circular columns 

with lap splices at the critical section, but its effect on shear strength has not yet been quantified. 

It is, however, expected to be beneficial. An advantage of this technique is that it has little effect 

on the flexural strength and stiffness of the column. Reliable anchorage of the wire ends is 

essential for an effective field application. 

Initial attempts to design a machine to wrap prestressing wire around a column proved not to be 

economical due to the size of the machine required to produce the desired tension in the strand. 

Another method of stressing uses standard 15 mm (0.6 inch) diameter prestressing strand that is 

anchored in special anchorages originally designed for prestressed water tanks. This method 

uses existing materials and equipment, and has been successfully tested in the laboratory (Lin et 

al., 1994). These anchors are shown in figure 9-14. Subsequent tests (Hawkins et al., 1999) 

showed that columns retrofitted with prestressing strand lose strength at dilation strains beyond 

0.001 as a result of high prestress losses that occur due to stressing the strands around a typical 

column, and the penetration of the strand into the concrete during cyclic loading. This 

observation is used to set the strain limit in the design procedures that follow. 

1/4” Concrete Screw

2-3/4” to 3-1/2” Long

(69.9 mm to 88,9 mm Long)

If Required

2-1/2”

63.5 mm

14”

Figure 9-14. Anchorages for prestress strand retrofit. 

Another method of stressing that has been developed and successfully tested involves placing 

unstressed strand around the column (Swanson, 1999). A hand-held machine is used to stress the 

strand by pulling a portion of the strand radially away from the column and inserting a steel 

wedge between the strand and the column, as shown in figure 9-15. This procedure is repeated 

around the circumference of the column until the strand is adequately stressed and provides a 

somewhat uniform confining stress. This stressing technique has the added advantage of 

minimizing prestress losses due to friction. 
372

Threaded semicircular steel bars may be used instead of prestressing wire to achieve a similar 

effect. Originally designed to prevent the splitting of a concrete column under service conditions, 

these bars are placed around a column and tightened to produce a prestressing force to prevent 

nonductile column failure under cyclic loading (Lin et al., 1994). Similar improvements may be 

obtained from a system using semicircular threaded reinforcing bars tightened by threaded swage 

couplers, as shown in figure 9-16 (Coffman et al., 1991). 

Figure 9-15. Prestressing retrofit by wedging between column and strand. 

Steel on the outside of a column must be protected from corrosion. This can be done by encasing the wire with a concrete jacket of nominal thickness that stops short of adjacent members 

(footings or cap beams) to avoid increasing the flexural strength or stiffness of the column. 

Since no data is available on the performance of rectangular or square columns using this 

technique, applications should be limited to circular columns at this time. 

Performance of Lap Splices – The confinement stress required to prevent splice failure, fl, was 

discussed in section 9.2.1.3(a). The free body diagram of half of a circular column retrofitted by 

a wire wrap method is shown in figure 9-17. Equilibrium requires that 

 2 T = fl D (9-24) 

and if Es = 200 GPa (29,000 ksi): 

( ) p

i

2A

f 200 f D (MPa units)

s

+ = A (9-25a) 
373

( ) p

i

2A

f 29 f D (ksi units)

s

+ = A (9-25b) 

where Ap is the cross-sectional area of wire equal to 0.785 dp

2

, dp is the diameter of wire, s is the 

spacing of wire, and fi is the tensile stress in wire after losses (MPa, ksi). 

.

.

.

.

. .

.

.

.

.

. .

.

.

.

. .

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

Shotcrete Cover

Chip Concrete

for Turnbuckle

Existing Concrete

Column

Reinforcing Hoops

Tensioned with

Special Turnbuckle

Cross Section

Special

Turnbuckle Shotcrete

Elevation

Threaded

Reinforcing

Bar

Turnbuckle Detail

Figure 9-16. Semi-circular reinforcing bars with couplers. 
374

D

f

d at P spacing s

T T

after Priestley et al., 1992 

Figure 9-17. Free body diagram of column retrofitting with external prestressing. 

Solving equation 9-25 for the required spacing gives: 

2A f 200 2A (f 29) p i pi ( ) s (MPa units) (ksi units) fD fD+ +

≤ =

A A

 (9-26) 

The required level of confinement, fl, can be determined from equation 9-1. Alternatively, the 

confinement stress can be taken as 2.07 MPa (300 psi), which will be adequate to prevent splice 

failure in most columns (Chai et al., 1991), and equation 9-26 then becomes: 

A f 200 6.7A f 29 pi pi ( ) ( ) s MPa units (ksi units) D D+ +

≤ = (9-27) 

Strand spacing, s, should not exceed six times the diameter of the main reinforcing bar (Hawkins 

et al., 1999). 

Flexural Confinement – Taking account of differences in the shape of the stress-strain curves for 

prestressing wire and mild structural steel, and the configuration of wire and strand versus that of 

a shell, equation 9-5 can be rewritten as: 

p pu su

cu '

cc

4A f

0.004

Dsf

ε

ε= + (9-28) 

where fpu and εsu are the ultimate design stress and strain, respectively, for prestressing wire, and 

f'cc is as follows: 
375

' ' p pu p pu

cc c ' '

c c

15.88A f 4A f

f f 1.254 2.254 1

sDf sDf

⎛ ⎞

=− + + − ⎜ ⎟

⎝ ⎠

 (9-29) 

Rewriting equation 9-28 leads to the following requirement for prestressed wire wraps: 

( ) ' p cc cu

pu su

A Df 0.004

s 4f

≥ ε−

ε

 (9-30) 

Once the value of εcu at the target displacement has been determined from a moment-curvature 

curve and an inelastic static analysis, using the methods described above for steel shells, it will 

be possible to solve for Ap/s using an iterative approach. Assuming εcu = 0.002, and f'cc = 1.7 f'c, 

will give a conservative initial value for Ap/s. 

In the case of prestressing strand, the above design approach can be simplified by assuming a 

confinement stress, fl, equal to 1.72 MPa (250 psi) and a passive steel strain of 0.004. This 

should be sufficient to keep the strand on the linear portion of the stress strain curve 

(i.e., fs = 0.8 fpu) provided moderate levels of active stress are used. This results in the following 

simplified design equations: 

p

pu pu

A D D (MPa units) (ksi units) s 0.92f 6.4f

≥ = (9-31) 

This simplified requirement for prestressed wire wraps will be adequate for most columns, 

although it may be overly conservative in some cases. 

Shear Strength Enhancement – The shear strength provided by external prestressing wire may be 

determined in the same manner as conventional circular hoops or spirals. Therefore, the shear 

capacity contribution of this steel is given by equation 9-32: 

ps pu

sp

A (0.8f )D V cot

2 s

π = θ (9-32) 

A design example of a column wrap using external prestressing wire is given in example 9-4. 

9.2.1.3(d). Concrete Jacketing 

The addition of a jacket of reinforced concrete around an existing circular or rectangular column 

was discussed earlier in section 9.2.1.2(a). The application is straightforward, following the 

rules for reinforced concrete design. However, adding a concrete jacket requires holes to be 

drilled through the existing column and placement of supplemental ties through these holes. 

These ties must be adequately anchored on both sides of the existing column. 
376

EXAMPLE 9.4: EXTERNAL PRESTRESS 

Given: the single column pier in example 9-3. The column is 6.1 m (20 ft) high, and has a diameter of 

1.68 m (5.5 ft). It is assumed to be rigidly connected to the footing. The dead load of the superstructure 

is 9.9 MN (2250 k). The structure is located in a site where FvS1 = 0.5. 

The main reinforcement in the column is 28 #42 (metric) (#14 cus) grade 40 reinforcing bars that are not 

lap spliced and are confined by #13 hoops at 305 mm (12 in) spacing. The concrete cover over the #13 

hoops is 50 mm (2 in). The concrete strength is assumed to be approximately 35 MPa (5000 psi) and the 

effective moment of inertia of the section is 0.194 m4

 (22.5 ft4). 

The column details are not ductile, and it is proposed to use a steel wire wrap to retrofit the column. 

Calculate the diameter and spacing of the wire using the simplified procedure for flexural confinement, 

and check shear strength requirements. 

Mn = 13.75 MNm 

Prestress strand fpu = 1862 MPa (270 ksi) 

STEP 1. SIMPLIFIED DYNAMIC ANALYSIS 

4

eff c 3 33

3EI 3(27800 MPa) 0.194 m k 71.3 MN/m

L 6.10 m

⋅ == = (4140 k/ft) 

Mass: == = 2

2

W 9.90 MN m 1.010 MNsec /m

g 9.8 m / sec (69.2 k sec2 /ft) 

Period: 

2

c

m 1.010MN sec /m T 2 2 0.748 sec

k 71.3MN/m

=π =π =

v 1 a

F S 0.5 S 0.668 g T .748 == =

V S W 0.668 9.90 MN 6.61 MN EQ a == ⋅ = (1486 k) 

M V L 6.61MN (6.10 m) 40.34 MNm EQ EQ == = (29,745 kft) 

EQ

EQ

c

V 6.61MN 0.093 m

k 71.3 MN/m

∆= = = (0.3 ft) 

STEP 2. SIMPLIFIED PROCEDURE FOR FLEXURAL CONFINEMENT 

p 2 2

pu

A D 1.68 0.00098 (0.04 ) m in

s 0.92f 0.92(1862) m in == = (Eq. 9-31) 
377

i.e., minimum wire wrap area per unit length is required, which is approximately a 12.7 mm strand at 75 

pitch (0.5 in strand at 3” pitch). 

== = < ⇒ eq

eff

n

M 40.34 MNm R 2.97 4 OK

M 13.57 MNm 

where Mn is derived from section analysis. 

STEP 3. SHEAR STRENGTH ENHANCEMENT 

== = n P

1.4M 1.4(13.57 MNm) V 3.11 MN

L 6.10 m (699 k) 

' 2 V 0.8A 0.083 2 f 0.85(0.8) 2.21m (0.083) 2 35 MN/m or MPa 1.48 MN cg c =φ ⋅ = ⋅ ⋅ = (333 k); 

i.e., the concrete confined within the wire wrap is 100 percent effective in resisting shear. 

P c

sj

V V 3.11 1.48 V 1.92 MN

0.85

− − == = φ (432 k) 

Solving for 

Aps

s

from equation 9-32: 

( ) ≥ = =< ⇒

π θ

ps sj

pu

A 2V 2 (1.92 MN) 0.000489 0.00098 OK

s 3.142(0.8)(1862 MPa)(1.68 m)(1.0) 0.8f Dcot

No additional wire is required for shear strength enhancement, but wire wrap should be full height. 

Spacing of the wire wrap above the plastic hinge zone may be increased to reflect only the shear 

requirements. 
378

Concrete jackets are usually constructed as overlays and may be of two types. 

• Shotcrete overlays and mild steel reinforcement (Rodriguez and Park, 1992; Amari et al., 

1994; Pardoen et al., 1998), and 

• Fiber reinforced concrete overlays, comprised of a steel fiber mat wound around a column 

and infiltrated with slurry (Hawkins et al., 1999). 

Concrete jackets have been used to force yielding away from the location of starter bar splices, 

and in such cases, the need to improve the performance of the lap-splice has been avoided. 

Nevertheless, it is recommended that sufficient transverse reinforcement be provided in the 

concrete jacket to develop sufficient passive confinement to produce the pressures given by 

equation 9-1. Alternatively, a design pressure of 2.06 MPa (300 psi) can be used for typical 

columns. Flexural confinement and shear strength may be determined by methods similar to 

those used for design of new reinforced concrete members. 

A concrete jacket will increase the flexural strength and stiffness of a column more than a steel 

jacket, composite or wire wrap, with potentially undesirable effects on bridge performance. 

These increases in strength and stiffness need to be included in the analytical model that is used 

to determine the effectiveness of any retrofit scheme using these jackets. 

9.2.1.4. Supplemental Column Shear Walls 

Infill shear walls are walls cast-in-place between the columns of a multi-column pier. They have 

been used successfully to increase transverse shear capacity, as shown in figure 9-18. These 

walls prevent the formation of plastic hinges in the columns during transverse loading, and will 

help overcome deficiencies in the flexural or shear strength of the pier cap. For single column 

piers, a buttress wall can be used on one side of the column. Strengthening in the longitudinal 

direction by this means is usually difficult, due to geometric constraints, but the need can often 

be avoided in relatively short bridges by relying on the abutments to carry the longitudinal 

earthquake forces. 

 

To be effective, infill shear walls should be designed to act compositely with the existing 

members. This is usually done by providing a sufficient number of drilled and bonded dowels in 

the columns and bottom of the pier cap, so that shear is transferred at the interfaces through a 

shear friction mechanism. This may require the existing concrete surfaces to be roughened to 

minimize the number of dowels required. It may also be necessary to use a forming system that 

can be sealed so that concrete can be placed under pressure, in order to achieve a satisfactory 

joint along the underside of the cap beam. Small vent holes, on the order of 100 mm (4 in), 

through the existing cap may also be useful in this regard. 

A footing, tied into the existing column footings, should be provided under the infill wall. The 

design should provide sufficient reinforcement to transfer all seismic forces, and should also 

consider the potential for differential settlement between the existing and new footings. 
379

New Infill Wall - Tie to Existing

Cap Beam, Columns

with Drilled and Grouted

and New Footing

Dowels

Existing Columns

Existing Footing

New Footing - Tie to Existing

Footing and New Infill Wall

with Drilled and Grouted Dowels

Figure 9-18. Infill shear wall in multi-column bents. 

An infill wall constructed in this manner will also provide a small contribution to the 

longitudinal strength of the pier, which could be beneficial. 

9.2.1.5. Preservation of the Vertical Load Capacity of Columns 

Not all vertical load-carrying members in a bridge require strengthening or ductility 

improvement for lateral load, since these loads can be carried by adjacent members. However, it 

is necessary to ensure that the vertical load capacity of all members is preserved, at the 

displacements to which they will be subjected during an earthquake. Although a full retrofit of 

all columns would accomplish this result, it may also require that connecting elements, such as 

footings and pier caps, be strengthened as well. 

To keep retrofitting costs to a minimum, some columns can be allowed to fail at splices or other 

weak points, provided their vertical load capacity is maintained. One retrofitting technique that 

achieves this purpose (called a ‘Type P’ retrofit), places a relatively thin layer of expansion joint 

material around the existing column before grout is placed between the column and a steel jacket 

(figure 9-19). The expansion joint material is flexible and will prevent flexural strengthening of 

the member through either composite action with the shell or enhanced performance of the 
380

Figure 9-19. Steel shell retrofit for vertical capacity preservation. 

confined concrete. As the column concrete fails, the resulting rubble will be retained within the 

steel shell and thus continue to support vertical load. The cost of this approach is typically only 

a fraction of the cost of a full column retrofit, particularly if the strengthening of footings and cap 

beams can be avoided. 

9.2.1.6. Limitation of Column Forces 

Instead of retrofitting columns to increase their capacity, an alternative approach is to reduce the 

forces to be resisted. This is a particularly useful approach when column strengthening or 

ductility improvement is impractical. Two methods for doing this are discussed below. Any 

method that uses a weakened section or ‘fuse’ to protect the substructure falls into this category. 

9.2.1.6(a). Isolation Bearings 

Isolation bearings can be used as a mechanism to limit inertial forces transmitted to the 

substructure. The bearings will change the dynamic properties of the bridge and add damping 

(section 8.3.2.2). The Sierra Point Overhead on US 101 near San Francisco was the first bridge 

in the U.S. to be retrofitted using this approach (figure 9-20). In this skewed, multi-span, steel 

beam bridge, the columns and footings were particularly weak and non-ductile. Isolation 

bearings were placed at the top of each column and at the abutment seats, to modify the response 
381

and limit the forces that can be transferred to the columns. The result was a retrofit approach in 

which the columns did not require strengthening. 

It is not always necessary to use custom-designed isolation bearings to limit the forces being 

transmitted to the substructure, nor is global response modification always required. Use of a 

standard sliding bearing at a vulnerable location, such as on top of a particularly stiff or brittle 

pier, could protect the pier from damage if the superstructure is continuous and can redistribute 

lateral loads to adjacent ductile supports. 

Figure 9-20. Sierra Point overhead, US 101, near San Francisco, California. 

9.2.1.6(b). Flexural Strength Reduction 

In certain cases, the flexural strength of a column can be reduced, in conjunction with ductility 

improvement, and thereby limit the shear forces transmitted to columns. This is because the 

ultimate shear demand on a column is directly proportional to the plastic moment that can be 

developed. Columns retrofitted according to section 9.2.1.3 may have sufficient capacity (as a 

result of a retrofit for flexure) to satisfy this shear demand. In some cases, it may not be practical 

or desirable to use column jacketing over the full height of the column – for example, when a 

large architectural column flare exists. Such flares are difficult to confine with a jacket, and it 

may be preferable to leave them in place for their architectural effect. If inadequate capacity 

exists to resist the shear demand, it may be possible to reduce the plastic moment by cutting 

longitudinal reinforcement near the base of the column; this will reduce the plastic shear 

demand. Because the ductility demand at the weakened section is likely to increase, it will also 

usually be necessary to provide some form of ductility improvement over the affected portion of 

the column. This part of the column is usually prismatic below the architectural flare and may 

even be below or near ground level. 

 

A retrofit in which the primary column reinforcement is cut, allowing the column to act as a pin 

by rocking on steel bearing plates that are attached to a concrete retaining collar, is illustrated in 

figure 9-21. 
382

Existing Column

Concrete Collar

Column Base Cut

Buffer

Steel Plate on Steel Plate 

Displacement Restrainer

Existing Footing

Figure 9-21. Column retrofit by flexural strength reduction. 

9.2.2. STEEL COLUMNS, FRAMES AND COMPRESSION MEMBERS 

Although steel is a ductile material, steel substructures and steel compression members are often 

vulnerable to damage during an earthquake due to inelastic buckling. Once local buckling 

occurs, steel can fracture due to low cycle fatigue. This type of behavior can cause a rapid loss 

in strength that will effectively limit the ductility of a steel member. Therefore, the retrofitting of 

steel compression members usually focuses on preventing or delaying inelastic buckling. The 

following sections describe retrofitting techniques that have been used or proposed for steel 

compression members. 

9.2.2.1. Braced Frames 

Although more common in building design, steel braced frames have been used as piers in many 

bridges. These frames rely on diagonal members subjected to both tension and compression to 

carry lateral load, and vertical members to carry gravity load. Seismic retrofitting may include 

strengthening of the steel members, or increasing the ductility of the frame. 

The diagonal members in existing braced frames are usually not strong enough to resist 

earthquake loads elastically. When these members are subjected to compression loading, they 

will usually experience global or local inelastic buckling, which will reduce the overall ductility 

of the frame. 
383

In the case of concentrically braced frames, the diagonals are usually not relied upon for 

compression. Therefore, only one set of diagonals, acting in tension, is considered effective in 

resisting lateral loads. Yielding of these members should be limited to prevent excessive 

permanent displacements of the frame, and to prevent tension diagonals from experiencing 

excessive buckling during the reverse cycle of loading. This may be done by strengthening the 

diagonal bracing, through either member replacement or adding supplemental diagonal members. 

An example of diagonal strengthening on a light steel structure is illustrated in figure 9-22. In 

designing such a retrofit, capacity design principles should be used to assure all other members 

and their connections behave elastically. The diagonal members may be designed in tension 

using a force-based approach. It is suggested that an R-factor of two be used, and that ultimate 

capacities be determined using the provisions of the AASHTO LRFD Specifications (AASHTO, 

1998). Alternatively, a displacement-based approach using pushover analysis may be used if 

member elongation is limited to a reasonable level (e.g., on the order of one percent). 

In the case of chevron bracing (K-bracing), diagonal members are designed to participate in both 

compression and tension. Earthquake loading will cause global or local inelastic buckling of the 

compression members if they are not designed to carry the loads elastically. Because member 

capacities that are sufficient to assure elastic behavior cannot generally be economically 

achieved, retrofits of these types of frames often seek to limit the force levels in the diagonal 

members by other means. 

One example is the retrofit of the Richmond-San Rafael Bridge, a major structure spanning San 

Francisco Bay. Existing chevron braces in the bridge were replaced with eccentric braces that 

included a shear link (Itani, 1996). Vertical steel members were also strengthened. This retrofit 

was designed to increase the ductility of the pier, while forcing all yield to occur in the specially 

designed shear link. The existing pier and the proposed retrofit are shown in figure 9-23. 

9.2.2.2. Built-up Compression Members 

Steel bridge columns occasionally consist of members that are built-up from standard rolled 

shapes and plates. These members usually do not meet the criteria for compact sections and are 

prone to local inelastic buckling when plastic hinges form. Such buckling can lead to fracture of 

the steel due to low cycle fatigue and a corresponding decrease in the ductility capacity of the 

bent. 

Steel columns of this type that can be retrofitted by the addition of steel plates are shown in the 

cross-section of figure 9-24 (Holombo et al., 1994). In addition to strengthening the column, 

these plates are intended to reduce local inelastic buckling of the section. Another method for 

retrofitting this column is to use an elliptical grouted steel shell, similar to that used for concrete 

columns, to prevent local inelastic buckling, as shown in figure 9-25. 

Because many existing steel substructure members do not meet current design standards, it is 

often difficult to predict how they will behave under seismic loading conditions. When this is 

the case, laboratory testing involving cyclic loading should be considered to determine the 

behavior of as-built columns and to validate proposed retrofit concepts. 
384

Figure 9-22. Retrofit of X-braced steel bent. 
385

Mud Line

Concrete Pier

Mud Line

Steel Casing

Tremie-Filled

Precast Pile

Cap and Added

Shear Piles

Sea Level

Precast Concrete

Jacket

Steel Tower

Double-Deck

Trusses

Dual EBF Frames

(with Shear Links)

H-Piling

Sea Level

Strengthened

Tower Legs

Strengthening

of Selected

Truss Members

Existing Intermediate Towers of RSRB Proposed EBF for Intermediate Towers

Figure 9-23. Richmond-San Rafael bridge bent retrofit. 
386

Figure 9-24. Built-up steel columns stiffened by plates. 
387

Figure 9-25. Built-up steel column retrofitted with steel shell. 

In the case of single-column piers, the moment connection between the column and the footing 

may also require retrofitting, particularly if the column has been strengthened. This connection 

should be designed to behave elastically, based on capacity-design principles. 

9.2.3. CONCRETE WALL PIERS 

Concrete wall piers are very stiff and very strong in the transverse direction. In general, they 

will not require retrofitting in this direction, but the designer should make sure that the 

foundations are adequate for the lateral loads transmitted from the piers. These foundations 

should not experience displacements that could jeopardize the stability of the structure. 

In the weak (longitudinal) direction, pier walls are typically treated like conventional reinforced 

concrete columns. However, even without retrofitting, walls often exhibit high ductility capacity 

in this direction because the axial concrete stresses are distributed over a larger area due to the 

large transverse width and low longitudinal reinforcement ratios. As a result, a force-based 

design is appropriate using R-factors of four or greater, even for walls with nominal levels of 

transverse reinforcement (Haroun et al., 1994; Abo-Shadi et al., 2000). Therefore, retrofitting of 

pier walls is frequently unnecessary unless inadequate starter bar splices are present within the 

potential plastic hinge zone. When they are present, the wall may be retrofitted by the placement 

of a steel plate over the height of the plastic hinge region of the pier (Haroun et al., 1994). Bolts, 

anchored by nuts and heavy steel and plate washers, are drilled through the pier at relatively 

close spacing to brace the steel plate. The bolts are placed so as not to interfere with the existing 

vertical reinforcing steel, as shown in figure 9-26. 
388

Figure 9-26. Retrofit for wall pier with starter bars. 

9.3. RETROFIT MEASURES FOR CAP BEAMS AND COLUMN-TO-CAP BEAM 

 JOINTS 

The preferred location for plastic hinging is at the ends of columns. In the past, cap beams were 

designed without consideration for earthquake effects, and little if any positive moment 

reinforcement was provided in the cap beam adjacent to an exterior column. Negative moment 

reinforcement at this location may also be inadequate. This will lead to plastic hinging in the pier 

cap, which most probably will have only limited capacity for ductility and retrofitting the beam 

may be necessary. Cap beam retrofitting may also be necessary for unusual pier configurations, 

such as outrigger frames, ‘C’ frames, and double-deck bridges. 

Retrofitting methods used for concrete cap beams must consider both the longitudinal and 

transverse response of the bridge, which is dependent on the configuration of the pier. Several 

typical configurations for multi-column piers are shown in figure 9-27. 

Because cap beams are usually cast integrally with reinforced concrete columns, transverse 

loading will subject the cap beams to flexural and shear stresses. Column-to-cap beam joints 

will also be subjected to large stresses and are vulnerable to damage during an earthquake 

transverse to the bridge. Because existing cap beams and joints were not usually designed to 

behave in a ductile manner, the retrofit design must ensure that these elements are either capable 
389

of accommodating the ductility demands placed on them, or are capable of elastically resisting 

the forces that will result from plastic hinging in the columns. 

The longitudinal response of a pier with bearings, as shown in figure 9-27(a), does not usually 

result in significant cap beam stresses, although some torsion and weak axis bending will occur. 

This is because longitudinal moments at the tops of the columns are small. A similar situation 

exists in a cap beam that is integral with the superstructure when the columns are pinned at the 

top, as indicated in figure 9-27(d). Therefore, retrofit designs for these cap beams will generally 

be expected to resist stresses and strains resulting from transverse response only. An exception 

is for the cap beam of an outrigger pier, for which the longitudinal response of the bridge can 

cause significant weak-axis bending and shear in the beam. 

Longitudinal response is most significant when the superstructure is integral with the pier cap 

and the columns are fixed at the top, as shown in figures 9-27(b) and (c). When the columns are 

located within the width of the superstructure, as in figure 9-27(b), the superstructure itself helps 

to resist the moments and shears generated at the tops of the columns (Priestley, 1993). 

Figure 9-27. Typical cap beam configurations for boxgirder bridges. 
390

Although research has shown that the pier cap can become overstressed and heavily cracked due 

to longitudinal response, a collapse mechanism is seldom formed in the cap and retrofitting will 

usually not be required unless a higher level of seismic performance is desired. However, when 

the pier cap is of an outrigger type, as shown in figure 9-27(c), the torsional and weak axis 

bending stresses in the cap and column-to-cap joint resulting from longitudinal response will be 

large. In this case, it is often necessary to strengthen the cap beam to resist these forces, as well 

as those resulting from transverse loading. 

When cap beams can fail, the bridge is in danger of collapse and retrofitting the cap beams is 

essential. The following sections describe several methods for retrofitting concrete cap beams 

and column-to-cap beam joints. 

9.3.1. PIER CAP REPLACEMENT 

Total or partial pier cap replacement should be considered whenever column replacement is 

being contemplated. 

9.3.1.1. Partial Replacement at a Joint 

Because of the large amount of retrofitting required to make an outrigger cap-to-column joint 

seismically resistant, it is sometimes more practical to simply replace the joint. Design criteria 

are the same as for joints in new bridges. This method was widely used on knee joints in various 

San Francisco viaducts after the 1989 Loma Prieta earthquake. Many of these retrofits involved 

the replacement of existing columns or the construction of supergirders (see section 9.3.4) that 

made joint replacement necessary. These new joints must tie into existing or strengthened pier 

caps, which will require splicing into existing cap beam reinforcement. The remainder of the cap 

may also require strengthening by the methods discussed in section 9.3.2 and illustrated in figure 

9-28. 

9.3.1.2. Total Replacement 

Total cap beam replacement is rare, but was used in San Francisco after the Loma Prieta 

earthquake where the decision to replace, rather than strengthen, was based on savings in time 

and cost. Significant shoring will be necessary if a cap beam is replaced and the bridge kept open 

to traffic. 

9.3.2. PIER CAP STRENGTHENING 

The flexural strength of a cap beam is usually less than that of the columns framing into it. This 

is frequently the case for positive moment in the cap beam (i.e., tension on the lower face) as a 

result of a small amount of bottom reinforcement anchored in the joint region. Negative moment 

capacity may also be insufficient to force plastic hinging into the columns, particularly when the 

top reinforcement is prematurely terminated. Both cases are a consequence of the design 

specifications used for older bridges, which were designed for full dead load but a reduced, or 

no, seismic load. 
391

Figure 9-28. Pier cap strengthening. 

There are at least two approaches to correcting these problems. One is to allow yielding to occur 

in the pier cap, but to make sure that the beam is capable of meeting the ductility demands placed 

upon it. This is often difficult to achieve, and will result in considerable damage to the cap beam 

that may be difficult to repair. It is generally preferable to adopt a second approach, in which the 

cap beam is strengthened in flexure to force plastic hinging into the columns. If necessary, the 

columns can then be retrofitted using one of the methods discussed earlier in this chapter. 

Capacity design principles are used to design the beam retrofit in the second approach. 

In the case of a pier which uses bearings to support the superstructure (figure 9-27(a)), 

retrofitting can be done by adding reinforced or prestressed concrete bolsters to the sides of the 

cap beam, after roughening the interface, as shown in figure 9-29. Bolsters must act compositely 

with the existing bent cap to provide sufficient flexural and shear strength to prevent yielding of 

the cap. Because the column moment is transmitted into the existing cap, a sufficient number of 

dowels must be provided to transfer the moment capacity of the bolster into the existing cap at 

the face of the column. The number of required dowels is determined by calculating the force 
392

New

Concrete

Bolster

Prestressing Tendons

Through-Dowel

hc

Existing

Cap

Beam

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after Priestley et al., 1992 

Figure 9-29. Flexural and shear retrofit of free standing multi-column bents. 

required to transfer the capacity of the bolster reinforcement across the vertical face of the 

existing cap by shear friction. For the bottom reinforcement, the vertical surface available for 

shear transfer may be taken as the lower half of the cap beam, taken between the centerline of the 

column and the midpoint between columns. Similarly, the capacity of top reinforcement in the 

bolster must be transferred across the corresponding upper half of the cap beam. 

Prestressing the bolsters, either alone or in conjunction with additional reinforcing, will increase 

the flexural strength of a pier cap, as shown in figure 9-29. External prestressing may also be 

used, provided it is protected from corrosion. In both cases, the prestressing steel should be 

anchored at the end of the cap beam. It will not be necessary to transfer the capacity of the 

prestressing steel across the vertical interface of the existing cap beam. 

Adding conventional stirrups within the bolsters, as shown in figure 9-29, may also increase the 

shear strength of the cap beam. Prestressing will also increase the shear capacity of the bolsters 

themselves. Shear design of a retrofitted bent cap should follow design procedures required for 

new bridges. 

 

Prestressed bolsters were tested in the laboratory on part of a 30-year-old reinforced concrete 

frame, which had been salvaged from a bridge being demolished in upstate New York (figure 

9-30 and Mander et al., 1996a). This retrofit also included an enlarged column-to-cap beam joint 

to improve the anchorage of existing reinforcing steel. The technique was effective in forcing 

plastic hinging to occur in the columns, which provided limited ductility for the frame as a 

whole. In lower seismic zones, this may be the only retrofitting that is required. Retrofitting 

columns, as discussed earlier in this chapter, could further enhance frame ductility for higher 

seismic zones. 

The use of fiber composite materials has also been considered as a means of increasing the 

flexural and shear strength of a bent cap (Priestley et al., 1996). This method is more effective if 

the material can be wrapped completely around the cap (Pulido et al., 2002). Refer to ACI 

440.2R-02 for guidance. 
393

51 279 51

51

279

51

Existing Cap Beam

D4@102 Stirrup

10-13 mm Diameter

Threaded Bar

(Unbonded)

2-#3 Bar

F = 56 kN / Tendon

Total PS Force = 560 kN

Section C-C

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Figure 9-30. Pier cap retrofit tested at University at Buffalo. 

Enhancing the flexural capacity of cap beams that are integral with the longitudinal girders is 

more difficult because of the geometric constraints presented by the existing girder stems on the 

sides of the cap. Reinforced bolsters may be added to the lower face of the cap beam to increase 

positive moment capacity, but it is more difficult to rectify inadequate negative moment 

capacity. Removing concrete from the top of the beam and adding additional reinforcement can 

increase this capacity, but this will require closure to traffic and most probably shoring of the cap 

beam to prevent failure under gravity loads. External prestressing placed in grouted galvanized 

ducts or conventional prestressing within bolsters placed between beam stems will generally be 

the most economical means for increasing both positive and negative moment capacity, provided 

elastic behavior in the cap can be assured through capacity design principles. This retrofit is 

shown in figure 9-31. 

Replace, Add Rebar

Dowels

External

Prestressing Tendon

Overlay

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Internal

Prestressing Tendon

after Priestley et al., 1992 

Figure 9-31. Flexural and shear retrofit of integral cap beams. 
394

9.3.3. REDUCTION OF PIER CAP FORCES 

It may be possible to reduce bent cap forces by using isolation bearings as force limiting devices, 

in a manner similar to that described earlier for columns. However, if the force reduction is not 

sufficient to prevent column yielding, it will not limit forces in the cap beam. An alternative 

method to alleviate cap beam problems is to construct a ‘link beam’ below the existing pier cap, 

as shown in figure 9-32. 

The link beam is cast around the existing column and creates a new critical section in the column 

just below the link beam. This limits shear forces in the column to the plastic shear achieved in 

the portion of the column below the link beam. If the vertical distance between the link beam 

and existing cap beam is sufficiently small, then the moments in the section of the column 

between these two beams will also be small since they are a function of the plastic shear and the 

distance between the beams. Moment equilibrium at the column-to-cap beam joints then dictates 

that cap beam forces due to seismic actions are limited to relatively small values, and no further 

retrofit is needed. The link beam must be designed according to capacity design principles to 

ensure that plastic hinges form in the column, and not in the link beam. 

Link beams can be very effective in retrofitting tall piers. Judicious choice of the position of the 

link beam will result in protection for the existing cap beams, coupled with a substantial increase 

in lateral strength and stiffness of the bent. The technique can also be used to advantage at 

ground level, by linking columns transversely or longitudinally to alleviate footing problems. 

Existing Cap Beam

Link Beam

F

Critical

Section

V

V

Moment

col

col

after Priestley et al., 1992 

Figure 9-32. Horizontal link beams for pier cap force reduction. 
395

The resulting location for column hinge formation must be checked for adequate ductility 

capacity. Shear forces in the columns will also increase due to the shorter column length below 

the link beam. Column retrofit measures, as described previously in this chapter, will need to be 

considered when the requirements for column ductility or the shear force demands become 

excessive as a result of adding the link beams. 

9.3.4. STRENGTHENING OF COLUMN AND BEAM JOINTS 

Inadequate shear strength is a common problem in column-to-cap beam joints in many older 

bridges. The anchorage of reinforcement within these joints and stresses resulting from torsion 

are also of concern. Techniques developed for the retrofit of several San Francisco double-deck 

viaducts following the 1989 Loma Prieta earthquake may also be applied to bridge piers of the 

types shown in figures 9-27(a) and 9-27(d). Generally, both vertical and horizontal shear 

reinforcement is needed in the joint. The most satisfactory solution will be the complete 

replacement of the joint, as described in section 9.3.1. This also enables deficiencies in the 

anchorage of column and beam reinforcement to be rectified. However, adequate performance 

could also be ensured by reinforced concrete jacketing, which is added to the sides of the joint 

region, as shown in figure 9-33. 

Existing Concrete New Concrete Jacket

Doweled to Existing

Cap Beam

after Priestley et al., 1992 

Figure 9-33. Joint retrofit with external concrete jacket. 

Since the shear resistance at the joint is provided by reinforcement external to the original joint, 

the new concrete jacket must be doweled into the existing concrete to transfer the shear force by 

friction. This is because shears develop within the original joint as a consequence of column 
396

flexural actions, and these must be transferred across the joint by the external reinforcement. 

Referring to figure 9-34, the interface between new and old concrete is simultaneously subjected 

to friction forces in the vertical and horizontal directions. The shear forces are found from 

equilibrium with the flexural forces in the cap beam and column. Thus, in figure 9-34: 

 Vjh = Tb (9-33) 

and b

jv jh

c

h V V

h ≈ ⋅ (9-34) 

after Priestley et al., 1992 

Figure 9-34. Free body diagram of shear forces on a knee joint. 

Assuming that all joint shear resistance is provided by reinforcement in the new concrete on the 

side of the joint, the horizontal and vertical interface shear stresses will be: 

jh

ih

b c

V

v

2h h ≈ (9-35) 

and 

jv

iv

b c

V

v

2h h ≈ (9-36) 

where the two in the denominator is required because 50 percent of the shear is transmitted 

across each of the two interfaces between new and old concrete, one on each side of the joint. 

The maximum interface shear is the vector combination of νih and νiv. That is: 

2 2

i ih iv v vv = + (9-37) 
397

Substituting Vjh and Vjv from equations 9-33 and 9-34 into equations 9-35 and 9-36, respectively, 

and further substituting into equation 9-37 yields: 

2

jh b

i

bc c

V h v 1

2h h h

⎛ ⎞

= + ⎜ ⎟ ⎝ ⎠ (9-38) 

A shear friction clamping stress equal to νi must be provided across the interface, and it follows 

that the dowels must satisfy: 

d y

i 2

A f

v

s

≥ (9-39) 

where Ad is the area of the dowel bar, fy is the yield stress of the dowel, and s is the spacing of 

dowels on a square grid. 

Wherever possible, the jacket should extend over the top of and underneath (at the corners) the 

cap beam, as shown in figure 9-33, to improve the connection to the cap beam. It is also 

recommended that the interface friction stress given by equation 9-38 be limited to: 

νi ≤ 0.2 '

cf ≤ 6.9 MPa (1000 psi) (9-40) 

where '

cf is based on the weaker of the existing and new concrete strengths. Increasing the size 

of the joint overlay will increase the effective bond lengths for existing column and cap beam 

reinforcement within the joint, thus mitigating anchorage problems. 

It is noted that transverse prestressing reduces the need for horizontal shear reinforcement of the 

joint and should also improve the transfer of the vertical shear through the joint. 

Steel jackets bonded to the sides of the joint and anchored to the bent cap are also an effective 

retrofit measure (Thewalt and Stojadinovic, 1995). This technique, which also includes column 

retrofitting to increase pier ductility, is illustrated in figure 9-35. 

Column-to-cap beam joints that are integral with the superstructure are less likely to require 

retrofitting for longitudinal or transverse response. Despite the fact that these joints are subject 

to damage during an earthquake, this damage is unlikely to cause the collapse of the bridge 

(Priestley, 1993). When retrofitting is required to strengthen these joints for longitudinal 

response, transversely prestressed bolsters, as shown in figure 9-36, may be used. 

9.3.5. SUPERGIRDERS 

If an outrigger pier cap is integral with the superstructure, as shown in figure 9-27(c), the 

longitudinal response of the bridge produces flexural actions in the superstructure and torsion in 
398

Figure 9-35. Knee joint retrofit with steel plates. 
399

Existing Cap Beam

Hole Cored Through

Cap Beam

New Concrete Bolster

Prestress Rod

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Figure 9-36. Retrofit of an integral cap beam joint for longitudinal loading. 

the cap beam. These effects will not be significant in short bridges where superstructure 

displacements in the longitudinal direction are limited by the abutments. But in long, viaducttype structures, comprising two or more segments, these effects will be important. In these cases, 

cap beam torsional moments will develop, equal to the column’s longitudinal plastic moment 

capacity. Because of the torsional flexibility of the cap beam, this torsional moment is not 

uniformly distributed to the longitudinal beams in the superstructure. As a consequence, the 

moment demand on the beams closest to the columns will be highest, while beams near the 

bridge centerline will be subjected to very little moment. This effect is accentuated when 

torsional cracking of the cap beam occurs, since the torsional stiffness of a beam, after cracking, 

quickly reduces to about 10 percent of the uncracked value. 

The retrofit of outrigger frames on several San Francisco double-deck viaducts involved the 

addition of an edge beam (termed a supergirder) in the plane of the columns from pier to pier to 

reduce the torsional effect, as shown in figure 9-37. Provided that the edge beam is sufficiently 

stiff and strong, inadequacies of cap beam torsional strength and superstructure flexural capacity 

become largely irrelevant, since satisfactory longitudinal response can be assured even when the 

cap beam has zero torsional capacity. Under these conditions, the design approach is to force 

plastic hinges into the column rather than the supergirder, since plastic hinges in the supergirder 

allow large rotations of the cap beam and can result in degradation of gravity load-carrying 

capacity. If the supergirder is protected against plastic rotation by a capacity design approach, 

torsional rotations of the cap beam will remain small, and will be dictated by rotational 

compatibility with the supergirder. Torsional cracking of the cap beam, if it occurs, reduces the 

torsional moment rather than increasing the torsional rotation. 

Whereas the supergirder plays an important role in the longitudinal response of a long bridge, it 

is largely ineffective in the transverse direction. In this case, the cap beam is required to 

participate and simultaneously support gravity loads by beam action. Deficient cap beams may 

retrofitted using one of the methods described earlier in this section. 
400

Figure 9-37. “Supergirders” added between piers to reduce torsion in pier caps. 
CHAPTER 10: RETROFIT MEASURES FOR ABUTMENTS, 

FOOTINGS, AND FOUNDATIONS 

10.1. GENERAL 

Abutments, footings, and foundations connect a bridge to the earth. They are the means by which 

a bridge feels the effects of an earthquake. Most abutment, foundation and pier failures that 

occur during earthquakes can be attributed to the instability of the supporting soil due to 

liquefaction, lateral spreading, fault movement, or landslide. Very few bridges have collapsed 

due to structural failure of an abutment, piles or footings. Most failures have been soil failures. 

For this reason, and the fact that foundation retrofitting can often be very expensive, the designer 

should think twice before deciding to retrofit a foundation. It often makes more sense to tolerate 

a certain amount of abutment or foundation damage or movement, as long as the overall stability 

of the bridge is not compromised. 

Nevertheless, there are instances when it may be desirable to retrofit abutments and foundations. 

For important bridges, immediate access to a bridge may be critical, and the potential for lateral 

movement of an abutment, or settlement of an abutment fill, may justify retrofitting. In addition, 

the use of restrainers to limit relative displacement at the abutment may result in much larger 

abutment forces than expected when the abutment was designed. Therefore, situations will exist 

in which abutment strengthening should be considered. Stiffening or strengthening an abutment 

can also provide an alternative load path for inertial forces, and thus reduce the load on other 

more critical components. 

Footings that support columns may be structurally unable to resist the forces transmitted from 

those columns. This usually occurs when there is a lack of reinforcement in the top of the 

footing. Structural strengthening of the footing will be necessary to force plastic hinging into the 

column, which will be an important consideration if the columns are also to be retrofitted. 

There are also instances where the capacity of pier foundations should be increased. This could 

be the case when movements of existing footings can result in overall instability of the pier. This 

becomes more likely when the piers are to be strengthened. Instability caused by liquefaction or 

lateral spreading can also occasionally be addressed by providing stronger foundations. 

This chapter presents methods for abutment, footing and foundation retrofitting. Although many 

of these methods were proposed in the early 1980s, most were never implemented until after the 

1989 Loma Prieta earthquake, when substructure retrofitting became commonplace. 

10.2. RETROFIT MEASURES FOR ABUTMENTS 

Retrofit measures depend to a large degree on the type of abutment. For seismic purposes, 

abutments may be divided into two types: seat-type abutments and integral abutments, as shown 

in figure 10-1. 

401
Seat-type Abutments Integral Abutments 

Seat (Stub) Diaphragm Diaphragm 

 (on piles) (on spread 

 footings)

Cantilever Strutted Rigid Frame

Figure 10-1. Abutment types. 

Seat-type abutments are probably the most common form. Even though significant longitudinal 

movement of the superstructure can occur at these abutments, loss of support does not usually 

result from this movement, unless the seat is particularly short or another supporting member, 

such as an adjacent pier, fails. In fact, unrestrained movement is often assumed in new design. 

Strengthening of seat-type abutments in both the longitudinal and transverse directions can, 

however, reduce forces on the piers in short-span bridges, and should be considered as part of an 

overall retrofit strategy. 

Integral or diaphragm abutments are relatively stable during an earthquake. When earthquake 

loads are particularly large, the flexural and shear strength of the diaphragm wall may be an issue 

and strengthening may be necessary. As with seat-type abutments, strengthening should be 

considered as part of an overall retrofit strategy. 

402
Closed abutments with high walls such as cantilever, strutted, or rigid frame abutments, that 

retain deep backfills, are likely to move outwards due to active pressure of the backfill. This can 

result in rotation of the bottom of the wall, causing cantilever and strutted abutment walls to 

become unstable and rigid frame abutments to become overstressed. Supplemental tiebacks may 

be required to prevent this type of failure. In these cases, the overall stability of the backfill, 

acting together with the abutment, must be considered. 

The potential for settlement of the backfill often requires the use of approach slabs. These and 

other retrofit measures for abutments are discussed in the following sections. 

10.2.1. APPROACH SLABS 

Approach slabs, which are sometimes called settlement slabs, are intended to provide continuity 

between the bridge deck and the approach pavement, should the approach fill settle. An 

approach slab should be tied to the abutment to prevent it from pulling away and becoming 

ineffective. To minimize the discontinuity at the abutment, these slabs should have a minimum 

length of 3 m (10 ft). They are designed as simple span, reinforced concrete slabs spanning their 

full length. 

Ties between the slab and the abutment should be capable of resisting the slab dead load, forces 

applied through friction between the slab and the abutment, and forces resulting from the spectral 

acceleration of the bridge in the longitudinal direction. That is: 

 FD = (μ + Sa) DLs (10-1) 

where FD is the design force, μ is the coefficient of friction between soil and concrete (~ 0.6), Sa

is the spectral acceleration for longitudinal response (figure 1-8), and DLs is the dead load of slab 

and any overlaying fill. 

Note that this connection should be free to rotate so that moment will not be transferred to the 

abutment backwall as the approach fill settles. 

At least two types of approach slabs have been used in the past. Figure 10-2 shows the type of 

slab commonly used in California. Its sole function is to provide limited access between the 

approach fill and the bridge abutment. If the anticipated fill settlements are large, consideration 

should be given to lengthening the slab to reduce rotational discontinuity that will occur at each 

end of the slab. Some form of traffic control will be required to construct this retrofit, but staged 

construction can be used to minimize disruption to traffic. Many other states use variations of 

this type. 

403
404

Drill Hole and Grout in Anchor

Concrete Settlement Slab

Anchor Slab to Bridge

Figure 10-2. California style settlement slabs. 

Another type of settlement slab is shown in figure 10-3. Because this slab is buried in the 

approach fill, it will deform and settle with the fill during an earthquake, providing a smooth 

ramp onto the bridge. It will also act as a friction slab and dissipate energy during an earthquake. 

It is therefore reasonable to assume some reduction in longitudinal design forces with these 

slabs, on the order of 10 to 20 percent. However, because these slabs are buried, they will be 

more difficult to construct under traffic than a California slab. 

Remove and Replace

Portion of Backwall

Concrete Settlement

Slab

Existing Reinforcement

Left in place

Chip in Shear Key

Figure 10-3. New Zealand style settlement slabs. 
405

10.2.2. ANCHOR SLABS 

Anchor slabs are used to increase the ability of abutments to carry both longitudinal and 

transverse loads, and are used to resist displacement in either of these directions. They can also 

serve as approach slabs, by providing access to the bridge following an earthquake. Anchor 

slabs are generally used on short, continuous bridges where the superstructure is capable of 

transferring loads away from the piers and into the abutments. 

Two anchor slab designs are shown in figures 10-4 and 10-5. Both of these slabs take advantage 

of the passive resistance of the approach fill soils. The ‘waffle’ slab design in figure 10-4 

provides a number of surfaces for passive resistance, in addition to the lateral resistance provided 

by the cast-in-drilled hole (CIDH) piles. It is recommended that the vertical stems in this design 

be spaced no closer than twice their depth, h, below the bottom of the horizontal slab. Passive 

resistance on the vertical surfaces can be assumed as: 

 pp = 0.37h / 2.44 = 0.15h MPa (10-2a) 

or pp = 7.7h / 8 = 0.96 h ksf (10-2b) 

A standard 0.40 m (16 in) diameter CIDH pile can be assumed to have an ultimate lateral 

capacity of 178 KN (40 kips). 

The CIDH pile anchor slab shown in figure 10-5 uses large diameter piles as the mechanism for 

lateral resistance. The vertical capacity of these piles is of no concern, so they need only to be 

deep enough to provide lateral stability. The stiffness and ultimate lateral capacity of large 

diameter piles is dependent on the type of soil and can be determined using the p-y curve 

approach (Reese and Allen, 1977). The slab acts as a beam on a continuous foundation, and 

computer programs are available to solve for design forces and displacements 

Anchor slabs must be securely connected to the superstructure to be effective, and the 

connections must be strong enough to resist the forces determined from analysis. These forces 

can be substantial, and the bridge must be able to transfer them to the slab. This will often 

require the abutment diaphragm or backwall to be strengthened. 

Because movement through the approach fill is necessary to develop the passive pressures 

needed for lateral resistance, these slabs dissipate a considerable amount of energy. This can be 

accounted for during analysis by increasing the equivalent viscous damping. Damping ratios of 

10 percent can be assumed for systems that include these types of anchor slabs. This in turn 

reduces longitudinal design forces by approximately 20 percent, as compared to those obtained 

from five percent damped design spectra. 

These retrofits will be disruptive to traffic, although staged construction can minimize this 

disruption. Problems can also develop during construction due to the excavation subsiding under 

the waffle slabs or into the holes drilled for CIDH piles. Concrete pours must be made against 

undisturbed soil, which requires excavations with near-vertical faces. In the past, pea gravel was 

used as a fill material behind bridge abutments. When encountered, this material cannot be 
406

Figure 10-4. “Waffle” slabs. 
407

Figure 10-5. Anchor slabs with CIDH piles. 
408

excavated with a vertical edge, which can cause the size of the excavation to become 

unacceptably large unless special precautions are taken to stabilize the fill, or it is removed and 

replaced. 

10.2.3. DIAPHRAGM WALLS 

It may be necessary to strengthen the end diaphragm of an integral abutment to prevent a shear 

or flexural failure. When an abutment is driven into an approach fill during an earthquake, large 

passive pressures on the back of the abutment wall are generated. This can result in structural 

failure of the wall that will, in turn, reduce the passive resistance at the back of the wall. If 

displacements are sufficiently large, support can be lost at the abutment, severely damaging the 

bridge and rendering it unusable. To avoid this type of failure, it may be necessary to strengthen 

the wall. This can be done by constructing a reinforced concrete overlay on the face of the 

abutment wall, as shown in figure 10-6. If the old and new concrete can be made to act as a 

composite member, then the entire section may be considered effective in resisting the forces 

generated by the passive soil pressure. To achieve composite action, the existing concrete 

surface should be roughened and the overlay must be connected to the existing backwall with a 

sufficient number of dowels to provide for shear transfer through friction at the interface between 

the existing and new concrete. In addition, flexural reinforcement in the overlay must be 

continuous or, as a minimum, adequately spliced. 

This retrofit method is very disruptive to traffic, as well as to utilities that may be located on the 

back face of the wall. Staging is often difficult unless shoring is used. For these reasons, this 

method is easier to use when constructed on the front face of the wall, but construction may be 

limited on the front face due to restricted horizontal clearances under the bridge. 

10.2.4. TRANSVERSE ABUTMENT SHEAR KEYS 

Existing bearings and shear keys at seat-type abutments may be incapable of transferring 

transverse inertial forces from the superstructure to the abutments. While loss of support due to 

movement at the abutments may not be an issue, effective transfer of these forces can reduce 

forces on the piers and should be considered as part of an overall retrofit strategy. Two methods 

of strengthening the transverse shear transfer mechanism are shown in figures 10-7 and 10-8. 

Other methods are available. The goal of this approach is to design a transverse shear 

mechanism that remains essentially elastic (i.e., yielding is not permitted). The abutment must, 

itself, be able to resist the forces being transferred. 

10.2.5. TRANSVERSE ABUTMENT ANCHORS 

Other sources of transverse resistance at abutments include: 

• Lateral capacity of piling, 

• Passive soil pressure on wingwalls, and 

• Friction on the bottom of a spread footing. 
409

Figure 10-6. Diaphragm abutment overlay. 

When these mechanisms are inadequate, the ability of the abutment to transfer transverse load 

into the approach embankments can be improved by retrofitting. In addition to the anchor slabs 

discussed in section 10.2.2, other methods are available as described in the following sections. 

In all cases, the displacement compatibility of the resisting mechanisms must be accounted for 

during design. 

10.2.5.1. Soil Shear Keys 

Soil shear keys are essentially short supplemental walls that project from the back face of the 

abutment wall into the fill, and resist load through passive soil pressure. An example is shown in 

figure 10-9. Construction of this type of retrofit can disrupt traffic and utilities, and the 

precautions discussed earlier should be applied here. To be fully effective, the minimum clear 

spacing between soil shear keys should be approximately equal to twice the distance they project 

into the fill. The passive pressure that can be developed is given by equation 10-2. When 

relying on passive pressure in the longitudinal direction, it is advisable to limit the resulting 

pressure to the value given by these equations. 
410

Figure 10-7. Concrete block abutment shear key. 
411

Figure 10-8. Pipe shear key at abutment. 

These keys can be designed for limited ductility demands. An R-factor of two is acceptable for 

flexural design of the key, based on cantilever bending. The shear capacity should be designed 

for full passive pressure, to allow for the possibility of flexural overstrength and deviation from 

pure cantilever bending. The connection to the abutment wall should be capable of resisting the 

overstrength moment and shear capacity of the wall at this location. These keys will also 

introduce transverse bending moments and forces into the abutment wall, which may then 

require that the wall be strengthened. 

10.2.5.2. Wingwall Strengthening 

When the structural capacity of a wingwall limits its contribution to the transverse resistance of 

the abutment, structural strengthening of the wingwall is an option. This may be accomplished 
412

Figure 10-9. Abutment seismic shear walls. 

by constructing composite concrete overlays on the wingwall, as shown in figure 10-10. The 

design criteria in this case are similar to those for supplementary shear keys discussed above. 

Another method of increasing wingwall capacity that can be used in conjunction with wingwall 

strengthening is to place cross-ties between two wingwalls, as shown in figure 10-11. The crossties will help redistribute forces in the wingwalls, resulting in increased resistance to transverse 

loads. This method will cause disruption to traffic and utilities during construction. 
413

Figure 10-10. Wingwall overlay. 
414

Figure 10-11. Wingwall cross-tie. 

10.2.5.3. Cast-In-Drilled Hole Piles 

A popular method for increasing the transverse capacity of abutments is to use supplemental 

large diameter CIDH piles at the edges of the abutment. These piles are usually designed to 

carry lateral load only, and their design therefore does not need to consider compression or uplift 

capacity. Lateral capacity can be determined using the p-y method discussed in section 6.2.2.2. 

Because these piles are more effective for lateral loads when they have a fixed-head condition, it 

is often desirable to provide a moment-resistant connection at the abutment. This will not always 

be practical if the bridge must be free to accommodate large longitudinal temperature 

movements at the abutment. In the case of seat-type abutments, an extension of these piles can 

also be used as a transverse shear key. An example of this type of retrofit is shown in figure 

10-12. 

10.2.6. SOIL AND GRAVITY ANCHORS 

Large horizontal displacements at the abutments may make the bridge inaccessible or, in some 

cases, result in bridge instability. Abutment displacements can be minimized by adding soil or 

gravity anchors. Various types of soil anchors may be used to transfer longitudinal loads into the 

approach embankments. These can resist inertial forces from the superstructure or they can 

stabilize the abutments and prevent rotation or sliding. 

Various types of soil anchors have been used to retrofit abutments. Because the backfill may 

move during an earthquake, the anchors should extend a sufficient distance into the backfill to 

avoid being affected by this movement. 
415

Figure 10-12. CIDH pile shear keys. 

Gravity anchors, as shown in figure 10-13, consist of tie rods running between the superstructure 

or abutment and a ‘dead man,’ which may be a large diameter CIDH pile or a heavy beam buried 

in a trench some distance behind the abutment. This method is intended to provide longitudinal 

resistance against movement toward the mid-span of the bridge. Passive soil pressure and 

friction mechanisms on the dead man provide this resistance. The tie rods are usually embedded 

in a concrete trench to protect them from corrosion or other damage. 

Soil anchors have been used for many years to stabilize retaining walls. Many of these systems 

are proprietary and require licenses for their use. Typically, a hole that is slightly inclined to the 

horizontal is drilled into the soil embankment. Prestressing rods or strands are placed in the hole 

and the hole is pressure-grouted. Anchorage for the strand is usually provided by a mechanical 

end-plate or toggle. Soil anchorage is provided through friction with the soil or formation of a 

grout bulb at the end of the hole. A portion of the steel rod is left unbonded so it can be 

prestressed. When it is stressed, it provides a test of the reliability of the anchor. If acceptable, 
416

Figure 10-13. Gravity or “Deadman” anchors. 

the steel is locked off at the face of the wall. The entire length of prestressing steel is then 

grouted to provide protection against corrosion. An abutment retrofit using soil anchors is 

shown in figure 10-14. 

A modified friction slab anchor, as shown in figure 10-15, can be used in embankments with 

granular soils where little or no settlement is expected. In this case, flexible steel piles that can 

resist uplift are used to prevent vertical movement at the end of the slab. Because the slab is 

sloped at approximately 15 degrees to the horizontal, the soil under the slab is compressed as the 

wall is pulled toward the bridge. This generates friction between the slab and the soil, as well as 

increasing the shear strength of the soil under the slab, stabilizing it against failure. The slab, 

piles, and pile anchorages must be designed to resist the increased soil bearing forces. This 

method allows large tieback forces to be generated, resulting in tiebacks that are capable of 

resisting significant inertia forces from the superstructure. 
417

Figure 10-14. Abutment retrofit using soil anchors. 

Figure 10-15. Modified abutment friction anchor. 
418

The ultimate capacity of each of the above anchor types should be greater than or equal to the 

inertial forces transferred to the abutment from the superstructure, or the earth pressures acting 

against the abutment wall during the design earthquake. 

10.3. RETROFIT MEASURES FOR FOOTINGS 

This section discusses various approaches for retrofitting bridge footings. The approaches 

address structural deficiencies in footings as well as global deficiencies in bridge foundation 

capacity. As noted earlier, footing and foundation retrofitting is expensive, and a collapse 

resulting from structural failure in a footing or foundation is relatively rare. The cost of 

retrofitting footings must therefore be weighed against the consequences of not doing so. It is 

sometimes advisable to accept the risk of damage, unless there is a potential for loss of life. 

10.3.1. FOOTING REPLACEMENT 

Total footing replacement is an extreme measure, but was used on a number of bridges in San 

Francisco following the 1989 Loma Prieta earthquake. This type of retrofitting is usually 

necessary when columns are replaced, existing footings are inadequate and strengthening of the 

footing is not feasible due, in part, to the connections required between the new columns and the 

footing. 

The structure must be supported during reconstruction of the foundation. This is further 

complicated if the bridge must continue to carry traffic during retrofitting. It is often necessary 

to drive new piles in this situation and if so, they must be driven with limited headroom under 

the existing superstructure and in the proximity of temporary supports. A discussion of piles 

used in bridge retrofitting is given in section 10.4.1. 

10.3.2. STRENGTHENING OF FOOTINGS 

Rocking of a spread footing or piled footing is not necessarily an undesirable response. If 

neither the column nor footing are ductile, rocking of the footing may provide a form of isolation 

that can prevent structural damage to the bridge. However, if rocking displacements are 

expected to be large or if failure of the footing is likely, retrofit will be necessary. 

Overturning resistance may be improved by increasing the footing area, adding more piles 

(generally in conjunction with an increase in plan area), or by adding supplemental soil or rock 

anchors. The existing tensile capacity of the piles is often inadequate because of poor 

connections between the piles and footings, and the lack of continuous tensile reinforcement in 

the piles. Even if the pile’s tensile capacity is adequate, the mobilization of this capacity will 

require vertical displacement of the pile and thus rocking of the footing. The designer should 

consider the rocking flexibility of the footing when performing seismic vulnerability analysis of 

the bridge, in order to assess the stability of the structure under seismic loads. 

An increase in footing area is usually required if the stability of the footing itself is an issue. 

This is always accompanied by structural strengthening of the footing (figure 10-16). 
419

Figure 10-16. Global footing strengthening. 

 
420

In many cases, the overall size of the footing is adequate, but the footings fail before the pier can 

develop its full plastic capacity. This is most often due to the absence of a top layer of 

reinforcement and vertical ties in existing footings. As a consequence, these footings have little 

negative moment capacity and diminished shear strength for resisting uplift forces. During an 

earthquake, this can result in flexural cracking in the footing or shear failure in the column-tofooting joint. There may also be a loss of anchorage for the longitudinal reinforcement in the 

column as a result of the damage to the footing. This failure mode results in a degradation of 

moment capacity, effectively creating a pinned condition at the column-to-footing connection. 

This can be a serious problem in single-column piers and is more likely to occur if the column is 

on a pile footing. These footings may be retrofitted with a concrete overlay, as shown in figure 

10-17. 

Although a concrete overlay can be easily designed to provide increased negative moment 

capacity in the footing, it is very difficult to add positive moment reinforcement. It is therefore 

desirable to use the existing positive moment reinforcing and rely on the added footing depth, 

provided by the overlay, to increase the positive moment capacity. If this is insufficient and the 

size of the footing must be increased, it will be necessary to splice the existing positive moment 

steel as illustrated in figure 10-18. 

Alternatively, prestressing may be used to increase the positive moment capacity. This is done 

by drilling horizontal holes through the footings and adding new prestressing bars, as shown in 

figure 10-19. It should be noted that many existing footings were cast against undisturbed soil. 

Therefore, it is likely that footings will not have planar surfaces and will not conform exactly to 

the as-built plans. In this case, new anchor blocks will be required to accommodate the 

prestressing. 

10.3.3. RETROFIT DESIGN OF FOOTINGS 

Footing retrofits must be designed to prevent flexural, shear, and joint shear failure within the 

footing. Retrofit design is therefore similar in many ways to the design of new footings. Since it 

is often necessary to rely on resistance mechanisms that vary from those of new footings, the use 

of strut-and-tie models and yield line theory may be employed in addition to conventional beam 

theory. There are seven steps in the design of a footing retrofit: 

Step 1. Determine footing design forces – Forces (moments and shears) are applied to the 

footing through the column. Unless the footing is assumed to be rock, these forces will 

be limited to the flexural capacity of the column at the level of applied axial load. 

Axial loads should include dead load and any additional axial load resulting from 

seismic response. The potential for flexural overstrength in the column must be 

considered. Increased moment capacity can result from material properties that are 

greater than nominal, confinement of the concrete, and strain hardening of the 

reinforcing steel. One method for determining moment overstrength capacity is to 

calculate the nominal moment capacity based on actual or estimated material strengths, 

and to apply a factor of 1.4 to account for the various overstrength factors. Therefore, 

the column overstrength is given by: 
421

Figure 10-17. Footing retrofit with concrete overlay. 
422

 Mp = 1.4 Mn (10-3) 

Alternatively, a moment curvature analysis that accounts for concrete confinement and 

steel strain hardening can be used. If expected material strengths are used, the moment 

at the design displacement is increased by 15 percent to provide the overstrength 

moment. The moment applied to the footing need not be more that the unreduced 

moment obtained from a dynamic analysis of the bridge subjected to the design 

earthquake. 

Step 2. Check the footing’s overturning capacity – The overturning capacity of a spread 

footing is based on the ultimate bearing capacity of the soil, and on the ultimate tension 

and compression capacity of the piles in the case of a pile footing. If this capacity is 

exceeded, the stability of the footing, using a rocking analysis, should be checked. If 

the footing is stable against overturning and unacceptable permanent displacements in 

the footing can be avoided, it may not be necessary to increase the size of the footing, 

nor add additional piles or tie-down anchors. Rocking action may also reduce the 

forces in the column and the footing, eliminating or reducing the need for retrofitting 

these members. 

Figure 10-18. Bottom reinforcing splice detail. 
423

Figure 10-19. Footing strengthening by horizontal prestressing. 

If an increase in footing size or the number of piles is necessary, follow conventional 

footing design procedures. Existing piles, which are carrying the dead load, may reach 

their ultimate capacity before the new piles. If this occurs, there may still be some 

capacity remaining in the footing as a whole. Therefore, existing piles can exceed their 

ultimate capacity as long as there is no excessive movement. If excessive movement 

can occur, the capacity of these piles should be ignored or limited. The same reasoning 

can be applied to the soil under spread footings. 

Step 3. Design for shear – Shear strength deficiencies in footings are difficult to retrofit. In 

many existing footings, the shear between the column compressive stress resultant and 

the foundation or pile reaction may be carried by a diagonal compression strut. This, 

however, requires a dependable tie force in the bottom steel, which can be relied upon 

only if the bottom steel is anchored by a 90 degree hook, as shown in figure 10-20(a). 

If the compression strut cannot be developed because either the bottom tensile 

reinforcement is inadequately anchored or the reinforcement has insufficient strength, 

the shear strength can be increased by: 

1. Increasing the footing depth through the addition of a concrete overlay made 

composite with the existing footing, which will increase the shear capacity of the 

concrete. 
424

after Priestley et al., 1992 

Figure 10-20. Footing shear strength retrofitting. 

2. Adding prestressing bars placed in holes drilled longitudinally through the footing, 

as recommended for increasing flexural strength, or 

3. Placing additional reinforcement in vertical holes drilled into the footing, to act as 

additional shear reinforcement. Unless the bars pass through the footing and are 

anchored beyond the bottom layer of reinforcement, they will not be as effective as 

conventionally anchored shear reinforcement. Access to the underside of the 

footing is difficult, but is required to properly anchor this reinforcement. Vertical 

reinforcement in the overlay will be effective in resisting shear if there are two 

layers of reinforcement in the overlay. Vertical dowels that are anchored in drilled 

holes extending to the bottom layer of reinforcement but are not hooked around this 

reinforcement will have reduced effectiveness. In this case, it is recommended that 

they be considered to have 50 percent of the effectiveness of properly anchored 

stirrups (Priestley et al., 1996). 

Note that the addition of shear reinforcement in a newly widened portion of a footing, 

as shown in figure 10-20(b), is unlikely to be effective because of its distance from the 

source of the shear force, which is primarily the resultant of the column’s compression 

forces. 

Step 4. Design for flexure – Even those footings with adequate tensile connections to piles 

will frequently be deficient in flexural strength, due to a lack of top reinforcing steel. 

Bottom steel may also be inadequate, particularly in wide footings where the 

reinforcing steel furthest from the column is likely to be ineffective. 

Retrofitting to increase flexural strength may involve the addition of a reinforced 

concrete overlay connected by dowels to the existing footing, as shown in figures 10-16 
425

and 10-17. The top reinforcement should be located so that more than half of it is 

within a distance of hf from the column faces, where hf is the depth of the retrofitted 

footing. 

Increasing the depth of the footing will also increase the positive moment capacity due 

to the increased section depth. If this is still insufficient, horizontal prestressing, as 

noted earlier, can improve the positive moment capacity. This may be used alone or in 

conjunction with an overlay. Prestressing ducts should be kept as close as possible to 

the column. 

Step 5. Design dowels for connecting existing and new concrete – Dowels between new and 

existing concrete should be capable of transferring the shear stress at the interface 

between existing and new concrete, using a shear friction approach and a coefficient of 

friction, μ, of 1.0. This assumes that the surface of the existing footing has been 

adequately roughened prior to casting the new concrete. The thickness of the overlay 

should be sufficient to develop the tensile capacity of the dowels, which are often 

hooked at the top end. If the dowels are also to be used to improve shear strength, they 

should pass through the full depth of the existing footing and be properly anchored at 

the bottom. 

Step 6. Check column-to-footing joint shear – Experimental studies (Xiao et al., 1994; 

McLean et al., 1995) have shown that a principal mode of failure in existing footings is 

due to joint shear. Figure 10-21 presents a diagram illustrating the forces that generate 

joint shear. The vertical joint shear force, Vjv, can be calculated by subtracting the 

tensile forces in the piles, Rt, from the total tensile force in the column reinforcement, 

Tc, 

i.e., Vjv = Tc – Rt (10-4) 

M

V

T C

C

R

T

R

V

V

c c

c c

c

j

jv

h

f

f

t

P

Figure 10-21. Footing joint shear forces. 
426

The joint shear stress, vjv, can be calculated from: 

jv

jv

eff f

V

v

b d = (10-5) 

where beff is the Bc + Dc for rectangular columns, Bc is the column width, Dc is the 

column depth, beff is 1.414 D for circular columns, D is the column diameter, and df is 

the depth of the retrofitted footing. 

The effective axial compressive stress within the joint in the vertical direction is given 

by: 

 fa = P / Aeff (10-6) 

where: 

P = axial load (i.e., dead load + earthquake) at the base of the column, and 

Aeff = effective bearing area: (Bc + df)(Dc + df) for rectangular columns, and 

π (D + df)

2

 / 4 for circular columns. 

Thus, the principle tensile stress in the joint region is given by: 

( )2 2

t a a jv f f2 f2 v =− + + (10-7) 

If ' '

tc c f 0.42 f MPa ( 5 f psi) < , no further design is required. Otherwise, proceed to 

the next step. 

Step 7. Check the capacity of the footing overlay - When ' '

tc c f 0.42 f MPa (5 f psi) ≥ , the 

ability of the overlay to participate in the resistance of joint shear stresses must be 

checked. The strut and tie model shown in figure 10-22 illustrates how the anchorage 

forces in the column tensile reinforcement are resisted. Two compression struts, one 

acting toward the interior of the column, Cin, and the other acting toward the exterior of 

the column, Cex, act at the midpoint of the anchorage length, yb, of the column 

reinforcement. The vertical component of Cin is balanced by the column compression 

block, but Cex must be resisted by the footing overlay. The horizontal component of 

Cex is resisted by membrane action, which stresses the horizontal reinforcement in the 

overlay. In addition, the overlay and the existing footing must be capable of resisting 

the vertical component of Cex perpendicular to the overlay, as given by: 

 Pv = cos (45) Cex = 0.5Tc (10-8) 
427

Column C.L.

z=y x

T C

V

e e

e e

e

e

45 45 o o

y =y /2 b

by

Cex Cin

Figure 10-22. Strut and tie model for joint design. 

The effective depth of an assumed concrete plate will be equal to the depth of the 

overlay plus the embedment of the dowels (i.e., this assumes a partial delamination 

plane at the bottom of the dowels). This results in both negative and positive bending 

forces in the assumed plate that is best resisted by two layers of horizontal reinforcing 

steel in the overlay. The ultimate capacity of the assumed plate to resist Pv can be 

determined using yield line theory. The computer program FT-YIELD, which employs 

a virtual work method, was developed for this purpose (Xiao et al., 1994). The 

assumed yield line pattern for a typical footing is shown in figure 10-23(a). If the plate 

assumes the displaced shape shown in figure 10-23(b), the virtual work done by Pv

must be balanced by virtual work done by plate bending forces along the assumed plate 

fold lines. Increasing the thickness of the overlay, increasing the dowel embedment 

depth, or adding additional overlay reinforcement can increase the flexural capacity of 

the assumed plate. 

However, the simplest approach is to increase the overlay thickness until vjv is less than 

' '

c c 0.42 f MPa (5 f psi) , unless there is limited space available for doing this. 

An example of a retrofit design of a bridge footing is presented in example 10-1. 
428

EXAMPLE 10.1: RETROFIT DESIGN OF A FOOTING 

The footing shown in the figure below supports a 1.22 m (4 ft) diameter reinforced concrete column with 

21 # 32 (metric) (#10 cus) reinforcing bars grade 40. The column is fixed for moment into the footing. The 

footing does not have a top layer of reinforcement but it is supported on nine 0.381 m (1.25 ft) diameter 

concrete piles that are adequately connected to the footing to resist uplift forces. 

Evaluate the capacity of the footing and design an appropriate overlay retrofit. 

21 #32 Bars

.150 m (6 in) clr

0.623 MN (70 ton)

Concrete Piles

11 #32 Bars Each Way

1.07 m (3.5 ft)

.45 (1.5 ft) .45 (1.5 ft)

.45 m (1.5 ft)

.45 m (1.5 ft)

1.22 m 1.22 m

1.22 m (4 ft)

1.22 m (4 ft)

Elevation Plan

1.22 m (4 ft) Column

(4 ft) (4 ft)

Assume the following data: 

PDL = 3.56 MN (800 k) 

PLL = 1.56 MN (350 k) 

f 'c = 35 MPa (5 ksi) (assumed existing strength and strength of new concrete) 

Ppile = 0.623 MN (70 tons) (design capacity) 

Pultc = 1.25 MN (140 tons) (pile ultimate capacity in compression) 

Pultt = 0.623 MN (70 tons) (pile ultimate capacity in tension) 

STEP 1. DESIGN COLUMN MOMENTS USING CAPACITY DESIGN 

Mn = 4.3 MN-m (nominal moment capacity at dead load axial force, from interaction diagram in 

handbook, etc.) 

Mp = 1.4 Mn = 1.4 (4.3 MN-m) = 6.02 MN-m (4437 k-ft) 

STEP 2. EVALUATE FOOTING CAPACITY 

Npiles = 9 

⋅ = ==

2 22

piles

2 x #piles /row x y 2(3) 1.22 m S 7.32 m

c 1.22 m (24.02 ft) 

DL p

comp 3

piles piles

P M 3.56 MN 6.02 MN/m P 1.22 MN/pile 1.25 MN (140 tons) OK NS 9 7.32 m

=+= + = < ⇒
429

DL p

ten 3

piles piles

ten ultt

P M 3.56 MN 6.02 MN/m P 0.427 MN tension

NS 9 7.32 m

P P

0.43 MN (48.3 tons) 0.623 MN (70 tons) OK

= − = − =−

<

> − ⇒

Therefore, the existing piles are adequate. 

STEP 3. CHECK SHEAR CAPACITY OF EXISTING FOOTING 

( )

u piles

u 2

'

c c

c u

V 3 (1.22 MN/pile) v 1.19 MPa (173 psi) bd 3.34 0.915 m

V 0.168 f 0.85 (0.168) 35 MPa 0.845 MPa (123 psi)

V V

0.845 MPa (123 psi) 1.193 MPa (173 psi) NG i.e., not satisfactory

== = ⋅

=φ = =

<

= < ⇒

Increase the depth of footing 0.305 m (1.0 ft) and add #16 (metric) stirrups at 0.305 m (12 in) spacing to 

accommodate the shear demand. Stirrups are assumed to be only 50 percent effective due to lack of 

proper anchorage. 

vu = 3piles (1.22 MN/pile) = 3.66 MN (822 k) 

Vu = vc bd = 0.845 MPa (3.34 m) 1.22 m = 3.44 MN (778 k) 

s 0.305 m =

2

s by b

d 1.22 m V 0.5 A F N 0.85(0.5) 0.0002 m (414) MPa (11 bars) 1.55 MN (348 k) s 0.305 m =φ ⋅ = ⋅ =

V V 3.46 MN 1.55 MN 4.99 MN (1122 k) 3.66 MN (822 k) OK c s += + = > ⇒

Therefore, #16 stirrups @ 0.305 m (12 in) spacing are more than adequate for vertical shear. 

STEP 4. CHECK MOMENT CAPACITY AND DESIGN TOP REINFORCEMENT FOR PILES IN 

TENSION 

Distance from equivalent face of column to pile centerline. 

Xpile = D – 0.5 (Ag)

0.5

2

pile

(0.61 ) x 1.22 1.22 0.54 0.68 m (2.23 ft) 2π =− =−=
430

( ) ( ) T ten p pile piles m

0.43 MN M P MN N x 3 0.68 0.88 MN/m (649 kft) pile == =

Try 8 #19 (metric) bars. 

( ) = s y

'

c

A f

a

.85 f b

( ) == =

2 2

b by

'

c

N bars A m F MPa 8 (0.000283 m ) 414 MPa a 0.0094 m (0.37 in)

0.85f MPa b 0.85 (35 MPa) 3.34 m

u sy

a M Af d

2

⎛ ⎞ =φ − ⎜ ⎟ ⎝ ⎠

u b by

2

a M N A F (d ) 2

0.0094 0.9 (8bars) 0.000283 m (414 MPa) 1.287 2

 1.08 MN/m (796 kft) 0.88 MN/m OK

=φ −

⎛ ⎞ = − ⎜ ⎟ ⎝ ⎠

= > ⇒

M M OK u t >

Check moment on compression side (#32 @ 0.3 m [1 ft] spacing) 

M P N x 1.22 (3) 0.68 2.49 MN/m (1835 kft) C comp p pile = = ⋅=

( ) == = s y

'

c

A f 11 x .000818 x 276 a .025

.85 f b 3.34 x 35 x .85

u b by

a .025 M N A F (d ) 0.9 11 0.000818 276 (1.22 ) 2 2

 2.70 MN/m (1990 k/ft) 2.49 MN/m OK

=φ − = ⋅ ⋅ ⋅ ⋅ −

= > ⇒

M M OK u c >

STEP 5. CALCUATE HORIZONTAL SHEAR AT INTERFACE OF OVERLAY AND EXISTING FOOTING 

Area of new concrete in footing 

A = d (b) ol

⎛ ⎞ == ⋅ − = ⎜ ⎟ ⎝ ⎠

3 3

Q

1.375 m 0.305 m Q Ad 0.305 m 3.34 m 0.54 m (19.3 ft ) 2 2
431

⋅ == =

3 3 3 bh 3.354 m 1.372 m 4 4 I 0.528 m (83.6 ft ) 12 12

( )

⋅ == ⋅ =

3 u

h 4

V Q 3.44 MN 0.54 m 1 v 1.06 MPa (154 psi) l b 0.528 m 3.34 m

φψ ⋅ ⋅ == = s y

sf 2 2

A F 0.85(1.0) 0.0002 414 v 0.757 MPa (110 psi) NG

s 0.305

V V NG sf h <

Try #16 @ 0.250 m Vsf (0.82 ft) spacing (#16 and 35 MPa [5000 psi] concrete required to achieve 

anchorage in 0.3 m [1 ft] overlay) 

φψ ⋅ ⋅ == =

2

s y

sf 2

A F (0.85) 1.0(0.000283 m ) 414 MPa v 1.126 MPa (135 psi) OK

s 0.250

Vsf > Vh OK 

STEP 6. CHECK JOINT SHEAR 

( ) ⋅ + ( ) =π + = =

2

2 2 2

eff f

3.1416 1.22 m 1.22 m

A D d 4 4.68 m (50.3 ft ) 4 (Eq. 10-6) 

 

b D 2 1.414(1.22) 1.73 m (5.67 ft) eff == =

T Af col s y =

Assume grade 60 steel; therefore fy = .414 MPa 

2 T 15 bars (0.000818 m bar) 414 MPa 5.08 MN (1142 k) col = ⋅= ⇒ if approximately 15 bars in tension 

V T N P 5.08 3 0.43 3.79 MN (852 k) jv col piles ten = − = −⋅ = (Eq. 10-4) 

V joint shear jv = 

== = jv

jv

eff f

V 3.79 MN v 1.80 MPa (261 psi) b d 1.73 m (1.22 m) (Eq. 10-5) 

== = DL

a 2

eff

P 3.56 MN f 0.76 MPa (110 psi) A 4.68 m

=− + + =− + + = ( )

2

2 2 2

t a a jv

0.76 MPa (0.76 MPa) f f 2 f 2 v (1.80 MPa) 1.46 MPa (212 psi) 2 2 (Eq. 10-7) 
432

( ) ' F 0.42 f 0.42 35 2.48 MPa (360 psi) 1.46 OK t c = = ⋅= > ⇒

Therefore, check of overlay capacity by yield line theory is not required. The following is a sketch of the 

footing retrofit. 
433

Loading Direction

Free Edges

Load Point No

Displacement

C D

B A

E H

F G

8.61 m

8.61 m

23 m

23 m

5.83 m

7.15 m 8.7 m 7.15 m

4.27 m

No

Displacement

C D

B A

E

E

P

F

F

G

H v

o

o

Figure 10-23. Yield line analysis of footing overlay. 

10.3.4. LIMITING FORCES TRANSMITTED TO THE FOOTINGS 

Link beams similar to those used to reduce moments in pier caps, as described in section 9.3.3, 

can be used to reduce the bending forces transmitted to the footings. Link beams are placed near 

the footing and can be constructed below grade against undisturbed soil. The design objective is 

to force a plastic hinge in the column just above the link beam. This limits the shear in the 

column below the link beam to the plastic shear that is generated in the column above the link 

beam. This, in turn, reduces the moment in the column below the link beam, which limits the 

moment transmitted to the foundation, and could therefore eliminate the need to retrofit the 

foundations. An example of such a retrofit is shown in figure 10-24. 

10.4. RETROFIT MEASURES FOR PILES AND PILE-TO-FOOTING CONNECTIONS 

During an earthquake, piles must carry compression, tension, and lateral loads. If they are 

inadequate for these loads, retrofitting is required. The most common method is to use 

supplemental piles. 
434

Figure 10-24. Footing link beam retrofit. 

When piles are located in competent soils, their ultimate compression capacity is between two 

and four times the pile capacity based on allowable stress. If the soils are not competent, such as 

in a soft clay or sand susceptible to liquefaction, pile capacity will be limited by the structural 

capacity of the pile acting as a column. This may require supplemental piles that are stiffer and 

less likely to buckle. Supplemental piles will also limit the lateral displacements of the pile cap, 

reduce P-∆ effects, and increase the effectiveness of the existing piles. 

The tensile capacity of piles is often limited by their connection to the footing. Timber piles 

have little or no capacity for resisting uplift forces. Standard connections for steel H piles are 

also inadequate for large tensile forces. The longitudinal reinforcing steel in reinforced concrete 

piles may be terminated too close to the top of the pile, resulting in insufficient tensile capacity 

of the pile. In each of these cases, and in others where pile uplift capacity is absent, it may be 

desirable to add hold-down anchors or supplemental tension piles. 

The lateral capacity of piles in competent soil is generally not a problem. Such is not the case if 

lateral loads on the pile are exceptionally high or if the soil cannot provide adequate lateral 

support. In this case, supplemental piles with high lateral resistance can be added. 

The following sections discuss considerations for the addition of piles as a seismic retrofitting 

approach, and details for seismic hold-down anchors. 
435

10.4.1. PILE TYPE CONSIDERATIONS (NEW PILES) 

It is often necessary to drive new piles when footing strengthening or replacement is required. 

Frequently, these piles must be driven under the existing superstructure, where there will be 

limited headroom. This requires that the piles be driven in short lengths and spliced at frequent 

intervals, effectively eliminating the use of precast concrete or timber piles. Steel H piles may be 

too flexible for some retrofitting projects, and they require complete butt-welding between 

sections to obtain moment continuity. Two types of piling that have received increased use for 

retrofitting are discussed below. 

Steel pipe piles, often constructed with a cast-in-place concrete core, have been popular for 

retrofitting because they are relatively easily spliced with complete penetration butt welds. 

These piles come in 400-mm and 600-mm (16-in and 24-in) diameters and are stiffer than a 

steel H-pile of equal cross-sectional area. This makes them more efficient at resisting lateral 

loads, particularly if the surface soil is relatively weak. These shells can be driven with a closed 

bottom, or they can be driven with an open end and drilled out afterwards. The shells are usually 

filled with concrete and a reinforcing bar cage long enough to resist pile bending and axial 

forces. This allows the cage to be anchored at its top in new footing concrete, thus developing 

significant resistance to pile pullout. Drilling out the inside of steel shells results in loss of some 

skin friction. 

CIDH piles with diameters between 400 mm and 2500 mm (16 in and 96 in) are also popular 

where site conditions allow. Equipment exists to allow small diameter piles of this type to be 

drilled with relatively little headroom. Large diameter shafts have considerable stiffness and are 

used where large axial or lateral loads must be developed in a relatively small horizontal area. 

CIDH piles are easiest to construct if the hole is dry and not subject to caving in. If this is not 

the case, methods are available that will allow these piles to be installed under wet conditions or 

when cave-in is likely. These piles also have a reinforcing steel cage that is deep enough to 

resist axial forces and bending moments. If the soil is weak, the cage must extend deeper, which 

may require splicing of the longitudinal reinforcing when there is not enough headroom. The 

cage also provides a strong connection between the top of the pile and the new footing. 

10.4.2. PILE TIE-DOWNS 

For piles in tension (uplift), tie-downs can be added to an existing footing if overturning of the 

footing can occur. Several pile tie-down details have been developed and tested, such as that 

shown in figure 10-25. To be effective in providing sufficient tensile stiffness, tie-down anchors 

are usually prestressed, which also acts to proof-load the tie-down. The amount of prestressing 

is limited, however, by the compressive capacity of the remaining piles. This may allow 

significant rotation of the footing to take place before tie-downs are fully effective. These 

rotations must be checked against acceptable limits because large footing rotations can aggravate 

P-∆ effects in tall columns. For this reason, the use of tie-downs is often limited to footings with 

relatively short columns. 
436

Figure 10-25. Prestressed tie-down anchor. 
437

CHAPTER 11: RETROFIT MEASURES FOR BRIDGES ON 

HAZARDOUS SITES 

11.1. GENERAL 

A hazardous site is one that has the potential for extreme permanent displacement during an 

earthquake, leading to large forces and/or differential displacements in the structural members of 

a bridge. Such sites include bridges that cross or are immediately adjacent to: 

 

• Active faults. 

• Steep, unstable slopes. 

• Liquefiable sands or silty sands. 

This chapter discusses issues and approaches to be considered when retrofitting a bridge on such 

a site. 

11.2. BRIDGES ACROSS OR NEAR ACTIVE FAULTS 

Bridges crossing or immediately adjacent to active faults may be subjected to large differential 

displacements between adjacent piers and/or abutments due to surface faulting. Although the 

probability of this occurring at a given location during the life of a bridge will be low, the 

possibility should be considered when choosing retrofit measures for the bridge. 

 

Methods for determining the likelihood of surface fault rupture are discussed in chapter 3. A 

conservative design approach should be adopted if surface faulting is possible. For example, 

adding extra confinement in the plastic hinge zones of the substructure might be used to provide 

the maximum displacement capacity. 

 

If a bridge is a series of simply supported spans, the relative merits of making the structure 

continuous should be carefully evaluated. This may require increases in the section modulus at 

the ends of the simple beams due to the creation of negative moments. On the other hand, simple 

spans can tolerate large relative movements, but it will be difficult to ensure that the spans do not 

become unseated. To minimize this risk, very generous support lengths should be provided. 

The additional redundancy in continuous superstructures that are integral with their substructures 

will reduce the probability of total collapse. There is, however, a practical limit to the amount of 

relative displacement across a fault that can be accommodated in a monolithic structure. One 

alternative is to support a continuous superstructure on elastomeric bearings at each pier and 

abutment. These bearings can accommodate relatively large displacements and still provide an 

elastic restoring force to the superstructure. Restrainers may also be provided if gross 

movements are expected. Note that acceleration records from recent earthquakes indicate 
438

vertical accelerations in excess of 1.0 g in the near-field of the fault. In these situations, integral 

construction is preferred, but if elastomeric bearings are used, vertical restrainers should be 

provided to limit the uplift. 

11.3. BRIDGES ON OR NEAR UNSTABLE SLOPES 

Many bridges in mountainous regions cross steep-sided valleys. Detailed geotechnical 

investigations should be made to evaluate the potential for slope instability under seismic 

conditions. 

For major structures, these investigations should include geological and geomorphic studies, 

including expert study of aerial photographs for evidence of bank movement during recent 

earthquakes, as well as material testing and extensive bore hole and trenching investigations to 

check for unstable layers and vertical fissures. Particular attention should be paid to drainage to 

prevent infiltration of surface water and increased porewater pressures in potential failure 

regions. Special studies should be made to investigate the practicality of increasing the factor of 

safety against slope failure by such means as unloading the banks by removal of overburden. 

For important structures, it may be advisable to move the abutments back from the top of the 

slope, and reconstruct each end span accordingly. It may also be prudent to tie back footings 

located on the bank, with rock anchors or other techniques. 

Methods for evaluating the potential for earthquake-induced slope stability and the magnitude of 

slope deformations, together with suggested retrofit measures to improve slope stability, are 

described in more detail in Part 2 of this manual. 

11.4. BRIDGES ON LIQUEFIABLE SOILS 

The vulnerability of bridges to liquefaction-induced ground failure has been clearly 

demonstrated in past earthquakes (1964 Niigata, 1964 Alaska, 1990 Luzon, 1991 Costa Rica, 

1995 Kobe). 

Based on past experience, three liquefaction-related hazards should to be considered: 

1. Flow slides (large translational or rotational movements) mobilized by existing static 

stresses. 

2. Limited cyclic or permanent lateral spreads (displacements) triggered and sustained by the 

earthquake’s ground motion. 

3. Post-liquefaction ground settlement. 

Each of these hazards can cause major structural damage (figure 11-1) and should be considered 

during the development of a bridge retrofit plan. Damage modes include: 
439

Abutment Pile Cap Pier Deck

Dense Sand

Piles

Liquefiable Loose

Sand & Silt Layers

Dense Underlying Soil

Approach

Embankment

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Figure 11-1. Elevation view of a representative bridge on a liquefiable stratum. 

• Lateral deformation of abutments and piers due to liquefaction-induced flow or lateral 

spreads, leading to pile damage and potential span collapse. 

• Differential ground lurching between adjacent piers with the potential for span collapse. 

• Differential settlement of footings and abutments leading to damage in continuous 

superstructures and pile damage due to downdrag. 

• Loss of pile stiffness and capacity leading to structural damage during shaking. 

Methods to assess the potential for liquefaction at a bridge site are discussed in chapter 3, 

together with recommended procedures to estimate post-liquefaction settlement. Refer to 

appendix B (figure B-1) for evaluation of liquefaction potential. The calculation of the amount 

of lateral ground deformation to be expected in the free-field (i.e., in the absence of a bridge) is 

also described in chapter 3. Since this calculation involves considerable uncertainty, and is the 

subject of ongoing research, current practice uses the Newmark sliding block approach on an 

assumed dominant failure plane within the liquefied zone. Free-field displacements are defined 

by an estimated lateral displacement on a failure surface, with the displacement linearly 

distributed over the failure zone, as shown in figure 11-2. 

Mitigation methods should be considered when any of the above damage modes are likely to 

occur. Aside from relocating the bridge to a less vulnerable site, two basic options are available: 

1. Structural retrofit to accommodate the predicted liquefaction and related ground 

displacement demands. This requires an analysis of the soil-structure interaction (SSI) to 

determine if the deformation and load capacity of the existing foundation system can 

accommodate the ground deformation demands without collapse, or can meet other 

prescribed performance criteria. If it is inadequate, mitigation methods to strengthen the 

foundation should be evaluated, and costs compared to the ground modification option 

described below. Evaluation methods that may be used in this approach are discussed in 

section 11.4.1. 
440

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .

Critical Failure Surface (k min)

Probable Displacement

Profile

Liquefiable

Zone

Lowest FOS

Dense Sands

Clay

Fill

Lateral

Spread

y

Figure 11-2. Pseudo-static stability analysis. 

2. Site remediation to stabilize the soil and prevent or inhibit liquefaction, and minimize ground 

displacement demands. Remediation methods include: 

• In-situ densification of liquefiable soils in zones surrounding bridge piers, and 

underneath the toe of approach fills; this will reduce ground deformations to tolerable 

levels. 

• Deep soil mixing using cement, which creates stabilizing zones in the soil similar to 

those developed by ground densification techniques. 

• Dewatering, installation of gravel drains or permeation grouting, and similar techniques. 

A summary of ground improvement methods for liquefaction remediation at existing bridges is 

given in table 11-1. Overviews of the state-of-practice are documented in several publications 

(ASCE, 1997; Andrus and Chung, 1995). A comprehensive report on ground remediation 

measures for liquefaction at existing bridge sites has also been developed (Cooke and Mitchell, 

1999). 

In-situ densification, using either the vibro-replacement technique or compaction grouting, is the 

most widely used ground remediation techniques for both new and existing bridges. 

11.4.1. STRUCTURAL RETROFITTING OF PILED FOUNDATIONS 

An initial assessment should be made to see if the shear and moment capacity of pile foundations 

is sufficient to prevent collapse of the bridge under the displacement demands imposed on the 

foundation system. This is done by using analytical results that give the magnitude of the lateral 
Table 11-1. Summary of ground improvement methods for liquefaction remediation at existing bridges 

Abutment Applicability* Method Principle Suitable 

Soil Types 

Treated Soil 

Properties 

Relative

Costs 

Stub Full-Height/ 

Wingwalls 

Pier Applicability* Comments 

Compaction Grout 

Highly viscous grout acts as spherical hydraulic jack when pumped under high pressure resulting in densification. 

Compressible soils with some fines 

Increased Dr 

SPT: (N1)60 25 to 30 CPT: qcl = 80 

to 150 tsf (Kg/cm2) 

Low material cost; high injection cost. 

Particulate Grouting 

Penetration grouting: fill soil pores with cement, soil and/or clay. 

Clean, medium to coarse sand and gravel 

Cementgrouted soil: high strength 

Lowest of grouting systems 

Chemical Grouting 

Solutions of two or more chemicals react in soil pores to form a gel or solid precipitate. 

Silts and sands 

Low to high strength 

High to very high 

Jet Grouting High speed jets at 

depth excavate, inject and mix stabilizer with soil to form column or panels. 

Sands, silts and clays 

Solidified columns and walls 

High 

1. High. Treat anywhere between abutment and embankment toe; treat under and around abutment if excessive settlement expected. 2. High. Treat around pile groups. 

1. Generally high. Treat under and around footing. 2. High. Treat around pile groups. 

High for solid wall, multicolumn, and hammerhead piers. High to moderate for circular single-column piers. 1. Treat under and around footing. 2. Treat around pile groups. 

Must control heave and/or hydraulic fracture of soil. Particulate and chemical grouting: verify size and strength of grouted soil mass. Jet grouting: stage work to limit settlements. Evaluate potential damage to piles from jetting pressure. 

Vibratory Probe 

Densification by vibration, liquefaction-induced settlement underwater. 

Sand (< 15% passing No. 200 sieve) 

Dr: up to 80+% 

Ineffective in some sands. 

Moderate 

VibroCompaction Densification by vibration and 

compaction of backfill at depth. 

Sand (< 20% passing No. 200 sieve) 

Dr: up to 85+% 

SPT: (N1)60 

25 to 30 CPT: qcl = 80 

to 150 tsf (Kg/cm2) 

Moderate 

1. Moderate for lateral spreading; low for settlement. Treat at embankment toe to reduce risk of construction settlement. 2. Low. Treating around piles difficult due to access problems. 

1. Low. Potential for excessive settlement and vibrations of bridge. Overhead clearance limitations. 2. Low. Treating around piles difficult due to access problems. 

1. Low. Potential for excessive settlement and vibrations of bridge. Overhead clearance limitations. 2. Moderate to high. Treat around pile groups. 

Overhead clearance limitations will restrict use. Monitor bridge for excessive vibrations. Construction in water requires special procedures. 

(continued) 

441
Table 11-1. (continued) Summary of ground improvement methods for liquefaction remediation at existing bridges 

Abutment Applicability* Method Principle Suitable 

Soil Types 

Treated Soil 

Properties 

Relative

Costs 

Stub Full-Height/ 

Wingwalls 

Pier Applicability* Comments 

Vibroreplacement/ Stone Columns 

Densely compacted gravel columns provide densification, reinforcement, and drainage 

Sands and silts 

Increased Dr 

SPT: (N1)60 25 to 30 CPT: qcl = 80 

to 150 tsf (Kg/cm2) 

Moderate to high 

Surcharge/ Buttress Fill 

Weight of surcharge increases liquefaction resistance by increasing effective stresses. Buttress fill increases stability by reducing overturning moment and increasing resisting moment. 

Any soil surface provided it will be stable 

Increase in strength 

Low 1. High for 

slope stability and low for settlement. Place at embankment toe. 2. Low. Ineffective in increasing soil stresses at piles. 

1 & 2. Moderate. Place buttress fill in front of wall. 

1 & 2. Moderate to low. Place surcharge around pier. 

Need large area. Evaluate loads and settlement imposed on bridge. 

Drains:  Gravel  Sand  Wick 

Relief of excess pore water pressure to prevent liquefaction. Intercept and dissipate excess pore water pressure plumes from adjacent liquefied soil. 

Sands, Silt Improved 

drainage 

Low to moderate 

1 & 2. Moderate. Install drains around zone improved by other method(s). 

1 & 2. Moderate. Install drains around zone improved by other method(s). 

1 & 2. Moderate. Install drains around zone improved by other method(s). 

Topography and space limitations may restrict use. 

Mini-Piles Provide piles to carry 

loads through liquefiable soils to firm stratum. 

All soils - High 1 & 2. High to 

moderate. 

1 & 2. High to moderate. 

1 & 2. High to moderate. Must provide means of 

tying mini-piles into existing foundation. 

Note: * Item No. 1 indicates applicability of improvement method for foundations over or in liquefiable soils. Item No. 2 indicates applicability for pile (or drilled shaft) foundations 

extending through liquefiable soils. 

Cooke and Mitchell, 1999 

442
443

deformations in a liquefied layer (normally the layer with the lowest factor of safety against 

liquefaction) and the geometry of the most likely failure surfaces based on Newmark analyses. 

These analyses assume that the effects of ground displacement can be uncoupled from the effects 

of structural inertial loading. In most cases this is a reasonable assumption, because peak 

vibration response is likely to occur in advance of maximum ground displacement, and 

maximum values of displacement-induced moment and shear will generally occur at deeper 

depths than the depths at which maximum moments and shears occur due to structural inertial 

loading. 

The magnitude of moments and shears induced in pile foundations by ground displacements may 

be computed using soil-pile interaction computer programs such as LPILE (Wang and Reese, 

1998), where the assumed displacement field is applied to interface springs using p-y curves. 

Many examples of such analyses are described in the literature1

. In the liquefied zone, the soil is 

normally treated as being soft and cohesive. When calculating the characteristics of the lateral 

springs (i.e., the p-y curves), the maximum soil cohesion is assumed equal to the residual 

strength of the liquefied soil. In some cases, large ground deformations may slide past the 

foundation system, exerting full passive pressures in the process (Berrill et al., 1997). 

A refinement of the above approach is to consider the reinforcement or ‘pinning’ effect that a 

pile or pile group may have on lateral displacements. This is done by representing the pile shear 

forces at the location of the failure plane as an equivalent shear strength in the calculation of 

yield accelerations used in the Newmark analyses. This becomes an iterative approach, since 

shear forces are a function of displacements, which in turn are reduced as the shear forces 

increase. 

The impact of lateral spread on an existing pile foundation depends on a number of geotechnical 

factors, such as the thickness of the crustal layer overlying the liquefied zone, the thickness of 

the liquefied layer, and the magnitude of lateral displacement. Pinning action becomes less 

effective as the thickness of liquefied soil increases. A simplified analysis using this technique is 

described below. 

If the pile foundations cannot tolerate the displacement demands, one option is to install 

additional piles to increase the pinning action and reduce the displacement demands. In such a 

case, the footing will need to be extended and possibly overlayed. Another option is to use pinch 

piles, driven through the liquefiable layer and failure surface, to provide additional pinning 

action. The former option is particularly relevant if additional piles are required to resist high 

overturning moments or minimize downdrag effects. Before deciding on this approach, the cost 

of additional piles and extended footings should be compared with the cost of ground 

remediation as described in section 11.4.2. 

11.4.1.1. Simplified Analysis 

A simplified analytical method for designing the foundations of new bridges is described in 

ATC/MCEER 2003 and is applicable to existing foundations. It involves four basic elements: 

 

1

 Jakura and Abghari (1994); O’Rourke et al. (1994); Soydemir et al. (1997); Ishihara and Cubrinovski (1998). 
444

1. A pseudo-static stability analysis to establish seismic coefficients that cause yield. 

2. Newmark sliding block analysis. 

3. Calculation of foundation forces due to soil movement in the liquefied zone. 

4. Determination of plastic hinge mechanisms that are likely to develop in the foundations. 

The rationale behind this approach is to determine the likely magnitude of lateral soil movement, 

and to evaluate the ability of the pile foundation and the bridge above, to accommodate this 

movement. 

Allowing a plastic mechanism to form in the foundation, under the action of lateral spreading, is 

tantamount to accepting substantial damage in the foundation. This is a departure from the 

design philosophy of most seismic codes where plastic hinging is only acceptable if it occurs 

above the ground where it can be inspected and repaired. This exception is reasonable because it 

is unlikely that a plastic hinge in the foundation will lead to overall structural collapse. This is 

because lateral spreading is essentially a displacement-controlled process. Thus, the estimated 

soil displacements represent a limit on the structure displacement, while excluding the 

phenomena of buckling of the piles or shafts below grade and the continued displacement that 

could be produced by large P-∆ effects. Buckling should be checked and methods that include 

the residual resistance of the soil should be used. A method for checking buckling is described 

in O’Rourke et al. (1994). 

A flowchart for estimating lateral spreading is given in figure 11-3 and the methodology is 

described below. This approach also allows the benefits of passive piles or pin piles to be 

determined. The steps involved are as follows: 

Step 1. Identify the soil layers that are likely to liquefy. 

Step 2. Conduct a stability analysis to determine the likelihood of soil movements and the 

extent of such movements. 

This includes finding the depths of soil likely to move and the extent of the soil block 

that is likely to fail. The effect of these movements on the bridge can then be 

determined by considering the proximity of the failure block to the foundation system. 

Step 3. Estimate the maximum lateral spread displacement of the soil. 

This may be done using Newmark displacement charts, or by conducting a site-specific 

Newmark time history analysis. 

Step 4. Decide whether the soil will continue to flow around the footing and/or piles or whether 

they will move with the soil. 

This assessment requires a comparison between the estimated passive soil forces that 

can be exerted on the foundation and the ultimate resistance that can be developed by 

the bridge. In cases where a crust of non-liquefied material exists at the ground surface, 
445

the structural resistance of the bridge may be less than the displacement-induced 

passive forces. If so, the substructure is then likely to move with the soil. In many 

situations, it will be immediately obvious which condition is more likely to occur. 

Step 5. Design the foundation to withstand the passive pressures created by the flowing soil. 

Step 6. Evaluate the adequacy of the bridge at the maximum expected displacement if analysis 

indicates that the foundation is likely to move with the soil. 

The implication of this assessment is that, for relatively large ground movements, soil 

displacements are likely to cause similar movements of the foundation. In this context, 

‘large’ is assessed relative to the structural resistance at yield. The resulting movement 

of the foundation may produce substantial plasticity or hinge zones in the piles, and 

may induce large reactions in the superstructure. 

Step 7. Limit the pile and substructure forces to values less than yield if the deformations 

determined in Step 6 are unacceptable. 

This can be done by either of two methods. The first is to retrofit the foundations to 

resist the forces that accompany passive flow of the soil around them. The second is to 

limit ground movement by providing either soil or structural remediation. Structural 

remediation makes use of the pinning or dowel action that piles and foundation shafts 

contribute when they cross the potential failure plane of the moving soil mass. This can 

be very effective in reducing the magnitude of the lateral movement. 

Step 8. Determine the plastic mechanism that is likely to develop in the presence of spreading. 

This should be done in a rational manner. Due to the number of uncertainties in the 

process, precision in the calculation of the various mechanisms may not be justified and 

may not produce more accurate results. Simplified approximations for the lateral 

capacity of various mechanisms may be adequate. 

Such approximations could be based on hinge development in the stable soil zones at 

two pile diameters above and below the liquefiable layer. The maximum pile shear 

could then be assumed equal to 2Mp/L, where Mp is the plastic moment and L is the 

distance between hinges. Note that this assumes the load transfer in the liquefied zone 

is negligible. The lateral shear that produces the plastic mechanism can be adjusted 

downward to account for the influence of the P-∆ effect. Figure 11-4 illustrates firstorder corrections for P-∆ effects. 

A more precise method of determining the plastic hinge mechanism would be to ensure 

compatibility of deformations between the soil and piles, as in the computer program 

LPILE. Another refinement is to account for the plastic deformations in the piles 

themselves, as in the program BSTRUCT (O’Rourke et al., 1994). 
446

ATC/MCEER, 2003 

Figure 11-3. Design flowchart. 
447

Pier on Pile Footings and an Integral or

Pinned Connection to the Superstructure

Figure 11-4. Corrections for P-∆ effects in piles with plastic hinges. 

Step 9. Evaluate the foundation system for a displacement field that represents the likely 

deformations due to lateral spreading. From this analysis, an estimate of the likely shear 

resistance that the foundation can provide is determined, which can then be included in 

the stability analysis. 

Step 10. Re-evaluate the stability of the soil block taking into account the additional resistance 

provided by the piles. 

Steps 11 and 12. Recalculate the overall displacement using the revised resistance levels. 

Once a realistic displacement is calculated, the foundation and structural system can be 

analyzed for this movement. It is at this point that larger displacements than those 
448

normally allowed in substructure design can be used. This implies that large plastic 

rotations are allowed to occur in the foundation under these conditions. 

Step 13. If the response of the bridge is acceptable, then the retrofit design is complete, but if not 

determine if the behavior can be improved by providing additional piles or shafts. 

Note that the additional piles do not need to connect to the footing. Alternatively, 

ground improvement approaches may be considered, such as stone columns. The 

selection of structural or geotechnical remediation methods is based on the relative 

economics of the system being used. 

This process is repeated by returning to Step 8 and modifying the available resistance until the 

slope is stabilized. 

11.4.2. SITE REMEDIATION USING GROUND IMPROVEMENT 

To prevent or inhibit liquefaction and reduce lateral ground deformation to acceptable levels, site 

remediation such as ground densification is commonly used in selected liquefiable soil zones. 

The most widely used densification method is the vibro-replacement technique, which has been 

shown to be effective in past earthquakes (Mitchell et al., 1995). This method involves the 

repeated insertion and withdrawal of a large vibrating probe into the soil to a desired depth of 

densification, as shown in figure 11-5. As vibration-induced liquefaction occurs, crushed stone 

backfill is placed around the vibrator, leading to the development of a stone column that is 

approximately 1 m (3 ft) in diameter. The stone column increases the effectiveness of vibration 

transmission, and facilitates drainage of excess pore water pressures as densification occurs. The 

Figure 11-5. Vibro-replacement equipment and process. 
449

procedure is repeated at a grid spacing of 2.4 to 3.7 m (8 to 12 ft) to form a densified zone. 

Relative densities of the order of 80 percent can be achieved by this method. 

Vibro-replacement is effective if the sands to be densified contain less than 15 to 20 percent 

fines. In addition, the use of wick drains placed at the midpoints of stone column grid points will 

aid drainage and can potentially help densify sandy silts (Luehring et al., 1998). Details on 

design approaches and equipment applications can be found in many publications2

. 

A representative application of vibro-replacement is shown in figure 11-6, where a densified 

stone column buttress has been constructed at the toe of an embankment. The effect of the 

buttress is to increase values of the lateral yield acceleration computed using the Newmark 

approach to evaluate embankment stability. This, in turn, reduces lateral spreading. Hence, the 

buttress width is determined based on acceptable displacement-performance criteria for bridge 

piles located in the failure zone. An example of this approach is described by Egan et al. (1992). 

A further application is shown in figure 11-7, where a densified soil zone has been placed around 

a pile-supported bridge pier. The placement of such a zone (normally within a radius of 150 to 

200 percent of the thickness of the liquefiable layer) would be used to eliminate post-liquefaction 

downdrag on the piles, and the potential effects of cyclic ground lurch. 

Cooke and Mitchell, 1999 

Figure 11-6. Vibro-system (stone column) treatment at toe of approach embankment. 

 

2

 Baez (1995), Baez and Martin (1995), Hayden and Baez (1994); Martin (1998b) 
450

For sites where vibratory techniques may be impractical, compaction grouting can be used. 

Shown schematically in figure 11-8, compaction grouting involves pumping a stiff mix of soil, 

cement, and water into the ground under high pressure to compress or densify the soil. A very 

stiff soil-cement and water mixture is injected into the soil forming a grout bulb, which displaces 

and potentially densifies the surrounding ground, without penetrating the soil pores. A grid or 

network of grout columns that is formed by grouting from the bottom up, results in improved 

liquefaction resistance over the required surface area, similar to a network of stone columns 

(Boulanger and Hayden 1995). A theoretical study of the mechanics of ground improvement in 

sands (Mace and Martin, 2000) has shown that increased liquefaction resistance arises primarily 

from increased lateral stresses rather than from densification. Consequently, increases in 

liquefaction strength are best measured using cone penetration test (CPT) correlations (Salgado 

et al., 1997). In this respect, some uncertainties still remain as to the permanence of increases in 

the lateral stress. 

Bridge

Stone Columns Pile Cap Radius of Improved Zone

Liquefiable

Soil

Non-liquefiable Soil

Piles

Densified Soil

Jackura and Abghari, 1994 

Figure 11-7. Vibro-system (stone column) treatment around a pile-supported bridge pier.

11.4.3. SUPERSTRUCTURE IMPROVEMENTS 

In addition to site stabilization, retrofitting the superstructure will often be necessary to improve 

its tolerance for large differential displacements. Required strengthening methods will depend 

on the configuration of the structure and the components most susceptible to damage. These will 

usually involve tying superstructure sections together and providing a positive connection 

between the superstructure, piers and abutments. In some cases, column retrofitting may also be 
451

required. Attempts to stabilize the abutments through the use of anchors would probably not be 

very effective. Because abutment tilting usually does not result in collapse, this type of failure is 

not considered to be critical. The use of settlement slabs may be in order, however, if immediate 

access to the bridge is important. 

Longitudinal restrainers should be provided at all bearings to prevent a loss of support. If the 

superstructure is not tied to its piers, foundation movements may pull the support out from under 

the bearings, as shown in figure 11-9. It would be preferable to fail the column in flexure rather 

than have the span fall off the pier. 

Transverse and vertical restrainers at expansion joints tend to prevent the superstructure from 

buckling (sideways) and should be used along with longitudinal restrainers. When expansion 

joints occur at the piers, these restrainers should provide a positive tie to the substructure. 

Figure 11-8. Compaction grout bulb construction. 
452

The superstructure should be anchored to the pier, and the design load in the anchors should be 

greater than the capacity of the pier. Care should be taken to provide a sufficient gap in the 

restrainers so that normal temperature movement or moderate earthquakes will not result in a 

column failure. 

Because ductile failures of the piers are required to accommodate large movements, pier 

retrofitting may be necessary to ensure that a brittle failure does not occur. Suitable methods for 

retrofitting piers, joints, and foundations are described in chapters 9 and 10. 

An alternative to adding ductility is to increase the stiffness of the structure so as to uncouple it 

from the long period motions associated with liquefiable layers. Stiffening may be accomplished 

by improving the lateral resistance of the foundations, adding infill walls between columns, or 

even increasing the number of substructure units. These measures will be costly, but may be 

justified for critical bridges. 

Figure 11-9. Effect of restrainers at bent during liquefaction failure. 
453

APPENDIX A: GUIDELINES FOR CONDUCTING 

SITE-SPECIFIC GEOTECHNICAL INVESTIGATIONS AND 

DYNAMIC SITE RESPONSE ANALYSES 

As indicated in section 2.4 and table 2-4, site coefficients Fa and Fv are not provided for Site 

Class F soils. Instead, site-specific geotechnical investigations and dynamic site response 

analyses are required for these soils. Guidelines are given below for conducting site-specific 

investigations and site response analyses for Site Class F soils. These guidelines are also 

applicable if it is desired to conduct site response analyses for other soil types. Additional 

guidance on topics addressed below is presented in CSABAC (1999). 

A.1. SITE-SPECIFIC GEOTECHNICAL INVESTIGATION 

For purposes of obtaining data to conduct a site response analysis, site-specific geotechnical 

investigations should include borings with sampling, standard penetration tests (SPTs), cone 

penetrometer tests (CPTs), and/or other subsurface investigative techniques and laboratory soil 

testing to establish the soil types, properties, layering and the depth to rock or rock-like material. 

Shear wave velocities should be measured in all soil layers unless they can be confidently 

estimated from shear wave velocity data for similar soils in the same area. 

A.2. DYNAMIC SITE RESPONSE ANALYSIS 

Components of a dynamic site response analysis include modeling the soil profile; selecting rock 

motions as input to the soil profile; and conducting site response analysis and interpreting results. 

A.2.1. MODELING THE SOIL PROFILE 

Typically, a one-dimensional soil column extending from the ground surface to bedrock is 

adequate to capture first-order site response characteristics. However, two- or three-dimensional 

models may be considered for critical projects when two- or three-dimensional wave propagation 

effects may be significant (e.g., in basins). The soil layers in a one-dimensional model are 

characterized by their total unit weights, shear wave velocities from which low-strain 

(maximum) shear moduli may be obtained, and by relationships defining the nonlinear shear 

stress-strain relationships of the soils. The required relationships for analysis are often in the 

form of curves that describe the variation of shear modulus with shear strain (modulus reduction 

curves) and by curves that describe the variation of damping with shear strain (damping curves). 

In a two- or three-dimensional model, compression wave velocities or moduli, or Poisson’s 

ratios, are also required. In an analysis to estimate the effects of liquefaction on soil site 

response, the nonlinear soil model must also incorporate the buildup of soil pore water pressures 

and the consequent reduction in soil stiffness and strength. Typically, modulus reduction curves 

and damping curves are selected on the basis of published relationships for similar soils1

. Site-

 

1

 Seed and Idriss, 1970; Seed et al., 1986; Sun et al., 1988; Vucetic and Dobry, 1991; Electric Power Research 

Institute, 1993; Kramer, 1996 
454

specific laboratory dynamic tests on soil samples to establish nonlinear soil characteristics can be 

considered where published relationships are judged to be inadequate for the types of soils 

present at the site. The uncertainty in soil properties should be estimated, especially in the 

selected maximum shear moduli and modulus reduction and damping curves. 

A.2.2. SELECTING INPUT ROCK MOTION 

Acceleration time histories that are representative of horizontal rock motions at the site are 

required as input to the soil model. Unless a site-specific analysis is carried out to develop the 

design rock response spectrum at the site, the design rock spectrum for Site Class B rock can be 

defined using the general procedure described in section 2.5. For hard rock (Site Class A), the 

spectrum may be adjusted using the site factors in table 2-4. For profiles having great depths of 

soil above Site Class A or B rock, consideration can be given to defining the base of the soil 

profile and the input rock motions at a depth at which soft rock or very stiff soil of Site Class C 

is encountered. In such cases, the design rock response spectrum may be taken as the spectrum 

for Site Class C defined using the site factors in table 2-4. 

Several acceleration time histories, usually four or more, recorded during earthquakes having 

magnitudes and distances that significantly contribute to the site seismic hazard should be 

selected for analysis. The USGS results for deaggregation of seismic hazard (website address: 

http://eqhazmaps.usgs.gov) can be used to evaluate the dominant magnitudes and distances 

contributing to the hazard. Prior to analysis, each time history should be scaled so that its 

spectrum is at the approximate level of the design rock response spectrum in the range of periods 

of structural significance. The average of the response spectra of the suite of scaled input time 

histories should be approximately at the level of the design rock response spectrum in the period 

range of interest. Because rock response spectra are defined at the ground surface rather than at 

depth below a soil deposit, the rock time histories should be input in the analysis as rock outcrop 

motions rather than at the soil-rock interface. 

A.2.3. SITE RESPONSE ANALYSIS AND RESULTS INTERPRETATION 

Analytical methods may be linear, using equivalent linerarized properties, or explicity nonlinear. 

Frequently used computer programs for one-dimensional analysis include the equivalent linear 

program SHAKE (Idriss and Sun, 1992) and various nonlinear programs (DESRA-2 (Lee and 

Finn, 1978), MARDES (Chang et al., 1991), SUMDES (Li et al., 1992), D-MOD (Matasovic, 

1993), TESS (Pyke, 1992), and DESRA-MUSC (Qiu, 1998)). If the soil response is highly 

nonlinear (e.g., high acceleration levels and soft clay soils), nonlinear programs are generally 

preferable to equivalent linear programs. For analysis of liquefaction effects on site response, 

computer programs incorporating pore water pressure development (effective stress analyses) 

must be used (e.g., DESRA-2, SUMDES, D-MOD, TESS and DESRA-MUSC). Response 

spectra of output motions at the ground surface should be calculated and the ratios of response 

spectra of ground surface motions to input rock outcrop motions should be calculated. Typically, 

an average of the response spectral ratio curves is obtained and multiplied by the design rock 

response spectrum to obtain the design soil response spectrum. Sensitivity analyses to evaluate 

effects of soil property uncertainties should be conducted and considered in developing the 

design response spectrum. 
455

APPENDIX B: EVALUATION OF GEOTECHNICAL HAZARDS 

B.1. GENERAL 

This appendix describes procedures for the evaluation of seismically-induced geotechnical 

hazards, which include soil liquefaction, soil settlement, surface fault rupture, and flooding. 

Slope failures (landslides and rockfalls) are treated in Part 2 of this manual. Hazard evaluation 

should be carried out by engineers and earth scientists (e.g. geotechnical engineers, geologists, 

and seismologists) who have expert knowledge of the hazard and experience with its evaluation. 

B.2. EVALUATION OF SOIL LIQUEFACTION HAZARD 

Procedures given in this appendix for the assessment of liquefaction are based in part on three 

major source documents: 

• Proceedings of the1996 NCEER Workshop (now MCEER) on evaluating liquefaction 

resistance (Youd and Idriss, 1997; Youd et al., 2001). 

• Recommended LRFD guidelines for the seismic design of highway bridges (ATC/MCEER, 

2003). 

• Procedures for implementing guidelines for analyzing and mitigating liquefaction in 

California (SCEC, 1999). 

B.2.1. FIELD EXPLORATION FOR LIQUEFACTION HAZARD ASSESSMENT 

A number of factors must be considered during the planning and conduct of the field exploration 

phase of the liquefaction investigation. These include: 

• Location of liquefiable soils. 

• Location of groundwater level. 

• Depth of liquefaction. 

These factors are discussed below. 

B.2.1.1. Location of Liquefiable Soils 

During the field investigation, the limits of unconsolidated deposits with liquefaction potential 

should be mapped within and beyond the footprint of the bridge. Typically, this will involve an 

investigation at each pier location and at a sufficient number of locations away from the 

approach fill to establish the spatial variability of the material. The investigation should establish 
456

the thickness and consistency of liquefiable deposits from the ground surface to the depth at 

which liquefaction is not expected to occur. The ‘zone of influence’ where liquefaction could 

affect a bridge approach fill will generally be located within a 2H:1V (horizontal to vertical) 

projection from the bottom of the approach fill. 

B.2.1.2. Location of Groundwater Level 

The permanent groundwater level should be established during the exploration program. Shallow 

groundwater may exist for a variety of reasons, some of which are of natural and some of manmade origin. Groundwater may be shallow because the ground surface is only slightly above the 

elevation of the ocean, a nearby lake or reservoir, or the sill of a basin. Another concern is manmade lakes and reservoirs that may create a shallow groundwater table in young sediments that 

were previously unsaturated. If uncertainty exists in the location of the groundwater level, 

piezometers should be installed during the exploration program. The location of the groundwater 

level should be monitored in the piezometers over a sufficient duration to establish seasonal 

fluctuations that may be due to rainfall, river runoff, or irrigation. 

Usually, soils located below the groundwater level are fully saturated. However, at locations 

where fluctuations in groundwater occur, soil can be in a less-than-fully saturated condition. The 

liquefaction resistance of the soil is affected by the degree of saturation, with the resistance 

increasing significantly as the degree of saturation decreases. If the groundwater level fluctuates 

due to tidal action or seasonal river fluctuations, then the zone of fluctuation will often have a 

lower degree of saturation, making the soil more resistant to liquefaction. Unless the seasonal 

fluctuation is in place for an extended period of time, say weeks at a higher level, it is usually 

acceptable to use a long-term groundwater level as a basis for design. 

B.2.1.3. Depth of Liquefaction 

Field exploration should be conducted to the maximum depth of liquefiable soil. A depth of 

about 15 m (50 ft) has often been used as the depth of analysis for the evaluation of liquefaction. 

However, the EERI Monograph (Seed and Idriss, 1982) does not recommend a minimum depth 

for evaluation, but notes 12 m (40 ft) as a depth to which some of the numerical quantities in the 

‘simplified procedure’ can be estimated reasonably. Liquefaction has been known to occur 

during earthquakes at depths greater than 15 m (50 ft) given the proper conditions such as lowdensity granular soils, presence of groundwater, and sufficient cycles of earthquake ground 

motion. For example, liquefaction occurred to depths in excess of 25 m (80 ft) during the 1964 

Alaska earthquake. 

For this reason it is recommended that a minimum depth of 25 m (80 ft) below the existing 

ground surface or lowest proposed finished grade (whichever is lower) be investigated for 

liquefaction potential. For deep foundations (e.g. shafts or piles), the depth of investigation 

should extend to a depth that is a minimum of 6 m (20 ft) below the lowest expected foundation 

level (e.g. shaft bottom or pile tip) or 25 m (80 ft) below the existing ground surface or lowest 

proposed finished grade, whichever is deeper. However, an investigation need not extend below 

a depth at which geologic deposits that are clearly non-liquefiable are present. 
457

If, during the investigation, the indices to evaluate liquefaction indicate that the potential for 

liquefaction may extend to a greater depth, the exploration should be continued until a significant 

thickness (e.g., at least 3 m (10 ft) if possible) of non-liquefiable soils is encountered. 

B.2.1.4. Field Exploration Methods 

Two field exploration methods are normally used during the evaluation of liquefaction potential: 

the Standard Penetration Test (SPT) method, and the Cone Penetrometer Test (CPT) method. 

Other methods should be used as required. For example, the measurement of shear wave 

velocities or the use of the Becker Hammer Test (BHT) method may be useful in some 

assessments of liquefaction potential (section B.2.2.1). A geologic reconnaissance and review of 

the available geotechnical information for the site should supplement any field investigation. 

B.2.1.4(a). SPT Method

Procedures for evaluating liquefaction potential using SPT methods are described in detail by 

Youd and Idriss (1997), Youd et al. (2001) and SCEC (1999). These procedures include 

consideration of correction factors for drilling method, hole diameter, drive-rod length, sampler 

type, energy delivery, and spatial frequency of tests. 

Information presented in Youd and Idriss (1997), Youd et al. (2001) and SCEC (1999) indicate 

that the results of SPT explorations are affected by small changes in measurement method. 

Therefore, it is critical for these tests that standard procedures are followed and that all 

information regarding the test method and equipment used during field work be recorded. The 

energy of the SPT hammer system should also be established for the equipment, as this energy 

directly affects the determination of liquefaction potential. The variation in hammer energy can 

be as much as a factor of 2, which can easily cause a liquefiable site to be identified as being 

non-liquefiable, if a correct hammer calibration factor is not introduced. 

Primarily because of their inherent variability, sensitivity to test procedure, and uncertainty, SPT 

N-values have the potential to provide misleading assessments of liquefaction hazard, if the tests 

are not performed carefully. The engineer who wishes to use the results of SPT N-values to 

estimate liquefaction potential should become familiar with the details of SPT sampling as given 

in ASTM D 1586 (ASTM, 1998) and the recommendations in SCEC (1999) in order to avoid, or 

at least reduce, some of the major sources of error. 

B.2.1.4(b). CPT Method

The CPT method is gaining recognition as the preferred method of evaluating liquefaction 

potential in many locations. Methods for assessing liquefaction potential from CPT results are 

given in Youd and Idriss (1997) and Youd et al. (2001). The primary advantages of the CPT 

method are: 

• It provides an almost continuous penetration resistance profile that can be used for 

stratigraphic interpretation, which is particularly important in determining the potential for 

lateral spreading, lateral flows, and significant differential post-liquefaction settlements. 
458

• Repeatability of the test is very good. 

• Test is fast and economical compared to drilling and laboratory testing of soil samples. 

The limitations of the method are: 

• It does not routinely provide soil samples for laboratory tests. 

• It provides approximate, interpreted soil behavior types and not the actual soil types 

according to ASTM Test Methods D 2488 (Visual Classification) or D 2487 (USCS 

Classification) [ASTM, 1998]. 

• Tests cannot be performed in gravelly soils and sometimes the presence of hard/dense crusts 

or layers at shallow depths makes penetration to desired depths difficult. 

The CPT method should be performed in accordance with ASTM D 5778 (ASTM, 2003. Unless 

site-specific experience is available on the correlation of CPT resistance and soil type, it is 

recommended that at least one boring be drilled to confirm soil types and obtain samples for 

laboratory testing. 

B.2.2. EVALUATION OF LIQUEFACTION POTENTIAL 

Two basic procedures are used to evaluate the potential for liquefaction at a site. These are: 

• Simplified procedure that is based on empirical correlations to observations of liquefaction. 

• More rigorous numerical modeling. 

The decision between the two procedures should be made after careful review of conditions at 

the site and the risks associated with liquefaction, and with the concurrence of the owner. 

For most projects, the simplified procedure will be acceptable. However, for some projects, 

more rigorous modeling using equivalent linear and nonlinear computer codes may be 

appropriate. Conditions warranting the use of more rigorous methods include: 

• Sites where liquefiable soils extend to depths greater than 25 m (80 ft). 

• Sites that have significant interlayering, particularly where interlayers comprise highly 

permeable soils or soft clay layers. 

• Sites where the cost of ground remediation methods to mitigate liquefaction is great. 

Results of most site-specific ground response analyses result in lower estimates for ground 

acceleration and shearing stresses within the soil profile, because the energy dissipative 

mechanisms occurring during liquefaction are explicitly considered in this approach. 
459

B.2.2.1. Simplified Procedure 

The most basic procedure used in engineering practice for assessment of site liquefaction 

potential is that of the ‘Simplified Procedure’1

. The procedure compares the cyclic resistance 

ratio (CRR, the cyclic stress ratio required to induce liquefaction for a cohesionless soil stratum 

at a given depth) with the earthquake-induced cyclic stress ratio (CSR) at that depth from a 

specified design earthquake (defined by a peak ground surface acceleration and an associated 

earthquake magnitude). CRR and CSR are ratios of cyclic shear stress to initial (pre-earthquake) 

effective overburden pressure on a horizontal plane at a given depth in the soil deposit. 

B.2.2.1(a). Cyclic Resistance Ratio, CRR

CRR from SPT Blowcount Data: Values of CRR for the Simplified Procedure were 

originally established from databases for sites that did or did not liquefy during past 

earthquakes and where values of the normalized SPT value, (N1)60, could be correlated with 

liquefied strata. The current version of the baseline chart defining values of CRR as a function 

of (N1)60 for magnitude 7.5 earthquakes is shown in figure B-1. This chart was established by a 

consensus at the1996 MCEER Workshop (Youd and Idriss, 1997; Youd et al., 2001). The 

determination of CRR must consider the fines content of the soil, the energy of the hammer, 

the effective overburden pressure, and the magnitude of the earthquake (Youd and Idriss 1997; 

Youd et al., 2001). 

The effect of the earthquake magnitude on CRR (i.e. the effect of magnitude on the ordinates of 

figure B-1) was re-evaluated during the 1996 MCEER Workshop (Youd and Idriss, 1997; Youd 

et al., 2001). The resulting range of magnitude scaling factors is shown in figure B-2. It is 

recommended that the factors defined by the curve at the lower-bound of the recommended 

range (cross-hatched area) in figure B-2 be used unless different factors can be adequately 

justified. Note that for magnitudes greater than 7.5, the corresponding recommended curve is 

the second highest curve shown in figure B-2. 

Adjustments for changes in water table and overburden condition should be made during the 

simplified analyses. The following provides guidance when making these adjustments. 

• Overburden Corrections for Differing Water Table Conditions To perform analyses of 

liquefaction potential, liquefaction settlement, seismically induced settlement, and lateral 

spreading, it is necessary to develop a profile of SPT blowcounts that have been normalized 

using the effective overburden pressure. Normalization factors are presented in Youd and 

Idriss (1997) and Youd et al. (2001). 

This normalization should be performed using the effective stress profile that existed at the 

time the SPT testing was performed. These normalized values are then held constant 

throughout the remainder of the analyses, regardless of whether or not the analyses are 

performed using higher or lower water-table conditions. Although the possibility exists that 

softening effects due to soil moistening can influence SPT results if the water table 

 

1

Seed and Idriss (1982), Seed et al. (1983) and (1985), Seed and De Alba (1986), Seed and Harder (1990), Youd 

and Idriss (1997) and Youd et al. (2001) 
460

after Youd and Idriss, 1997; Youd et al., 2001 

Figure B-1. Simplified base curve recommended for determination of CRR from SPT 

data for magnitude 7.5 along with empirical liquefaction data. 

fluctuates, it is commonly assumed that the only effect that changes in the water table have 

on the results is due to changes in the effective overburden stress. 

• Overburden Corrections for Differing Fill Conditions Approach fills, and other increases in 

overburden pressure, should be handled similarly to that described above for changes in 

groundwater location. It is necessary to develop a profile of SPT blowcounts that have been 

normalized using the effective overburden pressure existing before the fill is placed. These 

normalized values are then held constant throughout the remainder of the analyses, regardless 

of whether or not the analyses are performed using a higher fill condition. 
461

after Youd and Idriss, 1997; Youd et al., 2001 

Figure B-2. Magnitude scaling factors derived by various investigators. 

CRR from CPT Data: Figure B-3 shows the Robertson and Wride chart (Youd and Idriss, 

1997) for determining liquefaction strength (CRR) for clean sands (fines content, FC, less than 

or equal to 5 percent) from CPT data. The chart, which is only valid for magnitude 7.5 

earthquakes, shows calculated cyclic stress ratios plotted as a function of corrected and 

normalized CPT resistance qC1N, from sites where liquefaction effects were or were not 

observed following past earthquakes. A curve separates regions of the plot with data indicative 

of liquefaction from regions indicative of non-liquefaction. Dashed curves showing 

approximate cyclic shear strain potential as a function of qC1N are shown to emphasize that 

cyclic shear strain and ground deformation potential of liquefied soils decrease as penetration 

resistance increases. 

As for the SPT method, the determination of CRR using the CPT method must account for 

fines content of the soil, the effective overburden pressure, and the magnitude of the 

earthquake. Factors to account for fines content and effective overburden pressure are 

presented in Youd and Idriss (1997) and Youd et al. (2001). Comments given above regarding 

use of normalized (for effective overburden pressure) SPT blowcounts in analyses are valid for 

use of normalized CPT data. Magnitude scaling factors for CRR are recommended to be the 

same for SPT and CPT and are shown in figure B-2. 

CRR from other Types of Data: The CRR can also be obtained using shear wave velocity 

data and Becker Hammer Test (BHT) data, as discussed in Youd and Idriss (1997) and Youd et 

al. (2001). These procedures are potentially useful especially for gravelly soils, because of the 

difficulty of obtaining meaningful SPT blowcounts or CPT resistances in gravelly soils (due to 

the large size [relative to sand] of gravel particles in comparison to the diameter of the SPT 

sampler and the cone penetrometer). 
462

after Youd and Idriss, 1977 

Figure B-3. Robertson and Wride chart. 

B.2.2.1(b). Cyclic Stress Ratio, CSR

For estimating values of the earthquake-induced cyclic shearing stress ratio, CSR, the MCEER 

workshop (Youd and Idriss, 1997; Youd et al., 2001) recommended essentially no change to 

the original simplified procedure (Seed and Idriss, 1971), where a mean soil flexibility factor, 

rd, (figure B-4) is used to define the reduction in CSR with depth. Idriss (1999) proposed a 

quantification of rd as a function of earthquake magnitude that differs from the rd chart in figure 

B-4. As an alternative to using rd charts, a site-specific response analysis of the ground 

motions can be performed, as discussed in section B.2.2.2. 

With the Simplified Procedure, CSR is calculated using the following equation: 

CSR = (τav/σ’vo) = 0.65(amax/g)(σvo/σ’vo)rd (B-1) 

where τav/σ’vo is the average earthquake induced shearing stress divided by the effective 

overburden stress, amax is the peak ground acceleration in units of g (acceleration due to 

gravity), σvo/σ’vo is the ratio of total overburden stress to effective overburden stress, and rd is a 

soil flexibility factor. 
463

after Youd and Idriss 1977 

Figure B-4. Soil flexibility factor (rd) versus depth 

curves developed by Seed and Idriss (1971) with 

added mean value lines. 

B.2.2.1(c). Liquefaction Potential

After values of CRR and CSR are established for a soil stratum at a given depth, the factor of 

safety against liquefaction (i.e. FS = CRR/CSR) can be computed. The ratio of CRR to CSR 

should be greater than 1.0 to preclude the development of liquefaction. As the ratio drops 

below 1.0, the potential for liquefaction increases. Even when the ratio of CRR to CSR is as 

high as 1.5, increases in pore water pressure can occur. The potential consequences of these 

increases should be considered during design. 

B.2.2.2. Numerical Modeling Methods 

For important projects, the use of equivalent linear or nonlinear site-specific, one-dimensional 

ground response analyses may be warranted to assess the liquefaction potential at a site. For 

these analyses, acceleration time histories representative of the seismic hazard at the site are 

used to define input ground motions at a hypothetical outcrop of an appropriate rock or firmground interface at depth. 

One common approach to numerical modeling is to use the equivalent linear total stress 

computer program SHAKE (Idriss and Sun, 1992) to determine maximum earthquake-induced 

shearing stresses at depth for use with the simplified procedure described above, in lieu of using 

equation B-1. Note that, as in equation B-1, a factor of 0.65 should still be applied to maximum 

values of CSR obtained using SHAKE to obtain average values of CSR for comparison with 
464

CRR values. Another alternative involves the use of nonlinear effective stress methods to 

directly determine developed pore water pressures. 

In general, equivalent linear analyses are considered to have reduced reliability as ground 

shaking levels increase to values greater than about 0.4g in the case of softer soils, or where 

maximum shearing strain amplitudes exceed 1 to 2 percent. For these cases, true nonlinear 

effective stress site response programs should be used, where nonlinear shear stress-shear strain 

models (including failure criteria) can replicate the hysteretic, degrading soil response and pore 

pressure development over the full time history of earthquake loading. The computer program 

DESRA 2, originally developed by Lee and Finn (1978), was perhaps the first widely 

recognized nonlinear, effective stress, one-dimensional site response program. Since the 

development of DESRA 2, a number of other nonlinear, effective stress, one-dimensional 

programs have been developed, including D-MOD (Matasovic, 1993), SUMDES (Li et al., 

1992), TESS (Pyke, 1992) and DESRA-MUSC (Martin and Qiu, 2000). 

B.2.3. LIQUEFACTION HAZARDS ASSESSMENT 

Results of the liquefaction assessment are used to evaluate the potential severity of three 

liquefaction-related hazards to the bridge: 

1. Flow failures, which involve large translational or rotational slope failures mobilized by 

existing static stresses (i.e., the site static factor of safety drops below 1.0 due to low 

strengths of liquefied soil layers). 

2. Limited lateral spreads, which involve a progressive accumulation of deformations during 

ground shaking with eventual deformations that can range from a fraction of a meter to 

several meters. 

3. Ground settlement. 

The potential for these hazards can be determined initially on the basis of the factor of safety 

calculated from the ratio of CRR to CSR. If the ratio is less than 1.3, the hazard should be 

evaluated following guidelines given below. No single factor of safety value constitutes an 

appropriate criterion in all situations, and considerable judgment is needed in weighing the 

many factors involved in a decision as to whether to perform additional analyses. These factors 

include: 

• The type of structure and its vulnerability to damage. Structural mitigation solutions may be 

more economical than ground remediation. 

• Levels of risk accepted by the owner regarding design for life safety, limited structural 

damage, or essentially no damage. 

• Damage potential associated with the particular liquefaction hazards. Flow failures or major 

lateral spreads pose more damage potential than differential settlement. Hence, factors of 

safety could be adjusted accordingly. 
465

• Damage potential associated with design earthquake magnitude. A magnitude 7.5 event is 

potentially far more damaging than a 6.5 event. 

• Damage potential associated with SPT values, i.e. low blowcounts are indicative of a greater 

cyclic strain potential than higher blowcounts. 

• Uncertainty in SPT- or CPT- derived liquefaction strengths used for evaluations. Note that a 

change in silt content from 5 to 15 percent could change a factor of safety from say 1.0 to 

1.25 (see figure B-1). 

• For high levels of design ground motion, factors of safety may be indeterminate. For 

example, if (N1)60 = 20, M = 7.5 and fines content = 35 percent, liquefaction strengths 

cannot be accurately defined due to the vertical asymptote on the empirical CRR curve 

(figure B-1). 

Factors of safety of about 1.1 may be acceptable for some locations, where the potential for 

lateral spreading is very low and differential settlement is the hazard of concern. On the other 

hand, factors of safety of 1.3 may be more appropriate for assessing hazards related to flow 

failure potential or loss of foundation-bearing capacity. The final choice of an appropriate factor 

of safety must reflect the particular conditions associated with a specific site and the 

vulnerability of site related structures. 

Sections B.2.3.1 and B.2.3.2 describe an evaluation methodology for liquefaction-induced flow 

failures and lateral spreading, respectively. Methodology for the evaluation of liquefactioninduced settlement is described in section B.3, which also provides a methodology for 

estimating settlements of dry or unsaturated soils. 

B.2.3.1. Flow Failures 

Flow failures are the most catastrophic form of ground failure that may be triggered when 

liquefaction occurs. These large translational or rotational flow failures are produced by existing 

static stresses when average shearing stresses on potential failure surfaces exceed the average 

residual strength of the liquefied soil. 

To assess the potential for flow failure, the static strength properties of the soil in a liquefied 

layer are replaced with the residual strength of liquefied soil. A widely used correlation to 

estimate residual strength is given in figure B-5. A conventional slope stability check is then 

conducted. No seismic coefficient is used during this evaluation, thus representing conditions 

after completion of earthquake shaking. The resulting factor of safety defines the potential for 

flow failures. If the factor of safety is less than 1.0, lateral flow is predicted. 

The estimation of deformation associated with lateral flow cannot be easily made. The 

deformations can be on the order of tens of meters, depending on the geometry of the flowing 

ground and the types and layering of soil. In the absence of reliable methods for predicting 

deformations, it is usually necessary to assume that the soil will undergo unlimited 
466

deformations. If the loads imposed by these movements exceed those that can be tolerated by 

the structure, some type of ground remediation will likely be required. This situation should be 

brought to the attention of the owner and a strategy for dealing with the flow problem agreed 

upon. 

Valuable commentary on the lateral flow problem may be found in several publications2

. The 

topic of post-liquefaction shear strength of granular soils was also the subject of an NSFsponsored MCEER workshop at the University of Illinois in 1997 (Stark et al., 1998). The 

complexities of the problem have also been illustrated in centrifuge tests3

. 

after Seed and Harder, 1990 

Figure B-5. Relationship between residual strength (Sr) and corrected ‘clean 

sand’ SPT blowcount (N1)60 from case histories. 

 

2

 NRC (1985), Seed (1987), Seed and Harder, (1990), Dobry (1995), and Kramer (1996) 

3

 Arulandan and Zeng (1994), Fiegel and Kutter (1994) 
467

The most difficult determination in the lateral flow problem is the determination of the residual 

strength of the soil. The most common procedure for evaluating the residual strength involves 

an empirical correlation between SPT blowcounts and apparent residual strength, backcalculated from observed flow slides. This relationship is shown in figure B-5. Mean or lowerbound values in the data range shown are often adopted. Some experimental work suggests that 

residual strength is related to confining pressure (Stark and Mesri, 1992). Steady state undrained 

shear strength concepts based on laboratory tests have also been used to estimate post 

liquefaction residual strengths4

. Due to the difficulties of test interpretation and corrections for 

sample disturbance, the empirically based correlations are normally used rather than laboratory 

tests. 

B.2.3.2. Lateral Spreading 

The degradation in undrained shearing resistance arising from liquefaction can lead to limited 

lateral spreads induced by earthquake inertial loading. Such spreads can occur on gently sloping 

ground or where nearby drainage or stream channels can lead to static shearing stress biases on 

essentially horizontal ground (Youd, 1995). 

The lateral spreading mechanism is a complex process involving the post-liquefaction strength 

of the soil, coupled with the additional complexities of potential pore water pressure redistribution and the nature of earthquake loading on the sliding mass. At larger cyclic shearing strains, 

the effects of dilation can also significantly increase post-liquefaction undrained shearing 

resistance of the liquefied soil. Incremental permanent deformations will still accumulate during 

portions of the earthquake load cycles when low residual resistance is available. Such low 

resistance will continue even while large permanent shearing deformations accumulate through 

a ratcheting effect. These effects have recently been demonstrated in centrifuge tests to study 

liquefaction-induced lateral spreads, as described by Balakrishnan et al. (1998). Once 

earthquake shaking has ceased, the effects of dilation under static loading can mitigate the 

potential for a flow slide. 

Four approaches can be used to assess the magnitude of deformations from lateral spreading as 

follows: 

1. Youd et al. empirical approach. 

2. Newmark time history analyses. 

3. Simplified Newmark charts. 

4. Numerical modelling. 

Each method is described in the following sections. 

 

4

 Poulos et al., 1985, and Kramer, 1996 
468

B.2.3.2.1. Youd et al. Empirical Approach

Using regression analyses and a large database of lateral spread case histories from past 

earthquakes, Bartlett and Youd (1992) developed empirical equations relating lateral-spread 

displacements to a number of site and earthquake parameters. A refined version of this approach 

was presented by Youd et al. (2002). Generally, this approach should be used only for screening 

of the potential for lateral spreading, as the uncertainty associated with this method of 

estimating displacement is generally assumed to be too large for bridge design. 

The Youd et al. empirical approach uses a variety of earthquake and site parameters, including 

earthquake magnitude, geometry, and soil grain size in an empirical equation to estimate 

displacement. Two cases, a sloping ground model and a free-face model, are used. This 

prediction method is the least reliable in the small displacement range with the level of accuracy 

probably no better than 1 meter. However, it does allow a relatively straightforward screening to 

be accomplished to identify the potential severity of lateral spreads. 

B.2.3.2.2. Newmark Time History Analyses

The simplest of the numerical methods is the so-called Newmark sliding block analysis5 where 

deformation is assumed to occur on a well-defined failure plane and the sliding mass is assumed 

to be a rigid block. This approach requires: 

1. An initial pseudo-static stability analysis to determine the critical failure surface and 

associated yield acceleration coefficient (ky) corresponding to a factor of safety of 1.0. 

2. A design earthquake acceleration record at the base of the sliding mass. 

Cumulative displacements of the sliding mass generated when accelerations exceed the yield 

acceleration are computed using computer programs such as described by Houston et al. (1987). 

These methods are most appropriate when local site effects modify the ground motion as it 

propagates though the soil profile and when the database for the chart method described below 

is not adequate. This latter consideration generally involves sites where the source mechanism 

will be from a magnitude 8 or higher earthquake. 

The Newmark method has been used extensively to study earthquake-induced displacements in 

dams (e.g., Makdisi and Seed, 1978) and natural slopes (e.g., Jibson, 1993). This approach 

involves the double integration of earthquake records above the yield acceleration. The yield 

acceleration (ky) is determined by finding the seismic coefficient that causes the factor of safety 

in a slope stability assessment to be 1.0. During the stability analyses, the liquefied layer is 

modeled with the residual strength of the soil. Other layers with partial buildup in porewater 

pressure can also be degraded in strength for the evaluation. 

The earthquake records must be selected from the available catalogue of records, such that they 

are representative of the source mechanism, magnitude, and distance for the site. A minimum of 

three records from three independent earthquakes should be selected for the Newmark analyses. 

 

5

 Newmark, 1965, and Kramer, 1996 
469

Often it is necessary to modify these records for local site effects, as the ground motion 

propagates through soil to the base of the sliding block. 

A number of uncertainties are inherent in the approach due to the assumptions involved. In 

particular, for liquefaction-induced lateral spreads, uncertainties include: 

• The point in the time history when cyclic strength degradation or liquefaction is triggered. 

• The magnitude of the apparent post-liquefaction residual resistance as discussed above. 

• The influence of the thickness of liquefied soil on displacement. 

• Changes in values of yield acceleration (ky) as deformations accumulate. 

• The influence of a non-rigid sliding mass. 

• The influence of ground motion incoherence over the length of the sliding mass. 

B.2.3.2.3. Simplified Newmark Charts

Charts have been developed by a number of individuals6 using large databases of earthquake 

records and the Newmark Time History Analysis method. These charts allow deformations 

during seismic loading to be estimated using relationships between the acceleration ratio (i.e. 

ratio of yield acceleration (ky) to the peak ground acceleration (kmax) occurring at the base of the 

sliding mass) to ground displacement. The Martin and Qiu (1994) charts are recommended for 

most cases, as they include both peak ground acceleration and peak ground velocity as 

regression parameters. Figures B-6 and B-7 show the relationships developed by Martin and 

Qiu (1994). 

The Franklin and Chang (1977) procedure, which was given in earlier editions of the AASHTO 

Specifications, is now thought to overestimate displacements, partly because it was developed 

by bounding all data and partly because the database had some records that produced unusually 

large displacements. The Hynes and Franklin (1984) charts used the same database as did 

Martin and Qiu, and therefore the mean values from the Hynes and Franklin chart are normally 

similar to the values estimated by the Martin and Qiu method. The Wong and Whitman (1982) 

chart provides the smallest estimate of displacements, and appears to be unconservative in some 

cases. 

To use these charts, the yield acceleration is determined by finding the seismic coefficient that 

causes the factor of safety in a slope stability assessment to be 1.0. As noted for the Newmark 

Time History Analyses, the liquefied layer is modeled with the residual strength of the soil. 

 

6

 Franklin and Chang, 1977, Hynes and Franklin, 1984, Wong and Whitman, 1982, and Martin and Qiu, 1994 
470

after Martin and Qiu, 1994 

Figure B-6. Simplified displacement chart for velocity-acceleration ratio of 30. 

K 0.1

0.2

0.3

0.4

0.5

0.6

V = 60 k (in/sec)

1000

100

10

1.0

0.1

0.01

0.01 0.1 1

Acceleration Ratio (k /k ) y max

Displacement (inches)

max

max

after Martin and Qiu, 1994

Figure B-7. Simplified displacement chart for velocity-acceleration ratio of 60. 
471

Other layers with partial buildup in pore water pressure can also be degraded in strength during 

the evaluation. With the yield acceleration (ky) and the peak ground acceleration at the base of 

the failure surface (kmax), it is a simple matter to enter the chart and determine the estimated 

amount of displacement. 

These simplified chart methods are limited by the database used in their development. Typically 

few records greater than magnitude 7.5 were available for analysis and therefore, use of the 

methods for larger magnitudes must be done with caution. Other limitations are similar to those 

presented for the Newmark Time History Analyses. 

B.2.3.2.4. Numerical Modeling

The most rigorous approach to assessing liquefaction-induced lateral spread or slope 

deformations entails the use of dynamic finite element / finite difference programs coupled with 

effective stress-based soil constitutive models. However, the use of such programs is normally 

beyond the scope of routine bridge design projects. Finn (1991) gives a summary of such 

approaches, and a recent case history has been described by Elgamal et al. (1998). Various twodimensional, nonlinear computer programs have been used to perform these analyses. For 

realistic modeling, these programs must be able to account for large displacements, nonlinear 

soil properties, and changes in effective stress during seismic modeling. One computer program 

seeing increasing use for this type of modeling is FLAC7

. This program has been used on a 

number of bridge-related projects, including the Alaska Way Viaduct in downtown Seattle, 

Washington. 

As with any rigorous modeling method, considerable experience and judgment are required 

when using a program such as FLAC. Good practice when using these methods is to compare 

their output to results obtained from empirically-based simplified methods or to laboratory 

experimental data, such as those produced in the centrifuge. 

If lateral spreading is anticipated at a site, the geotechnical engineer should meet with the owner 

and decide what approach offers the most appropriate method of estimating the magnitude of 

lateral spread. 

B.3. EVALUATION OF SOIL SETTLEMENT HAZARD 

Another consequence of liquefaction resulting from an earthquake is the post-earthquake 

volumetric strain caused by the dissipation of excess pore water pressures generated in saturated 

granular soils by the earthquake shaking. The volumetric strain, in the absence of lateral flow or 

spreading, results in settlement. Liquefaction-induced settlement could lead to collapse or 

partial collapse of a structure, especially if there is significant differential settlement between 

adjacent structural elements. Even without collapse, significant settlement could result in 

damage or render the bridge deck impassable to traffic. 

 

7

 Itasca, 1998, Wang and Makdisi, 1999 
472

In addition to the settlement of saturated soil deposits, the settlement of dry or unsaturated 

granular deposits due to earthquake shaking should also be considered in estimating the total 

seismically induced settlements. 

The Tokimatsu and Seed (1987) methodology for saturated and dry/unsaturated sands is the 

most common procedure currently used to estimate the magnitude of settlement. Figure B-8 

shows the relationship between the cyclic stress ratio (τav/σ’o) and volumetric strain for 

saturated clean sands for different values of (N1)60. It is noted that the settlement estimates are 

valid only for level-ground sites that have no potential for lateral spreading. If lateral spreading 

is likely at a site and is not mitigated, the settlement estimates using the Tokimatsu and Seed 

method will likely be less than the actual values. 

The settlement of saturated silty sand and silt requires adjustments to the cyclic strength for 

fines content. Ishihara (1993) recommends increasing the cyclic shear strength of the soils if the 

Plasticity Index (PI) of the fines is greater than 10. This increases the factor of safety against 

liquefaction and decreases the seismically-induced settlement estimated using the Ishihara and 

Yoshimine procedure. Field data suggest that use of the Tokimatsu and Seed procedure without 

correcting the SPT values for fines content could result in an overestimation of seismicallyinduced settlements8

. The use of an appropriate fines-content correction will depend on whether 

the soil is dry/unsaturated or saturated and, if saturated, whether it is completely liquefied (i.e. 

post-liquefaction), on the verge of becoming liquefied (initial liquefaction), or not liquefied. 

SCEC (1999) suggests that for 15 percent fines, the SPT correction value ranges from 3 to 5 and 

for 35 percent fines it ranges from 5 to 9. 

Although the Tokimatsu and Seed procedure for estimating seismically-induced settlements in 

saturated sand is applicable for most level-ground cases, caution is required when using this 

method for stratified subsurface conditions. Martin et al. (1991) demonstrated that for stratified 

soil systems, the SPT-based method of liquefaction evaluation9 could over-predict 

(conservative) or under-predict (unconservative) excess pore water pressures developed in a soil 

layer depending on the location of the soil layer in the stratified system. Given the appropriate 

boundary conditions, Martin et al. (1991) shows that thin, dense layers of soils could liquefy if 

sandwiched between liquefiable layers. The estimated settlement using the Tokimatsu and Seed 

procedure (which is based on the SPT values and excess pore pressures generated in the 

individual sand layers) may therefore be over-predicted or under-predicted. 

The Tokimatsu and Seed (1987) method can be used to estimate settlement in layered deposits 

of saturated granular soils by accounting for settlement of non-liquefiable layers. One approach 

to estimate the settlement of such a non-liquefiable soil layer is to use figure B-8, in 

combination with figure B-9, to determine if the layer will be affected by the layer below. If 

Hb < Hc (see figure B-9), then the settlement of the layer should be estimated by assuming 

settlement in the non-liquefiable layer (Hb) will be approximately 1.0 percent, provided that the 

non-liquefiable layer meets all of the following criteria: 

 

8

 O’Rourke et al., 1991; Egan and Wang, 1991 

9

 Seed et al. (1983) and (1985) 
473

after Tokimatsu and Seed, 1987 

Figure B-8. Relationship between cyclic stress ratio, blowcount (N1)60, and 

volumetric strain for saturated clean sands and magnitude 7.5. 

• Thickness of the layer is less than or equal to 1.5 m (5 ft). 

• Corrected SPT value (N1)60 is less than 30 or CPT tip resistance (qc1N) normalized to 100 kPa 

(15 psi) is less than 160. 

• Soil type is sand or silty sand with fines content less than or equal to 35 percent. 

• Magnitude of design earthquake is greater than or equal to 7.0. 

The logic for using these four criteria is that the migration of pore water pressure into, and 

subsequent settlement of, the non-liquefiable layer depends on factors such as the thickness, 

density (SPT or CPT tip value), permeability (soil type) of the layer, and the duration of 

earthquake shaking (magnitude). It is noted that these criteria are only guidelines to allow the 

designer to be aware of the potential settlement contributions from certain non-liquefiable soil 

layers present in a layered system. 
474

Loose cohesionless soil above the water table (dry or unsaturated soils) will tend to densify 

during the period of earthquake ground shaking. This potential should be considered when 

evaluating the potential for differential displacement between the bridge abutment and the 

closest central pier or between central piers in a multiple span bridge. These settlements are 

additive to the settlements of soils below the water table discussed above (assuming settlementsusceptible soils are located beneath foundation depths). 

Procedures described by Tokimatsu and Seed (1987) can be used to estimate the amount of 

settlement of dry or unsaturated sand. The Tokimatsu and Seed procedure for estimating 

seismically-induced settlements in these soils requires that the settlement estimates be 

multiplied by a factor of 2.0 to account for the effect of multidirectional shaking, as discussed 

by Tokimatsu and Seed (1987). 

H

H

H

H

1

a

b

c

Liquefiable Soil

Liquefiable Soil

Non-liquefiable Soil

Existing Ground Surface

Water

Table

Non-liquefiable Soil

If H H : then H = H + H + H b c 2 abc 

If H > H : then H = H bc 2a

after Isihara, 1985 

Figure B-9. Schematic diagram for determination of H1 and H2. 

B.4. EVALUATION OF SURFACE FAULT RUPTURE HAZARD 

After a site has been evaluated by the screening criteria presented in chapter 3 and either there is 

insufficient information to rule out a surface fault rupture hazard, or there is seismic, 

geomorphic, and/or geologic data that suggests active fault(s) might be present at or near the site, 

the following information is required to refine definition of the hazard: 
475

1. The location of fault traces (if any) with respect to the site. 

2. The timing of most recent slip activity on the fault. 

3. The ground rupture characteristics for a design earthquake on the fault (e.g. type of faulting 

(figure B-10), amount of slip and distribution into strike-slip and dip-slip components, and 

width of the zone of ground deformation). 

Figure B-10. Types of earthquake faults. 
476

B.4.1. FAULT LOCATION 

There are several steps that can be taken to confirm and define the location of faults. Further 

assessments are not required if it can be shown, on the basis of the procedures outlined below, 

that there are no faults passing beneath the site. 

B.4.1.1. Interpretation of Aerial Photographs 

Aerial photographs can be an excellent supplementary resource to geologic and topographic 

maps of the site and vicinity for identifying faults. Older photographs are particularly useful if 

they depict the site and/or its environs prior to development activities that would have altered or 

destroyed landforms that indicate the presence of faults. For many parts of the country, stereo 

photographic coverage is available as far back as the 1920s or 1930s. Aerial photographs are 

usually available from several sources including private companies and from various 

governmental agencies including the U.S. Geological Survey (USGS), U.S. Department of 

Agriculture (Soil Conservation Service), Bureau of Land Management, and the Forest Service. 

The USGS maintains the repository for federal photographic resources at its EROS Data Center, 

Sioux Falls, South Dakota 57198. 

B.4.1.2. Contacting Knowledgeable Geologists 

Geologists and other earth scientists familiar with geologic and tectonic conditions in the site 

vicinity may be willing to share their knowledge. These geologists might work for governmental 

agencies (federal, state, and local), teach and conduct research at nearby colleges and 

universities, or practice as consultants. 

B.4.1.3. Ground Reconnaissance of Site and Vicinity 

A walk-down of the site and its vicinity should be conducted to observe unusual topographic 

conditions and evaluate any geologic relationships visible in cuts, channels, or other exposures. 

Features requiring a field assessment might have been previously identified during the geologic 

and topographic map review, aerial photographic interpretation, and/or during conversations with 

geologists. 

B.4.1.4. Surface Exploration 

Faults obscured by overburden soils, site grading, and/or structures can potentially be located by 

one or more techniques. Geophysical techniques such as seismic refraction surveying provide a 

remote means of identifying the location of steps in a buried bedrock surface and the 

juxtaposition of earth materials with different elastic properties. Geophysical surveys require 

specialized equipment and expertise, and their results may sometimes be difficult to interpret. 

Trenching investigations are commonly used to expose subsurface conditions to a depth of 4.5 to 

6 m (15 to 20 feet). While expensive, trenches have the potential to locate faults precisely and 

provide exposures for assessing their slip geometry and slip history. Borings can also be used to 

assess the nature of subsurface materials and to identify discontinuities in material type or 

elevation that might indicate the presence of faults. 
477

B.4.2. FAULT ACTIVITY 

If it is determined that faults pass beneath the site, it is essential to assess their activity by 

determining the timing of the most recent slip(s) as discussed below. If it is determined, based 

on the procedures outlined below, that the faults are not active faults (see section 3.4.2), further 

assessments are not required. 

B.4.2.1. Assess Fault Relationship to Young Deposits/Surfaces 

The most definitive assessment of the recent history of fault slip can be made in natural or 

artificial exposures of the fault where it is in contact with earth materials and/or surfaces of 

Quaternary age (last 1.8 million years). Deposits might include native soils, glacial sediments 

like till and loess, alluvium, colluvium, beach and dune sands, and other poorly consolidated 

surficial materials. Surfaces might include marine, lake, and stream terraces, and other erosional 

and depositional surfaces. A variety of age-dating techniques, including radiocarbon analysis 

and soil profile development, can be used to estimate the timing of the most recent fault slip. 

B.4.2.2. Evaluate Local Seismicity 

If stratigraphic data are not available for assessment of fault activity, historical seismicity 

patterns might provide useful information. Maps and up-to-date plots depicting historical 

seismicity surrounding the site and vicinity can be obtained from the USGS at its National 

Earthquake Information Center in Golden, Colorado. Additional seismicity information may be 

obtained from state geologic agencies and from colleges and universities that maintain a network 

of seismographs (e.g. California Institute of Technology, University of California, Berkeley, 

University of Nevada, Reno, University of Washington, and the Multidisciplinary Center for 

Earthquake Engineering Research, Buffalo, New York). If the fault(s) that pass beneath the site 

are spatially associated with historical seismicity, and particularly if the seismicity and fault 

trends are coincident, the faults should probably be considered active. 

B.4.2.3. Evaluate Structural Relationships 

In the absence of both stratigraphic and seismological data, an assessment of the geometric/ 

structural relationships between faults at the site and faults of known activity in the region could 

be useful. Although less definitive than the two prior criteria, the probability that the site fault is 

active increases if it is structurally associated with another active fault, and if it is favorably 

oriented relative to stresses in the current tectonic environment. 

B.4.3. FAULT RUPTURE CHARACTERISTICS 

If the evaluation indicates one or more active faults are present beneath the site, the 

characteristics of future slip on the faults can be estimated. Several methods can be used to 

estimate the size of future fault displacements. These include: 
478

1. Observations of the amount of displacement during past surface-faulting earthquakes. 

2. Empirical relations that relate displacement to earthquake magnitude or to fault rupture 

length. 

3. Calculated values based on the cumulative fault displacement or fault slip rate. 

The most reliable displacement assessments are based on past events. Observations of historical 

surface ruptures and geologic evidence of paleoseismic events provide the most useful indication 

of the location, nature, and size of future events. Where the geologic conditions do not permit a 

direct assessment of the size of past fault ruptures, the amount of displacement must be estimated 

using indirect methods. Empirical relations between displacement and earthquake magnitude 

based on historical surface-faulting earthquakes (e.g. Wells and Coppersmith, 1994) provide a 

convenient means for assessing the amount of fault displacement. An example of such a 

relationship is shown in figure B-11. In this example, maximum displacement along the length 

of a fault rupture is correlated with earthquake magnitude. Maximum displacement typically 

occurs along a very limited section of the fault rupture length. Relationships are also available 

for the average displacement along the rupture length. Data from well-documented historical 

earthquakes indicate that the ratio of the average displacement to the maximum displacement 

ranges between 0.2 and 0.8 and averages 0.5 (Wells and Coppersmith, 1994). Other methods for 

calculating the average size of past displacements include dividing the cumulative displacement 

by the number of events that produced the displacement, and multiplying the geologic slip rate 

by the recurrence interval. 

Predicting the width of the zone and the distribution of slip across the zone of surface 

deformation associated with a surface faulting event is more difficult than predicting the total 

displacement. The best means for assessing the width of faulting is site-specific trenching that 

crosses the entire zone. Historical records indicate that the width of the zone of deformation is 

highly variable along the length of a fault. No empirical relationships having general 

applicability have been developed that relate the size of the earthquake or the amount of 

displacement on the primary fault trace to the width of the zone or to the amount of secondary 

deformation. The historical record indicates, and fault modeling shows, that the width of the 

zone of deformation and the amount of secondary deformation tend to vary as a function of the 

dip of the fault and the sense of slip. Steeply dipping faults, such as vertical strike-slip faults, 

tend to have narrower zones of surface deformation than shallow-dipping faults. For dipping 

faults, the zone of deformation is generally much wider on the hanging wall side than on the foot 

wall side. Low-angle reverse faults (thrust faults) tend to have the widest zones of deformation. 

Probabilistic methods for assessing the hazard of fault rupture have been developed that are 

similar to the probabilistic seismic hazard analysis (PSHA) methods used to assess earthquake 

ground motions (e.g. Coppersmith and Youngs, 1990, 2000). A PSHA for fault rupture defines 

the likelihood that various amounts of displacement will be exceeded at a site during a specified 

time period. For critical bridges, such analyses could be considered to assess whether the 

likelihood of surface fault rupture is high enough to warrant design for the hazard and to aid in 

quantifying design values of displacement. 
479

100

10

1

10-1

10-2

10-3

4.5 5.5 6.5 7.5 8.5

Maximum Displacement, MD (meters)

Magnitude, Mw

1 meter = 3.3 feet

Strike-Slip faulting

Oblique faulting, primarily Strike-Slip

Log(MD) = -7.03+1.03 M

Standard deviation of Log(MD) = 0.34

*

after Wells and Coppersmith, 1994 

Figure B-11. Relationship between maximum surface fault displacement 

and earthquake moment magnitude for strike-slip faulting. 

B.5. EVALUATION OF FLOODING HAZARD 

If a bridge site has the potential for earthquake-induced flooding based on the initial screening 

described in section 3.5.2, then further evaluations should be directed at assessing the potential 

for, and severity, consequences, and likelihood of the hazard. The evaluation of the potential for 

landslides into, or within, a body of water uses methodologies described for evaluating 

liquefaction hazards (section B.2 of this appendix) and landslide hazards (chapter 3 of Part 2 of 

this manual). 

Evaluation of the height of waves that could be produced by a tsunami, seiche, or landslide 

requires special expertise in fields such a fluid dynamics and coastal engineering as well as 

seismological, geophysical, and earthquake engineering expertise in characterizing the 

earthquakes and ground shaking that cause these phenomena. Similarly, geological, 

seismological, and geophysical expertise is required to assess tectonic movements such as uplift 

or tilting that could cause flooding. Such studies of hazard potential and severity should be 

considered unless it can be concluded that the effects of flooding on the bridge are tolerable 
480

considering the performance objective, or the probability of occurrence of the hazard is 

sufficiently low that the risk can be accepted. 

If a bridge has possible exposure to flooding from failure of a dam or other water retention 

structure, the agencies having jurisdiction over these facilities should be contacted to determine 

whether the structure has been evaluated or designed for appropriate ground shaking using 

modern seismic analysis and design methods. The potential effects of the flooding at the site 

should also be evaluated. 
481

APPENDIX C: FRAGILITY CURVE THEORY 

For a given bridge, it is not possible to precisely determine the level of ground shaking necessary 

to cause a target level of response or damage state to be achieved. In addition to assuming 

material properties and certain other structural attributes that affect the overall capacity of a 

bridge, such a deterministic assessment requires certain assumptions to be made about the 

ground motion and site conditions. These are both factors that also affect seismic demand. 

Naturally, values of these parameters are not exact. Instead, they invariably have a measure of 

both randomness and uncertainty associated with them. An increasingly popular way of 

characterizing the probabilistic nature of these phenomena is through the use of so-called 

fragility curves1

. 

Figure C-1 shows how the inherent uncertainty and randomness of bridge capacity versus ground 

motion demand can be used to establish fragility curves. The pushover capacity of a bridge has 

been superimposed on the acceleration-displacement spectra in this figure. In a deterministic 

analysis, the intersection of the two curves gives the expected level of performance. However, 

probability distributions are drawn over both the capacity and demand curves to indicate the 

associated uncertainty and randomness of performance. From this graph, it is evident there is a 

wide range of possible performance outcomes. In other words, there is not a unique outcome. 

Figure C-1. Capacity-demand acceleration-displacement spectra showing 

randomness and/or uncertainty in structural behavior and ground motion response. 

If structural capacity and seismic demand are random variables that roughly conform to either a 

normal or log-normal distribution, then by following the central limit theorem, it can be shown 

that the composite performance outcome will be log-normally distributed. Therefore, the 

 

1

 Ayyub and McCuen, 1997; Benjamin and Cornell, 1970 
482

probabilistic distribution is expressed in the form of a so-called fragility curve given by lognormal cumulative probability density function. Fortunately, only two parameters are needed to 

define such a curve: a median value (the 50th percentile) and a normalized logarithmic standard 

deviation. 

Figure C-2 presents the form of a normalized fragility curve for bridges. The cumulative 

probability function is given by: 

⎥

⎦

⎤ ⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

β = Φ

i

a

c

a A

S ln 1 F (C-1) (S )

where Sa is the spectral acceleration at a period T = 1.0 sec, Ai is the median spectral acceleration 

necessary to cause the ith damage state to occur, and βc is the normalized composite log-normal 

deviation which incorporates aspects of uncertainty and randomness for both capacity and 

demand. This parameter is sometimes referred to as either the coefficient of variation or the 

coefficient of dispersion. It has been calibrated theoretically by several researchers2, and 

validated against experiential fragility curves obtained from the 1989 Loma Prieta and 1994 

Northridge earthquakes (Basoz and Mander, 1999). Based on these investigations, it is 

recommended that βc = 0.6, and Φ is the standard log-normal cumulative distribution function. 

Figure C-2. A normalized fragility curve based on a lognormal probability 

distribution with a coefficient of dispersion βc = 0.6. 

 

2

 Pekcan (1998); Dutta and Mander (1998); Dutta (1999) 
483

Unfortunately, evaluating the cumulative distribution function is not trivial, as considerable 

numerical integration is usually necessary. However, the fragility curve can be established using 

intrinsic functions found in most computer software for spreadsheet calculations. For example, 

LOGNORMDIST(Sa , ln (Ai) , βc ) is a function which returns the cumulative normal distribution of Sa, where ln(Sa) is normally distributed with mean ln(Ai) and standard deviation βc. 

 

As an alternative to using standard spreadsheet software, equation (C-1) can be evaluated using 

the following approximation: 

[ ] c

1.7

i

a

i a

A

S 1

1 P D DS | S

β

−

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

+

> = (C-2) 

where [ ] i Sa P D > DS | is the probability of being in damage state D that is equal to or greater 

than DSi for a given spectral acceleration Sa at a structural period T = 1.0 sec, and DSi is the ith

damage state described as follows: 

DS1 = no damage (pre-yield), 

DS2 = slight damage, 

DS3 = moderate damage, 

DS4 = extensive damage, 

DS5 = collapse, 

and all other terms are defined as above. 
 
485

APPENDIX D: CAPACITY/DEMAND RATIOS FOR BRIDGE 

MEMBERS AND COMPONENTS 

D.1. GENERAL 

Capacity/demand (C/D) ratios are used to quantify the likely performance of a bridge during an 

earthquake and can be thought of as that fraction of the design earthquake which will cause a 

member or component to be damaged. The consequences of this damage must then be assessed 

in terms of its effect on the stability and usability of the bridge following an earthquake. 

This appendix presents a methodology for calculating C/D ratios for various member and 

component limit states based on a combination of analysis, testing, and engineering judgment. 

Eleven ratios are defined in four categories as follows: 

1. Support length and restrainer C/D ratios: 

rad displacement C/D ratio for abutment 

rbd displacement C/D ratio for bearing seat or expansion joint 

rbf force C/D ratio for bearing or expansion joint restrainer 

2. Column C/D ratios: 

rca anchorage length C/D ratio for column longitudinal reinforcement 

rcc confinement C/D ratio for column transverse reinforcement 

rcs splice length C/D ratio for column longitudinal reinforcement 

rcv shear force C/D ratio for column 

rec bending moment C/D ratio for column 

3. Footing C/D ratios: 

ref bending moment C/D ratio for footing 

rfr rotation C/D ratio for footing 

4. Soil C/D ratio: 

rsl acceleration C/D ratio for liquefaction potential 

The relative magnitudes of these ratios may be used to sequentially upgrade a deficient bridge. 

D.2. MINIMUM BEARING OR RESTRAINER FORCE DEMANDS 

When determining the minimum bearing or restrainer force demands for the evaluation of an 

existing bridge, the minimum equivalent horizontal force given in section 5.2 should be used. 

This minimum is either 10 percent or 25 percent of the tributary dead load as described in 

section 5.2. 
486

Bearing or restrainer force demands are generally obtained from an analysis of the bridge. 

However, bearing or restrainer forces derived from an elastic analysis do not include the effects 

of nonlinear response of the structure or variations in motions at the supports due to traveling 

surface waves. Because a linear analysis of a bridge often results in relatively low bearing or 

restrainer forces, minimum force demands are specified to account for uncertainties in the 

analysis and to identify bearings that have unreasonably low force capacities. These minimum 

forces are intended for evaluation and should not be confused with minimum design forces for 

bearing restrainers. Different minimum forces for evaluation and design are consistent with other 

requirements of this manual in which evaluation and design are treated differently. Minimum 

force demands are not applicable to devices specifically designed to limit the transfer of forces. 

The engineer performing the evaluation may use simplified methods to determine the portion of 

the minimum equivalent horizontal force carried by the bearings and restrainers. As an example, 

the minimum equivalent horizontal force may be distributed to each horizontal force-resisting 

element based on the portion of the total dead load included within the plan area of the bridge 

bounded by imaginary lines midway between adjacent horizontal force-resisting elements. When 

the ultimate force capacity of a ductile horizontal force-resisting element is insufficient to resist 

its share of the minimum equivalent horizontal force, then the excess of that force should be 

distributed to adjacent bearings or restrainers. 

D.3. MINIMUM SUPPORT LENGTHS 

The supports at the abutments, columns, and expansion joints must be of sufficient length to 

accommodate anticipated relative displacements. Minimum support lengths are specified because 

an elastic analysis does not account for the effects of nonlinear response of the structure or 

variation in motions at the support due to traveling surface waves. 

Minimum support lengths, N(d), for bearing seats supporting the unrestrained expansion ends of 

girders, as shown by the dimension N in figure D-1, are used to calculate bearing displacement 

C/D ratios, rbd by Method 1, as described in section D.4. These support lengths are measured 

normal to the face of abutment, pier, or mid-span joint. The values for minimum support length 

are given by the equations below: 

2 B (1 1.25F S ) v 1 N(d) 100 1.7L 7.0H 50 H 1 2 L cos

⎡ ⎤ ⎛ ⎞ +

=++ + + ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ α ⎣ ⎦ (D-1a) 

where: 

 N(d) is the minimum seat width (mm), 

 L is the distance between joints, or (L1 + L2) for a hinge seat within a span (m), 

 H is the tallest pier between the joints (m), and 

 B is the width of the superstructure (m). 
487

Or, in customary units: 

2 B (1 1.25F S ) v 1 N(d) 4.0 0.02L 0.08H 1.1 H 1 2 L cos

⎡ ⎤ ⎛ ⎞ +

=+ + + + ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ α ⎣ ⎦ (D-1b) 

where: 

 N(d) is the minimum seat width (in), 

 L is the distance between joints, or (L1 + L2) for a hinge seat within a span (ft), 

 H is the tallest pier between the joints (ft), and 

 B is the width of the superstructure (ft). 

In both equations, α is the angle of skew (zero for a right bridge). The ratio of B/L need not be 

taken greater than 3/8. 

Figure D-1. Minimum support length requirements. 
488

D.4. CAPACITY/DEMAND RATIOS FOR EXPANSION JOINTS AND BEARINGS 

D.4.l. GENERAL 

Bridge superstructures are often constructed with intermediate expansion joints to accommodate 

anticipated superstructure movements, such as those caused by temperature variation or to allow 

for the use of incompatible materials. Joints necessitate the use of bearings, which provide for 

rotational and/or translational movement. During earthquakes, bridge bearings have proven to be 

one of the most vulnerable of all bridge components. 

In major earthquakes, the loss of support at bearings has been responsible for several bridge 

failures. Although many of these failures resulted from permanent ground displacements, several 

were caused by vibration effects alone. Some recent examples of earthquakes in which bridge 

collapse resulted from bearing failure include the Loma Prieta, California, earthquake of 1989, 

and the Scotts Mills earthquake in Oregon of 1993, and the Kobe, Japan, earthquake in 19951. 

Even relatively minor earthquakes have caused failure of anchor bolts, keeper bar welds, and 

concrete shear keys. In many of these cases, the collapse of the superstructure was believed 

imminent and would have occurred had the ground motion been slightly more intense or longer 

in duration.

The dynamic behavior of bridge bearings is often very nonlinear and difficult to analyze using 

conventional linear-elastic analysis techniques. Elastic bearing forces obtained from a 

conventional analysis are likely to be lower than those actually experienced by bearings during 

an earthquake. This is because bearings, which are non-ductile components, often do not resist 

loads simultaneously. This has been demonstrated in past earthquakes by the failure of anchor 

bolts or keeper bars in some, but not all, of the bearings at a support. In addition, the yielding of 

ductile members, such as columns, can transfer load to the bearings. This phenomenon has been 

observed in the results from nonlinear analytical case studies of several bridge structures. For 

these reasons, it is necessary to increase elastic analysis force results when evaluating the force 

demand on non-ductile motion-restraining components. 

In the case of differential horizontal displacements at expansion joints during earthquakes, elastic 

response spectrum analysis results yield displacements that are often below those intuitively 

expected based on observed bridge behavior during past earthquakes. In addition to the nonlinear 

behavior of expansion joints, possible independent movement of different parts of the 

substructure and out-of-phase movement of abutments and columns resulting from traveling 

surface-wave motions also tend to result in larger displacements. As noted in the previous 

section, minimum support lengths are required in the AASHTO Specifications to allow for this 

possibility. These support lengths are useful in evaluating the girder seats of existing bridges at 

unrestrained expansion joints. 

When retrofitting expansion joints, however, it is often difficult or impossible to increase the 

existing support length. In these cases, longitudinal restrainers or other displacement-limiting 

devices may be the only feasible means of preventing a loss of support at the bearings. To 

 

1

 Fung et al., 1971; Imbsen, 1981; Governor’s Board of Inquiry, 1990; EERI, 1993 
489

evaluate the effectiveness of these devices in reducing displacements, it is necessary to more 

accurately analyze the movement at bearings. To obtain a reasonable estimate of the actual 

displacements, a multi-mode spectral method of analysis including the effect of foundation 

flexibility should be performed. 

When evaluating the effect of seismic displacements, it is necessary to remember that the entire 

seat width will not be available during an earthquake. Shortening of the bridge superstructure 

due to shrinkage, temperature, or creep may reduce the effective support width. In addition, the 

pounding of adjacent superstructure sections during strong seismic shaking is likely to cause 

localized damage of the expansion joints. This damage will involve crushing of concrete and a 

probable loss of concrete cover, which will further reduce the available seat width. This is shown 

schematically in figure D-2. 

Figure D-2. Effective seat width. 

A bridge with a sloping vertical alignment may have a tendency to shift downhill during an 

earthquake, leaving some expansion joints closed and others open. This same tendency to move 

downhill may also result from other causes, such as temperature movement, traffic vibrations, 

and vehicle breaking forces. This latter phenomenon should also be considered in determining 

the available support length. 

In determining the force capacity of bearings, consideration should be given to the following 

shortcomings of bridge bearings: 

• Grout pads under bearing masonry plates have traditionally given trouble during and after 

construction and have been one of the main sources of trouble in minor earthquakes. Failure 

of a grout pad will allow the bearing assembly to move, subjecting the anchor bolts to 

combined bending and shear. 
490

• Anchor bolts which pass through an elastomeric bearing pad will be subjected to combined 

bending and shear. 

• Anchor bolts are frequently installed with threads that extend below the top surface of the 

pier or abutment seat. This gives a reduced area for shear and may reduce the flexural 

capacity of the bolt due to notch sensitivity at the root of the threads leading to brittle 

fracture. 

• Anchor bolts may have insufficient uplift capacity unless provided with an embedded anchor 

plate. 

• Anchor bolts may be too close to the edge of the bearing seat and may break through the 

concrete when subjected to horizontal loads. 

• All of the bearings supporting one end of a span do not resist horizontal forces equally or 

even simultaneously. Because keeper bars or other devices are not set with exactly the same 

clearances, the bearings will not be equally effective in resisting load. It is quite common for 

bearings on the same support line to be damaged to different degrees during an earthquake. 

• Bridge bearings may not be what they are represented to be on "as-built" plans or 

maintenance records. Adjustments to keeper bars and other details are occasionally made 

after construction is completed. The details and workmanship in such cases may be inferior 

to the original construction. 

D.4.2. DISPLACEMENT CAPACITY/DEMAND RATIOS 

The displacement C/D ratios, rbd should be calculated for restrained and unrestrained expansion 

joints and for bearings at which movement can occur due to the absence of fixity in a horizontal 

direction. The displacement C/D ratio is the lesser of the values calculated using the following 

two methods, except in the case where displacement-limiting devices such as restrainers are 

provided. In that case, Method 2 should be used. 

Method 1: 

( )

( ) bd

N c

r

N d = (D-2) 

where N(c) is the actual support length provided, measured normal to the expansion joint or 

bearing line, and N(d) is the minimum support length defined in section D.3. 

Method 2: 

() ()

( ) s bd

eq

c d

r

d

∆ −∆ = ∆ (D-3) 

where: 

∆s(c) is the available capacity of the expansion joint or bearing for movement; 

for structures in SRC D, cover concrete should not be included in determining the 
491

allowable movement, 

∆i(d) is the maximum possible movement resulting from temperature, shrinkage, and 

creep shortening (if field measurements of available capacity in older bridges are 

used for ∆s(c), then only temperature effects need to be considered here), and 

∆eq(d) is the maximum calculated relative displacement due to earthquake loading for 

the load cases described in section 7.4. 

D.4.3. FORCE CAPACITY/DEMAND RATIOS 

The force C/D ratio for bearings and expansion joint restrainers are evaluated as follows: 

( )

( )

b

bf

b

V c

r

V d = (D-4) 

where Vb(c) is the nominal ultimate capacity of the component in the direction under 

consideration, and Vb(d) is the seismic force acting on the component. This force is the elastic 

force determined from an analysis in accordance with section 7.4, multiplied by 1.25. The 

minimum bearing force demand, as specified in section D.2, is used when an analysis is not 

performed, or when it exceeds the force demand obtained from an analysis. 

D.5. CAPACITY/DEMAND RATIOS FOR REINFORCED CONCRETE COLUMNS, 

WALLS, AND FOOTINGS 

It is expected that reinforced concrete columns, walls, and even footings may yield and form 

plastic hinges during a strong earthquake. The interaction between these components will 

determine the probable mode of failure. To evaluate these components, it is first necessary to 

determine the location of potential plastic hinges. Plastic hinges form at locations of peak 

bending moment and are therefore likely to occur in the end regions of columns or within a 

footing near the column face. An effect similar to a plastic hinge may also develop due to 

yielding of the soil or pilings. Walls, which are defined as supports having a height-to-width 

ratio of 2.5 or less in the strong direction, may develop plastic hinges in the end regions about the 

strong, as well as the weak, axes. 

Once potential plastic hinges have been located, it is necessary to investigate the potential modes 

of column and/or footing failure associated with the location and type of plastic hinging. A 

ductility indicator is used to account for the ability of the columns and/or footings to resist 

certain modes of failure controlled by the amount of yielding. The ultimate moment capacity / 

elastic moment demand ratios are multiplied by ductility indicators to enable elastic analysis 

results to be used for determining the C/D ratios of components subject to yielding. 

The following procedure should be used to determine the C/D ratio for columns, walls, and 

footings as illustrated in the flowchart in figure D-3. This procedure includes a systematic 

method for locating plastic hinges and evaluating the capacity of the columns and / or footings to 

due to plastic hinging. Sections D.5.1 through D.5.5 describe detailed procedures for 

investigating different column and/or footing failure modes sometimes associated with plastic 

hinging. 
492

Figure D-3. Procedures for determining capacity/demand ratios for columns, piers and footings.
493

Step 1: Determine the elastic moment demands at both ends of the column or wall for the seismic 

load cases described in section 7.4. Moment demands for both the columns and footings 

should be determined. The elastic moment demand may be taken as the sum of the 

absolute values of the earthquake and dead load moments. 

Step 2: Calculate nominal ultimate moment capacities for both the column and the footing at 

axial loads equal to the dead load plus or minus the seismic axial load resulting from 

plastic hinging in the columns, walls, or footings as discussed in Article 7.2.2 of the 

AASHTO Standard Specifications (AASHTO, 2002). 

Step 3: Calculate the set of moment C/D ratios (nominal ultimate moment capacity / elastic 

moment demand), rec and ref, for each combination of capacity and demand, assuming 

first that the column will yield and the footing will remain elastic, and second that the 

footing will yield and the column will remain elastic. 

Step 4: Calculate the C/D ratios for the anchorage of longitudinal reinforcement, splices in the 

longitudinal reinforcement, and/or transverse confinement reinforcement at the base of 

the column, and/or footing rotation or yielding for the most severe possible cases of 

plastic hinging as indicated by each set of rec and ref. The following cases describe the 

C/D ratios that should be investigated based on the location and extent of plastic hinging. 

• Case I: Both rec and ref exceed 0.8 It may be assumed that neither the footing nor the 

column will yield sufficiently to require an evaluation of their ability to withstand 

plastic hinging. In this case, only the column C/D ratios for anchorage of longitudinal 

reinforcement (section D.5.1) and splices in longitudinal reinforcement (section 

D.5.2) need to be calculated. 

• Case II: ref is less than 0.8 and rec either exceeds 0.8 or exceeds ref by 25 percent The 

footing will require an evaluation of its ability to rotate and/or yield unless an 

anchorage or splice failure will occur and prevent footing rotation. Anchorage or 

splice failures may be assumed when either the C/D ratio for anchorage of 

longitudinal reinforcement (section D.5.1) or for splices in longitudinal reinforcement 

(section D.5.2) is less than 80 percent of ref. When this is not the case, only the C/D 

ratio for rotation and/or yielding of the footing should be calculated. 

• Case III: rec is less than 0.8 and ref either exceeds 0.8 or exceeds rec by 25 percent It 

may be assumed that only the column will yield sufficiently to require an evaluation 

of its ability to withstand plastic hinging. In this case, the column C/D ratios should 

be calculated for anchorage of longitudinal reinforcement (section D.5.1), splices in 

longitudinal reinforcement (section D.5.2), and column transverse confinement 

(section D.5.4). 

• Case IV: rec and ref are both less than 0.8 and within 25 percent of one another It may 

be assumed that both the column and footing have the potential to yield sufficiently to 

require further evaluation. Since yielding of the footing will be prevented by a 

column failure prior to column yield, column C/D ratios for anchorage of longitudinal 
494

reinforcement (section D.5.1), splices of longitudinal reinforcement (section D.5.2), 

and column transverse confinement (section D.5.4) should be calculated first. When 

all of these C/D ratios exceed 80 percent of ref, then the C/D ratio for rotation and/or 

yielding of the footing (section D.5.5) should also be calculated. 

Step 5: Calculate the column C/D ratios for anchorage of longitudinal reinforcement (section 

D.5.1) and splices in longitudinal reinforcement (section D.5.2) at the top of the column. 

If the moment C/D ratio, rec, of the column is less than 0.8, the C/D ratio for column 

transverse confinement (section D.5.4) should also be calculated. 

Step 6: Calculate the column C/D ratios for column shear (section D.5.3). 

C/D ratios for anchorage of longitudinal reinforcement (rca), longitudinal reinforcement splice 

lengths (rcs), column shear capacity (rcv), column confinement reinforcement (rcc), and rotation 

and/or yielding of the footing (rfr) are dependent on the amount of flexural yielding in the column 

or footing. In columns with poorly detailed transverse reinforcement, one of the most critical 

consequences of flexural yielding is the spalling of cover concrete. Such spalling is followed by 

a rapid degradation in the effectiveness of the transverse steel that can lead to column failure. 

The procedure for calculating C/D ratios for column confinement reinforcement is based on the 

assumption that spalling will begin at a ductility indicator of 2. The effectiveness of poorly 

detailed transverse reinforcement is assumed to begin to degrade at the onset of spalling. This 

type of transverse reinforcement is considered totally ineffective beyond a ductility indicator of 

5. Figure D-4 shows the relationship between the ductility indicator and the effectiveness factor, 

k3, for poorly detailed transverse reinforcement. The effectiveness factor gives the decimal 

fraction of the transverse steel reinforcing that can be considered effective. 

Figure D-4. Effectiveness of poorly anchored transverse 

reinforcement as a function of ductility indicator. 
495

Commentary 

Reinforced concrete columns or walls and the footings to which they are attached form a group 

of interacting components that are among the most vulnerable to earthquake damage. During 

high levels of ground shaking, it is likely that one of these components will be subjected to 

yielding. Because of the interaction between yielding in one component and the response of the 

remaining components, the columns, walls, and footings should be considered as a group. The 

weakest of these components will determine the type of failure that is likely to occur. 

In quantitatively evaluating the strength of the columns and walls, four failure modes should be 

considered. These are: pullout of main reinforcement, splice failures in the main reinforcement, 

sudden shear failure, and loss of flexural capacity due to insufficient confinement. Each of these 

failure modes is a function of the level of column yielding that takes place in the column and 

depends on the amount of transverse confinement of the main longitudinal reinforcing steel. 

Although some useful research has been performed with respect to the behavior of bridge 

columns under cyclic loading, the state-of-the-art is such that column evaluation must rely 

heavily on engineering judgment, especially in the case of existing columns which may have vulnerable details.2

 The methods proposed for evaluating the C/D ratios are based on the latest 

research related to the behavior of reinforced-concrete columns, but still reflect considerable 

judgment on the part of the researchers. 

Most existing bridge columns not only have an insufficient quantity of transverse reinforcing 

steel, but the details with regard to the placement of this steel make it less effective than new 

construction in resisting cyclic column loading. Evaluation of the effectiveness of this 

reinforcement is necessary if a reasonably accurate analysis of seismic capacity is to be made. 

The effectiveness of this steel will be greatly reduced when the concrete cover in the vicinity of 

the plastic hinge spalls. Transverse steel in the region of spalling will then be partly exposed, 

which will greatly reduce anchorage. To some extent, the reduction of efficiency of lap splices in 

transverse reinforcement depends on the degree of spalling. It is assumed that spalling of cover 

concrete will commence at a ductility indicator of approximately 2 and, at this ductility indicator, 

the efficiency of the lap splice drops to approximately 50 percent. At higher ductility indicators, 

a greater amount of spalling of the cover concrete is assumed, and the efficiency of a lap splice is 

assumed to be reduced linearly, eventually reaching zero at a ductility indicator of approximately 

4. These estimates of the efficiency of lap splices in transverse steel are based mostly on 

engineering judgment, although observed column behavior during past earthquakes lends support 

to the conclusions drawn. 

When a column failure occurs due to insufficient transverse reinforcement in any of the four 

potential failure modes, it is likely that poorly anchored transverse reinforcement will unravel 

and become totally ineffective. Therefore, this reinforcement should not be considered in 

calculating C/D ratios for the remaining indicators above the level where the initial column 

failure occurred. 

 

2

 Priestley and Park, 1979; Jirsa, 1979 
496

D.5.1. ANCHORAGE OF LONGITUDINAL REINFORCEMENT 

A sudden loss of flexural strength can occur if longitudinal reinforcement is not adequately 

anchored. The following terms are used to calculate the C/D ratio for anchorage of longitudinal 

reinforcement, rca: 

La(c) is the effective anchorage length of longitudinal reinforcement as shown in figure D-5, and 

La(d) is the required effective anchorage length of longitudinal reinforcement. 

Figure D-5. Effective anchorage length of 

longitudinal reinforcement. 

For straight anchorage, the effective anchorage length, in mm or in, is given by: 

( ) ( )

( ) ( ) s b

a b ' b tr c

2.626 k d

L d 30 d mm, kPa

1 2.5c / d k f

= ≥

+ +

 (D-5a) 

( ) ( ) s b

b ' b tr c

k d 30 d in, psi

1 2.5c / d k f

= ≥

+ +

 (D-5b) 

where: 

 ks = constant for reinforcing steel with a yield stress of fy (kPa or psi), i.e., 

 y (f - 75,845) kPa

33.1 

 or y (f - 11,000) psi 4.8 , 

 db = nominal bar diameter (mm or in), 

'

cf = concrete compression strength (kPa or psi), 
497

 c = lesser of the clear cover over the bar, or half the clear spacing between adjacent 

 bars, 

 ktr = tr yt

b

(A (c)f ) 2.5 (mm, kPa) or (4137 sd )

≤ tr yt

b

(A (c)f ) 2.5 (in, psi), (600 sd )

≤ (D-6) 

Atr(c) = area of transverse reinforcing normal to potential splitting cracks (when splitting 

will occur between several bars in a row, Atr(c) is the total of the transverse steel 

crossing the potential crack divided by the number of longitudinal bars in the row), 

 fyt = yield stress of transverse reinforcement (kPa or psi), and 

 s = spacing of transverse reinforcement (mm or in). 

Note that the value for c/db should not be taken more than 2.5. 

For anchorage with 90° standard hooks, the effective anchorage length, in mm, is: 

( ) ( ) ( ) y

a mb b

c

2.626 f

L d 1200k d 15 d mm, kPa 60000 f

⎛ ⎞

= > ⎜ ⎟ ′ ⎝ ⎠ (D-7a) 

( ) ( ) y

a mb b

c

f

L d 1200k d 15 d in, psi 60000 f

⎛ ⎞

= > ⎜ ⎟ ′ ⎝ ⎠ (D-7b) 

where km is 0.7 for #11 bars or smaller, when the side cover (normal to plane of the hook) is not 

less than 63 mm (2.5 in), and the cover on the bar extension beyond the hook is not less than 50 

mm (2 in), and 1.0 for all other cases. 

The procedure for calculating the C/D ratio for the anchorage of the longitudinal reinforcement, 

rca, is shown in figure D-6. Methods for calculating rca will depend on the adequacy of the 

effective anchorage length provided and the reinforcing details at the anchorage. These methods 

are described in the two cases that follow. 

Case A: If the effective development length provided is insufficient (la(c) < la(d)), then the C/D 

ratio for anchorage of the longitudinal reinforcement rca is given by: 

( )

( )

a ca ec

a

L c

r r

L d = (D-8) 

Case B: If the effective development length is sufficient (La(c) ≥ La(d)), the C/D ratio will 

depend on the reinforcing details at the anchorage. The six possible details and 

corresponding methods for calculating the C/D ratios are as follows: 
498

• Detail 1: When no flexural tensile reinforcement is present in the top of the footing and 

column bar development is by straight anchorage, i.e., no hooks are present at the bottom of 

the footings, 

ca ef r = r (D-9) 

unless 1.25 times the soil overburden and/or pile anchorage is insufficient to overcome the 

negative moment capacity of the footing based on the modulus of rupture of the concrete, in 

which case, rca = 1.0. This negative moment capacity will be used to calculate ref for this and 

the following two detail types. 

• Detail 2: when no flexural tensile reinforcement is present in the top of the footing and the 

column bars are anchored with 90° or greater standard hooks, turned away from the column 

towards the edges of the footing, 

ca ef r = 1.3 r 1.0 ≤ (D-10) 

• Detail 3: When no flexural tensile reinforcement is present in the top of the footing and the 

column bars are anchored with 90° or greater standard hooks turned toward the vertical 

centerline of the column, 

ca ef r = 2r 1.0 ≤ (D-11) 

• Detail 4: When the top of the footing contains adequately anchored flexural tensile 

reinforcement so that ref can be reliably computed from the flexural strength of the top 

reinforced footing section, and the column bar development is by straight anchorage only, 

ca ef r = 1.5 r 1 0 ≤ (D-12) 

unless soil overburden and/or pile anchorage is insufficient to overcome the negative moment 

capacity of the footing, in which case rca = 1.0. 

• Detail 5: When the top of the footing contains adequately anchored flexural reinforcement, as 

for the above detail, and the column bars have been provided with 90° standard hooks, the 

C/D ratio for anchorage should be taken as 1.0. 

• Detail 6: When the anchorage is in a bent cap, the C/D ratios for anchorage should also be 

taken as 1.0. 

Commentary 

The pullout of longitudinal reinforcement can occur at the footings or at the bent cap. This may 

result either due to an inadequate anchorage length or as a result of bond degradation due to 

flexural or shear cracking of the concrete in the footing or cap. In either case, a sudden loss of 

flexural capacity may result. 
499

Figure D-6. Procedure for determining capacity/demand ratios for anchorage 

of longitudinal reinforcement. 
500

If inadequate anchorage length is provided for the reinforcing steel, the ultimate capacity of the 

steel cannot be developed and failure will occur below the ultimate moment capacity of the 

column. 

If anchorage failure results from bond degradation that accompanies the flexural cracking of 

footing concrete, the load level at which failure occurs will depend on the amount of yield in the 

footing. This is accounted for by multiplying the moment C/D ratio for the footing, ref, by a 

ductility indicator. Since most existing footings are not reinforced to resist flexural cracking 

resulting from footing uplift, failure may occur as a result of the negative moments developed in 

the footing due to overturning. Usually this will not be a problem in spread footings, since they 

are not sufficiently restrained by the soil overburden to develop high tensile stresses in the 

concrete. On the other hand, pile footings are usually anchored, although only nominally, to the 

piles. This will allow high tensile stresses to be developed due to overturning of the footing and 

the resulting flexural cracking of the concrete could cause anchorage failures. 

The ductility indicator that is applied to the footing moment C/D ratio to evaluate anchorage 

failure due to flexural cracking in a footing depends on the details of the anchorage and the 

extent to which flexural cracking will occur. Straight anchorage in a footing without a top layer 

of reinforcement may fail rather suddenly when flexural cracking occurs, and therefore a 

ductility indicator of 1.0 is used. Failure will be delayed somewhat when anchored bars are 

hooked. When the hooks are bent away from the centerline of the column, the concrete in the 

vicinity of the hook may eventually be subjected to flexural cracking, and therefore a ductility 

indicator of 1.3 is specified. A greater ductility indicator is allowed when hooks project toward 

the centerline of the column because concrete in the vicinity of the hook will be in compression, 

which will tend to mitigate an anchorage failure when nonstandard hooks are present, the 

required anchorage length will be determined by interpolating between equation D-5 and D-7, 

based on the ratio of the actual length of the hook extension to the length of a standard hook 

extension. 

When a top layer of footing flexural reinforcement is provided, flexural cracking may occur if 

the reinforcement is inadequate, but will progress more slowly and allow a larger ductility 

demand indicator to be used. When straight anchorage is provided, anchorage failure may still 

occur, although the ductility indicator related to this detail is specified as 1.5. If hooks are 

provided, the performance of the splice is assumed to be dependent on the nominal adequacy of 

the anchorage. 

It must be stressed again that the procedures for evaluating loss of anchorage in a footing are 

based largely on engineering judgment. 

This type of failure can also occur in pier shafts if bars are not extended below the level of fixity 

a sufficient distance to develop the ultimate stress in the reinforcement. Similarly, if splices 

occur in a pier shaft, sufficient confinement of the shaft must exist within the area of potential 

yielding to provide for a transfer of stresses in the reinforcing steel. 

Development lengths used to evaluate columns for retrofitting were determined by research 

carried out at the University of Texas (Orangun et al., 1975) The failure hypothesis presented as 
501

a result of this research assumes that the radial component of reactions on the lugs of an 

anchored bar will produce stresses analogous to bursting stresses on a thick-walled hollowconcrete cylinder as shown in figure D-7. The resistance to bursting is a function of the wall 

thickness of the hypothetical concrete cylinder taken as the lesser of the clear bar cover or half 

the clear bar spacing. In addition, bursting will be prevented by transverse reinforcement 

crossing a potential splitting crack in the hypothetical cylinder wall. In some cases, the proposed 

equation for development lengths will result in lengths significantly below those specified by 

previous design codes. In the case where the clear bar cover is much larger than half the clear bar 

spacing, such as in footings, the confining effect of this large cover may be considered by 

assuming the cover to be equivalent to transverse steel of equal tensile strength. 

In circular columns, potential splitting cracks may occur between adjacent bars, resulting in all 

bars failing in anchorage as a group and pulling out of the footings as a plug. In this case, the 

amount of transverse steel, Atc(c), can be assumed to be twice the cross-sectional area of a single 

hoop divided by half the number of anchored bars. A similar group anchorage failure can occur 

with columns having cross-sections of different shapes. 

Figure D-7. Radial stresses developed due to bar anchorage. 

D.5.2. SPLICES IN LONGITUDINAL REINFORCEMENT 

Columns that have longitudinal reinforcement spliced near or within a zone of flexural yielding 

may be subject to a rapid loss of flexural strength at the splice unless sufficient closely spaced 

transverse reinforcement is provided. The minimum area of transverse reinforcement required to 

prevent a rapid splice failure due to reversed loading below the yield strength of the spliced bars 

is given by: 

( ) y

tr b

s yt

s f

Ad A

L f = (D-13) 
502

where s is the spacing of transverse reinforcement, Ls is the splice length, fy is the yield stress of 

the longitudinal reinforcement, fyt is the yield stress of the transverse reinforcement, and Ab is 

area of the spliced bar. 

If the clear spacing between spliced bars is greater than or equal to 4db, where db is the diameter 

of the spliced reinforcement, Atr(c) will be the cross-sectional area of the confining hoop. If the 

clear spacing is less than 4db, then Atr(c) will be the area of the transverse bars crossing the 

potential splitting crack along a row of spliced bars divided by the number of splices. Extra 

splice length by itself does not significantly improve the inelastic response of splices, but splice 

lengths should not be less than 4885 db/ cf′ ) mm (1860 db/ cf′ ) in). 

The procedure for calculating the seismic C/D ratio for splices in longitudinal reinforcement, r, is 

shown in figure D-8. This C/D ratio should be determined only when splices occur within 

locations potentially subject to column flexural yielding unless minimum splice lengths are not 

provided. This includes splices located outside the center half of columns with height-to-depth 

ratios greater than 3 and all splices located within columns with height-to-depth ratios less than 

or equal to 3. The following two cases will apply to these splices. 

Case A: When splice length, transverse reinforcement amount, or transverse reinforcement 

spacing is inadequate [Ls < 4885 db/ cf′ mm (1860 db/ cf′ ) in); Atr(c) < Atr(d); or s > 150 mm 

(6 in)], then the C/D ratio for splices in longitudinal reinforcement, rcs, is given by: 

 

( )

( )

( )

( ) ( ) s tr tr cs ec ec

tr tr

b

c

150 L Ac Ac s r r r mm and kPa

Ad Ad 4885 d

f

⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ = ≤ ⎜ ⎟ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ′ ⎝ ⎠ ⎝ ⎠

 (D-14a) 

 

( )

( )

( )

( ) ( ) s tr tr cs ec ec

tr tr

b

c

6 L Ac Ac s r r r in and kips Ad Ad 1860 d

f

⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ = ≤ ⎜ ⎟ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ′ ⎝ ⎠ ⎝ ⎠

 (D-14b) 
503

Figure D-8. Procedure for determining capacity/demand ratios for splices in longitudinal 

reinforcement (mm and kPa). 
504

where the ratio 150/s (6/s) should not be taken larger than 1 and 4885/ db/ cf′ ) (1860/ db/ cf′ ) 

should not be taken less than 30. The C/D ratio for splices, rcs, need not be taken as less than 

0.75rec when the minimum splice length is provided. 

Case B: When the splice is sufficient [Ls > 4885 db/ cf′ mm (1860 db/ cf′ ) in); Atr(c) ≥ (d); and 

s ≤ 150 mm (6 in)], then the C/D ratio for splices in longitudinal reinforcement, rcs is given by: 

( )

( )

tr cs ec ec

tr

A c

r r 2r

A d

= ≤ (D-15) 

Commentary 

Stress is transferred between spliced bars by the longitudinal component of diagonal compressive 

stresses that are developed in the concrete between the bars. The transverse component of this 

concrete stress acts against the spliced bars and may cause a longitudinal split to form in the 

concrete between the bars unless sufficient reinforcement is provided across the potential 

splitting surface. Splitting cracks may also develop between adjacent sets of spliced bars if 

sufficient spacing is not provided between bars. In the absence of sufficient reinforcement, 

failure will be initiated by splitting at the ends of the splice. This splitting will propagate along 

the splice under progressive cyclic reversed loading, eventually causing the splice to "unzip." 

Therefore, additional splice length will not necessarily prevent failure. The key to preventing a 

splice failure in a bridge column is the presence of sufficient, closely spaced transverse 

reinforcement that will prevent the initiation of splitting. It is necessary, however, to provide a 

minimum splice length. 

The provisions for evaluating the potential for a splice failure are based on the results of 

experimental research conducted at Cornell University and the University of Canterbury, New 

Zealand3

. The research has been directed primarily at splices in building columns, which are 

typically subjected to stress reversals slightly below yield stress. This research was successful in 

identifying the amount and maximum spacing of transverse reinforcement required to prevent a 

splice failure under these loading conditions. In addition, minimum splice lengths were 

determined based on concrete strength. 

For the case where splices could be expected to yield, as might be the case in the zones of 

maximum moment in a bridge column, testing showed that rapid degradation in the stiffness and 

strength of the splice would occur when transverse reinforcement equal to or less than that 

required for unyielding splices was provided. Further testing indicated, however, that an 

improvement in splice performance resulted when additional transverse reinforcement was used. 

If approximately twice the transverse reinforcement required for the unyielding case was used, 

splices were shown to be capable of withstanding reversed loading with displacement ductilities 

as high as six in some cases, although at these extreme ductilities, tensile fracture of the spliced 

bars occurred. Other tests did not result in bar fracture, but indicated strength losses at somewhat 

lower ductility demands. In evaluating splice performance for the yielding case a conservative 

 

3

 Sivakuman et al., 1983; Lukose et al., 1982; Paulay et al., 1981 
505

estimate of the maximum allowable ductility is proposed in the guidelines. This is because of the 

small amount of testing done for the yielding case and the poor transverse reinforcing anchorage 

details typical of most existing bridge columns. Based on judgment, the allowable ductility was 

assumed to be linearly related to the amount of transverse reinforcement in excess of that 

required for the non-yielding case. 

When spliced bars are not stressed beyond 75 percent of the yield capacity, the splice will not 

degrade when subjected to reversed loading. Therefore, the C/D ratio for splices should not be 

less than 0.75 rec when sufficient splice length is provided. 

D.5.3. COLUMN SHEAR 

Column shear failure will occur when shear demand exceeds shear capacity. This may occur 

prior to flexural yielding or during flexural yielding due to the degradation of shear capacity. The 

following terms are used to calculate the C/D ratio for column shear, rcv: 

Vu(d) = the maximum column shear force resulting from plastic hinging at both the 

top and bottom of the column (if both ends are fixed or at one end if the other 

end is pinned) due to yielding in the column or footing (Vu(d) = 1.3 ΣMu/Lc), 

or due to an anchorage or splice failure in the column, whichever occurs first 

(see Note 1 below), 

 Ve(d) = maximum calculated elastic shear force, 

Vi(c) = initial shear resistance of the undamaged column (includes the resistance of 

the gross concrete section and the transverse steel, see Note 2 below), and 

Vf(c) = final shear resistance of the damaged column (includes the resistance of the 

concrete core of the column and only that transverse steel which is effectively 

anchored). When the axial stress is greater than or equal to 0.10 cf′ , an 

allowable shear stress of 5.2 cf′ kPa (2 cf′ psi) may be assumed for the core 

of the concrete column; otherwise use a value of zero. 

Note 1: Procedures for calculating shear forces resulting from column hinging are given in 

Article 7.2.2 of the AASHTO Standard Specifications (AASHTO, 2002). These procedures may 

be extended to consider nominal moment capacities of the footings or a reduced column nominal 

moment capacity due to an anchorage or splice failure below the nominal ultimate column 

moment. The shear force associated with a pinned end should also be included. Few pin 

connections are frictionless, which will cause a shear demand during member rotation. 

Note 2: The shear resistance of concrete columns is calculated using the provisions of Article 

8.16.6 of Division I of the AASHTO Standard Specifications4

, or Article 5.8 of the LRFD 

Specifications (AASHTO 1998), except that capacity reduction factors are not used, i.e., φ = 1.0. 

The procedure for calculating the C/D ratio for column shear is shown in figure D-9. 

 

4

 This reference is to Art. 8.16.6 in Division I of the AASHTO Standard Specifications for Highway Bridges, not 

Division I-A of the same Specifications. 
506

Figure D-9. Procedure for determining capacity/demand ratios for column shear. 
507

When columns do not experience flexural yielding (rec ≥ 1.0), the C/D ratio for column shear 

should be calculated using the initial shear capacity, Vi(c), and the elastic shear demand, Ve(d). 

In columns subject to yielding (rec < 1.0), the C/D ratio for column shear, rcv is calculated 

according to the procedure outlined in figure D-9. Each of three possible cases are described 

below. 

Case A: If the initial shear resistance of the undamaged column is insufficient to withstand the 

maximum shear force due to plastic hinging, [Vi(c) < Vu(d)], a brittle shear failure may occur 

prior to formation of a plastic hinge and the C/D ratio, rcv must be calculated using elastic shear 

demands, i.e., 

( )

( )

i cv ec

e

V c

r r

V d

= ≤ (D-16) 

Case B: If the initial shear resistance of the column is sufficient to withstand the maximum shear 

force due to plastic hinging, but the final shear resistance of the column is not, [Vi(c) ≥ Vu(d) > 

Vf (c)], then the C/D ratio for column shear will depend on the amount of flexural yielding, 

which will cause a degradation in shear capacity from Vi(c) to Vu(d). The C/D ratio is given by: 

cv ec r = r μ (D-17) 

where: 

 μ = () ()

() () c i u

ci f

L Vc V d

2 0.75

b Vc V c

⎛ ⎞ − + ⎜ ⎟ ⎝ ⎠ − , (D-18) 

 Lc = the height of the column, and 

 bc = the width of the column in the direction of shear. 

The column height-to-width ratio should not be taken to be greater than 4 in equation D-18. 

Case C: If the final shear resistance of the column is sufficient to withstand the maximum shear 

force due to plastic hinging, [Vf(c) > Vu(d)], then the C/D ratio for column shear is given by: 

c cv ec

c

L r 2 0.75 r

b

⎛ ⎞

= + ⎜ ⎟ ⎝ ⎠

 (D-19) 

where the terms are defined above. As with Case B, the column height-to-width ratio should not 

be taken to be greater than 4. 

Commentary 

Column shear failure is critical because it results in a comparatively sudden loss of shear 

strength. when this occurs, the resulting excessive deformations may cause disintegration of the 

column and the loss of vertical support. This happened to the Route 5 (Truck Lane) / 405 
508

Separation (California Bridge No.53-1548) during the San Fernando earthquake. Several other 

bridges in the San Fernando earthquake were in various stages of this type of failure and 

probably would have collapsed had the intensity of the ground motion been higher or longer in 

duration. 

The method proposed for evaluating a column for shear failure is based on engineering judgment 

and assumes an idealized model of column behavior. This method may be visualized by 

examining the assumed relationship between shear capacity and shear demand as shown in figure 

D-10. Three possible cases are considered in evaluating C/D ratios for column shear. Case A 

occurs when the column cannot achieve flexural yielding because of a low initial shear capacity. 

In this case, column C/D ratios for shear are calculated by dividing the initial shear capacity of 

the column by the elastic shear demand. This is possible because the initial shear strength of the 

column is not expected to degrade in the absence of plastic hinging, although a brittle shear 

failure can be expected when the initial shear capacity is exceeded. Case B will result when a 

shear failure is expected to occur due to shear capacity degradation resulting from plastic hinging 

of the column. In this case, column C/D ratios for shear are calculated by multiplying the column 

moment C/D ratio, rec, by the ductility indicator corresponding to the amount of yielding at 

which the column shear demand is assumed to exceed the column shear capacity. Case C is 

assumed when the degradation in column shear capacity is not expected to result in a shear 

failure. In this case, the column C/D ratio for shear will be calculated by multiplying the column 

moment C/D ratio by the ductility indicator corresponding to an assumed maximum allowable 

level of flexural yielding. 

The assumed relationship between shear demand and shear capacity in reinforced concrete 

columns is used to identify which of the three cases applies and to determine the ductility 

indicator for Case B. Both demand and capacity are assumed to be dependent on the level of 

flexural yielding as measured by the ductility indicator. 

The relationship between column shear demand and the ductility indicator is based on the 

observation that column behavior will be linear-elastic at a ductility indicator of 1.0 or less. At a 

ductility indicator above 1.0, the shear demand is assumed to be constant and may be determined 

from statics assuming that, where possible, plastic hinges have formed in the column end 

regions. The moments developed in the plastic hinges are assumed to be the maximum ultimate 

column moments adjusted for the possibility of overstrength. Actual shear demands at a ductility 

indicator above 1.0 will vary due to variation in the column axial load, strain hardening of the 

column flexural reinforcement, degradation of column ultimate moment capacity, failure of the 

column to form plastic hinges at both ends simultaneously, and other factors. The proposed 

model of shear demand was selected because it provides a simple yet conservative method for 

relating shear demand to flexural yielding. 

The assumed relationship between shear capacity and flexural yielding, as measured by a 

ductility indicator, is based on observations of column shear behavior during experimental 

investigations and past earthquakes. These observations have established a qualitative 

relationship between shear capacity and flexural yielding, but the quantification of this 

relationship, as proposed in the guidelines, is based largely on the judgment of specialists in 

reinforced concrete column behavior. 
509

Figure D-10. Resolution of shear demand and shear capacity. 

Writers of design codes have found it convenient to subdivide the shear capacity of reinforced 

concrete columns into two parts. The first part is the resistance provided by shear reinforcement, 

such as hoops or spirals. The assumed resistance provided by the reinforcement has been derived 

from a logical model of shear behavior that is based on a truss analogy. 
510

The shear resistance of the concrete portion of the column is assumed to provide the second part 

of the total column shear capacity. However, attributing all shear resistance other than that 

provided by shear reinforcement to the concrete portion of the column is an oversimplification of 

what actually occurs. This shear resistance in reality is composed of shear resistance of concrete 

within the zone of flexural compression stress, dowel action of the flexural reinforcement, and 

aggregate interlock along diagonal cracks. In present design codes, these mechanisms of shear 

resistance are usually lumped into a single empirically derived effective shear stress that 

conservatively approximates the actual shear capacity not provided by shear reinforcement. This 

simplified approach is usually adequate when designing for static loads, since the mechanisms of 

shear resistance will remain intact at these load levels. Such an approach is probably also 

adequate for low levels of seismic loading that will not result in excessive flexural yielding of the 

columns. 

The sum of the shear resistance provided by shear reinforcement and the concrete portion of the 

column is termed the initial shear capacity in this manual. It is assumed that the initial shear 

capacity will not be significantly affected prior to the commencement of spalling of the cover 

concrete and, therefore, column shear capacity is assumed constant at a ductility indicator of 2 or 

less. 

With reversed cyclic loading beyond the elastic limit, many of the mechanisms of shear 

resistance will begin to break down. This breakdown in shear resistance, which is assumed to 

commence at a ductility indicator of 2, is more rapid in columns with a low height-to-width ratio 

because shear demands are typically higher than in more slender columns. For this reason, this 

manual considers height-to-width ratios when evaluating columns for a Case B or Case C shear 

failure. 

Shear reinforcement will not be seriously affected by moderate flexural yielding provided it is 

adequately anchored into the core of the column. However, many existing columns were built 

prior to 1973, when transverse reinforcement anchorage requirements were first included in the 

AASHTO Specifications. Reinforcement that is not adequately anchored into the core of the 

column will be subject to a rapid loss of effectiveness when cover concrete spalls. 

Reversed cyclic flexural yielding will usually have a detrimental effect on the shear resistance of 

the column that is not provided by shear reinforcement. Flexural and diagonal cracks may open 

under loading in one direction and never close on subsequent stress reversals. The mechanism of 

aggregate interlock will be affected as small transverse movements occur along these crack 

interfaces. The shear resistance due to the doweling action of flexural reinforcement will also be 

reduced as the concrete cover spalls and transverse confinement either fails or yields. 

Some experimental research has shown that this degradation of the concrete shear resistance 

seems to be mitigated by increased column axial loads. Although the relationship between axial 

load and concrete shear resistance under reversed cyclic loading has not been precisely 

quantified, many researchers have suggested that the concrete shear resistance be ignored in 

columns with an average axial stress below 0.10 cf′ ) , and considered totally effective for 

columns with greater average stresses. while this is a rather crude treatment of axial load effects, 
511

it has been used in other seismic design provisions and is therefore adopted in this manual until a 

more precise relationship can be derived. 

For purposes of simplicity, it is assumed that a linear degradation of shear capacity will occur 

between a ductility indicator of 2 and a ductility indicator based on the maximum allowed level 

of flexural yielding. At the maximum level of yielding, the column shear capacity, termed the 

"final shear capacity" in this manual, will be assumed to consist of the shear resistance provided 

by adequately anchored shear reinforcement and the effective concrete shear resistance based on 

the magnitude of the axial loads. This method seems appropriate until research establishes a 

more precise relationship between shear capacity degradation and flexural yielding. 

D.5.4. TRANSVERSE CONFINEMENT REINFORCEMENT 

Inadequate transverse confinement reinforcement in the plastic hinge region of a column will 

cause a rapid loss of flexural capacity due to buckling of the main reinforcement and crushing of 

the concrete in compression. The following equation may be used to calculate the C/D ratio for 

transverse confinement, rcc: 

cc ec r = r μ (D-20) 

where: 

 μ = 1 2

3

k k 24 k

2

⎛ ⎞ +

+ ⎜ ⎟ ⎝ ⎠ , (D-21) 

 K1 = ( )

( ) c

c g

c

1

1.25P d 0.5

f A

ρ ≤ ⎛ ⎞

ρ + ⎜ ⎟ ′ ⎝ ⎠

, (D-22) 

 k2 = 6db/s ≤ 1 or 0.2 bmin/s ≤ 1, whichever is smaller, 

 k3 = effectiveness of transverse bar anchorage. This will be 1.0 unless transverse bars 

are poorly anchored, in which case figure D-4 shall be used to determine k3. 

Note that when this is the case, an iterative solution of equation D-21 will be 

required, 

ρ(c) = volumetric ratio of existing transverse reinforcement, 

ρ(d) = required volumetric ratio of transverse reinforcement determined in accordance 

with the provisions of Article 7.6 of the AASHTO Standard Specifications 

(AASHTO, 2002), or Article 5.10.11 of the AASHTO LRFD Specifications 

(AASHTO 1998), 

 Pc = axial compressive load on the column, 

cf′ = compressive strength of the concrete, 

 Ag = gross area of column, 

 s = spacing of transverse steel, 

 db = diameter of longitudinal reinforcement, and 

 bmin = minimum width of the column cross-section. 
512

Commentary 

Transverse confinement reinforcement is required to prevent strength degradation in a column 

subjected to reversed cycles of flexural yielding. Degradation is prevented because confinement 

increases the capability of the concrete core to develop significant stress at high compressive 

strains and prevents buckling of longitudinal compressive reinforcement by providing lateral 

restraint for the reinforcing bars. The degree to which degradation will be prevented is dependent 

on the amount and spacing of transverse reinforcing and the adequacy of the anchorage of this 

reinforcing. 

Current requirements for transverse confinement used in the AASHTO Specifications were 

developed by calculating the amount of reinforcement required to prevent a loss of axial strength 

in a reinforced concrete column due to the loss of cover concrete. Although this approach is 

simple and will result in column designs that can withstand high ductility demands, it is based on 

an inappropriate criteria for column performance and is of limited use for evaluating existing 

columns. 

A more rational approach to calculating the effect of confinement was initially suggested by 

Priestley and Park (1979). This approach uses the calculated moment curvature relationships of a 

concrete column based on the assumed stress strain behavior of reinforcing steel and concrete at 

various levels of confinement. The available curvature ductility of a column would be assumed 

at a curvature that corresponds to a predetermined reduction (e.g. 80 percent) in the column 

moment capacity. This approach was subsequently used to develop the transverse confinement 

requirements for the New Zealand Concrete Design Code: NZS 3101 (Standards Association of 

New Zealand, 1982). The confinement provisions of NZS 3101 are based on the confinement 

requirements used in the AASHTO Specifications modified to account for the effect of axial load 

level. For low axial loads, NZS 3101 results in as much as a 50 percent savings over the amount 

of confinement reinforcing required by the AASHTO Specifications. The New Zealand code 

requires that maximum spacing of transverse steel for adequate concrete confinement be 20 

percent of the minimum cross-section dimension or 6 times the longitudinal bar diameter, 

whichever is less. Testing of near full-scale columns demonstrated the validity of the New 

Zealand transverse confinement requirements (Park et al., 1980). 

Despite the work mentioned in the previous paragraphs, the evaluation of transverse confinement 

in existing columns must be tempered with judgment based on experience gained from past 

earthquakes. It is assumed that spalling of cover concrete commences at a ductility indicator of 2 

and that even poorly confined columns can withstand yielding up to this level because the cover 

concrete provides some confinement. Columns with transverse reinforcement complying with 

the New Zealand code are assumed to be capable of withstanding cyclic yielding corresponding 

to a ductility indicator of 6. Most existing columns have deficiencies in transverse reinforcement 

and are assumed to be able to withstand a limited level of yielding corresponding to a ductility 

indicator between 2 and 6. The equation developed to determine the appropriate ductility 

indicator uses three factors for assessing the relative effectiveness of transverse reinforcement. 

These factors are intended to account for reduction in the efficiency of confinement due to 

deficiencies in the amount, spacing, and anchorage of reinforcement. Factors for amount and 

spacing are averaged because they affect the efficiency of confinement in parallel but separate 
513

ways. A product of these factors would yield results that are too conservative. Deficiencies in 

anchorage, however, will effect the overall efficiency of transverse reinforcement and therefore 

the factor for anchorage is multiplied by the average of the first two factors to obtain the overall 

confinement efficiency. Although this approach is based largely on engineering judgment, it will 

allow for a reasonably accurate evaluation of the C/D ratio for transverse confinement. 

D.5.5. FOOTING ROTATION AND/OR YIELDING 

Column footings may rotate and/or yield before columns can yield. This can occur due to any 

one of several failure modes. The amount of rotation and/or yielding allowed in the footing will 

depend on the mode of failure. The seismic C/D ratio for these types of footing failures, rfr, are 

calculated as follows: 

fr ef r = r μ (D-23) 

where μ the ductility indicator, is taken from table D-1 and depends on the type of footing and 

mode of failure. See the commentary below for a discussion on the method for calculating the 

nominal ultimate capacity of the footing. 

Table D-1. Footing ductility indicators. 

Type of Footing Factor Limiting the Capacity μ

Spread Footing 

Pile Footing 

Soil Bearing Failure 

Reinforcing Steel Yielding in the Footing 

Concrete shear or Tension in the Footing 

Pile overload (Compression or Tension) 

Reinforcing Steel Yielding in the Footing 

Pile Pullout at Footing 

Concrete Shear or Tension in the Footing 

Flexural Failure of Piling 

Shear Failure of Piling 

4 

4 

1 

3 

4 

2 

1 

4 

1 

Commentary 

Footing failures may be classified in one of two ways. The first type of failure involves large 

displacements of the foundation material resulting from instabilities generated within the soil by 

the earthquake ground motion. Liquefaction or slope instability would fall into this category. 

More discussion of these types of failures is included in section D.7. 

The second type of failure, which will be discussed in this section, involves the yielding or 

rupture of foundation elements due to excessive seismic forces transmitted from the structure 

itself. This would include steel and/or concrete failure, bearing failure of the soil, footing failure 

due to sliding or overturning, and pile failure. These failures may result in ductile behavior or in 

sudden brittle failure. 
514

Ductile yielding in the footing is avoided in the design of new bridges because of the difficulties 

involved in inspecting and repairing foundations. Such yielding results in structural damage, but 

will not usually result in structure collapse unless yielding is particularly extensive. Therefore, in 

the case of existing structures, the prospects of yielding in the footings is generally not sufficient 

grounds to justify seismic retrofitting. In fact, from the standpoint of preventing collapse, footing 

yielding may have a beneficial effect, since it can limit shear and flexure in the columns and thus 

decrease the chances of a brittle column failure. 

A sudden brittle failure of the footing, on the other hand, could have serious consequences in 

terms of the ability of the structure to remain standing. The chances of total collapse will depend 

on the configuration of the structure and the nature of the footing failure. For example, the 

sudden loss of flexural capacity in the footings supporting a multi-column bent would probably 

not result in a structure collapse, since the bent would remain stable. However, similar failure in 

a structure with single-column bents would be much more serious. Structure collapse due to a 

sliding failure of the footing is difficult to imagine unless the movement is extensive and the 

structure is discontinuous and supported on narrow bearing seats. In summary, therefore, 

structures with single-column bents are most threatened by a footing failure. 

In evaluating a structure, it is important to determine the capacity of the footings even if footing 

failure will not result in the collapse of the bridge. Footing failure modes will depend, to a 

certain extent, on the type of footing that is being examined. The following sections contain 

recommended procedures for determining the capacities of the two major types of footings used 

in bridge construction: spread and pile footings. 

Spread Footings. The capacity of the footing to resist the loads transmitted from the column or 

pier should be determined. There is an interaction between vertical load and moment capacity, 

which may be governed by the following types of footing failures, as shown in figure D-11: 

• Tilting of the footing due to a soil bearing failure. 

• Flexural yielding of footing reinforcing. 

• Concrete shear failure of the footing. 

• Bond failure of the main column steel. 

The last two failure modes could have serious consequences that, in some cases, could 

potentially result in a structure collapse. Bond failure will be the most critical and should be 

evaluated based on the strength of the anchorage of the column main reinforcement in the 

footing, as discussed in section D.5.1. Insufficient anchorage indicates that the yield capacity of 

the reinforcing cannot be developed and that failure will occur before the column reaches its 

ultimate capacity. A reduction in the effectiveness of the anchorage due to flexural cracking of 

the footing is usually not a problem for unanchored spread footings because the tensile strength 

of the concrete is usually sufficient to prevent cracking. 
515

A concrete shear failure in the footing could be serious because it could result in a fairly sudden 

loss of overturning resistance. In determining the possibility of a shear failure, the shear capacity 

at the critical section determined according to the AASHTO Specifications should be sufficient to 

resist a uniform pressure equal to 1.3 times the ultimate soil bearing capacity. 

Figure D-11. Modes of failure for spread footings. 

Flexural yielding of the footing is also possible, but will not result in a rapid loss of overturning 

resistance as is the case with shear failure. Flexural capacity should be checked at the critical 

section according to the AASHTO Specifications. This capacity should be sufficient to resist 

uniform footing pressures of 1.3 times the ultimate soil bearing capacity. Flexural yielding of the 

footing will cause the column shear force to be limited in order to satisfy static equilibrium. 

If neither shear nor flexural failure will occur in the footing, then the footing capacity will be 

governed by a soil bearing failure. The interaction between axial force and moments at the yield 

capacity of the footing may be calculated by assuming various areas of the footing to be loaded 

with a uniform pressure equal to the ultimate soil pressure. This will produce an interaction 

surface that will indicate the possibility of bearing failure only at the locations where this surface 

falls within the column interaction surface factored for overstrength. Ultimate soil bearing 

pressures can generally be taken at three times the design allowable value. The actual ultimate 

capacity should be provided by the geotechnical engineer. 

This mode of "failure" is considered acceptable by Caltrans because it generally does not lead to 

structure collapse. Retrofitting would only be considered if the structure was required to perform 

to a higher level, i.e., to meet certain functionality criteria immediately following an earthquake. 

Pile Footings. Possible failure modes for pile footings are shown in figure D-12 and may be 

classified as follows: 

• Tilting of the footing due to uplift or compression failures in the piling. 

• Pullout of a pile from the footing. 
516

• Flexural yielding of footing reinforcing. 

• Concrete shear failure of the footing. 

• Bond failure of the main column steel. 

• Flexural or shear failure of the piling. 

Unlike spread footings, the tensile stress in the concrete of footings unreinforced for uplift may 

be insufficient to prevent flexural cracking of the footing and a subsequent loss of column-steel 

anchorage. This type of failure is accounted for in section D.5.1. 

Figure D-12. Modes of failure for pile footings. 

A concrete shear failure in the footing could be serious because it could result in a fairly sudden 

loss of overturning resistance. In determining the possibility of a shear failure, the shear capacity 

at the critical section determined according to the AASHTO Specifications should be sufficient to 

resist the shear produced by 1.3 times the ultimate capacity of the piles. 

A flexural failure of the footing is also possible, but will not result in a rapid loss of overturning 

resistance as is the case with shear failure. Flexural capacity should be checked at the critical 

section according to the AASHTO Specifications. This capacity should be sufficient to resist the 

moment produced by the piles acting at 1.3 times their ultimate capacity. Flexural yielding of the 

footing will cause the column shear force to be limited in order to satisfy static equilibrium. 

If neither shear nor flexural failure will occur in the footing, then the footing capacity will be 

governed by pile failure. The interaction surface for axial force and moments at the yield 

capacity of the footing may be produced by assuming the ultimate compression or uplift in 

various combinations of piles. Pile uplift may be limited by pullout of the pile from the footing 
517

or by the pile withdrawal force. In the case where piles will pull out of the footing, a lower 

ductility indicator is proposed because of the more brittle nature of this type of failure. Ultimate 

pile compression capacity for friction piles can generally be taken at four times the design 

allowable capacity. The actual capacity should be provided by the geotechnical engineers. 

This mode of ‘failure' is considered acceptable by Caltrans because it generally does not lead to 

structure collapse. Retrofitting would only be considered if the structure was required to perform 

to a higher level, i.e., to meet certain functionality criteria immediately following an earthquake. 

Geotechnical specialists should be consulted for the ultimate capacity of the soils, especially with 

respect to the uplift capacity. At poor soil sites, the potential degradation of soil strength should 

be evaluated. The soil capacities can then be compared with the capacities of the connection 

details and the pile members based on the pile type and the expected failure mode, i.e., pile 

overload or pile pullout at the footings. 

D.6. CAPACITY/DEMAND RATIOS FOR ABUTMENTS 

Failure of abutments during earthquakes usually involves tilting or shifting of the abutment, 

either due to inertia forces transmitted from the bridge superstructure or to seismically induced 

earth pressures. Usually these types of failures alone do not result in collapse or impairment of 

the ability of the structure to carry emergency traffic loadings. However, these failures often 

result in loss of access, which can be critical in certain important structures. 

Large horizontal movement at the abutments is often the cause of large approach fill settlements 

that can prevent access to the bridge. Therefore, when required, abutment C/D ratios are based 

on the horizontal abutment displacement. The displacement demand, D(d), will be the elastic 

displacements at the abutments obtained by properly modelling the abutment stiffness. The 

displacement capacity, D(c), is taken as 75 mm (3 in) in the transverse direction and 150 mm (6 

in) in the longitudinal direction, unless determined otherwise by a more detailed evaluation. 

Therefore: 

( )

( ) ad

D c

r

D d = (D-24) 

Commentary 

Abutment displacement capacities are limited to those that are likely to cause problems with 

accessibility to the bridge. Based on experience from past earthquakes, displacement capacities 

of 75 mm (3 in) in the transverse direction and 150 mm (6 in) in the longitudinal direction were 

chosen. These values are based largely on engineering judgment and are likely to be modified as 

more experience is gained. 
518

D.7. CAPACITY/DEMAND RATIOS FOR LIQUEFACTION INDUCED FOUNDATION 

FAILURE 

Many foundation failures that occur during earthquakes are the result of loss of foundation 

support due to liquefaction. A C/D ratio should be calculated when the preliminary screening 

indicates the potential exists for major or severe liquefaction-related foundation damage. To 

determine the C/d ratio for liquefaction failure, rsl a two-stage procedure is necessary. First, the 

depth and areal extent of soil liquefaction required for foundation failure and associated damage 

must be assessed. Second, the level of seismic shaking that will produce liquefaction of the 

above foundation soils must be evaluated. The C/D ratio is obtained by dividing the effective 

peak ground acceleration at which liquefaction failure is likely to occur by the design 

acceleration coefficient: 

( )

( )

L

sl

L

A c

r

A d = (D-25) 

where AL(c) is the effective peak ground acceleration at which liquefaction failures are likely to 

occur, AL(d) is the effective acceleration coefficient for the site equal to 0.4 Ss/g, and Ss is the

‘short-period’ spectral acceleration defined in section 1.5.1. 

Although a great deal of work has been done with respect to determining earthquake-induced 

liquefaction potential of soils, the parameter AL(c) is difficult to determine precisely. Selection of 

a realistic value for AL(c) will require considerable engineering judgment. For example, whereas 

a sand seam may liquefy, its influence on a pile foundation may be minimal. Significant lateral 

foundation displacement leading to damage may require a 3 m (10 ft) depth of liquefied soil to 

occur near the pile head followed by continued ground shaking. 

The amount of movement at a given site due to soil liquefaction is a function of the intensity and 

duration of shaking, the extent of liquefaction, and also the relative density of the soil, which 

controls post-liquefaction undrained or residual strength. In addition, different bridges will be 

able to sustain different amounts of movement. Therefore, when determining AL(c), both the site 

and the bridge characteristics must be taken into consideration. 

The references to bridge-related liquefaction failures noted in the commentary may be of 

assistance in evaluating this problem, as well as references related to assessing liquefaction 

potential of soils. Finally, it is recommended that geotechnical specialists participate in the 

determination of AL(c) at a specific bridge site and assist in the evaluation of the subsequent 

foundation displacement and damage potential. 

Commentary 

Bridge failures resulting from seismic activity have often been classified as failures resulting 

from permanent displacement of the foundations or from structural failures arising from dynamic 

loading. The majority of severe seismic bridge failures have resulted from liquefaction-induced 

permanent displacement of the foundation systems. Despite this fact, the emphasis in both 
519

research and design has been on preventing structural failures. This perhaps reflects the problem 

that foundation failures are difficult to treat quantitatively, whereas structural response is more 

amenable to analysis and generally represents a preventable type of failure. 

Designers have generally approached the problem of liquefaction by attempting to select bridge 

sites at which such failures are unlikely. In many cases, however, the use of such sites is 

unavoidable. In the case of existing bridges, vulnerable sites may have been used without a full 

understanding of the consequences. Designers faced with improving the earthquake resistance of 

such bridges should take advantage of knowledge gained from the performance of bridges in past 

earthquakes to identify collapse mechanisms and evaluation procedures. 

A qualitative description of mechanisms of foundation failure or displacement arising from 

liquefaction is provided in chapter 3 and appendix B. Bridge failures in recent Alaskan and 

Japanese earthquakes are probably the best documented examples5

. In many other earthquakes 

where bridge damage has been reported as a result of liquefaction, modes of foundation failure 

have been similar to the Japanese and Alaskan earthquake case histories. Multispan bridges with 

unrestrained simply supported spans have usually suffered the most damage. 

Foundation conditions which are susceptible to liquefaction are common to bridges that cross 

waterways where foundation soils have been deposited over the years by flowing water. These 

soils are often loose, saturated cohesionless deposits, and are most susceptible to liquefaction. It 

is noteworthy that liquefaction is a combination of earthquake intensity and duration. In the 1964 

Alaskan earthquake, it is estimated that maximum ground accelerations as low as 0.lg to 0.2g 

were responsible for the extensive and widespread bridge foundation failures (Ross et al., 1973). 

The duration of strong shaking was rather long, however, lasting more than 90 seconds. 

Therefore, bridge sites located some distance from a major fault could still be subjected to 

liquefaction failure if the necessary soil conditions are present. 

Methods for assessing the liquefaction potential of site soils are provided in Appendix B. Two 

basic approaches are typically used, namely empirical methods based on blowcount correlations 

for sites that have not liquefied, and analytical techniques based on the laboratory determination 

of liquefaction strengths and dynamic site-response analyses. A rough indication of the potential 

for liquefaction may be obtained by making use of empirical correlations between earthquake 

magnitude and epicenter distance, as described in chapter 3. 

D.8. SUMMARY 

This appendix presented a methodology for determining the capacity/demand ratios for 

displacements and forces for a number of bridge components, including expansion joints and 

bearings, reinforced concrete columns and walls, and foundations. A summary of these C/D 

ratios is given in table D-2, where they listed according to member or component. 

 

5

 Ross et al., 1973; Iwasaki et al., 1972 
520

Table D-2. List of capacity/demand ratios. 

Notation Definition Equation Page 

SUPPORT LENGTH and RESTRAINER C/D RATIOS 

rad Displacement C/D ratio for abutment D-24 

rbd

Displacement C/D ratio for bearing seat or 

expansion joint D-2, D-3 

rbf

Force C/D ratio for bearing or expansion joint 

restrainer D-4 

COLUMN C/D RATIOS 

rca

Anchorage length C/D ratio for column 

longitudinal reinforcement D-8 through D-12 

rcc

Confinement C/D ratio for column transverse 

reinforcement D-20 

rcs

Splice length C/D ratio for column longitudinal 

reinforcement D-14, D-15 

rcv Shear force C/D ratio for column D-16 through D-19 

rec Bending moment C/D ratio for column (Section D.5, Steps 1-4) 

FOOTING C/D RATIOS 

ref Bending moment C/D ratio for footing (Section D.5, Steps 1-4) 

rfr Rotation C/D ratio for footing D-23 

SOIL C/D RATIO 

rsl Acceleration C/D ratio for liquefaction potential D-25 
521

APPENDIX E: EXAMPLE PROBLEM 5.1: 

COMPONENT CAPACITY/DEMAND EVALUATION OF A

4-SPAN, REINFORCED CONCRETE BOX GIRDER BRIDGE 

E.1. INTRODUCTION

This example problem illustrates how some of the recommendations of this Manual are applied 

to a reinforced concrete box girder bridge. In particular, the Seismic Rating of the bridge is 

calculated using the indices method for the Vulnerability Rating, as described in section 4.2. In 

addition, a detailed evaluation of the bridge is performed using the Component 

Capacity/Demand Method described in section 5.4 (Method C). This evaluation is then used to 

identify and evaluate potential seismic retrofitting measures. 

E.2. DESCRIPTION OF THE EXAMPLE BRIDGE

The bridge to be examined is a typical freeway overcrossing of the type that was being 

constructed in California prior to improved seismic design provisions. It carries a major city 

street over an urban freeway in the region and is classified as ‘essential’. It has an anticipated 

service life of 55 years (ASL 3, table 1-1). Soils are Site Class C (table 1-3). The short and longperiod spectral accelerations for the site are 1.00 and 0.40 respectively for the upper level event. 

The Seismic Retrofit Category (SRC) is D (section 1.6). 

The superstructure is a concrete box girder. One portion of this right bridge is prestressed and 

the other is conventionally reinforced. An in-span expansion joint is located between the 

prestressed and reinforced sections. The 143 meter (470 foot) long superstructure is divided into 

four spans, which are continuous over three 2-column bents as shown in figure E-1. The 

diaphragm type abutments are cast monolithically with the superstructure and the entire structure 

is supported on spread footings. 

As is the case with most existing bridges of this vintage and type, the expansion joint is 

unrestrained and supported on a relatively narrow bearing seat. The details of the expansion 

joint are shown in figure E-2. Concrete columns are confined by steel hoops, which are 

inadequate for seismic resistance. At two of the bents, column steel is spliced within a zone of 

potential plastic hinging. The column details are shown in figure E-3. 

E.3. SEISMIC RATING

A seismic rating system is used to identify those bridges that are in greatest need of retrofitting. 

This process is sometimes called preliminary screening because it quickly identifies structures 

that are at risk and deserve closer examination. In practice, all bridges in a region should be 

rated and the results compared in order to develop a prioritized list of bridges requiring detailed 

evaluation. 
522

Figure E-1. Example bridge. 

Figure E-2. Expansion joint detail. 

As described in section 4.2, this rating system requires the calculation of a priority index for each 

bridge. This index is based on an assessment of bridge rank, importance, non-seismic 

deficiencies, and other factors such as network redundancy, political, and economic 

considerations. One of the critical elements in this calculation is the bridge rank (R), which is 

obtained from the vulnerability rating of the structure (V) and the seismic hazard rating of the 

site (E), as follows (equation 4-2): 

 R = V E 
523

Figure E-3. Column details. 
524

Both of these ratings are determined for the example bridge in the following sections. 

E.3.1. VULNERABILITY RATING, V 

The procedure suggested in section 4.2.1.1 is used to calculate the vulnerability rating. Because 

the structure is classified as SRC D, all components will be considered. 

A. Bearings 

Step 1: Because the bridge superstructure is discontinuous at the expansion joint, the bridge does 

not have satisfactory bearing details. 

Step 2: Although the concrete shear keys are subject to failure, the bearing seat is continuous in 

the transverse direction and therefore not subject to a serious failure resulting from 

transverse movement; i.e., VT = 5. 

Step 3: In the longitudinal direction, calculate the minimum required support length at the hinge 

seat. Given: 

L = L1 + L2 = 143 m (470 ft) 

H = (12.2 + 0) ÷ 2 = 6.1 m (20 ft) 

B = 19.8 m (65 ft) 

S1 = 0.40 

Fv = 1.4 (Site Class C, table 1-4(b)) 

SD1 = Fv S1 = 0.56 

α = 00 (no skew) 

Then from equation 5-1a: 

2 B (1 1.25S ) D1 N(d) 100 1.7L 7.0H 50 H 1 2 L cos

⎡ ⎤ ⎛ ⎞ +

=++ + + ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ α ⎣ ⎦ 

 = [100 + 243 + 43 + 128] 1.70 

 = 874 mm 

and N(c)

N(d)

 = 

203

874

 = 0.23 < 0.50 

Since 0.5 ND > Nc, 

Therefore, VL = 10 and the overall rating for connections, bearings, and seatwidths: 

V1 = maximum of VL and VT

 = 10 
525

B. Columns, Piers, and Footing

Step 1: Does not apply 

Step 2: Does not apply 

Step 3: Does not apply 

Step 4: Calculate the value for Q, for the shortest and most heavily reinforced columns which are 

the columns in bent 2. 

Q = 13 - 6 c

s max

L

P Fb

⎛ ⎞ ⎜ ⎟ ⎝ ⎠

 = 13 - 6 12.19

4.6(2)(1.219)

⎛ ⎞ ⎜ ⎟ ⎝ ⎠

 = 6.5 

Because the support skew is less than 200

, the maximum reduction that can be taken is 2 

(table 4-1), i.e.: 

CVR = Q - 2 = 6.5 - 2 = 4.5 ≈ 5 

Step 5: Does not apply 

Step 6: Does not apply 

Therefore column vulnerability rating, CVR = 5 

C. Abutments

Step 1: Does not apply 

Step 2: Does not apply because the freeway passing under the bridge is in a cut. 

Step 3: Does not apply 

 

Therefore abutment vulnerability rating, AVR = 0 

D. Liquefaction

Step 1: The soil at the site is dense to very dense unsaturated sand and gravel. Therefore, the site 

has a low susceptibility to liquefaction. 
526

Step 2: Low liquefaction-related damage is likely. 

Step 3: Does not apply 

 

Step 4: Does not apply 

Step 5: Does not apply 

Step 6: Therefore liquefaction vulnerability rating, LVR = 0 

E. Vulnerability Rating for Components Other Than Bearings, V2

V2 = CVR + AVR + LVR (Eq. 4-4) 

 = 5 + 0 + 0 = 5 

F. Overall Bridge Vulnerability (V)

The overall bridge vulnerability rating is the maximum of V1 and V2, i.e.: 

V = 10. 

E.3.2. SEISMIC HAZARD RATING, E 

The seismic hazard rating, E, is determined by the seismic coefficient at 1.0 sec period (SD1) as 

defined in equation 4-8. Thus: 

E = 10 SD1

 = 10 (0.56) 

 = 5.6 

E.3.3. BRIDGE RANK, R 

Bridge rank is given by: 

R = V E (Eq. 4-2) 

 = 10 (5.6) 

 = 56 

E.3.4. PRIORITY INDEX 

A priority index is assigned once all of the bridges are listed in order of their bridge rank, R. 

This process will require considerable judgment since this prioritized list must also take into 

account such factors as structure importance, non-seismic deficiencies, remaining useful life, 

network redundancy and the like. 
527

E.4. DETAILED EVALUATION

The existing bridge is evaluated in detail to identify structural weaknesses and potential 

retrofitting measures. The Component Capacity/Demand method is used for this evaluation 

(Method C, section 5.4). 

E.4.1. CAPACITY/DEMAND RATIOS - EXISTING BRIDGE 

Capacity/Demand ratios are calculated for the applicable components shown in table 5-2. 

Analysis Procedure (Section 5.4.2)

A multi-modal spectral analysis, is used to calculate the elastic demands on the bridge. 

Although any one of a number of computer programs can be used for such an analysis, the 

SEISAB computer program was used for this problem. SEISAB is a user-oriented program 

specifically developed for the seismic analysis of bridges and the implementation of the seismic 

design provisions in Division I-A of the AASHTO Standard Specifications (AASHTO 2002). It 

may also be used in accordance with the AASHTO LRFD Specifications (AASHTO 1998). The 

program automatically combines orthogonal elastic forces as described in section 7.4.2. 

Foundation stiffnesses at the abutments were selected using the procedure outlined in chapter 6. 

The following pages include the input coding and the applicable portion of the output from this 

program for the example bridge. It will be seen that the version of SEISAB used for this analysis 

does not permit the direct input of S1 and Fv. Instead an Acceleration Coefficient A = 0.4 is used 

and Soil Type II (S=1.2) is specified. The product AS (= 0.48) is then approximately the same as 

the FvS1 product of 0.56. Output from SEISAB is consistent with the input, which means that in 

this case the units are kilonewtons and meters. Forces and moments should be interpreted 

according to the convention shown in figures E-4 and E-5. 

Minimum Bearing Force Demands (Appendix D.2)

The minimum force demand for the transverse shear key at the expansion joint is calculated by 

considering the equivalent static load to be acting only on the suspended portion of the first span. 

Superstructure Weight = 8.44 (23.56) + 35.90 = 234.7 kN/m 

Minimum Force Demand = 0.25(234.7)(44.81 ÷ 2) = 1,315 kN 

Minimum Support Length (Appendix D.3)

N(d) = [100 + 1.7(143) + 7(6.10) +50(2.56)] 1.70 = 874 mm 

Capacity/Demand Ratio at the Expansion Joints and Bearings (Appendix D.4)

Displacement C/D Ratio (Appendix D.4.2) - Expansion joint 

Method 1: 

N(c) = 203 mm 

rbd = N(c) 203 = = 0.23

N(d) 874
528

Figure E-4. Positive sign convention for abutment, hinge and joint forces. 

Figure E-5. Positive sign convention for column forces and moments. 
529

Method 2: 

Assume that of the 203 mm of total seat length, 76 mm may be considered ineffective because it 

is the cover on the expansion joint reinforcement. 

∆s(c) = 203 - 76 = 127 mm 

∆i(d) = 84 mm (temperature, other environmental effects) 

∆eq(d) = 135 mm (from computer output) 

rbd = (127 - 84) ÷ 135 = 0.32 
530
531
532
533
534
535

Shear Force C/D Ratio (Appendix D.4.3) 

Vb(d) = 2926 x 1.25 = 3658 kN > 1,315 kN 

Vb(c) = 6 x 12 x 6

100

10

1 x 400 x 103 = 2,880 kN 

Vb(c) is the shear resistance provided by six, 762 mm long shear keys with twelve No. 4 bars in 

each key. 

bf

2880

r = = 0.79

3658

Capacity/Demand Ratios at Columns, Piers, and Footings (Appendix D.5)

Step 1: Elastic Moment Demands

The elastic moment demands are calculated by combining the moments about the principal axes 

of the columns to obtain the maximum moments. In most cases, Load Case 2 has the highest 

demands. Dead load moments, which are also included in the calculations, have been obtained 

from a separate analysis. Moments at the base of the footing were obtained by adding the 

moment created by the shear at the top of the footing to the moment at the top of the footing. 

Elastic moments of highest demands are summarized in table E-1. 

 Table E-1. Maximum elastic moment demands (kN m). 

Trans. Moment Long. Moment 

Location Component 

EQ DL EQ DL 

Elastic 

Moment 

Demand 

B-2 (C-2) Top Column 22,685 1,786 4,011 107 24,815 

B-2 (C-2) Bottom Column 23,728 895 4,281 14 24,995 

B-2 (C-2) Bottom Footing 30,248 1,268 5,458 27 31,990 

B-3 (C-2) Top Column 11,436 1,243 3,251 228 13,148 

B-3 (C-2) Bottom Column 12,021 619 3,381 168 13,129 

B-3 (C-2) Bottom Footing 13,534 737 3,815 193 14,823 

B-4 (C-2) Top Column 1,949 1,315 7,889 33 8,568 

B-4 (C-2) Bottom Column 1,992 660 8,222 68 8,704 

B-4 (C-2) Bottom Footing 2,247 784 9,269 74 9,822 
536

Step 2: Ultimate Moment Capacities

Ultimate moment capacities for the columns are obtained from the computer generated column 

interaction diagrams shown in figures E-6 and E-7. Ultimate moment capacities for the footing 

are obtained from interaction diagrams for the footings, which are also shown in figures E-6 and 

E-7. Ordinates for the footing interaction diagram for bent 2 are calculated in figure E-8. 

Interaction diagrams for the footings at bents 3 and 4 are similar. 

Figure E-6. Bent 2 column and footing interaction diagrams. 

Because the elastic moment demands are primarily in the plane of the bent, moment capacities 

will be calculated for bending in this plane. This requires a consideration of the variation in 

axial load due to bent overturning as outlined in the iterative procedure presented in article 4.8.2 

of Division I-A of the AASHTO Standard Specifications (AASHTO, 2002). The steps of this 

procedure are as follows. 
537

Figure E-7. Bents 3 and 4 column and footing interaction diagrams. 

Step 2.1 Overstrength Moment Capacities at Axial Load Corresponding to Dead Load

Table E-2 summarizes the overstrength column and footing moment capacities taken from the 

interaction diagrams. An example for the bottom of bent 2 is shown in figure E-8. Bents 3 and 4 

have identical capacities. 

Table E-2. Column and footing overstrength moments. 

1.3 Mu

Bent End 

Axial Force 

Due to Dead 

Load Column Footing 

2 Top 5,293 11,796 - 

2 Bottom 5,694 11,796 10,223 

3 & 4 Top 4,270 8,379 - 

3 & 4 Bottom 4,715 8,528 7,213 
538

Figure E-8. Development of footing interaction diagram at bent 2. 

Step 2.2 Column Shear Forces*

Bent 2: Vu = (11769 + 10223) ÷ 14.08 = 1,562 kN 

 

Bent 3 & 4: Vu = (8379 + 7213) ÷ 16.25 = 960 kN 

* Because the ultimate moment is less at the footing than at the column base, the footing 

moments and the distance between the superstructure soffit and the base of the footing are used 

to calculate column shears. 
539

Step 2.3 Axial Forces Due to Overturning in the Transverse Direction

Bent 2: Axial Force = 2(1562)(14.08) ÷ 9.45 = + 4,655 

Bent 3 & 4: Axial Force = 2(960)(16.25) ÷ 9.45 = + 3,302 

Step 2.4 Revised Overstrength Moment Capacities

The axial loads due to overturning calculated in Step 3 are used to obtain new overstrength 

moment capacities from the interaction diagrams. Table E-3 summarizes these revised moment 

capacities. 

These moment capacities are used to calculate revised shear forces at the bent. 

Bent 2: Shear = (11036 + 2088) ÷ 14.08 + (11796 + 11674) ÷ 14.08 = 2,599 kN 

Bents 3 & 4: Shear = (6996 + 2996) ÷ 16.25 + (8786 + 7132) ÷ 16.25 = 1,594 kN 

These bent shears are not within 10 percent of the bent shears (twice the column shear) 

calculated in Step 2. Therefore the axial forces due to overturning must be recalculated. 

Bent 2: Axial Force = 2599(14.08) ÷ 9.45 = 3,872 kN 

Bent 3 & 4: Axial Force = 1594(16.25) ÷ 9.45 = 2,741 kN 

These axial loads are used to recalculate the overstrength moments, which are summarized in 

table E-4. 

Table E-3. Revised column and footing overstrength moments (iteration 1).

1.3 Mu

Bent End 

Axial Force 

Due to Dead 

Load 

+ 

Overturning Column Footing 

2 Top 638 11,036 - 

2 Top 9,948 11,796 - 

2 Bottom 1,039 11,145 2,088 

2 Bottom 10,349 11,741 11,674 

3 & 4 Top 968 6,996 - 

3 & 4 Top 7,572 8,786 - 

3 & 4 Bottom 1,413 7,769 2,996 

3 & 4 Bottom 8,017 8,813 7,132 
540

 Table E-4. Revised column and footing overstrength moments (iteration 2). 

1.3 Mu

Bent End 

Axial Force 

Due to Dead 

Load 

+ 

Overturning Column Footing 

2 Top 1,421 11,240 - 

2 Top 9,165 11,836 - 

2 Bottom 1,822 11,321 3,715 

2 Bottom 9,566 11,823 11,972 

3 & 4 Top 1,529 7,728 - 

3 & 4 Top 7,011 8,759 - 

3 & 4 Bottom 1,974 7,850 3,864 

3 & 4 Bottom 7,456 8,786 7,389 

New shear forces at the bents are calculated using these moments. 

Bent 2: Shear = (11240 + 3715) ÷ 14.08 + (11836 + 11972) ÷ 14.08 = 2,753 kN 

Bents 3 & 4: Shear = (7728 + 3864) ÷ 16.25 + (8759 + 7389) ÷ 16.25 = 1,707 kN 

The bent shears are now within 10 percent of the previously calculated shears and therefore no 

further iteration is needed. 

Step 3: Ultimate Moment Capacity/Elastic Moment Demand Ratios

The most critical combinations of the unfactored nominal ultimate moment capacities (Mu) and 

elastic moment demands are used to calculate rec and ref at each bent. Values of rec and ref are 

summarized in table E-5. 
541

Table E-5. Ultimate moment capacity/elastic moment demand ratios. 

Column Footing

Bent End Axial 

Load Demand Capacity rec Demand Capacity ref

 2 Top Min. 24,815 8,650 0.35 - - 

 2 Top Max. 24,815 9,111 0.37 - - 

 2 Bottom Min. 24,995 8,704 0.35 31,990 2,861 0.09 

 2 Bottom Max. 24,995 9,098 0.36 31,990 9,206 0.29 

 3 Top Min. 13,148 5,938 0.45 

 3 Top Max. 13,148 6,738 0.51 

 3 Bottom Min. 13,129 6,033 0.46 14,823 2,969 0.20 

 3 Bottom Max. 13,129 6,752 0.51 14,823 5,681 0.38 

 4 Top Min. 8,568 5,938 0.69 

 4 Top Max. 8,568 6,738 0.79 

 4 Bottom Min. 8,704 6,033 0.69 9,822 2,969 0.30 

 4 Bottom Max. 8,704 6,752 0.76 9,822 5,681 0.58 

Step 4: Calculate C/D Ratios for Possible Plastic Hinging Cases at the Bottom of the Columns

Bent 2 - Case II (rec = 0.35 and ref = 0.09): 

1. Anchorage (appendix D.5.1) - Straight anchorage 

la(c) = 1880 - 76 = 1,804 mm 

For anchorage in the footing, assume the large cover (1575 mm) has a confining 

effect equal to transverse steel with equivalent tensile strength. In this case, twice the 

area of the cover divided by half the number of longitudinal bars is considered. 

Concrete tensile strength = 19.69 = 2947 kPa f c

′

ktr = 2947(1575)(2)/[21÷2)(4137)(55)] = 3.89 > 2.5 

( )

3

a 4

((400 x - 75842) 33.1) 55 10 (d) = 

2.24 x / 2.63 (1 + 2.5(55 55) + 2.5) 10

A

÷

÷

 = 1,578 mm 
542

Therefore, Case B applies. Calculate the negative moment capacity of the footing 

using a concrete tensile strength of 2,947 kPa (19.69√f′

c). 

Negative Moment Capacity = 2947(4.267(1.88)2

÷ 6) 

 = 7,407 kN m 

This capacity is sufficient to resist the weight of the overburden. Therefore: 

rca = 1.0 

2. Splices (appendix D.5.2) - Does not apply 

3. Footing Rotation (appendix D.5.5) 

Because anchorage or splice failures will not prevent footing rotation: 

rfr = μref = 4(0.09) = 0.36 

Bent 2 - Case II (rec = 0.36 and ref = 0.29): 

1. Anchorage - Same as before 

rca = 1.0 

2. Splices - Does not apply 

3. Footing Rotation 

rfr = μref = 4(0.29) = 1.16 

Bent 3 - Case II (Two possible combinations of rec and ref must be investigated: 

rec = 0.46 and ref = 0.20 plus rec = 0.51 and ref = 0.38). 

1. Anchorage - Hooked Anchorage 

la(c) = 838 mm 

la(d) = 0.7(1200) (35) ÷ 4 ( 2.24 x / 2.63) 10 = 517 mm 

Case B applies. 

Negative Moment Capacity = 2947(3.658(0.991)2

÷ 6) 

 = 1,764 kN m 

Because this capacity is sufficient to resist the weight of the overburden, 

rca = 1.0 
543

2. Splices (appendix D.5.2) 

Because the clear spacing between splices = 38 mm < 4(35) 

Atr(c) = 2(100)/(33 ÷ 2) = 12.1 

Atr(d) = 236

1422

 1000 = 166 

ls = 1422 > b

c

4885

d

f ′

⎛ ⎞ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

2 = 1,144 

Therefore, Case A applies. 

rcs = 0.75 (0.46) = 0.35 

rcs = 0.75 (0.51) = 0.38 

Notice that the minimum value for rcs controls. 

3. Footing 

0.8 ref = 0.8(0.38) = 0.30 < 0.38 

0.8 ref = 0.8(0.20) = 0.16 < 0.35 

Therefore a splice failure cannot be assumed to prevent footing rotation. The 

minimum C/D ratio for the footing is given by 

rfr = 4(0.20) = 0.80 

Bent 4 - Case II (rec = 0.69 and ref = 0.30): 

1. Anchorage - Same as bent 3 

rca = 1.0 

2. Splices 

rcs = 0.75 (0.69) = 0.52 

3. Footing 

0.8 ref = 0.8(0.30) = 0.24 < 0.52 

Bent 4 - Case II (rec = 0.76 and ref = 0.58): 

1. Anchorage 

rca = 1.0 
544

2. Splices 

rcs = 0.75 (0.76) = 0.57 

3. Footing 

0.8 ref = 0.8(0.58) = 0.46 < 0.57 

Therefore, a splice failure cannot be assumed to prevent footing rotation. The 

minimum C/D ratio for the footing is given by: 

rfr = 4(0.30) = 1.20 

Step 5: Calculate C/D Ratio at the Top of the Column

Bent 2

1. Anchorage 

la(c) = 1,676 mm 

la(d) = 1200 (55) ÷ 4 2.24 x / 2.63 = 1160 mm 10

Therefore Case B applies: 

rca = 1.0 

2. Splices - Does not apply 

3. Confinement (appendix D.5.4) 

ρ(c) = 100(π)(1118) ÷ [π(610)2

(236)] = 0.0013 

ρ(d) = 

2 4

2 5

 (610 2.24 x ) 10 0.45 - 1 

 (559) 10 4 x 

⎛ ⎞ π

⎜ ⎟

⎝ ⎠ π

 = 0.0048 

c

c g

P

f A′ = 

4 2 2

9165 = 0.35

(2.24 x ) (610 /(1000 10 π ) )

k1 = 

0.0013

0.0048 (0.5 + 1.25(0.35))

 = 0.29 

k2 = 

0.2 = 0.8

(305 1219) ÷
545

Because transverse steel is poorly anchored, an iterative solution for μ is required. 

Try k3 = 0.35 (corresponds to μ = 2.7) 

 μ = 

0.29 + 0.80 2 + 4 0.35

2

⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = 2.8 ok 

 rcc = μrec = 2.8(0.35) = 0.98 

Bent 3

1. Anchorage 

la(c) = 1,676 mm 

la(d) =

3

4

((400 x - 75842) 33.1) 35 10

36 ( 2.24 x / 2.63) (1 + 2.5 ) 10 35

÷

 = 1,687 mm 

Case A applies 

rca = 

1676 (0.45) = 0.45 1687

2. Splices - Does not apply 

3. Confinement 

k1 =

0.0013

0.0048 (0.5 + 1.25(0.270))

 = 0.32 

k2 =

6 = 0.69

305/ 35

Try k3 = 0.35 (corresponds to μ = 2.7) 

μ =

0.32 + 0.69 2 + 4 0.35

2

⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = 2.7 ok 

rcc = 2.7 (0.45) = 1.22 
546

Bent 4

1. Anchorage 

rca = 

1676 (0.69) = 0.69 1687

2. Splices - Does not apply 

3. Confinement 

rcc = 2.7(0.69) = 1.86 

Step 6: Calculate C/D Ratios for Column Shear (appendix D.5.3)

Bent 2 - Transverse bending - Footing rotation will govern the maximum shear. 

Therefore use the nominal footing overstrength moment plus an effective length measured to the 

base of the footing. 

Vu(d) =11836 + 11972

14.08 = 1,691 kN 

Vc(d) = 3468 + 198 = 3,666 kN 

Vi(c) = tr yt

c

A f d

v db + 

s

 =

5

2 2

787(1059)(1219) (2 x 100)(4 x )(1059) 10 + 

(1000 236(1000 ) )

 = 1,375 kN 

Because column axial stress may fall below 0.10f c

′

and transverse steel is ineffective: 

Vf(c) = 0 

Therefore, Case A applies; i.e.: 

rcv = 1375 = 0.38 > 0.36

3666 (the value of rec) 

rcv = 0.36 

Bent 3 - An anchorage failure at the top of the column and rotation of the footing at the bottom 

of the column will limit the maximum shear. 
547

Vu(d) = (1676/1687) 8759 + 7389

16.25

 = 990 kN 

Ve(d) = 1527 + 119 = 1,646 kN 

Vi(c) = 1,375 kN 

Vf(c) = 0 

Therefore, Case B applies; i.e. 

μ =

1375 - 990 2 + (0.75)(4) 1375 - 0

⎛ ⎞ ⎜ ⎟ ⎝ ⎠ = 2.8 

rcv = 2.8(0.45) = 1.26 

Bent 4

rec = 0.69 

Ve(d) = 671 + 125 = 796 kN 

μ = 2.8 

rcv = 2.8 (0.69) = 1.93 

Capacity/Demand Ratio for Abutments (Appendix D.6)

Abutment C/D ratios are based on the displacements from the analysis. 

Transverse Displacement

d(c) = 75 mm 

Abutment 1: 

d(d) = 1 mm 

rad = 

75 = 75

1

Abutment 5: 

d(d) = 23 

rad = 

75 = 3.3

23

Longitudinal Displacement

d(c) = 150 mm 
548

Abutment 1: 

d(d) = 9 mm 

rad = 

150 = 16.7

9

Abutment 5: 

d(d) = 119 mm 

rad = 

150 = 1.26

119

Capacity/Demand Ratio for Liquefaction (Appendix D.7)

Because the preliminary screening (Seismic Rating System) indicated that the likelihood of 

liquefaction-related damage was low, a C/D ratio for liquefaction is not determined. 

E.4.2. IDENTIFICATION AND ASSESSMENT OF POTENTIAL RETROFIT MEASURES 

Table E-6 summarizes the C/D ratios that are less than 1.0 for the bridge. 

Table E-6. Capacity/demand ratios for the as-built bridge. 

Component Notation 

C/D Ratios 

for As-Built 

Bridge 

rbd 0.23 

Expansion Joint 

rbf 0.79 

Bent 2 (Overall) rcv 0.36 

Bent 2 (Bottom) rfr 0.36 

Bent 3 (Bottom) rfr 0.80 

Bent 3 (Top) rca 0.45 

Bent 4 (Top) rca 0.69 

The expansion joint displacement is critical because it has the lowest C/D ratio and may result in 

a partial collapse of the bridge. This may be economically corrected by retrofitting the joint with 

longitudinal expansion joint restrainers. Because the transverse shear keys are also inadequate 

as indicated by the C/D ratio for bearing force, transverse pipe restrainers should also be 

included in the retrofit. 

A potentially serious failure is indicated by the C/D ratio for shear in bent 2. Shear failure could 

be sudden and result in rapid disintegration of the column’s ability to support axial load. The 

seriousness of this particular failure is compounded because the column is located adjacent to the 
549

expansion joint, which increases the probability of a partial collapse. Therefore, the consequences of a shear failure in bent 2 are unacceptable and this deficiency should be corrected. 

Because this failure mode is caused by forces transverse to the centerline of the bridge, an infill 

shear wall at bent 2 would be a relatively economical way to retrofit the bridge. This type of 

retrofit would also reduce the potential for a rotational failure of the footing at bent 2. 

The next lowest C/D ratio occurs at the column steel anchorage in bent 3. There are several 

factors that make this potential failure of less concern. The primary effect of anchorage loss will 

be a reduction in the flexural strength at the top of the column. But because the bent is 

redundant, this will not result in the formation of a collapse mechanism. Therefore, this kind of 

failure in itself is not considered unacceptable. However, the rotation failure of the footing at 

bent 3 would threaten the stability of this bent, particularly when combined with the possibility 

of anchorage failure. But as noted, if the bent is stiffened with an infill shear wall, the likelihood 

of this happening is greatly reduced and additional retrofitting is not proposed. 

The top column steel anchorage of bent 4 has a C/D ratio (rca) of 0.69. Since the C/D ratio of 

footing rotation (rfr = 1.20) is larger than 1.0, failure of steel anchorage would not cause a 

collapse mechanism to occur, and is considered an acceptable failure. Therefore, it is not 

proposed to retrofit bent 4. 

Because the infill shear wall at bent 2 would significantly affect the dynamic response of the 

structure, another analysis is required. Computer input and output files for the retrofitted bridge 

are included on the following pages. An abbreviated reevaluation of the C/D ratios for the most 

critical components in the retrofitted bridge shows that the C/D ratios at bent 3 are greatly 

improved by the modified response. 

Capacity/Demand Ratio at the Retrofitted Expansion Joint

Displacement C/D Ratio

Method 2: 

∆s(c) = 127 mm 

∆i(d) = 84 mm 

∆eq(d) = 24 mm 

rbd = (127 - 84) = 1.79 > 1 ok

24

Capacity/Demand Ratios at the Columns, Piers, and Footing

Bent 3 (Bottom) 

ref ≈ 2969 ÷ 6052 = 0.49 

rfr = 4(0.49) = 1.96 ok 
550

Bent 3 (Top) 

rec ≈ 5938 ÷ 5058 = 1.17 ok 

rca = 

1676 (1.17) = 1.16 1687

 ok 
551
552
553
554
555
556
557

APPENDIX F: EXAMPLE PROBLEM 5.2: 

COMPONENT CAPACITY-DEMAND EVALUATION OF A 

MULTISPAN SIMPLY-SUPPORTED STEEL GIRDER BRIDGE

F.1. INTRODUCTION

This example problem illustrates the evaluation of a multiple span, simply-supported steel girder 

bridge using the Component Capacity-Demand method described in section 5.4 (Method C). 

Two cases are analyzed for the bridge: Case 1 assumes the bridge is located where the longperiod spectral acceleration for the upper level earthquake (S1) is 0.16g, and Case 2 assumes this 

acceleration is 0.26g. 

F.2. DESCRIPTION OF THE EXAMPLE BRIDGE

The example bridge is part of the L.R. 767-5 bridge over L.R. 22019 in Dauphin County, 

Pennsylvania. The bridge contains several simple steel beam spans and several continuous 

spans. A plan and elevation of the portion of the bridge studied in this example is shown in 

figure F-1. Pier 2 is evaluated in detail in this example, and is shown in figure F-2. 

The bridge was built in 1968 and rehabilitated 10 years ago to give an anticipated service life of 

at least 50 years. It is considered to be an essential bridge. The bearings are of the steel rocker 

type. Two 32 mm anchor bolts per bearing are used to connect the bearing shoes to the bent cap. 

Both piers 1 and 2 are founded on rock. 

Figure F-1. Plan and elevation of example bridge. 
558

Figure F-2. Pier 2 bent details. 
559

F.3. BRIDGE EVALUATION 

F.3.1. SEISMIC RETROFIT CATEGORIES 

Based on the estimated 50-year service life, the Service Life Category is ASL 3 (table 1-1). 

Since the bridge is ‘essential’, the performance level is PL2 (table 1-2), for the upper earthquake 

level. The Site Class is B, for which Fv = 1.0 (table 1-4(b)). The Seismic Retrofit Category is 

determined using the procedure given in section 1.6, and the results are summarized in table F-1. 

F.3.2. COMPONENTS FOR CAPACITY/DEMAND RATIO EVALUATION 

Case 1 

Based on table 5-2 the components that must be evaluated are the bearings (support length and 

forces) and the potential for liquefaction. 

Case 2

Based on table 5-2, the components that must be evaluated include the bearings (support length 

and forces), the reinforced concrete column bents and their footings (anchorage, splices, shear, 

and confinement), and the potential for liquefaction. 

Table F-1. Seismic retrofit categories for upper level earthquake (Fv=1.0, PL2). 

Case 

Long-Period 

Spectral 

Acceleration, 

Upper Level 

Earthquake, S1 

Seismic 

Coefficient 

SD1 =FvS1

Seismic 

Hazard Level 

SHL (table 1-5) 

Seismic 

Retrofit 

Category 

SRC (table1-6) 

1 0.16 0.16 II B 

2 0.26 0.26 III C 

 

 

F.3.3. ANALYSIS PROCEDURE 

Case 1 

Based on table 1-9, analysis is not required for this case. Minimum capacity checks are 

performed instead. 

Case 2

Based on table 1-9 and recommendations in section 5.1.1, Method C is chosen (Component 

Capacity-Demand Method) and since the bridge is regular, the Uniform Load Method is used to 

obtain elastic demands. 
560

F.3.4. UNIFORM LOAD METHOD FOR CASE 2 

F.3.4.1. Structural Data - Pier 2 

Column diameter D = 915 mm 

Column area A = 660 x 103 mm2

Column moment of inertia I = 34.5 x 109

 mm

4

Bent cap area Ac = 1,100 x 103 mm

2

Bent cap moment of inertia Ic = 103 x 109 mm

4

Modulus of elasticity E = 26 x 103 MPa 

Weight of span 2 Wsp2 = 1,920 kN 

Average weight of spans 2+3 Wsp2,3 = 1,530 kN 

F.3.4.2. Longitudinal Earthquake Loading 

The expansion rocker bearings provide very little resistance to longitudinal movements and most 

of the longitudinal load will be transferred to the fixed end of the span. Span 2 will most likely 

move as a rigid body and its movement will be resisted by the longitudinal stiffness of pier 2. 

The weight distribution along the span is uniform and, if the response of each simple span is 

assumed to be independent of the other spans, the natural period in the longitudinal direction 

may be computed as follows: 

Longitudinal Stiffness of Bent 2 Acting as a Cantilever: 

Note that uncracked section properties are used due to an anticipated overdesign of the columns. 

Fundamental Period of Pier 2 in the Longitudinal Direction:

Seismic Coefficient (figure 1-8):

Cs = SD1/T 

Thus CS = 0.16/0.81 = 0.20 for Case 1 (SD1 = 0.16) 

 CS = 0.26/0.81 = 0.32 for Case 2 (SD1 = 0.26) 

3 9 4

L 3 3

L

3E I (3)(26 x MPa) (34.5 x ) 10 10 mm K = (2) = (2) = 11.8 kN/mm

H (7,700 mm)

L

L 2

L

W 1,920 kN T = 2 = 2 = 0.81 sec

K g (11.8 kN/mm) (9,810 mm/ ) sec

π π
561

F.3.4.3. Transverse Earthquake Loading 

Since the superstructure comprises several simple spans sitting on rocker type bearings, the inplane continuity effect of the spans is neglected. Thus the natural period of pier 2 in the 

transverse direction is computed as a stand-alone pier based on the stiffness of the pier and the 

tributary mass of the superstructure from spans 2 and 3. 

Transverse Stiffness of Pier 2 Acting Alone:

The transverse stiffness of pier 2 was obtained from a plane frame model of analysis by 

computing the horizontal displacement of the cap due to a unit horizontal load and then taking its 

inverse. 

 KT = 54 kN/mm 

The transverse stiffness of the bent may also be approximated by assuming a rigid cap and fixed 

end conditions: 

which yields a value about 9 percent higher. 

Fundamental Period of Pier 2 in the Transverse Direction: 

Seismic Coefficient (figure 1-8): 

Cs = SD1/T 

Thus CS = 0.16/0.34 = 0.47 for Case 1 (SD1 = 0.16) 

 CS = 0.26/0.34 = 0.76 for Case 2 (SD1 = 0.26) 

F.4. CAPACITY/DEMAND RATIO FOR BEARINGS FOR CASE 1

F.4.1. DISPLACEMENT CAPACITY/DEMAND RATIO (CASE 1) 

The available support length is: 

 N(c) = 430 mm (This is the length from the end of the girder to the edge of the cap.) 

3 9 4

T 3 3

T

12 EI (12)(26 x MPa) (34.5 x ) 10 10 mm K = (2) = (2) = 58 kN/mm

H (7,200 mm)

′

T

T 2

T

W 1.53 kN

T = 2 = 2 = 0.34 sec

K g (54 kN/mm) (9,810 mm/ ) sec

π π
562

The minimum support length using equation 5-1a, for Case 1 (SD1 = 0.16) is given by: 

 N(d) = (100 + 41 + 53 + 166) 1.2 = 432 mm 

The displacement capacity/demand ratio is: 

 bd

N(c) 430

r = = = 1.0

N(d) 432

 ************ 

F.4.2. FORCE CAPACITY/DEMAND RATIO (CASE 1) 

Bearing failure may be caused by shearing of the anchor bolts, sliding along the rotation pin in 

the transverse direction, or shifting over the rotation pin in the longitudinal direction. 

Shearing of Anchor Bolts

The shear capacity of the anchor bolts is: 

 Vb(c) = (2)(4)(800 mm2

)(186 MPa) = 1,200 kN 

The minimum bearing force demand is 25 percent of the dead load: 

 Vb(d) = (0.25)(1,920 kN) = 480 kN 

 The anchor bolts C/D ratio is: 

 b

bf

b

V (c) 1,200

r = = = 2.5

V (d) 480

 ************* 

Sliding of Bearing Along the Rotation Pin

If a coefficient of friction of 0.25 is assumed, the C/D ratio is given by: 

 b

bf

b

V (c) 0.25

r = = = 2.5

V (d) 0.47

 ********************** 

F.5. POTENTIAL FOR LIQUEFACTION IN CASE 1

Since pier 2 is founded on rock, there is no potential for liquefaction. 

Thus, the governing force C/D ratio is 0.53 for Case 1. 

F.6. CAPACITY/DEMAND RATIO FOR BEARINGS FOR CASE 2

The rocker type bearings are vulnerable to earthquakes and they may suffer damage even during 

a moderate earthquake. 
563

F.6.1. DISPLACEMENT CAPACITY/DEMAND RATIO (CASE 2) 

Method 1

The available support length is: 

 N(c) = 430 mm (This is the length from the end of the girder to the edge of the cap.) 

The minimum support length using equation 5-1a, for Case 2 (SD1 = 0.26) is: 

 N(d) = (100 + 41 + 53 + 166) 1.325 = 477 mm 

The displacement C/D ratio is: 

 bd

N(c) 430

r = = 0.9

N(d) 477 ≈ *************** 

Method 2

The available capacity of the expansion bearing for movement is: 

 ∆s(c) = 430 mm 

The maximum possible movement resulting from temperature changes is: 

 ∆i(d) = (11.7 x 10-6)(420

C)(24,300 mm) = 1 mm 

The maximum calculated relative displacement in the longitudinal direction for Case 2 

(SD1 = 0.32) is: 

 ∆eq(d) ≈ (2)(0.32)(1,920 kN)/(11.8 kN/mm) = 104 mm 

This is conservatively estimated by assuming completely out-of-phase motion at the two 

adjacent piers; i.e., ∆eq(d) has been increased by a factor of 2. 

The displacement C/D ratio is: 

 s i

bd

eq

 (c) - (d) 430 - 13

r = = = 4.0 > 1.0

 (d) 104

∆ ∆

∆

*************** 

Thus, the lowest displacement C/D ratio is 0.9 by Method 1, but may be replaced by the more 

realistic result from Method 2, to obtain the governing value (4.0). 
564

F.6.2. FORCE CAPACITY/DEMAND RATIO (CASE 2) 

Bearing failure may be caused by shearing of the anchor bolts, sliding along the rotation pin in 

the transverse direction, or shifting over the rotation pin in the longitudinal direction. 

Shearing of Anchor Bolts

The longitudinal seismic force demand per anchor bolt for Cs = 0.32 is: 

 Vb(d)l = (1.25)(0.32)(1,920 kN)/8 = 96 kN 

The transverse seismic force demand per anchor bolt for Cs = 0.76 is: 

 Vb(d)t = (1.25)(0.76)(1,530 kN)/16 = 91 kN 

The combined seismic force demand per anchor bolt for this case is: 

 2 2

b c V (d = (96 + [(0.3)(91) = 100 kN ) ) ] ************** 

The anchor bolt C/D ratio is: 

 b

bf

b c

V (c) 150

r = = = 1.5

V (d 100 )

****************** 

Sliding of Bearing Along the Rotation Pin

If a coefficient of friction of 0.25 is assumed the C/D ratio is given by: 

 b

bf

b

V (c) 0.25

r = = = 0.33

V (d) 0.76

 ********************************** 

Thus, the governing force C/D ratio is 0.33 for Case 2. 

F.7. CAPACITY/DEMAND RATIOS FOR PIER 2 FOR CASE 2

The C/D ratio computations for pier 2 are shown in the following pages. The computations 

include derivation of the column interaction surface, the footing interaction surface in the 

transverse and the longitudinal direction (see figures F-3, F-4, and F-5), plastic hinge analysis for 

the transverse direction, seismic elastic load analysis, ultimate moment capacity calculations 

based on seismic axial loads, and a C/D ratio evaluation for the bent. 

Note that these results have been computed using SEISAB, which does not allow the direct input 

of SD1, nor does it use the spectral shape specified in figure 1-8. Instead an Acceleration 

Coefficient, A, is used and the seismic coefficient is calculated using Cs =1.2 AS/T2/3 for the 

spectral shape. For rock, S = 1 and therefore Cs at T = 1.0 sec = 1.2 A. Hence values of A= 0.1 
565

Figure F-3. Column interaction diagram. 

Figure F-4. Foundation interaction diagram. 
566

Figure F-5. Foundation interaction diagram - transverse direction. 

and 0.2, give seismic coefficients very close to the SD1 values of 0.16 and 0.26 respectively. Thus 

values of A = 0.1 and 0.2 have been used to obtain the attached output. 

The plastic hinge analysis in the transverse direction shows that the capacity of the bent is 

limited by uplift of the column footing sitting on rock foundation, before any hinge mechanism 

can form. Low axial compression load in a column can significantly reduce the C/D ratio at the 

base of the footing since the resistance to rotation provided by the foundation rock is dependent 

on the existence of compression stresses at the interface. 

In the longitudinal direction, the acceleration coefficient (A) of 0.1 yields C/D ratios of greater 

than 1.0. When A is 0.2, C/D ratios less than 1.0 are obtained. The limiting factor is also the 

resistance to rotation provided by the foundation rock. This condition is caused by the relatively 

large moment arm to foundation width ratio. Thus, for an acceleration coefficient of 0.2, the 

assumption of column base fixity used in the computation of the natural period is not adequate. 

A partial column base fixity assumption would be more appropriate for both the longitudinal and 

the transverse direction. This would increase the fundamental periods, lower the seismic 

demand, and further increase the column C/D ratios. For all cases, the seismic elastic loads are 

relatively low. Due to the limited capacity for uplift and rotation provided by the foundation 

rock, flexural yielding of the columns is not expected. Thus, the assumption of uncracked 

column section properties used is applicable in this case. In addition, only the C/D ratios for 

anchorage, splices of longitudinal reinforcement, and column shear need to be evaluated (see 

figure D-3). The anchorage of the reinforcing bars is satisfactory. The minimum splice length 

for the longitudinal reinforcement is provided; however, the C/D ratio for the splices is less than 

0.75 rec and therefore a value of 0.75 rec is applicable. Also, since the columns do not yield, the 
567

C/D ratio for column shear is given by the ratio of initial shear resistance of the undamaged 

column to the maximum calculated elastic shear force, which is greater than 1.0. 

F.8. POTENTIAL FOR LIQUEFACTION IN CASE 2

Since pier 2 is founded on rock, there is no potential for liquefaction. 
568
569
570
571
572
573
574
575
 
577

Glossary 

Anticipated Service Life (ASL) – An estimate of the remaining life of a bridge based on current 

age, structural condition, specification used for design, and capacity to handle current and future 

traffic. The ASL may be taken as the expected life of a new bridge (75 years, AASHTO 1998) 

less the age of the bridge under consideration. 

Arching – One of the mechanisms by which a bridge deck distributes lateral loads to the 

abutments, by acting as an arch in the plane of the deck spanning from one abutment to the other. 

If expansion joints occur within the superstructure, arching action cannot develop until all of 

these joints are closed. 

Bolster – A structural addition to the face of a cap beam or abutment seat that extends the 

support length for the superstructure. Bolsters are usually dowelled to beam or seat to assist the 

transfer of forces and moments across the interface. In some cases a bolster may be used as an 

overlay. See Overlay. 

Bridge Rank – A numerical index which indicates the seismic vulnerability of a bridge. It is 

based on the structural vulnerability of the bridge (seismic deficiencies) and the site hazard. The 

rating ranges from 0 to 100; the higher the number the greater the vulnerability. 

Bumper Block – A steel bracket attached to the underside of a steel stringer that will restrict 

longitudinal movement of the bridge superstructure in the event of excessive motion. The 

superstructure will be free to move until the block impacts the abutment. 

Capacity / Demand (C/D) Ratio – The ratio of the capacity of a member (or bridge system) to 

the seismic demand on that member (or bridge system). Ratios less than 1.0 indicate members or 

systems needing consideration for retrofitting. In this Manual, C/D ratios are developed for 

bending moments, shear and axial forces, displacements, and rotations. 

Capacity Protected Design – A design methodology that protects selected components and/or 

members of a bridge from excessive seismic forces by the yielding of adjacent components 

and/or members. For example, the maximum shear that can be transmitted to a footing by a 

column is determined by the flexural capacity of the column, i.e., the column’s yield strength. In 

other words, the footing is protected by the capacity of the column, and need only be designed 

for the forces that can be transmitted by the column when at yield (including the effect of any 

overstrength in the column). See Capacity Protected Element. 

Capacity Protected Element – An element of a bridge that is protected from excessive seismic 

forces by the yielding of an adjacent element in the same load path. See Capacity Protected 

Design. 

Capacity Spectrum Design – A design procedure that combines demand and capacity analyses 

in one step, usually through the use of a capacity spectrum which plots spectral acceleration 

against spectral displacement (e.g., total base shear against structure displacement). 
 578

Central and Eastern United States – The Central and Eastern U.S. is generally considered to 

be those states east of the Rocky Mountains. 

Collateral Seismic Hazard – A seismic hazard other than direct ground shaking e.g. ground 

failure (slumping, settlement), liquefaction, fault rupture, landslide and the like. 

Complete Quadratic Combination (CQC) – A statistical rule for combining individual modal 

responses of a bridge subject to earthquake loads so as to obtain the maximum response to be 

used for design. See Multimode Spectral Analysis Method. 

Design Strength – This is the strength of a member after making allowance for approximations 

in calculations and variations in material strengths, workmanship, and dimensions. Each of these 

factors may be within tolerable limits but in combination may result in a capacity that is less than 

the nominal strength. A strength reduction factor, which is always less than unity, is used to 

relate the design strength to the nominal strength. 

Dowel – A short bar of steel used to assist the transfer of shear between a structural overlay or 

bolster and an existing concrete member. 

Ductile Elements – Members or components of a bridge that are expected to undergo significant 

inelastic deformations while maintaining their strength and stability. 

Energy Dissipation Device – A mechanical device that dissipates energy during an earthquake 

and limits bridge movements to acceptable levels. Energy is typically dissipated by plastically 

deforming a metal such as lead or steel, employing friction, or moving a viscous fluid through an 

orifice. 

Essential Bridge – A bridge that is needed to continue functioning immediately after an 

earthquake, or one which crosses a route that is expected to remain open immediately following 

an earthquake. Any bridge not classified as essential is said to be a standard bridge. 

Exempt Bridge – A bridge may be exempt from retrofitting if it is located in the lowest seismic 

zone, or has limited remaining useful life. Temporary bridges and those closed to traffic, may 

also be exempt. 

Expected Damage Method - One of three methods used to determine the seismic vulnerability 

rating of a bridge in the screening and prioritization of bridges for retrofitting. This method 

compares the severity of expected damage for each bridge in the inventory, for the same 

earthquake. Severity of damage is measured either by the damage state(s) sustained or by the 

estimated direct economic losses. 

Expected Strength – This is the strength of a member or component after making allowance for 

the fact that actual material strengths are generally greater than the specified strengths. For 

example, the yield strength of steel may be 20 percent higher than the specified strength, and 

concrete strength may be 30 percent higher the specified strength, or even greater with age 

and/or confinement. Expected strengths can be found from routine testing of material samples 

taken from existing bridges, but if these are not available, they are based on previous experience 
579

with the same materials in other bridges. An expected strength factor (φe) is used to relate the 

expected strength to the nominal strength, where φe is always greater than unity. 

Fragility Curve – A graph that indicates the probability of a bridge sustaining a particular 

degree of damage when subject to a given level of ground shaking. 

Indices Method – One of three methods used to determine the seismic vulnerability rating of a 

bridge in the screening and prioritization of bridges for retrofitting. Indices for both structure 

vulnerability and hazard level range from 0 to 10, and their product is used to give the Bridge 

Rank. These indices are based on conservative, semi-empirical rules. See Bridge Rank. 

Lateral Ground Movement – This seismically induced movement is a permanent offset in the 

ground and usually occurs due to one or more of the collateral hazards. See Collateral Seismic 

Hazards, and Liquefaction-Induced Lateral Flow.

Liquefaction – A seismically induced loss of shear strength in loose, cohesionless soil that 

results from a build-up of pore water pressure in the soil as it tries to consolidate during strong 

ground shaking. 

Liquefaction-Induced Lateral Flow – A lateral displacement of a relatively flat slope that 

occurs under the combination of gravity load and excess pore water pressure due to liquefaction. 

This flow can occur without earthquake inertial loading and often continues well after the 

cessation of ground shaking. 

Lower Level Earthquake Ground Motion– A small earthquake that has a reasonable 

probability of occurrence within the life of the bridge (assumed to be 75 years). This ground 

motion may also be called the frequent earthquake, the expected earthquake (NCHRP 12-49 - 

ATC/MCEER 2003), or the functional evaluation earthquake (FEE) (Caltrans Seismic Design 

Methodology (Caltrans 1999)). Rather than assign a probability of occurrence, it is common 

practice to use a probability of exceedance to characterize the motion. Accordingly, the lower 

level motion has a relatively high probability of exceedance within the life of a bridge and a 

figure of 50 percent is recommended for retrofit design, which corresponds to a return period of 

about 100 years. 

Minimum Seat Width – The minimum prescribed width of a beam seat that must be provided to 

avoid loss of support to the superstructure. 

Multi-mode Spectral Analysis Method – A linear dynamic analysis of a two- or threedimensional model of a bridge, that is used when more than one mode of vibration contributes 

significantly to the dynamic response of the bridge. The method gives peak design values for 

moment, shear, and displacement by combining modal maxima. See CQC Method and SRSS 

Method. 

Network Redundancy - A measure of route availability that considers the presence of alternate 

routes and detour lengths. Network redundancy is used in the prioritization of bridges for 

retrofitting (along with Bridge Rank), and for defining bridge importance. 
 580

Nominal strength – This strength is obtained from failure theory for a section of the member or 

component and is based on assumed section geometry and specified material strengths. Material 

resistance factors are taken as 1.0 for the calculation of this strength. Other strength levels are 

usually referenced to the nominal strength. 

Overlay – A reinforced concrete slab added to the face of an existing structural element to 

increase its strength or stiffness, such as a footing overlay. Overlays are usually dowelled to the 

existing element to assist transfer of forces and moments across the interface. In some cases an 

overlay may be used as a bolster. See Bolster. 

Overstrength – The strength of a member that takes into account all possible factors 

contributing to an increase in strength above the nominal value. These include but are not limited 

to: steel strength higher than the specified yield strength; additional steel strength due to strain 

hardening at large deformations; concrete strength higher than specified; section sizes larger than 

assumed; axial compression in flexural members due to lateral restraint; additional reinforcement 

placed for construction purposes and not accounted for in design; and conservative assumptions 

made in the derivation of the equations for nominal strength. An overstrength factor (φo), is used 

to relate the overstrength to the nominal strength, and is always greater than unity. 

Performance-Based Design – A design philosophy that explicitly provides for different levels 

of bridge performance for earthquakes of different sizes, according to the importance of the 

bridge and its anticipated service life. 

Performance Level (PL) or Performance Criteria – A level of performance, expressed in 

terms of post-earthquake service and damage, that is expected to be achieved during and 

immediately following an earthquake of a specified size. 

Plastic Hinge – The region of a structural member, which undergoes flexural yielding and 

plastic rotation without loss of strength or stability. 

Probability of Exceedance – The likelihood that an earthquake, larger than the one under 

consideration, will occur within a specific time frame, usually taken to be the life of a bridge. 

This same likelihood can also be described by an annual probability of occurrence, or a return 

period expressed in years. 

R-Factor (Response Modification Factor) – Factors used to modify the element demands from 

an elastic analysis to account for ductile behavior and obtain design demands. 

Reference Bridge – A bridge that is a ‘long’ structure with no appreciable three-dimensional 

effects present; used for creating bridge fragility curves. 

Restrainer – A cable or steel rod used to limit the relative displacement at a movement joint in a 

bridge superstructure to decrease the likelihood of the superstructure becoming unseated at that 

joint during an earthquake. 

Retrofit Priority – The priority assigned to a bridge for detailed evaluation and possible 

retrofitting, based on bridge rank and such qualitative factors as importance, network 
581

redundancy, non-seismic deficiencies, remaining useful life, and relevant socio-economic issues. 

See Bridge Rank. 

Seismic Hazard Level (SHL) – Used to describe the severity of the seismic and geotechnical 

hazard at a bridge site for the purpose of detailed evaluation and retrofitting. Four levels are 

defined in this Manual based on the short- and long-period spectral accelerations (Ss and S1) for 

the upper level earthquake and corresponding site factors. 

Seismic Hazard Rating (E) – Used to describe the severity of the seismic and geotechnical 

hazard at a bridge site for the purpose of screening and prioritization. The rating is based on the 

long-period spectral acceleration (S1) for the upper level earthquake and the corresponding site 

factor, and ranges from low (0) to high (10). See Bridge Rank. 

Seismic Isolator – A mechanical device that uncouples a bridge from earthquake ground 

motions by adding flexibility and lengthening the period of vibration. Seismic forces are reduced 

due to the longer period, but isolator displacements may be very large. To limit these 

displacements, energy dissipation devices are sometimes used in parallel with seismic isolators. 

Typical isolators include elastomeric and sliding systems. See Energy Dissipation Devices. 

Seismic Retrofitting Category (SRC) – Used to recommend minimum screening requirements, 

evaluation methods and retrofitting measures for deficient bridges. Four categories are defined in 

this Manual, which are determined by the anticipated service life, importance, and the seismic 

and geotechnical hazards at the site. 

Seismic Risk Assessment Method - One of three methods that may be used to determine the 

seismic vulnerability rating of a bridge. Explicit analysis of a highway network is performed for 

a given hazard level and the resulting damage states used to estimate system performance as 

measured by traffic flow (e.g., increased travel times). Sensitivity of this performance to bridge 

condition is subsequently used to determine bridge retrofit needs and priorities. 

Shock Transmission Unit (STU) – A device installed across a movement joint in a 

superstructure that allows very slow movement to occur, such as that caused by temperature 

change, but ‘locks up’ under rapid motion. An STU is typically composed of a piston and 

cylinder filled with a viscous fluid. 

Site Class – Used to classify a site according to the properties of the soil as measured by shear 

wave velocities, blow counts, layer depths and thicknesses. 

Site Factors – Two site factors are defined: Fa and Fv. Fa is determined by the site class and the 

short-period spectral acceleration (Ss) for the upper level earthquake. Fv is determined by the site 

class and the long-period spectral acceleration (S1) for the upper level earthquake. These factors 

are used to account for soil amplification of bedrock ground motions when determining the 

design response spectrum at the surface. 

Square Root of the Sum of the Squares (SRSS) Combination – This classical statistical 

combination rule is used in three ways in this Manual: (1) for the combination of individual 

modal responses to obtain design maxima in a multimode spectral analysis; (2) for the 

combination of forces resulting from two or three orthogonal ground motion components; and 
 582

(3) for establishing orthogonal moments in the biaxial design of a column. See Multimode 

Spectral Analysis Method. 

Starter Bar – Reinforcing steel that extends from a footing a short distance into a column. The 

main longitudinal reinforcing steel in the column is then lapped (spliced) with these starter bars. 

Lapping or splicing rebar in plastic hinge zones is not recommended and is a common seismic 

deficiency in many older bridges. It is another name for a footing dowel. 

STRAHNET – The defense highway network. This network is a subset of the National Highway 

System and provides connecting routes to military installations, industries and resources. 

Uniform Risk – Used to describe the methodology for characterizing the seismic hazard in the 

United States. In this approach, the same probability of exceedance (return period) is used from 

one region to another to specify the hazard. In a country where the seismicity varies from low to 

high, this approach is considered to be more rational than using the maximum historical event for 

each region, which may have a very low probability of occurrence. 

Upper Level Earthquake Ground Motion – A large earthquake that has a finite, but remote, 

probability of occurrence within the life of the bridge. This ground motion may also be called the 

rare earthquake, maximum considered earthquake (MCE) (NCHRP 12-49 - ATC/MCEER 2003) 

or the safety evaluation earthquake (SEE) (Caltrans Seismic Design Methodology (Caltrans 

1999)). In this manual, the upper level motion has a 7 percent probability of exceedance in 75 

years, which corresponds to a return period of about 1,000 years. 

Vulnerability Rating (V) – Used to describe the severity of the seismic deficiencies in a bridge 

for the purpose of screening and prioritization. The rating is based on structural and geotechnical 

deficiencies and ranges from low (0) to high (10). See Bridge Rank. 
583

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Uniform Hazard Response Spectra for the United States 

The main body of this report reference a CD-ROM prepared 

by the U.S. Geological Survey (USGS) that contains seismic 

hazard maps and curves issued by the USGS in 1996. The 

CD-ROM is not included with this report as these hazard maps 

have been superseded and reissued with updated versions by 

the USGS. Users of this report are encouraged to use the most 

current version of the U.S. seismic hazard maps issued by the 

USGS, which are available at 

http://earthquake.usgs.gov/research/hazmaps/ 

However, if you would like a copy of the CD-ROM with the 

out-of-date 1996 seismic hazard maps, please write to: 

Seismic Hazard Map CD – HRDI-01 

Federal Highway Administration 

Office of Infrastructure R&D 

6300 Georgetown Pike 

McLean, VA 22101 
Recycled

Recyclable HRDI-07/01-06(1M)E