Research, Development, and Technology

Turner-Fairbank Highway Research Center

6300 Georgetown Pike

McLean, VA 22101-2296

PUBLICATION NO. FHWA-HRT-20-055 NOVEMBER 2020 

Steel Truss Retrofits to Provide Alternate 

Load Paths for Cut, Damaged, or 

Destroyed Members 
FOREWORD

This study was part of the Federal Highway Administration Hazard Mitigation Research 

Program addressing bridge vulnerabilities to single or multiple hazards, developing 

countermeasures for mitigation and adaptation, and developing analysis and design tools and 

methodologies. This report presents the results of a study on load-path redundancy and its 

importance in redistributing loads and maintaining stability, which may become necessary to 

prevent collapse at the onset of critical member failure. It focuses on long-span steel truss 

bridges of interest to the bridge engineering community and design and practicing engineers. 

This research extensively investigated load-path redundancy, including quantification of 

alternate load path (ALP), which is defined as the spectra of the surrounding members 

undergoing load redistribution after sudden damage to bridge members. Bridge members were 

compared before and after potential retrofit to judge the effectiveness of retrofit schemes in 

improving the ALP. An integrated framework to quantify the ALP of long-span truss bridges in 

terms of demand-to-capacity ratio (DCR) for the linear elastic analysis and strain ratio (SR) for 

the nonlinear dynamic analysis was developed. Different performance levels in terms of DCR

and SR are also presented for practicing engineers to use for the retrofit of long-span bridges to 

protect against progressive collapse.

Cheryl Allen Richter, P.E., Ph.D. 

Director, Office of Infrastructure

Research and Development 

Notice

This document is disseminated under the sponsorship of the U.S. Department of Transportation 

(USDOT) in the interest of information exchange. The U.S. Government assumes no liability for 

the use of the information contained in this document.

The U.S. Government does not endorse products or manufacturers. Trademarks or 

manufacturers’ names appear in this report only because they are considered essential to the 

objective of the document. 

Quality Assurance Statement

The Federal Highway Administration (FHWA) provides high-quality information to serve 

Government, industry, and the public in a manner that promotes public understanding. Standards 

and policies are used to ensure and maximize the quality, objectivity, utility, and integrity of its 

information. FHWA periodically reviews quality issues and adjusts its programs and processes to 

ensure continuous quality improvement.
TECHNICAL REPORT DOCUMENTATION PAGE

1. Report No.

FHWA-HRT-20-055

2. Government Accession No. 3. Recipient’s Catalog No.

4. Title and Subtitle

Steel Truss Retrofits to Provide Alternate Load Paths for

Cut, Damaged, or Destroyed Members

5. Report Date

November 2020

6. Performing Organization Code

7. Author(s)

A.K. Agrawal (ORCID: 0000-0001-6660-2299),

M. Ettouney, X. Chen, H. Li (ORCID: 000-0002-0858-

2258), and H. Wang

8. Performing Organization Report No.

9. Performing Organization Name and Address

The City College of New York

160 Convent Avenue

New York, NY 10031

10. Work Unit No.

11. Contract or Grant No.

DTFH61-14-D-00010/0004

12. Sponsoring Agency Name and Address

Office of Infrastructure Research and Development

Federal Highway Administration

6300 Georgetown Pike

McLean, VA 22101

13. Type of Report and Period Covered

Final Report; August 2014–June 2018

14. Sponsoring Agency Code

HRDI-20

15. Supplementary Notes

The Federal Highway Administration (FHWA) Study Managers were Eric Munley (FHWA, retired) and Sheila

Rimal Duwadi, P.E. (HRDI-20).

16. Abstract

Continued stability and performance of long-span truss bridges after loss of a critical member is broadly

attributed to “redundancy” because of alternate load paths (ALPs). This report presents an extensive investigation

on the load-path redundancy of long-span truss bridges, including quantification of ALP, defined as the spectra of

surrounding members undergoing load redistribution to prevent bridge collapse after sudden damage to a member

or members. This research developed an integrated framework to quantify ALP of long-span truss bridges in

terms of demand-to-capacity ratio (DCR) for linear elastic analysis and strain ratio for nonlinear dynamic (NLD)

analysis. ALP of long-span truss bridges was investigated through finite-element simulations of two example

long-span truss bridges. Results of the member removal analysis showed that the three-dimensionality of truss

bridges, stemming from upper and lower braces, side trusses, floor beam trusses, and the deck, plays a primary

role in protecting the bridge from collapse after removal of a member or members. Simulation results showed

that the stress contribution to DCR changes from primarily axial to predominantly moment (both in-plane and

out-of-plane) for truss members affected by sudden removal of another truss member. This change occurs

because the superstructure tends to undergo torsional motion about its longitudinal axis due to the asymmetrical

geometry created after removal of a member. Upper and lower braces and floor truss systems resist this torsional

motion, thereby redistributing the load among truss members. Various retrofit approaches were investigated to

improve ALP of two example bridges. The typical member strengthening approach used during seismic retrofit

had limited effectiveness in improving ALP of long-span truss bridges; however, retrofits that involved member

strengthening as well as adding new members as braces or parts of floor trusses (i.e., members that enhance the

three-dimensionality of the bridge) were the most effective and added the least amount of additional weight.

NLD analysis using LS-DYNA software (Hallquist 2014) resulted in more cost-effective retrofit than linear

dynamic analysis using SAP2000 software (Computers and Structures, Inc.). Performance levels are presented

for practicing engineers to use for the retrofit of long-span bridges to protect against progressive collapse.

Experimental and theoretical needs for investigating ALP of long-span truss bridges are also discussed.

17. Key Words

Alternate load path, redundancy, long-span truss

bridges, progressive collapse

18. Distribution Statement

No restrictions. This document is available to the public

through the National Technical Information Service,

Springfield, VA 22161.

http://www.ntis.gov

19. Security Classif. (of this report)

Unclassified

20. Security Classif. (of this page)

Unclassified

21. No. of Pages

216

22. Price

N/A

Form DOT F 1700.7 (8-72) Reproduction of completed page authorized.
ii
iii

TABLE OF CONTENTS

CHAPTER 1. INTRODUCTION ................................................................................................ 1

Background ............................................................................................................................. 1

Bridges That Survived Local Member Damage ................................................................. 2

Bridges That Collapsed After Local Member Damage ...................................................... 4

Past Studies on Progressive Collapse of Buildings............................................................. 7

Past Studies on Progressive Collapse of Bridges ................................................................ 7

Research Motivation and Needs ............................................................................................ 8

Research Objectives and Outcome ........................................................................................ 8

Summary of the Chapters of This Report ............................................................................ 9

CHAPTER 2. ALP AND REDUNDANCY ............................................................................... 11

Introduction ........................................................................................................................... 11

Proposed ALP Definition and Redundancy ....................................................................... 12

DCR Using Linear Static Analysis ....................................................................................... 15

SR Using NLD Analysis ........................................................................................................ 17

Loads and Load Combinations ............................................................................................ 18

ASD Versus LRFD ................................................................................................................ 18

CHAPTER 3. FINITE-ELEMENT MODELING ................................................................... 21

Introduction ........................................................................................................................... 21

Finite-Element Modeling in LS-DYNA ............................................................................... 21

Beam Element Formulation .............................................................................................. 21

Simulation of Loads in LS-DYNA ................................................................................... 24

Material Models ................................................................................................................ 24

Verification of Behavior of the B-S Beam Element ........................................................... 26

Example 1: Cantilever Beam ............................................................................................ 26

Example 2: Toggle Truss and Williams Toggle Frame .................................................... 28

Example 3: Aby Bridge .................................................................................................... 30

Example 4: Simulation of MR .......................................................................................... 33

CHAPTER 4. ALP ANALYSIS OF THE I-35W BRIDGE .................................................... 39

Introduction ........................................................................................................................... 39

Finite-Element Modeling of the Bridge............................................................................... 41

Basic Model Details .......................................................................................................... 41

Validation of FEMs in SAP2000 and LS-DYNA ............................................................. 48

Damping Effect ................................................................................................................. 52

As-Built Bridge and Nonseismic Retrofits .......................................................................... 54

ALP Analysis of the Design Bridge ..................................................................................... 57

MR Event .......................................................................................................................... 57

Dual Member Removal Event .......................................................................................... 89

CHAPTER 5. RETROFITS FOR ENHANCING ALP ........................................................ 109

Seismic Retrofits.................................................................................................................. 109

Conventional Seismic Retrofit Schemes ......................................................................... 109

Seismic Retrofit of the I-35W Bridge ............................................................................. 113
iv

PBD and Retrofit for Sudden Member Loss .................................................................... 116

Hazard Events ..................................................................................................................... 118

DCR-Based Retrofit Schemes............................................................................................. 118

Retrofits for MR .............................................................................................................. 118

Retrofits for DMR ........................................................................................................... 125

SR-Based Retrofit Schemes ................................................................................................ 128

Retrofits for MR .............................................................................................................. 129

Retrofits for DMR ........................................................................................................... 132

Dynamic Load Amplification ............................................................................................. 135

Critical Evaluation of Retrofit Performance .................................................................... 137

CHAPTER 6. IKITSUKI BRIDGE MODELING, ALP ANALYSIS, AND 

PERFORMANCE-BASED RETROFITS .............................................................................. 139

Introduction ......................................................................................................................... 139

Finite-Element Modeling of the Bridge............................................................................. 139

Loads and Load Combinations ....................................................................................... 141

Verification of the As-Built Bridge Model ..................................................................... 142

Behavior of the As-Built Bridge......................................................................................... 144

ALP Analysis of the Design Bridge ................................................................................... 147

ALP (MR) Analysis for Design Bridge: DCR Metric .................................................... 147

ALP (MR) Analysis for Design Bridge: SR Metric ........................................................ 152

Performance-Based Retrofits ............................................................................................. 156

Performance of Retrofits in Terms of DCR .................................................................... 156

Performance of Retrofits in Terms of SR ........................................................................ 161

CHAPTER 7. LABORATORY AND FIELD TESTING RECOMMENDATIONS .......... 163

System Behavior .................................................................................................................. 163

Modeling Methods .......................................................................................................... 163

ALP Quantification ......................................................................................................... 164

Component Behavior .......................................................................................................... 164

Joint Simulation .............................................................................................................. 164

Hazard Simulation .............................................................................................................. 165

Single MR ....................................................................................................................... 165

Blast Simulation: ALP Modeling.................................................................................... 166

Blast Simulation: Member Strengthening Modeling ...................................................... 166

Performance-Based Design ................................................................................................ 167

Retrofit Methods and Technologies .................................................................................. 168

CHAPTER 8. CONCLUSIONS, LIMITATIONS, AND FUTURE WORK ....................... 169

Summary and Conclusions ................................................................................................ 169

Limitations and Future Work............................................................................................ 171

APPENDIX A. SR CALCULATION ...................................................................................... 173

3D Beam Element Transformation Matrix ...................................................................... 173

Displacement Vector of 3D Beam in Global Coordinates .............................................. 173

Rotation Matrix From Global to Local ........................................................................... 173

Displacement Vector of 3D Beam in Local Coordinates ................................................ 175

Strain-Displacement Relations........................................................................................ 176

Bending in the Local s-r, t-r Plane.................................................................................. 176
v

Strains From 3D Beam Element Internal Forces Calculation ........................................ 178

Axial Force...................................................................................................................... 178

Bending Moments ........................................................................................................... 178

Torsional Moment ........................................................................................................... 179

APPENDIX B. CROSS-SECTIONAL AREA OF TRUSS MEMBERS FOR THE 

I-35W BRIDGE ......................................................................................................................... 181

APPENDIX C. THE CALCULATION OF DCR FOR COMPRESSION AND 

TENSION MEMBERS ............................................................................................................. 185

Compression Member ........................................................................................................ 185

Tension Member ................................................................................................................. 186

ACKNOWLEDGMENTS ........................................................................................................ 189

REFERENCES .......................................................................................................................... 191
vi

LIST OF FIGURES

Figure 1. Chart. Long-span truss bridges in the United States by decade built. ............................. 1

Figure 2. Photos. Fractured upper chord member of the Delaware River Bridge 

(FHWA 2017). ...................................................................................................................... 3

Figure 3. Photo. Blast-induced fracture of the bottom chord member in the 

Milton–Madison Bridge (FHWA 2013). ............................................................................... 4

Figure 4. Photo. Silver Bridge collapse in 1967 (NTSB 1968). ..................................................... 4

Figure 5. Photo. Collapse of suspended span of the Sung Soo Grand Bridge. ............................... 5

Figure 6. Photo. I-35W Bridge collapse (NTSB 2008d). ............................................................... 5

Figure 7. Illustrations. Co-rotational coordinate system for the B-S beam element..................... 23

Figure 8. Graph. Typical stress–strain curve for steel. ................................................................. 25

Figure 9. Graph. Measured and calibrated stress–strain relationships for A992 steel and 

Mat 98 in LS-DYNA. .......................................................................................................... 26

Figure 10. Illustration. Section of the example cantilevered beam. .............................................. 27

Figure 11. Illustration. Basic loading cases for the cantilevered beam. ....................................... 27

Figure 12. Illustrations. Typical truss systems.............................................................................. 28

Figure 13. Graphs. Constitutive stress–strain relationships for steel. ........................................... 29

Figure 14. Graph. Load-displacement curves for the toggle truss system. ................................... 29

Figure 15. Graph. Load-displacement curves for Williams toggle truss system. ......................... 30

Figure 16. Photo and Illustration. Application of LL on the Aby Bridge. .................................... 31

Figure 17. Graph. Load-displacement curves of the Aby Bridge. ................................................ 32

Figure 18. Illustration. 3D FEM of Aby Bridge in LS-DYNA. .................................................... 33

Figure 19. Photo. Approach span of the Milton–Madison Bridge looking northwest 

(FHWA 2013). .................................................................................................................... 34

Figure 20. Illustration. FEMs of the Milton–Madison Bridge in LS-DYNA with sand 

loading. ................................................................................................................................ 34

Figure 21. Photo. Sand loading on the Milton–Madison Bridge (FHWA 2013). ......................... 35

Figure 22. Graph. Calculated total stress time history after sudden removal of a member of 

the Milton–Madison Bridge. ............................................................................................... 36

Figure 23. Graph. Measured stress time history after fracture of a member of the 

Milton–Madison Bridge by blast (FHWA 2013). ............................................................... 37

Figure 24. Photo. I-35W Bridge before collapse (NTSB 2008d). ................................................ 39

Figure 25. Flowchart. Framework for the investigation of ALP in long-span truss bridges. ....... 40

Figure 26. Illustrations. 2D and 3D models of the I-35W Bridge in SAP2000. ........................... 41

Figure 27. Illustrations and graph. Comparison between deformed shapes and vertical 

displacements from 2D models in SAP2000 and LS-DYNA. ............................................ 48

Figure 28. Graphs. Displacement time history of node 3 for various Ds values. .......................... 53

Figure 29. Illustration. DCR spectra for the as-built bridge subjected to service loads. .............. 55

Figure 30. Illustration. Members of the as-built bridge retrofitted by member strengthening 

to obtain the Design Bridge for the ALP analysis. .............................................................. 56

Figure 31. Illustration. Location of critical members for removal in one-quarter of the 

bridge on the west side truss. .............................................................................................. 57

Figure 32. Illustration. DCR before and after MR case 1 near the damaged location in west 

and east side trusses. ............................................................................................................ 60
vii

Figure 33. Illustration. Contributions to DCR before and after MR case 1 near the damaged 

location in west and east side trusses. ................................................................................. 60

Figure 34. Illustration. DCR spectra for the Design Bridge for diagonal member removal 

(MR case 1 in table 20). ...................................................................................................... 63

Figure 35. Illustration. DCR spectra for the Design Bridge for lower chord MR 

(MR case 2 in table 20). ...................................................................................................... 64

Figure 36. Illustration. DCR spectra for the Design Bridge for lower chord MR 

(MR case 3 in table 20). ...................................................................................................... 65

Figure 37. Illustration. DCR spectra for the Design Bridge for diagonal member removal 

(MR case 4 in table 20). ...................................................................................................... 66

Figure 38. Illustration. DCR spectra for the Design Bridge for upper chord MR 

(MR case 5 in table 20). ...................................................................................................... 67

Figure 39. Illustration. DCR spectra for the Design Bridge for upper chord MR 

(MR case 6 in table 20). ...................................................................................................... 68

Figure 40. Illustration. DCR spectra for the Design Bridge for diagonal member removal 

(MR case 7 in table 20). ...................................................................................................... 69

Figure 41. Illustration. DCR spectra for the Design Bridge for diagonal member removal 

(MR case 8 in table 20). ...................................................................................................... 70

Figure 42. Illustration. DCR spectra for the Design Bridge for diagonal member removal 

(MR case 9 in table 20). ...................................................................................................... 71

Figure 43. Illustration. DCR spectra for the Design Bridge for diagonal member removal 

(MR case 10 in table 20). .................................................................................................... 72

Figure 44. Illustration. Complete envelope of DCR for the Design Bridge representing 

40 MR cases (10 removal cases in each quarter of the bridge). .......................................... 73

Figure 45. Graph. Nodal vertical displacement in MR case 1. ..................................................... 74

Figure 46. Graphs. Axial forces in lower chord members in west and east side trusses of 

the Design Bridge for MR case 1. ....................................................................................... 76

Figure 47. Graphs. In-plane and out-of-plane moments in lower chord members in west 

and east side trusses of the Design Bridge for MR case 1. ................................................. 77

Figure 48. Illustration. SR spectra for the Design Bridge for diagonal member removal 

(MR case 1 in table 20). ...................................................................................................... 78

Figure 49. Illustration. SR spectra for the Design Bridge for lower chord MR 

(MR case 2 in table 20). ...................................................................................................... 79

Figure 50. Illustration. SR spectra for the Design Bridge for lower chord MR 

(MR case 3 in table 20). ...................................................................................................... 80

Figure 51. Illustration. SR spectra for the Design Bridge for diagonal member removal 

(MR case 4 in table 20). ...................................................................................................... 81

Figure 52. Illustration. SR spectra for the Design Bridge for upper chord MR 

(MR case 5 in table 20). ...................................................................................................... 82

Figure 53. Illustration. SR spectra for the Design Bridge for upper chord MR 

(MR case 6 in table 20). ...................................................................................................... 83

Figure 54. Illustration. SR spectra for the Design Bridge for diagonal member removal 

(MR case 7 in table 20). ...................................................................................................... 84

Figure 55. Illustration. SR spectra for the Design Bridge for diagonal member removal 

(MR case 8 in table 20). ...................................................................................................... 85
viii

Figure 56. Illustration. SR spectra for the Design Bridge for diagonal member removal 

(MR case 9 in table 20). ...................................................................................................... 86

Figure 57. Illustration. SR spectra for the Design Bridge for diagonal member removal 

(MR case 10 in table 20). .................................................................................................... 87

Figure 58. Illustration. Complete envelope of SR for the Design Bridge representing 

40 MR cases (10 removal cases in each quarter of the bridge). .......................................... 88

Figure 59. Illustration. Cases of critical DMR in one-quarter of the bridge on the west side 

truss. .................................................................................................................................... 89

Figure 60. Illustration. DCR before and after DMR case 1 near the damaged location in 

west and east side trusses. ................................................................................................... 90

Figure 61. Illustration. Contributions to DCR before and after DMR case 1 near the 

damaged location in west and east side trusses. .................................................................. 91

Figure 62. Illustration. DCR spectra for the Design Bridge for DMR case 1. .............................. 93

Figure 63. Illustration. DCR spectra for the Design Bridge for DMR cases 2, 6, and 7. ............. 94

Figure 64. Illustration. DCR spectra for the Design Bridge for DMR cases 3, 5, and 8. ............. 95

Figure 65. Illustration. DCR spectra for the Design Bridge for DMR case 4. .............................. 96

Figure 66. Illustration. DCR spectra for the Design Bridge for DMR case 9. .............................. 97

Figure 67. Illustration. DCR spectra for the Design Bridge for DMR case 10. ............................ 98

Figure 68. Illustration. Complete envelope of DCR for the Design Bridge representing 

40 DMR cases (10 DMR cases in each quarter of the bridge). ........................................... 99

Figure 69. Illustrations. Moment diagram of the west side truss (assuming the truss as a 

continuous beam) before any damage and after blast damage at a certain panel. ............. 101

Figure 70. Illustration. SR spectra for the Design Bridge for DMR case 1. ............................... 102

Figure 71. Illustration. SR spectra for the Design Bridge for DMR cases 2, 6, and 7. ............... 103

Figure 72. Illustration. SR spectra for the Design Bridge for DMR cases 3, 5, and 8. ............... 104

Figure 73. Illustration. SR spectra for the Design Bridge for DMR case 4. ............................... 105

Figure 74. Illustration. SR spectra for the Design Bridge for DMR case 9. ............................... 106

Figure 75. Illustration. SR spectra for the Design Bridge for DMR case 10. ............................. 107

Figure 76. Illustration. Complete envelope of SR for the Design Bridge representing 40 

DMR cases (10 DMR cases in each quarter of the bridge). .............................................. 108

Figure 77. Photo. Retrofit of the Brooklyn Bridge with six arch systems. ................................. 110

Figure 78. Illustration. Schematic of seismic retrofits to the Golden Gate Bridge. .................... 111

Figure 79. Photo. Auburn-Foresthill Bridge, California’s tallest bridge. ................................... 112

Figure 80. Photo. I-40 Mississippi River Bridge (steel tied-arch bridge). .................................. 112

Figure 81. Graph. Seismic response spectra for 2500-year earthquake in San Diego, CA. ....... 113

Figure 82. Illustration. Members of the Design Bridge retrofitted for SD2500 by member 

strengthening to obtain the Base Bridge for the ALP analysis. ........................................ 115

Figure 83. Illustration. Complete envelope of DCR for the Base Bridge representing 

40 MR cases (10 removal cases in each quarter of the bridge). ........................................ 119

Figure 84. Illustration. Member strengthening retrofit of the Base Bridge for MR based on 

DCR ≤ 1.67 limit state. ...................................................................................................... 120

Figure 85. Illustration. Addition of bracings in the floor truss systems and member 

strengthening retrofit of the Base Bridge for MR based on DCR ≤ 1.67 limit state. ........ 121

Figure 86. Illustration. Addition of DMs and member strengthening retrofit of the Base 

Bridge for MR based on DCR ≤ 1.67 limit state. .............................................................. 122
ix

Figure 87. Illustration. Addition of bracings in the floor truss systems, addition of DMs, 

and member strengthening retrofit of the Base Bridge for MR based on DCR ≤ 1.67 

limit state. .......................................................................................................................... 123

Figure 88. Graphs. Increase in weight of steel and DispRs for different MR retrofit 

strategies. ........................................................................................................................... 125

Figure 89. Illustration. Complete envelope of DCR for the Base Bridge representing 

40 DMR load cases (10 DMR cases in each quarter of the bridge). ................................. 127

Figure 90. Graphs. Increase in weight of steel and DispRs for different DMR retrofit 

strategies. ........................................................................................................................... 128

Figure 91. Graph. Subjective relationships between DCR and SR metrics. ............................... 129

Figure 92. Illustration. Complete envelope of SR for the Base Bridge representing 40 MR 

cases (10 removal cases in each quarter of the bridge). .................................................... 130

Figure 93. Chart. Percentage increase in the weight of steel for different MR retrofits based 

on SR limit states. .............................................................................................................. 132

Figure 94. Illustration. Complete envelope of SR for the Base Bridge representing 40 DMR 

cases (10 DMR cases in each quarter of the bridge). ........................................................ 133

Figure 95. Chart. Percentage increase in the weight of steel for different DMR retrofits 

based on SR limit states. .................................................................................................... 135

Figure 96. Photo. Ikitsuki Bridge connecting Ikitsuki and Hirado Island in Japan. ................... 139

Figure 97. Illustrations. Overall layout drawing of Ikitsuki Bridge (unit: mm). ........................ 140

Figure 98. Illustration. Comparison between mode shapes of the Ikitsuki Bridge based on 

Kubota et al. (2004) and the LS-DYNA model. ............................................................... 143

Figure 99. Illustration. DCR spectra for the as-built bridge subjected to service loads. ............ 145

Figure 100. Illustration. Members of the as-built bridge retrofitted by member 

strengthening to obtain Design Bridge for the ALP analysis. ........................................... 146

Figure 101. Illustration. DCR spectra for the Design Bridge for upper chord MR 

(MR case 1). ...................................................................................................................... 149

Figure 102. Illustration. DCR spectra for the Design Bridge for upper chord MR 

(MR case 2). ...................................................................................................................... 150

Figure 103. Illustration. Complete envelope of DCR for the Design Bridge representing 

eight MR cases (two removal cases in each quarter of the bridge). .................................. 151

Figure 104. Illustration. SR spectra for the Design Bridge for upper chord MR 

(MR case 1). ...................................................................................................................... 153

Figure 105. Illustration. SR spectra for the Design Bridge for upper chord MR 

(MR case 2). ...................................................................................................................... 154

Figure 106. Illustration. Complete envelope of SR for the Design Bridge representing eight 

MR cases (two removal cases in each quarter of the bridge). ........................................... 155

Figure 107. Illustration. Member strengthening retrofit of the Design Bridge for MR based 

on 1.0 ≤ DCR ≤ 1.67 limit state. ........................................................................................ 157

Figure 108. Illustration. Addition of floor truss members and member strengthening retrofit 

of the Design Bridge for MR based on 1.0 ≤ DCR ≤ 1.67 limit state. .............................. 158

Figure 109. Illustration. Addition of DMs and member strengthening retrofit of the Design 

Bridge for MR based on 1.0 ≤ DCR ≤ 1.67 limit state. ..................................................... 159

Figure 110. Illustration. Addition of floor truss members and DMs and member 

strengthening retrofit of the Design Bridge for MR based on 1.0 ≤ DCR ≤ 1.67 limit 

state. ................................................................................................................................... 160
x

Figure 111. Illustrations. Representations of joints. ................................................................... 164

Figure 112. Illustration. Detailed FHWA global model of the I-35W Bridge with fine 

detailed model of U10W and L11W joints (Schulthesiz et al. 2008)................................ 165

Figure 113. Graph. Typical blast damage monograph. ............................................................... 166

Figure 114. Illustrations. Illustration of blast load on a bridge. .................................................. 167

Figure 115. Illustration. 3D beam global and local coordinate specification. ............................ 174

Figure 116. Illustration. Output locations for strain for a beam section. .................................... 178
xi

LIST OF TABLES

Table 1. Proposed limit states for ALP analysis using DCR metric. ............................................ 15

Table 2. Proposed limit states for ALP analysis using SR metric. ................................................ 15

Table 3. Calibrated parameters for material model Mat 98. ......................................................... 26

Table 4. Sectional and material properties for the example cantilevered beam. .......................... 27

Table 5. Comparison between analytical and FEM solutions for the cantilevered beam. ............ 28

Table 6. Material and sectional properties of Williams toggle truss element. .............................. 30

Table 7. Material properties of steel in truss components. ........................................................... 32

Table 8. Comparison between finite-element modeling approaches for the Aby Bridge. ............ 33

Table 9. DLs on the I-35W Bridge................................................................................................ 42

Table 10. Main truss vertical reaction forces for DL condition. ................................................... 43

Table 11. Axial forces in upper chord members for DL1 case. .................................................... 44

Table 12. Axial forces in DMs for DL1 case. ............................................................................... 45

Table 13. Axial forces in lower chord members for DL1 case. .................................................... 46

Table 14. Axial forces in vertical members for DL1 case. ........................................................... 47

Table 15. Comparisons between 2D models in SAP2000 and LS-DYNA. .................................. 49

Table 16. Comparison between displacements from 3D models of the bridge under steel 

truss self-weight in SAP2000 and LS-DYNA. .................................................................... 50

Table 17. Comparison between displacements from 3D models of the bridge under DL1 

(1965) in SAP2000 and LS-DYNA. ................................................................................... 51

Table 18. Weights of various portions of 3D full bridge models. ................................................ 51

Table 19. Damping ratios for different values of Ds. .................................................................... 54

Table 20. MR cases. ...................................................................................................................... 57

Table 21. Percentage change in the reaction force for corresponding MR cases. ...................... 100

Table 22. Percentage change in the reaction force for corresponding DMR cases. ................... 100

Table 23. Number of elements in each range of ALP analysis: DCR metric. ............................ 100

Table 24. Number of members in each range for different DCR limit states under MR 

scenarios for the Design Bridge and Base Bridge. ............................................................ 114

Table 25. Number of members in each range for different DCR limit states under DMR 

scenarios for the Design Bridge and Base Bridge. ............................................................ 114

Table 26. Allowable indicators for various performance levels using DCR metric. .................. 116

Table 27. Allowable indicators for various performance levels using SR metric. ...................... 116

Table 28. Retrofit strategies for MRs. ........................................................................................ 118

Table 29. Increase in the weight of steel and DispRs for different MR retrofit strategies. ........ 124

Table 30. Increase in the weight of steel and DispRs for different DMR retrofit strategies. ..... 128

Table 31. SR-based retrofit levels for an assumed blast intensity. .............................................. 129

Table 32. Percentage increase in the weight of steel for different MR retrofit strategies 

based on DCR and SR limit states. .................................................................................... 131

Table 33. Percentage increase in the weight of steel for different DMR retrofit strategies. ...... 134

Table 34. DLA for different MR analysis cases and scenarios. .................................................. 136

Table 35. DLA for different DMR analysis cases and scenarios. ............................................... 136

Table 36. DispRs for MR cases of ALP-retrofitted bridge with the elastic deflection limit 

L ∕ 800 for the four retrofit strategies. ................................................................................ 137

Table 37. DispRs for DMR cases of ALP-retrofitted bridge with the elastic deflection limit 

L ∕ 800 for the four retrofit strategies. ................................................................................ 137
xii

Table 38. DLs on the Ikitsuki Bridge. ......................................................................................... 141

Table 39. Support reactions for the Ikituski Bridge under different load combinations. ........... 141

Table 40. Comparison between natural frequencies of the Ikitsuki Bridge from the FEM by 

Kubota et al. (2004) and the SAP2000 model. .................................................................. 142

Table 41. Comparison between natural frequencies of the Ikitsuki Bridge from the FEM by 

Kubota et al. (2004) and the LS-DYNA model. ............................................................... 143

Table 42. Increase in the weight of steel and DispRs for different MR retrofit strategies 

based on DCR limit state. .................................................................................................. 161

Table 43. Increase in the percentage weight of steel for different MR retrofit strategies 

based on DCR and SR limit states. .................................................................................... 162

Table 44. Cross-sectional area of upper chord members for I-35W Bridge model. ................... 181

Table 45. Cross-sectional area of lower chord members for I-35W Bridge model. ................... 182

Table 46. Cross-sectional area of vertical members for I-35W Bridge model. .......................... 183

Table 47. Cross-sectional area of DMs for I-35W Bridge model. .............................................. 184

Table 48. Cross-sectional properties of the compression member. ............................................ 186

Table 49. Demand of the compression member. ........................................................................ 186

Table 50. Cross-sectional properties of the tension member. ..................................................... 186

Table 51. Demand of the tension member. ................................................................................. 187
xiii

LIST OF ABBREVIATIONS AND SYMBOLS

Abbreviations 

2D two-dimensional 

3D three-dimensional 

AASHTO American Association of State Highway and Transportation Officials

AISC American Institute of Steel Construction

ALP alternate load path

APM alternate path method

ASCE American Society of Civil Engineers

ASD allowable strength design 

B-S Belytschko–Schwer

BRB buckling-restrained brace

CSA Canadian Standards Association

DispR displacement ratio

DLA dynamic load amplification

DM diagonal member 

DMR dual member removal

DOF degree of freedom

DR dynamic relaxation

FEM finite-element model

FHWA Federal Highway Administration

FRP fiberglass reinforced plastic

FS floor truss system 

GSA General Services Administration

H-L Hughes–Liu

LRFD load and resistance factor design 

MR member removal

NLD nonlinear dynamic 

NLS nonlinear static

PBD performance-based design

Symbols

DCR demand-to-capacity ratio

SR strain ratio

DL dead load

LL live load

BL blast

S structural system

rm number of components 

M components 

W set of individual components (members) 

i subscript denoting ith component 

D demand

αDL dead load factor

αLL live load factor
xiv

DM design metric

LIM(DM) limit state of design metric

σy yield stress

ε strain

εy yield strain

0 subscript denoting base system/condition/set/metric

1 subscript denoting new system/condition/set/metric 

j subscript denoting jth limit state

Pr axial force demand

Pc axial force capacity

Mrx bending moment demand with respect to the x-axis

Mcx bending moment capacity with respect to the x-axis

Mry bending moment demand with respect to the y-axis

Mcy bending moment capacity with respect to the y-axis

Ac member cross-sectional area

Sxx section modulus with respect to the x-axis

Syy section modulus with respect to the y-axis

σrp axial stress

σMx stress due to moments with respect to the x-axis

σMy stress due to moments with respect to the y-axis

Ω safety factor

σallow allowable stress obtained from σy divided by Ω based on the allowable strength 

design approach 

α design coefficient for load and resistance factor design or allowable strength 

design method 

Pe elastic critical buckling strength of member

Cm reduction coefficient 

M1 smaller moment

M2 larger moment

EQ seismic load

R subscript denoting response 

I node 

J node 

quantity defined in local element coordinate system 

local x-axis

local y-axis

local z-axis

e1, e2, e3 unit vectors associated with , , and 

quantity defined in global element coordinate system 

global x-axis

global y-axis

unit vector along global axis 

unit vector along global axis 

unit vector along local axis in initial configuration

unit vector along local axis in initial configuration
xv

unit vector along local axis 

unit vector along local axis 

displacement matrix

δIJ change in length (from nodes I to J) 

torsional deformation 

bending rotational deformations (first subscript is direction; second subscript is 

node) 

A, B, C, n input constants 

εp effective plastic strain

ε* normalized effective strain rate

E Young’s modulus 

G shear modulus 

Ixx moment of inertia (xx component) 

Iyy moment of inertia (yy component) 

ρ density 

Fx concentrated force along x direction

Fy concentrated force along y direction

Fz concentrated force along z direction 

Mx concentrated moment along x direction

My concentrated moment along y direction

Mz concentrated moment along z direction

ux displacement in x direction

uy displacement in y direction

uz displacement in z direction

F force

h height 

L length 

σu ultimate stress

K effective length factor

Δ difference

Ds global damping factor 

δe logarithmic decrement

q number of cycles

u1 peak amplitude of the first peak

uq+1 peak amplitude of the (q+1)th peak

ζ damping ratio 

Min in-plane moment 

Mout out-of-plane moment 

Ss short period 

S1 1-second period

{U}i 3D beam global displacements and rotations vector of the ith beam

{U}i1 vector of the six global displacement and rational measures of the first node of the

beam

{U}i2 vector of the six global displacement and rational measures of the second node of

the beam
xvi

u1X displacement in the global X direction for node 1 

u1Y displacement in the global Y direction for node 1 

u1Z displacement in the global Z direction for node 1 

u2X displacement in the global X direction for node 2 

u2Y displacement in the global Y direction for node 2 

u2Z displacement in the global Z direction for node 2 

θ1X rotation along the global X direction for node 1 

θ1Y rotation along the global Y direction for node 1 

θ1Z rotation along the global Z direction for node 1 

θ2X rotation along the global X direction for node 2

θ2Y rotation along the global Y direction for node 2 

θ2Z rotation along the global Z direction for node 2 

ujk displacement in the global axis for the jth node and kth global degree of freedom 

θjk rotation in the global axis for the jth node and kth global degree of freedom 

r member 3 × 3 rotation matrix

p position vector 

IX unit vector in the global X direction

IY unit vector in the global Y direction

IZ unit vector in the global Z direction

ir unit vector in the local x(r) direction

is unit vector in the local y(s) direction

it unit vector in the local z(t) direction

[T]i transformation matrix

o a 3 × 3 null matrix

{UL}i 3D beam local displacements and rotations vector of the ith beam

{UL}i1 vector of the six local displacement and rational measures of the first node of the 

beam

{UL}i2 vector of the six local displacement and rational measures of the second node of 

the beam 

u1r displacement in the local x(r) direction for node 1 

u1s displacement in the local y(s) direction for node 1 

u1t displacement in the local z(t) direction for node 1 

u2r displacement in the local x(r) direction for node 2 

u2s displacement in the local y(s) direction for node 2 

u2t displacement in the local z(t) direction for node 2 

θ1r rotation along the local x(r) direction for node 1 

θ1s rotation along the local y(s) direction for node 1 

θ1t rotation along the local z(t) direction for node 1 

θ2r rotation along the local x(r) direction for node 2 

θ2s rotation along the local y(s) direction for node 2 

θ2t rotation along the local z(t) direction for node 2 

us displacement in the local y(s) direction 

ut displacement in the local z(t) direction

εt (r, s) strain due to bending about the local t axis

εs (r, t) strain due to bending about the local s axis

εts (r, t, s) total bending strain along the beam 
xvii

εr axial strain due to the axial deformation 

ε(r, t, s) total strain due to bending and axial along the beam 

[α] constant matrix

[β] constant matrix

F1r internal axial force

δ axial elongation 

Ka axial stiffness

M1s bending moment in the local r-t plane at node 1 of the beam 

M2s bending moment in the local r-t plane at node 2 of the beam 

M1t bending moment in the local r-s plane at node 1 of the beam 

M2t bending moment in the local r-s plane at node 2 of the beam 

Ks

b flexural stiffness in the s direction

Kt

b flexural stiffness in the t direction

φs shear factor in the s direction

φt shear factor in the t direction

As effective area in shear in the s direction

At effective area in shear in the t direction

Is moment of inertia about the s-axis

It moment of inertia about the t-axis

M1r torsional moment 

Kt torsional stiffness

Jt torsion constant 

Fe Euler stress

rg radius of gyration 

Fcr critical stress

Pn nominal axial force using allowable strength design

1 

CHAPTER 1. INTRODUCTION

BACKGROUND

Bridges are considered crucial links in the transportation network of highways and railways in 

the country and are expected to perform as designed throughout their functional life. Based on 

analysis of the National Bridge Inventory database, 752 (0.12 percent) of 614,387 highway 

bridges in the United States have a maximum span longer than 122 m (400 ft) (FHWA n.d.). 

Approximately 300 (~40 percent) of these bridges are either deck-truss or through-truss types. 

Figure 1 shows a histogram of these long-span truss bridges by decade built. As the figure 

demonstrates, most long-span truss bridges in the United States were built from the 1920s to the 

1970s. 

Source: FHWA.

Figure 1. Chart. Long-span truss bridges in the United States by decade built.

A truss bridge system consists of simple members that can support a large amount of weight, and 

these bridges can be considerably long. Theoretically, the members in a simple truss system are 

subjected only to axial force (tension or compression) rather than bending. However, the actual 

behavior of truss bridges could be significantly more complex and may involve axial forces and 

moments (both in-plane and out-of-plane).

Many truss bridges built between the 1920s and 1970s are aging and may have critical members 

with significant deterioration. Lee et al. (2013) analyzed bridge failures in the United States from

1980 to 2012 and observed that, although girder bridges are the most common structural type of 

failed bridges, truss bridges are much more vulnerable to failure than girder bridges. Even 

though truss bridges account for only 29 percent of the failures, they make up less than 4 percent

of the total number of bridges. 

In the event of loss of critical members of a truss bridge because of corrosion or terrorist events, 

such as intentional member cutting or blast loading, continued stability and performance of the 

bridge can be broadly attributed to “redundancy.” Unfortunately, the redundancy of long-span 

truss bridges, which is quantified based on the behavior of these bridges after failure of a 
2 

member or members, is not well understood and was generally not considered in the design of 

existing long-span truss bridges. Many bridges with failed members have survived without 

collapse, whereas many bridges with local damage to a critical member have collapsed. The next 

sections discuss various examples of survival as well as collapse of truss bridges after failure of a 

critical member or members to highlight the challenges associated with understanding the 

complex behavior of truss bridges. 

Bridges That Survived Local Member Damage

On May 7, 1975, the main girder of the Lafayette Street Bridge over the Mississippi River in St.

Paul, MN, was found to have a crack that originated at the lateral bracing gusset to the transverse 

stiffener weld area due to the lack of fusion (Fisher et al. 1977). Brittle failure of the girder in the 

central span occurred after the crack’s penetration through the web thickness of the girder. 

However, collapse of one girder did not lead to the collapse of the entire bridge. 

The Yukon River Bridge, located approximately 32 km (20 mi) south of the city of Whitehorse 

in Canada and carrying the Alaska Highway, is a two-span Warren through-truss bridge. It was 

damaged because of an impact by an overheight vehicle in 1982. The impact damaged many 

tension members, including a bottom chord member near the midspan, which opened by 

approximately 50 mm (2 inches) after the accident (Beauchamp et al. 1984). Significant vertical 

deflection and horizontal displacement at the roller support were also observed. However, the 

bridge survived without collapse. The bridge was restored to fully functional condition by 

replacing all damaged members. Interestingly, Beauchamp et al. (1984) observed that the bridge 

did not collapse because the floor system, acting as an equivalent bottom chord, took over most 

of the broken truss’s dead load (DL). Similarly, one of the web verticals near the midspan of the 

East Brough’s Bridge, a Pratt through-truss bridge in London, Ontario, Canada, was severed 

because of an impact by a bus in 2000 (Jelinek and Bartlett 2002). Although the bridge was 

heavily damaged, it escaped complete collapse.

On January 20, 2017, a resident engineer on an active painting job noticed a fractured truss 

member (U19-19′) in the north truss of the Delaware River Bridge, as shown in figure 2. The 

bridge continued to perform under full live load (LL) during the period between when the failure 

occurred and when it was discovered. This is another example that shows that certain truss 

systems can carry a full traffic load even after the loss of a member. 
3 

Source: FHWA. 

A. Elevation view of the fractured truss member.

Source: FHWA. 

B. Close-up photo of the fractured member. 

Figure 2. Photos. Fractured upper chord member of the Delaware River Bridge 

(FHWA 2017). 

The Federal Highway Administration (FHWA) (2013) investigated the after-fracture 

performance of a two-line simple-span truss bridge that was part of the Milton–Madison Bridge 

and was slated for explosive demolition. During the testing, a built-up bottom chord member at 

the midspan of the bridge was completely severed during a controlled blast test. Figure 3 shows 

half of the lower chord severely damaged. It was observed that the total removal of the bottom 

chord member did not cause collapse of the bridge. In fact, the analyses presented in FHWA 

(2013), Diggelmann et al. (2013), and Cha et al. (2014) show that the bridge likely could have 

remained functional under normal service loads even after complete loss of the bottom chord 

member.
4 

Source: FHWA. 

Figure 3. Photo. Blast-induced fracture of the bottom chord member in the

Milton–Madison Bridge (FHWA 2013). 

Bridges That Collapsed After Local Member Damage

Silver Bridge, which collapsed in 1967, is considered the earliest case of collapse of an entire 

bridge structure because of local damage to a member. In the evening hours of December 15, 

1967, the Silver Bridge across the Ohio River collapsed because of a single eyebar corrosion 

cracking and stress concentration, resulting in the deaths of 46 people (Lichtenstein 1993). 

Figure 4 shows a photograph of the collapsed bridge. The collapse of this bridge led to the 

establishment of the current National Bridge Inspection Standards (Lichtenstein 1993). 

Follow-up study on this bridge collapse led to further understanding of redundancy and 

fracture-critical bridges.

Source: NTSB.

Figure 4. Photo. Silver Bridge collapse in 1967 (NTSB 1968). 

The suspended span of the Sung Soo Grand Bridge, a Gerber-type truss bridge across the 

Han River in Seoul, Korea, collapsed on October 21, 1994, killing 32 and injuring 17 people 

(Cho et al. 2001). The suspended trusses were connected to anchor trusses by vertical hangers 

acting as pins or hinges between piers. The structural failure was caused by improper welding of 

the steel trusses of the suspension structure beneath the concrete slab roadway. Figure 5 shows a 

photograph of the bridge after the collapse of the suspended span. 
5 

© 2014 Choi Kwangmo, Wikimedia Commons (CC BY-SA-4.0

(https://creativecommons.org/licenses/by-sa/4.0)).

Figure 5. Photo. Collapse of suspended span of the Sung Soo Grand Bridge. 

Collapse of the I-35W Bridge in Minneapolis, MN, in 2007 is the most recent example of 

collapse of a steel deck-truss bridge because of local damage (Hao 2010). Collapse of this 

bridge, shown in figure 6, resulted in the loss of 13 lives and injuries to 145 people. Per 

investigation by the National Transportation Safety Board (NTSB), the bridge collapsed because 

of undersized gusset plates U10 and U11, which were half as thick (12.5 mm (0.5 inches)) as a

gusset plate in the U12 joint (NTSB 2008a–d). Investigation into the causes of this bridge failure 

led to a greater emphasis on not only the design of gusset plates in truss bridges but also the 

insufficiency of conventional one-dimensional influence line models compared to detailed 

three-dimensional (3D) models for design. In this bridge, gusset plates were the pivots 

responsible for transferring loads from the deck and truss members to the supporting piers. This 

load path resulted in force flow with high amplitude in the diagonal members (DMs) attached to 

these gusset plates. However, a conventional one-dimensional influence line model probably 

gave zero or very low amplitude of the bending moment in this area because of the model’s 

transition from a positive to a negative sign. This model result could have led to the undersized 

design of the gusset plates in this bridge. The investigation also highlighted the significant role 

played by redundancy and alternate load paths (ALPs) in preventing such failures (Hao 2010). 

Source: NTSB. 

Figure 6. Photo. I-35W Bridge collapse (NTSB 2008d). 
6 

From the study of bridges that survived local member damage, Liu et al. (2013) noted that both 

the East Brough’s Bridge and the Lewes Bridge would have collapsed if the load path in these 

bridges was from the floor system to the panel point of the main trusses, and joints were pin 

connections that structurally isolated the main trusses from each other and from the floor system, 

assumptions that are made in typical truss bridge design. In reality, structural features in these 

two bridges facilitated ALPs to enhance their redundancy. Some of these prominent features 

were as follows: (1) truss connections that were designed as pinned were actually rigid joints 

consisting of the gusset plates with many fasteners to transfer moments; (2) trusses that were

designed to carry loads independently were interconnected by lateral and diagonal bracings, floor 

trusses, and sway frame members; and (3) the floor system that was assumed not to contribute to 

stiffness, strength, or load sharing between the trusses was rigidly connected to the trusses at the 

floor beams. Isolated studies have also documented the effects of these structural features on the 

behavior of trusses. For example, Nagavi and Aktan (2003) showed that 3D finite-element

models (FEMs) with rigid joints more accurately simulated the behavior of steel through-truss 

bridges with riveted gusset plates, especially the inelastic response, than conventional 

two-dimensional (2D) and 3D truss models. DelGrego et al. (2008) observed floor system–truss 

interaction during the monitoring of a railroad through-truss bridge. 

Redundancy is defined as “the quality of a bridge that enables it to perform its design function in 

the damaged state” (AASHTO 2014, sec 6.2: 6–7). In general, redundancy in truss bridges can

be classified as internal (member) redundancy, structural redundancy, and load-path redundancy 

(FHWA 2013). A structural system can have internal redundancy because of multiple parallel 

elements within a member, such as a built-up member made from many different plates and other 

structural shapes that are bolted or riveted together. Fracture in one part of such members may be 

arrested by other parts of the member. Structural redundancy is directly related to the static 

indeterminacy of the structure. For example, continuous-span structures would have structural 

redundancy. Load-path redundancy is based on available paths for load redistribution in the 

event of failure of a critical member. All three forms of redundancy may play a role in 

preventing the collapse of a bridge after the failure of a critical member. However, the 

contribution of load-path redundancy to the overall redundancy of a long-span truss bridge is not 

very well understood. 

Barth et al. (2014) mentioned that a sound explanation of load-path redundancy could lead to a 

more efficient manner of designing and rating highway bridges, thereby avoiding structure 

collapse and potential disasters. More recently, the American Association of State Highway and 

Transportation Officials (AASHTO) Guide Specifications for Analysis and Identification of 

Fracture Critical Members and System Redundant Members was published to address the issue 

of load-path redundancy in different types of bridges, including truss bridges (AASHTO 2018a). 

These guide specifications provide prescriptive recommendations on the selection of members to 

be removed during redundancy analysis. They also provide load factors for the Redundancy-I

case, when loads are applied before the failure (so that the effect of dynamic amplification can be 

captured), and the Redundancy-II case (for normal use of the bridge without wind after a 

member failure). Both the linear method with dynamic amplification factor and the dynamic 

method (where member removal (MR) can be simulated) are recommended in these guide 

specifications. However, they do not provide an approach to quantify redundancy. Rather, they

provide guidance on the design and evaluation of a bridge after the damage of a fracture-critical 

member. The AASHTO Guide Specifications for Internal Redundancy of Mechanically-Fastened 
7 

Built-up Steel Members provides guidance on evaluating internal redundancy in built-up 

members traditionally designed as fracture-critical members (AASHTO 2018b). These guide 

specifications also provide guidance on special inspection intervals for fracture-critical members.

Past Studies on Progressive Collapse of Buildings 

In general, bridge collapse originating from the failure of a critical member, section, or 

connection is quite similar to progressive collapse in building structures. Blandford (1997) noted 

that progressive failure of a structure involves analyzing the structure’s response to the failure or 

damage of one or more of its members. The loss or failure of a member or members causes force 

redistribution to the remaining structural members. Based on building collapses, the definition of 

progressive collapse that has been widely accepted in the engineering profession was given by 

Ellingwood (2006): “A progressive collapse initiates as a result of local structural damage and 

develops, in a chain reaction mechanism, into a failure that is disproportionate to the initiating 

local damage.” Local structural damage can be initiated by causes, such as blast, impact, and 

others. 

Past research on progressive collapse advanced in the aftermath of high-profile building failures. 

For example, the Ronan Point building collapse in England in 1968 triggered the development 

and improvement of building codes and specifications in many countries, such as Canada, 

Denmark, France, the Netherlands, Sweden, the United Kingdom, and West Germany. Based on 

general lessons learned from the progressive collapse events and other numerous failures, 

Prendergast (1995) presented several prescriptive guidelines to increase resistance against 

progressive collapse, such as the following: (1) local resistance should be increased in key 

regions to inhibit initiation of the collapse process, (2) structural redundancy in the building 

structural system must be provided so that the system can seek ALPs when needed, and (3) all 

structural and nonstructural components should be interconnected to minimize debris projectiles.

Terrorist attacks on the Alfred P. Murrah Federal Building in 1995 and the World Trade Center 

towers in 2001 led to standardized design approaches for buildings to protect against progressive 

collapse in the United States (GSA 2003; DOD 2009). Improvements in building practices to 

address low-probability high-consequence events and to lessen building vulnerability to 

progressive collapse currently are based on the fundamental considerations and assessment of 

general structural integrity, ALP method, prescriptive design provisions, critical member design, 

and MR methods. 

Past Studies on Progressive Collapse of Bridges 

Unlike buildings, bridges are much more vulnerable to collapse in the event of localized failure 

because they have less or no redundancy and unidentified ALPs. Woodward and Zoli (2005) 

noted that long-span bridges are generally not designed to be resistant to progressive collapse. 

The development and adoption of comprehensive AASHTO load and resistance factor design 

(LRFD) provisions in 2012 and 2014 represent explicit consideration of redundancy, ductility, 

and operational importance in the form of load-factor modifiers (AASHTO 2012, 2014). 

However, consideration of these three factors may have a negligible effect on the design, 

particularly for bridges with single-point vulnerabilities and the potential for abrupt member loss. 

In the current code, nonredundant elements must be designed to resist an added 5 percent of the 

factored design loads used to design all other members. However, no instructions are given about 
8 

how to perform a proper analysis or the level of LL that the bridge should be able to carry in its 

damaged condition. A nonredundant bridge may be vulnerable to collapse because of sudden loss 

of a critical member even though the member may have been designed to resist a higher factored 

design load. Load factors that can be applied to the analysis of the damaged structure are also 

currently lacking (Grubb et al. 2015). The current AASHTO approach also does not provide 

discussion on the dynamic aspects of member loss that have been reported in FHWA (2013). 

Hence, unlike the progressive collapse of buildings, there has been no advancement in the 

development of design guidelines or provisions to protect against progressive collapse of bridges 

in the United States even though a number of high-profile progressive bridge failures have been 

observed during the last several decades. Although two recent AASHTO guide specifications do 

provide prescriptive guidance for the analysis of progressive collapse vulnerabilities of bridges, 

they do not address the role of ALP in long-span truss bridges (AASHTO 2018a, b). 

RESEARCH MOTIVATION AND NEEDS

The main objective of this study was to investigate the load-path redundancy of long-span truss 

bridges in the event of loss or failure of one or more of their critical members. Although the 

load-path redundancy of truss bridges has been discussed in case studies, such as that by 

Beauchamp et al. (1984), no studies have investigated factors contributing to the load-path 

redundancy of truss bridges. In current practice, a bridge system is designed by ensuring that the 

capacity of each of the structural members and components stays below the load effects caused 

by a set of pertinent design load combinations. However, such a design approach does not 

guarantee sufficient levels of redundancy to withstand an accidental single point of failure or 

regional damage resulting from intentional threats or other hazards. Because of their operational 

importance for economic, social, and security requirements and their high replacement costs,

major truss bridges should have sufficient load-path redundancy and the capability to survive 

extraordinary events beyond the scope of conventional design criteria. These extraordinary 

events are generally loss of a member or members because of intentional member cutting, blast 

loads, vehicular impacts, strong wind loads, and so forth. 

RESEARCH OBJECTIVES AND OUTCOME

A wide variety of factors that affect the stability and load-carrying capacity of long-span truss 

bridges need to be studied. The overall objective of this research was to develop an 

understanding of the behavior of long-span steel truss bridges in the event of sudden loss of a 

member or members. Specific objectives of this study included the following: 

• Development of an integrated framework to quantify the redundancy or existing ALP

before and after a postulated event involving long-span steel truss bridges. 

• Development of an evaluation procedure to understand the relationship between the 

ALPs, system performance, and overall bridge stability, ensuring sufficient reserve 

capacity.

• Investigation of retrofit approaches that can improve the ALP of long-span truss bridges 

cost-effectively. 
9 

In summary, the main outcome of this research is to prescribe a process that can objectively 

identify ALPs in a steel truss bridge after an abnormal event. Using an objective identification of 

ALPs in producing an optimal retrofit strategy after such an event is demonstrated through 

detailed investigation of two example long-span truss bridges. 

SUMMARY OF THE CHAPTERS OF THIS REPORT

The detailed work performed to achieve the research’s objectives is described in the chapters of 

this report as follows. 

Chapter 2 describes the conventional definition of ALP and redundancy and proposes a novel 

approach to quantify ALP. The definition and properties of ALP are outlined, and the ALP 

methodology is introduced through demand-to-capacity ratio (DCR) based on linear static 

analysis and strain ratio (SR) based on nonlinear dynamic (NLD) analysis.

Chapter 3 describes the finite-element modeling process. This chapter describes in detail both the 

linear static and the NLD analysis approaches. For the linear static method, conventional beam 

elements of the popular SAP2000 computer code were used for modeling truss members

(Computers and Structures, Inc. 2014). Because NLD analysis for MR may require significant 

computational resources, an efficient element type that is built in computer-coded LS-DYNA for 

the NLD analysis method was explored (Hallquist 2014). The accuracy of using such an element 

type was validated through several examples with varying degrees of complexity. 

Chapter 4 presents detailed modeling of the I-35W Bridge in as-built condition to understand its 

behavior before collapse. FEMs of the bridge for both the linear static and the NLD analysis 

were validated through comparison with existing models and research on this bridge. Then, the 

resulting ALPs were identified objectively for dynamic MR events. ALPs were identified for two 

distinct approaches: the linear static approach using DCR as a metric and the NLD approach 

using SR as a metric.

Chapter 5 investigates various retrofit approaches for improving ALP of the I-35W truss bridge. 

For this purpose, the bridge was first seismically retrofitted to resist a hypothetical earthquake of 

the magnitude that may be experienced in a high seismic zone, such as San Diego, CA. Typical 

seismic retrofit primarily involves strengthening truss members by changing the I-shaped 

sections to box sections and increasing the cross-sectional area. Results showed that the seismic 

retrofit of strengthening a member led to limited improvement in ALP of the bridge. Therefore, 

other retrofit strategies were investigated based on the information gained from the identified 

ALPs. This information was used to choose the retrofit procedure or strategy that resulted in 

addition of members at vulnerable locations so that these members contributed to the 

three-dimensionality of the bridge. While retrofitting, a performance-based design (PBD) 

paradigm was followed by introducing several performance criteria in terms of limit states. It 

was shown that (1) following an ALP-based retrofit strategy produces more efficient (in terms of 

added retrofit weight) designs, (2) using a limit state based on the NLD approach produces more 

efficient retrofits than using a limit state based on the linear static analysis, and (3) allowing for a 

more relaxed performance criteria or limit state (in terms of accepting more ductile behavior) 

results in lesser increase in retrofit weight. Thus, bridge engineers can make decisions between 
10 

various retrofit strategies by balancing the acceptance of higher risk and lower performance 

versus savings on extra retrofit weights or costs.

Chapter 6 repeats the analyses in chapter 5 for the Ikitsuki Bridge in Japan, which is a 

through-truss bridge. It is shown that the observations and conclusions of chapter 5 are still 

applicable for this different type of bridge. 

Chapter 7 presents some suggestions for future work, both experimental and analytical, to 

address some of the knowledge gaps identified in this research. 

Chapter 8 presents a summary, conclusions, and recommendations.
11 

CHAPTER 2. ALP AND REDUNDANCY

INTRODUCTION

Conventionally, ALPs are defined as changes in the load paths and load distribution experienced 

by other members in the event of loss of a critical member. The concept of ALP is more 

advanced for buildings than for bridges. For buildings, the General Services Administration 

(GSA) (2003) criteria use the alternate path method (APM) to prevent progressive building 

collapse and allow the use of both linear static and NLD analysis to identify structural members 

in the alternate path structure. Redundancy of a structure tends to promote an overall more robust 

structure and helps to ensure that ALPs are available in the case of local failure of a critical 

member. Additionally, redundancy of a structure generally provides multiple locations for 

yielding to occur, which increases the probability that the damage will be constrained. In 2016, 

GSA upgraded their 2003 progressive collapse requirements of explicit design for loss of vertical 

load-bearing elements by adopting a threat-based approach and APM in Unified Facilities 

Criteria 04-023-03 (GSA 2003; DOD 2009; GSA 2016). This method requires the building to be 

able to compensate for vertical load-bearing elements that are notionally removed one at a time 

at specific plan and elevation locations. 

A detailed framework for addressing issues related to low-probability high-consequence events 

during progressive collapse of buildings can be found in Ellingwood (2006). This work also 

summarizes measures for progressive collapse risk mitigation and identifies challenges for 

implementing general provisions in national standards, such as American Society of Civil 

Engineers (ASCE) Standard 7: Minimum Design Loads for Buildings and Other Structures

(ASCE/SEI 2010). To reduce the probability of progressive collapse, it may be necessary to 

design key elements in a structural system to withstand stipulated “abnormal” loads that are far 

in excess of what would be required under a normal design condition in order to facilitate the 

development of ALPs. Redundancy, as a key to preventing progressive collapse through ALPs, 

means that no critical element’s failure should initiate a series reaction of successive loss of 

members that would take down the building. For each critical element, one or more redundant 

counterparts should take over the critical load scenario in case the critical member fails

(Ellingwood and Dusenberry 2005). El-Tawil et al. (2007) investigated the progressive collapse 

of steel frame structures using a macromodel-based simulation approach and noted that the 

collapsing system continually seeks ALPs in order to survive in the dynamic process. Their work 

showed that a nonlinear APM is useful for judging the ability of a system to absorb the loss of a 

critical member. Compared to an elastic APM analysis, inelastic APM simulation provides more 

resolution, shows failure progression, and provides information on the likelihood of complete 

versus partial collapse. Izzuddin et al. (2008) investigated the APM for buildings and showed

that ALPs are beneficial in improving redundancy and structural robustness in buildings. 

For bridges, although load-path redundancy may currently be considered in the classification of 

fracture-critical members for design and fabrication (Lwin 2012), structural and internal 

redundancy considerations have not yet been codified in AASHTO LRFD specifications. 

However, two recent AASHTO guide specifications provide prescriptive guidance on selection 

of members to be removed, analysis methods, evaluation of bridges for different types of 
12 

analysis following a member loss, and internal redundancy evaluation of built-up members 

(AASHTO 2018a, b). 

Codes, such as the Canadian Standards Association (CSA) CAN/CSA-S6-00, have been 

permitting engineers to consider redundancy through system behavior (CSA 2006). In a previous 

study on ALP, for example, Sanders et al. (1975) tested the load capacity of the single-lane truss 

Hubby Bridge over the Des Moines River (which was scheduled for removal) by cutting its 

vertical member at one cross section. The test results showed that cutting the member did not 

result in a significant decrease in the load-carrying capacity of the bridge. Sanders et al. (1975) 

attributed this capability of redistributing loads to the frame action inherent in trusses and the 

semicontinuous nature of the deck truss. Csagoly and Jaeger (1979) analyzed the failure of a 

number of bridges to support the existence of multiple load paths where load redistribution took 

place after damage to a critical member. They also noted that the construction of single-load-path 

bridges was prohibited by the 1979 Ontario Highway Bridge Design Code. Frangopol and Curley 

(1987) proposed various indices to quantify redundancy in terms of strengths of intact and 

damaged systems without identifying ALPs that contribute to redundancy. Sirisak (1996) 

investigated the structural integrity evaluation of truss bridges. They noted that the joints in these 

bridges may be more rigid than pinned joints and that a bridge truss that is assumed to be 

statically determinate may be highly indeterminate and show significant reserve strength when 

loaded beyond the classical limit load determined by the limit state of a single member. If the 

joints are treated as rigid, the failure of a member may not lead to an immediate collapse of the 

bridge; ALPs may be found. In such a situation, it may be more prudent to evaluate the system’s

strength without exclusively concentrating on component or member strength to identify ALPs. 

Williamson et al. (2010) investigated the safety of reinforced concrete bridges against blast loads 

and recommended increasing the redundancy of reinforced concrete bridges by proving multiple 

load paths. This includes, but is not limited to, decreasing the spacing of longitudinal girders and 

stringers and decreasing the deck beam spacing. Although local damage to the bridge deck, the 

supporting bridge girders, or both is undesirable, redundancy and ductility will often allow 

internal forces to redistribute when damage to these components occurs, thus allowing an ALP to 

be realized to maintain the global stability. 

The detailed discussion in this section shows that although efforts have been made to quantify 

redundancy in bridges, which depends on multiple load paths in the event of damage to a critical 

member, ALP is generally defined qualitatively. One of the objectives of this research is to 

define ALP in terms of quantifiable load redistribution among members affected by the removal 

of a critical member. It is assumed that a truss bridge structure is load-path redundant if the 

structural system has sufficient ALPs after local damage. Otherwise, the truss bridge is 

nonredundant. 

PROPOSED ALP DEFINITION AND REDUNDANCY

Based on the qualitative description of ALP described in the previous section, major shifts in the 

behavior of a structural system after a change (such as sudden local damage in a member) in an 

otherwise functionally acceptable structural system constitute an ALP within the system.

However, an unacceptable structural performance or an unstable system after this dynamic 

process of local damage may not produce an ALP in the system. The change in the system can be 

attributed to the following two scenarios: 
13 

1. A hypothetical situation that is needed for design, such as an MR in a framed structure 

(including but not limited to trusses, frames, suspension systems, etc.). 

2. An abnormal loading demand such as a blast, earthquake, tornado, shock or impact, and 

so forth. 

The change is manifested in either complete loss of or degradation of the performance of one or 

more of the system components. However, this qualitative vision of ALPs is difficult for 

engineers to put into design practice. 

One of the goals of this research is to objectively use ALP to attempt to find optimal design and 

retrofit measures for long-span truss bridges undergoing two specific abnormal hazards: MR and 

blast (BL) (which refers to blast demands, hazards, or threats). To achieve this goal, a concise 

and quantitative definition of ALP for these types of structural systems is introduced in the 

following. 

For a structural system S with rm components M, it is assumed that W is a set of individual 

components (members). The ith component M (Mi) is thus Mi ∈ W, i = 1, 2, 3, ... rm. 

Conventional design practices for the system state that under a base (normal) loading condition 

or demand (D) (e.g., D = (αDL DL + αLL LL), where αDL and αLL are DL and LL factors, 

respectively), the state of design metric in the ith component (DMi), satisfies the system’s 

acceptance criteria or design limit state (LIM(DM)), as shown in equation 1: 

(1)

In equation 1, DMi could be the stress in the ith member, which should be less than or equal to 

the maximum allowable stress (which will be LIM(DM)). The performance of the system can be 

measured by any of the possible design metrics, such as the following: 

• Internal forces, stresses.

• Reactions.

• Displacements, velocities, accelerations, deformations, strains. 

• Other capacity or demand metrics.

• Combinations of any of the previous bulleted items. 

Among these metrics, the following two were deemed reasonable for the work on ALPs in this 

research:

1. For linear elastic behavior, a popular design metric is the DCR, or DCRi for the ith 

component based on linear elastic analysis. For this metric, the demand is calculated as 

the stress in the ith member, and the capacity is calculated as the maximum allowable 

stress (0.6σy) (where σy is the yield stress of the member), which is used for the design of 

steel truss members.

2. For nonlinear behavior, a reasonable design metric is the SR, or SRi for the ith 

component, where SRi is defined as shown in equation 2: 
14 

(2)

In equation 2, Max(εi) is the maximum strain in the ith component and εy is the pertinent yield 

strain. Max(εi) needs to be obtained based on detailed NLD analysis simulating hazards causing 

the change, such as loss of a member.

For the purpose of this study, it is assumed, without any loss of generality, that the system is a 

base system. As such, the base system is renamed as S0, the base loading condition as D0, the 

base ith component as M0i, the base structural set as W0, and the base design metrics as DM0i. 

Note that M0i ∈ W0. 

It is assumed that the values of DM0i have changed into a new set of values, DM1i. Such a change 

can occur for numerous reasons, including the following: 

1. A change in the topology of S from S0 to a new topology S1, which results in a change of 

the composition of W from W0 to a new component set W1. Such a change can occur if 

one or more members in W0 are removed from the set (i.e., because of sudden member 

loss).

2. A change in D0 to a new demand D1 (e.g., D1 = (αDL DL + αLL LL + BL). 

Many other reasons for such changes exist; however, they are beyond the scope of this report. 

The new set of design metrics, DM1i, is not equal to DM0i, as shown in equation 3: 

(3)

Note that, by their definitions, the states and distribution of DMi identify the load path given S

and its companion W. It can then be concluded that the base load path can be illustrated by the 

distribution of DM0i. Similarly, it can be concluded that the new load path can be illustrated by 

the distribution of DM1i. Hence, ALPs for the structural system can be quantified as the changes 

in the spectra of DM1i (with respect to the spectra of DM0i) due to the changes in the system, 

such as the loss of a member. This definition of an ALP needs to meet the following three 

important properties: 

1. Appropriateness. For an ALP to be appropriate, it needs to accurately illustrate the flow 

of the loading throughout S: the applied loads all the way to the boundary conditions. 

Clearly, the choice of design metrics, DM1i (such as stress or strain in the ith component 

of the changed system), meets this criterion. 

2. Generality. For an ALP to be general, it needs to satisfy any steel truss structural system. 

The ALP as defined by DM1i meets this criterion because no limit was placed on the 

applicable systems.

3. Usefulness. For an ALP to be useful, it needs the ability to be used in retrofit and initial 

design situations. The ALP, as defined by DM1i, can be used in retrofit and initial design 

situations if equation 1 is developed further. Furthermore, identifying new load paths 

with limit state distributions that are beyond the LIM(DM) is also of interest. A 
15 

reasonable set of jth and (j+1)th limit states can then be identified, as shown in 

equation 4: 

(4)

By identifying different desired limit states (LIM(DM)j) appropriately, a useful ALP spectra of 

M1i, as illustrated by DM1i, will emerge. Using these ALP spectra for retrofit or initial designs 

can be helpful in attaining optimal retrofits. In equation 1, one could assume LIM(DM)j, for 

example, as 0.6σy and LIM(DM)j+1 as σy for truss members. Then, all members with stresses 

between these two limits can be identified to characterize the ALP for the particular member loss 

scenario.

DCR USING LINEAR STATIC ANALYSIS

The DCR of each truss member can be used to determine the potential for progressive collapse. 

This research uses SAP2000 to calculate DCR. In this software, a steel frame design/check 

module calculates axial, flexural, and shear forces or stresses at several locations along the length 

of a member and then compares those calculated values with acceptable limits. This comparison 

produces DCR for each of the members, which typically should not exceed a value of 1.0 if code 

requirements are to be satisfied. Table 1 and table 2 show the proposed limit states for DCR and 

SR, respectively, which can be used during the analysis of ALPs for long-span truss bridges. 

Table 1. Proposed limit states for ALP analysis using DCR metric.

Limit State Metric Range 

Design limit DCR ≤ 1.0

Elastic limit 1.0 < DCR ≤ 1.67

Beyond elastic limit DCR > 1.67 

Table 2. Proposed limit states for ALP analysis using SR metric.

Limit State Metric Range 

Elastic limit SR ≤ 1.0

Limited ductility 1.0 < SR ≤ 2.0

Medium ductility 1.0 < SR ≤ 4.0

High ductility SR > 4.0 

These limit states are based in general engineering practice. For linear elastic analysis, DCR is 

defined as the ratio of member stress to the maximum allowable stress in steel truss members

(0.6σy). Hence, the limit state of DCR ≤ 1.0 is commonly used to ensure that the stresses in steel 

members of a truss bridge are less than 0.6σy. The limit state of 1.0 < DCR ≤ 1.67 may be used to 

evaluate whether the stresses in steel members are more than 0.6σy but less than σy. The value of 

DCR corresponding to σy in a truss member is 1 ÷ 0.6 = 1.67. A limit state of DCR > 1.67 implies 

potential inelastic behavior in the member.

For NLD analysis, SR is defined for individual local members and has a similar meaning as the 

ductility ratio (which is a representation of global system behavior in terms of displacement). 
16 

Hence, values of SR ≤ 1.0 imply completely elastic behavior at the element level, whereas values 

of SR > 1.0 imply nonlinear material member behavior responsible for the ductile behavior of the 

bridge system. 

The DCR is obtained according to AASHTO (2010) and the American Institute of Steel 

Construction (AISC) (2010), which define the resistance by axial and flexure interaction 

equations. For doubly and singly symmetric members subject to axial force and flexure, DCR is 

calculated by equations 5 and 6: 

(5)

(6)

Where: 

Pr = axial force demand. 

Pc = axial force capacity. 

Mrx = bending moment demand with respect to the x-axis.

Mcx = bending moment capacity with respect to the x-axis. 

Mry = bending moment demand with respect to the y-axis.

Mcy = bending moment capacity with respect to the y-axis.

Equation 5 can be rewritten as equation 7 as follows: 

(7)

Where: 

Ac = member cross-sectional area. 

Sxx = section modulus with respect to the x-axis.

Syy = section modulus with respect to the y-axis.

σrp = axial stress. 

σMx = stress due to moments with respect to the x-axis.

σMy = stress due to moments with respect to the y-axis.
17 

Ω = safety factor.

σallow = allowable stress obtained from σy divided by Ω based on the allowable strength 

design approach. 

Safety factors for compression, flexure, and tensile yielding are 1.67. The tensile rupture case 

with a safety factor of 2.0 is not considered in this study. Examples of calculating DCR for 

compression and tension members are in appendix C. 

Both allowable strength design (ASD) and LRFD designs consider a second-order analysis by 

adjusting the first-order analysis (AISC 2010), where the factor (1 − αPr ÷ Pe) is recognized as 

an amplification factor accounting for the second-order moment caused by the P-δ effect (Pe is 

the elastic critical buckling strength of the member, and the design coefficient α is 1.0 for LRFD 

and 1.6 for ASD designs, respectively). For a steel truss bridge system, because the members are 

rigidly connected at truss joints that cause lateral moments, moments are not uniform along the 

length of a member and are largest at member ends. This nonuniform moment is accounted for 

by the reduction coefficient Cm (equation A-8-4 in AISC 2010). For a member subject to 

compression, M1 and M2 in the equation for Cm are the smaller and larger moments, respectively, 

calculated from a first-order analysis at the ends of an unbraced portion of the member in the 

plane of bending under consideration when no transverse loading is present between supports in 

this plane. The ratio of these moments, M1 ÷ M2, is positive when the member is bent in a reverse 

curvature and negative when the member is bent in a single curvature. However, this adjustment 

may not be needed when the second-order analysis is automatically included in the computer 

model. For DCR calculated in this research, the SAP2000 model is programmed to perform the 

second-order analysis during the calculation of DCR. 

SR USING NLD ANALYSIS

SR is calculated from the results of NLD analysis in LS-DYNA and is obtained by dividing the 

total strain of the member section by the yield strain of the steel. As will be described in 

chapter 3, examples of long-span bridges in this study are modeled in LS-DYNA using 

Belytschko–Schwer (B-S) beam elements and a Mat 98 material model to represent material 

nonlinearities and strain hardening in steel. Strains are not explicitly exported from the analysis 

using LS-DYNA; rather, they are calculated from displacement, rotation, and strain relationship. 

The displacements can be obtained from the analysis directly. Detailed formulation on the 

calculation of the SR from the output of the beam element in LS-DYNA can be found in 

appendix A. Various limit states for using the SR metric are shown in table 2. 

After characterizing the bridge members using DCR and SR, redundancy can be quantified by 

ALP using the following steps: 

• Identify whether the structure needs retrofit after a triggering hazard such as MR. 

• Identify components requiring retrofits. 

• Determine the level of retrofit. 

• Identify load redistribution among members through MR analysis.
18 

LOADS AND LOAD COMBINATIONS

Models of example bridges may need to be analyzed for combinations of DL, LL, and seismic 

load (EQ). For design purposes, the load conditions in equations 8, 9, and 10 can be used: 

(8)

(9)

(10)

Based on literature review, load combinations for progressive collapse analysis are available 

only for buildings. For example, Ellingwood et al. (2007) presented load combinations for 

progressive collapse of buildings. This report noted that many standards, such as ASCE/SEI 

(2010), have recommended an LL factor of 0.5 because of a small probability of the accidental 

and design loads occurring jointly. 

Superstructures of highway bridges are designed using load factors recommended in tables 

3.4.1-1 and 3.4.1-2 of the AASHTO LRFD Specifications (2014). These AASHTO load factors 

were recommended based on extensive research and calibration. Some general principles for

selecting load factors can be found in table 3.4.1-1 of AASHTO (2014). This report recommends 

using 0.5 as the load factor for the LL during extreme events because it is likely that a reduced 

LL will be present on the bridge during an extreme event when considering the likelihood of 

simultaneous load effects. Based on these observations, following the AASHTO LRFD (2017) 

load combination for extreme hazard events shown in equation 11 is recommended:

(11)

ASD VERSUS LRFD

The DCR in this research was calculated based on an ASD approach where the demand is 

calculated using the load combination in equation 11 and the capacity is calculated as 0.6σy. 

Table 3.4.1-2 in the current AASHTO LRFD bridge design specifications presents load 

combinations and load factors for Extreme Event II (AASHTO 2017). Following this table and 

table 3.10.1.1-2 in FHWA (2015), load combinations for MR in LRFD can be expressed as

shown in equation 12: 

(12)

The resistance factor for the capacity in LRFD is 0.9 (i.e., the design stress is 0.9σy). Hence, the 

DCR for the LRFD case will be as shown in equation 13: 

(13)

The DCR for the ASD case (by using equation 11) will be as shown in equation 14: 
19 

(14)

In equations 13 and 14, (DL)R and (LL)R are the response of a member because of DL and LL, 

respectively, divided by σy. Generally, LL is significantly smaller than the total DL of bridges. 

For example, for the I-35W Bridge (table 15 in chapter 4), LL is 22 percent of the DL. Therefore, 

for this bridge, (LL)R = 0.22(DL)R. Using this, the DCR for ASD and LRFD can be approximated 

as 1.85(DL)R and 1.79(DL)R, respectively. Hence, DCR calculated using ASD is slightly higher 

than DCR calculated using LRFD. The major difference between ASD and LRFD is in terms of 

limit states. For ASD, the primary limit states are DCR = 1 (allowable stress limit state) and 

DCR = 1.67 (yield stress limit state). For LRFD, these limit states are DCR = 1 for design and 

DCR = 1.11 for yield stress. However, because the main objective of the research is to 

investigate effective retrofit strategies, the proposed ALP approach can be applied to a specific 

bridge using the LRFD approach without any loss of generality. 

SR in this research is calculated as the strain in a member for the load combination in equation 11 

divided by εy. This SR can be interpreted similarly to ductility (i.e., SR = 1 represents the limit of 

elastic behavior). However, this value is based on the assumption that the total load on the bridge 

is represented by the load combination in equation 11. Hence, calculation of SR does not follow 

the ASD or the LRFD approach.

21 

CHAPTER 3. FINITE-ELEMENT MODELING

INTRODUCTION

As discussed in chapters 1 and 2, few studies (e.g., Liu et al. 2013) have investigated the 

progressive collapse of truss bridges using an elastic FEM with a truss member modeled as a 

frame element to account for secondary moment effects. Fiorillo et al. (2015) investigated the 

redundancy of truss bridges by considering the model of the Aby truss bridge, a 33.5-meter

(110-foot) simply supported bridge. This bridge, as a prototype, was tested to failure to study its 

remaining load-carrying capacity by Blanksvärd et al. (2014). Fiorillo et al. (2015) modeled this 

bridge in ABAQUS (Dassault 2013a, b) by four-node 2D shell elements to analyze the behavior 

of the superstructure and to determine the reserve capacity after a critical member failure.

Although the bridge has only 175 elements and 98 nodes, it was modeled by 43,000 shell 

elements with 172,000 nodes so that behavior such as buckling and yielding after MR could be 

simulated. The computational time for each run of this model was estimated to be longer than 

6 h. Modeling the long-span bridges in this study by shell elements in this way would have 

resulted in millions of elements and nodes. Simulation of such a large model would have 

required computational resources and time significantly beyond those available for this research.

In this research, a two-part modeling approach was adopted to investigate the ALP of truss 

bridges. To start, all truss bridges were modeled in SAP2000 and verified against available 

design, analysis, and modeling data to confirm the bridge’s elastic behavior. This model was

used to investigate the ALP of bridges using the DCR approach discussed in chapter 2. Detailed 

information on the modeling of example bridges in SAP2000 will be presented in later chapters. 

However, the elastic model of the bridge may not be sufficient for designing cost-effective 

retrofit measures to protect against progressive collapse. Software such as SAP2000 cannot 

simulate the removal of members dynamically for the redundancy analysis. 

Therefore, the second modeling approach involved simulation of the NLD behavior of truss 

bridges due to loss of a critical member in LS-DYNA, which is a general-purpose multiphysics 

simulation software package used widely for simulations involving dynamical effects. 

LS-DYNA’s explicit analysis capability provides options for modeling truss elements by beam 

element formulations that can be selected based on the geometry, loading, boundary conditions, 

and material properties of the structure being analyzed.

FINITE-ELEMENT MODELING IN LS-DYNA

Beam Element Formulation

Truss members in LS-DYNA are best modeled by beam or shell elements. As discussed in the 

previous section, using shell elements for long-span truss bridges is computationally 

cost-prohibitive. Two types of beam element formulations exist in LS-DYNA: Hughes–Liu 

(H-L) and B-S. Degenerated from the isoparametric eight-node solid element into the two-node 

beam geometry, the H-L beam element requires cross-section integrations to account for 

different shapes, and it generates a constant moment along the length. Using this element 

requires considerably fine element mesh to achieve desired accuracy. The B-S beam element, a 

resultant element in LS-DYNA, is formulated explicitly. This element produces a linearly 
22 

varying moment along a member. The bending effect in truss members may not be negligible 

after the sudden loss of a member. Thus, it is important to consider variations in the end 

moments of truss members. Moreover, considering the convergence issue related to sophisticated 

integration across the section, mesh accuracy, and iterations involved in multistage analysis, the 

B-S beam element seems to be more efficient in achieving this study’s objectives.

Based on a classical resultant beam formulation, the B-S beam element with 6 degrees of 

freedom (DOFs) for each node was developed by Belytschko et al. (1977). This beam element is 

capable of treating large rotation using a co-rotation technique in the element formulation. This 

element is somewhat faster in computation because no numerical integration is performed over 

the cross section. However, it may be less accurate for elastoplastic analysis when partial section 

yielding is important. Because it is a resultant-based formulation, any type of cross section may 

be easily modeled by specifying its cross-sectional area and area moments of inertia. The B-S 

beam element differentiates between flexure about strong and weak axes by the assignment of a 

third node, n3. Potential reductions in strength due to local buckling are not considered in this

formulation. 

The co-rotation technique, as shown in figure 7, uses two types of coordinate systems: one 

associated with each element to show the deformed shape, excluding the rigid body rotations, 

and the other embedded in the nodes to display the rigid body rotations. The element coordinate 

system is defined to have the local x-axis ( ) originating at node I and terminating at node J, and

the local y-axis ( ) and the local z-axis ( ) are constructed normal to and . The element 

coordinate system ( , , ) and associated unit vector triad (e1, e2, e3) are updated at every time 

step by the same technique used to construct the initial system. One advantage of the 

co-rotational formulation is the ease with which existing small displacement element 

formulations can be adapted to a larger displacement formulation having small deformations in 

the element system. In figure 7, and are the x- and y-axes in the global coordinate system, 

respectively; and are the x- and y-axes in the local coordinate system, respectively; and 

 are the unit vectors along the global axes and , respectively; and are the unit 

vectors along the local axes and in the initial configuration, respectively; and and are 

the unit vectors along the local axes and , respectively.
23 

Source: FHWA.

A. Initial configuration. 

Source: FHWA.

B. Rigid rotational configuration. 

Source: FHWA.

C. Deformed configuration. 

Figure 7. Illustrations. Co-rotational coordinate system for the B-S beam element.

The displacements ( ) represent the deformation in the B-S beam

element formulation, where δIJ isthe change in the length, is the torsional deformation, and 

 are the bending rotational deformations. Here, the superscript emphasizes that 

these quantities are defined in the local element coordinate system and I and J are the nodes at 

the ends of the beam. The internal forces and moments are then related to displacement and 

curvatures at the resultant element level. Using such a formulation, only forces and moments are 

calculated, not stresses (Hallquist 2014). This method of calculation is a major difference 

between this approach and approaches using shell or solid elements, where the calculation of 

internal forces and moments based on each single node is controlled by certain stress criteria. 

However, for truss bridges, the resultant force of each member is more crucial than the stress at a 

single point. 
24 

Simulation of Loads in LS-DYNA

The analysis for a long-span bridge involves two steps. In the first step, gravity loads should be 

applied to the structure. The application of gravity is important for the simulations during the 

second step where the added LL or extreme loading scenario, such as MR, is applied. Dynamic 

relaxation (DR) is needed during the first step, and the “restart” algorithm is used for the

transition between the two steps. For the dynamic simulations using an explicit solver, all the 

loads, including the gravity loads, are considered dynamic, which is not the actual behavior of 

these loads. Because the structural response to these loads in LS-DYNA will experience dynamic 

magnification, the DR method is adopted to filter or remove undesirable dynamic effects. The 

second step is basically a “restart” from the first step and is a continuation of LS-DYNA run 

after the convergence of the DR phase. 

Material Models

Material properties play an important role in controlling structural behavior under external loads. 

Accurate structural response requires a sound and effective stress–strain or constitutive 

relationship to describe the material behavior. Researchers have been trying to implement a 

proper material model that can successfully incorporate the typical response, such as elastic, 

post-yield (tension or compression), elastic and inelastic post-buckling, and stress reversal 

(in buckled or yielded members), of truss members (Schmidt and Gregg 1980; Papadrakakis

1983; Mueller and Wagner 1984; Hill et al. 1989; Murtha-Smith 1988; Blandford 1996). 

The nonlinear behavior of the B-S element in LS-DYNA can be simulated by material models 

Mat 171 and Mat 98. Although the Mat 171 model was intended to represent both cyclic 

buckling and tensile yielding of steel concentric braces, which is similar to the behavior of truss 

members, it cannot be used for simulation of nonlinear behavior during MR. Hence, material 

model Mat 98 was used to investigate the behavior of bridges after the loss of a critical member.

The Johnson–Cook material model (Mat 98) is expressed by the stress–strain relationship for 

beam members as shown in equation 15: 

(15)

Where: 

A, B, C, n = input constants. 

εp = effective plastic strain. 

ε* = normalized effective strain rate.

As noted in equation 15, this material model considers the strain rate effect through the term ε*, 

although this effect may not be significant during the removal of truss bridge members. This 

general-purpose material model considers material hardening nonlinearity by including the 

user-calibrated realistic material stress–strain curves. In addition, the model enables users to 

calibrate the constitutive model parameters via regression from laboratory data without using a 

sophisticated numerical algorithm. Users can verify the selected parameters to replicate the data 

used to obtain these parameters and reproduce the test results using simple numerical runs. A 

failure strain is defined for the material as well.
25 

Mat 98 can be used during both DR and MR phases. However, it is not capable of simulating the 

reduction in the strength of a buckled member. When using this material model, buckling of a 

truss member due to sudden loss of a critical member can be identified through separate buckling 

analysis of the damaged bridge. If any member is likely to buckle, it will need to be retrofitted 

before performing dynamic MR analysis. Because the main objective of the research is to 

investigate cost-effective retrofitting of truss members to protect against progressive collapse, 

Mat 98 can be used to model inelastic tensile and compressive behavior while avoiding buckling 

behavior through retrofit as members prone to buckling are identified. 

When compression buckling is not a concern during the MR analysis, it is sufficient to consider 

the typical stress–strain relationship in figure 8 during both compression and tension. In the 

elastic region, stress increases linearly with an increase in the strain, and it is quantified by the 

Young’s modulus, E = 2 × 105

 MPa (29,000 ksi). Steel is a relatively ductile material and can

undergo large strains before rupture. Low-carbon structural steel, such as A36 and A572, 

exhibits a well-defined yield point followed by a flat yield plateau. When high-strength steel is 

used, the yield plateau becomes less obvious. For a specific A992 steel in this research, 

parameters A, B, C, and n in equation 15 were calibrated based on the actual stress–true strain 

relationships obtained by Arasaratnam et al. (2011) from a standard uniaxial tensile test. The 

comparison between calibrated and experimental stress–strain results is shown in figure 9, and 

the calibrated parameters for the material model are presented in table 3. 

Source: FHWA. 

Figure 8. Graph. Typical stress–strain curve for steel. 
26 

Source: FHWA. 

1 MPa = 0.145 ksi.

Figure 9. Graph. Measured and calibrated stress–strain relationships for A992 steel and 

Mat 98 in LS-DYNA.

Table 3. Calibrated parameters for material model Mat 98. 

Parameter Unit (MPa) Unit (ksi)

A 345 50 

B 448 65 

C 0.0162 0.0162 

n 0.328 0.328 

Failure strain 0.1 0.1 

VERIFICATION OF BEHAVIOR OF THE B-S BEAM ELEMENT

To develop confidence in the capability of the B-S element with the Mat 98 material model, 

verification studies were performed for structures with single and multiple elements, as described 

in the following sections. 

Example 1: Cantilever Beam 

In this example, a beam that is cantilevered at one end was modeled by a single B-S beam 

element in LS-DYNA. Table 4 presents the sectional and material properties of this cantilever 

beam, which were obtained from real truss member U8-L8 (the vertical member at one of the 

piers, with a length of 18.288 m (60 ft)) of the I-35W Bridge that collapsed in 2007 (Hao 2010). 

This member had the largest area moment of inertia among truss members in the bridge. 

Figure 10 shows a section of the member, and figure 11 shows the six basic loading cases 

studied. Table 5 shows loading cases and results from analytical formulations and finite-element 

analysis using a single B-S element. Because LS-DYNA performs analysis by considering static 

loads as dynamic, damping is applied to achieve the stable (or steady-state) solution in the 

column “End B Translation From FEM” of table 5. This table demonstrates that the elastic 

behavior of a cantilever beam with concentrated force and displacement acting at the ends of the 

beam can be accurately represented by a single B-S element.
27 

Table 4. Sectional and material properties for the example cantilevered beam.

Property Parameter Value

Sectional Sectional dimensions 2 plates: 81.28 cm (32 inches) × 6.0325 cm 

(2.375 inches)

2 cover plates: 50.8 cm (20 inches) × 1.905 cm 

(0.75 inches)

1 web plate: 41.275 cm (16.25 inches) × 1.905 cm 

(0.75 inches)

Sectional Cross-sectional area (Ac) 1.2528 × 10−1 m2 (194.11 inches2) 

Sectional Shear modulus (G) 1.2224 × 10−4 m4 (293.7 inches4) 

Sectional Moment of inertia (Ixx) 8.7476 × 10−3 m4 (21,016.9 inches4) 

Sectional Moment of inertia (Iyy) 6.0443 × 10−3 m4 (14,521.5 inches4) 

Material Density (ρ) 7,860 kg/m3 (490.68 lb/ft3) 

Material Young’s modulus (E) 210 GPa (3.05 × 104 ksi) 

Material Poisson ratio 0.3 

Material Yield stress 345 MPa (50 ksi) 

Material Ultimate stress 448 MPa (65 ksi) 

Source: FHWA. 

Figure 10. Illustration. Section of the example cantilevered beam. 

Source: FHWA. 

Note: Fx, Fy, and Fz are concentrated forces along the x, y, and z directions, respectively, at the end of the 

beam, and Mx, My, and Mz are concentrated moments along the x, y, and z directions, respectively, at the end 

of the beam. 

Figure 11. Illustration. Basic loading cases for the cantilevered beam.
28 

Table 5. Comparison between analytical and FEM solutions for the cantilevered beam.

Load Case Number

Applied Force (N) 

or Moment (N·m) at 

End B

Analytical Solution 

(m)

End B Translation 

From FEM 

(m)

Fx 2.073 × 107 ux = 0.0144 0.0144 

Fy 2.312 × 105 uy = 0.372 0.372 

Fz 1.861 × 105 uz = 0.207 0.207 

Mx 1.080 × 104 ux = 0 0 

My 3.400 × 106 uz = 0.310 0.310 

Mz 4.229 × 106 uy = 0.557 0.557 

1 m = 3.281 ft; 1 N = 0.225 lb; 1 N·m = 0.738 lb·ft. 

Note: ux, uy, and uz are the displacements in the x, y, and z directions, respectively. 

Example 2: Toggle Truss and Williams Toggle Frame

As typical examples of the snap-through phenomenon, the toggle truss and Williams toggle 

frame shown in figure 12 were modeled by the B-S element with the Mat 98 material model. The

instability in these structures could be caused at both the elemental and the structural levels. The 

structure could collapse when it reaches structure-level instability. In the toggle truss, two 

members are connected by a pin, whereas in the Williams toggle frame, two beam members are 

connected by a rigid connection. Both of these shallow structures exhibit typical snap-through 

response when loaded by a force F at the apex. 

Source: FHWA.

1 m = 3.281 ft.

A. Toggle truss. 

Source: FHWA.

1 m = 3.281 ft.

B. Williams toggle frame.

Figure 12. Illustrations. Typical truss systems.

For the toggle truss case, values of the Young’s modulus and of yield stress were taken as 

206 GPa (29,877.83 ksi) and 2.35 × 105 kPa (34.08 ksi), respectively. Figure 13 shows the 

constitutive stress–strain relationships for elastic and elastic-kinematic plastic materials. 

Figure 14 shows the comparison between LS-DYNA results and analytical solution results for 

both elastic and kinematic plastic systems based on Papadrakakis (1983). Finite-element and

analytical results are in excellent agreement.
29 

Source: FHWA.

A. Elastic material. 

Source: FHWA. 

B. Elastic-kinematic material. 

Figure 13. Graphs. Constitutive stress–strain relationships for steel.

Source: FHWA. 

1 m = 3.281 ft; 1 kN = 0.225 kips.

Figure 14. Graph. Load-displacement curves for the toggle truss system.

In the Williams toggle frame (Williams 1964), the frame is fully fixed against translation and 

rotation at both ends. The frame is loaded by force F at the apex. Table 6 shows material and 

sectional properties of members for this structure. Figure 15 shows force-deformation plots for 

the structure based on an analytical formulation by Williams (1964), finite-element formulations 

by Davalos (1989) and Warren (1998), and finite-element formulations with the B-S beam 

element in this research when frame elements are modeled by one and two elements. In the 

elastic region, good agreement is observed among force-deformation curves using different 

formulations. In the immediate inelastic region, the maximum difference between the 

force-deformation curve for the B-S element and the other curves is approximately 10 percent
30 

when the frame element was modeled by one B-S beam element and is much smaller when two 

B-S beam elements were used to model the frame element. This example again shows the 

reliability of the modeling using the simplified B-S beam element.

Table 6. Material and sectional properties of Williams toggle truss element.

Description Value

Young’s modulus (E) 1.28 × 105 MPa (1,855 ksi) 

Section area (Ac) 1.18 × 10−4 m2 (0.183 inches2) 

Moment of inertia (Ixx) 3.75 × 10−10 m4 (9.0 × 10−4 inches4) 

Moment of inertia (Iyy) 3.60 × 10−9 m4 (8.65 × 10−3 inches4) 

Length (L) 0.329 m (12.936 inches) 

Height (h) 9.80 × 10−3 m (0.386 inches) 

Source: FHWA. 

1 N = 0.225 lb; 1 mm = 0.0394 inches.

Figure 15. Graph. Load-displacement curves for Williams toggle truss system. 

Example 3: Aby Bridge 

Aby Bridge, constructed in 1957, was a simply supported truss bridge superstructure carrying the 

Swedish northern mainline from Stockholm over the Aby River 80 km (49.7 mi) southwest of 

Luleå (Blanksvärd et al. 2014). In 2013, after its replacement by a new steel bridge, the old Aby 

Bridge was tested to failure to study its remaining load-carrying capacity. The bridge was 33.5 m 

(110 ft) long and 5.5 m (18 ft) wide. The truss’s vertical members and the main transverse beams 

were spaced 4.125 m (13 ft) from each other. The truss height was 4.7 m (15 ft). The longitudinal 
31 

deck stringers were evenly spaced in the width. The southwest supports had four pins, whereas

the northeast supports had four rollers. The steel truss members were built-up sections. 

The loading on the bridge was applied by two hydraulic jacks that pulled the bridge downward, 

as shown in figure 16. The jacks were anchored to the underlying bedrock by two injected 

cables. Several test runs were made with different maximum loads to monitor strains and 

deflections under different conditions. Figure 17 shows the load-displacement plot of the bridge 

based on the test results. The bridge had substantial yielding deformations for a load of 

approximately 11,000 kN (2,472.9 kips). 

© 2014 Thomas Blanksvärd, Lulea University of 

Technology, Sweden.

A. Loading arrangement in field test. 

© 2014 Thomas Blanksvärd, Lulea University of 

Technology, Sweden. 

B. Loading arrangement in FEM analysis.

Figure 16. Photo and Illustration. Application of LL on the Aby Bridge.
32 

Source: FHWA.

1 kN = 0.225 kips; 1 mm = 0.0394 inches. 

Figure 17. Graph. Load-displacement curves of the Aby Bridge. 

Blanksvärd et al. (2014) performed finite-element modeling for a quarter of the bridge because 

of symmetry in two directions. Table 7 shows the material properties of the steel in the truss 

components. Figure 18 shows the model of Aby Bridge in LS-DYNA developed in this research 

using B-S beam elements. The loading from hydraulic jacks was applied through distributed 

beams in the FEM. Table 8 compares the computational effort required to model the truss by 

shell and B-S beam elements. Blanksvärd et al. (2014) used 43,000 four-node shell elements 

with 172,000 nodes, compared to 175 beam elements with 98 nodes in this research. The 

calculation time decreased from more than 6 h to 10 to 15 min. 

Table 7. Material properties of steel in truss components.

Steel Member Type Material

Yield Stress σy

(MPa)

Ultimate Stress σu

(MPa)

Main beams SIS 1411 250–260 440 

Verticals, diagonals, 

secondary systems 

SIS 1311 200–220 370 

1 MPa = 0.145 ksi.
33 

Source: FHWA. 

Figure 18. Illustration. 3D FEM of Aby Bridge in LS-DYNA.

Table 8. Comparison between finite-element modeling approaches for the Aby Bridge.

Parameter

ABAQUS

(Blanksvärd 2014) 

LS-DYNA

Simplified Model

Element type Shell Beam

Number of elements 43,000 175 

Number of nodes 172,000 98 

Computation time Greater than 6 h 10–15 min 

Because the test-to-failure results indicated the buckling of one top girder followed by yielding, 

material model Mat 171 was used to simulate both tensile and compressive behavior in the 

finite-element simulation. For this material model, the effective length factor K in the 

slenderness ratio was selected as 1.2 based on a parametric study. Figure 17 shows the 

load-displacement plots from the field test and the finite-element simulations by ABAQUS and 

LS-DYNA. The displacement was chosen from the maximum deformation point. Figure 17

shows that the load-displacement plots by ABAQUS and LS-DYNA agree well, although 

modeling and computation time using the B-S element was significantly shorter. Differences 

between these plots and that from the field test are probably because of loading facilities. The 

decreased slope in the load-displacement plot from LS-DYNA indicates buckling of the beam 

element.

Example 4: Simulation of MR

Although the study of redundancy of truss bridges has been a topic of research for quite some 

time, not many laboratory or field studies on the removal of members have been conducted to 

investigate redundancy. FHWA (2013) and Diggelmann et al. (2013) performed a full-scale 

testing of MR in a span of a bridge that was scheduled to be demolished. The tested single span 

was part of the US-421 Milton–Madison Bridge, a 19-span continuous truss bridge that 

connected Milton, KY, to Madison, IN. This bridge was constructed over the Ohio River in 1921 

and was replaced in 2014. FHWA (2013) and Diggelmann et al. (2013) tested the northernmost 

approach span of this bridge for redundancy. 

The tested bridge was a Pratt truss type 45.415 m (149 ft) in length with a center-to-center 

bearing. Figure 19 shows the bridge looking northwest. The Pratt truss was symmetric and 
34 

consisted of seven panels, each 6.4 m (21 ft) in length. The superstructure width was 7.32 m 

(24 ft), and the truss height was 6.70 m (22 ft). The deck system was located 1.45 m (4 ft and 

9 inches) below the top chord of the truss. The deck system consisted of six lines of stringers 

spaced at 1.1176 m (3 ft and 8 inches). Floor transverse beams were attached to the verticals 

below the deck. Redundancy of this bridge was investigated through field testing by damaging a 

fracture-critical bottom chord member.

Source: FHWA. 

Figure 19. Photo. Approach span of the Milton–Madison Bridge looking northwest 

(FHWA 2013). 

In LS-DYNA, the deck system was modeled by shell elements with elastic material. Nodal rigid 

body constraints were applied for the nodal sets shared by the deck, transverse stringers, and 

floor system to account for the connection between these three systems. In addition, to highlight 

the sand-loaded zone in the field test, figure 20 indicates the location of the sand-loading zone in 

three bridge deck panels.

Source: FHWA. 

Figure 20. Illustration. FEMs of the Milton–Madison Bridge in LS-DYNA with sand 

loading. 

In modeling the bridge, Young’s modulus for all steel members was assumed to be 200,000 MPa 

(29,000 ksi). Because of insufficient data on other material properties, they were assumed based 
35 

on the typical steels available during the period of construction of the bridge (design drawings 

simply implied that the steel was made of “carbon steel” with a minimum yield stress and 

ultimate tensile stress of 248.21 MPa (36 ksi) and 400 MPa (58 ksi), respectively). The original 

deck system was removed from the bridge in 1996 and replaced with an exodermic deck system 

with an overlay. The weight of the exodermic deck combined with the 38.1-millimeter (1.5-foot) 

concrete overlay was assumed to be 3.73 kPa (5.41 psi). A concrete deck of equivalent weight is 

158.75 mm (6.25 inches) in thickness. 

Information on overall bridge geometry and section properties in the LS-DYNA model 

(as shown in figure 20) was based on shop drawings and reports downloaded from the online 

database DataHub (DataHub n.d.). Before the controlled damage to selected bridge truss 

members, the bridge was loaded with 645 kN (2,870.25 kips) of sand spread across the three 

central panels of the deck width (figure 21). This loading magnitude represented approximately 

two-thirds of the original design LL (although it was distributed over only the three central 

panels rather than the full length of the bridge). This loading was maintained during controlled 

damage to selected bridge members. In the redundancy evaluation, the lower chord member

L3US-L4US of the bridge was fractured in two steps using shape charges. One-half of the lower 

chord member cross section was fractured first. The structure was allowed to reach static 

equilibrium, and then a second blast caused complete fracture of the member.

Source: FHWA. 

Figure 21. Photo. Sand loading on the Milton–Madison Bridge (FHWA 2013). 

Because both DL and sand load are static loads, DR was used to bring the LS-DYNA model to 

the initial condition for the dynamic member fracture test. Although the actual field test 

consisted of two-step demolition of the member, as described in the previous paragraph, this 

process was simulated in LS-DYNA by complete fracture of the entire lower chord member 

L3US-L4US in one step because the B-S beam element is a resultant member where damage can 

be simulated only in the entire member.

The results from FEM analysis indicating the fundamental characteristics of the dynamic 

member loss process are shown in figure 22. The stress time history in the lower chord member 
36 

L4DS-L3DS, which is downstream of the removed upstream lower chord member L4US-L3US, 

included three stages. Total stress (including contributions of axial and bending) in this member 

was 41.69 MPa (6.05 ksi) after the application of gravity loading. This stress increased by 

14.52 MPa (2.11 ksi) during the sand loading. Hence, total stress in the member was 56.21 MPa

(8.16 ksi) before the MR, and the truss was in static condition. After the MR, the bridge vibrated 

for a few seconds while load redistribution occurred among truss members. Eventually, free 

vibration died down, and the bridge entered into another static equilibrium status without 

collapse. Increase in stress in the downstream member L4DS-L3DS because of the loss of its 

symmetric member L4US-L3US in the upstream truss was 15.77 MPa (2.28 ksi). The final axial 

stress in L4DS-L3DS after the removal of L4US-L3US was 71.98 MPa (10.44 ksi). 

Source: FHWA. 

Figure 22. Graph. Calculated total stress time history after sudden removal of a member of 

the Milton–Madison Bridge.

Figure 23 shows the time history plot of stresses measured during the field test. The increase in 

the steady-state stress in member L4DS-L3DS during the field test was approximately 

16.55 MPa (2.4 ksi). This value was calculated as 15.77 MPa (2.28 ksi) from the LS-DYNA 

model. These values are reasonably close to each other. The minor difference could be due to the 

effect of the two successive blasts used to cut the member, which could not be simulated in 

LS-DYNA using the B-S beam element. Comparison of figure 22 and figure 23 shows dynamic 

amplification in stresses and vibration in truss members after the removal of a member both 

during the test and in the LS-DYNA simulation. These results also show that the B-S element 

with the Mat 98 material model in LS-DYNA can be used for realistic simulation of the dynamic 

effects of MR in a truss bridge.
37 

Source: FHWA. 

1 ksi = 6.89 MPa.

Figure 23. Graph. Measured stress time history after fracture of a member of the

Milton–Madison Bridge by blast (FHWA 2013).

39 

CHAPTER 4. ALP ANALYSIS OF THE I-35W BRIDGE 

INTRODUCTION

The I-35W Bridge, an eight-lane steel truss bridge carrying I-35W across Saint Anthony Falls of 

the Mississippi River in Minneapolis, MN, collapsed on August 1, 2007, killing 13 and injuring 

145 people. The cause of this failure was determined to be an underdesigned gusset plate. 

Figure 24 shows the bridge before collapse. The collapse of this bridge heightened safety and 

redundancy concerns about truss bridges nationwide. This bridge is used for investigating ALPs

and redundancy in long-span truss bridges because of the detailed information available on its 

design and inspection and the detailed investigation on its collapse by FHWA and NTSB. 

Source: NTSB. 

Figure 24. Photo. I-35W Bridge before collapse (NTSB 2008d). 

Past studies on this bridge focused mainly on the inadequate capacity of undersized gusset plates 

at U10 and the connected vertical (Ocel and Wright 2008; Astaneh-Asl 2008; Hao 2010; 

Liao et al. 2011; Higgins et al. 2012). However, the failure of the gusset plate near one joint was 

only a localized phenomenon. Unlike the past research on the I-35W Bridge collapse, this 

research focuses on the global behavior of the bridge in the event of local damage, such as 

sudden loss of a member. In addition to axial forces, the effects of strong and weak axes 

moments were included in calculating total stress in truss members. The bridge was at significant 

risk of failure even with gusset plates of sufficient thickness because of increased deck loading 

from rehabilitations during the 1970s and the 1980s, insufficient capacity of major truss 

members, and lack of sufficient alternate load transfer paths.

Figure 25 shows a detailed framework for investigating ALP and retrofit strategies for this 

bridge. As noted in the figure, this research on the ALP was divided into three parts. In the first 

part, modeling and validation of FEMs of the bridge were performed based on the original design 

details of the bridge in 1965 and the as-built status of the bridge before collapse in 2007. The 

performance of the bridge under the service load was also checked. Collapse of the bridge based 

on DCR was investigated. Finite-element analysis results indicated that the DCR for many 
40 

members before the collapse was in excess of 1.0 (i.e., combined stresses due to DL and LL were 

greater than allowable stress). Hence, several members of a before-collapse model of the bridge 

were strengthened by increasing cross-sectional shape, areas, or both such that DCR ≤ 0.9 under 

the DL + LL load combination for all truss members. This model of the bridge was termed 

“Design Bridge” because the bridge has sufficient capacity to carry DLs and LLs. Behavior of 

this bridge after the removal of a member was investigated extensively to identify ALPs. DCR of 

truss members in this bridge was reduced to 0.9 or below so that the change in DCR after sudden 

removal of a member could be easily mapped to investigate the ALP (0.9 is an illustrative value; 

any value from 0.9 to 1.0 could have been considered). 

Source: FHWA. 

Figure 25. Flowchart. Framework for the investigation of ALP in long-span truss bridges.

Unlike during extreme hazard events such as blast or member cutting, behavior of bridges during 

seismic loads is well understood by engineers, and bridges are retrofitted regularly to meet 

seismic demands. In the second part of this research framework, the Design Bridge was 

retrofitted to investigate the effects of seismic retrofits on ALP of a truss bridge. The seismic 

hazard considered was based on a 2,500-year earthquake in the San Diego area (termed the 

SD2500 hazard). It should be mentioned that the Design Bridge was in Minneapolis, MN, which 

is a low seismic region. It was found to have sufficient seismic resistance for a seismic hazard in 

Minnesota. The bridge retrofitted for the SD2500 hazard was termed “Base Bridge” and was 

considered for investigating retrofits for improving ALP of the bridge. 
41 

In the third part, the adaptability of various retrofit strategies on ALP and progressive collapse 

was examined extensively. Because of the differences between seismic events and sudden 

member loss, it is expected that seismic retrofits may not be sufficient to withstand member loss 

demands. Hence, effective retrofit strategies were investigated for mitigating sudden member 

loss demands on critical bridge components. 

FINITE-ELEMENT MODELING OF THE BRIDGE

Basic Model Details

Detailed FEMs of the bridge were developed in SAP2000 and LS-DYNA. As shown in 

figure 26, the bridge had 29 nodes along the upper chord (denoted by “U”) and 27 nodes along 

the bottom chord (denoted by “L”). Traffic on the bridge was in north and south directions, and 

two trusses were on the east and west sides. As shown in figure 26, nodes 1 (pier 5), 20 (pier 7), 

and 27 (pier 8) had roller supports, whereas node 8 (pier 6) had fixed constraint bearings. 

Source: FHWA.

A. 2D model (one side truss). 

Source: FHWA. 

B. 3D model. 

Figure 26. Illustrations. 2D and 3D models of the I-35W Bridge in SAP2000. 

The model of the bridge in LS-DYNA included main side trusses, floor trusses, floor stringers, 

diaphragms, and secondary bracing members. All truss members, bracings, and stringers were 

modeled by B-S beam elements with the Mat 98 material model to account for hardening in 

steel. Each element of the truss was modeled by a single beam element. Overall, the LS-DYNA 

model had 2,910 beam elements. The deck system in LS-DYNA was modeled by shell elements 

with elastic material for imposing the DL of the concrete deck on the truss system. The Young’s

modulus of the concrete in the deck was reduced such that the elasticity of the deck did not affect 

the truss behavior, and vertical deflection profiles of the bridge in LS-DYNA and SAP2000 

agreed well. (The deck was not modeled in SAP2000. Rather, deck weights were applied at truss 

joints appropriately to include the effect of deck weight on the truss behavior). Nodal rigid body 
42 

constraints were applied for the nodal group shared by the deck, transverse stringers, and floor 

system to account for the connection between these three systems. All the joints in the SAP2000 

and LS-DYNA models were treated as rigid.

The model in SAP2000 was similar to that in LS-DYNA except the equivalent nodal weight was 

assigned at deck stringers in SAP2000 and shell elements in LS-DYNA. All beam elements in 

SAP2000 were assigned elastic material properties. Geometrical, sectional, and material property 

data for the bridge models were obtained from shop drawings and bridge retrofit reports.

Because of the complete symmetry of the bridge’s west and east sides, cross-section areas for the 

main truss members in only the west side truss are listed in appendix B. 

Dead Loads 

After the bridge collapse, Ocel and Wright (2008) performed a detailed estimation of DLs on the 

bridge. They considered the bill of materials for both design drawings and shop drawings to 

estimate the steel weight and estimated the concrete deck weight based on original construction 

as well as rehabilitations. In addition, the DL consisted of vertical point loads from approach 

spans added at the ends of the main truss cantilevers. The original average thickness of the 

concrete deck was 165.1 mm (6.5 inches), and it increased by 50.8 mm (about 2 inches) after 

rehabilitations in 1977 and again substantially in 1998. Table 9 shows the DLs for the main span 

of the bridge based on calculations by Ocel and Wright (2008). DL1 represents the DL after 

construction of the bridge, and DL2 represents the DL in 2007 (before collapse of the bridge but 

not including construction load at the time of collapse). The DL on the truss span increased by 

approximately 24 percent from 1965 (time of construction) to 2007 (before collapse). 

Table 9. DLs on the I-35W Bridge. 

Component

DL1 (1965) 

(kN)

DL2 (2007) 

(kN)

Steel members 28,364 28,364 

Concrete members 53,248 72,889 

Approach span 5,585 6,989 

Sum 87,197 108,239 

1 kN = 0.225 kips.

Verification of the FEM of the Bridge Using DL

The accuracy of the FEM in SAP2000 was investigated by comparing vertical reactions for DL

conditions (both DL1 and DL2) and axial forces in members for DL1. Table 10 shows vertical 

reactions for DL1 and DL2 from the FEM of the bridge in SAP2000. These results were also 

compared with those from the FHWA detailed as-built case available in Ocel and Wright (2008). 

The DL reactions from the SAP2000 model agreed very well with those in Ocel and Wright 

(2008); the maximum difference was less than 5 percent. The differences in reactions could be 

attributed to the modeling approach as well as points of application of loads. 
43 

Table 10. Main truss vertical reaction forces for DL condition. 

DL

Condition 

Pier 

Location 

Detailed As-Built

Condition in 2007 per 

Ocel and Wright 

(2008) (D) 

(kN) 

SAP2000 (S) 

(kN) 

Difference: 

(S − D) ∕ D 

(%)

DL1 (1965) Pier 5 east 4,408 4,600 4.36 

DL1 (1965) Pier 6 east 16,605 16,595 −0.06 

DL1 (1965) Pier 7 east 16,298 16,354 0.34 

DL1 (1965) Pier 8 east 6,023 5,975 −0.79 

DL1 (1965) Pier 5 west 4,577 4,783 4.49 

DL1 (1965) Pier 6 west 16,632 16,572 −0.36 

DL1 (1965) Pier 7 west 16,338 16,352 0.08 

DL1 (1965) Pier 8 west 5,996 6,000 0.06 

DL1 (1965) Sum 86,878 87,230 0.41 

DL2 (2007) Pier 5 east 5,663 5,821 2.81 

DL2 (2007) Pier 6 east 20,360 20,349 −0.05 

DL2 (2007) Pier 7 east 19,901 20,056 0.78 

DL2 (2007) Pier 8 east 7,540 7,464 −1.00 

DL2 (2007) Pier 5 west 5,907 5,885 −0.38 

DL2 (2007) Pier 6 west 20,351 20,337 −0.07 

DL2 (2007) Pier 7 west 20,021 20,046 0.12 

DL2 (2007) Pier 8 west 7,393 7,554 2.18 

DL2 (2007) Sum 107,135 107,512 0.35 

1 kN = 0.225 kips.

Table 11 through table 14 show comparisons between axial forces obtained from the SAP2000 

model and those reported in Ocel and Wright (2008) for the FHWA detailed as-built DL1 case. 

The axial forces in the main truss members in the central span, the upper chords, the lower 

chords, and the diagonals agreed well with those from the design plans; the maximum difference 

was less than 10 percent. Because the loading assignment may vary when it is applied to the 

approach spans in different models, discrepancies exist in some members of the side spans. 

Members in the side spans have a lesser magnitude of axial force than those in the central span. 

Hence, even a small difference in the magnitude results in a larger percentage difference between 

results from Ocel and Wright (2008) and the SAP2000 model in table 11 and table 13. These 

percentage differences can be up to 61.40 and 46.80 percent in table 11 and table 13, 

respectively. All vertical members, except for those at the two end nodes, were modeled by 

adding two internal nodes for the connection of the vertical member to the floor truss members in 

the transverse direction. Therefore, verticals were subjected to biaxial forces due to the 

interaction with the floor truss system (FS), which may cause larger discrepancies in results 

because the design plan results may simply represent axial force for an entire member. However, 

the maximum discrepancy was still less than 23 percent for vertical members. Hence, the FEM 

of the bridge in SAP2000 can be considered to represent the I-35W Bridge reasonably well. It 

should be noted that a similar level of discrepancy was observed in models by FHWA and other 

consultants (Ocel and Wright 2008). 
44 

Table 11. Axial forces in upper chord members for DL1 case. 

Upper Chord 

Member

Design Plans (1965) (D) 

(kN) 

SAP2000 (S) 

(kN) 

Difference: 

(S − D) ∕ D 

(%)

U0-U1 1,953 1,708 −12.50 

U1-U2 1,953 1,689 −13.50 

U2-U3 −1,005 −1,087 8.10 

U3-U4 −1,005 −1,087 8.10 

U4-U5 2,295 2,121 −7.60 

U5-U6 2,295 2,133 −7.10 

U6-U7 7,838 8,071 3.00 

U7-U8 7,838 8,091 3.20 

U8-U9 6,899 7,097 2.90 

U9-U10 6,899 7,086 2.70 

U10-U11 −2,162 −2,060 −4.70 

U11-U12 −2,162 −2,070 −4.20 

U12-U13 −8,447 −8,514 0.80 

U13-U14 −8,447 −8,116 −3.90 

U14-U15 −8,447 −8,209 −2.80 

U15-U16 −8,447 −8,608 1.90 

U16-U17 −2,162 −2,323 7.50 

U17-U18 −2,162 −2,314 7.00 

U18-U19 6,899 6,758 −2.00 

U19-U20 6,899 6,770 −1.90 

U20-U21 7,838 7,911 0.90 

U21-U22 7,838 7,894 0.70 

U22-U23 2,295 2,440 6.30 

U23-U24 2,295 2,431 5.90 

U24-U25 −138 −55 −60.20 

U25-U26 −138 −53 −61.40 

U26-U27 3,541 3,484 −1.60 

U27-U28 3,541 3,487 −1.50 

1 kN = 0.225 kips.
45 

Table 12. Axial forces in DMs for DL1 case. 

DM

Design Plans (1965) (D) 

(kN) 

SAP2000 (S) 

(kN) 

Difference: 

(S − D) ∕ D 

(%)

U0-L1 −2,491 −2,031 18.50 

U2-L1 −2,945 −2,606 11.50 

U2-L3 854 523 38.70 

U4-L3 1,428 1,315 7.90 

U4-L5 −2,847 −3,118 9.50 

U6-L5 3,928 3,900 0.70 

U6-L7 −5,222 −5,407 3.50 

U8-L7 5,409 5,146 4.90 

U8-L9 6,939 6,711 3.30 

U10-L9 −7,473 −7,605 1.80 

U10-L11 6,370 6,285 1.30 

U12-L11 −5,405 −5,325 1.50 

U12-L13 3,710 3,470 6.50 

U14-L13 −952 −1,055 10.80 

U14-L15 −952 −962 1.00 

U16-L15 3,710 3,387 8.70 

U16-L17 −5,405 −5,244 3.00 

U18-L17 6,370 6,240 2.00 

U18-L19 −7,473 −7,560 1.20 

U20-L19 6,939 6,700 3.40 

U20-L21 5,409 4,898 9.50 

U22-L21 −5,222 −5,071 2.90 

U22-L23 3,928 3,529 10.20 

U24-L23 −2,847 −2,687 5.60 

U24-L25 979 832 15.00 

U26-L25 1,294 954 26.30 

U26-L27 −3,430 −3,091 9.90 

U28-L27 −4,510 −4,081 9.50 

1 kN = 0.225 kips.
46 

Table 13. Axial forces in lower chord members for DL1 case. 

Lower Chord 

Member

Design Plans (1965) (D) 

(kN) 

SAP2000 (S) 

(kN) 

Difference: 

(S − D) ∕ D 

(%)

L1-L2 356 516 45.10 

L2-L3 356 522 46.80 

L3-L4 −80 −79 1.50 

L4-L5 −80 −83 3.80 

L5-L6 −4,835 −5,004 3.50 

L6-L7 −4,835 −5,005 3.50 

L7-L8 −11,267 −11,460 1.70 

L8-L9 −11,312 −11,528 1.90 

L9-L10 −2,487 −2,691 8.20 

L10-L11 −2,487 −2,693 8.30 

L11-L12 5,832 5,371 7.90 

L12-L13 5,832 5,374 7.90 

L13-L14 9,057 8,312 8.20 

L14-L15 9,057 8,312 8.20 

L15-L16 5,832 5,505 5.60 

L16-L17 5,832 5,503 5.60 

L17-L18 −2,487 −2,467 0.80 

L18-L19 −2,487 −2,466 0.80 

L19-L20 −11,312 −11,246 0.60 

L20-L21 −11,267 −11,232 0.30 

L21-L22 −4,835 −5,117 5.80 

L22-L23 −4,835 −5,117 5.80 

L23-L24 −609 −749 22.90 

L24-L25 −609 −745 22.30 

L25-L26 −845 −836 1.10 

L26-L27 −845 −839 0.80 

1 kN = 0.225 kips.
47 

Table 14. Axial forces in vertical members for DL1 case. 

Vertical Member

Design Plans (1965) (D) 

(kN) 

SAP2000 (S) 

(kN) 

Difference: 

(S − D) ∕ D 

(%)

U1-L1 −1,437 −1,524 6.10 

U2-L2 1,183 955 19.30 

U3-L3 −1,423 −1,123 21.10 

U4-L4 1,041 1,046 0.50 

U5-L5 −1,415 −1,192 15.70 

U6-L6 1,223 1,147 6.20 

U7-L7 −1,481 −1,190 19.60 

U8-L8 −11,241 −11,420 1.60 

U9-L9 −1,472 −1,225 16.80 

U10-L10 1,205 1,138 5.60 

U11-L11 −1,197 −1,155 3.40 

U12-L12 1,201 1,049 12.70 

U13-L13 −1,468 −1,202 18.10 

U14-L14 1,085 1,013 6.60 

U15-L15 −1,468 −1,202 18.10 

U16-L16 1,201 1,048 12.80 

U17-L17 −1,197 −1,155 3.50 

U18-L18 1,205 1,136 5.70 

U19-L19 −1,472 −1,226 16.70 

U20-L20 −11,241 −11,206 0.30 

U21-L21 −1,481 −1,191 19.60 

U22-L22 1,223 1,145 6.40 

U23-L23 −1,415 −1,189 16.00 

U24-L24 1,041 1,049 0.70 

U25-L25 −1,423 −1,109 22.10 

U26-L26 1,183 969 18.10 

U27-L27 −1,437 −1,573 9.50 

1 kN = 0.225 kips.

Live Loads 

According to AASHTO (2014), the LL is a combination of lane load with a magnitude of 

0.87 kN/m (0.64 klf) and standard truckload. Because the I-35W Bridge is a long-span bridge 

where the truck effect may be negligible, only the lane load with multilane presence factor was

applied on the bridge for the redundancy analysis. This was done to simplify the loading scenario 

on the bridge because of the large number of simulations required. 

Material Properties

The behavior of the steel in the truss members was modeled by the stress–strain behavior in

figure 8 in chapter 3, assuming a Young’s modulus of 200 GPa (29,000 ksi). 
48 

Validation of FEMs in SAP2000 and LS-DYNA

A detailed validation of FEMs in SAP2000 and LS-DYNA was performed by comparing 2D and 

3D models of the bridge. For example, figure 26 shows both 2D and 3D models of the bridge in 

SAP2000. The geometry and nodal details of the bridge are the same in LS-DYNA. One 

comparison is the DL of the two main trusses. Based on the layout and the cross-sectional areas 

provided in the drawings, the net weight of each of the two main trusses was calculated to be 

5,622.95 kN (1,264.09 kips), which compares well with the value of 5,604.75 kN (1,260 kips) 

reported by Liao et al. (2009). Additional DLs, such as the concrete slab, floor trusses, sway 

frames, and so forth, were applied to each node according to Liao et al. (2009). Figure 27-A and 

figure 27-B show the deformation profile of 2D trusses in SAP2000 and LS-DYNA under 

self-weight. The self-weight of one side truss was 5,622.95 kN (1,264.09 kips) in SAP2000 and 

5,610.16 kN (1,261.22 kips) in LS-DYNA (0.23-percent difference). Figure 27-C shows the plot 

of the deflection profile using the displacement extracted from the lower node of the truss. The 

difference in displacement between the two models was less than 5 percent, except for at nodes 2 

and 3, where this difference was higher because of their proximity to boundary regions. Other 

comparisons, such as weight, total support reaction for DL + LL, maximum displacement at node 

14 (the middle node) for DL + LL cases, and external force percentage for 2D models in 

SAP2000 and LS-DYNA, are shown in table 15. 

Source: FHWA.

A. 2D model in LS-DYNA.

Source: FHWA.

B. 2D model in SAP2000. 

Source: FHWA. 

C. Vertical displacement shape from FEM under steel self-weight of one side truss. 

Figure 27. Illustrations and graph. Comparison between deformed shapes and vertical 

displacements from 2D models in SAP2000 and LS-DYNA.
49 

Table 15. Comparisons between 2D models in SAP2000 and LS-DYNA.

Item SAP2000 LS-DYNA Difference Δ

Weight (one side truss) (kN) 5,623 5,610 0.2% 

Total reaction force (kN) (DL + LL) 67,625 63,977 5.4% 

Maximum displacement at node 14 (the 

middle node) for DL + LL case (mm) 

371.53 374.68 0.85% 

External force percentage (DL ∕ (DL + LL)) 82% 80% 2.3% 

1 kN = 0.225 kips; 1 mm = 0.0394 inches.

Table 16 and table 17 show the comparison between displacements at nodes of 3D models of the 

bridge in SAP2000 and LS-DYNA under the loading cases of steel-truss self-weight and DL1, 

respectively. The maximum difference in vertical displacement between the two models was 

around 5 percent. Table 18 shows the contribution of steel and concrete to the total DL. The 

concrete deck contributed approximately 63 percent to the total bridge weight, almost three times 

as much as the steel members. 
50 

Table 16. Comparison between displacements from 3D models of the bridge under steel 

truss self-weight in SAP2000 and LS-DYNA. 

Node Number

SAP2000 (S)

(mm)

LS-DYNA (L)

(mm)

Difference: 

(L − S) ∕ S 

(%)

1 0 0 0 

2 −6.32 −6.31 −0.14 

3 −8.26 −8.22 −0.50 

4 −5.96 −5.9 −1.03 

5 −1.65 −1.62 −2.0 

6 −0.35 −0.35 −0.32 

7 −0.37 −0.37 −1.07 

8 0 0 0 

9 −19.70 −19.64 −0.29 

10 −38.83 −38.76 −0.18 

11 −58.24 −58.11 −0.23 

12 −75.53 −75.48 −0.06 

13 −87.11 −87.07 −0.05 

14 −92.18 −93.26 1.17 

15 −87.15 −87.28 0.15 

16 −75.59 −75.68 0.12 

17 −58.32 −58.32 −0.01 

18 −38.90 −38.71 −0.50 

19 −19.74 −19.72 −0.09 

20 0 0 0 

21 −0.29 −0.28 −4.72 

22 −0.19 −0.18 −4.79 

23 −1.40 −1.43 2.03 

24 −5.65 −5.69 0.74 

25 −7.92 −7.96 0.52 

26 −6.02 −6.15 2.24 

27 0 0 0 

1 mm = 0.0394 inches.
51 

Table 17. Comparison between displacements from 3D models of the bridge under DL1 

(1965) in SAP2000 and LS-DYNA.

Node Number

SAP2000 (S) 

(mm)

LS-DYNA (L) 

(mm)

Difference: 

(L − S) ∕ S 

(%)

1 0 0 0 

2 −30.2 −28.7 −4.97 

3 −44.8 −42.5 −5.13 

4 −43.4 −41.1 −5.30 

5 −30 −28.5 5 

6 −17.8 −16.9 −5.06 

7 −11.3 −10.8 −4.42 

8 0 0 0 

9 −44.1 −42 −4.76 

10 −83.8 −79.8 −4.77 

11 −127.5 −121.5 −4.71 

12 −167.9 −159.7 −4.88 

13 −195 −185.5 −4.87 

14 −208.6 −198.3 −4.94 

15 −195.4 −185.9 −4.86 

16 −168.6 −160.4 −4.86 

17 −128.4 −122.3 −4.75 

18 −84.6 −80.6 −4.73 

19 −44.5 −42.5 −4.49 

20 0 0 0 

21 −11.6 −11.1 −4.31 

22 −18.2 −17.3 −4.95 

23 −30.4 −28.9 −4.93 

24 −43.5 −41.4 −4.83 

25 −44.5 −42.3 −4.94 

26 −30 −28.5 5 

27 0 0 0 

1 mm = 0.0394 inches.

Table 18. Weights of various portions of 3D full bridge models. 

Item Weight (kN) Percentage

Steel members 25,391 22 

Concrete members 72,886 63 

DL additional 17,510 15 

Sum 115,787 100 

1 kN = 0.225 kips.
52 

Damping Effect

In NLD analysis, the damping ratio affects the dynamic response of a structural system 

significantly. In general, long-span truss bridges likely have damping ratios of less than 

2 percent. For example, Sarraf (1998) suggested a 2-percent damping ratio for an 80-meter

(262.5-foot) span deck-truss bridge. Kanaji et al. (2003) suggested a damping ratio of 2 percent

for the 980-meter (3,215.2-foot) Minato Bridge, a cantilever truss bridge with two 235-meter

(771-foot) side spans and a 510-meter (1,673.2-foot) main span. Frangopol and Imai (2004) used 

a damping ratio of 5 percent for the Akashi Kaikyo Bridge, a suspension bridge with side spans 

of 960 m (3,149.61 ft) and a main span of 1,991 m (6,532.2 ft). Casciati et al. (2008) and 

Dyke et al. (2003) adopted a 3-percent critical damping ratio for the Bill Emerson Memorial 

Bridge, which is a fan-type medium-span cable-stayed bridge with a main span length of 

350.6 m (1,150.3 ft). Mustafa et al. (2018) investigated damping in a Warren-type steel truss 

bridge consisting of five simply supported main spans, each having a length of 70.77 m 

(232.19 ft) and a width of 6.01 m (19.72 ft), through field monitoring and found that the damping 

in the first mode was approximately 0.93 percent. 

Two approaches are available for dealing with damping in LS-DYNA. One is to control the 

global damping factor and the other is to control the material damping factor. Only limited types 

of material models have the option to input a selected damping factor. The elastic material 

model, for example, allows the use of damping ratios for axial and bending behaviors separately. 

Material model Mat 98 used to model the bridge in LS-DYNA does not allow the use of a 

material damping factor; rather, it allows the use of a global damping factor (Ds).

To select a reasonable value for the Ds to give a damping ratio between 1 and 2 percent for the 

first mode of the bridge, a parametric study was performed by varying the Ds in the Mat 98 

material model. Normally, when a truss member is removed suddenly and the material model is 

not assigned a Ds, bridge members will continue to vibrate indefinitely because of the dynamical 

effect of the MR. When the bridge is assigned a certain value of Ds, the vibration of bridge 

members is damped out gradually. For long-span bridges, the damping ratio in the first mode is 

generally less than 2 percent and can be estimated by the logarithmic decrement method on the 

member vibration time history, as shown in equation 16: 

(16)

Where:

δe = logarithmic decrement. 

q = number of cycles between 1st and (q+1)th peak amplitude. 

u1 = peak amplitude of the first peak. 

uq+1 = peak amplitude of the (q+1)th peak. 

ζ = damping ratio. 

In the parametric study, vibration signatures for nodes of the truss due to dynamic effects of MR

were generated by considering different values for Ds in the simulation. The displacement of 

randomly picked node 3 is plotted for different values of Ds in figure 28. When Ds is assigned 

values of 1.5 or 3.0, vibration decays more rapidly than when Ds is assigned values of 0.75 and 
53 

0.5. The converged value of displacement in all four cases is −0.0326 mm (−0.00128 inches), 

although the peak magnitude varies with damping. Table 19 shows damping values for four 

cases of Ds calculated from these vibration signatures by using equation 16. The damping ratio 

decreased from approximately 9 to 1 percent as Ds was reduced from 3 to 0.5. Based on the 

results in figure 28 and table 19, the value of Ds was selected to be 0.75 to achieve a damping 

value from 1.57 to 1.63 percent. Table 19 demonstrates that the difference in peak values of 

displacement for a damping ratio in the range of 3.90 to 1.14 percent was negligible. Hence, both 

peak dynamic and steady-state displacements due to MR can be obtained reliably by using the 

0.75 value for Ds. 

Source: FHWA.

1 mm = 0.0394 inches. 

A. Displacement time history for Ds values of 3.0 and 1.5. 

Source: FHWA. 

1 mm = 0.0394 inches. 

B. Displacement time history for Ds values of 0.5 and 0.75. 

Figure 28. Graphs. Displacement time history of node 3 for various Ds values. 
54 

Table 19. Damping ratios for different values of Ds. 

Global Damping 

Factor 

(Ds) 

Damping Ratio 

Range 

(ζ) 

Steady-State Value

(mm)

Peak Value 

(mm)

3 9.26–7.74% −0.0326 −0.0367

1.5 3.89–3.95% −0.0326 −0.0373

0.75 1.57–1.63% −0.0326 −0.0377 

0.5 1.14–1.18% −0.0326 −0.0378 

1 mm = 0.0394 inches. 

AS-BUILT BRIDGE AND NONSEISMIC RETROFITS

Figure 29 shows the DCR for the as-built bridge with loading conditions in 2007 (before collapse 

and without construction load) under the service load condition in equation 11. In this figure, 

bridge members with DCR less than 1.0 are shown in black solid lines and those with DCR

greater than 1.0 are shown in red dotted lines. DCR of 1.0 implies that the stress in the member is 

at the allowable limit corresponding to 0.6σy. DCR of 1.67 implies that the stress in the member 

is equal to σy. Figure 29 shows that diagonal and lower chord members near supports have DCRs 

larger than 1.0, which implies that the stress due to service loads (DL + LL) in these members 

was more than the allowable stress. This probably happened because the DL on the bridge 

increased by 24 percent from 1965 to 2007 because of rehabilitations. However, none of the 

members of the truss had DCR greater than 1.67 (i.e., all members of the truss were in elastic 

range under the combination of DLs and LLs). 

For a bridge that has been designed for service loads, DCR for all members must be less than 1.0. 

Because the objective of this research was to study ALP in a well-designed bridge, retrofit of 

members with DCR greater than 1.0 was performed by strengthening the member (by changing 

I-shaped sections to box sections and increasing the cross-sectional area) so that DCR of all 

members became 0.9. This was done so that change in DCR of members after the removal of a 

critical member or members could be quantified easily (with respect to a uniform value of 0.9). 

This bridge model was designated the Design Bridge. Figure 30 shows the Design Bridge where 

retrofitted members are indicated by red dotted lines. This member strengthening resulted in a 

2.65-percent increase in the weight of the steel in the bridge. Although target DCR for this bridge 

was 0.9, ALP analysis (i.e., after removal of a member) was performed by considering limit 

states of allowable (1.0) and yield (1.67) stresses.
55 

Source: FHWA. 

Figure 29. Illustration. DCR spectra for the as-built bridge subjected to service loads.
56 

Source: FHWA. 

Figure 30. Illustration. Members of the as-built bridge retrofitted by member strengthening to obtain the Design Bridge for 

the ALP analysis. 
57 

ALP ANALYSIS OF THE DESIGN BRIDGE

An exhaustive ALP analysis of the Design Bridge was performed by modeling the sudden loss of 

various members. The MR was investigated using the SAP2000 model for DCR and the 

LS-DYNA model for SR. Elastic static analysis was performed using SAP2000 after removing a 

target member from the model. DCR for all truss members was calculated from the results of the 

SAP2000 model. In LS-DYNA, MR was performed by assigning a specific part number and time 

interval in the card of *Load_remove_part. SR for all truss members was calculated from 

LS-DYNA results for different MR cases.

MR Event

MR analysis was performed by removing one critical member at a time. Members with the 

highest values of DCR under service load (DL + LL) were identified for removal. Analysis 

results indicated that vertical members were not critical in this specific bridge, so they were not 

considered for removal. On the other hand, diagonal, lower chord, and upper chord members 

were found to be critical. Considering the double symmetry of the bridge geometry, 10 members 

were chosen for removal from one-quarter of the bridge (the southern part of the west side truss), 

as shown in figure 31. Table 20 shows the 10 MR cases that were investigated. The MR scenario 

in SAP2000 was implemented by analyzing two separate states of the structure: before and after 

the loss of the member. 

Source: FHWA. 

Figure 31. Illustration. Location of critical members for removal in one-quarter of the 

bridge on the west side truss.

Table 20. MR cases.

MR Case Number Location 

1 DM

2 Lower chord

3 Lower chord

4 DM

5 Upper chord 

6 Upper chord 

7 DM

8 DM

9 DM

10 DM
58 

ALP (MR) Analysis for Design Bridge: DCR Metric

The objective of the ALP analysis was to identify both the local and global effect of the specific 

local damage (MR) introduced. First, to investigate the stress variations due to loss of a member 

or members, DCR results before and after an MR event were compared in three panels in the east 

and west trusses when a member was removed in one of the panels in the west truss, as shown in 

figure 32. Because of the geometric and loading symmetry, the DCRs before MR were the same 

for symmetric members in the three panels of the east and west trusses. After DM U18-L19 in 

the west truss was removed, upper chord, lower chord, and vertical members near the removed 

member in the west truss experienced a significant increase in their internal forces compared to 

those in the east truss. The damage propagated to other members around the removed member. 

For vertical members, there were two internal nodes (in addition to two nodes at the ends of the 

member) in the FEM, whereas there was only one continuous physical member in the real bridge 

(which was connected to the FS through gusset plates). The internal nodes were connected to the 

floor truss members in the transverse direction. Because of the interaction with the FS, verticals 

were subjected to biaxial forces, which caused variation in DCR along the length of the vertical 

member.

In the intact bridge, overall torsional resistance of the bridge about its longitudinal axis is 

generally designed to keep the structure in equilibrium. Two major mechanisms contribute to the 

entire torsional resistance to sustain the balanced or unbalanced traffic loads: 

• The two side trusses together with the deck, stringers, and upper braces as well as the 

lower braces could be treated as a hollow closed box section that provides torsional 

resistance.

• FSs in the transverse direction act as diaphragms to provide the hollow closed box section 

more rigidity.

The removal of a member undermines the structural integrity and symmetry of the bridge. By 

calculating the contributions of axial and flexural interactions to the DCR of a member, the 

impact of biaxial bending stress on the member and system can be investigated. For each 

member in the three panels, the contributions of axial force, in-plane moment (Min), and 

out-of-plane moment (Mout) to its DCR were calculated and are expressed as percentages 

(i.e., contribution to DCR divided by the total DCR) in figure 33. For a typical truss member in

figure 33, the numbers above and below a member show results for before and after the MR, 

respectively. For example, for member U17-U18, “56 42 2” above the member represents 56-, 

42-, and 2-percent contributions of axial, Min, and Mout to DCR, respectively, before the MR. 

Likewise, “3 90 7” below the member represents 3-, 90-, and 7-percent contributions of axial, 

Min, and Mout to DCR, respectively, after the MR. For this member, the major contribution to 

DCR changed from 56-percent axial and 44-percent moment (total of Min and Mout) before the 

MR to 3-percent axial and 97-percent moment after the MR. 

Figure 33 also demonstrates that the contribution of axial forces to DCR is generally more than 

that of the total moment in all main truss members before the removal of a member, except for a 

few vertical members. Because of the unbalanced load introduced into the vertical members 

through the connections with the FS in the same panel, the moment component in verticals may 
59 

be more dominant than the axial force component even before the removal of a member. After 

the removal of a member, the major contributor to DCR in the upper and lower chords on the 

west truss changed from axial to total moment. In the east truss, lower chords in the nearby 

region experienced a similar change. However, the deck, stringers, and upper braces helped the 

upper chords maintain the shape without distortion. Hence, axial contribution to DCRs was 

dominant in upper chords. This discussion demonstrates that an analysis that treats truss 

members as primarily axial is not accurate and not reliable for evaluating the capacity of the truss 

members in the event of MR. 
60 

Source: FHWA. 

Figure 32. Illustration. DCR before and after MR case 1 near the damaged location in west and east side trusses. 

Source: FHWA. 

Figure 33. Illustration. Contributions to DCR before and after MR case 1 near the damaged location in west and east side 

trusses.
61 

The results presented in this section so far illustrate the localized view of the effect of damage to 

nearby members. To investigate the global view of the impact of damage to a member, 

distribution of DCR on 3D and 2D models of the bridge and its subcomponents (upper and lower 

braces and floor systems) is shown for 10 cases of MR (table 20) in figure 34 to figure 43. The 

six views (3D, east truss, upper braces, west truss, lower braces, and FS) of the bridge in each of 

these figures show the load path and can be defined as ALP spectra for DCR in order to quantify 

ALP. For MR case 1 in figure 34, the DM U18-L19 in the west side truss was removed. This led 

to load redistribution in the damaged system and an increase in the demand on the main truss 

members near the damaged zone on the same (west) side; the effect was less significant on 

members of the truss on the opposite (east) side. Because the member loss event is a localized 

phenomenon, damage to other members was concentrated around the damaged member. In this 

condition, the removed DM, as the web member, disrupted the normal load path. Compared with 

upper chord members, lower chord members were more dominant and effective in transferring 

the load to supports because the load always travels in the shortest path. If the load traveled to 

the upper chord, it needed to travel through vertical members before being transferred to the 

support, which is a longer path. Hence, the other three lower chords near support 8 were also 

affected even though they are far from the damaged zone. On the other hand, the increased 

demand in the #18 and #19 floor systems, as well as the upper brace and lower brace systems,

indicated the 3D effect in load redistribution from west to east truss. This again shows the role 

these members play in resisting the bridge’s tendency to undergo global torsional motion 

because of removal of a member from one side truss. 

Figure 34 demonstrates that in addition to members near the damaged member, many members 

of the upper braces, lower braces, and FS have DCR greater 1.67 (i.e., these members probably 

yielded). MR case 2 in figure 35 shows the results for the removal of lower chord L18-L19. The 

loss of the lower chord member had less impact on both trusses than the removal of the DM in 

case 1. Members of the main truss remained in the elastic range, although a few members of the 

lower and upper braces and floor trusses probably yielded (i.e., they had a DCR larger than 1.67). 

MR cases 3 and 4 simulated the loss of lower chord and diagonal members, respectively, in the 

side span. Results for these two cases are plotted in figure 36 and figure 37, respectively. 

Figure 36 shows that removal of the lower chord member L20-L21 in the side span led to 

redistribution of loads in both the side span and the central span. However, significant load 

redistribution occurred in the truss members on the other side (east side). There was no damage 

to members in the north side span. Figure 37 shows that removal of the DM L21-U22 in the 

south side span limited the damage to other members in the same span, although load 

redistribution occurred in the truss members on the other side (east side). In both cases 3 and 4, 

some members of the upper and lower braces probably yielded (i.e., their DCR was greater than 

1.67). 

Cases 5 and 6 simulated the loss of upper chord members on the right and left sides of 

support #20, respectively. Results for these two cases are presented in figure 38 and figure 39, 

respectively. These figures show that these two cases caused minor effects on other members of 

the truss, and there was significantly less load redistribution after the loss of these two members

than in previous cases. This is because the deck, longitudinal stringers, transverse diaphragm 

between stringers, and upper braces effectively compensated for the loss of upper chord 

members. Hence, the loss of an upper chord member does not seem to create a potential threat to 
62 

the stability of the load-carrying system in this bridge. For both these cases, no members in the 

lower braces and floor trusses yielded, and fewer than three members in the upper braces 

yielded. In MR cases 7 to 10, the effect of the loss of a DM was similar to that in case 1. Results 

for these cases are shown in figure 40 to figure 43, respectively.

Results in figure 34 to figure 43 correspond to 10 MR cases in one-quarter of the bridge. Because 

of the symmetry of the west and east side trusses, an envelope can be drawn based on these 

10 cases to show the ALP for the entire bridge. DCR values for truss members of the bridge for 

the 10 cases were tabulated, and the DCR value selected for a particular member was the largest 

DCR for that member among the 10 MR cases. This process was mirrored in both longitudinal 

and transverse directions to consider the MR events at the symmetric spot for the north end of 

the east side truss. The boundary conditions of roller and fixed support at joints 8 and 20, 

respectively, had a negligible effect on these results. Figure 44 shows the complete envelope for 

the entire bridge, which can be used to identify members to retrofit to provide effective ALP in 

case of loss of any member of the bridge. Some of the members illustrated by dash-dotted (red) 

lines in this figure are part of the ALP that are likely to yield after the removal of any member of 

the truss. After retrofit of members highlighted by dash-dotted (red) lines, all members of the 

truss will stay in the elastic zone after the removal of any member of the retrofitted bridge. 

Hence, the development of ALP spectra in figure 44 can be considered the quantification of ALP 

of the truss bridge from an effective retrofit point of view.
63 

Source: FHWA. 

Figure 34. Illustration. DCR spectra for the Design Bridge for diagonal member removal (MR case 1 in table 20).
64 

Source: FHWA. 

Figure 35. Illustration. DCR spectra for the Design Bridge for lower chord MR (MR case 2 in table 20).
65 

Source: FHWA. 

Figure 36. Illustration. DCR spectra for the Design Bridge for lower chord MR (MR case 3 in table 20).
66 

Source: FHWA. 

Figure 37. Illustration. DCR spectra for the Design Bridge for diagonal member removal (MR case 4 in table 20).
67 

Source: FHWA. 

Figure 38. Illustration. DCR spectra for the Design Bridge for upper chord MR (MR case 5 in table 20).
68 

Source: FHWA. 

Figure 39. Illustration. DCR spectra for the Design Bridge for upper chord MR (MR case 6 in table 20).
69 

Source: FHWA. 

Figure 40. Illustration. DCR spectra for the Design Bridge for diagonal member removal (MR case 7 in table 20).
70 

Source: FHWA. 

Figure 41. Illustration. DCR spectra for the Design Bridge for diagonal member removal (MR case 8 in table 20).
71 

Source: FHWA. 

Figure 42. Illustration. DCR spectra for the Design Bridge for diagonal member removal (MR case 9 in table 20).
72 

Source: FHWA. 

Figure 43. Illustration. DCR spectra for the Design Bridge for diagonal member removal (MR case 10 in table 20).
73 

Source: FHWA. 

Figure 44. Illustration. Complete envelope of DCR for the Design Bridge representing 40 MR cases (10 removal cases in each 

quarter of the bridge).
74 

ALP (MR) Analysis for Design Bridge: SR Metric

The dynamic response of the structure at the instant of MR can be captured from the time history 

of forces and displacements in LS-DYNA. For example, figure 45 shows the vertical 

displacement of node 18 on the west side truss and node 18′ on the east side truss for MR case 1 

in table 20. The load combination in this case was DL + 0.5LL, and the vertical displacement of 

nodes 18 and 18′ at the end of DR (about 6 s, when the response of the intact structure because of 

DLs and LLs reaches steady state) was 107 mm (4.21 inches). After the DM U18-L19 was 

removed at 6 s, the entire structure underwent dynamic vibration with a peak dynamic amplitude 

of 613 mm (24.13 inches) at node 18. However, this vibration was damped out after a few 

seconds, and the damaged structure (with member U18-L19 removed) gradually reached a new 

steady state with a steady-state vertical displacement of 504 mm (19.84 inches) at node 18. 

Although the east side truss also underwent dynamic vibration, the magnitude of vertical 

displacement was significantly lower, with steady-state displacement of 156 mm (6.14 inches) at 

node 18′ (compared to 107 mm (4.21 inches) before the MR). Larger permanent displacement on 

the damaged side (west side) implies that the dynamic effect was significantly larger in the truss 

with the damaged (removed) member. Larger peak and steady-state vertical displacement in the 

west side truss (compared to those in the east side truss) also show the bridge’s tendency toward

torsional rotation about its longitudinal axis. 

Source: FHWA. 

1 mm = 0.0394 inches.

Figure 45. Graph. Nodal vertical displacement in MR case 1. 

Figure 46-A and figure 46-B show axial force time histories for lower chord members L17-L18 

and L18-L19 in the west side truss and corresponding members L17′-L18′ and L18′-L19′ in the 

east side truss, respectively. Similar to the displacement time histories in figure 45, the axial 

forces in members also had dynamic effects. For member L17-L18, the compressive force due to 
75 

DL + 0.5LL was 3,556 kN (799.1 kips) before the removal of DM U18-L19. After the removal, 

this force increased to a peak dynamic value of 10,312 kN (2,317.3 kips) before reaching a 

steady-state value of 6,601 kN (1,483.37 kips). Likewise, the axial force in member L18-L19 

increased from 3,556 kN (799.1 kips) to a peak dynamic value of 10,744 kN (2,414.38 kips) 

before reaching a steady-state value of 7,516 kN (1,689.66 kips). However, axial forces in these 

members were below the yield limit, although total stress from combined axial and bending 

stresses was above the yield limit. Figure 46-B shows axial force time histories for lower chord 

members L17′-L18′ and L18′-L19′ on the east side truss. These two members had a compressive 

force of 3,664 kN (819.20 kips) before the removal of DM U18-L19 on the west side. Both of 

these members became unloaded because of the diagonal member removal on the west side. 

Because the removal of the DM caused a substantial amount of unsymmetrical deformation, 

in-plane and out-of-plane bending effects cannot be neglected. Figure 47-A and figure 47-B 

show time history plots of in-plane and out-of-plane moments for lower chord members on the 

west and east side trusses, respectively. This figure shows that the lower chord members of both 

trusses experienced a significant amount of moment, especially out-of-plane moment. The 

combined effect of axial forces and moments caused SR to increase beyond 1.0 (i.e., total strain 

became more than yield strain) around the damaged member, which is discussed further in 

figure 48. 

Figure 48 to figure 57 present the SR metric for the 10 MR cases discussed in table 20. The 

detailed approach for calculating SR using the results from LS-DYNA can be found in 

appendix A. Truss members that had DCR values in the range of 1.0 to 1.67 in figure 34 have 

SRs less than 1 in figure 48. These members might have experienced overloading or unloading 

and may exhibit some hardening rather than linear elastic behavior. However, they are still in the 

elastic range. Members with DCR > 1.67 in figure 34 have SRs between 1 and 2 in figure 48, 

implying that these members underwent yielding. For MR case 2, figure 35 demonstrates that all 

members of the bridge stayed in elastic range. This is further confirmed from the SR metric plots 

in figure 49 (except for a few braces that show DCR > 1.67 in figure 35 but SR < 1.0 in 

figure 49). This observation for MR case 2 is also valid for MR cases 3, 5, and 6. For MR case 4 

and cases 7 to 10, members around the damaged member and some of the upper braces, lower 

braces, and floor truss members have SR > 1.0, which implies yielding in these members. 

Figure 58 shows the complete envelope for the entire bridge based on the SR metric. Similar to

figure 44, the complete envelope can be used to identify members to be retrofitted to improve 

ALP of the entire bridge in the event of loss of any member. Members with SR > 1.0 in this 

figure underwent yielding and may need to be retrofitted to ensure no damage in the event of 

failure of any member of the bridge. 
76 

Source: FHWA.

1 kN = 0.225 kips.

A. Lower chord members in the west truss. 

Source: FHWA. 

1 kN = 0.225 kips.

B. Lower chord members in the east truss.

Figure 46. Graphs. Axial forces in lower chord members in west and east side trusses of the 

Design Bridge for MR case 1. 
77 

Source: FHWA.

1 kN·m = 23.73 kips·ft. 

A. Lower chord members in the west truss. 

Source: FHWA. 

1 kN·m = 23.73 kips·ft.

B. Lower chord members in the east truss.

Figure 47. Graphs. In-plane and out-of-plane moments in lower chord members in west 

and east side trusses of the Design Bridge for MR case 1.
78 

Source: FHWA. 

Figure 48. Illustration. SR spectra for the Design Bridge for diagonal member removal (MR case 1 in table 20).
79 

Source: FHWA. 

Figure 49. Illustration. SR spectra for the Design Bridge for lower chord MR (MR case 2 in table 20).
80 

Source: FHWA. 

Figure 50. Illustration. SR spectra for the Design Bridge for lower chord MR (MR case 3 in table 20).
81 

Source: FHWA. 

Figure 51. Illustration. SR spectra for the Design Bridge for diagonal member removal (MR case 4 in table 20).
82 

Source: FHWA. 

Figure 52. Illustration. SR spectra for the Design Bridge for upper chord MR (MR case 5 in table 20).
83 

Source: FHWA. 

Figure 53. Illustration. SR spectra for the Design Bridge for upper chord MR (MR case 6 in table 20).
84 

Source: FHWA. 

Figure 54. Illustration. SR spectra for the Design Bridge for diagonal member removal (MR case 7 in table 20).
85 

Source: FHWA. 

Figure 55. Illustration. SR spectra for the Design Bridge for diagonal member removal (MR case 8 in table 20).
86 

Source: FHWA. 

Figure 56. Illustration. SR spectra for the Design Bridge for diagonal member removal (MR case 9 in table 20).
87 

Source: FHWA. 

Figure 57. Illustration. SR spectra for the Design Bridge for diagonal member removal (MR case 10 in table 20).
88 

Source: FHWA. 

Figure 58. Illustration. Complete envelope of SR for the Design Bridge representing 40 MR cases (10 removal cases in each 

quarter of the bridge).
89 

Dual Member Removal Event

Actual simulation of blast loads requires finer mesh of the FEM and simulating blast loading 

using either *Mat-High-Explosive-Burn card in LS-DYNA or software such as ConWep (which 

is labeled distribution statement C) (Kingery and Bulmash 1984; Randers-Pehrson and Bannister

1997). Use of this software was beyond the scope of this project. Hence, an alternate approach 

was used to simulate the effect of the blast load.

Generally, a blast load results in the loss of several members of a truss within a certain zone, 

depending on the location of the blast and the amount of TNT used. In this research, blast load 

effect was represented by dual member removal (DMR), which is defined here as loss of one 

diagonal and one lower chord member because it has been observed that the loss of vertical and 

upper chord members has a lesser effect on the safety of the truss system. However, this 

approach is only a crude approximation to simulate the effect of blast loading. More research is 

needed to find the correlation between the blast load parameters (location, amount of TNT used), 

number of members lost, and impact on the number of bays in any truss. 

Figure 59 shows the six cases for DMR. For the sake of comparison with MR cases, these six 

cases are numbered as 1, 2, 3, 4, 9, and 10, as illustrated in figure 59. In each case, one diagonal 

and one lower chord member was selected for removal in one-quarter of the bridge. Effects of 

these six cases on ALP of the bridge were investigated by both DCR and SR metrics. MR cases 5 

and 8 in figure 31 were combined as case 3 of DMR because removal of upper chord members 

had negligible effect. Similarly, MR cases 6 and 7 in figure 31 were combined as case 2 of 

DMR. Hence, there were 6 unique cases of DMR in total for comparison to the 10 cases of MR 

investigated previously. 

Source: FHWA. 

Figure 59. Illustration. Cases of critical DMR in one-quarter of the bridge on the west side 

truss.

ALP (DMR) Analysis for Design Bridge: DCR Metric

Figure 60 shows the DCR before and after the removal of two members in DMR case 1. This 

figure demonstrates that the DCR after DMR increased significantly in west side truss members 

(where the dual members were removed). The increase in DCR was less in the east side truss

members. Figure 60 also shows that the load path in the lower chord was completely disrupted 

because of DMR. Thus, loads designed to be transferred through the lower chord members to the 

support were detoured through the upper chord members to the FS via verticals.
90 

Source: FHWA. 

Figure 60. Illustration. DCR before and after DMR case 1 near the damaged location in 

west and east side trusses. 

Figure 61 shows the contributions of axial force and in-plane and out-of-plane moments to the 

DCR for DMR case 1. For example, for member L17-L18 in the west side truss, “1 83 16” 

implies 1-, 83-, and 16-percent contributions of axial force, Min, and Mout to the DCR, 

respectively. Numbers above the member denote these contributions before DMR, and numbers 

below the member represent contributions after DMR. It is observed that the axial force 

contributed predominantly (86 percent) to the DCR of member L17-L18 before DMR. However, 

after DMR, Min contributed predominantly (83 percent) to the DCR, and the contribution of axial 

force was reduced to 1 percent. This effect (change from axial to bending) was more prominent 

in lower chord members in the west side truss. Hence, lower chord members in the west side 

truss may be seriously distorted because of the DMR. Because the west side truss lost greater 

load-carrying capacity, members of the east side truss started carrying more load through axial 

force rather than bending. As a result, the increase in bending in the lower chord members of the 

east side truss after MR was less than that for the single MR case 1 in table 20. These results 

show that the load redistribution in the east side truss occurred over a broader range.
91 

Source: FHWA. 

Figure 61. Illustration. Contributions to DCR before and after DMR case 1 near the damaged location in west and east side 

trusses.
92 

Figure 62 through figure 67 present DCR metric for six DMR cases. Although both single and 

dual member removals involve hypothetical damage imposed on the system, a global pattern of 

load redistribution can be identified by tracking the change in the reactions at each support. 

Table 21 and table 22 show the net change in the magnitude of the reaction force for MR and 

DMR cases, respectively. There are three key findings based on table 21 and table 22: 

1. Because damage occurred in the west side truss near support #20, there was a net 

decrease in the reaction at support #20 in the west side truss and a net increase in the 

reaction at support #20 on the east side truss for all MR and DMR cases. 

2. For the west side truss, the sum of the other three support reaction forces (i.e., at supports 

#1, #8, and #27) increased. 

3. Support #1 (at the very far end of the damaged region on both the east and west side 

trusses) experienced the least change in the reaction. This could imply that the damage 

does not cause extra loading on support #1. 

Because dual member damage hinders the original load path to transfer the load to its nearest 

support, the bridge seeks ALPs such that the load will be transferred to either the opposite side or 

the supports on the same side. However, additional load taken by the support will decrease as the 

support’s distance from the damage zone increases. The load carried by the opposite side truss is 

transferred by the three-dimensionality of the bridge because of the upper and lower bracing 

systems, although these systems were designed to maintain the stability and geometrical shape of 

the bridge rather than to carry load. When damage is introduced into the bridge, loss of 

geometrical symmetry forces the bracing systems to carry redistributed loads. The impacted zone 

based on DCR plots for a DMR case is larger than that for an MR case.

Figure 68 shows the envelope of the DCR metric for the full bridge based on figure 62 through 

figure 67 for one-quarter of the bridge. The role of three-dimensionality is more prominent in 

DMR cases than the MR cases shown in figure 44 because most members of the lower braces, 

upper braces, and floor truss yielded during DMR case 1 (i.e., DCR > 1.67). Table 23 lists the 

number of truss members with DCR ≤ 1.0, 1.0 < DCR ≤ 1.67, and DCR > 1.67 in envelopes for 

MR and DMR. The number of members with DCR ≤ 1.67 decreased from 1,766 for MR to 1,689 

for DMR, whereas the number of members with DCR > 1.67 increased from 118 for MR to 202 

for DMR. These results show that DMR has significantly more impact than MR, as expected. It 

is conceivable that the effect of actual blast loading (instead of simplified representation by 

DMR) on the bridge may be even more significant. 

Figure 62 through figure 67 also demonstrate that the west side truss between supports #8 and 

#20 started behaving like a cantilevered beam after two members near support #20 were 

damaged. This cantilevered portion of the truss had a tendency to rotate clockwise, which caused 

an increased net negative moment at support #8 to resist this tendency to rotate, as illustrated in 

figure 69. DCR of all west side truss members increased because of this net negative moment.
93 

Source: FHWA. 

Figure 62. Illustration. DCR spectra for the Design Bridge for DMR case 1. 
94 

Source: FHWA. 

Figure 63. Illustration. DCR spectra for the Design Bridge for DMR cases 2, 6, and 7. 
95 

Source: FHWA. 

Figure 64. Illustration. DCR spectra for the Design Bridge for DMR cases 3, 5, and 8. 
96 

Source: FHWA. 

Figure 65. Illustration. DCR spectra for the Design Bridge for DMR case 4. 
97 

Source: FHWA. 

Figure 66. Illustration. DCR spectra for the Design Bridge for DMR case 9.
98 

Source: FHWA. 

Figure 67. Illustration. DCR spectra for the Design Bridge for DMR case 10. 
99 

Source: FHWA. 

Figure 68. Illustration. Complete envelope of DCR for the Design Bridge representing 40 DMR cases (10 DMR cases in each 

quarter of the bridge).
100 

Table 21. Percentage change in the reaction force for corresponding MR cases.

Support 

Number Direction

MR 

Case 1 

MR 

Case 2 

MR 

Case 3 

MR 

Case 4 

MR 

Case 9 

MR 

Case 10 

1 East −0.3 −0.4 −0.1 −0.5 0.0 −0.2 

1 West −0.3 −0.5 −0.1 −0.5 0.0 −0.2 

8 East −1.1 −1.5 −0.7 −2.0 −0.2 −0.9 

8 West 2.1 3.6 1.0 4.5 0.2 2.3 

20 East 1.9 3.1 1.2 4.6 0.8 3.3 

20 West −2.5 −5.4 −1.2 −7.6 −0.9 −5.5 

27 East −0.4 −0.6 −0.4 −1.2 −0.6 −1.5 

27 West 0.6 1.7 0.2 2.7 0.7 2.7 

Table 22. Percentage change in the reaction force for corresponding DMR cases.

Support 

Number Direction

DMR 

Case 1

DMR 

Case 2

DMR 

Case 3

DMR 

Case 4

DMR 

Case 9

DMR 

Case 10 

1 East 0.0 −0.1 −0.3 −0.3 −0.2 −0.3 

1 West 0.0 −0.1 −0.3 −0.4 −0.3 −0.3 

8 East −0.1 −0.3 −0.9 −1.1 −0.7 −0.8 

8 West 0.4 1.4 1.9 2.7 1.8 1.9 

20 East 1.3 2.3 1.2 1.7 1.1 1.1 

20 West −1.7 −4.0 −1.8 −3.2 −2.0 −1.7 

27 East −1.1 −1.4 −0.2 −0.2 0.0 −0.1 

27 West 1.3 2.3 0.3 0.8 0.4 0.2 

Table 23. Number of elements in each range of ALP analysis: DCR metric.

Loading Condition DCR ≤ 1.0 1.0 < DCR ≤ 1.67 DCR > 1.67 

MR 1,766 214 118 

DMR 1,689 207 202 
101 

Source: FHWA.

A. West side truss.

Source: FHWA. 

B. Moment diagram of the west side truss. 

Figure 69. Illustrations. Moment diagram of the west side truss (assuming the truss as a 

continuous beam) before any damage and after blast damage at a certain panel. 

ALP (DMR) Analysis for Design Bridge: SR Metric

Figure 70 through figure 75 show ALP spectra in terms of the SR metric corresponding to six 

DMR cases. The results in terms of SR are consistent with those in terms of DCR. Figure 76

shows the envelope of the SR metric for the full bridge based on SR in figure 70 through 

figure 75 for one-quarter of the bridge. This envelope is similar to that for DCR in figure 68.
102 

Source: FHWA. 

Figure 70. Illustration. SR spectra for the Design Bridge for DMR case 1. 
103 

Source: FHWA. 

Figure 71. Illustration. SR spectra for the Design Bridge for DMR cases 2, 6, and 7. 
104 

Source: FHWA. 

Figure 72. Illustration. SR spectra for the Design Bridge for DMR cases 3, 5, and 8. 
105 

Source: FHWA. 

Figure 73. Illustration. SR spectra for the Design Bridge for DMR case 4. 
106 

Source: FHWA. 

Figure 74. Illustration. SR spectra for the Design Bridge for DMR case 9. 
107 

Source: FHWA. 

Figure 75. Illustration. SR spectra for the Design Bridge for DMR case 10. 
108 

Source: FHWA. 

Figure 76. Illustration. Complete envelope of SR for the Design Bridge representing 40 DMR cases (10 DMR cases in each 

quarter of the bridge).
109 

CHAPTER 5. RETROFITS FOR ENHANCING ALP 

SEISMIC RETROFITS

Most existing long-span truss bridges undergo seismic retrofits during major rehabilitation to 

address deterioration and other vulnerabilities because they were not designed according to 

current design standards and may be vulnerable to damage during earthquakes. Currently, ALP 

vulnerability is not addressed as part of this major rehabilitation process. Hence, a detailed 

investigation was performed on the retrofit methods that could be effective during both seismic 

and sudden member loss scenarios.

When the seismic resistance of a bridge is found to be insufficient, a retrofit may be needed to 

reduce the demand experienced by bridge components. Such retrofits include seismic isolation 

and energy dissipation systems (Pan et al. 2010a, b; Kandemir et al. 2011; Agrawal et al. 2012; 

Agrawal and Amjadian 2016), member strengthening (Imbsen and Liu 1993; Matson and 

Buckland 1995; Liu et al. 1997; Imbsen and Schamber 1999), buckling-restrained braces (BRBs)

(Usami et al. 2005; Chen et al. 2011; Uang and Bruneau 2018), and replaceable fuses (El-Bahey 

and Bruneau 2011; Saravanan et al. 2018). During an earthquake, seismic demand is transferred 

from the substructure to the superstructure. Hence, most seismic retrofits focus on the 

substructure for effective control and dissipation of input seismic energy. For member loss 

hazards that initiate at the superstructure level, load is transferred from the superstructure to the 

substructure because of load redistribution. Moreover, the loading during these hazards is of very 

short duration and must be countered within the superstructure through effective redistribution 

among members such that the safety of the bridge system is guaranteed. An even higher standard 

(no damage or minimal damage to the bridge system and its components) may be required, 

depending on the importance of the bridge. Because of significant differences between seismic 

and MR demands, common seismic retrofit measures for substructure elements such as bearings, 

seats, columns, piers, cap beams, column-to-cap joints, abutments, and foundations are not 

applicable to sustain demand during sudden member loss. Only the member strengthening 

approach that is generally used for seismic retrofit of truss bridges (Imbsen and Liu 1993; 

Matson and Buckland 1995; Liu et al. 1997; Imbsen and Schamber 1999) may also be useful in 

enhancing the resistance against sudden loss demands. 

Conventional Seismic Retrofit Schemes

Commonly used seismic retrofits, such as isolation bearings and supplemental damping systems, 

are designed to dissipate input energy without affecting the ALPs. However, retrofits such as 

member strengthening (e.g., adding cover plates to increase the effective area), lateral bracing 

enhancement, and stiffening of bracing systems may be effective in resisting both seismic and 

MR or blast load demands. 

For major truss members, the capacity to sustain member loss can be improved by increasing the 

cross-sectional area, changing the sectional shape, or both. An optimized sectional shape, such as 

a box shape rather than an I shape, may improve the radius of gyration, which in turn could 

increase the capacity of compressive members. BRBs consist of a hollow box section filled with 

core material and can provide stable energy dissipation capacity under dynamic excitations with 
110 

similar behaviors in both tension and compression because global buckling of the core member 

of BRBs is restrained (Usami et al. 2005). Both these retrofits (increasing the cross-sectional area 

and BRBs) are effective in increasing the buckling capacity of compression members. In 

long-span truss bridges, horizontal or lateral bracing systems are significant secondary members. 

Horizontal or lateral bracings are installed between girders to prevent lateral buckling of 

compression flange of girders. These bracings can also participate in load redistribution from one 

truss to another and need to be designed for extra loading induced by sudden MRs. Another 

potential retrofit approach is modification of the bridge system. For example, the Brooklyn 

Bridge was retrofitted nonseismically by adding six arches to reduce loading on the stiffening 

truss, as shown in figure 77. 

Source: FHWA.

Figure 77. Photo. Retrofit of the Brooklyn Bridge with six arch systems. 

Several examples of retrofit of bridges where retrofit involved measures other than member 

strengthening, such as addition of bracings and strengthening of FSs, have been observed from 

the literature. The seismic retrofit measures for the Golden Gate Bridge included replacement 

and addition of top and bottom lateral bracings and strengthening of vertical truss members and 

truss connections (Golden Gate Bridge Authority 2014). Other measures were also taken to 

minimize the effect of ground motions on structures, such as converting five simply supported 

truss spans into a continuous truss system, installing seismic expansion joints at the north and 

south ends of the viaduct truss, and installing isolator bearings atop the new steel support towers. 

Figure 78 shows the schematic of various seismic retrofits to this bridge.
111 

© 2014 Golden Gate Bridge Highway and Transportation District. 

Figure 78. Illustration. Schematic of seismic retrofits to the Golden Gate Bridge.
112 

During seismic retrofit of the Auburn-Foresthill Bridge (figure 79), which is California’s tallest

bridge, analysis revealed that the bracing members near the abutments carried much of the 

seismic load (Reno and Pohll 2010). Large BRBs, capable of handling up to 450 kN (2,003 kips), 

were installed at these locations to absorb some of the input seismic energy and reduce 

deformations. Replacement of bolted connections and gusset plates was also performed

for increased strength. 

© 2008 Moiseiko, English Wikipedia (CC BY-SA 3.0

(https://creativecommons.org/licenses/by-sa/3.0)).

Figure 79. Photo. Auburn-Foresthill Bridge, California’s tallest bridge. 

The Tennessee Department of Transportation performed seismic retrofit of the I-40 Mississippi 

River Bridge, which is a steel tied-arch bridge (figure 80). Retrofits in this bridge included 

diaphragm or cross frame replacement, bottom lateral retrofit, bearing replacement, and 

expansion joints replacement. 

© 2018 TRC Companies, Inc.

Figure 80. Photo. I-40 Mississippi River Bridge (steel tied-arch bridge). 
113 

Based on the case studies of seismic retrofits discussed in this section, effective retrofits that 

could also enhance the three-dimensionality of a truss system for greater distribution of member 

force and robust ALP (in addition to seismic resistance) can be summarized as follows: 

1. Vertical cross bracing, diaphragm or cross frame replacement. 

2. Longitudinal truss enhancement. 

3. Top or bottom lateral bracing enhancement. 

These retrofit strategies are investigated in detail in the following section. 

Seismic Retrofit of the I-35W Bridge

To investigate the effectiveness of seismic retrofits on ALP, it is important to ensure that the 

Design Bridge analyzed in the previous chapter has sufficient capability to withstand seismic 

loads. The I-35W Bridge was located in Minnesota, which is a low seismic hazard region. 

Hence, seismic demands on the bridge were investigated for a high seismic region 

(San Diego, CA). Seismic analysis of the bridge was performed for a 2,500-year earthquake

(SD2500) with short-period (Ss) and 1-second-period (S1) ground accelerations of 0.979 and 

0.377 g, respectively, using the response spectrum approach in AASHTO LRFD 

(AASHTO 2014). Figure 81 shows the seismic response spectra for SD2500. Because the 

fundamental period of the I-35W Bridge was about 2.23 s, the spectral acceleration during this 

period was about 0.2 g. A vertical seismic component was not included because the foundation 

and the soil interaction were not considered in evaluating seismic demand on the bridge. Seismic 

load effect was combined with DLs and LLs using equation 10. 

Source: FHWA. 

Figure 81. Graph. Seismic response spectra for 2500-year earthquake in San Diego, CA. 
114 

Analysis results indicated that the DCR of all members of the bridge for DL, LL, and seismic 

load combinations was between 0.9 and 1.0. Hence, the bridge was in elastic range when 

subjected to SD2500 and did not require seismic retrofit. However, similar to the investigation of 

the Design Bridge in the previous chapter, some members of this bridge were strengthened so

that the DCR of all members for DL, LL, and seismic load combinations was about 0.9. Although 

this DCR value may be conservative during the actual seismic design of bridges, the DCR of all 

members was limited to 0.9 so that the change in DCR due to MR could be easily evaluated to 

identify ALPs. 

The increase in the weight of steel in the bridge because of this retrofit was approximately 

3.42 percent. This increase in steel weight was approximately 0.98 percent for low seismic 

regions such as Minnesota and New York City. The seismically retrofitted bridge for SD2500 

was termed “Base Bridge” for further MR and retrofit analysis. Figure 82 shows detailed views 

of the bridge and its components after retrofit. Retrofitted members of the bridge in this figure 

are shown as red dotted lines. The figure demonstrates that some members of the trusses, the 

upper and lower braces, and the FS were retrofitted to limit the DCR to 0.9. 

Sudden member loss analysis was conducted for the DL and LL combination in equation 11 to 

check the effectiveness of seismic retrofit for load redistribution after the removal of a member. 

Results were compared with those for the Design Bridge (the bridge without seismic retrofit) by 

counting truss members falling into different DCR limit states, as shown in table 24 and table 25. 

These two tables show that the number of truss members with DCR > 1.67 (i.e., member 

possibly undergoing inelastic deformation) decreased from 118 for the Design Bridge to 75 for 

the Base Bridge for MR scenarios. Likewise, the number of truss members with DCR > 1.67 

decreased from 202 for the Design Bridge to 117 for the Base Bridge for DMR scenarios. These 

results demonstrate the limited effectiveness of seismic retrofit (through member strengthening 

only) during sudden member loss because of the lack of new load paths created. The Base Bridge

was used to investigate the effectiveness of other retrofit measures that involved changing the 

three-dimensionality of the bridge. 

Table 24. Number of members in each range for different DCR limit states under MR 

scenarios for the Design Bridge and Base Bridge.

Bridge DCR ≤ 1.0 1.0 < DCR ≤ 1.67 DCR > 1.67 DCR > 1.0 

Design Bridge 1,766 214 118 332 

Base Bridge 1,845 178 75 253 

Table 25. Number of members in each range for different DCR limit states under DMR 

scenarios for the Design Bridge and Base Bridge.

Bridge DCR ≤ 1.0 1.0 < DCR ≤ 1.67 DCR > 1.67 DCR > 1.0 

Design Bridge 1,689 207 202 409 

Base Bridge 1,768 213 117 330 
115 

Source: FHWA. 

Figure 82. Illustration. Members of the Design Bridge retrofitted for SD2500 by member strengthening to obtain the Base 

Bridge for the ALP analysis.
116 

PBD AND RETROFIT FOR SUDDEN MEMBER LOSS

PBD is a rational approach to designing a new structure or evaluating an existing structure. By 

identifying specific structural performance objectives, an engineer can use PBD to design a 

structure that satisfies desired serviceability or strength-level targets. By accounting for the 

significance of inelastic structural response, the overall performance of the structure is controlled 

during the PBD process.

There are a variety of performance-based criteria and design parameters based on the type of 

structure, the materials, the hazards to which the structure is exposed, and the analysis approach 

adopted. The criteria can be either qualitative or quantitative for different performance levels. 

PBD framework was first adopted in earthquake engineering for seismic design and retrofit of 

structures. Seismic retrofit philosophy is based on the concept of minimizing loss of life and 

serious injury to the public through acceptable bridge performance. A similar PBD approach has 

been considered for retrofitting truss bridges to enhance their ALP. 

The PBD approach in this research considered two types of design criteria for long-span steel 

truss bridges: DCR through elastic analysis and SR through inelastic analysis using an LS-DYNA 

model that includes both geometrical and material nonlinearities. For each design criterion, 

various limit states that correspond to different damage levels are presented in table 26 and 

table 27. 

Table 26. Allowable indicators for various performance levels using DCR metric. 

DCR Limit State (SAP2000) Performance Level

DCR ≤ 1.0 No damage: allowable stress level (elastic behavior), 

no impact on traffic mobility.

1.0 < DCR ≤ 1.67 Minor damage: intermediate state between allowable 

(elastic) and plastic, controlled traffic mobility, minor 

damage at yield point. 

DCR > 1.67 Major damage: some members of the bridge system 

may experience yielding. Collapse prevention. 

Table 27. Allowable indicators for various performance levels using SR metric. 

SR Limit State (LS-DYNA) Performance Level

SR ≤ 1.0 No damage: elastic range. 

1.0 < SR ≤ 2.0 Minor damage: low ductility. 

2.0 < SR ≤ 4.0 Moderate damage: moderate ductility, repairable 

damage. 

SR > 4.0 Significant damage: minimum risk of collapse. 

Significant differences exist between seismic and MR or DMR demands. Perhaps the most 

obvious difference is the broadness of the affected zone or region. Seismic demands are global 

(i.e., they affect the whole bridge and excite lower modes of the bridge). MR or DMR demands 

involve localized damage initiation events and very short time durations (compared to 

earthquake excitations) and so affect higher modes. Based on these characteristics of MR or 

DMR scenarios, it is expected that seismic retrofits may not be comprehensive enough to resist
117 

MR or DMR demands and that the three-dimensionality characteristics of the structure may be 

more effective at resisting MR or DMR demand than seismic demand, even for bridges with 

perfectly symmetric geometry (Ettouney et al. 2006; Ettouney and Alampalli 2014, 2016).

Because of these differences between seismic and MR or DMR demands, it is expected that the 

analysis performed for the seismic retrofit may not be sufficient to withstand member loss 

demands. A detailed investigation is needed to identify the shortfalls of the seismic retrofits in 

MR or DMR scenarios. These limitations could be identified by observing the shortcomings of 

total bridge performance (displacements) and the shortcomings of member-based performance 

metrics (strain). Hence, further retrofits of the Base Bridge for MR or DMR scenarios were

performed based on the demands of the corresponding acceptance limits.

A typical long-span truss bridge consists of side trusses and horizontal and vertical 

load-supporting systems, such as horizontal top and bottom bracing systems and FSs. When 

these subsystems function together as a whole, they contribute to the three-dimensionality of the 

bridge in load redistribution after MR or DMR events. When designing a bridge for DLs, LLs, 

and seismic loads, the relative contribution of seismic load to the total stress in a member is 

negligible compared to those from DL and LL (i.e., major loading on the bridge is in the vertical 

direction only). Adding a few extra members in the horizontal system could improve the 

horizontal stiffness of the bridge and thus the seismic resistance. However, results show that 

horizontal (top or bottom) bracing enhancement is less efficient than enhancements in the 

vertical system, such as FSs or DM systems, for improving vertical resistance against 

deformation induced during MR or DMR events. Nevertheless, enhancing horizontal (top or 

bottom) bracings does contribute to the three-dimensionality of the bridge, which is important for 

resisting the torsion that the bridge may undergo about its longitudinal axis after MR or DMR 

events. Although the bracing system is a secondary subsystem in the main truss structure, it is

essential because it provides stability to the main girders during construction. It also contributes 

to the distribution of load effect as well as provides restraint to compression flanges or chords 

where they would otherwise be free to buckle laterally.

In addition to the performance levels presented in table 26 and table 27, the cost of retrofits and 

the practical issues concerning increase in truss weight and complexity of the joint should also be 

considered in evaluating the effectiveness of retrofits. Because effective retrofits to enhance ALP 

may depend on the characteristics of a bridge, retrofits presented in this chapter are based on a 

trial and error approach. For the I-35W truss bridge, the following four strategies for retrofit 

against MR or DMR demands were investigated: 

1. Member strengthening only. 

2. Adding bracings in the FS plus member strengthening. 

3. Adding DMs plus member strengthening. 

4. Adding DMs plus adding bracings in the FS plus member strengthening. 

Table 28 shows the cases of MR retrofit strategies to be investigated for different DCR limits. 
118 

Table 28. Retrofit strategies for MRs.

DCR Range Retrofit Type Case MR Retrofit Strategy Description

1.0 < DCR ≤ 1.67 1-1 Member strengthening only 

1.0 < DCR ≤ 1.67 2-1 Adding bracings in the FS + member 

strengthening 

1.0 < DCR ≤ 1.67 3-1 Adding DMs + member strengthening 

1.0 < DCR ≤ 1.67 4-1 Adding DMs + adding bracings in the FS 

+ member strengthening 

0 < DCR ≤ 1.0 1-2 Member strengthening only 

0 < DCR ≤ 1.0 2-2 Adding bracings in the FS + member 

strengthening 

0 < DCR ≤ 1.0 3-2 Adding DMs + member strengthening 

0 < DCR ≤ 1.0 4-2 Adding DMs + adding bracings in the FS 

+ member strengthening 

HAZARD EVENTS

The hazard events used in this chapter for the Base Bridge follow the same types (MR or DMR) 

used in chapter 4 for the Design Bridge. 

DCR-BASED RETROFIT SCHEMES

The DCR-based retrofit implies that the bridge was retrofitted following the retrofit strategies in 

table 28 to achieve DCR limits of 1.0, 1.67, and 5.0. A DCR limit of 1.67 implies initiation of 

yielding in some members (which can be considered essentially elastic behavior), and a DCR

limit of 5.0 implies significant damage to the truss members without causing collapse. The 

rationale for the 1.67 limit is that the bridge member should be able to use the maximum capacity 

up to the yield limit for the extreme event scenarios of MR or DMR. However, depending on the 

importance of the bridge, a bridge owner can also select a DCR limit of 5.0 to economize the 

design. 

Retrofits for MR 

Similar to the MR analysis in chapter 4 on the Design Bridge, MR analysis was performed by 

considering 10 cases of MR to obtain the complete envelope shown in figure 83. Figure 84

through figure 87 show the distribution of retrofitted members for retrofit types 1-1, 2-1, 3-1, and 

4-1 in table 28, respectively, for a desired performance level of 1.0 ≤ DCR ≤ 1.67. Increase in the 

weight of steel in the bridge for these retrofit types was 26, 24, 15, and 11 percent, respectively, 

as shown in table 29. Likewise, increase in the weight of steel for retrofit types 1-2, 2-2, 3-2, and 

4-2 in table 28 for DCR ≤ 1.0 was 70, 65, 27, and 24 percent, respectively.
119 

Source: FHWA. 

Figure 83. Illustration. Complete envelope of DCR for the Base Bridge representing 40 MR cases (10 removal cases in each 

quarter of the bridge).
120 

Source: FHWA. 

Figure 84. Illustration. Member strengthening retrofit of the Base Bridge for MR based on DCR ≤ 1.67 limit state.
121 

Source: FHWA. 

Figure 85. Illustration. Addition of bracings in the floor truss systems and member strengthening retrofit of the Base Bridge 

for MR based on DCR ≤ 1.67 limit state.
122 

Source: FHWA. 

Figure 86. Illustration. Addition of DMs and member strengthening retrofit of the Base Bridge for MR based on 

DCR ≤ 1.67 limit state.
123 

Source: FHWA. 

Figure 87. Illustration. Addition of bracings in the floor truss systems, addition of DMs, and member strengthening retrofit of 

the Base Bridge for MR based on DCR ≤ 1.67 limit state.
124 

Table 29. Increase in the weight of steel and DispRs for different MR retrofit strategies.

DCR

Value

MR Retrofit 

Strategy 

Additional Increased 

Weight 

(kN)

Increased 

Weight 

(%) DispR

≤1.67 1-1 7,924 26 0.74 

≤1.67 2-1 7,232 24 0.76 

≤1.67 3-1 4,408 15 0.77 

≤1.67 4-1 3,493 11 0.80 

≤1.0 1-2 21,351 70 0.63 

≤1.0 2-2 19,517 64 0.64 

≤1.0 3-2 8,251 27 0.66 

≤1.0 4-2 7,255 24 0.68 

1 kN = 0.225 kips. 

Note: The total steel weight of the Base Bridge was 30,378 kN. 

To compare the effectiveness of different retrofit strategies in terms of displacement reduction, 

maximum vertical displacements corresponding to MR cases were normalized with respect to a 

reference maximum allowable displacement limit. According to section 2.5.2.6.2 of AASHTO 

(2014), the deflection limits for steel vehicular bridges can be considered as span length ∕ 800 for 

general conditions and span length ∕ 300 for the vehicular load on cantilever arms. For the 

abnormal loading conditions (MR or DMR) in this study, a higher level of deflection could be 

considered because of the rare probability of occurrence of MR or DMR events. In this research, 

the deflection limit used for the elastic analysis was span length ∕ 800 and for the nonlinear 

analysis in LS-DYNA was span length ∕ 300. It should be noted that these deflection limits are 

empirical and are based on very old criteria for short-span bridges. Generally, long-span bridges, 

including long-span truss bridges, can tolerate much higher levels of displacement without 

inelastic stress in any of the members. For example, the displacement limit where any member of 

the Base Bridge (without retrofit) will reach the yield stress is 507 mm (19.96 inches), whereas 

the displacement limits by span length ∕ 800 and span length ∕ 300 are 406 mm (15.98 inches) and 

1,082 mm (42.6 inches), respectively. 

Table 29 shows the magnitude of percentage weight increase and displacement ratio (DispR) for 

the four groups of retrofit strategies for two performance levels (i.e., 1.0 ≤ DCR ≤ 1.67 and 

DCR ≤ 1.0). Figure 88 shows plots of weight increase and DispRs for the four groups of retrofit 

strategies. DispR was selected as the indicator of the PBD and was defined as the peak of the 

maximum vertical displacement of the retrofitted bridge among 10 MR cases in table 20 for each 

retrofit strategy divided by the allowable deflection limit discussed in the previous paragraph.
125 

Source: FHWA.

A. Increase in weight of steel.

Source: FHWA. 

B. Displacement ratio.

Figure 88. Graphs. Increase in weight of steel and DispRs for different MR retrofit 

strategies.

Table 29 and figure 88 demonstrate that the percentage increase in steel weight decreased from 

26 to 11 percent for 1.0 ≤ DCR ≤ 1.67 and from 70 to 24 percent DCR ≤ 1.0 for retrofit strategies 

1 to 4, whereas DispRs were similar for all retrofit strategies. This shows that all retrofit 

strategies had approximately similar performance. However, the member strengthening only 

retrofit was the least efficient, and the member strengthening combined with adding DMs and FS 

members retrofit was the most efficient from both cost (or weight of steel) and performance 

(displacement) points of view. This is important because an increase in the weight of the bridge 

would result in added seismic demand on the bridge member, thereby increasing seismic 

vulnerability while addressing vulnerability due to MR or DMR loads.

It should be noted that the addition of members (e.g., diagonal, FS, bracings) to improve ALP of 

the bridge can be formulated as an optimization problem with performance criteria in terms of 

DCR and DispR. Such optimization could yield much more efficient retrofit strategies than 4-1 

and 4-2 that could enhance the ALP of the bridge much more with less corresponding increase in 

the weight of steel. However, such optimization for a complex long-span bridge would be very 

computationally expensive but could be investigated in future research. In this context, retrofit 

strategies 4-1 and 4-2 may not be the most optimized but can be considered to illustrate the 

potential of the three-dimensionality of the entire bridge system in improving ALP of long-span 

truss bridges. Retrofit strategies 4-1 or 4-2 also show that improvement in ALP of long-span 

truss bridges requires 3D interaction of added members with different vertical systems rather 

than a single vertical system of the bridge. 

Retrofits for DMR

Similar to the DMR analysis in chapter 4 on the Design Bridge, DMR analysis was performed by 

considering six cases of DMR to obtain the complete envelope shown in figure 89. The bridge 

was retrofitted by four different retrofit strategies for performance levels 1.0 ≤ DCR ≤ 1.67 and 

DCR ≤ 1.0 in table 28. Table 30 shows the percentage increase in the weight of steel and the 

DispRs for eight cases of retrofits. Similar to the MR case, retrofit strategy 4 (i.e., 4-1 and 4-2) 

was the most efficient in terms of increase in the weight of steel, and the displacement 
126 

performance was similar among all four retrofit strategies. Figure 90 shows plots of the increase 

in the weight of steel and the DispRs for the four types of retrofits. The trend in this figure is 

similar to that for the MR retrofit cases in figure 88. The increase in weight of the steel was 

higher for DMR than for MR.
127 

Source: FHWA. 

Figure 89. Illustration. Complete envelope of DCR for the Base Bridge representing 40 DMR load cases (10 DMR cases in each 

quarter of the bridge).
128 

Table 30. Increase in the weight of steel and DispRs for different DMR retrofit strategies. 

DCR Value

DMR 

Retrofit 

Strategy 

Additional 

Increased Weight 

(kN)

Increased 

Weight 

(%) DispR

≤1.67 1-1 10,014 33 0.77 

≤1.67 2-1 8,734 31 0.78 

≤1.67 3-1 36,796 21 0.81 

≤1.67 4-1 35,738 18 0.83 

≤1.0 1-2 55,988 84 0.61 

≤1.0 2-2 13,463 77 0.62 

≤1.0 3-2 2,422 39 0.64 

≤1.0 4-2 4,079 35 0.67 

1 kN = 0.225 kips.

Note: The total steel weight of the Base Bridge was 30,378 kN. 

Source: FHWA.

A. Increase in weight of steel. 

Source: FHWA. 

B. Displacement ratio.

Figure 90. Graphs. Increase in weight of steel and DispRs for different DMR retrofit 

strategies.

SR-BASED RETROFIT SCHEMES

For SR-based retrofit, three levels of performance in terms of SR (which are correlated to levels 

of ductility) are defined in table 31. When the SR is equal to or less than 1.0, the bridge system is 

in the elastic range. When SR ≤ 2.0 (low ductility), some members are allowed to have strain 

beyond the yield limit because of the ductility of steel. For this level of performance, no retrofit 

is needed for an assumed MR or DMR loading scenario. When bridge members are allowed a 

ductility of medium level, SR is assumed to be less than or equal to 4.0. Allowable bridge 

deflection for the three levels of SR and ductility are shown in table 31. Figure 91 illustrates the 

subjective relationship between DCR and SR metrics.
129 

Table 31. SR-based retrofit levels for an assumed blast intensity.

Performance Level SR

Displacement Ratio 

(≤1.0) Comment

Elastic range SR ≤ 1.0 L ∕ 800 Linear elastic range

Low ductility SR ≤ 2.0 L ∕ 300 No retrofit needed 

Medium ductility SR ≤ 4.0 L ∕ 300 Not observed 

L = bridge span length.

Source: FHWA. 

Figure 91. Graph. Subjective relationships between DCR and SR metrics. 

During inelastic analysis in LS-DYNA, both peak dynamic and steady-state results were used for 

comparison. The four types of retrofit strategies in table 28 were investigated for both MR and 

DMR loading scenarios.

Retrofits for MR

Figure 92 shows the complete envelope of truss members of the bridge for MR scenarios in 

terms of the three levels of SR. This figure demonstrates that, although many truss members had 

SRs in the range of 1.0 < SR ≤ 2.0, none of the truss members had an SR > 2.0. Hence, the only 

level of retrofit required was for the performance level SR ≤ 1.0, when a retrofit strategy will 

reduce the SR of members shown in red dotted lines to SR less than or equal to 1.0. Hence, there 

are only four retrofit strategies for the first performance level and no retrofit requirements for 

performance levels 2 and 3. Table 32 shows the increase in the weight of steel for the SR ≤ 1.0 

limit state for the four retrofit strategies.
130 

Source: FHWA. 

Figure 92. Illustration. Complete envelope of SR for the Base Bridge representing 40 MR cases (10 removal cases in each 

quarter of the bridge).
131 

Table 32. Percentage increase in the weight of steel for different MR retrofit strategies 

based on DCR and SR limit states.

Performance 

Level Limit State

1: Member 

Strengthening 

Only

2: Adding 

Bracings in FS

+ Member 

Strengthening

3: Adding 

DMs + 

Member 

Strengthening

4: Adding 

DMs + Adding 

Bracings in FS

+ Member 

Strengthening

Factor of 

safety (>1.0)

DCR ≤ 1.0 70 64 27 24 

Elastic limit 

state

1.0 < DCR ≤ 

1.67 

26 24 15 11 

Elastic limit 

state

(peak strain)

SR ≤ 1.0a 23 20 13 9 

Elastic limit 

state

(steady-state 

strain)

SR ≤ 1.0b 18 16 9 7 

Low ductility DCR ≤ 5.0 NA NA NA NA

Low ductility SR ≤ 2.0 0 0 0 0 

Medium 

ductility

SR ≤ 4.0 NO NO NO NO

a

MR demands using the peak of NLD strain. b

MR demands using the peak of steady-state strain. 

NA = not applicable; NO = not observed.

After removal of a member, bridge members undergo peak dynamic strain (or displacement) 

before damping out to steady-state strain (or displacement), as shown in the time history 

response in figure 45. Hence, the performance level SR ≤ 1.0 could be imposed on either the 

peak dynamic strain or the steady-state strain. The percentage increase in steel weight for these 

two types of strain is presented in separate rows of table 32 corresponding to the elastic limit 

state for peak strain and the elastic limit state for steady-state strain. Because SR ≤ 1.0 is an 

elastic limit state, the percentage increase in weight corresponding to DCR ≤ 1.0 and 

1.0 ≤ DCR ≤ 1.67, which are also elastic limit states, are shown in table 32 for comparison. 

Table 32 demonstrates that the percentage increase in the weight of steel considering inelastic 

behavior of the bridge during MR was much smaller than that based on DCR (which considers 

elastic behavior of the bridge). Furthermore, the percentage increase in weight considering 

SR ≤ 1.0 for peak dynamic strain was slightly higher than that considering SR ≤ 1.0 for 

steady-state strain. For example, for retrofit strategy 4, which was found to be the most 

cost-effective based on DCR, the percentage increase in weight for 1.0 ≤ DCR ≤ 1.67 was 

11 percent, whereas it was 9 and 7 percent for SR ≤ 1.0 for peak dynamic and steady-state 

strains, respectively. These results are illustrated in a bar chart in figure 93. 
132 

Source: FHWA. 

Figure 93. Chart. Percentage increase in the weight of steel for different MR retrofits based 

on SR limit states.

The results illustrated in figure 93 can have significant implications on the design of retrofit 

strategies. For a given retrofit strategy, for example retrofit strategy 3, performing inelastic 

analysis in LS-DYNA instead of elastic analysis in SAP2000 could result in cost savings of 

6 percent of the weight of steel if SR ≤ 1.0 is based on steady-state strain. However, if it is 

important to consider SR ≤ 1.0 based on peak dynamic strain, this cost savings in the weight of 

steel could be just 2 percent. This cost savings of 2 percent could be significantly more than the 

savings in analysis costs for performing elastic analysis. Generally, factors for labor and other 

costs are significantly more than material cost factors. Analysis models that consider material 

and geometric nonlinearities also give more reliable retrofit performance results and are able to 

provide a more realistic picture of actual damage in the event of sudden member loss. 

Retrofits for DMR

Figure 94 shows the complete envelope of truss members of the bridge for DMR scenarios in 

terms of the three levels of SR. Figure 94 demonstrates that many truss members had SRs in the 

range of 1.0 < SR ≤ 2.0 and a few bracings had SR > 2.0. Hence, two levels of retrofit were 

required: (1) reducing members with 1.0 < SR ≤ 2.0 to SR ≤ 1.0, and (2) reducing members with 

2.0 < SR ≤ 4.0 to SR ≤ 2.0. For performance level 1 (SR ≤ 1.0), four retrofit scenarios were 

investigated for DMR cases. For performance level 2 (SR ≤ 2.0), because only eight secondary 

bracings and a few members in the FSs needed retrofits, the member strengthening retrofit 

strategy (strategy 1) was efficient in reducing SR to ≤ 2.0. The weight increase in this case was 

3 percent. Table 33 shows increase in weight of steel for SR ≤ 1.0 for the four retrofit strategies.
133 

Source: FHWA. 

Figure 94. Illustration. Complete envelope of SR for the Base Bridge representing 40 DMR cases (10 DMR cases in each 

quarter of the bridge).
134 

Table 33. Percentage increase in the weight of steel for different DMR retrofit strategies.

Performance 

Level Limit State

1: Member 

Strengthening 

Only

2: Adding 

Bracings in 

FS + Member 

Strengthening

3: Adding 

DMs + 

Member 

Strengthening

4: Adding 

DMs + 

Adding 

Bracings in 

FS + Member 

Strengthening

Factor of 

safety (>1.0)

DCR ≤ 1.0 84 77 40 35 

Elastic limit 

state

1.0 < DCR ≤ 

1.67 

33 31 23 18 

Elastic limit 

state

(peak strain)

SR ≤ 1.0a 33 31 19 15 

Elastic limit 

state

(steady-state 

strain)

SR ≤ 1.0b 25 22 16 13 

Low ductility DCR ≤ 5.0 NA NA NA NA

Low ductility SR ≤ 2.0 3 NA NA NA

Medium 

ductility

SR ≤ 4.0 NO NO NO NO

a

DMR demands using the peak of NLD strain. b

DMR demands using the peak of steady-state strain. 

NA = not applicable; NO = not observed.

The percentage increase in the weight of steel for SR ≤ 1.0, where SR is calculated by peak 

dynamic and steady-state strains, is presented in separate rows of table 33 corresponding to 

elastic limit state for peak strain and elastic limit state for steady-state strain). The percentage 

increase in the weight of steel corresponding to DCR ≤ 1.0 and 1.0 ≤ DCR ≤ 1.67, which are also 

elastic limit states, are shown in table 33 for comparison. Table 33 demonstrates that the 

percentage increase in the weight of steel by considering inelastic behavior of the bridge during 

DMR was much smaller than that based on DCR for retrofits 3 and 4, which are effective in 

improving ALP. 

Furthermore, the percentage increase in weight considering SR ≤ 1.0 for peak dynamic strain was 

slightly higher than that considering SR ≤ 1.0 for steady-state strain, particularly for retrofit 

strategy 4. For example, for retrofit strategy 4, which was found to be the most cost-effective 

based on DCR, percentage increase in the weight of steel for 1.0 ≤ DCR ≤ 1.67 was 18 percent, 

whereas it was 15 and 13 percent for SR ≤ 1.0 for peak dynamic and steady-state strains, 

respectively. These two increases in weight of steel were 9 percent and 7 percent, respectively, 

for the MR scenario (table 32). These results are illustrated in a bar chart in figure 95. The trend 

of results for DMR scenarios was the same as that for MR scenarios, although the increase in the 

weight of steel was more in DMR cases because they represent more extreme loading scenarios.
135 

Source: FHWA. 

Figure 95. Chart. Percentage increase in the weight of steel for different DMR retrofits 

based on SR limit states.

DYNAMIC LOAD AMPLIFICATION

Three types of displacements can be obtained for each of the four retrofit strategies. Analysis of 

MR and DMR cases in SAP2000 gives linear static displacement, whereas NLD analysis in 

LS-DYNA gives both peak dynamic and steady-state displacements. For example, as shown in 

the time history plot of vertical displacement at node 18 on the west side truss in figure 45 in 

chapter 3, the magnitude of the peak dynamic displacement was 613 mm (24.13 inches), whereas

the steady-state displacement was 504 mm (19.84 inches) after the vibration damped out and the 

bridge gradually reached steady-state equilibrium. From the results of simulations, three 

displacements for the lower nodes of trusses of the bridge were recorded to calculate nonlinear 

static (NLS) and NLD factors. The NLS factor is calculated as the steady-state displacement 

from LS-DYNA divided by the linear static displacement from SAP2000. Likewise, the NLD 

factor is calculated as the peak dynamic displacement from LS-DYNA divided by the linear 

static displacement from SAP2000. Table 34 and table 35 show the maximum NLS and NLD 

factors for 10 MR and 10 DMR cases, respectively. These tables also show the dynamic load 

amplification (DLA) factor, which is NLD divided by NLS. The NLS factors for both MR and 

DMR cases were about 1, which was expected. The steady-state displacement from LS-DYNA 

was expected to be slightly higher than the linear static value from SAP2000 because of 

modeling of geometrical and material nonlinearities and finite-element formulation (B-S beam 

element in LS-DYNA versus frame element in SAP2000). The NLD factor varied from 1.06 to 

1.75 for MR cases and from 1.18 to 2.02 for DMR cases, depending on the severity of the MR or 

DMR loading. The DLA factor, which could be considered the amplification in response because 

of dynamic loading, varied from 1.05 to 1.60 for MR cases and from 1.14 to 1.71 for DMR 

cases. For a mass attached to an undamped spring, the DLA is 2 (Chopra 2000).
136 

Table 34. DLA for different MR analysis cases and scenarios. 

MR Case NLS Factor NLD Factor DLA

1 1.10 1.75 1.60 

2 1.09 1.30 1.19 

3 1.09 1.29 1.18 

4 1.01 1.06 1.05 

5 1.03 1.19 1.16 

6 1.02 1.13 1.11 

7 1.02 1.31 1.28 

8 1.01 1.12 1.11 

9 1.09 1.66 1.51 

10 1.01 1.30 1.29 

Range 1.01–1.10 1.06–1.75 1.05–1.60 

Table 35. DLA for different DMR analysis cases and scenarios. 

DMR Case NLS Factor NLD Factor DLA

1 1.11 1.79 1.62 

2 1.19 2.02 1.71 

3 1.08 1.28 1.18 

4 1.04 1.18 1.14 

5 1.08 1.28 1.18 

6 1.19 2.02 1.71 

7 1.19 2.02 1.71 

8 1.08 1.28 1.18 

9 1.12 1.78 1.59 

10 1.03 1.43 1.39 

Range 1.03–1.19 1.18–2.02 1.14–1.71 

The recent AASHTO Guide Specifications for Analysis and Identification of Fracture Critical 

Members and System Redundant Members recommended a DLA of 1.4 for long-span truss 

bridges (AASHTO 2018a). Table 34 shows that a DLA factor of 1.4 is conservative for eight 

cases of MR and slightly less conservative for MR cases 1 and 9. Hence, considering some 

conservatism in the design practice (i.e., DCR may be slightly less than 1 instead of being just 1), 

the DLA factor of 1.4 recommended in AASHTO (2018a) seems to be sufficient when using 

simplified design using commercial software such as SAP2000. It should be noted that the DLA 

factor for DMR for some cases was slightly higher than that for MR, which was expected 

because the dynamic effect of sudden DMR was expected to be higher than that for MR.

Table 36 and table 37 show linear static displacement from SAP2000 and peak dynamic and 

steady-state displacements from LS-DYNA for MR and DMR cases, respectively, normalized by 

the elastic deflection limit of span length ∕ 800 in AASHTO (2014). These ratios were calculated 

for MR and DMR loads for the four retrofit strategies investigated to achieve performance levels 

in terms of SR and DCR. Only results for performance levels representing elastic behavior of the 

retrofitted bridge in the event of MR or DMR (i.e., SR ≤ 1.0 and 1.0 ≤ DCR ≤ 1.67) are presented 

in table 36 and table 37. All displacements stayed below the allowable displacement limit. In 
137 

addition, the steady-state, peak, and linear static displacements all stayed below or around the 

displacement limit of span length ∕ 800, which was 406 mm (15.98 inches) for the I-35W Bridge. 

Although the added percentage weight of steel because of retrofit decreased from retrofit 

strategies 1 to 4, the maximum DispRs for MR and DMR loads were similar for retrofit 

strategies 1 to 4. This shows that the increased displacement because of added steel to the bridge 

was counterbalanced by the increased stiffness because of the retrofit.

Table 36. DispRs for MR cases of ALP-retrofitted bridge with the elastic deflection limit 

L ∕ 800 for the four retrofit strategies. 

MR 

Case

NLD, Steady State 

(SR ≤ 1.0) 

NLD, Peak Response 

(SR ≤ 1.0) 

Linear Static 

(1.0 ≤ DCR ≤ 1.67) 

1 2 3 4 1 2 3 4 1 2 3 4 

Case 1 0.98 0.77 0.8 0.83 0.82 0.91 0.62 0.68 0.69 0.68 0.60 0.60

Case 2 0.89 0.98 0.92 0.94 0.96 1.00 0.76 0.83 0.74 0.72 0.66 0.67

Case 3 0.84 0.94 0.94 0.96 0.92 0.94 0.78 0.87 0.7 0.68 0.68 0.68

Case 4 0.66 0.71 0.78 0.81 0.62 0.71 0.6 0.65 0.58 0.58 0.59 0.59

Case 5 0.74 0.85 0.93 0.95 0.8 0.85 0.81 0.88 0.66 0.65 0.68 0.68

Case 6 0.69 0.79 0.89 0.93 0.71 0.76 0.76 0.82 0.62 0.61 0.66 0.66

Case 7 0.78 0.73 0.79 0.82 0.71 0.85 0.61 0.66 0.62 0.63 0.59 0.59

Case 8 0.64 0.70 0.77 0.81 0.6 0.70 0.59 0.65 0.56 0.56 0.58 0.58

Case 9 0.97 0.78 0.8 0.83 0.77 0.88 0.62 0.68 0.69 0.68 0.6 0.6

Case 10 0.91 0.78 0.8 0.83 0.88 0.99 0.61 0.67 0.72 0.72 0.6 0.6

Maximum 0.98 0.98 0.94 0.96 0.96 1.00 0.81 0.88 0.74 0.72 0.68 0.79

Weight 

increase 

percentage

18 16 9 7 23 20 13 9 26 24 15 11

Table 37. DispRs for DMR cases of ALP-retrofitted bridge with the elastic deflection limit 

L ∕ 800 for the four retrofit strategies.

DMR

Case

NLD, Steady State 

(SR ≤ 1.0) 

NLD, Peak Response 

(SR ≤ 1.0) 

Linear Static 

(1.0 ≤ DCR ≤ 1.67) 

1 2 3 4 1 2 3 4 1 2 3 4 

Case 1 0.83 0.84 0.93 0.96 0.93 0.83 0.94 0.91 0.79 0.70 0.71 0.70

Case 2 0.96 0.97 0.89 0.92 1.05 0.95 0.88 0.85 0.84 0.75 0.68 0.67

Case 3 0.61 0.66 0.96 0.99 0.88 0.79 0.99 0.97 0.73 0.65 0.70 0.70

Case 4 0.56 0.6 0.93 0.96 0.82 0.72 0.96 0.93 0.70 0.62 0.7 0.69

Case 9 0.77 0.72 0.77 0.81 0.85 0.74 0.75 0.74 0.75 0.66 0.61 0.60

Case 10 0.74 0.75 0.77 0.81 0.92 0.82 0.74 0.74 0.80 0.70 0.61 0.60

Maximum 0.96 0.97 0.96 0.99 1.05 0.95 0.99 0.97 0.84 0.75 0.71 0.70

Weight 

increase 

percentage

25 22 16 13 33 31 19 15 33 31 23 18

CRITICAL EVALUATION OF RETROFIT PERFORMANCE

Results of this detailed investigation showed that the member strengthening only or the section 

strengthening was the least cost-effective approach for improving the resistance of the bridge 

against MR or DMR events. Rather, strengthening in combination with improvement in the 

bridge truss system by adding new members was significantly more efficient at enhancing ALP 

of the bridge. Among the four retrofit strategies investigated, the fourth retrofit strategy was the 
138 

most sophisticated because it involved three major subcomponents of the bridge. Results 

presented in this chapter show that the design of retrofits for ALPs is an optimization problem 

that, when solved through complex and time-consuming finite-element simulations, can result in 

a retrofit strategy that can potentially improve the ALP of long-span truss bridges without any 

significant increase in the cost of the retrofit.
139 

CHAPTER 6. IKITSUKI BRIDGE MODELING, ALP ANALYSIS, AND 

PERFORMANCE-BASED RETROFITS

INTRODUCTION

In previous chapters, the ALP of the I-35W Bridge, which was a deck-truss bridge, was

investigated. Through-truss bridges, such as the Ed Koch (Queensboro) Bridge in New York 

City and the Commodore Barry Bridge over the Delaware River between Pennsylvania and New 

Jersey, are another prevalent type of long-span truss bridges. In this chapter, the ALP of a longspan through-truss bridge was investigated following the approach developed for the I-35W 

Bridge. The Ikitsuki Bridge, shown in figure 96, connecting Ikitsuki to Hirado Island in Japan, is 

the longest continuous truss bridge in the world, with a main span of 400 m (1,311.48 ft) and two 

side spans of 200 m (655.74 ft) each. Construction of the bridge was completed in July 1991. 

Previous research on this bridge focused on the nonlinear seismic response properties of the 

bridge (Kubota et al. 2004). In an inspection in December 2009, a critical crack was found in a 

DM near an intermediate support (Nakamura et al. 2016; Wang et al. 2017). This crack began

and propagated as a fatigue crack due to wind-induced vibration. Because fatigue cracks are 

localized structural damage, progressive, and often lead to loss of a critical member, this bridge 

is a good candidate for investigation of ALP in long-span through-truss bridges. This bridge was 

selected for study in this research because of the availability of bridge plans, section details, and 

monitoring data. Results of this investigation demonstrate that the approach in this research is 

applicable to similar types of long-span bridges in the United States. 

© 2018 Shozo Nakamura, Nagasaki University, Japan. 

Figure 96. Photo. Ikitsuki Bridge connecting Ikitsuki and Hirado Island in Japan.

FINITE-ELEMENT MODELING OF THE BRIDGE

The finite-element modeling approach for the Ikitsuki Bridge was similar to that for the I-35W 

Bridge using LS-DYNA and SAP2000. The FEM of the bridge in LS-DYNA included main side 

trusses, floor trusses, floor stringers, and secondary bracing members. All truss members, 

bracings, and stringers were modeled by B-S beam elements using the Mat 98 material model in 

LS-DYNA to account for both geometric and material nonlinearity and strain hardening in steel. 

Each element of the truss was modeled by a single beam element. In addition, the deck system 

was modeled by shell elements with elastic material for imposing DL of the concrete deck on the 

truss system. Overall, the LS-DYNA model had 3,230 beam elements and 128 shell elements. 
140 

The model in SAP2000 was similar to that in LS-DYNA except the equivalent nodal mass was 

assigned at deck stringers instead of shell elements. The deck in LS-DYNA was assigned a low 

Young’s modulus of elasticity so that the stiffness of the deck did not add to the stiffness of the 

truss (i.e., the deck was modeled just to transfer DL and LL from the deck to the truss). All beam 

elements in SAP2000 were assigned elastic material properties. The two trusses of the bridge are 

on the north and south sides, and traffic on the bridge is in east and west directions. Figure 97

shows the elevation, plan, and side views of the bridge. Piers P4, P5, and P7 of the bridge have 

roller supports, and pier P6 has fixed support. 

© 2018 Shozo Nakamura, Nagasaki University, Japan. 

1 mm = 0.0394 inches.

A. Elevation view. 

© 2018 Shozo Nakamura, Nagasaki University, Japan. 

1 mm = 0.0394 inches.

B. Plan view. 

© 2018 Shozo Nakamura, Nagasaki University, Japan. 

1 mm = 0.0394 inches.

C. Side view. 

Figure 97. Illustrations. Overall layout drawing of Ikitsuki Bridge (unit: mm). 
141 

Loads and Load Combinations 

Analysis of the two FEMs (SAP2000 and LS-DYNA) was performed for DL and LL. The 

seismic load scenario has not been investigated because the bridge was built in 1991 when 

seismic design criteria were well established. DL and LL were applied per load combinations in 

equations 8 and 9 in chapter 2. Although the bridge was designed according to Japanese code, 

the LL applied in the study follows the AASHTO LRFD specifications (2014). 

The deck system of the bridge is composed of a concrete-filled I-beam grid floor 0.16 m 

(0.63 inches) in thickness. The design compressive strength of the concrete was 24 MPa 

(3.48 ksi). The weight of steel in the entire bridge is approximately 64,200 kN (14,432.73 kips), 

which consists of the main structure (43,700 kN (9,824.15 kips)), floor system (8,780 kN 

(1973.82 kips)), and lateral bracings and cross frames (11,720 kN (2,634.76 kips)). Unit weights 

of steel and reinforced concrete are 77 kN/m3

 (494.58 lb/ft3) and 24.5 kN/m3 (157.37 lb/ft3), 

respectively. According to Kubota et al. (2004), the main truss members, accessories, and 

concrete deck weigh 64,200 kN (14,432.73 kips), 3,100 kN (696.91 kips), and 48,800 kN 

(10,970.68 kips), respectively. The sum of these is 113,000 kN (25,403.41 kips). In the FEM, the 

total weight of steel members was 88,948 kN (19,996.31 kips), and the weight of the concrete 

deck was 23,224 kN (5,220.96 kips). The sum of these is 112,172 kN (25,217.27 kips), which is 

approximately 0.7-percent different from the total weight calculated by Kubota et al. (2004). 

This comparison is shown in table 38. Table 39 shows the eight support reactions for different 

load combinations on both sides of the bridge. The LL is approximately 13.5 percent of the DL. 

The two supports in the central span carry approximately 96.2 percent of the DL. 

Table 38. DLs on the Ikitsuki Bridge.

Component

Kubota et al. (2004) 

(kN)

FEM 

(kN)

Steel members 64,200 88,948 

Accessories 3,100 Not applicable 

Concrete deck 48,800 23,224 

Sum 113,000 112,172 

1 kN = 0.225 kips.

Table 39. Support reactions for the Ikituski Bridge under different load combinations.

Truss Support

Boundary 

Condition 

Steel System 

Only

(kN)

DL

(kN)

DL + LL

(kN)

North side P4 Roller 848 1,340 1,658 

P5 Roller 21,393 26,783 30,261 

P6 Pin 21,386 26,777 30,254 

P7 Roller 847 1,339 1,656 

South side P4 Roller 848 1,340 1,658 

P5 Roller 21,393 26,783 30,261 

P6 Pin 21,386 26,777 30,254 

P7 Roller 847 1,339 1,656 

1 kN = 0.225 kips.
142 

Verification of the As-Built Bridge Model

Kubota et al. (2004) obtained the natural frequencies of the first 10 modes of the bridge using a 

finite-element formulation. Table 40 and table 41 show comparisons between the natural 

frequencies presented by Kubota et al. (2004) and those obtained from the SAP2000 and 

LS-DYNA models in this research, respectively. Figure 98 shows comparisons between mode 

shapes from Kubota et al. (2004) and those from the LS-DYNA model. The maximum difference 

between the natural frequencies from Kubota et al. (2004) and those from the SAP2000 and 

LS-DYNA models was approximately 1.67 and 2.16 percent, respectively. Mode shapes from the 

LS-DYNA model also matched very well to those from Kubota et al. (2004). Because the FEM

by Kubota et al. (2004) was based on as-built drawings and the models in this research are based 

on information received from Nagasaki University in developing the model in Kubota et al. 

(2004), the FEMs in SAP2000 and LS-DYNA represent the structural behavior of the Ikitsuki 

Bridge well. It should be noted that Kawabata et al. (2011) measured the natural frequencies of 

the first 10 modes of the bridge through ambient vibration. Measured frequencies are generally 

10-percent higher than those from FEMs because of the nonidealized condition of actual bridge

connections and components. If bridge-specific results are desired, a calibrated FEM based on

monitoring data may be more suitable for ALP analysis. However, updating the FEM using

monitoring data for such a long-span bridge is a complex optimization task and is beyond the

scope of this research.

Table 40. Comparison between natural frequencies of the Ikitsuki Bridge from the FEM by 

Kubota et al. (2004) and the SAP2000 model. 

Mode Number

Natural Frequency 

From FEM by 

Kubota et al. (2004) 

Natural Frequency 

From SAP2000 

Model

Difference

(%)

1 0.237 0.239 0.75 

2 0.352 0.356 1.04 

3 0.496 0.491 −1.05

4 0.584 0.576 −1.39

5 0.614 0.614 0.05 

6 0.630 0.621 −1.49

7 0.820 0.828 0.24 

8 0.882 0.867 −1.67

9 0.932 0.940 0.88 

10 1.036 1.034 −0.16
143 

Table 41. Comparison between natural frequencies of the Ikitsuki Bridge from the FEM by 

Kubota et al. (2004) and the LS-DYNA model.

Mode Number

Natural Frequency 

From FEM by 

Kubota et al. (2004) 

Natural Frequency 

From LS-DYNA 

Model

Difference

(%)

1 0.237 0.241 1.77 

2 0.352 0.359 2.07 

3 0.496 0.507 2.16 

4 0.584 0.592 1.38 

5 0.614 0.622 1.35 

6 0.630 0.635 0.72 

7 0.820 0.834 0.93 

8 0.882 0.877 −0.61 

9 0.932 0.947 1.59 

10 1.036 1.048 1.18 

Source: FHWA. 

Figure 98. Illustration. Comparison between mode shapes of the Ikitsuki Bridge based on 

Kubota et al. (2004) and the LS-DYNA model.
144 

BEHAVIOR OF THE AS-BUILT BRIDGE

Figure 99 shows DCR for the as-built bridge with a service load condition of DL + 1.0LL. In this 

figure, truss members with DCR less than 1.0 are shown in solid black lines and those with DCR

greater than 1.0 but less than 1.67 are shown in blue dotted lines. Stress in members shown in 

blue dotted lines is more than allowable stress under the service load condition DL + 1.0LL, 

probably because of differences in LL standards in the AASHTO LRFD specifications (2014) 

and Japanese design specifications. However, none of the members of the truss had DCR

exceeding 1.67. 

For a bridge that has been designed for service loads, DCR for all members must be less than 1.0. 

To study ALP in a well-designed bridge, retrofit of members with DCR greater than 1.0 was 

performed by member strengthening so that DCR of all members became less than 0.9. This 

model of the bridge was designated as the “Design Bridge.” The member strengthening to 

modify the as-built bridge into the Design Bridge resulted in an increase in the weight of the 

steel by 16.03 percent. Figure 100 shows the distribution of members in the bridge with 

retrofitted members indicated in red dotted lines. Although target DCR during member 

strengthening for this bridge was 0.9, DCR spectra for ALP analysis in the next section were

plotted for limit states of allowable (1.0) and yield (1.67) stress limits.
145 

Source: FHWA. 

Figure 99. Illustration. DCR spectra for the as-built bridge subjected to service loads.
146 

Source: FHWA. 

Figure 100. Illustration. Members of the as-built bridge retrofitted by member strengthening to obtain Design Bridge for the 

ALP analysis.
147 

ALP ANALYSIS OF THE DESIGN BRIDGE

For the Ikitsuki Bridge, ALP was investigated for only two cases of MR. After a series of trial 

and error studies, the most critical members in the Ikitsuki Bridge were found to be the upper 

chord members. Hence, two upper chord members on the east side were selected for removal: 

one near P6 (case 1 of MR) and the other in the central span (case 2 of MR). The MR analysis 

was performed in SAP2000 for DCR and in LS-DYNA for SR metrics. The MR analysis in 

SAP2000 was elastic static analysis and was performed by executing the SAP2000 model of the 

bridge after removing a target member from the model. DCR for all truss members was 

calculated from the results of this analysis. The MR analysis in LS-DYNA was dynamic and 

considered geometrical and material nonlinearities and stress hardening in steel.

ALP (MR) Analysis for Design Bridge: DCR Metric

Figure 101 and figure 102 show ALP in terms of DCR for MR cases 1 and 2, respectively. For 

MR case 1 in figure 101, the upper chord U98-U99 in the side span on the east side truss was 

removed. This led to load redistribution in the damaged system and increase in demand in truss 

members on the east side near the damaged zone; the increase in demand was much less 

significant in truss members on the opposite (west) side. As observed in the I-35W Bridge, 

damage (or increase in demand because of load redistribution) was concentrated around the 

damaged member. The increased demand in the upper brace system around the damaged panel 

indicates the three-dimensionality effect in load redistribution from one side truss to the other. If 

one imagines the lower chord and upper chord members acting as a force couple, the height of 

the FS can be considered the moment arm. Near central supports, tension in upper chord 

members and compression in lower chord members of the same panel create a resisting moment 

to balance the external moment produced by the DL and LL. Because of the long span of this 

continuous bridge, there were significant negative moments near central supports P5 and P6. 

Increasing the height of the FSs near these central supports means increasing the moment arm to 

control the force increase in the truss members. As shown in figure 101, there are five levels of 

braces with a total height of 45 m (147.64 ft) in the FS near the central supports. The height of 

the FSs is 19 m (62.34 ft) in the middle of the central span. Multiple levels of transverse braces 

provide lateral support to the vertical members to reduce the risks of buckling. In addition, the 

transverse braces can provide more torsional stiffness to the virtual hollow closed box section 

created by the two side trusses, the deck system, and the upper brace system. The enhanced 

torsional stiffness can ensure structural stability when some damage exists. For MR case 1, 

13 members had DCR greater than 1.67. 

Figure 102 shows the results for MR case 2, in which upper chord member U65-U66 near the 

center of the main span in the east side truss was removed. Figure 102 demonstrates that the loss 

of an upper chord member far from the supports has less impact than loss of a member near the 

supports (as in MR case 1 in figure 101). The number of members with DCR of more than 1.0 

decreased from 209 in MR case 1 to 145 in MR case 2. The impacted region was concentrated 

near the damaged panel. Both upper and lower braces nearby participated in the load 

redistribution. 

Figure 103 shows the complete envelope for the entire bridge considering the symmetry of the 

bridge in the west and east sides. This envelope was created by applying DCR results for the two 
148 

MR cases to other locations of the bridge (i.e., applying MR case 1 results to symmetric upper 

chord members near pier P5 on the east side truss and piers P5 and P6 on the west side truss, and 

applying MR 2 case results to a symmetric upper chord member near the central span in the west 

side truss). The boundary conditions of roller support and fixed support at P5 and P6, 

respectively, had a negligible effect on creating this envelope based on symmetric geometry of 

the truss. A total of 575 members exceeded the allowable stress. Among these members, 

68 members had DCR greater than 1.67 (i.e., these members yielded).
149 

Source: FHWA. 

Figure 101. Illustration. DCR spectra for the Design Bridge for upper chord MR (MR case 1).
150 

Source: FHWA. 

Figure 102. Illustration. DCR spectra for the Design Bridge for upper chord MR (MR case 2).
151 

Source: FHWA. 

Figure 103. Illustration. Complete envelope of DCR for the Design Bridge representing eight MR cases (two removal cases in 

each quarter of the bridge).
152 

ALP (MR) Analysis for Design Bridge: SR Metric

Figure 104 and figure 105 present the SR metric for two MR cases. Several members had SRs 

between 1.0 and 2.0 and between 2.0 and 4.0 limit states. These two limit states correspond to 

minor and moderate damage in the bridge, respectively. Similar to figure 103, figure 106 shows 

the envelope of SR metric for the entire bridge. Among all members of the bridge, 86 members 

had SR between 1.0 and 2.0, and 10 members had SR greater than 2.0 (i.e., these members 

yielded and had repairable damage). 
153 

Source: FHWA. 

Figure 104. Illustration. SR spectra for the Design Bridge for upper chord MR (MR case 1).
154 

Source: FHWA. 

Figure 105. Illustration. SR spectra for the Design Bridge for upper chord MR (MR case 2).
155 

Source: FHWA. 

Figure 106. Illustration. Complete envelope of SR for the Design Bridge representing eight MR cases (two removal cases in 

each quarter of the bridge).
156 

PERFORMANCE-BASED RETROFITS

Performance of Retrofits in Terms of DCR

Similar to the investigation of retrofit of the I-35W Bridge in chapter 5, the four retrofit 

strategies in table 28 were investigated for the two performance levels of DCR ≤ 1.0 and 

1.0 ≤ DCR ≤ 1.67 for the Ikitsuki Bridge. These four retrofit strategies are as follows: 

(1) member strengthening only, (2) member strengthening plus adding transverse braces for the 

FS at the supports, (3) member strengthening plus adding DMs at two panels near the supports, 

and (4) member strengthening plus adding members to the FS and DMs as described in 

strategies 2 and 3. Distribution of retrofitted members for these four retrofit strategies for 

performance level 1.0 ≤ DCR ≤ 1.67 is shown in figure 107 to figure 110. Because of the long 

span of the Ikitsuki Bridge, it is not cost-effective to add members along the entire length of the 

bridge. In this case, the weight of the bridge may increase significantly when extra diagonals and 

braces are added in the central span even though the ALP of the bridge may not improve 

significantly. On the other hand, adding members around the supports is effective in improving 

the ALP. Member strengthening is the most effective for members in the central span.
157 

Source: FHWA. 

Figure 107. Illustration. Member strengthening retrofit of the Design Bridge for MR based on 1.0 ≤ DCR ≤ 1.67 limit state.
158 

Source: FHWA. 

Figure 108. Illustration. Addition of floor truss members and member strengthening retrofit of the Design Bridge for MR

based on 1.0 ≤ DCR ≤ 1.67 limit state.
159 

Source: FHWA. 

Figure 109. Illustration. Addition of DMs and member strengthening retrofit of the Design Bridge for MR based on 

1.0 ≤ DCR ≤ 1.67 limit state.
160 

Source: FHWA. 

Figure 110. Illustration. Addition of floor truss members and DMs and member strengthening retrofit of the Design Bridge for 

MR based on 1.0 ≤ DCR ≤ 1.67 limit state.
161 

The DispR is calculated as the maximum of the maximum vertical displacements of the 

retrofitted bridge divided by the allowable deflection limit, as shown in table 42, for each retrofit 

strategy. For the Ikitsuki Bridge, the deflection limit is taken as span length ∕ 300 because the 

bridge can tolerate a much higher level of displacement than span length ∕ 800 without causing 

inelastic stress in any of the members. 

Table 42 shows the percentage increase in steel weight and DispR for eight retrofit cases (MR 

case 1 and case 2 for each of the four retrofit strategies). Members added in retrofit strategies 2, 

3, and 4 are shown in blue dotted lines in figure 108 to figure 110, respectively. Retrofit 

strategy 4 was again the most efficient in terms of additional weight increase, whereas the 

displacement performance was similar for all retrofit cases. The percentage increase in steel 

weight decreased from 15 to 9.1 percent for 1.0 ≤ DCR ≤ 1.67 and from 36.7 to 21.4 percent for 

DCR ≤ 1.0 for retrofit strategies 1 to 4, whereas DispRs were similar for all retrofit strategies. 

These results demonstrate that all retrofit strategies have approximately similar performance but 

retrofit strategy 4 is the most cost-effective because of the more significant three-dimensionality 

effect created by the addition of braces and DMs.

Table 42. Increase in the weight of steel and DispRs for different MR retrofit strategies 

based on DCR limit state.

DCR Value

MR Retrofit 

Strategy

Additional 

Increased Weight

(kN)

Increased 

Weight 

(%) DispR

≤1.67 1-1 15,595 15.0 0.62 

≤1.67 2-1 14,065 13.6 0.62 

≤1.67 3-1 11,735 11.3 0.63 

≤1.67 4-1 9,419 9.1 0.64 

≤1.0 1-2 38,061 36.7 0.61 

≤1.0 2-2 33,613 32.4 0.61 

≤1.0 3-2 28,196 27.2 0.63 

≤1.0 4-2 22,233 21.4 0.64 

1 kN = 0.225 kips.

Note: The total steel weight of the Base Bridge was 103,708 kN. 

Performance of Retrofits in Terms of SR

The complete envelope of the SR metric for the entire bridge in figure 106 indicates that 

10 members have SR ≥ 2.0. The increase in the weight of steel to change these members to

SR ≤ 2 is approximately 2 percent. Table 43 shows results for the four retrofit strategies when 

performance requirements are set in the elastic zone (corresponding to SR ≤ 1.0). Distribution of 

retrofitted members for these four cases of retrofits is similar to that for 1.0 ≤ DCR ≤ 1.67. 

Similar to observations from the I-35W Bridge and the DCR-based retrofit in table 43, the 

percentage of added steel weight decreases from 13 to 7 percent for retrofits 1 to 4. Retrofit 

strategy 4 is the most effective and the most economical because of the increased 

three-dimensionality from added members. This strategy also shows that both member 

strengthening in the central span and member additions near supports are effective in improving 

ALP of the bridge. 
162 

Table 43. Increase in the percentage weight of steel for different MR retrofit strategies 

based on DCR and SR limit states.

Performance 

Level Limit State

1: Member 

Strengthening 

Only

2: FS + 

Member 

Strengthening

3: DM + 

Member 

Strengthening

4: FS + DM + 

Member 

Strengthening

Elastic limit 

state

DCR ≤ 1.0 37 32 27 21 

Elastic limit 

state

1.0 < DCR ≤ 

1.67 

15 14 11 9 

Elastic limit 

state

SR ≤ 1.0 13 12 8 7 

Low ductility DCR ≤ 5.0 NA NA NA NA

Low ductility SR ≤ 2.0 2 NA NA NA

Medium 

ductility

SR ≤ 4.0 0 0 0 0 

NA = not applicable.
163 

CHAPTER 7. LABORATORY AND FIELD TESTING RECOMMENDATIONS

To investigate ALP and effective retrofit strategies of long-span truss bridges, several important 

assumptions were made. These assumptions were based on experience and the best judgment of 

the research team members. This chapter reviews some of those assumptions and makes 

recommendations for improving them via experimental and analytical verifications. All future 

experimental verifications should focus on three targets: system (full bridge) performance, 

component behavior, and joint performance (by considering connected components at a typical 

joint). 

SYSTEM BEHAVIOR

Modeling Methods

In chapter 3, the FEM used in this research was compared with well-known accurate baselines, 

and their similarity was verified and demonstrated. However, numerous factors might affect the 

overall system behavior during and after an MR event. The effects of those factors on the overall 

system need to be tested. Numerous truss bridges were built decades ago, and now they are 

suffering from aging and deterioration. Field tests could be performed before the demolition of 

one such bridge. For example, Azizinamini (2002) performed ultimate load tests on a steel truss 

bridge (the Rock Creek Bridge). Before demolition, the Rock Creek Bridge was disassembled by 

disconnecting the floor cross beams and moved to the laboratory of the University of 

Nebraska–Lincoln. To understand the behavior of an old steel truss bridge with forging tension 

members, tests were performed before and after the retrofit of the forged tension components. 

However, as mentioned in chapter 3, not many laboratory or field studies on the removal or 

fracture of members have been performed to investigate the ALP. Field testing of ALPs through 

destructive testing of decommissioned bridges can be an attractive solution to challenges 

involving progressive collapse testing in laboratories. For the full bridge test, testing scales 

should be carefully checked to balance the cost and accuracy.

Similar to the work done by Azizinamini (2002), full-scale bridge testing with appropriate 

validation by numerical processes for MR is recommended. This full-scale test can use a 

decommissioned bridge. A single member of such a bridge can be removed using mechanical 

means. Proper instrumentations for measuring resulting strains and displacements need to be part 

of the experiment. The loading on the bridge can be increased carefully while monitoring the 

bridge’s responses. Validation of different analytical modeling methods, including but not 

limited to the methods presented in this research, can be performed during this large-scale field 

test. This type of experiment will help to find optimal modeling processes and to identify any 

source of errors in such modeling techniques. 

After completion of the field study, a typical joint of the bridge should be carefully taken out and 

tested in the laboratory to understand the behavior and modeling of the joints of truss bridges. 

This exercise should be performed for as many typical truss joints as possible. If joints from 

decommissioned bridges are not available, typical joints in critical truss bridges should be 

fabricated in the laboratory based on original design details of the bridge to develop empirical 

models for these joints that can be used in FEMs. 
164 

ALP Quantification

The full-scale test can also be used to validate the ALP methodology presented in this research. 

This validation will require extensive sensor arrays throughout the bridge to keep track of the 

changes in load paths before and after MR. Component testing might be necessary to obtain 

properties of a single component, but it is not always sufficient to address the adequacy of ALPs. 

Because an ALP is a load path from the location of damage to the bridge supports, such a load 

path may need to be simulated in full. This would require modeling of a whole system, or most 

of the system, rather than a single component. 

COMPONENT BEHAVIOR

Joint Simulation 

In the physical bridge model, joint performance is not clearly understood because of the complex 

stresses involved around the joint region. In a typical truss bridge, a joint in a truss may connect 

compression and tension members in the longitudinal direction through gusset plates. Any 

damage to a truss member may cause significant bending stresses at the joint connecting the 

member with the gusset plate (Hao 2010). Numerous tests on gusset plates after the collapse of 

the I-35W Bridge have shown that the behavior of stresses in the gusset plates is very complex. 

This behavior could be even more complex at joints connecting transverse floor trusses with the 

main longitudinal truss. 

During finite-element modeling of the bridges in this research, it was assumed that joints of truss 

bridges are totally rigid and infinitesimal in size (i.e., their physical size was ignored in the FEM) 

and remain intact during the bridge’s response to hazards (e.g., MR). Even though these

assumptions are consistent with the current state of practice, their accuracies and effects on the 

retrofit results are not well documented. Therefore, a robust set of experimental and analytical 

tests are recommended to investigate all or some of these assumptions. Figure 111 illustrates 

finite-element representation of a physical joint.

Source: FHWA.

A. Physical joint. 

Source: FHWA. 

B. Analytical representation.

Figure 111. Illustrations. Representations of joints.

In two example bridges investigated in LS-DYNA, truss members were modeled by B-S beam 

elements connected through rigid joints. During the investigation of the collapse of the I-35W 

Bridge, Schultheisz et al. (2008) studied the global model of the bridge (consisting of beam and 

shell elements) by building a very detailed model of two affected joints (U10 and L11) to 

examine the fine details of stress distribution of these joints. The FHWA global model modeled 

truss members and gusset plates by shell elements, whereas the fine detailed model used 
165 

solid-element representations, which was partly motivated by a goal of considering stress 

concentrations associated with the riveted connections. This detailed model had approximately 

2.7 million DOFs, as illustrated in figure 112. For investigating alternate path in truss bridges, a 

similar study can be performed by developing a model of the bridge where truss elements can be 

modeled by B-S beam elements and truss joints in the region affected by sudden loss of a 

member can be modeled by either shell or solid elements. However, the level of model 

complexity and simulation time will be drastically higher in this case, although results are likely

to be a better representation of the behavior of joints after sudden loss of members.

Source: NTSB. 

Figure 112. Illustration. Detailed FHWA global model of the I-35W Bridge with fine 

detailed model of U10W and L11W joints (Schulthesiz et al. 2008). 

HAZARD SIMULATION

Single MR

This research provides a simple process for retrofitting bridges that are subject to single MR. The 

choice of the single member to be removed, as argued in previous chapters, should be based on 

the objective judgment of the bridge engineer. Currently, no objective process is available to aid 

bridge engineers in selecting the members to be removed. An obvious process would be to 

remove all members one by one. Such a process could be resource-intensive. In this research, 

both DCR and SR were used as objective criteria for MR. However, an optimal process for MR

that considers the risk exposure of different members in addition to quantities such as DCR or SR

is needed. Such an approach may be more practical and may aid bridge engineers in making 

efficient selections.
166 

Blast Simulation: ALP Modeling

An assumed blast’s effects on components of the bridge under certain conditions 

(i.e., simultaneous failure of two members) was simulated. For simplicity, such an assumption 

ignored the following: 

1. An exact location of the blast source. 

2. An exact weight of the blast event. 

3. The dynamic blast wave propagation that might affect nearby bridge components. 

The simplifying assumptions used in this research were justified by the limited resources of this 

project. However, the relationships between the blast source location, the blast weight, and the 

dynamic effects on bridge members and the resulting degradations of those members need 

detailed analytical investigation and experimental testing. The results of those tests can be 

documented in a monograph that can be used by bridge engineers to accurately predict damage 

in bridge members due to blast loading without having to model the blast event dynamically. 

Figure 113 shows a general illustration of such a monograph. The availability of such a 

monograph will not only improve the accuracy of the process described in this report but also 

save immense computational resources. Because of their sensitive nature, such monographs can 

be made available on a need-to-know basis. 

Source: FHWA. 

Figure 113. Graph. Typical blast damage monograph. 

Blast Simulation: Member Strengthening Modeling

Blast loading on bridges is frequently modeled based on simplifying assumptions. For example, 

the role of bridge decks on blast pressure distribution on structural members of the bridge 

systems is ignored, or the blast source is assumed to be on top of the bridge deck (I-35W Bridge) 

or within the bridge portal (Ikitsuki Bridge). 
167 

The first assumption is a conservative assumption because it ignores the energy dissipated by the 

damage to the bridge deck. It also ignores the role of the bridge deck in redistributing the blast 

pressure, as shown in figure 114-A. The second assumption ignores many other possible 

locations of the blast pressure, such as near a bridge support or directly under the bridge deck, as 

shown in figure 114-B. 

Source: FHWA.

A. Blast source near the deck.

Source: FHWA.

B. Blast source near the supports.

Figure 114. Illustrations. Illustration of blast load on a bridge. 

If bridge engineers find these simplifying assumptions to be either too conservative or too 

permissible, one possible solution would be to use a coupled direct blast-bridge analysis and 

design process. Such a process would be resource-intensive and difficult to perform. Another 

approach would be to perform physical tests supported by analytical experiments. These tests 

should be designed to address these assumptions and make it possible to use and modify the ALP 

process from this research to produce results that are more accurate.

PERFORMANCE-BASED DESIGN

Two alternate methodologies for the design or retrofit of bridges based on MR or DMR were

presented in chapters 5 and 6. For these abnormal demands, both linear static analysis and NLD

analysis processes were investigated. For the linear static process, DCR was used as the metric 

for acceptance limits, whereas for the NLD process, SR was used as the metric for acceptance 

limits. Different performance levels for each analysis process and metric, as shown in figure 91

(in chapter 5), have been provided. 

The acceptance limits used in figure 91 are based on the personal judgment and experience of the 

authors of this report. It has been shown that acceptance limits are related to acceptable risk by 

the stakeholders (Ettouney and Alampalli 2010a, b). Therefore, it is recommended to investigate 

the relationship between acceptable risk levels during MR and blast hazards and the different 

acceptance limits proposed in this research. Bridge engineers can then adjust those acceptance 

limits depending on desired acceptable risk levels for different bridges. 
168 

RETROFIT METHODS AND TECHNOLOGIES

During the retrofit processes described in chapters 5 and 6, manual adjustments of member size 

or shape were performed to reach the required acceptance limits. Retrofit options did not involve 

the use of advanced materials or technologies. Examples of advanced materials or technologies 

for retrofitting bridges include but are not limited to the use of fiberglass reinforced plastic (FRP) 

wrapping, FRP sections, and so forth (Ettouney and Alampalli 2010b). It is believed that using 

advanced materials and technologies in retrofits in addition to the ALP-based strategies 

illustrated in chapters 5 and 6 might result in even more cost-benefit results. The use of such a 

combined strategy is beyond the scope of this project. However, it is recommended as a future 

research and development effort.
169 

CHAPTER 8. CONCLUSIONS, LIMITATIONS, AND FUTURE WORK

SUMMARY AND CONCLUSIONS

An extensive investigation on ALP of long-span bridges in the event of loss of a member or 

members was performed for two example long-span truss bridges: the I-35W Bridge in 

Minneapolis, MN, which collapsed in 2007, and the Ikituski Bridge in Japan, which is the 

longest continuous steel truss bridge in the world. Although ALP has been defined qualitatively 

in the literature, a quantitative interpretation of ALP was developed in this research. Bridge 

designers commonly use DCR to design truss bridge members within the allowable stress limit, 

whereas they use SR as a limit between elastic and inelastic behaviors. Using these two design 

metrics, ALP is defined as the spectrum of the surrounding member undergoing load 

redistribution to prevent collapse after sudden damage to a member or members of a bridge 

because of abnormal events such as member loss or blast loading. Comparison of such spectra of 

bridge members before and after retrofit can also provide information on the effectiveness of a 

retrofit scheme.

The work performed in this research included calibration of long-span bridge models using 

available data in the literature, validation of the material model based on available analytical 

results, simulation of sudden MRs, and development of a performance-based methodology to 

retrofit the bridge against abnormal events, such as sudden loss of members. Key highlights and 

conclusions of this research are as follows: 

• Generally, investigation of sudden MR or progressive collapse is conducted by using 

either an elastic model or a very fine model of a truss bridge in software such as 

LS-DYNA, where elements are meshed using shell elements. The former approach limits 

the work to elastic behavior, whereas the latter approach is computationally intensive and 

not feasible for long-span bridges. In this research, the suitability of modeling a truss 

element using a B-S beam element in LS-DYNA was validated through several known 

examples. These validation examples as well as comparisons between models of bridges 

in SAP2000 and LS-DYNA demonstrated that the B-S element with material model Mat 

98 can be used to investigate progressive collapse of long-span bridges even when the 

bridge’s behavior becomes nonlinear. However, the B-S element with the Mat 98 

material model cannot simulate strength deterioration in the event of buckling. For 

long-span bridges, buckling of members can be evaluated separately by local or global 

buckling analysis before and after MR scenarios. A FEM in LS-DYNA using a B-S beam 

element and the Mat 98 material model can provide information on nonlinear behavior of 

the bridge in the event of MR and is computationally more efficient than models using 

shell elements to model the bridge elements.

• Simulation of blast loading on long-span bridges is very complicated, requires finer 

meshing of elements, and may require the use of software such as ConWep, which is 

labeled distribution statement C and may not be available to all bridge engineers freely. 

Hence, the effects of blast loading were simulated by the loss of two members 

simultaneously. More research is needed on quantifying the effects of blast loads on truss 

bridge members. 
170 

• Single and dual member removal analysis demonstrated that the bridge members that 

were not designed to carry structural loads, such as the lower and upper braces, actually 

participated in the redistribution of loads among members of trusses on two sides of the 

bridge and the transverse FS. However, the zone of affected members in the event of 

damage to a critical member was limited to the vicinity of the damaged member. After

damage to a critical member, geometrical asymmetry in the bridge tended to cause global 

torsional motion of the bridge about its longitudinal axis. This motion was resisted by 

(1) a virtual 3D geometrical box section (hollow closed boxed section) created by trusses 

on two sides together with the deck, stringers, and upper and lower braces, and (2) the 

FSs that act as diaphragms to supply rigidity to this hollow closed box section. These 

observations imply that the three-dimensionality of long-span truss bridges has a 

significant effect on their ability to survive after the loss of a critical member or 

members.

• In both the SAP2000 and LS-DYNA models, joints in truss bridges were modeled as 

rigid. Stresses in truss members connected through these joints were primarily axial 

because of DLs and LLs. However, in the event of sudden loss of a member or members, 

contributions of axial stress to DCR decreased significantly, whereas contributions of 

stresses due to in-plane and out-of-plane bending moments to DCR increased 

significantly. Hence, truss bridge analysis that considers truss members as primarily axial 

load-carrying members is not valid in situations of sudden member loss. In addition, 

simulations of the I-35W Bridge, which is a deck-truss bridge, showed that lower chord 

members were more dominant and effective in transferring loads to supports than upper 

chord members. On the other hand, upper chord members were more dominant in 

through-truss bridges, such as the Ikitsuki Bridge. 

• Generally, seismic retrofit of truss bridges is done either by isolating or dissipating input 

seismic energy through seismic isolation and energy dissipation devices or by 

strengthening deficient members (i.e., increasing the cross-sectional areas of members by 

adding an angle or cover plate or changing the sectional shape). Simulation results have 

shown that these seismic retrofits have a limited effect on the ALP of truss bridges after 

sudden loss of a member because of the way demands are transmitted during seismic and 

member loss events. During a seismic event, loads are transmitted from supports to the 

superstructure, where mitigation measures such as isolation, energy dissipation, and 

member strengthening are effective. However, during sudden member loss events, loads 

are transmitted from the superstructure to supports. In this case, load redistributions must 

occur in the superstructure before support components can participate in transmitting

loads to the ground. The three-dimensionality of the bridge cross section plays a 

significant role in this load redistribution within the superstructure.

• Another significant difference between demands because of seismic and sudden member 

loss events is the broadness of the affected zone or region. Demands during sudden 

member loss are limited to near the damaged member (because of the local initiation of 

the event and the short time duration (compared to earthquake excitations)) and affect 

much higher modes than those during seismic loads. Effects of seismic loads are global 

and affect the entire bridge system.
171 

• The effectiveness of retrofit measures, such as member strengthening and addition of 

extra members as diagonal or FSs and bracings, was investigated through simulation of 

both linear and nonlinear FEMs. Results showed that using member strengthening had 

only limited effectiveness in improving the ALP of the bridge, whereas retrofit measures 

that improved the three-dimensionality of the bridge (i.e., adding DMs, floor truss 

members, and upper or lower bracing members) were significantly more cost-effective in 

improving the ALP of the bridge while minimizing the increase in the weight of added 

steel (because of retrofit). These results indicated that retrofitting to improve ALP is an 

optimization problem where bridge truss members can be added in a certain orientation to 

improve performance during member loss while minimizing any increase in the weight of 

steel (i.e., cost). This type of retrofitting will also be effective in addressing seismic 

demands on the bridge. 

• A PBD approach, similar to the seismic design and retrofit of structures, was considered 

for retrofitting truss bridges to enhance their ALP. By accounting for the significance of 

the inelastic structural response, the overall performance of the structure is controlled in 

the PBD process. Two types of design criteria were considered for long-span steel truss 

bridges: DCR through elastic analysis and SR through inelastic analysis using an 

LS-DYNA model that modeled both geometrical and material nonlinearities. The 

increase in the weight of retrofit for the same level of performance (i.e., when 

DCR ≤ 1.67 and SR ≤ 1.0, which are both elastic limits) was less when analysis was 

performed using the inelastic model in LS-DYNA. This savings in costs was equivalent 

to approximately 2 percent less weight of added steel needed for the most effective 

retrofit strategy. However, overall savings in dollars will be much higher after factoring 

in labor and other costs (such as transportation) associated with 2-percent savings in steel 

weight. The savings in the weight of steel could be more if an optimization process is 

implemented when designing retrofits. Hence, performing inelastic analysis when 

designing retrofit strategies for ALP is recommended over linear elastic analysis.

• Compared with the single MR scenario, DMR (to represent blast loading) is significantly 

more severe. However, a real blast scenario may cause more damage to the bridge 

system. Further detailed investigation is needed on this issue. 

LIMITATIONS AND FUTURE WORK

The following represent some limitations of this research and possible suggestions for future 

work: 

• The joints simulated in the analytical models were assumed to be rigid and of 

infinitesimal size. In addition, joints were assumed to be intact during the bridge response 

to hazards. However, because the physical joints have more complicated stress behavior, 

the simulation of joints and the interaction of beam elements with the joint region should 

be investigated in detail. A dominating joint failure rather than a member failure may 

lead to collapse of the bridge. A possible compromise to detailed joint simulation may be 

to model selected critical truss joints and elements in the affected zone by finer shell 

elements to examine stress distribution while still modeling other joints and elements by 
172 

B-S elements. This approach could significantly optimize computational effort while 

significantly improving the reliability of the modeling. 

• When modeling the two example bridges, the deck system was modeled in LS-DYNA by 

shell elements with elastic material only for imposing DL of the concrete deck on the 

truss system. In SAP2000, the effect of the concrete deck was included as weight at truss 

joints (without explicit modeling of the deck). The Young’s modulus of the deck in 

LS-DYNA was reduced such that the displacement profile of the truss in LS-DYNA 

matched that in SAP2000. Hence, elastic behavior of the concrete deck itself did not 

affect the behavior of members affected by sudden loss of a critical member. However, a 

more realistic modeling of the deck by considering nonlinear concrete material 

(without reducing its Young’s modulus) should be considered in future studies to 

investigate the effect of the deck on load redistribution during sudden member loss. 

• The blast hazard simulated by a DMR scenario was only a crude approximation of the 

effect of member loss due to blast loads. More research, including laboratory testing 

(mentioned in chapter 7), is recommended to investigate the correlation between the blast 

load parameters (location, amount of TNT) and the number of members and number of 

bays that may be affected.

• Only four types of retrofit strategies were explored in the study, including member 

strengthening and addition of extra members (e.g., diagonal, FS, bracings), to improve 

ALP of the bridge. Among these four strategies, member strengthening only was found to 

be the least efficient approach. However, advanced materials or technologies can be used 

to design more efficient strategies than those investigated in this research. Other retrofit 

strategies that enhance the three-dimensionality of the bridge, such as several levels of 

transverse bracings near a central support or adding bracings for the DMs in the main 

truss girders, should also be investigated. 

• For long-span bridges, such as the Ikitsuki Bridge, retrofit of panels around supports was

found to be more effective than retrofit of members in the central span. Because of 

retrofit, stiffness around the supports was increased, which eventually helped to control 

the maximum displacement in the central span. However, further research is needed for 

other types of long-span bridges to develop prescriptive recommendations that can be 

used by bridge engineers.
173 

APPENDIX A. SR CALCULATION

3D BEAM ELEMENT TRANSFORMATION MATRIX

A 3D beam has two nodes (N1 (beginning node, X1, Y1, and Z1) and N2 (ending node, X2, Y2, and 

Z2)), with six DOFs, three translation displacements, and three rotation displacements in each 

node. Normally, a beam could be oriented either by its angles of roll or by means of the global 

coordinates of an arbitrarily chosen reference point that lies in one of the principal planes of the 

member but not on its centroid axis. In LS-DYNA, a reference node N3 (X3, Y3, and Z3) is 

defined. 

Displacement Vector of 3D Beam in Global Coordinates

3D beam displacements and rotations of the ith beam are defined as the vector in equation 17: 

(17)

Where the six displacement and rotational measures of the first and second nodes are {U}i1 and 

{U}i2. They can be defined as in equations 18 and 19: 

(18)

(19)

Note that ujk and θjk represent the displacement (in mm) and rotation (in rad) in the global axis 

for the jth node and kth global DOF, respectively. Note that for a two-node beam j = 1, 2 and for 

a 3D geometry k = X, Y, Z. The order of the vector {U}i1 and {U}i2 is 6 × 1, whereas the order of 

the vector {U}i is 12 × 1. 

Rotation Matrix From Global to Local 

Global displacements in the XYZ coordinate system (as shown in figure 115), as an output file of 

nonlinear finite-element calculation from LS-DYNA, are transformed into local rst (xyz) system 

by the member rotation matrix r determined in the following steps. 
174 

Source: FHWA. 

Figure 115. Illustration. 3D beam global and local coordinate specification. 

The position vector p, directed from member begin N1 to the reference node N3, lies in the local 

st (xy) plane and can be written as in equation 20: 

(20)

Here, IX, IY, and IZ are the unit vectors in the X, Y, and Z directions, respectively. Because the 

direction cosines of the local r(x) axis are already known from the global coordinates of the 

member ends, the direction cosines of the local t(z) axis (it) can be conveniently established 

using the following relationship: cross product of ir (the direction cosines of the local r (x) axis)

and p divided by the magnitude of the vector that results from the cross product, as shown in 

equation 21: 

(21)

With both now known, the direction cosines of the local s(y) (is) can be obtained via equation 22: 

(22)

Here, ir, is, and it are unit vectors in local x, y, and z directions. The member 3 × 3 rotation matrix 

r can now be obtained by arranging the components using the formula in equation 23: 
175 

(23)

The total beam transformation 12 × 12 matrix [T]i is formed as in equation 24: 

(24)

In which o represents a 3 × 3 null matrix.

Displacement Vector of 3D Beam in Local Coordinates

Thus, the nodal displacement or rotational vector in local coordinates is as in equation 25:

(25)

In which 3D beam local displacements and rotations of the ith beam are defined as the vector in 

equation 26: 

(26)

Where the six displacement and rotational measures of the first and second nodes are {U}i1 and 

{U}i2, and they can be defined as in equations 27 and 28: 

(27)

 (28)
176 

Strain-Displacement Relations

For Euler beam, strain due to bending about the local t-t axis can be expressed as in equation 29: 

 (29)

Similarly, strain due to bending about the local s-s axis can be expressed as in equation 30: 

(30)

Equations 29 and 30 assume that the t-t and s-s axes are the principal axes of inertia of the beam. 

The total bending strain along the beam is as in equation 31: 

(31)

The axial strain due to the axial deformation is as in equation 32: 

(32)

Note that L is the total length of the beam.

Combining the axial strain resulting from bending moments and axial force, the strain 

distribution in a general 3D beam is obtained from equation 33: 

(33)

Bending in the Local s-r, t-r Plane

The beam displacement is assumed as the cubic displacement function in equation 34: 

(34)

The constant αi in matrix [α] = [α1 α2 α3 α4] can be evaluated by applying displacement and 

rotation boundary conditions at the nodal ends of the beam as in equation 35: 
177 

(35)

The strains in the s-r plane are expressed as in equation 36: 

(36)

To express the strains in the s-r plane as a function of nodal displacements as in equation 37: 

(37)

Where [β] is calculated using equation 38: 

(38)

For bending in the t-r plane and using a similar process as before, strains can be evaluated as in 

equation 39: 

(39)

In general, the maximum strains occur at either end of the beam (i.e., r = 0 or r = L). Based on 

the methodology developed in the previous equations, strains are produced at four corner points 

for both ends of each beam element, as shown in figure 116. 
178 

Source: FHWA. 

Figure 116. Illustration. Output locations for strain for a beam section.

STRAINS FROM 3D BEAM ELEMENT INTERNAL FORCES CALCULATION

The other possible approach to verify the strain is a force-based method. For the type 2 resultant 

beam in LS-DYNA, the element forces and moments are output at the first node of the beam 

connectivity. Based on the beam equilibrium, the second nodal forces are yielded. The B-S beam 

uses the classical Timoshenko beam formulation. Thus, for the internal forces of B-S beam 

element formulation, the functional forms relating the overall response of the beam are adopted 

(e.g., moment–curvature relations). 

Axial Force

The internal axial force (F1r) can be calculated from the elongation (δ) of the beam and the axial 

stiffness (Ka

), as in equation 40: 

(40)

Ka can be calculated by equation 41: 

(41)

Where L is the original length of the beam. 

Bending Moments

Based on the deformation rotations, the bending moments are obtained by equations 42 and 43: 

(42)

(43)
179 

Equation 42 is for bending moments M1s and M2s in the local r-t plane, and equation 43 is for 

bending moments M1t and M2t in the r-s plane. The bending constants Ks

b

 and Kt

b

 and shape 

factors φs and φt can be determined by equations 44, 45, 46, and 47, respectively: 

 (44)

(45)

(46)

(47)

Where: 

Ks

b = flexural stiffness in the s direction. 

Kt

b = flexural stiffness in the t direction. 

φs = shear factor in the s direction. 

φt = shear factor in the t direction. 

As = effective area in shear in the s direction. 

At = effective area in shear in the t direction. 

Is = moment of inertia about the s-axis. 

It = moment of inertia about the t-axis. 

Torsional Moment

The torsional moment is produced based on the torsional deformation rotation (θ1r) and the 

torsional stiffness (Kt

) as in equation 48: 

(48)

Kt can be calculated by equation 49: 

(49)

Where Jt is the torsion constant for the section. 

The remaining internal force components can be found from beam equilibrium.

181 

APPENDIX B. CROSS-SECTIONAL AREA OF TRUSS MEMBERS FOR THE 

I-35W BRIDGE

Table 44 through table 47 detail the cross-sectional areas of truss members for the I-35W Bridge. 

Table 44. Cross-sectional area of upper chord members for I-35W Bridge model. 

Upper Chord Member Cross-Sectional Area (m2) 

U0-U1 3.55 × 10−2

U1-U2 3.55 × 10−2

U2-U3 4.90 × 10−2

U3-U4 4.90 × 10−2

U4-U5 3.55 × 10−2

U5-U6 3.55 × 10−2

U6-U7 7.16 × 10−2

U7-U8 7.16 × 10−2

U8-U9 6.48 × 10−2

U9-U10 6.48 × 10−2

U10-U11 4.90 × 10−2

U11-U12 4.90 × 10−2

U12-U13 9.13 × 10−2

U13-U14 9.13 × 10−2

U14-U15 9.13 × 10−2

U15-U16 9.13 × 10−2

U16-U17 4.90 × 10−2

U17-U18 4.90 × 10−2

U18-U19 6.48 × 10−2

U19-U20 6.48 × 10−2

U20-U21 7.16 × 10−2

U21-U22 7.16 × 10−2

U22-U23 3.55 × 10−2

U23-U24 3.55 × 10−2

U24-U25 4.90 × 10−2

U25-U26 4.90 × 10−2

U26-U27 4.68 × 10−2

U27-U28 4.68 × 10−2

1 m2 = 10.764 ft2. 
182 

Table 45. Cross-sectional area of lower chord members for I-35W Bridge model. 

Lower Chord Member Cross-Sectional Area (m2) 

L1-L2 4.29 × 10−2

L2-L3 4.29 × 10−2

L3-L4 4.68 × 10−2

L4-L5 4.68 × 10−2

L5-L6 6.03 × 10−2

L6-L7 6.03 × 10−2

L7-L8 1.16 × 10−1

L8-L9 1.16 × 10−1

L9-L10 4.68 × 10−2

L10-L11 4.68 × 10−2

L11-L12 6.48 × 10−2

L12-L13 6.48 × 10−2

L13-L14 8.97 × 10−2

L14-L15 8.97 × 10−2

L15-L16 6.48 × 10−2

L16-L17 6.48 × 10−2

L17-L18 4.68 × 10−2

L18-L19 4.68 × 10−2

L19-L20 1.16 × 10−1

L20-L21 1.16 × 10−1

L21-L22 6.03 × 10−2

L22-L23 6.03 × 10−2

L23-L24 4.68 × 10−2

L24-L25 4.68 × 10−2

L25-L26 4.29 × 10−2

L26-L27 4.29 × 10−2

1 m2 = 10.764 ft2. 
183 

Table 46. Cross-sectional area of vertical members for I-35W Bridge model. 

Vertical Member Cross-Sectional Area (m2) 

U1-L1 2.74 × 10−2

U2-L2 1.45 × 10−2

U3-L3 2.74 × 10−2

U4-L4 1.45 × 10−2

U5-L5 2.90 × 10−2

U6-L6 1.45 × 10−2

U7-L7 3.26 × 10−2

U8-L8 1.25 × 10−1

U9-L9 3.26 × 10−2

U10-L10 1.45 × 10−2

U11-L11 2.74 × 10−2

U12-L12 1.45 × 10−2

U13-L13 2.74 × 10−2

U14-L14 1.45 × 10−2

U15-L15 2.74 × 10−2

U16-L16 1.45 × 10−2

U17-L17 2.74 × 10−2

U18-L18 1.45 × 10−2

U19-L19 3.26 × 10−2

U20-L20 1.25 × 10−1

U21-L21 3.26 × 10−2

U22-L22 1.45 × 10−2

U23-L23 2.90 × 10−2

U24-L24 1.45 × 10−2

U25-L25 2.74 × 10−2

U26-L26 1.45 × 10−2

U27-L27 2.74 × 10−2

1 m2 = 10.764 ft2. 
184 

Table 47. Cross-sectional area of DMs for I-35W Bridge model. 

Diagonal Member Cross-Sectional Area (m2) 

U0-L1 3.95 × 10−2

L1-U2 5.16 × 10−2

U2-L3 1.79 × 10−2

L3-U4 2.29 × 10−2

U4-L5 4.19 × 10−2

L5-U6 3.72 × 10−2

U6-L7 6.65 × 10−2

L7-U8 4.77 × 10−2

U8-L9 5.98 × 10−2

L9-U10 9.00 × 10−2

U10-L11 5.68 × 10−2

L11-U12 6.65 × 10−2

U12-L13 3.72 × 10−2

L13-U14 3.18 × 10−2

U14-L15 3.18 × 10−2

L15-U16 3.72 × 10−2

U16-L17 6.65 × 10−2

L17-U18 5.68 × 10−2

U18-L19 9.00 × 10−2

L19-U20 5.98 × 10−2

U20-L21 4.77 × 10−2

L21-U22 6.65 × 10−2

U22-L23 3.72 × 10−2

L23-U24 4.19 × 10−2

U24-L25 2.29 × 10−2

L25-U26 1.93 × 10−2

U26-L27 5.16 × 10−2

L27-U28 5.74 × 10−2

1 m2 = 10.764 ft2. 
185 

APPENDIX C. THE CALCULATION OF DCR FOR COMPRESSION AND TENSION 

MEMBERS

COMPRESSION MEMBER

To illustrate the calculation of DCR, a typical compression member (member L8-L9 of the 

I-35W Bridge) is used as an example. This member has a box section, and the corresponding 

cross-sectional properties and demand are shown in table 48 and table 49, respectively. The 

detailed calculation of DCR for this member can be expressed as follows in equation 50 through 

equation 56: 

(50)

(51)

(52)

(53)

(54)

(55)

(56)

Where:

Fe = Euler stress.

rg = radius of gyration. 

Fcr = critical stress.

Pn = nominal axial force using ASD. 
186 

Table 48. Cross-sectional properties of the compression member.

Parameter Value 

Section area (Ac) 0.1161 m2

Sectional moment of inertia about 

x-axis (Sxx) 

0.0206 m3

Sectional moment of inertia about 

y-axis (Syy) 

0.0160 m3

Member length (L) 11.880 m 

1 m = 3.281 ft; 1 m2 = 10.764 ft2; 1 m3 = 35.315 ft3. 

Table 49. Demand of the compression member.

Parameter Value 

Axial force (Pr) 1.539 × 106 N 

Moment about x-axis (Mrx) 285.5 N·m 

Moment about y-axis (Mry) 1.075 × 106 N·m 

1 N = 0.225 lb; 1 N·m = 0.738 lb·ft. 

TENSION MEMBER

Likewise, the calculation of DCR for a typical tension member (member L11-L12 of the I-35W 

Bridge) is detailed in this section. This member has a box section, and its cross-sectional 

properties and demand are shown in table 50 and table 51, respectively. The detailed calculation 

of DCR for this member can be expressed as follows in equation 57 through equation 60: 

(57)

(58)

(59)

(60)

Table 50. Cross-sectional properties of the tension member.

Parameter Value 

Section area (Ac) 0.06484 m2

Sectional moment of inertia about 

x-axis (Sxx) 

0.0133 m3

Sectional moment of inertia about 

y-axis (Syy) 

0.0105 m3

Member length (L) 11.597 m 

1 m = 3.281 ft; 1 m2 = 10.764 ft2; 1 m3 = 35.315 ft3. 
187 

Table 51. Demand of the tension member.

Parameter Value

Axial force (Pr) 7.564 × 106 N 

Moment about x-axis (Mrx) 1.262 × 104 N·m 

Moment about y-axis (Mry) 1.131 × 105 N·m 

1 N = 0.225 lb; 1 N·m = 0.738 lb·ft. 

189 

ACKNOWLEDGMENTS

The authors sincerely acknowledge the help of Professor Shozo Nakamura from Nagasaki 

University (Japan) with the FEM of the Ikitsuki Bridge.

191 

REFERENCES

AASHTO. (2010). AASHTO LRFD Bridge Design Specifications, 5th Ed., American Association 

of State Highway and Transportation Officials, Washington, DC. 

AASHTO. (2012). AASHTO LRFD Bridge Design Specifications, 6th Ed., American Association 

of State Highway and Transportation Officials, Washington, DC. 

AASHTO. (2014). AASHTO LRFD Bridge Design Specifications, 7th Ed. American Association 

of State Highway and Transportation Officials, Washington, DC. 

AASHTO. (2017). AASHTO LRFD Bridge Design Specifications, 8th Ed. American Association 

of State Highway and Transportation Officials, Washington, DC. 

AASHTO. (2018a). Analysis and Identification of Fracture Critical Members and System 

Redundant Members, American Association of State Highway and Transportation 

Officials, Washington, DC. 

AASHTO. (2018b). Guide Specifications for Internal Redundancy of Mechanically-Fastened 

Built-up Steel Members. American Association of State Highway and Transportation 

Officials, Washington, DC. 

Agrawal, A.K., Ghosn, M., Alampalli, S., and Pan, Y. (2012). “Seismic Fragility of Retrofitted 

Multispan Continuous Steel Bridges in New York.” Journal of Bridge Engineering,

17(4), pp. 562–575, American Society of Civil Engineers, Reston, VA. 

Agrawal, A.K. and Amjadian, M. (2016). “Seismic Component Devices,” in Innovative Bridge 

Design Handbook: Construction, Rehabilitation and Maintenance, edited by Alessio

Pipinato, pp. 531–553, Butterworth-Heinemann, Boston, MA.

AISC. (2010). Specifications for Structural Steel Buildings, American Institute of Steel 

Construction, Chicago, IL. 

Arasaratnam, P., Sivakumaran, K.S., and Tait, M.J. (2011). “True Stress-True Strain Models for 

Structural Steel Elements.” ISRN Civil Engineering, pp. 1–11, Hindawi Publishing 

Corporation, London, UK. 

ASCE/SEI 7-10. (2010). Minimum Design Loads for Buildings and Other Structures. American 

Society of Civil Engineers, Reston, VA. 

Astaneh-Asl, A. (2008). Progressive Collapse of Steel Truss Bridges, the Case of I-35W 

Collapse. In Proceedings of the 7th International Conference on Steel Bridges, 

Guimarăes, Portugal.

Azizinamini, A. (2002). “Full Scale Testing of Old Steel Truss Bridge.” Journal of 

Constructional Steel Research, 58(5–8), pp. 843–858, Elsevier, Amsterdam, Netherlands. 
192 

Barth, K., Michaelson, G., and Gonano, D. (2014). “Assessment of Redundancy Protocols for 

Short-Span Steel Truss Bridges.” Bridge Structures, 10(2–3), pp. 105–114, IOS Press, 

Amsterdam, Netherlands. 

Beauchamp, J.C., Chan, M.Y.T., and Pion, R.H. (1984). “Repair and Evaluation of a Damaged 

Truss Bridge-Lewes, Yukon River.” Canadian Journal of Civil Engineering, 11(3), 

pp. 494–504, Canadian Science Publishing, Ottawa, Canada. 

Belytschko, T., Schwer, L., and Klein, M.J. (1977). “Large Displacement, Transient Analysis of 

Space Frames.” International Journal for Numerical Methods in Engineering, 11(1), 

pp. 65–84, John Wiley & Sons, Hoboken, NJ. 

Blandford, G.E. (1996). “Large Deformation Analysis of Inelastic Space Truss Structures.” 

Journal of Structural Engineering, 122(4), pp. 407–415, American Society of Civil 

Engineers, Reston, VA. 

Blandford, G.E. (1997). “Review of Progressive Failure Analyses for Truss Structures.” Journal 

of Structural Engineering, 123(2), pp. 122–129, American Society of Civil Engineers, 

Reston, VA. 

Blanksvärd, T., Häggström, J., Nilimaa, J., Sabourova, N., Grip, N., Täljsten, B., Elfgren, L., 

Carolin, A., Paulsson, B., and Tu, Y. (2014). “Test to Failure of a Steel Truss Bridge: 

Calibration of Assessment Methods,” in Bridge Maintenance, Safety, and Life Extension, 

edited by Arong Chen, Dan Frangopol, and Xin Ruan, pp. 1076–1081, CRC Press, Taylor 

& Francis Group, London, UK. 

Casciati, F., Cimellaro, G.P., and Domaneschi, M. (2008). “Seismic Reliability of a CableStayed Bridge Retrofitted with Hysteretic Devices.” Computers & Structures, 86(17–18), 

pp. 1769–1781, Elsevier, Amsterdam, Netherlands. 

Cha, H., Lyrenmann, L., Connor, R.J., and Varma, A.H. (2014). “Experimental and Numerical 

Evaluation of the Post-Fracture Redundancy of a Simple Span Truss Bridge.” Journal of 

Bridge Engineering, 19(11), American Society of Civil Engineers, Reston, VA. 

Chen, X., Ge, H., and Usami, T. (2011). “Seismic Demand of Buckling-Restrained Braces 

Installed in Steel Arch Bridges under Repeated Earthquakes.” Journal of Earthquake and 

Tsunami, 05(02), pp. 119–150, World Scientific, Singapore. 

Cho, H.N., Lim, J.K., and Choi, H.H. (2001). “Reliability-Based Fatigue Failure Analysis for 

Causes Assessment of a Collapsed Steel Truss Bridge.” Engineering Failure Analysis, 

8(4), pp. 311–324, Elsevier, Amsterdam, Netherlands. 

Chopra, A.K. (2000). Dynamics of Structures: Theory and Applications, Pearson, Hoboken, NJ. 

Computers and Structures, Inc. (2014). SAP2000, Version 17.0, Berkeley, CA. 

CSA. (2006). Canadian Highway Bridge Design Code CAN/CSA-S6-06, Canadian Standards 

Association, Toronto, Ontario, Canada. 
193 

Csagoly, P.F. and Jaeger., L.G. (1979). “Multi-Load-Path Structures for Highway Bridges.” 

Transportation Research Record 711, pp. 34–39, National Academy of Sciences, 

Washington, DC. 

Dassault, S. (2013a). ABAQUS 6.13 Abaqus/CAE User’s Guide, Dassault Systemes Simulia, 

Providence, RI. 

Dassault, S. (2013b). ABAQUS 6.13 Theoretical Manual, Dassault Systemes Simulia, 

Providence, RI. 

DataHub. (n.d.). “Steel Bridge Database.” (website). Available online: 

http://datacenterhub.org/dv_dibbs/view/1024:dibbs/experiments_dv, last accessed

May 15, 2020. 

Davalos, J. F. (1989). Geometrically Nonlinear Finite Element Analysis of a Glulam Timber 

Dome, PhD thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA. 

DelGrego, M., Culmo, M., and DeWolf, J. (2008). “Performance Evaluation Through Field 

Testing of Century-Old Railroad Truss Bridge.” Journal of Bridge Engineering, 13(2), 

pp. 132–138, American Society of Civil Engineers, Reston, VA. 

DOD. (2009). Unified Facilities Criteria: Design of Buildings to Resist Progressive Collapse, 

Report No. UFC4-023-03, United States Department of Defense, Washington, DC. 

Diggelmann, L.M., Connor, R.J., and Sherman, R.J. (2013). Evaluation of Member and 

Load-Path Redundancy on the US-421 Bridge over the Ohio River, Federal Highway 

Administration, Washington, DC. 

Dyke, S.J., Caicedo, J.M., Turan, G., Bergman, L.A., and Hague, S. (2003). “Phase I Benchmark 

Control Problem for Seismic Response of Cable-Stayed bridges.” Journal of Structural 

Engineering, 129(7), pp. 857–872, American Society of Civil Engineers, Reston, VA. 

El-Bahey, S. and Bruneau, M. (2011). “Buckling Restrained Braces as Structural Fuses for the 

Seismic Retrofit of Reinforced Concrete Bridge Bents.” Engineering Structures, 33(3), 

pp. 1052–1061, Elsevier, Amsterdam, Netherlands. 

El-Tawil, S., Khandelwal, K., Kunnath, S., and Lew, H. S. (2007). Macro Models for 

Progressive Collapse Analysis of Steel Moment Frame Buildings. Presented at Structures 

Congress 2007, American Society of Civil Engineers, Long Beach, CA. 

Ellingwood, B.R. (2006). “Mitigating Risk from Abnormal Loads and Progressive Collapse.” 

Journal of Performance of Constructed Facilities, 20(4), pp. 315–323, American Society 

of Civil Engineers, Reston, VA. 

Ellingwood, B.R. and Dusenberry, D.O. (2005). “Building Design for Abnormal Loads and 

Progressive Collapse.” Computer-Aided Civil and Infrastructure Engineering, 20(3), 

pp. 194–205, Wiley, Hoboken, NJ. 
194 

Ellingwood, B.R., Smilowitz, R., Dusenberry, D.O., Duthinh, D., Lew, H.S., and Carino, N.J. 

(2007). Best Practices for Reducing the Potential for Progressive Collapse in Buildings, 

Report No. NISTIR 7396, National Institute of Standards and Technology, 

Gaithersburg, MD. 

Ettouney, M., Smilowitz, R., Tang, M., and Hapij, A. (2006). “Global System Considerations for 

Progressive Collapse with Extensions to Other Natural and Man-Made Hazards.” Journal 

of Performance of Constructed Facilities, 4(20), American Society of Civil Engineers, 

Reston, VA.

Ettouney, M. and Alampalli, S. (2010a). Infrastructure Health in Civil Engineering, Volume I: 

Theory and Components, CRC Press, Boca Raton, FL. 

Ettouney, M. and Alampalli, S. (2010b). Infrastructure Health in Civil Engineering, Volume II: 

Applications and Management, CRC Press, Boca Raton, FL. 

Ettouney, M. and Alampalli, S. (2014). Progressive Collapse Principles of Bridge Structures.

Presented at the 10th International Conference on Shock and Impact Loads on Structures, 

Singapore. 

Ettouney, M. and Alampalli, S. (2016). Risk Management in Civil Infrastructures, CRC Press, 

Boca Raton, FL. 

FHWA. (n.d.). “National Bridge Inventory.” (website). Available online: 

https://www.fhwa.dot.gov/bridge/nbi.cfm, last accessed May 14, 2020. 

FHWA. (2013). TechBrief: Evaluation of Member and Load Path Redundancy on the US-421 

Bridge over the Ohio River, Report No. FHWA-HRT-13-105, Federal Highway 

Administration, Washington, DC. 

FHWA. (2015). Load and Resistance Factor Design (LRFD) for Highway Bridge 

Superstructures, NHI Course Nos. 130081, 130081A, and 130081B, Federal Highway 

Administration, Washington, DC. 

Fiorillo, G., Miao, F., and Ghosn, M. (2015). “Direct Redundancy Evaluation of Bridges 

Designated as Fracture-Critical.” Journal of Performance of Constructed Facilities, 

30(3), American Society of Civil Engineers, Reston, VA. 

Fisher, J.W., Pense, A.W., and Roberts, R. (1977). “Evaluation of Fracture of Lafayette Street 

Bridge.” Journal of the Structural Division, 103(ST7), pp. 1339–1357, American Society 

of Civil Engineers, Reston, VA. 

Frangopol, D.M. and Curley, J.P. (1987). “Effect of Damage and Redundancy on Structural 

Reliability.” Journal of Structural Engineering, 113(7), pp. 1533–1549, American 

Society of Civil Engineers, Reston, VA. 
195 

Frangopol, D.M. and Imai, K. (2004). “Reliability of Long Span Bridges Based on Design 

Experience with the Honshu-Shikoku Bridges.” Journal of Constructional Steel 

Research, 60(3–5), pp. 373–392, Elsevier, Amsterdam, Netherlands. 

Golden Gate Bridge Authority. (2014). “Golden Gate Bridge Seismic Retrofit Project.” 

Available online: https://www.goldengate.org/assets/1/6/bridge-projects-seismic-retrofitschematic.pdf, last accessed April 9, 2020. 

Grub, M.A., Wilson, K.E., White, C.D., and Nickas, W.N. (2015). Load and Resistance Factor 

Design (LRFD) for Highway Bridge Superstructures–Reference Manual, Report No. 

FHWA-NHI-15-047, Federal Highway Administration, Washington, DC. 

GSA. (2003). Progressive Collapse Analysis and Design Guidelines for New Federal Office 

Buildings and Major Modernization Projects, General Services Administration, 

Washington, DC. 

GSA. (2016). Progressive Collapse Analysis and Design Guidelines for Progressive Collapse 

Resistance, General Services Administration, Washington, DC. 

Hallquist, J.O. (2014). LS-DYNA Keyword User’s Manual, Version 971, Livermore Software 

Technology Corporation, Livermore, CA. 

Hao, S. (2010). “I-35W Bridge Collapse.” Journal of Bridge Engineering, 15(5), pp. 608–614, 

American Society of Civil Engineers, Reston, VA. 

Higgins, C., Dusicka, P., and Scott, M. (2012). Experimental Tests and Numerical Analyses of 

Steel Truss Bridge Gusset Connections, Report No. OTREC-RR-12-03, Oregon 

Transportation Research and Education Consortium (OTREC), Portland, OR. 

Hill, C.D., Blandford, G.E., and Wang, S.T. (1989). “Post-Buckling Analysis of Steel Space 

Trusses.” Journal of Structural Engineering, 115(4), pp. 900–919, American Society of 

Civil Engineers, Reston, VA. 

Imbsen, R. and Liu. W. (1993). Seismic Evaluation of Benicia-Martinez Bridge. Presented at the 

First US Seminar on Seismic Evaluation and Retrofit of Steel Bridges, University of 

California at Berkeley.

Imbsen, R. and Schamber, R. (1999). “Seismic Retrofit of the North Approach Viaduct of the 

Golden Gate Bridge.” Transportation Research Record 1688, pp. 154–162, National 

Academy of Sciences, Washington, DC. 

Izzuddin, B.A., Vlassis, A.G., Elghazouli, A.Y., and Nethercot, D.A. (2008). “Progressive 

Collapse of Multi-Storey Buildings Due to Sudden Column Loss - Part I: Simplified 

Assessment Framework.” Engineering Structures, 30(5), pp. 1308–1318, Elsevier, 

Amsterdam, Netherlands. 
196 

Jelinek, J.J. and Bartlett, F. M. (2002). “Material Properties of Historical Bridge Steel,” in 

Proceedings of the 30th Annual Conference of the Canadian Society of Civil 

Engineering, pp. 1313–1322, Canadian Society for Civil Engineering, Montreal, Canada. 

Kanaji, H., Kitazawa, M., and Suzuki, N. (2003). Seismic Retrofit Strategy Using Damage 

Control Design Concept and the Response Reduction Effect for a Long-Span Truss 

Bridge. Presented at the 19th US–Japan Bridge Engineering Workshop Panel on Wind 

and Seismic Effects, US–Japan Cooperative Program in Natural Resources, Tsukuba 

Science City, Japan.

Kandemir, E.C., Mazda, T., Nurui, H., and Miyamoto, H. (2011). “Seismic Retrofit of an 

Existing Steel Arch Bridge Using Viscous Damper.” Procedia Engineering, 14, 

pp. 2301–2306, Elsevier, Amsterdam, Netherlands. 

Kawabata, S., Koga, T., Takahashi, K., Nakamura, S., and Wu, Q. (2011). “Verification of 

Analytical Model of the Ikitsuki Bridge by Ambient Vibration Measurement.” 

Proceedings of Constructional Steel, 19(11), pp. 1–6, Japanese Society of Steel 

Construction, Tokyo. 

Kingery, C.N. and Bulmash, G. (1984). Airblast Parameters from TNT Spherical Air Burst and 

Hemispherical Surface Burst, Report ARBRL-TR-02555, Army Ballistic Research

Laboratory, Aberdeen Proving Ground, MD. 

Kubota, K., Inuzuka, H., Wu, Q., Takahashi, K., and Nakamura, S. (2004). “Natural Vibration 

and Nonlinear Seismic Response Properties of Continuous Truss Bridge.” Journal of 

Constructional Steel Research, 12(11), pp. 1–8. 

Lee, G.C., Mohan, S., Huang, C., and Fard, B.N. (2013). A Study of US Bridge Failures 

(1980–2012), Technical Report 13-0008, Multidisciplinary Center for Earthquake 

Engineering Research (MCEER), Buffalo, NY. 

Liao, M., Okazaki, T., Ballarini, R., Schultz, A.E., and Galambos, T.V. (2009). Analysis of 

Critical Gusset Plates in the Collapsed I-35W Bridge, Structures Congress 2009, 

American Society of Civil Engineers, Reston, VA. 

Liao, M., Okazaki, T., Ballarini, R., Schultz, A.E., and Galambos, T.V. (2011). “Nonlinear 

Finite-Element Analysis of Critical Gusset Plates in the I-35W Bridge in Minnesota.” 

Journal of Structural Engineering, 137(1), pp. 59–68, American Society of Civil 

Engineers, Reston, VA. 

Lichtenstein, A.G. (1993). “The Silver Bridge Collapse Recounted.” Journal of Performance of 

Constructed Facilities, 7(4), pp. 249–261, American Society of Civil Engineers, 

Reston, VA. 

Liu, S., Bartlett, F.M., and Zhou, W. (2013). “Alternative Load Paths in Steel Through-Truss 

Bridges: A Case Study.” Journal of Bridge Engineering, 18(9), pp. 920−928, American 

Society of Civil Engineers, Reston, VA. 
197 

Liu, W. D., Nobari, F.S., Schamber, R.A., and Imbsen, R.A. (1997). “Performance Based 

Seismic Retrofit of Benicia-Martinez Bridge,” in Proceedings of the 2nd National 

Seismic Conference on Bridges and Highways, pp. 509–523, Sacramento, CA. 

Lwin, M. (2012). Clarification of Requirements for Fracture Critical Members, FHWA 

Memorandum HIBT 10, Federal Highway Administration, Washington, DC. 

Matson, D.D. and Buckland, P.G. (1995). “Experience with Seismic Retrofit of Major Bridges,” 

in Proceedings of the 1st National Seismic Conference on Bridges and Highways, San 

Diego, CA. 

Mueller, W.H. and Wagner, A.L. (1984). Plastic Behavior of Steel Angle Columns, Research 

Report, Civil/ Structural Engineering, Portland State University, Portland, OR. 

Murtha-Smith, E. (1988). “Alternate Path Analysis of Space Trusses for Progressive Collapse.” 

Journal of Structural Engineering, 114(9), pp. 1978–1999, American Society of Civil 

Engineers, Reston, VA. 

Mustafa, S., Matsumoto, Y., and Yamaguchi, H. (2018). “Vibration-Based Health Monitoring of 

an Existing Truss Bridge Using Energy-Based Damping Evaluation.” Journal of Bridge 

Engineering, 23(1), American Society of Civil Engineers, Reston, VA. 

Nagavi, R.S. and Aktan, A. E. (2003). “Nonlinear Behavior of Heavy Class Steel Truss Bridges.” 

Journal of Structural Engineering, 129(8), pp. 1113–1121, American Society of Civil 

Engineers, Reston, VA. 

Nakamura, S., Okumatsu, T., Nishikawa, T., and Okabayashi, T. (2016). “Fatigue Damage of a 

Diagonal Member in a Steel Truss Bridge Due to Wind-Induced Vibration,” in 

Developments in International Bridge Engineering, edited by Alp Caner, Polat Gülkan, 

and Khaled Mahmoud, pp. 211–220, Springer, Cham, Switzerland. 

NTSB. (1968). Collapse of US 35 Highway Bridge, Point Pleasant, West Virginia, December 15, 

1967, Highway Accident Report HAR-71-01, National Transportation Safety Board, 

Washington, DC. 

NTSB. (2008a). Safety Recommendation (H08-1), National Transportation Safety Board, 

Washington, DC. 

NTSB. (2008b). Structural Investigation Group Chairman Factual Report, NTSB Report No. 

08-015, National Transportation Safety Board, Washington, DC.

NTSB. (2008c). Data Report: State-by-State Bridge Counts, NTSB Data Report, Case Number 

HWY07MH024, National Transportation Safety Board, Washington, DC. 

NTSB. (2008d). Highway Accident Report Interstate 35W Collapse over the Mississippi River 

Minneapolis, Minnesota, August 1, 2007, NTSB/HAR-08/03, National Transportation 

Safety Board, Washington, DC. 
198 

Ocel, J.M. and Wright, W.J. (2008). Finite Element Modeling of I-35W Bridge Collapse - Final 

Report, Federal Highway Administration, Turner-Fairbank Highway Research Center, 

McLean, VA. 

Pan, Y., Agrawal, A.K., Ghosn, M., and Alampalli, S. (2010a). “Seismic Fragility of Multispan 

Simply Supported Steel Highway Bridges in New York State. I: Bridge Modeling, 

Parametric Analysis, and Retrofit Design.” Journal of Bridge Engineering, 15(5), 

pp. 448–461, American Society of Civil Engineers, Reston, VA. 

Pan, Y., Agrawal, A.K., Ghosn, M., and Alampalli, S. (2010b). “Seismic Fragility of Multispan 

Simply Supported Steel Highway Bridges in New York State. II: Fragility Analysis, 

Fragility Curves, and Fragility Surfaces.” Journal of Bridge Engineering, 15(5), 

pp. 462–472, American Society of Civil Engineers, Reston, VA. 

Papadrakakis, M. (1983). “Inelastic Post-Buckling Analysis of Trusses.” Journal of Structural 

Engineering, 109(9), pp. 2129–2147, American Society of Civil Engineers, Reston, VA. 

Prendergast, J. (1995). “Oklahoma City Aftermath.” Civil Engineering, 65(10), p. 42, American 

Society of Civil Engineers, Reston, VA. 

Randers-Pehrson, G. and Bannister, K. (1997). Airblast Loading Model for DYNA2D and 

DYNA3D, Report ARL-TR-1310, Army Ballistic Research Laboratory, Aberdeen 

Proving Ground, MD. 

Reno, M.L. and Pohll, M.N. (2010). "Seismic Retrofit of California’s Auburn-Foresthill Bridge.” 

Transportation Research Record, 2201, pp. 83–94, National Academy of Sciences, 

Washington, DC. 

Sanders Jr., W., Elleby, H.A., Klaiber, F.W., and Reeves, M.D. (1975). Ultimate Load Behavior 

of Full-Scale Highway Truss Bridges, Summary Report ISU-ERI-Ames-76035, Iowa 

Department of Transportation, Ames, IA. 

Saravanan, M., Goswami, R., and Palani, G.S. (2018). “Replaceable Fuses in Earthquake 

Resistant Steel Structures: A Review.” International Journal of Steel Structures, 18(3), 

pp. 868–879, Springer, New York. 

Sarraf, M. and Bruneau, M. (1998). “Ductile Seismic Retrofit of Steel Deck-Truss Bridges. II: 

Design Applications.” Journal of Structural Engineering, 124(11), pp. 1263–1271, 

American Society of Civil Engineers, Reston, VA. 

Schmidt, L.C. and Gregg, B.M. (1980). “A Method for Space Truss Analysis in the 

Post-Buckling Range.” International Journal for Numerical Methods in Engineering, 

15(2), pp. 237–247, Wiley, Hoboken, NJ.

Schultheisz, C.R., Kushner, A.S., Nakamura, T., Ocel, J., Wright, W., and Li, M. (2008). 

“Minneapolis I-35W Bridge Collapse–Engineering Evaluations and Finite Element 

Analyses.” Available online: http://www.simulia.com/download/pdf 

2010/Schultheisz_SCC2010.pdf, last accessed April 9, 2020. 
199 

Sirisak, S. (1996). Structural Integrity Evaluation of Truss Bridges, PhD thesis, Vanderbilt 

University, Nashville, TN. 

Uang, C.M. and Bruneau, M. (2018). “State-of-the-Art Review on Seismic Design of steel 

Structures.” Journal of Structural Engineering, 144(4), American Society of Civil 

Engineers, Reston, VA. 

Usami, T., Lu, Z., and Ge, H. (2005). “A Seismic Upgrading Method for Steel Arch Bridges 

Using Buckling-Restrained Braces.” Earthquake Engineering and Structural Dynamics, 

34(4–5), pp. 471–496, International Association for Earthquake Engineering, 

Tokyo, Japan. 

Wang, Q., Nakamura, S., Okumatsu, T., and Nishikawa, T. (2017). “Comprehensive 

Investigation on the Cause of a Critical Crack Found in a Diagonal Member of a Steel 

Truss Bridge.” Engineering Structures, 132, pp. 659–670, Elsevier, Amsterdam, 

Netherlands. 

Warren Jr., J.E. (1998). Nonlinear Stability Analysis of Frame-Type Structures with Random 

Geometric Imperfections Using a Total-Lagrangian Finite Element Formulation, PhD 

thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA. 

Williams, F.W. (1964). “An Approach to the Non-Linear Behavior of the Members of a Rigid 

Jointed Plane Framework with Finite Deflections.” The Quarterly Journal of Mechanics 

and Applied Mathematics, 17(4), pp. 451–469, Oxford University Press, London, UK. 

Williamson, E.B., Bayrak, O., Williams, G.D., Davis, C.E., Marchand, K.A., McKay, A.E., 

Kulicki, J., and Wassef, W. (2010). Blast-Resistant Highway Bridges: Design and 

Detailing Guidelines, NCHRP Report 645, National Cooperative Highway Research 

Program, Transportation Research Board, Washington, DC. 

Zoli, T. and Woodward, R. (2005). “Design of Long Span Bridges for Cable Loss,” in IABSE 

Symposium Report 90, pp. 17–25, International Association for Bridge and Structural 

Engineering, Zurich, Switzerland. 


HRDI-20/11-20(WEB)E Recycled

Recyclable