Post-Tensioned Box Girder

Design Manual

June 2016
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1. Report No.

FHWA-HIF-15-016

2. Government Accession No.

XXX

3. Recipient’s Catalog No.

XXX

4. Title and Subtitle

Post-Tensioned Box Girder Design Manual

Task 3: Post-Tensioned Box Girder Design Manual

5. Report Date

June 2016

6. Performing Organization Code

XXX

7. Author(s)

Corven, John

8. Performing Organization Report No.

XXX

9. Performing Organization Name and Address

Corven Engineering Inc.,

2864 Egret Lane

Tallahassee, FL 32308

10. Work Unit No.

XXX

11. Contract or Grant No.

DTFH61-11-H-00027

12. Sponsoring Agency Name and Address

Federal Highway Administration

Office of Infrastructure – Bridges and Structures

1200 New Jersey Ave., SE

Washington, DC 20590

13. Type of Report and Period Covered

XXX

14. Sponsoring Agency Code

HIBS-10

15. Supplementary Notes

Work funded by Cooperative Agreement “Advancing Steel and Concrete Bridge Technology to Improve Infrastructure Performance”

between FHWA and Lehigh University.

16. Abstract

This Manual contains information related to the analysis and design of cast-in-place concrete box girder bridges prestressed with posttensioning tendons. The Manual is targeted at Federal, State and local transportation departments and private company personnel that may be 

involved in the analysis and design of this type of bridge. The Manual reviews features of the construction of cast-in-place concrete box 

girder bridges, material characteristics that impact design, fundamentals of prestressed concrete, and losses in prestressing force related to 

post-tensioned construction. Also presented in this Manual are approaches to the longitudinal and transverse analysis of the box girder

superstructure. Both single-cell and multi-cell box girders are discussed. Design examples are presented in Appendices to this Manual. The 

document is part of the Federal Highway Administration’s national technology deployment program and may serve as a training manual.

17. Key Words

Box girder, Cast-in-place, Multi-cell, Concrete, Top Slab,

Cantilever Wing, Web, Bottom Slab, Prestressing, Posttensioning, Strand, Tendon, Duct, Anchorage, Losses,

Friction, Wobble, Elastic Shortening, Creep, Shrinkage,

Force, Eccentricity, Bending moment, Shear, Torsion, Joint

flexibilities, Longitudinal analysis, Transverse analysis

18. Distribution Statement

No restrictions. This document is available to the public 

online and through the National Technical Information 

Service, Springfield, VA 22161. 

9. Security Classif. (of this

report)

Unclassified

20. Security Classif. (of this

page)

Unclassified

21. No of Pages

355

22. Price

$XXX.XX
Visit http://www.fhwa.dot.gov/publications/convtabl.cfm for a 508 compliant version of this table.
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Post-Tensioned Box Girder Design Manual June 2016

Preface i 

Preface

This Manual contains information related to the analysis and design of cast-in-place concrete 

box girder bridges prestressed with post-tensioning tendons. The Manual is targeted at 

Federal, State and local transportation departments and private company personnel that may be 

involved in the analysis and design of this type of bridge. The Manual reviews features of the 

construction of cast-in-place concrete box girder bridges, material characteristics that impact 

design, fundamentals of prestressed concrete, and losses in prestressing force related to posttensioned construction. Also presented in this Manual are approaches to the longitudinal and 

transverse analysis of the box girder superstructure. Both single-cell and multi-cell box girders 

are discussed. Design examples are presented in Appendices to this Manual. The document is 

part of the Federal Highway Administration’s national technology deployment program and may 

serve as a training manual.
Post-Tensioned Box Girder Design Manual June 2016

ii

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Post-Tensioned Box Girder Design Manual June 2016

Table of Contents iii

Table of Contents

Chapter 1 – Introduction........................................................................................................... 1 

1.1 Historical Overview.............................................................................................................1

1.2 Typical Superstructure Cross Sections...............................................................................2

1.3 Longitudinal Post-Tensioning Layouts................................................................................3

1.4 Loss of Prestressing Force................................................................................................. 6

1.5 Post-Tensioning System Hardware ....................................................................................6

1.5.1 Basic Bearing Plates .......................................................................................6

1.5.2 Special Bearing Plates or Anchorage Devices.................................................7

1.5.3 Wedge Plates ..................................................................................................8

1.5.4 Wedges and Strand-Wedge Connection..........................................................8

1.5.5 Permanent Grout Caps....................................................................................8

1.5.6 Ducts ...............................................................................................................9

1.5.6.1 Duct Size...........................................................................................9

1.5.6.2 Corrugated Steel Duct.......................................................................9

1.5.6.3 Corrugated Plastic...........................................................................10

1.5.6.4 Plastic Fittings and Connections for Internal Tendons.....................11

1.5.6.5 Grout Inlets, Outlets, Valves and Plugs ...........................................11

1.5.7 Post-Tensioning Bars Anchor Systems..........................................................11

1.6 Overview of Construction ................................................................................................. 12

1.6.1 Falsework ......................................................................................................12

1.6.2 Superstructure Formwork ..............................................................................13

1.6.3 Reinforcing and Post-Tensioning Hardware Placement................................. 15

1.6.4 Placing and Consolidating Superstructure Concrete......................................15

1.6.5 Superstructure Curing....................................................................................16

1.6.6 Post-Tensioning Operations ..........................................................................17

1.6.7 Tendon Grouting and Anchor Protection........................................................19

Chapter 2 – Materials..............................................................................................................20

2.1 Concrete ..........................................................................................................................20

2.1.1 Compressive Strength ...................................................................................20

2.1.2 Development of Compressive Strength with Time .........................................21

2.1.3 Tensile Strength ............................................................................................22

2.1.4 Modulus of Elasticity......................................................................................23

2.1.5 Modulus of Elasticity Variation with Time.......................................................24

2.1.6 Poisson’s Ratio..............................................................................................25

2.1.7 Volumetric Changes ......................................................................................25

2.1.7.1 Coefficient of Thermal Expansion....................................................25

2.1.7.2 Creep..............................................................................................25

2.1.7.3 Shrinkage........................................................................................29

2.2 Prestressing Strands ........................................................................................................31

2.2.1 Tensile Strength ............................................................................................32

2.2.2 Modulus of Elasticity......................................................................................32

2.2.3 Relaxation of Steel ........................................................................................33

2.2.4 Fatigue ..........................................................................................................34

2.3 Reinforcing Steel..............................................................................................................34
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Table of Contents iv

Chapter 3 – Prestressing with Post-Tensioning ...................................................................36

3.1 Introduction ......................................................................................................................36

3.2 Cross Section Properties and Sign Convention ................................................................37

3.3 Stress Summaries in a Prestressed Beam .......................................................................37

3.4 Selection of Prestressing Force for a Given Eccentricity...................................................39

3.5 Permissible Eccentricities for a Given Prestressing Force................................................46

3.6 Equivalent Forces Due To Post-Tensioning and Load Balancing .....................................48

3.7 Post-Tensioning in Continuous Girders ............................................................................50

3.9 Tendon Profiles—Parabolic Segments .............................................................................54

Chapter 4—Prestressing Losses ...........................................................................................60

4.1 Instantaneous Losses ......................................................................................................60

4.1.1 Friction and Wobble Losses (AASHTO LRFD Article 5.9.5.2.2b)...................60

4.1.2 Elongation .....................................................................................................66

4.1.3 Anchor Set.....................................................................................................67

4.1.4 Two-End Stressing ........................................................................................69

4.1.5 Elastic Shortening (AASHTO LRFD Article 5.9.5.2.3b) ..................................71

4.2 Time-Dependent Losses ..................................................................................................72

4.2.1 General (AASHTO Article LRFD 5.9.5.4.1)....................................................72

4.2.2 Concrete Shrinkage (AASHTO Article LRFD 5.9.5.4.3a) ...............................73

4.2.3 Concrete Creep (AASHTO Article LRFD 5.9.5.4.3b)......................................75

4.2.4 Steel Relaxation (AASHTO Article LRFD 5.9.5.4.3c).....................................75

Chapter 5—Preliminary Design..............................................................................................76

5.1 Introduction ......................................................................................................................76

5.2 Establish Bridge Layout....................................................................................................77

5.2.1 Project Design Criteria...................................................................................77

5.2.2 Span Lengths and Layout..............................................................................78

5.3 Cross Section Selection ...................................................................................................79

5.3.1 Superstructure Depth.....................................................................................79

5.3.2 Superstructure Width.....................................................................................79

5.3.3 Cross Section Member Sizes.........................................................................80

5.3.3.1 Width and Thickness of Cantilever Wing .........................................80

5.3.3.2 Individual and Total Web Thickness................................................81

5.3.3.3 Top Slab Thickness.........................................................................82

5.3.3.4 Bottom Slab Thickness....................................................................85

5.3.3.5 Member Sizes for Example Problem ...............................................85

5.4 Longitudinal Analysis........................................................................................................87

5.4.1 Approach.......................................................................................................87

5.4.2 Analysis by Method of Joint Flexibilities.........................................................87

5.4.3 Span Properties and Characteristic Flexibilities .............................................87

5.4.4 Analysis Left to Right.....................................................................................88

5.4.5 Analysis Right to Left.....................................................................................88

5.4.6 Carry-Over Factors........................................................................................89

5.5 Bending Moments ............................................................................................................89

5.5.1 Effect of a Unit Uniform Load.........................................................................89

5.5.2 Dead Load—DC (Self Weight and Barrier Railing).........................................92

5.5.3 Dead Load—DW (Future Wearing Surface)...................................................92
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Table of Contents v 

5.5.4 Live Load—LL ...............................................................................................93

5.5.4.1 Uniform Load Component ...............................................................93

5.5.4.2 Truck—Positive Moment in Span 1 or 3 ..........................................93

5.5.4.3 Truck—Positive Moment in Span 2 .................................................94

5.5.4.4 Truck—Negative Moment over Piers...............................................95

5.5.4.5 Live Load Moment Totals ................................................................95

5.5.5 Thermal Gradient (TG) ..................................................................................97

5.5.6 Post-Tensioning Secondary Moments ...........................................................98

5.6 Required Prestressing Force After Losses .....................................................................101

5.7 Prestressing Losses and Tendon Sizing for Final Design (Pjack)...................................103

5.7.1 Losses from Friction, Wobble, and Anchor Set ............................................103

5.7.2 Losses from Elastic Shortening ...................................................................104

5.7.3 Losses from Concrete Shrinkage................................................................. 105

5.7.4 Losses from Concrete Creep.......................................................................107

5.7.5 Losses from Steel Relaxation ......................................................................107

5.7.6 Total of Losses and Tendon Sizing..............................................................107

5.8 Service Limit State Stress Verifications ..........................................................................107

5.8.1 Service Flexure—Temporary Stresses (DC and PT Only) ...........................108

5.8.2 Service Limit State III Flexure Before Long-Term Losses ............................109

5.8.3 Service Limit State III Flexure After Long-Term Losses ...............................109

5.8.4 Principal Tension in Webs after Losses .......................................................110

5.9 Optimizing the Post-Tensioning Layout ..........................................................................112

Chapter 6—Substructure Considerations ...........................................................................115

6.1 Introduction ....................................................................................................................115

6.2 Bending Moments Caused by Unit Effects......................................................................116

6.2.1 Effect of a Unit Uniform Load.......................................................................116

6.2.2 Effect of a Unit Lateral Displacement (Side-Sway Correction) .....................117

6.2.3 Effect of a Unit Contraction..........................................................................117

6.3 Dead Load—DC (Self Weight and Barrier Railing) .........................................................118

6.4 Dead Load—DW (Future Wearing Surface) ...................................................................118

6.5 Live Load—LL (Lane and Truck Components) ...............................................................119

6.5.1 Envelope of Uniform Load Component ........................................................119

6.5.2 Truck—Positive Moment in Span 1 or 3.......................................................119

6.5.3 Truck—Positive Moment in Span 2..............................................................120

6.5.4 Truck—Negative Moment over Piers ...........................................................120

6.6 Post-Tensioning Secondary Moments—Unit Prestressing Force....................................120

6.7 Thermal Gradient (TG)—20°F Linear .............................................................................122

6.8 Moments Resulting from Temperature Rise and Fall......................................................122

6.8.1 Temperature Rise—40°F Uniform Rise .......................................................122

6.8.2 Temperature Fall—40°F Uniform Fall ..........................................................123

6.9 Moments Resulting from Concrete Shrinkage ................................................................123

6.10 Moments Resulting from Concrete Creep.......................................................................125

6.11 Bending Moments Summaries........................................................................................127

6.12 Post-Tensioning Force Comparison (after all losses, with thermal effects) .....................128

6.12.1 Side Span Positive Bending.........................................................................128

6.12.2 Middle Span Positive Bending .....................................................................128

6.12.3 Negative Bending at Piers ...........................................................................129
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Table of Contents vi

Chapter 7—Longitudinal Analysis & Design.......................................................................130

7.1 Introduction ....................................................................................................................130

7.2 Modeling Concepts.........................................................................................................130

7.2.1 Straight Bridges Supported on Bearings......................................................131

7.2.1.1 Nodes ...........................................................................................131

7.2.1.2 Elements.......................................................................................132

7.2.1.3 Post-Tensioning ............................................................................134

7.2.2 Straight Bridges with Integral Piers..............................................................134

7.2.3 Curved Bridges............................................................................................136

7.2.4 Other Three-Dimensional Analyses .............................................................143

7.3 Strength Limit Verification—Flexure ...............................................................................145

7.3.1 Factored Loads for Longitudinal Flexure......................................................146

7.3.2 Flexural Resistance .....................................................................................148

7.3.2.1 Strain Compatibility .......................................................................149

7.3.2.2 Material Stresses and Internal Forces ...........................................150

7.3.2.3 Internal Equilibrium .......................................................................153

7.3.3 Resistance Factors (φ) .................................................................................155

7.3.4 Limits of Reinforcing ....................................................................................156

7.3.5 Procedure....................................................................................................157

7.4 Strength Limit Verification—Shear..................................................................................163

7.4.1 LRFD Design Procedures for Shear and Torsion.........................................163

7.4.2 General Requirements.................................................................................164

7.4.3 Sectional Model Nominal Shear Resistance ................................................165

7.4.3.1 Effective Web Width......................................................................166

7.4.3.2 Effective Shear Depth ...................................................................168

7.4.4 Shear Resistance from Concrete (Vc) ..........................................................169

7.4.4.1 Method 2 (Simplified MCFT) .........................................................169

7.4.4.2 Method 3 (Historical Empirical)......................................................173

7.4.5 Shear Resistance from Transverse (Web) Reinforcing Steel (Vs) ................174

7.4.6 Shear Resistance from Vertical Component of Effective Prestressing (Vp) ..175

7.4.7 Longitudinal Reinforcing ..............................................................................177

7.4.8 Torsion Reinforcing .....................................................................................178

Chapter 8—Transverse Analysis .........................................................................................179

8.1 Introduction ....................................................................................................................179

8.2 Methods of Analysis .......................................................................................................179

8.3 Applicable AASHTO LRFD Specifications ......................................................................180

8.3.1 Section 9—Deck and Deck Systems ...........................................................180

8.3.2 Section 3—Loads ........................................................................................182

8.3.3 Section 4—Analysis.....................................................................................185

8.3.4 Section 13—Railing .....................................................................................188

8.4 Strip Method Analysis for a Multi-Cell Box Girder Superstructure ...................................190

8.4.1 The Transverse Model.................................................................................191

8.4.2 Transverse Bending Moment Results ..........................................................192

8.4.3 Transverse Design Moments.......................................................................194

8.5 Top Slab Transverse Bending Moment Results for a Single-Cell Box Girder..................195

8.5.1 Introduction..................................................................................................195

8.5.2 Analysis for Uniformly Repeating Loads ......................................................197

8.5.3 Analysis for Concentrated Wheel Live Loads...............................................199

8.5.4 Live Load Moments in Cantilever Wings......................................................200
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Table of Contents vii

8.5.5 Negative Live Load Moments in the Top Slab..............................................201

8.5.6 Positive Live Load Moments at Centerline of the Top Slab. .........................204

8.6 Transverse Post-Tensioning...........................................................................................206

8.6.1 Transverse Post-Tensioning Tendon Layouts..............................................206

8.6.2 Required Prestressing Force .......................................................................206

8.6.3 Transverse Post-Tensioning Tendon Placement and Stressing...................208

Chapter 9—Other Design Considerations...........................................................................210

9.1 Effects of Curved Tendons .............................................................................................210

9.1.1 In-Plane and Out-of-Plane Forces ...............................................................211

9.1.2 AASHTO LRFD Design Approach ...............................................................213

9.1.3 Regional Effects—Transverse (Regional) Bending ......................................213

9.1.4 Local Shear and Flexure in Webs................................................................216

9.1.4.1 Shear Resistance to Pull-out.........................................................217

9.1.4.2 Cracking of Concrete Cover ..........................................................218

9.1.5 Out-of-Plane Force Effects ..........................................................................220

9.2 End Anchorage Zones....................................................................................................220

9.3 Diaphragms at Supports..............................................................................222

9.3.1 Single-Cell Box Girder Transfer of Vertical Shear to Bearings.......222

9.3.2 Single-Cell Box Girder Transfer of Torsion to Bearings .................224

9.3.3 Multi-Cell Box Girder Diaphragms .................................................225

Appendix A – Analysis of Two-Dimensional Indeterminate Structures by the Flexibility 

Method...................................................................................................................................227

Appendix B – Torsion...........................................................................................................260

Appendix C – Example 1: Multi-Cell Box Girder Bridge ....................................................280

Appendix D – Example 2: Curved Two-Cell Box Girder Bridge ........................................322 
Post-Tensioned Box Girder Design Manual June 2016

Table of Contents viii

List of Figures

Figure 1.1 Cast-in-Place Post-Tensioned Box Girder Bridge for MARTA................................ 1 

Figure 1.2 Typical Span Ranges for Prestressed Concrete Bridge Types .............................. 2 

Figure 1.3 Multi-Cell Box Girder Cross Section ...................................................................... 3 

Figure 1.4 Single-Cell Box Girder Cross Section.................................................................... 3 

Figure 1.5 Typical Post-Tensioning Tendon Layout for Simple Spans.................................... 4 

Figure 1.6 Tendon Layout for 4-Span Bridge, CIP on Falsework............................................ 4 

Figure 1.7 Tendon Locations within Box Girder Cross Section............................................... 5 

Figure 1.8 Possible Tendon Layout for Sequentially Cast Spans ........................................... 5 

Figure 1.9 Basic Bearing Plate Anchorage System ................................................................ 6 

Figure 1.10 Multi-Plane Anchorage System (Courtesy of VSL) ................................................ 7 

Figure 1.11 Anchorage System for Flat Duct Tendon (Courtesy of DSI)................................... 7 

Figure 1.12 Permanent Plastic Grout Caps (Courtesy of VSL)................................................. 9 

Figure 1.13 Corrugated Metal Duct ........................................................................................10

Figure 1.14 Corrugated Plastic Duct.......................................................................................10

Figure 1.15 Typical High-Point Grout Vent .............................................................................11

Figure 1.16 Post-Tensioning Bar Anchorage System (Courtesy of DSI)................................. 12

Figure 1.17 Modular Falsework Units for Cast-in-Place Construction.....................................12

Figure 1.18 Steel Pipe Support Towers for Cast-In-Place Construction..................................13

Figure 1.19 Web and Cantilever Wing Formwork for a Single-Cell Box Girder .......................14

Figure 1.20 Web Formwork for a Multi-Cell Box Girder ..........................................................14

Figure 1.21 Web and Bottom Slab Reinforcing, Tying Post-tensioning Ducts in Webs ...........15

Figure 1.22 Placing Deck Concrete and Finishing with a Roller Screed ................................. 16

Figure 1.23 Curing the Concrete Deck ...................................................................................17

Figure 1.24 Bundled Tendon Prepared for Pulling.................................................................. 18 

Figure 1.25 Stressing Post-Tensioning Tendons....................................................................18

Figure 2.1 Concrete strength gain with time ......................................................................... 22 

Figure 2.2 Typical Stress-Strain Curve for Concrete ............................................................ 23 

Figure 2.3 Concrete Modulus of Elasticity with Time ............................................................ 24 

Figure 2.4 Creep of Concrete............................................................................................... 25 

Figure 2.5 Creep of Concrete (with no long-duration transient loads)................................... 26 

Figure 2.6 Development of Concrete Creep with Time ......................................................... 29 

Figure 2.7 Development of Concrete Shrinkage with Time...................................................30

Figure 2.8 Rate of Concrete Shrinkage over Time................................................................ 31 

Figure 2.9 Stress-Strain Diagram for Prestressing Strand (Courtesy of PCI)........................ 33 

Figure 2.10 Comparison of Typical Stress-Strain Relationships for

Prestressing Strand and Mild Reinforcing............................................................ 35 

Figure 3.1 Prestressed Concrete Concepts.......................................................................... 36 

Figure 3.2 Cross Section Nomenclature and Sign Convention ............................................. 37 

Figure 3.3 Self Weight Flexure Stress in Simply-Supported Beam ....................................... 38 

Figure 3.4 Self Weight Plus Uniform Axial Compression ...................................................... 38 

Figure 3.5 Self Weight, Axial and Eccentric Prestress Stresses ........................................... 39 

Figure 3.6 Efficiencies of Various Cross Sections ................................................................40

Figure 3.7 Internal Equilibrium for Positive Bending .............................................................41

Figure 3.8 Internal Equilibrium for Negative Bending............................................................ 42 

Figure 3.9 Limiting Eccentricities for Zero Tension Under Axial Force Only.......................... 43 

Figure 3.10 Kern of a Cross Section for Bending About the Horizontal Axis ........................... 43 

Figure 3.11 Example Concrete I-Girders ................................................................................ 44 

Figure 3.12 Limiting Eccentricities for the Example Bridge..................................................... 48 

Figure 3.13 Parabolic Tendon Profile for a Simple Span Girder ............................................. 48 
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Table of Contents ix

Figure 3.14 Equivalent Forces Resulting from Prestressing ................................................... 49 

Figure 3.15 Restraining Moments in Continuous Girders .......................................................51

Figure 3.16 Prestressing Moments for a Two-Span Continuous Girder.................................. 52 

Figure 3.17 Total Prestressing Moments for a Two-Span Continuous.................................... 53 

Figure 3.18 Tendon Profile Parabolic Segment ...................................................................... 54 

Figure 3.19 Typical End Span Tendon Profile for Continuous Superstructures ...................... 55 

Figure 3.20 Typical Interior Span Tendon Profile for Continuous............................................ 55 

Figure 3.21 Example Tendon Profile Parabolic Segments...................................................... 56 

Figure 3.22 Curvature Diagram for Prestressing .................................................................... 57 

Figure 3.23 Curvature Diagram for Prestressing .................................................................... 57 

Figure 3.24 Loaded Conjugate Beam..................................................................................... 58 

Figure 4.1 Friction and Wobble ............................................................................................60

Figure 4.2 Section of Tendon with Radial Alignment ............................................................61

Figure 4.3 Cross Section of Superstructure for Design Example 1 ....................................... 63 

Figure 4.4 Tendon Profiles for Design Example 1 ................................................................ 64 

Figure 4.5 Tendon T2 Profile and Angular Deviations .......................................................... 64 

Figure 4.6 Tendon Loss Calculations—Friction and Wobble ................................................ 66 

Figure 4.7 Anchor Set .......................................................................................................... 68 

Figure 4.8 Tendon Force Diagram after Anchor Set at End A............................................... 68 

Figure 4.9 Tendon Force Diagram after Stressing from End B .............................................70

Figure 4.10 Final Tendon Force Diagram (After Anchor Set at End B) ...................................70

Figure 5.1 CIP Box Girder Bridge Preliminary Design Flow Chart ........................................ 76 

Figure 5.2 3-Span Box Girder Bridge for Preliminary Design................................................ 76 

Figure 5.3 Existing, At-Grade Highway Cross Section to Be

Spanned by Proposed Bridge.............................................................................. 77 

Figure 5.4 Span Layout for Preliminary Design Example...................................................... 78 

Figure 5.5 Bridge Width and Roadway Features ..................................................................80

Figure 5.6 Cantilever Wing Dimensions ...............................................................................80

Figure 5.7 Top Slab Span and Thickness............................................................................. 84 

Figure 5.8 Top Slab with Haunches...................................................................................... 84 

Figure 5.9 Example Cross Section Dimensions for Preliminary Design ................................ 86 

Figure 5.10 Model of 3-Span Bridge for Example 1................................................................ 87 

Figure 5.11 Moment Diagram for a Unit Uniform Load ...........................................................91

Figure 5.12 Moment Diagram for Dead Load (DC)................................................................. 92 

Figure 5.13 Moment Diagram for Dead Load (DW) ................................................................ 92 

Figure 5.14 Uniform Live Load Moment Envelope.................................................................. 93 

Figure 5.15 Simple Beam Rotations for a Concentrated Load................................................ 93 

Figure 5.16 Moment Diagram for HL-93 Design Truck in Span 1 (Positive Bending).............. 94 

Figure 5.17 Moment Diagram for HL-93 Design Truck in Span 2 (Positive Bending).............. 94 

Figure 5.18 Moment Diagram for Two HL-93 Design Trucks about

Pier 2 (Negative Bending) .................................................................................. 95 

Figure 5.19 Simple Beam Subjected to 20°F Positive Linear Gradient ................................... 97 

Figure 5.20 Moment Diagram for a 20°F Positive Linear Gradient.......................................... 98 

Figure 5.21 Possible Tendon Locations at Mid-Span at over the Piers................................... 99 

Figure 5.22 Center of Gravity Profile of Prestressing (End Spans)......................................... 99 

Figure 5.23 Center of Gravity Profile of Prestressing (Middle Span).....................................100

Figure 5.24 Conjugate Beam and Loads (End Spans) .........................................................100

Figure 5.25 Conjugate Beam and Loads (Main Span)..........................................................100

Figure 5.26 Secondary Prestressing Moments, M2(F) .........................................................101

Figure 5.27 Friction Diagram for the CG Profile Tendon.......................................................104 

Figure 5.28 Mohr Circle for Location of Maximum Shear in Middle Span .............................112 

Figure 5.29 Revised Center of Gravity Profile of Prestressing (End Spans) .........................113 
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Table of Contents x 

Figure 5.30 Revised Conjugate Beam and Loads (End Spans)............................................113 

Figure 5.31 Revised Secondary Prestressing Moments, M2(F)............................................114 

Figure 6.1 Example CIP Box Girder Bridge Elevation.........................................................115 

Figure 6.2 Bridge Cross Section at Piers............................................................................115 

Figure 6.3 Effect of a Unit Uniform Load ............................................................................116 

Figure 6.4 Effect of a Unit Lateral Displacement (Side-sway Correction)............................117 

Figure 6.5 Effect of a Unit Contraction................................................................................117 

Figure 6.6 Effect of Dead Load (DC) ..................................................................................118 

Figure 6.7 Effect of Dead Load (DW) .................................................................................118 

Figure 6.8 Uniform Live Load Moment Envelope ...............................................................119 

Figure 6.9 Moment Diagram for HS20 Truck in Span 1 or 3 ...............................................119 

Figure 6.10 Moment Diagram for HS20 Truck in Span 2 ...................................................... 120

Figure 6.11 Moment Diagram for Two HS20 Trucks about Pier 2......................................... 120

Figure 6.12 Secondary Prestressing Moments, M2(F) ......................................................... 121

Figure 6.13 Moment Diagram for a 20°F Positive Linear Gradient........................................122 

Figure 6.14 Moment Diagram for 40° Temperature Rise ......................................................122 

Figure 6.15 Moment Diagram for 40° Temperature Fall .......................................................123 

Figure 6.16 Moment Diagram for Concrete Shrinkage .........................................................124 

Figure 6.17 Moment Diagram for Concrete Creep................................................................127 

Figure 7.1 Example Straight Bridge on Bearings................................................................ 131

Figure 7.2 Box Girder Superstructure Cross Section.......................................................... 131

Figure 7.3 Two-Dimensional Analysis Model...................................................................... 132

Figure 7.4 Typical Element Stiffness Matrix for a Plane Frame

Member with 3DOF Nodes ................................................................................ 133

Figure 7.5 Cross Section Properties for the Box Girder shown in Figure 7.2 ......................133 

Figure 7.6 Cross Section of Design Example 1 Bridge at the Piers ....................................134 

Figure 7.7 Two-Dimensional Analysis Model with Integral Piers.........................................135 

Figure 7.8 Detail of Model at Pier.......................................................................................135 

Figure 7.9 Curved Bridge of Design Example 2..................................................................136 

Figure 7.10 Design Example 2 Bridge..................................................................................137 

Figure 7.11 3D Model for Bridge in Design Example 2 .........................................................137 

Figure 7.12 Cross Section of Design Example 2 Bridge at the Piers ....................................138 

Figure 7.13 3D Model for Bridge in Design Example 2 at the Piers ......................................138 

Figure 7.14 Torsion in a Single Cell Box Girder....................................................................139 

Figure 7.15 Torsion in a Two-Cell Box Girder....................................................................... 140

Figure 7.16 Cross Section of Bridge in Example 1, Appendix C ........................................... 142

Figure 7.17 Grillage Model Development for Design Example 2...........................................144 

Figure 7.18 Grillage Model Design Example 2 .....................................................................144 

Figure 7.19 Cross Section of Shell Element FEM Model for Design Example 2 ...................145 

Figure 7.20 Flexural Resistance by Strain Compatibility.......................................................149 

Figure 7.21 Rectangular Stress Block to represent Concrete Compression ......................... 150

Figure 7.22 Comparison of Typical Stress-Strain Relationships for Prestressing

Strand and Mild Reinforcing .............................................................................. 151

Figure 7.23 Stress-Strain Relationships for Prestressing Strand ..........................................153 

Figure 7.24 Flexural Resistance with Multiple Layers of Prestressing

Steel and Mild Reinforcing.................................................................................154 

Figure 7.25 Transition of Resistance Factors from Compression to

Tension Controlled ............................................................................................155 

Figure 7.26 Flow Chart for Verification of Flexure at the Strength Limit State.......................158 

Figure 7.27 Idealized Cross Section For Longitudinal Flexure..............................................159 

Figure 7.28 Location of Prestressing Reinforcing in Idealized Cross Section .......................159 

Figure 7.29 Web Width based on Horizontal Widths ............................................................165 
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Figure 7.30 Web Width based on Horizontal Widths ............................................................166 

Figure 7.31 Shear Flow in Single Cell Box Girder................................................................. 167 

Figure 7.32 Shear Stress and Shear Flow Around Ducts .....................................................168 

Figure 7.33 Effective Depth for Shear Calculations ..............................................................168 

Figure 7.34 Actual vs. MCFT Girders ................................................................................... 170

Figure 7.35 MCFT Forces and Longitudinal Strain ............................................................... 171

Figure 7.36 Types and Locations of Reinforced and Prestressed Girder Cracking............... 173

Figure 7.37 Contribution of Shear Reinforcing to Nominal Shear Resistance.......................174 

Figure 7.38 Simple Span Beam with Parabolically Draped Tendon......................................176 

Figure 7.39 Typical Tendon Profile for an End Span of a Continuous Unit ...........................176 

Figure 7.40 Typical Tendon Profile for an Interior Span of a Continuous Unit.......................177 

Figure 7.41 Tendon Profile Parabolic Segment ....................................................................177 

Figure 8.1 Concrete Box Girder Cross Sections and Loads ...............................................179 

Figure 8.2 AASHTO LRFD Design Truck and Design Tandem........................................... 182

Figure 8.3 Transverse Truck Placement.............................................................................183 

Figure 8.4 Tire Contact Area in the Transverse Direction...................................................184 

Figure 8.5 Alternate Vertical Loading for Overhang Design................................................184 

Figure 8.6 Perspective of Multi-cell Box Girder...................................................................185 

Figure 8.7 Transverse Strip for Approximate Design Method .............................................186 

Figure 8.8 Transverse Strip subjected to two Design Trucks..............................................186 

Figure 8.9 Critical Sections for Overhang Design to Develop Barrier Railing......................189 

Figure 8.10 Developing the Two-Dimensional Transverse Model......................................... 191

Figure 8.11 Transverse Self Weight Moments (ft-kip/ft)........................................................ 192

Figure 8.12 Transverse Barrier Railing Moments (ft-kip/ft) ................................................... 192

Figure 8.13 Transverse Wearing Surface Moments (ft-kips/ft).............................................. 192

Figure 8.14 Maximum Negative Design Truck Moment in Outer Web ..................................193 

Figure 8.15 Maximum Negative Design Truck Moment at Inner Web...................................193 

Figure 8.16 Maximum Positive Design Truck Moment in Top Slab.......................................193 

Figure 8.17 Typical Single-Cell Box Girder Cross Section Defined at Mid-Span...................195 

Figure 8.18 Typical Single-Cell Box Girder Span with Cross Section 

Defined at Mid-Span..........................................................................................196 

Figure 8.19 One-Foot Section of Typical Cross Section .......................................................196 

Figure 8.20 Developing the Two-Dimensional Transverse Model.........................................197 

Figure 8.21 Transverse Bending Moments for Uniformly Repeating Loads ..........................198 

Figure 8.22 Truck Loads on a Single-Cell Box Girder Span .................................................199 

Figure 8.23 Truck Location for Maximum Transverse Bending Moment

at Root of Cantilever..........................................................................................200

Figure 8.24 Loaded Influence Surface for the Cantilever Wing.............................................200

Figure 8.25 Distribution of Cantilever Live Load Moments in the Cross Section ...................201

Figure 8.26 Final Bending Moments for Live Load in Cantilever...........................................201

Figure 8.27 Truck Location for Maximum Transverse Bending Moment

at Middle of Top Slab ........................................................................................202

Figure 8.28 Influence Surface for Maximum Negative Bending at the

Left End of the Top Slab ...................................................................................202

Figure 8.29 Influence Surface for Maximum Negative Bending at the

Right End of the Top Slab .................................................................................202

Figure 8.30 Distribution of Fixed-End Live Load Moments for Maximum

Negative Moment Case.....................................................................................203 

Figure 8.31 Summed Live Load Moments for the Maximum Negative Moment Case...........203 

Figure 8.32 Live Load Position for Maximum Positive Bending ............................................204 

Figure 8.33 Maximum Positive Moment in the Top Slab for Fixed-End Conditions...............205 

Figure 8.34 Distribution of Fixed-End Live Load Moments for Maximum
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Negative Moment Case.....................................................................................205 

Figure 8.35 Summed Live Load Moments for the Maximum Positive Moment Case ............205 

Figure 8.36 Typical Transverse Tendon Layout ...................................................................206 

Figure 8.37 Transverse Duct Placement in Casting Machine ...............................................209 

Figure 8.38 Mono-Strand Stressing of a 4 Strand Tendon and Anchorage After

Stressing 2nd Strand.........................................................................................209 

Figure 9.1 Curved Tendon Deviations ................................................................................ 210

Figure 9.2 Tendons in Curved Superstructures .................................................................. 210

Figure 9.3 Cross Section of Multi-Cell Box Girder with Lateral Tendon Loads.................... 211

Figure 9.4 Tendon Plane of Curvature ............................................................................... 211

Figure 9.5 In-Plane and Out-of-Plane Tendon Forces ........................................................ 212

Figure 9.6 Hypothetical Concrete Member Completely Coincident with a Tendon..............213 

Figure 9.7 Post-Tensioning a Curved Plate........................................................................214 

Figure 9.8 Web Flexure Restrained by Top and Bottom Slabs ...........................................214 

Figure 9.9 Web Transverse (Regional) Bending Moments .................................................215 

Figure 9.10 Web Height for Equation 9.3 .............................................................................216 

Figure 9.11 Parameters for Local Shear and Flexure Design ...............................................216 

Figure 9.12 Effective Length of Failure Plane for Equation 9.6.............................................217 

Figure 9.13 Effective Length of Failure Plane for Equations 9.7 and 9.8 ..............................218 

Figure 9.14 Local Bending Moments for Evaluating Cracking of Concrete Cover.................219 

Figure 9.15 Details of End of Post-Tensioned Box Girder Bridge ......................................... 220

Figure 9.16 End Zone Design Development......................................................................... 221

Figure 9.17 Concentric Web/Bearing Orientation ................................................................. 222

Figure 9.18 Eccentric Web/Bearing Orientation .................................................................... 222

Figure 9.19 General Shear Friction and Localized Direct Tension........................................223 

Figure 9.20 Vertical Force Transfer with Inclined Webs........................................................223 

Figure 9.21 Transverse Post-Tensioning in Diaphragms......................................................224 

Figure 9.22 Shear Flow Resulting from Torsional Forces.....................................................224 

Figure 9.23 A-shaped Torsion Diaphragm............................................................................225 

Figure 9.24 V-shaped Torsion Diaphragm............................................................................225 

Figure 9.25 Possible Strut-and-Tie Layout for Diaphragm of Design Example 1 ..................226 

Figure 9.26 Strut-and-Tie Layout Considering Monolithic Column Connection .....................226 

Figure A.1 Continuous Beam Load, Shear and Moment Diagrams.....................................228 

Figure A.2 Bending Moment and Rotation Sign Convention ...............................................229 

Figure A.3 Equations for End Rotations of Simple Beams ..................................................229 

Figure A.4 Bending Moment Diagram for Unit Moment at Node i ........................................229 

Figure A.5 Bending Moment Diagram for Unit Moment at Node j........................................ 230

Figure A.6 Simple Span Beam Characteristics ................................................................... 230

Figure A.7 Span ij in a Continuous Unit .............................................................................. 231

Figure A.8 Isolating Span ij................................................................................................. 232

Figure A.9 Compatible rotations of adjacent members........................................................ 232

Figure A.10 Adjacent Member Flexibilities............................................................................ 233

Figure A.11 Member End Flexibility for Span hi....................................................................235 

Figure A.12 Carry Over Factor from j to i ..............................................................................237 

Figure A.13 Moment Equilibrium at Node i ...........................................................................239 

Figure A.14 Model of 3-Span Bridge for Example 1..............................................................242

Figure A.15 Moment Diagram for a Unit Uniform Load .........................................................245 

Figure A.16 Moment Diagram for a Unit Uniform Load .........................................................246 

Figure A.17 Cantilever Column.............................................................................................246 

Figure A.18 Column with Multiple Elements..........................................................................248 

Figure A.19 Column in a Rigid Frame...................................................................................249 

Figure A.20 Model of 3-Span Bridge for Example 1.............................................................. 250
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Figure A.21 Moment Diagram for a Unit Uniform Load .........................................................257 

Figure A.22 Moment Diagram for a Unit Lateral Side-Sway..................................................259 

Figure B.1 Circular Bar Subjected to Torsional Moment ..................................................... 260

Figure B.2 Segmented Circular Bar .................................................................................... 261

Figure B.3 Kinematic Study 1 ............................................................................................. 261

Figure B.4 Kinematic Study 2 ............................................................................................. 262

Figure B.5 Kinematic Study 3 ............................................................................................. 262

Figure B.6 Linear Twisting of the Circular Bar..................................................................... 263

Figure B.7 Shear Stresses and Shear Strains in the Circular Bar ....................................... 263

Figure B.8 Element of the Circular Bar ............................................................................... 264

Figure B.9 Closed Thin-Wall Shaft......................................................................................266 

Figure B.10 Segment of Closed, Thin-Wall Shaft..................................................................267 

Figure B.11 Equilibrium in the Cross Section of the Thin Wall Closed Shaft .........................268 

Figure B.12 Example Single Cell Box Girder ........................................................................ 271

Figure B.13 Idealized Thin-Walled Members ........................................................................ 272

Figure B.14 Two-Cell Box Girder Superstructure.................................................................. 274

Figure B.15 Example Two-Cell Box Girder ...........................................................................276 

Figure B.16 Example Four-Cell Box Girder...........................................................................278 

Figure C.1 Elevation of Example 1 Bridge........................................................................... 280

Figure C.2 Cross Section through Example 1 Bridge .......................................................... 280

Figure C.3 Cross Section Dimensions ................................................................................ 283

Figure C.4 Self Weight and Component Bending Moments ................................................288 

Figure C.5 Future Wearing Surface Bending Moments.......................................................288 

Figure C.6 Concrete Creep Bending Moments ...................................................................289 

Figure C.7 Concrete Shrinkage Bending Moments .............................................................289 

Figure C.8 Live Load Envelope Bending Moments ............................................................. 290

Figure C.9 Initial Post-Tensioning Bending Moments.......................................................... 290

Figure C.10 Bending Moments for Post-Tensioning Losses.................................................. 291

Figure C.11 Final Post-Tensioning Moments ........................................................................ 291

Figure C.12 Mohr’s Circle .....................................................................................................308

Figure C.13 State of Stress at Node 252 ..............................................................................308

Figure C.14 Overhang Design Sections................................................................................ 314

Figure C.15 Equilibrium at Strength Limit State .................................................................... 315 

Figure C.16 Length of Loaded Areas .................................................................................... 317 

Figure C.17 Live Load on Overhang ..................................................................................... 318 

Figure D.1 Curved Bridge of Design Example 2.................................................................. 322 

Figure D.2 Elevation of Example 2 Bridge........................................................................... 323 

Figure D.3 Cross Section through Example 2 Bridge .......................................................... 323 

Figure D.4 Cross Section Dimensions ................................................................................ 325 

Figure D.5 Frame Model for Transverse Analysis ............................................................... 327

Figure D.6 Load Location for Maximum Positive Flexure at Node 6.................................... 328
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List of Tables

Table 3.1 Limiting Eccentricities for Example Girder ........................................................... 47 

Table 4.1 Tendon Loss Calculations—Friction and Wobble ................................................ 65 

Table 4.2 Tendon Elongation .............................................................................................. 67 

Table 5.1 Moment Components of Service Level III Loading at Three Locations ..............102 

Table 5.2 Data for Friction Diagram for the CG Profile Tendon .........................................103 

Table 6.1 Bending Moments for Bridge on Bearings and Bridge with Fixed Piers .............128 

Table 7.1 Example Bridge 1 Bending Moments (ft-kips)....................................................148 

Table 7.2 Example Bridge 1 Bending Moments (ft-kips)....................................................148 

Table 8.1 Railing Loads for TL-4 Barrier (from AASHTO LRFD Table A13.2-1) ................189 

Table 8.2 Transverse Bending Moment Results from Frame Analysis ..............................194 

Table 8.3 Transverse Bending Moment Results from Frame Analysis ..............................207 

Table 8.4 Transverse Bending Moment Results from Frame Analysis ..............................208 

Table B.1 Coordinates of Points Defining the Thin-Walled Section.................................... 272

Table B.2 Thin-Walled Section Member Dimensions......................................................... 272

Table C.1 Service Limit State Load Factors....................................................................... 292

Table C.2 Service Limit State Flexural Verifications........................................................... 294

Table C.3 Strength Limit State Load Factors ..................................................................... 295

Table C.4 Flexural Strength Design Verifications...............................................................299 

Table C.5 Summary of Shear Design at Strength Limit State.............................................306

Table C.6 Principal Tensile Stress Summary.....................................................................309 

Table C.7 Load Factors for Overhang Design.................................................................... 313 

Table D.1 Fixed End Moments for Maximum Flexure at Node 6 ........................................ 328

Table D.2 Design Live Load Moments ............................................................................... 329

Table D.3 Service Limit State Load Factors....................................................................... 329

Table D.4 Top Slab Stresses at Stressing of PT................................................................ 330

Table D.5 Top Slab Stresses due to DC, DW, CR, SH after all losses............................... 331

Table D.6 Top Slab Stresses From Service I (Compression) and Service III (Tension)

Load Combinations ........................................................................................... 331

Table D.7 Strength I Load Factors for Transverse Analysis ............................................... 332

Table D.8 Ultimate Moments for Transverse Design.......................................................... 332

Table D.9 Ultimate Capacity of the Top Slab ..................................................................... 333

Table D.10 φMn vs. 1.33·Mu ................................................................................................ 333

Table D.11 Positive Bending Reinforcement, Bottom Slab and Webs.................................. 334

Table D.12 Negative Bending Reinforcement, Bottom Slab and Webs................................ 334

Table D.13 Extreme Event II Load Factors .......................................................................... 335

Table D.14 Number of Lanes per Web—Bending................................................................ 340

Table D.15 Number of Lanes per Web—Shear ................................................................... 341

Table D.16 Service Load Combinations............................................................................... 342

Table D.17 Change in Superstructure Stresses over Time .................................................. 344

Table D.18 Stress Summaries............................................................................................. 345

Table D.19 Strength Limit State Design Factors .................................................................. 346

Table D.20 Factored Moments ............................................................................................ 349

Table D.21 Verification of Torsion Considerations ............................................................... 351

Table D.22 Ultimate Shear at Node 27 ................................................................................ 353

Table D.23 Verification of Torsion Considerations ............................................................... 355

Table D.24 Shear at Node 26 .............................................................................................. 356

Table D.25 Ultimate Shear Forces....................................................................................... 357

Table D.26 Concrete Shear Capacity, Vci ............................................................................ 358

Table D.27 Concrete Shear Capacity, Vcw ........................................................................... 358

Table D.28 Required Web Reinforcing for Shear................................................................. 359
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Table D.29 Regional Web Bending Due to PT Moments and Required Reinforcing ............ 360

Table D.30 Shear Reinforcing Requirements....................................................................... 362

Table D.31 Shear Reinforcing Design Regions.................................................................... 362

Table D.32 Web Bending Reinforcing Requirements........................................................... 363

Table D.33 Final Web Reinforcing at Each Face of Each Web ............................................ 364

Table D.34 Results of Principal Tension Check ................................................................... 367

Table D.35 Verification of Longitudinal Shear Reinforcing ................................................... 368
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Chapter 1—Introduction

The objective of this manual is to present design methodologies for cast-in-place concrete box 

girder bridges post-tensioned with internal post-tensioning tendons, within the framework of the 

AASHTO LRFD Bridge Design Specifications (2012). The target audience for this manual is a 

graduate civil engineer with one year of bridge design experience. The manual presumes that 

the target audience has been exposed to prestressed concrete concepts, but does not 

necessarily have prestressed concrete design experience.

1.1 Historical Overview

The origin of reinforced concrete bridge construction in the United States dates back to 1889 

with the construction of the Alvord Lake Bridge in San Francisco, California. Though many 

advancements have been made, basic features of construction remain unchanged. The work

requires construction of formwork to contain and provide shape to the wet concrete. Formwork 

is supported by falsework either resting on the ground or on prepared foundations, until the 

structure itself is self-supporting and formwork and/or falsework can be removed. Unfortunately, 

bridges constructed with reinforced concrete are only economical for relatively short spans. 

Superstructure types include flat slabs, beam with slabs, and box girders. At the time, longer 

spans were achieved by using arch construction. 

Reinforced concrete box girder bridge construction flourished in the western part of the United 

States as a result of economy and local contractor experience. The California Department of 

Transportation (Caltrans) began constructing box girder bridges in the early 1950’s. With the 

popularization of prestressed concrete technology in the early 1960’s, Caltrans realized further 

economy through the construction of many post-tensioned concrete box girder bridges. 

Refinements to post-tensioned box girder bridge construction continued throughout the United 

States in the second half of the 20th century. Figure 1.1 shows two views of a cast-in-place 

post-tensioned box girder bridge.

Figure 1.1 – Cast-in-Place Post-Tensioned Box Girder Bridge for the Metropolitan Atlanta 

Regional Transit Agency (MARTA) – under construction (left), completed (right) 
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Today, cast-in-place post-tensioned box girder construction is used throughout the United 

States. The majority of this type of construction still occurs in western states, with much less 

frequency in other parts of the U.S. Reasons for this are varied, but stem from historical and 

regional developments—steel bridge construction in the northeast, precast prestressed beams 

in southern states, cast-in-place box girders in the west, etc. Though regional construction 

experience and expertise affect construction costs and consequently type selection, the need 

for further construction economy and alternate project delivery methods have led to a wider 

range of project specific bridge type evaluations.

Figure 1.2 shows a chart of applicable span ranges for the major types of prestressed concrete 

bridges. The span range for cast-in-place box girder construction is shown to vary from 100 feet 

to 250 feet. The lower end of the span range represents simple span bridges, shallow box 

girder bridges with depths restricted by vertical clearances, or bridges following highly curved 

alignments. The upper end of the span range represents continuous bridges, bridges with no 

restriction on box girder depth, or bridges on a tangent alignment. Longer span lengths can be 

achieved by using a variable depth structure, with deeper sections at piers to resist high 

negative moment demands.

The flexibility to accommodate a wide variety of span lengths and bridge geometries, over the 

most common range of highway bridge spans, is one of the excellent benefits of cast-in-place 

box girder construction. Other significant benefits include internal redundancy (multiple load 

paths), torsional stiffness and strength, and construction economy less sensitive to overall 

bridge size and aesthetics.

Figure 1.2 – Typical Span Ranges for Prestressed Concrete Bridge Types

1.2 Typical Superstructure Cross Sections 

The superstructure cross sections of post-tensioned box girder bridges are typically multi-cell or 

single-cell box girders. A typical cross section of multi-cell box girder bridge is shown in figure 

1.3. Figure 1.4 shows a typical cross section for a single-cell box girder superstructure.
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Figure 1.3 – Multi-Cell Box Girder Cross Section

Figure 1.4 – Single-Cell Box Girder Cross Section

The basic components of the cross section are:

• Top slab—the entire width of concrete deck, including the portions between the webs

and the overhangs outside of the webs.

• Overhangs (cantilever wings)—the overhanging portion of the top slab.

• Webs—vertical or inclined, exterior or interior.

• Bottom slab.

Multi-cell girder cross sections as shown in figure 1.3 can be used for bridges of nearly any 

width, by varying the spacing between, and/or, changing the number of webs. Widths of singlecell box girders typically range from 25 feet to 60 feet, though there are single-cell box girder 

cross sections as wide as 80 feet. This wide range of widths of single-cell box girders is 

achieved through the use of transverse post-tensioning within the top slab to control tensile 

stresses under the action of the permanent dead and live wheel loads plus impact effects. 

1.3 Longitudinal Post-Tensioning Layouts 

Cast-in-place box girder bridges are prestressed using post-tensioning tendons cast within the 

web concrete. These tendons are usually draped following parabolic profiles as shown in figure 

1.5. The tendon profiles are low in the cross section at the center of the span and rise in 

elevation at the ends of the span. The vertical distance from the neutral axis of the bridge to the 

centroid of a post-tensioning tendon is called the tendon eccentricity (e). The force in the 

tendon multiplied by the eccentricity forms the primary moment due to post-tensioning. The 

primary moment, along with the axial compression induced by the post-tensioning, work to 

offset the longitudinal tensile stresses in the superstructure resulting from bridge self weight and 
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other applied loads. Vertical components of the prestressing force can offset or add to the 

shear demand of the webs.

Tendons for simple span bridges are grouped closely together at the bottom of the bridge web 

at mid-span to maximize tendon eccentricity. The spacing of the tendons increases at the ends 

of the span to appropriately locate the post-tensioning anchorages. Post-tensioning anchorages 

are cast into diaphragms constructed at the ends of the spans. The diaphragms, which are 

solid concrete sections, transfer and distribute the tendon forces acting on the anchorages to 

the typical cross section of the box girder.

Figure 1.5 – Typical Post-Tensioning Tendon Layout for Simple Spans 

Continuous post-tensioned box girder construction is achieved by stressing long tendons that 

reach the full length of the continuous unit. The tendons are anchored at either end of the unit 

with geometry similar to the ends of simple spans. Within the spans of the continuous unit, the 

tendons drape with geometry similar to that shown in figure 1.6. Tendon profiles are low in the 

section within the span and high in sections over interior piers. Figure 1.7 shows the tendons in 

the webs in cross section view at mid-span and over the piers.

Figure 1.6 – Tendon Layout for 4-Span Bridge, CIP on Falsework 

Primary moments resulting from the post-tensioning are the same in both simply supported and

continuous structures. In a continuous superstructure, however, restraint of end rotations by 

adjacent spans and monolithic columns cause the development of secondary moments due to 

the post-tensioning. For tendon profiles similar to those shown in figure 1.6, the secondary 

moments reduce the effect of primary moments at mid-span sections and add to the effect of 

the primary moments over the piers. There are no secondary moments in a simply-supported 
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structure as the ends of the simple span are free to rotate and translate under the action of the 

post-tensioning.

Figure 1.7 – Tendon Locations within Box Girder Cross Section 

In very unique cases, an alternate to full length tendons in continuous spans is staged 

construction using shorter tendons that overlap at the piers. Figure 1.8 shows a concept of 

staged construction for the same four-span unit shown in figure 1.6. This approach can produce 

savings in falsework and formwork, but these savings may be offset by an increase in tendon 

and anchorage cost and by a slower rate of construction, as each span must gain sufficient 

strength prior to stressing the post-tensioning. The state of stress in bridges constructed in 

stages can be significantly different than those cast full length. Design calculations should 

consider the changing structural system as construction progresses and appropriate long-term 

bridge behavior.

Figure 1.8 – Possible Tendon Layout for Sequentially Cast Spans 
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1.4 Loss of Prestressing Force

Post-tensioning tendon forces are established in design to provide precompression to offset 

undesirable tensile stresses in the concrete box girder. The engineer conveys the tendon force 

requirements in the contract drawings as either the required jacking force at the end of the 

tendon or the final effective force at some point along the length of the tendon. The differences 

between jacking forces and effective forces are called prestressing force losses. Prestressing 

force losses can be grouped into two families: 1) losses related to the material properties of the 

concrete and prestressing steel, and 2) losses related to the mechanics of the post-tensioning 

system and tendon geometry. These losses, summarized below, are presented in detail later in 

this Manual.

Losses Related to Material Properties

• Elastic shortening of concrete 

• Shrinkage of concrete

• Creep of concrete

• Relaxation of prestressing steel

Losses Related to Physical Characteristics

• Duct friction due to curvature

• Wobble (unintentional friction)

• Wedge Set (or Anchor Set)

1.5 Post-Tensioning System Hardware

1.5.1 Basic Bearing Plates

A basic bearing plate is a flat plate bearing directly against concrete. This includes square, 

rectangular, or round plates, sheared or torch cut from readily available steel plate. Basic 

bearing plates are used in conjunction with galvanized sheet metal or plastic trumpets to 

transition from the strand spacing in the wedge plate to the duct (figure 1.9).

Basic bearing plate anchorages should comply with the requirements of section 10.3.2 of the 

AASHTO LRFD Bridge Construction Specifications (3rd Edition with Interims through 2015). 

Figure 1.9 - Basic Bearing Plate Anchorage System
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1.5.2 Special Bearing Plates or Anchorage Devices

A special bearing plate or anchorage device is any anchorage hardware that transfers tendon 

force into the concrete but does not meet normal analytical design requirements for basic 

bearing plates. Covered by this definition are devices having single or multiple plane bearing 

surfaces, and devices combining bearing and wedge plates in one piece. These anchorages 

typically require increased confinement reinforcement and should be accepted on the basis of 

physical tests. Figure 1.10 shows a cut-away view of a multi-plane anchorage system. Figure 

1.11 shows the components of an anchorage system for a four strand tendon in flat duct, 

commonly used in slabs.

Figure 1.10 – Multi-Plane Anchorage System (Courtesy of VSL) 

Figure 1.11 – Anchorage System for Flat Duct Tendon (Courtesy of DSI) 
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Use of a special bearing plate or anchorage device is acceptable if it complies with the testing 

requirements of section 10.3.2.3 of the AASHTO LRFD Bridge Construction Specifications.

1.5.3 Wedge Plates

Wedge plates, in conjunction with wedges, transfer the prestressing force in the strands to the 

anchorage. Wedge plates should comply with “Guide Specifications for Grouted PostTensioning, (PTI/ASBI M50.3-12, 2012)” section 4.3.2.

1.5.4 Wedges and Strand-Wedge Connection

Wedge performance is critical to the proper anchoring of strands. Different wedges have been 

developed for particular systems and applications such that there is no single standard wedge. 

However, wedges for post-tensioning systems should have the following characteristics:

• Wedge length at least 2.5 times the strand diameter. 

• Wedge angle of 5 to 7 degrees. 

• Internal serrated teeth for gripping the strand. 

• Case-hardened low carbon or alloy steel. 

• Two or three parts with a spring wire retainer clip or o-ring in a groove around the thick 

end of the wedge.

Wedges are case hardened with a ductile core to bite into the strand and conform to the 

irregularity between the strand and wedge hole. In so doing, the surface of the wedge may 

crack. This is normally acceptable and does not affect performance so long as wedge sections

do not break completely into separate pieces. Often, it is only the portion outside the retainer 

ring that cracks.

Wedges should comply with “Guide Specifications for Grouted Post-Tensioning, (PTI/ASBI 

M50.3-12, 2012)” section 4.3.2.

1.5.5 Permanent Grout Caps

Permanent grout caps similar to those shown in figure 1.12 should be provided in accordance 

with Protection Levels specified in section 3.0 of “Guide Specifications for Grouted PostTensioning, (PTI/ASBI M50.3-12, 2012).” Project specific documents should specify when and 

where caps are required.

Permanent grout caps should be made of a non-corrosive material such as fiber reinforced 

plastic or stainless steel. To ensure an enduring, maintenance-free, life of 75 years fiber 

reinforced plastic caps should contain an anti-oxidant additive with an environmental stress 

cracking endurance of 192 hours per ASTM D1693; stainless steel caps should meet the 

requirements of ASTM A240 Type 316.

Grout caps shall meet the requirements of “Guide Specifications for Grouted Post-Tensioning, 

(PTI/ASBI M50.3-12, 2012)” section 4.3.3.
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Figure 1.12 – Permanent Plastic Grout Caps (Courtesy of VSL) 

1.5.6 Ducts

Ducts are used to form a continuous void through the concrete for later placement of the posttensioning tendon steel. Originally, little attention was paid to the possible role of the duct as a 

barrier to corrosive agents. Today, strong emphasis is placed on the quality, integrity and 

continuity of the duct as a corrosion barrier in itself. This has resulted in a move toward the use 

of high density plastic ducts in some states. Nevertheless, more traditional metal ducts are still

used. 

1.5.6.1 Duct Size

Section 5.4.6.2 of the AASHTO LRFD Bridge Design Specifications calls for the inside crosssectional area of the duct to be at least 2.0 times the net area of the strand tendon. The one 

exception cited by AASHTO is in the case where the tendons are to be placed by the pullthrough method. In this case, the inside duct area should be 2.5 times the net area of the 

strand tendon. Section 4.3.5 of “Guide Specifications for Grouted Post-Tensioning, (PTI/ASBI 

M50.3-12, 2012)” standardizes the inside cross-sectional area of the duct to be at least 2.5 

times the net area of the strand tendon cross-sectional area.

Oval “flat” ducts are commonly used for transverse tendons in deck slabs of concrete box 

girders. These transverse tendons have typically been made of up to 4 strands of 15 mm (0.6

in) diameter, though there are systems that will accept up to 5 strands. The internal clear 

dimensions of oval duct for a four strand tendon should be a minimum of 25 mm (1 in) vertically 

and 75 mm (3 in) horizontally.

1.5.6.2 Corrugated Steel Duct

Ducts are spirally wound to the necessary diameter from strip steel with a minimum wall 

thickness of 0.45 mm (26-gauge) for ducts less than 66 mm (2-5/8 in) diameter or 0.6 mm (24-

gauge) for ducts of greater diameter. The strip steel should be galvanized to ASTM 

A653/A653M with a coating weight of G90. Ducts should be manufactured with welded or 

interlocking seams with sufficient rigidity to maintain the correct profile between supports during 

concrete placement (figure 1.13). Ducts should also be able to flex without crimping or 
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flattening. Joints between sections of duct and between ducts and anchor components should 

be made with positive, metallic connections that provide a smooth interior alignment with no lips 

or abrupt angle changes. 

Figure 1.13 – Corrugated Metal Duct

1.5.6.3 Corrugated Plastic

Corrugated plastic ducts, as shown in figure 1.14, are also used for tendons internal to the 

concrete. These ducts should be seamless and fabricated from polyethylene or polypropylene 

meeting the requirements of section 4.3.5.2 of “Guide Specifications for Grouted PostTensioning, (PTI/ASBI M50.3-12, 2012).”

Figure 1.14 – Corrugated Plastic Duct
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1.5.6.4 Plastic Fittings and Connections for Internal Tendons

All plastic duct splices, joints and connections to anchorages should be made with couplings 

and connectors that produce a smooth interior duct alignment with no lips or kinks. All fittings 

and connections between lengths of plastic duct and between ducts and steel components (e.g. 

anchors or steel pipe) should be made of materials compatible with corrugated plastic ducts. 

Plastic materials should contain antioxidant stabilizers and have an environmental stress 

cracking of not less than 192 hours as determined by ASTM D1693 “Standard Test Method for 

Environmental Stress-Cracking of Ethylene Plastics,” Condition C. Duct tape should not be 

used to join or repair ducts or make connections. See “Post-Tensioning Tendon Installation and 

Grouting Manual (2013),” available from the Federal Highway Administration, 

(http://www.fhwa.dot.gov/bridge/pt/) for further information on duct couplers.

1.5.6.5 Grout Inlets, Outlets, Valves and Plugs

Grout inlets, outlets, valves and plugs should be made of polypropylene or polyethylene meeting 

the requirements for plastic, corrugated ducts. Grout inlets, outlets, valves and plugs shall meet 

the requirements of “Guide Specifications for Grouted Post-Tensioning, (PTI/ASBI M50.3-12, 

2012)” sections 4.3.12 and 4.4.4. Figure 1.15 shows a graphic depiction of grout vents 

extending from an embedded duct.

Tubes for inlets and outlets for strand tendons should have a minimum inside diameter of 20

mm (3/4 in). For bar tendons and for tendons comprising up to 4 strands, tubes should be at 

least 10 mm (3/8 in) internal diameter. Inlets and outlets should be closeable with suitable 

valves or plugs. For grouting of long vertical tendons, dual mechanical shut-off valves are 

usually necessary to facilitate intermediate stages of grouting and venting.

Inlets and outlets should be arranged and attached to ducts, anchorages and grout caps in a 

manner that allows all air and water to escape in order to ensure that the system is completely 

filled with grout. (See chapter 4 of “Post-Tensioning Tendon Installation and Grouting Manual

(2013)” for examples of locations of inlets and outlets.)

Figure 1.15 – Typical High-Point Grout Vent

1.5.7 Post-Tensioning Bars Anchor Systems

Anchorage systems for post-tensioning bars are comprised of bearing plates and anchor nuts 

similar to the components shown in figure 1.16. The anchorage system should comply with 

“Guide Specifications for Grouted Post-Tensioning, (PTI/ASBI M50.3-12, 2012)” section 4.3.2.
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Figure 1.16 – Post-Tensioning Bar Anchorage System (Courtesy of DSI) 

1.6 Overview of Construction

1.6.1 Falsework

Falsework is the structural system that supports the formwork onto which the concrete of the 

box girder will be cast. Falsework systems can be comprised of prefabricated modular shoring 

towers comprising well-braced interlocking frames in a square or rectangular arrangement of 

four legs as shown in figure 1.17. Multiple towers are located as necessary to support the 

falsework deck which in turn supports the superstructure formwork. 

Figure 1.17 – Modular Falsework Units for Cast-in-Place Construction 
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Main vertical falsework supports may also consist of a series of individual columns made of 

structural steel sections or heavy section lumber. Poor ground conditions or the need to span 

an underlying roadway or waterway may limit the locations of falsework columns. Fewer but 

larger falsework columns, along with larger falsework beams, may be used to provide the 

needed clearances during construction. Figure 1.18 shows a network of falsework columns 

used to maintain navigational clearances during construction.

Figure 1.18 – Steel Pipe Support Towers for Cast-In-Place Construction

Depending on the nature of the site and the cost of temporary construction, falsework may be 

provided for the superstructure to be cast-in-place over the entire length of the bridge or 

continuous superstructure unit. If these costs are prohibitive, it may be necessary to move 

falsework from span to span as each is constructed and made self-supporting, provided that 

appropriate engineering analysis of intermediate states of construction is performed. In either 

case, careful consideration should be given to assuring that environmental clearances can be 

achieved for placing falsework. 

 

For further guidance with regard to the design and construction of falsework, refer to the 

following publications:

• “Guide Design Specifications for Bridge Temporary Works”, (2008 Interim, AASHTO). 

• “Construction Handbook for Bridge Temporary Works”, (2008 Interim, AASHTO). 

• “Falsework Manual” (January 1988, Caltrans). 

1.6.2 Superstructure Formwork

The falsework system provides supports for the superstructure formwork. Formwork may be 

made from lumber and plywood or prefabricated modular forming systems. Accuracy to line, 

level and thickness is essential to ensure the correct final shape and size of concrete members. 

External surfaces are usually formed of a high quality, smooth and dense finished plywood, 

metal or any required aesthetic texture, as necessary. Internal surfaces should be within 
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tolerance but are usually of a lesser quality finish and forming material. Figure 1.19 shows form 

components for a single-cell box girder bridge.

Figure 1.19 – Web and Cantilever Wing Formwork for a Single-Cell Box Girder

Box girder sections can be constructed in stages, beginning with the bottom slab, webs and 

finally the top slab, as shown in figure 1.20. In this Figure, the bottom slab for this portion of the 

bridge has been cast, web reinforcing and longitudinal ducts for post-tensioning tendons have 

been tied and portions of the webs have been cast. Supports for top slab forming are being 

placed in the portion where webs are complete. Many bridges combine the casting of the bottom 

slab and webs into one stage. 

Access to internal cells is usually necessary through diaphragms or manholes for future 

maintenance inspection which also provides a convenient way through which internal formwork 

can be removed after casting. Purpose-made, permanent, internal top slab soffit forms (lost 

deck forms) may remain in place provided that their weight and structural connectivity, if any, 

have been accounted for in the design.

Figure 1.20 – Web Formwork for a Multi-Cell Box Girder
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1.6.3 Reinforcing and Post-Tensioning Hardware Placement

Reinforcing steel is placed in stages to coincide with the casting of the cross section 

components—i.e., bottom slab, webs, and top slab. Reinforcing should be detailed accordingly, 

giving attention to the location of bar splices to meet structural requirements and also facilitate 

forming and casting. Reinforcing steel should be installed to project construction tolerances. 

All post-tensioning ducts, anchorage components and anchorage reinforcement should be 

installed in conjunction with the reinforcement. It is preferable that reinforcement and posttensioning be designed and detailed free of conflicts. However, conflicts are not always evident 

in advance. Whenever a conflict is encountered between reinforcement and post-tensioning in

the field, generally, the reinforcement should be adjusted locally as necessary to maintain the 

desired post-tensioning alignment. In cases of doubt, a decision should be sought from the 

Engineer of Record. 

Figure 1.21 – Web and Bottom Slab Reinforcing (left), Tying Post-tensioning Ducts in Webs (right) 

1.6.4 Placing and Consolidating Superstructure Concrete

Box girder superstructures can be cast in either two or three stages. When cast in two stages, 

the bottom slab and webs are poured at the same time. This is followed by the casting of the 

top slab. In three-stage casting the bottom slab, webs, and top slab are poured separately, with 

enough time between stages to permit sufficient concrete hardening. Longitudinal construction 

joints are normally located in the webs a few inches above the top of the bottom slab (threestage casting) and a few inches above the top of the webs in the top slab fillets (two-stage and 

three-stage casting). This is mainly for convenience of construction and to provide a clean joint 

between components. In order to ensure proper structural integrity and function, joints should be 

prepared, cleaned and roughened prior to the next pour. This is usually sufficient for shear 

transfer. However, construction keyways may be necessary and should be shown on the plans

where required. An approach to three-stage casting is presented in the following paragraphs.

The first stage of box girder construction is the casting of the bottom slab concrete. Placing 

typically commences at one side (usually the low side of grade and superelevation) and 

continues from there across the width of the slab and along the length of the bridge as 

necessary. The new open end face of the slab concrete face is kept fresh to facilitate 

consolidation with each new load of concrete. Concrete is consolidated using vibratory tools and 

then struck off to elevation by hand or mechanical screeds, followed by a float finish. 
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When the bottom slab concrete has set and sufficiently hardened the webs are formed. Web 

concrete is then placed and consolidated. The webs are placed in lifts of about two feet to 

control pressure on forms. Each fresh charge of concrete is consolidated and worked into the 

top of the previous lift—which must remain workable to receive the next load of concrete. 

Concrete in webs is typically consolidated by means of internal poker-type vibrators, though

external form vibrators can be used. In the latter case, formwork must be sufficiently robust and 

braced to withstand the heavy vibration. 

The top slab of the box section is cast last. Depending upon the profile of draped tendons, ducts 

may rise into the top slab at piers. All top slab reinforcing and, if necessary, transverse posttensioning tendons must be in place and set to correct elevations and required clear covers 

before concrete is placed. Ducts for longitudinal tendons should be checked for obstructions 

before the top slab concrete is placed. Concrete is usually placed working across and 

longitudinally up hill. Slab concrete is consolidated, struck off to levels with screeds (figure 1.22) 

and usually floated to a final finish. Longitudinal or transverse traffic tines may be brushed or 

groove-cut into the deck surface after curing to improve vehicular traction. 

The preceding paragraphs describe pouring the superstructure in three phases. Two-phased 

construction, in which bottom slab and webs are cast together, followed by the casting of the top 

slab, is also commonly used.

Longitudinally, vertical construction joints may be needed at various locations in a span or 

superstructure in order to keep the total volume of concrete placed to an amount which can be 

delivered, placed, consolidated and finished within a given work period. Vertical construction 

joints are typically not allowed in simple span bridges.

Figure 1.22 – Placing Deck Concrete and Finishing with a Roller Screed 

1.6.5 Superstructure Curing

Proper concrete set and sufficient strength is required prior to releasing forms before the next 

stage of casting and especially prior to imposing high local anchorage forces from posttensioning or releasing falsework. 
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For typical box girder construction, proper curing is accomplished using blankets, wet-burlap, 

moisture, fogging, and application of suitable curing compounds. Monitoring of internal concrete 

temperature using thermocouples or other devices at suitable locations over the curing period 

can be helpful in some cases, particularly for thick members and large pours. It provides a 

record of curing and can help avoid potential difficulties from a too rapid rise or fall from the heat 

of hydration. Protection of pours from adverse weather by enclosures and heating may be 

necessary in some situations (figure 1.23). 

Figure 1.23– Curing the Concrete Deck

1.6.6 Post-Tensioning Operations

Multi-strand tendons are the most frequent choice for main longitudinal tendons in bridges. All 

the strands of one longitudinal tendon are simultaneously tensioned using a multi-strand jack. 

The sequence of stressing tendons should be clearly shown on the contract plans or approved

shop drawings and must be followed on site. 

Post-tensioning strands may be pushed or pulled through ducts to make a tendon. Pushing 

should be done with care using a protective plastic or metal cap provided by the post-tensioning

system supplier so that the strands do not get caught by or introduce damage to the duct. 

Sometimes it may be easier to pull the entire tendon bundle of strands through the duct together 

using a special steel wire sock or other device securely attached to the end of the bundle (figure 

1.24). 
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Figure 1.24– Bundled Tendon Prepared for Pulling 

When a multi-strand tendon is stressed from one end it is often referred to as “single” or “oneend” stressing to distinguish it from tendons stressed from both ends. When a bridge has a 

number of similar, and often symmetrical, tendons that need only be stressed from one end,

“alternate end stressing” can be used to keep the overall post-tensioning effect as symmetric as 

possible. In this case, tendons are stressed from one end only, but from opposite, alternate, 

ends of the bridge for similar tendon profiles. 

When the tendons are very long, losses over the length of the tendon due to friction and wobble 

become large. Stressing the tendon from the second end results in a higher force in the tendon 

than if only stressed from one end. This is typically called “double” or “two-end” stressing. Also, 

for symmetrical tendons two-end stressing becomes effective when the effect of anchor set at 

the jacking end affects less than half of the tendon. Stressing from the second end should not 

be done if the calculated elongation is less that the length of the wedge grip. Re-gripping in a 

portion of the old grip length should be avoided.

It is important to also account for the staging of stressing across the width of the bridge. 

Individual tendon jacking forces (Pjack) must be selected to achieve a uniform distribution of 

stress across the width of the bridge.

Figure 1.25– Stressing Post-Tensioning Tendons 
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1.6.7 Tendon Grouting and Anchor Protection

After post-tensioning tendons have been installed and stressed, they must be properly grouted 

and anchorages sealed and protected to ensure long term durability. Grouting should proceed 

as soon as possible after installation and stressing of the tendons. Depending upon 

environmental conditions, temporary protection and sealing of open ducts may be necessary at 

anchorages, and temporary protection of the ends of the strands will be necessary. 

For comprehensive information on the installation, stressing, grouting and protection of posttensioning tendons and anchorages (including recommendations for the location of injection 

grout ports, vents, laboratory and field tests, quality control and records, etc.), refer to “PostTensioning Tendon Installation and Grouting Manual (2013)” available from the Federal 

Highway Administration (http://www.fhwa.dot.gov/bridge/pt/).
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Chapter 2—Materials

The primary materials needed for the design of cast-in-place post-tensioned box girders are: 

concrete, prestressing steel, and mild reinforcing. This chapter presents the material 

characteristics for these three materials with respect to the design of this bridge type.

2.1 Concrete

 

2.1.1 Compressive Strength

Compressive strength (f’c) is the characteristic that best gives an overall picture of the quality of 

a concrete. 

The basic components of concrete are Portland cement, aggregates (coarse and fine) of varied 

gradation, water, and admixtures. Concrete sets and gains strength as a consequence of a 

hardening of the cement/water gel through the chemical reaction of hydration. The ratio of 

water to cement (water/cement ratio) is an important factor of resulting concrete strength. If too 

little water is used, not all of the cement will undergo hydration and the desired strength will not 

be obtained. Excessive water leads to overly dispersed hardened cement particles, again 

leading to less than desired strength. Water/cement ratios often range from 0.35 to 0.40.

Freshly placed, unconsolidated concrete contains excessive and detrimental voids. 

Unconsolidated concrete, if allowed to harden, will be porous and will poorly bond to the 

reinforcement. The resulting hardened concrete will have low strength, high permeability, and 

poor resistance to deterioration. Freshly placed concrete should be consolidated if it is to have 

needed characteristics of structural concrete. 

Curing of the concrete is also important to producing high quality concrete. The main purpose 

of curing is to prevent unnecessary moisture loss, especially in the first few days of the initial 

hydration and strength development. In addition to moisture loss, control of the concrete 

temperature during curing is important. Hydration is an exothermic reaction, building up heat 

within the concrete member. This heat must be gradually dissipated in a controlled manner to 

offset excessive thermal gradients within the concrete that can lead to micro-cracking and 

diminished strength. 

Admixtures are incorporated into concrete mixes to enhance the qualities of the hardened 

concrete. A controlled percentage (4–6 percent) of well-dispersed, microscopic air bubbles 

introduced by air-entraining agents enhances durability against freeze-thaw and improves 

workability for placement and consolidation. Super-plasticizers improve workability, facilitating 

reduced water/cement ratios and enhanced strength. Supplementary cementitious materials, 

such as pozzolans (most typically fly ash, silica fume, or granulated blast furnace slag), can be 

used in conjunction with or as a replacement to part of the cement to contribute to the properties 

of the hardened concrete. While increasing certain characteristics, excessive or poorly matched 

admixtures can have a negative impact on the resulting concrete.

Concrete compressive strength is determined by physical testing in accordance with AASHTO 

T22 (ASTM C39). Tests are performed at a standardized age of 28 days by compression tests 

to failure of sample cylinders 6 inches in diameter and 12 inches in length.

AASHTO LRFD Article 5.4.2.1 specifies that prestressed concrete shall not have a compressive 

strength less than 4.0 ksi. Typical 28-day concrete compressive strength for cast-in-place posttensioned box girders range between 5.0 ksi to 6.0 ksi. Higher strength concrete can be used, 
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Chapter 2 – Materials and Post-Tensioning Components 21 of 389 

but local thickening of concrete members, such as bottom slabs near piers to control 

compressive stresses, are generally more cost effective that the increased cost of higher 

strength concrete throughout the bridge. In addition, higher strength concretes cast on site are 

more susceptible to environmental factors than high strength concrete produced in more 

controlled settings in casting yards.

Guidance as to the type or “Class of Concrete” for various applications is provided in AASHTO 

LRFD Article 5.4.2.1. Concrete mix characteristics, including strength, minimum cement content, 

maximum water cement ratio, range of air content and coarse aggregate per class of concrete, 

are given in AASHTO LRFD Table C5.4.2.1-1. 

2.1.2 Development of Compressive Strength with Time

Standardized testing classifies strength at an age of 28 days. However, concrete continues to 

increase in strength over time. The increase in concrete compressive strength acts to increase 

other material characteristics that are related to strength (tensile strength, modulus of elasticity, 

time-dependent effects, etc.). As a result, it is important to be able to consider the change in 

concrete strength with time. The AASHTO LRFD specifications do not specifically address this 

feature of concrete, but other codes do. The Comite Euro-Intermationale du Beton/Federation 

International de la Precontrainte (CEB-FIP) Model Code (1990) provides the following 

relationship for the change in concrete compressive strength over time:

(Eqn. 2.1) 

(Eqn. 2.2)

Where, fcm = 28-day compressive strength

fcm(t) = concrete compressive strength at time t

βcc = time-dependent coefficient dependent on age of concrete

t = age of concrete at which fcm(t) is computed (days)

t1 = 1 day

s = cement rate of hardening coefficient

(0.20 for rapid hardening high strength concretes, 0.25 for normal and rapid 

hardening cement, 0.38 for slow hardening cements)

Note that the 28-day compressive strength predicted by testing in accordance with the AASHTO 

LRFD specifications is used without modification in equation 2.1. This is as opposed to using 

the 28-day strength plus 8 MPa as in other CEB-FIP equations.

Figure 2.1 shows a plot of the ratio of concrete compressive strength to 28-day compressive 

strength (βcc) for normal hardening cement.
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Figure 2.1 – Concrete strength gain with time

2.1.3 Tensile Strength

Concrete tensile strength greatly impacts prestressed concrete design as it forms the basis of 

the criteria for which post-tensioning force and layout is typically chosen. AASHTO LRFD 

specifies concrete tensile stress in two different ways: Modulus of Rupture and Direct Tensile 

Strength. 

The Modulus of Rupture (fr) is defined as the tensile stress in concrete developed by concrete 

flexure. For concrete compressive strengths up to 15 ksi and for normal-weight concrete, 

AASHTO LRFD Article 5.4.2.6 specifies the modulus of rupture in ksi to be:

(Eqn. 2.3) 

The Modulus of Rupture is determined by standardized test AASHTO T97 (ASTM C78).

The Direct Tensile Strength, also designated as fr, is specified in AASHTO LRFD Article 5.4.2.7. 

The commentary of this section specifies that for normal-weight concrete of compressive 

strengths up to 10 ksi, the direct tensile strength in ksi is:

(Eqn. 2.4) 

Traditionally, testing for direct tensile strength is by the split cylinder tensile strength method 

performed in accordance with AASHTO T198 (ASTM C496). However, pull out methods 

specified in ASTM C900 may also be used.

The small difference in value between equations 2.3 and 2.4 is interesting to note. Most 

research indicates that, depending on the strength of the concrete, the modulus of rupture is 

significantly greater than the direct tensile stress (as much as two times as large). This is noted 

in the commentary to AASHTO LRFD Article 5.4.2.6., with rationale provided for the limit shown 

in equation 2.3. Of more practical importance are the allowable stresses established at service 

limit states for the particular behavior. For example, flexural tensile stresses in prestressed 
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Chapter 2 – Materials and Post-Tensioning Components 23 of 389 

concrete are most commonly limited to 40 percent of the modulus of rupture, where principal 

tensile stresses in webs of are limited to 50 percent of the direct tensile capacity of the concrete.

2.1.4 Modulus of Elasticity

The modulus of elasticity of a material, Ec, is the ratio of normal stress to corresponding strain in 

compression or tension. The modulus of elasticity is an integral feature of the stiffness based 

methods used to analyze structures. The modulus of elasticity is a part of the characteristic 

member stiffnesses which populate element stiffness matrices. Element stiffness matrices are 

transformed and assembled into a global stiffness matrix representing the entire structure. 

Member end (nodal) displacements are then computed by pre-multiplying the vector of applied 

fixed end member forces by the inverted stiffness matrix (or some more mathematically efficient 

matrix method). Member end forces are then back-computed from the nodal displacements.

Figure 2.2 shows a typical stress strain curve for concrete. The relationship of stress to strain 

as shown in this Figure is highly non-linear. However, within the range of stresses that define 

the service limit state for which bridges are designed, the relationship can be approximated by a 

linear relationship. Figure 2.2 also shows a line whose slope is the modulus of elasticity for the 

linear range. This modulus of elasticity is a secant modulus standardized by physical testing in 

accordance with ASTM C469. The two points that define the secant modulus are a strain of 

50x10-6 (ε1) and corresponding stress f1, and a strain ε2 that corresponds to a stress of 0.4f’c. 

Figure 2.2 – Typical Stress-Strain Curve for Concrete 

AASHTO LRFD Article 5.4.2.4 provides the equation for the concrete modulus of elasticity. This 

well-known equation, developed to fit physical testing, relates the modulus of elasticity of the 

concrete to the concrete unit weight and the concrete 28-day compressive strength. Equation 

2.5 provides the equation for the modulus of elasticity in ksi:

(Eqn. 2.5)
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where, f’c = the 28-day compressive strength of concrete (ksi)

wc = the unit weight of concrete (kcf)

Κ1 = correction factor for aggregate (equal to 1 unless determined by physical 

test and approved by authority of jurisdiction)

2.1.5 Modulus of Elasticity Variation with Time

Though the CEB-FIP Model Code 1990 is being used to estimate time-dependent 

characteristics, the AASHTO LRFD specification equation for Modulus of Elasticity as provided 

in Article 5.4.2.4 is used to estimate the 28-day Modulus of Elasticity.

The variation in Modulus of Elasticity with time is predicted by CEB-FIP Equations 2.1-57 and 

2.1-58:

 

(Eqn. 2.6) 

(Eqn. 2.7) 

Where, Eci = the modulus of elasticity of concrete at an age of 28 days

Eci(t) = the modulus of elasticity at time t

β e = coefficient depending on age of concrete, t (days)

βcc = coefficient defined by equation 2.2

Figure 2.3 shows a plot of the ratio of concrete modulus of elasticity to 28-day compressive 

strength (βE) for normal hardening cement. 

Figure 2.3 – Concrete Modulus of Elasticity with Time
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2.1.6 Poisson’s Ratio

Poisson’s ratio of lateral to axial strain (μ) is prescribed in AASHTO LRFD Article 5.4.2.5 to be 

equal to 0.20. 

2.1.7 Volumetric Changes

Volume changes in concrete arise from variations in temperature, shrinkage due to air-drying, 

and creep caused by sustained stress. These are influenced by environmental conditions such 

as temperature, humidity, and the maturity of the concrete, and by the time and duration of 

loading. Volume changes affect structural performance and must be properly accounted for 

when determining loss of prestress and long-term deflections.

2.1.7.1 Coefficient of Thermal Expansion

AASHTO LRFD Article 5.4.2.2 defines the coefficient of thermal expansion. For normal weight 

concrete the coefficient of thermal expansion (α) is 6x10-6 per OF.

2.1.7.2 Creep

The creep of concrete is defined as the increase in concrete strain under a sustained stress. 

Figure 2.4 shows the influence of concrete creep on increased strain under the loading and 

unloading of a sustained load on a concrete element.

Figure 2.4 – Creep of Concrete

The concrete element is loaded at time t0 and strains in proportion to the Modulus of Elasticity 

developed at that time. From time t0 to time t1, the concrete creeps under the action of the 

sustained stress. At time t1 the stress is removed and there is an elastic recovery in strain. The 

elastic recovery is less than the initial elastic strain because the Modulus of Elasticity has 

increased during the period of loading, as a function of concrete strength gain. Further 

reduction in strain is found at times greater than t1. This creep recovery strain is substantially 

smaller than the concrete creep, as it is proportional to the elastic recovery strain and occurs at 

a later age of the concrete’s life than the original loading. At some time, the effects of creep 

recovery stop and the concrete element is left with a permanent residual strain.

Creep characteristics can vary greatly as a function of the mix design, member thickness, 

environment, and nature of loading. Some important factors influencing concrete creep are:
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Chapter 2 – Materials and Post-Tensioning Components 26 of 389 

• Duration of Load—the longer the loading, the more creep realized.

• Initial Stress Level – the greater the level of initial stress, the more the concrete will 

creep.

• Concrete Age at Loading—the younger the concrete at loading, the more the 

concrete will creep.

• Relative Humidity—the higher the relative humidity of the loading environment, the 

lower the concrete will creep.

• Volume to Surface Ratio—the greater the volume to surface ratio, the less the 

concrete will creep.

• Concrete Strength—the greater the concrete strength the lower the creep. 

Generally speaking, highway bridge loads are divided primarily into two types: permanent and 

live loads. There are typically no live loads that remain on the bridge for extended durations. 

As a result the strain versus time relationship of figure 2.4 is modified to that of figure 2.5.

Figure 2.5 – Creep of Concrete (with no long-duration transient loads)

The AASHTO LRFD specifications permits three models for estimating time dependent behavior 

of these materials Article 5.4.2.3.1 permits three methods of computing the time-dependent 

creep behavior of concrete when mix-specific creep data is not available. The three methods 

permitted by code are:

• AASHTO LRFD Article 5.4.2.3.2.

• CEB-FIP Model Code.

• ACI 209.

The CEB-FIP Model Code 1990 is jointly produced by The Comite Euro-Intermationale du Beton 

(The Euro-International Committee for Concrete) and the Federation International de la 

Precontrainte (International Federation for Prestressing). The Model Code 1990 provides timedependent material characteristics for concrete and prestressing steel. Along with its 

predecessor, the Model Code 1978, the CEB-FIP Model Code has been used extensively over 

the last 30 years in the United States for estimating the effects of material time-dependent 

behaviors.

This Chapter provides the CEB-FIP Model Code (1990) time-dependent characteristics required 

for the analysis of prestressed concrete bridges. These characteristics have been used in both 

of the example problems presented in this Manual. The equation numbers presented in this 

Chapter are those of the CEB-FIP Model Code (1990).
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Chapter 2 – Materials and Post-Tensioning Components 27 of 389 

Within the elastic range, the creep strain at time t for a constant stress applied at time t0 is given 

as:

(Eqn. 2.8) 

Where, ε cc = the creep strain at time t > t0

Eci(t) = the modulus of elasticity of concrete at an age of 28 days

sc(t0) = compressive stress applied at time t0

f(t,t0) = the concrete creep coefficient

The creep coefficient is the product of the notional creep coefficient f0(t,t0) (sometimes called 

the ultimate creep coefficient) and a time dependent coefficient βc(t-t0). 

(Eqn. 2.9) 

Where, f(t,t0) = the creep coefficient0

f 0 = the notional creep coefficient

βc = the coefficient to describe the development of creep with time after 

loading

f(t,t0) = the concrete creep coefficient

The notional creep coefficient is a function of relative humidity, notional thickness, concrete 

strength, and age since loading, as per the following relationships:

(Eqn. 2.10) 

with,

(Eqn. 2.11) 

(Eqn. 2.12) 

(Eqn. 2.13) 

Where, RH = the relative humidity of the ambient environment (%)

RHo = 100 percent

fcm = the mean concrete compressive strength at 28 days (MPa)

fcmo = 10 MPa 

t0 = age of concrete at loading (days)

t1 = 1 day

h0 = 100 mm
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Chapter 2 – Materials and Post-Tensioning Components 28 of 389 

h = notional size of the member (mm) – see following equation where Ac is 

the cross section of the member and u is the perimeter in contact with the 

atmosphere.

In these expressions, the mean concrete compressive strength is a fixed number greater than 

the minimum compressive strength:

(Eqn. 2.14) 

where, fcm = mean compressive strength (MPa)

fck = 28-day concrete compressive strength (MPa)

∆f = 8 MPa

and the notional thickness is expressed as:

(Eqn. 2.15)

where, Ac = cross-sectional area

u = perimeter in contact with the atmosphere

For a closed box shape, where the environment inside the box girder can be markedly different 

than that outside the girder, standard practice has been to compute the perimeter in contact with 

the environment (u) as the outside perimeter plus one half of the interior perimeter.

The development of creep over time is given by:

(Eqn. 2.16) 

with,

(Eqn. 2.17) 

where, t1 = 1 day

RHo = 100 percent

h0 = 100 mm

Figure 2.6 shows a plot of the development of creep over time for a relative humidity of 75 

percent and a notional thickness of 319 mm (conditions of examples 1 and 2 in this manual).
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Chapter 2 – Materials and Post-Tensioning Components 29 of 389 

Figure 2.6 – Development of Concrete Creep with Time 

2.1.7.3 Shrinkage

The total shrinkage strain at time t is calculated by:

(Eqn. 2.18) 

Where, ε cso = the notional shrinkage strain

βs = coefficient to describe development of shrinkage with time

t = age of concrete (days)

ts = age of concrete at beginning of shrinkage (days)

The notional shrinkage coefficient is determined by:

(Eqn. 2.19)

with,

(Eqn. 2.20) 

where, fcm = the mean concrete compressive strength at 28 days (MPa)

fcmo = 10 MPa 

βsc = coefficient dependent on type of cement

(4 for slowly hardening cements, 5 for normal or rapid hardening 

cements, and 8 for rapid hardening high strength cements)
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Chapter 2 – Materials and Post-Tensioning Components 30 of 389 

For relative humidity between 40 percent and 99 percent,

(Eqn. 2.21) 

For relative humidity greater than 99 percent,

(Eqn. 2.22) 

where,

(Eqn. 2.23) 

The development of shrinkage with time is:

(Eqn. 2.24) 

Figure 2.7 shows a plot of the development of shrinkage over time for a relative humidity of 75 

percent and a notional thickness of 319 mm (conditions of examples 1 and 2 in this manual).

Figure 2.7 – Development of Concrete Shrinkage with Time 

Shrinkage is primarily a result of sustained drying of the concrete as it is exposed to its 

surrounding environment. The majority of shrinkage occurs early in the life of the concrete, 

gradually slowing over time until ultimate shrinkage values are approached. This is seen in 
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Chapter 2 – Materials and Post-Tensioning Components 31 of 389 

figure 2.8, which plots the Time Development Factor (ktd) used to predict concrete shrinkage (as 

well as creep) over time as per as per AASHTO LRFD Articles 5.4.2.3.2 and 5.4.2.3.3.

Figure 2.8 – Rate of Concrete Shrinkage over Time

As with concrete creep, AASHTO LRFD Article 5.4.2.3.1 permits three methods of computing 

time-dependent shrinkage behavior. It is important to note that the three methods should not 

be mixed. Inappropriately combined, the effects of for creep and shrinkage could be 

significantly miscalculated.

2.2 Prestressing Strands

Strands for post-tensioning are made of high tensile strength steel wire conforming to ASTM 

A416. A strand is comprised of 7 individual wires, with six wires helically wound to a long pitch 

around a center “king” wire. Strand is most commonly available in two nominal sizes, 0.5 inch 

and 0.6 inch diameter, with nominal cross-sectional areas of 0.153 in2 and 0.217 in2

, 

respectively. Though the majority of post-tensioning hardware and stressing equipment is 

based on these sizes, the use of 0.62 inch diameter strand has been increasing.

Strand size tolerances may result in strands being manufactured consistently smaller than, or 

larger than nominal values. Recognizing this, “Acceptance Standards for Post-Tensioning 

Systems” (Post-Tensioning Institute, 1998) refers to the “Minimum Ultimate Tensile Strength,” 

which is the minimum specified breaking force for a strand. Strand size tolerance may also 

affect strand-wedge action leading to possible wedge slip if the wedges and strands are at 

opposite ends of the size tolerance range.

Strand conforming to ASTM A416 is relatively resistant to stress corrosion and hydrogen 

embrittlement due to the cold drawing process. However, since susceptibility to corrosion 

increases with increasing tensile strength, caution is necessary if strand is exposed to corrosive 

conditions such as marine environments and solutions containing chloride or sulfate, phosphate, 

nitrate ions or similar. Consequently, ASTM A416 requires proper protection of strand 

throughout manufacture, shipping and handling. Protection during the project, before and after 

installation, should be specified in project drawings and specifications.
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Chapter 2 – Materials and Post-Tensioning Components 32 of 389 

2.2.1 Tensile Strength

All strands should be Grade 270 ksi low relaxation, seven-wire strand conforming to the 

requirements of ASTM A416 “Standard Specification for Steel Strand, Uncoated Seven Wire 

Strand for Prestressed Concrete.” ASTM A416 provides minimum requirements for mechanical 

properties (yield, breaking strength, elongation) and maximum allowable dimensional 

tolerances. The AASHTO LRFD specifications do recognize the use of Grade 250 ksi 

prestressing steel, but this material is almost never used in major bridge construction, and is not 

addressed in this manual.

2.2.2 Modulus of Elasticity

Figure 2.9 shows a stress-strain diagram for prestressing strand. Up to a loading of 

approximately 90 percent of the ultimate strength of a strand, the stress-strain relationship is 

near linear. The slope of this stress-strain curve in this region is the Modulus of Elasticity of the 

prestressing strand within the elastic limit. AASHTO LRFD Article 5.4.4.2 specifies that the 

Modulus of Elasticity for prestressing strands to be used for design (Ep) is equal to 28,500 ksi.

The AASHTO LRFD specifications do not provide a stress-strain relationship other than within 

the elastic range as defined by Ep. The Precast Concrete Institute presents the stress-strain 

relationship up to the ultimate capacity of the strand. For 270 ksi strand, the relationships 

presented by PCI are:

For εps ≤ 0.0086, 

(Eqn. 2.25) 

For εps > 0.0086,

(Eqn. 2.26)

Figure 2.9 is a plot of the stress-strain diagram for prestressing strand using equations 2.25 and 

2.26. The value of ultimate strain shown in figure 2.9 can vary with the source of the 

prestressing strand. Care should be used in selecting an appropriate value if excessive strains 

are required to provide ultimate capacity of a member.
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Chapter 2 – Materials and Post-Tensioning Components 33 of 389 

Figure 2.9 – Stress-Strain Diagram for Prestressing Strand (Courtesy PCI) 

2.2.3 Relaxation of Steel

Relaxation of steel stress is a result of an increase in elongation over time while under an 

applied stress. The loss in prestressing stress due to steel relaxation for low relaxation steel 

over a time interval from (t1) to (t) may be estimated as:

(Eqn. 2.27) 

With,

(Eqn. 2.28)

And,

(Eqn. 2.29) 

Where: fst=steel stress level at beginning of time interval from (t1) to (t) (psi)

fsu* = average stress in prestressed reinforcement at ultimate load (psi)

fy*=yield point stress of prestressing steel (psi)

t = end of time interval (days)

t1= beginning of time interval (days)
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Equations 2.27, 2.28 and 2.29 can be found in Article 10.4.1 of the Guide Specifications for 

Design and Construction of Segmental Concrete Bridges (2nd Edition 1999, with 2003 Interim 

Revisions). These equations do not appear in the AASHTO LRFD specifications in the form 

above, but are incorporated into predictions for prestressing force loss in AASHTO LRFD Article 

5.9.5.4.2c.

2.2.4 Fatigue

Tendons in prestressed concrete structures do not experience stress cycling significant enough 

to induce fatigue problems. Fatigue in concrete bridges is a concern only in certain applications 

such as cable-stays in cable-stayed bridges where traffic loads significantly affect stress 

variations.

2.3 Reinforcing Steel

Reinforcing steel shall be in accordance with AASHTO LRFD Article 5.4.3. For this manual only 

reinforcing steel with a yield stress of 60 ksi is considered. AASHTO LRFD Article 5.4.3.1 

permits the use of reinforcing steels with yield stresses greater than 60 ksi, up to 75 ksi, with the 

approval of the Owner. The modulus of elasticity of the reinforcing steel is 29,000 ksi as per 

AASHTO LRFD Article 5.4.3.2.

It is interesting to compare the stress-strain relationships for prestressing strands and mild 

reinforcing steel. Figure 2.10 provides a combined plot of stress-strain relationships used in this 

manual. 
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Figure 2.10 – Comparison of Typical Stress-Strain Relationships for Prestressing

Strand and Mild Reinforcing 
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Chapter 3 – Prestressing with Post-Tensioning 36 of 389 

Chapter 3—Prestressing with Post-Tensioning

3.1 Introduction

Prestressing is the application of compressive force to a concrete member in order to offset 

tensile stresses resulting from other applied loads. Figure 3.1 illustrates the concept of 

prestressing for a simple span beam. 

Figure 3.1 – Prestressed Concrete Concepts

Design of prestressed concrete involves optimizing member cross section and prestressing 

force and geometry to eliminate or offset concrete tension, enhance serviceability and reduce 

construction cost.

Prestressing is typically applied by means of tensioning high-strength steel strands or bars to 

react against and compress the concrete member. Prestressing concrete can be achieved in 

two ways:

• Pre-tensioning – In pre-tensioned members strands are installed along the length of a 

casting bed and tensioned against restraining bulkheads before the concrete is cast. 

After the concrete has been placed, allowed to harden and gain sufficient strength, the 

strands are released and their force is transferred to the concrete member by bond. 

• Post-tensioning – Post-tensioned construction involves installing and stressing strand or 

bar tendons only after the concrete has been placed, cured and hardened. Ducts are 

placed inside the concrete so that the tendons can be threaded through them after the 

concrete hardens. Once in place, the tendons are tensioned by jacks and anchored 

against the hardened member using anchorage devices cast into the concrete. 

The content of this manual deals with the application of prestressing through post-tensioning.
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Chapter 3 – Prestressing with Post-Tensioning 37 of 389 

3.2 Cross Section Properties and Sign Convention

Depending on the approach taken to analyze a multiple-cell box girder bridge, cross section 

properties are computed for either the full box girder or idealized “line girders”. Figure 3.2 

shows the girders for which cross section properties are typically computed: the full box girder, 

an internal “girder line,” and an external “girder line.”

Figure 3.2 – Cross Section Nomenclature and Sign Convention 

The cross section characteristics required for the analysis of the box girder are:

A = Area of Box Girder or Girder Line (ft2

) 

I = Moment of Inertia (ft4

) 

c1 = Distance from Neutral Axis to top Fiber (ft)

c2 = Distance from Neutral Axis to bottom fiber (ft)

The cross sections are assumed to be symmetrical about their vertical axes, and the 

prestressing is symmetrically applied so that there is no biaxial bending. Figure 3.2 also shows 

positive sign conventions for prestressing force (F), eccentricity of prestressing (e), and 

externally applied bending moments (M). For the conventions of this Manual, compressive 

stresses in the concrete are positive (+) and tensile stresses in the concrete are negative (-).

3.3 Stress Summaries in a Prestressed Beam 

Consider the flexure of a simply-supported concrete beam section under the action of its own 

self weight as shown in figure 3.3. Top and bottom stresses are determined according to elastic 

beam theory. The top of the beam is in compression; the bottom is in tension. For plain 

concrete, the tensile stress could exceed the modulus of rupture, crack, and possibly cause 

failure of the beam.
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Chapter 3 – Prestressing with Post-Tensioning 38 of 389 

Figure 3.3 – Self Weight Flexure Stress in Simply-Supported Beam 

If a purely axial compressive stress is applied through prestressing along the neutral axis of the 

beam as shown in figure 3.4, the sum of compressive stress at the top of the beam is increased 

while the tension at the bottom of the beam is reduced. The presence of a net bottom tension 

would indicate that the beam’s additional load carrying capacity is limited. 

Figure 3.4 – Self Weight Plus Uniform Axial Compression 

To improve the effects of the post-tensioning the elevation of the tendon is shifted vertically 

lower to make the prestressing force eccentric to the beam’s neutral axis (figure 3.5). In this 

case, in addition to the self-weight stress and the axial prestress effect, the eccentricity causes 

a resisting flexural moment that induces compression in the bottom of the beam and tension in 

the top.
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Chapter 3 – Prestressing with Post-Tensioning 39 of 389 

Figure 3.5 – Self Weight, Axial and Eccentric Prestress Stresses

The summation of these three load effects (i.e. self-weight with axial and eccentric prestress) as 

shown in figure 3.5, results in compressive stress throughout the depth of the beam. The 

additional compression at the bottom of the beam is available to offset tensile stresses caused 

by other load effects such as superimposed dead loads, highway traffic loadings, and 

temperature induced stresses.

Service limit state design of prestressed concrete bridges involves the summing of stresses at 

cross sections throughout the bridge, and verifying that the resulting stress totals are within 

allowable limits. In box girder bridges this also includes the summing of axial and shear 

stresses to verify principal tensile stresses in the webs are within limits.

3.4 Selection of Prestressing Force for a Given Eccentricity

Where service limit state flexural verification of prestressed members involves summing of 

stresses due to applied forces, it is often more convenient to work with internal forces and 

moments when designing prestressed members. This section develops expressions for 

prestressing based on force and eccentricity.

Stresses are determined at the extreme top and bottom fibers by the familiar equations:

(Eqn. 3.1) 

(Eqn. 3.2) 
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Chapter 3 – Prestressing with Post-Tensioning 40 of 389 

These equations can be rearranged to express the required prestressing force as a function of 

the other equation variables.

Consider first the prestressing requirements for the bottom stress. The limiting prestressing 

force would be that which satisfies equation 3.2 when the bottom stress fb is set to a permissible 

concrete stress fa. Making this substitution and multiplying equation 3.2 through by the cross 

section inertia and dividing by the distance from the neutral axis to the extreme bottom fiber the

equation becomes:

(Eqn. 3.3) 

This equation can be reduced further by noting that the left hand side of the equation is a 

bending moment that produces the permissible stress in the concrete at the bottom of the 

girder:

(Eqn. 3.4) 

Equation 3.3 now becomes:

(Eqn. 3.5) 

Further simplification is made by defining the dimensionless parameter: 

(Eqn. 3.6) 

The parameter ρ is termed the Efficiency of the cross section with regard to prestressing. Cross 

section efficiencies for three typical shapes are shown in figure 3.6. As seen in this figure, cross 

section efficiency increases as material is moved away from the neutral axis and is located in 

top and bottom flanges. 

Figure 3.6 – Efficiencies of Various Cross Sections
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Chapter 3 – Prestressing with Post-Tensioning 41 of 389 

Recognizing that:

(Eqn. 3.7) 

Equation 3.5 is now simplified to:

(Eqn. 3.8) 

Solving for the prestressing force: 

(Eqn. 3.9) 

The numerator of this equation is the bending moment at the cross section under study, 

adjusted by the moment causing allowable stress. The sign of Ma is established by the sign of 

the allowable stress at the section. A permissible tension would cause Ma to be negative, 

reducing the required prestressing force. A requirement for a residual compressive stress 

would cause Ma to be positive, increasing the required prestressing force.

When Ma is established by the minimum allowable stress, equation 3.9 becomes the expression 

for minimum required prestress force. When maximum permissible compressive stress is 

controlling, equation 3.9 becomes the expression for maximum permissible prestress force.

This exercise can be repeated for limiting stress control at the top of a cross section. In this 

case, equation 3.1 can be rearranged to find:

(Eqn. 3.10) 

It is interesting to study equations 3.9 and 3.10 for additional implications. Figure 3.7 shows the 

internal equilibrium expressed by equation 3.9 for a positive bending moment. Figure 3.8 shows 

a similar diagram for the equilibrium expressed by equation 3.10.

Figure 3.7 – Internal Equilibrium for Positive Bending. 
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Chapter 3 – Prestressing with Post-Tensioning 42 of 389 

Figure 3.8 – Internal Equilibrium for Negative Bending.

Now consider the case of zero bending moment acting on the cross section. Equation 3.9 

becomes:

(Eqn. 3.11)

This equation is satisfied in one of two conditions, either the prestressing force is equal to zero 

or:

(Eqn. 3.12) 

Equation 3.12 shows that any prestressing force can be applied at a distance of ρc1 above the 

neutral axis with the result being zero stress at the bottom of the cross section. Likewise, from 

equation 3.10: 

(Eqn. 3.13)

Equation 3.13 shows that any prestressing force can be applied at a distance of ρc2 below the 

neutral axis with the result being zero stress at the top of the cross section. Graphically, these 

two limiting eccentricities are seen in figure 3.9.

Together, these limiting eccentricities define the upper and lower locations of the cross section 

kern—that portion of the cross section through which no tension occurs if the resultant 

compressive force is located therein. Figure 3.10 shows the layout of the upper and lower kern 

for a cross section with bending about the horizontal axis. Further analysis of the cross section 

could be undertaken to define the kern limits in any direction.
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Figure 3.9 – Limiting Eccentricities for Zero Tension Under Axial Force Only 

Figure 3.10 – Kern of a Cross Section for Bending About the Horizontal Axis. 
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Chapter 3 – Prestressing with Post-Tensioning 44 of 389 

Example: Consider the line girder segment of a concrete box girder bridge shown in 

figure 3.11. The girder is simply supported, with a span length of 120’. In 

addition to the girder self weight, the girder is subjected to a uniformly 

distributed load of 1.0 kip/ft. The concrete strength of the girder is 5000 psi. 

Determine the limiting values of prestressing for maximum and minimum 

stress at the top and bottom of the girder. The maximum permissible 

compressive stress is 0.6f’c. The minimum permissible tensile stress is 3√f’c.

The section properties of the girder are:

Figure 3.11 –Example Concrete I-Girders

The efficiency of the cross section is:

The moment required to produce the permissible tensile stress at the bottommost fiber of the girder is:

The moment required to produce the permissible compressive stress at the 

bottom-most fiber of the girder with no axial force is:

The moment required to produce the permissible tensile stress at the topmost fiber of the girder is:
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Chapter 3 – Prestressing with Post-Tensioning 45 of 389 

The moment required to produce the permissible compressive stress at the 

top-most fiber of the girder is:

The mid-span bending moment due to the applied load is:

The minimum prestressing required to limit the bottom girder tension is:

The maximum prestressing permissible to not overstress the bottom of the 

girder is:

The maximum prestressing permissible to not exceed the minimum tension in 

the top of the girder is:

The minimum prestressing permissible to not over compress the top of the 

girder is:

Though the four limiting forces can be computed, the two limiting forces for 

top stress are not useful for this example. For the case of the maximum top 

tension, the limiting prestress force causes the bottom of the beam to exceed 

the maximum compressive stress. To reach the maximum top compression, 

the post-tensioning force would need to be negative. As a result, for this 

example, the prestressing force can vary between 1397.4 kips and 5819.2 

kips without exceeding allowable stresses.
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Chapter 3 – Prestressing with Post-Tensioning 46 of 389 

3.5 Permissible Eccentricities for a Given Prestressing Force

Equations for determining prestressing force for a given eccentricity were developed in the 

previous section. These equations can be used to solve for the prestressing force at a critical 

section. With the required prestressing force established, the tendon profile must be 

established such that the stress limitations are respected along the entire length of the member. 

Permissible ranges of eccentricity as a function of prestressing force can be determined in a 

fashion similar to those of the previous section.

Consider the condition of bottom stress in a simple span girder. As in the previous Section, 

equation 3.2 was simplified to equation 3.8, which is repeated here:

(Eqn. 3.8) 

Solving for the eccentricity results in:

(Eqn. 3.14) 

When the controlling bottom stress is minimum stress then equation 3.14 becomes the 

expression for minimum eccentricity. When the controlling bottom stress is compression, 

equation 3.14 is the expression for maximum eccentricity.

Likewise, the eccentricity limits established by allowable stress at the top of the girder can be 

expressed as:

(Eqn. 3.15) 

When the controlling top stress is minimum stress then equation 3.15 becomes the expression 

for maximum eccentricity. When the controlling bottom stress is compression, equation 3.15 is 

the expression for minimum eccentricity.

Using equations 3.14 and 3.15 along with maximum and minimum allowable stresses, we can 

define ranges of eccentricity. For stress control on the bottom of the girder:

(Eqn. 3.16) 

For stress control on the top of the girder:

(Eqn. 3.17) 

Where Mat represents the moment to cause the allowable tensile stress (or minimum 

compression), and Mac represents the moment to cause allowable compression.
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Chapter 3 – Prestressing with Post-Tensioning 47 of 389

Example: Consider the concrete line girder analyzed in the previous example. 

Determine the limits of permissible eccentricity at tenth points along the girder 

for a prestressing force of 1200 kips.

Recall from the previous example:

The kern of the cross section is defined by:

The limits of eccentricity with regard to bottom of girder stress are defined as:

The limits of eccentricity with regard to top of girder stress is defined as:

For simple spans subjected to a uniform load:

The resulting limiting eccentricities are:

Table 3.1 – Limiting Eccentricities for Example Girder

kL x M(x)

Bottom Stress Top Stress

emin emax emin emax

0 0 0 -2.6275 6.8508 -7.8714 1.5805

0.1 12 2737.8 -1.3237 8.1546 -6.5677 2.8842

0.2 24 4867.2 -0.3097 9.1686 -5.5537 3.8982

0.3 36 6388.2 0.4145 9.8928 -4.8294 4.6225

0.4 48 7300.8 0.8491 10.3274 -4.3949 5.0570

0.5 60 7605 0.9940 10.4723 -4.2500 5.2019
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Chapter 3 – Prestressing with Post-Tensioning 48 of 389 

Graphically, these limiting eccentricities of this example are shown in figure 

3.12. The tendon profile can lie within the shaded area in the sketch while 

respecting the minimum and maximum allowable flexural stresses in the 

girder.

Figure 3.12 – Limiting Eccentricities for the Example Bridge

3.6 Equivalent Forces Due To Post-Tensioning and Load Balancing

Figure 3.13 shows a tendon profile that would satisfy the requirements permissible eccentricity 

in the previous example. 

Figure 3.13 – Parabolic Tendon Profile for a Simple Span Girder

The profile of this tendon follows a parabolic trajectory, defined by the following equation for 

eccentricity:

(Eqn. 3.18) 

The moment caused by the post-tensioning, assuming a constant force along the length of the 

tendon, would also be parabolic:
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Chapter 3 – Prestressing with Post-Tensioning 49 of 389 

(Eqn. 3.19) 

The parabolic bending moment caused by the post-tensioning can be equated to the bending 

moment created by the application of a uniform load distributed over the span length. For a 

load p, the simple span bending moment is:

(Eqn. 3.20) 

Equations 3.19 and 3.20 can be equated and the equivalent uniform load determined:

(Eqn. 3.21) 

(Eqn. 3.22) 

(Eqn. 3.23) 

Figure 3.14 shows the parabolic tendon layout and the resulting equivalent forces acting on the 

concrete girder. In this layout the tendon has no eccentricity at the ends of the span. If there 

were eccentricities at the ends of the span, these would be represented by end moments equal 

to the prestressing force times the end eccentricity. Note that the equivalent horizontal force in 

the lower diagram is made equal to the prestressing force. This approximation is typically used 

for simplicity in hand calculation, as tendon profiles are flat and that at any location F ≈ Fcos(α), 

where α is the angle of the tendon relative to horizontal.

Figure 3.14 – Equivalent Forces Resulting from Prestressing
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Rearranging equation 3.22, we can find the prestressing force required to offset an applied 

uniformly distributed load. The state of stress all along the length of the girder would be uniform 

compression over the depth of the girder (Fbalancing/Agirder).

Eqn. 3.24

Load balancing has great merit in some prestressed concrete members. Most typically, 

however, the required quantity of prestressing is great, and cost effectiveness is found in 

reducing the applied prestressing to satisfy acceptable limits of tension in the concrete 

elements.

Example: For the girder of the previous examples, compare the prestressing force 

required for minimum bottom tension at mid-span and the prestressing 

required to balance the total applied load of 4.225 k/ft.

From Equation 3.9, the minimum prestressing force for the minimum bottom 

tension at mid-span is:

The prestressing force required to balance the applied load is found from 

equation 3.24

The post-tensioning force necessary to balance the applied loads is more 

than twice the force necessary to meet permissible tensile stress criteria.

3.7 Post-Tensioning in Continuous Girders

Up to this point we have been determining the values of prestressing force at cross sections and 

defining limits of eccentricities of tendons within simple span girders. These girders are free to 

rotate at their ends under the action of the prestressing. In continuous girders, adjacent spans 

restrain the rotations of each other. These restraining actions produce continuity moments 

along the length of the continuous girder.

Figure 3.15a shows a two-span girder, continuous over the middle support. The profile of the 

two-span post-tensioning tendon is comprised of two, simple span parabolic shapes described 

in figure 3.13. If the spans were simply supported, the two girders would deflect as in figure 

3.15b, with the end rotations shown. Figure 3.15c shows the continuity moments produced to 

resist the simple span rotations. 
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Figure 3.15 – Restraining Moments in Continuous Girders

The bending moment produced by the post-tensioning force acting at an eccentricity is called 

the primary moment due to the post-tensioning. The continuity moments produced in the 

continuous girders as a result of restraining the individual spans under the action of the posttensioning are called the secondary moments due to post-tensioning.

Figure 3.16 shows the primary, secondary, and the total moments caused by the posttensioning. The primary moment diagram is the result of the parabolically draped posttensioning profile. The secondary moments are a result of restraining the rotations of the 

primary moment beam end rotations at the middle supports. The Total post-tensioning moment 

is the sum of the primary and secondary moments at each section along the length of the 

bridge.

It is interesting to consider the secondary moments as an effective adjustment to the tendon 

profile. Dividing the secondary moment at any location by the post-tensioning force at that 

section establishes a change in eccentricity from the original tendon geometry. Likewise, 

dividing the total prestressing moment by the axial force results in the effective eccentricity of 

the tendon profile.
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Figure 3.16 – Prestressing Moments for a Two-Span Continuous Girder 

For the case of a structure symmetrical about the central support, the moments M21 and M23

shown in figure 3.15 would be those that would restore the cross section at the middle support 

to vertical (no net rotation). Considering the two span structure of the last section, the 

equivalent load of the tendon on the girder is a uniformly distributed load with a value of:

(Eqn. 3.25) 

The simple beam end rotations caused by this uniform load are:

(Eqn. 3.26) 
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The rotation at the end of a beam with constant cross section properties, under the action of an 

end couple, is:

(Eqn. 3.27) 

By setting equations 3.26 and 3.27 equal to each other, the value of the continuity moment at 

the central support of the two span structure can be determined:

(Eqn. 3.28) 

(Eqn. 3.29) 

The bending moment diagram for the combined effects of primary and secondary prestressing 

moments is shown in figure 3.17.

Figure 3.17 – Total Prestressing Moments for a Two-Span Continuous Girder 

The impact of the secondary moments on the effective eccentricity for this example is:

• The effective eccentricity at the mid-span of the spans is reduced by half to Femax/2.

• Though the tendon profile has no real eccentricity at the central support, the effective 

eccentricity is equal to the maximum eccentricity at the center of the spans.

Note: The preceding discussion considers a two span continuous bridge with a constant and

symmetric post-tensioning force. In actual design there would be losses along the length of the 

tendon, and most likely, tendons of a two-span bridge would only be stressed from one end. 

See chapter 4 for a further discussion of post-tensioning losses
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3.8 Tendon Profiles—Parabolic Segments

In section 3.6, a parabolic tendon was introduced to offset the effects of uniformly distributed 

applied forces. This is typically the case for cast-in-place concrete box girder bridges where the 

majority of the applied loads result from the uniformly distributed effects of girder self weight and 

barrier railing (DC), uniformly applied superimposed dead loads such as future wearing surfaces 

(DW) and the uniform load portion of the HL93 notional load. Rather than express the parabolic 

layout over its full length, the tendon geometry is typically subdivided into parabolic halfsegments as shown in figure 3.18.

Figure 3.18 – Tendon Profile Parabolic Segment 

The elevation of the tendon profile at a horizontal distance from the origin is given as:

(Eqn. 3.30) 

The slope of the tendon profile is:

(Eqn. 3.31) 

And the angle of the tendon profile at the end of the parabolic segment is:

(Eqn. 3.32) 

The tendon profiles used for the two-span continuous girder in this Chapter were parabolically 

draped tendons similar to a simple span girder. Though these profiles were useful to 

demonstrate principles of the development of secondary moments, they are not efficient with 

regard to design and construction of continuous box girder bridges. The secondary moment in 

the positive bending region is excessive, and though effective eccentricity was developed over 
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Chapter 3 – Prestressing with Post-Tensioning 55 of 389 

the middle support, typically more effective eccentricity is needed to optimize post-tensioning 

quantities.

Using parabolic segments, tendon profiles can be developed which improve post-tensioning 

effectiveness. Figures 3.19 and 3.20 show the more typical layout of tendon profile for 

continuous bridges. Figure 3.19 shows the profile for an end span of a continuous unit. Figure 

3.20 shows the profile for an interior span.

Figure 3.19 – Typical End Span Tendon Profile for Continuous Superstructures 

Figure 3.20 – Typical Interior Span Tendon Profile for Continuous Superstructures

When using parabolic segments to define tendons, it is required to maintain tendon slopes at 

junctions between the parabolic segments. Consider the common point of the parabolic 

segments within lengths b2 and b3 in figure 3.18. From Equation 3.32:

(Eqn. 3.33) 

And,

(Eqn. 3.34) 
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Equating these two tendon slopes, we find relationships between parabolic segment run and 

rise:

(Eqn. 3.35) 

Example: Using the two-span girder of the previous examples and the tendon profile 

shown in figure 3.21, compute the secondary moments for a post-tensioning 

force F.

Figure 3.21 – Example Tendon Profile Parabolic Segments 

The horizontal run of the tendon profiles are given. Complete the definition of 

the parabolic tendon segments by determining their rises. The first parabolic 

segment terminates at the neutral axis, so by inspection h1=2.5’. The rises of 

the second and third parabolic segments are found by observing that:

Rearranging equation 3.35 leads to:

Combining these expressions:

Solving for h2: 

And h3: 
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End rotations are found by the conjugate beam method, in which the end 

reactions of the conjugate beam, loaded with the curvature diagram 

(moment/EI), are equal to the end rotations. The bending moment for which 

rotations are to be computed is the primary prestressing moment. 

Prestressing force F, and beam stiffness, EI, are constant along the length of 

the structure in this example. Figure 3.22 shows curvature diagram due to 

the post-tensioning tendon.

Figure 3.22 – Curvature Diagram for Prestressing

The forces acting on the conjugate beam are found by concentrating the 

curvature diagram into sections that can be easily expressed geometrically. 

For the tendon layout of this problem the curvature diagram is concentrated 

into four conjugate beam loads—the three parabolic segments and a 

rectangular segment which accounts for the fact that the transition between 

the positive and negative curvatures do not occur at the neutral axis. Figure 

3.23 shows the four areas of concentrated curvature.

Figure 3.23 – Curvature Diagram for Prestressing

Working from left to right, the first load is the concentration of the first 

parabolic segment (A1):

 
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The second parabolic segment (A2) is concentrated to:

The third area, A3, is the rectangular area that when subtracted from A2

results in the positive curvature between A4 and the neutral axis.

The fourth load is:

With the loads computed, the reaction on the conjugate beam can be found. 

Figure 3.24 shows the loaded conjugate beam

Figure 3.24 – Loaded Conjugate Beam 

The reactions (end rotations) are:

 

 

 

 
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As in the previous example, the secondary moment at the interior support is 

equal to the moment that restrains the rotation, returning the section to 

vertical.

Solving for the secondary moment at the middle support (see Figure 3.16):

It is interesting to compare the total prestressing moments in terms of 

effective eccentricity for this example (with parabolic segments and 

eccentricity at the middle pier) and the tendon profile shown in figure 3.17 

(single parabola over the span lengths, with no eccentricity at the middle 

pier).

The effective eccentricities of this example at pier and mid-span are:

The effective eccentricities for the tendon profile shown in Figure 3.17 are:

 

The tendon profiles of this example are 20 percent more effective at the 

middle support and 50 percent more effective at mid-span

Secondary moments for the two-span girder in this Section were readily computed because of 

the structure’s symmetry. More complex structures require a generalized approach to 

determine secondary moments using hand calculations. Appendix A of this manual presents a 

generalized flexibility-based hand method for solving continuous bridges. The method begins 

with the determination of span end rotations under the action of the applied loads. These end 

rotations are then used to determine continuity moments at the end of that span. This 

procedure is performed for all loaded spans, and the results summed to complete the analysis.

 

 
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Chapter 4—Prestressing Losses

Post-tensioning tendon forces are established in design to provide precompression to offset 

undesirable tensile stresses in the concrete box girder. The engineer conveys the tendon force 

requirements in the contract drawings as either the required jacking force at the end of the 

tendon or the final effective force at some point along the length of the tendon. The differences 

between jacking forces and effective forces are the prestressing force losses. Prestressing 

force losses can be grouped into two families: 1) instantaneous losses related to the mechanics

of the post-tensioning system and tendon geometry, and 2) time-dependent losses related to 

the material properties of the concrete and prestressing steel. The components of prestressing 

losses addressed in this chapter are:

Instantaneous Losses

• Duct friction due to curvature

• Wobble (unintentional friction)

• Wedge Set (or Anchor Set)

• Elastic shortening of concrete

Time-Dependent Losses

• Shrinkage of concrete

• Creep of concrete

• Relaxation of prestressing steel

4.1 Instantaneous Losses

4.1.1 Friction and Wobble Losses (AASHTO LRFD Article 5.9.5.2.2b)

Friction between the strands and duct during stressing is related to intended angular changes in 

the tendon geometry. The top sketch in figure 4.1 shows the trajectory of a tendon within a 

desired duct profile. As the tendon is stressed, friction where the tendon contacts the duct wall 

reduces the force in the tendon. The friction coefficient (μ), defined to predict losses of this 

type, is a function of the duct material. 

Predicting friction losses along the length of a tendon using the friction coefficient alone has not 

proven to correlate well with field results. Another coefficient of friction loss, wobble (K), is used 

to account for unintended friction between strand and duct as a result of unintended duct 

misalignments. The concept of duct wobble is shown in the bottom sketch of figure 4.1.

Figure 4.1 – Friction and Wobble
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The equation to predict losses due to friction and wobble is found by considering a small section 

of tendon following a circular path as shown in figure 4.2. 

Figure 4.2 – Section of Tendon with Radial Alignment

The variables in figure 4.2 are defined as:

dα = angle change

dl = tendon length through angle dα

F = is the force at the stressing end of the section of tendon

dF = loss in tendon force P resulting from friction and wobble 

n = distributed radial force resisting the angle change of the tendon force.

N = resultant of distributed radial force, n

μ = friction coefficient

For an infinitesimally small element of tendon, the vertical equilibrium can be expressed as:

(Eqn. 4.1) 

Or,

(Eqn. 4.2)

For small angles, 

(Eqn. 4.3)

And,

(Eqn. 4.4) 

So that Equation 4.1 can be reduced to,

(Eqn. 4.5) 
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Summing the forces horizontally, 

(Eqn. 4.6) 

Again, for small angles, 

(Eqn. 4.7) 

Equation 4.6 then reduces to,

(Eqn. 4.8) 

Combining expressions 4.5 and 4.8,

(Eqn. 4.9) 

We define the wobble coefficient, K, as a function of the prestressing force, per unit length, l, 

along the tendon so that the total loss in prestress force becomes:

(Eqn. 4.10) 

The change in prestress force as a function of the original force is,

(Eqn. 4.11) 

Which can be integrated,

(Eqn. 4.12) 

To find, 

(Eqn. 4.13) 

which can be expressed as, 

(Eqn. 4.14) 

Or in terms of stress,

(Eqn. 4.15) 

Equation 4.15 relates tendon stress at a length along the tendon to the jacking force, tendon 

geometry, and the coefficients of friction and wobble. Expressions similar to equation 4.15 are 

found in AASHTO LRFD Article 5.9.5.2.2.
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Design example 1 (Appendix C) presents the design of a three-span cast-in-place box girder 

bridge with span lengths of 120’, 160’ and 120’. The cross section of the box girder 

superstructure, shown in figure 4.3, has five webs, each containing three post-tensioning 

tendons, each comprised of 19, 0.6” diameter prestressing strands. 

Figure 4.3 – Cross Section of Superstructure for Design Example 1

Figure 4.4 shows details of the tendon profiles of design example 1 in the end spans and center 

span. The profiles follow a series of parabolic segments as defined in the previous chapter. The 

tendon elevations shown take into account the location of the strands within the ducts as shown

in the detail of figure 4.3. For this tendon size, a 1” offset from the center of the duct to the 

center of gravity of the strands is specified (AASHTO LRFD Article 5.9.1.6). When the tendon is 

low at mid-spans, the strands are pulled to the top of the duct and are 1” above the center of 

gravity of the duct. Over the piers the strands are pulled to the bottom of the duct and their cg is 

modeled 1” below the cg of the duct. At points between the parabolic segments, the tendons 

are modeled at the cg of the duct.
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Figure 4.4 – Tendon Profiles For Design Example 1

Figure 4.5 shows the elevation of Tendon T2 only, highlighting the deviation angles that the 

tendon makes over the three spans. Assuming that the tendon is stressed from the left end, 

End A in figure 4.5, the calculation of the force along the length of the tendon can be made 

using equation 4.14. The coefficient of friction, μ, is assumed to be 0.25 (1/rad), and the wobble 

coefficient, k, is 0.0002 (1/ft).

Figure 4.5 – Tendon T2 Profile and Angular Deviations

The calculation of the force along the length of Tendon T2 is summarized in the table shown in 

table 4.1. The steps taken to compute the values in this table are:

1. Define points, in this case points 1 through 11, at the beginning and end of 

the tendon and at transitions between parabolic segments.
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2. Compute the lengths and heights of the parabolic segments from the

tendon profile with appropriate adjustments for strand location within the

ducts. The heights in this calculation are the absolute value of the

difference in dimensions from the bottom fiber to the tendon centroid.

3. Compute the angular deviation that the tendon makes through each

parabolic segment as computed using equation 3.32 from chapter 3.

4. Using equation 4.14 compute the incremental force coefficient as a fraction

of the force at the beginning of the parabolic segment. (Fi/F0).

5. Beginning with unit value at End A of the tendon, successively multiply the

incremental force coefficients to determine the cumulative force coefficients

at the end of each parabolic segment, in this case expressed as a fraction

of the jacking force.

6. Multiply each cumulative force coefficient by the jacking stress. For this

example the jacking stress was 75 percent of the ultimate strength of the

tendon.

7. Find the tendon force along its length by multiplying the stress at that

location times the area of the tendon. In this example the tendon is

comprised of 19, 0.6” diameter strands. The area of the tendon is 0.217 x

19 = 4.123 in2

.

Point b (ft) h (ft)

Angular 

Deviation

θ=2h/b (rad)

Angular 

Force 

Coefficient

Cumulative 

Force 

Coefficient

Tendon 

Stress 

(ksi)

Tendon 

Force 

(kips)

1 0 0.0000 0.00000 0.0000 1.0000 202.50 835

2 50 1.2500 0.05000 0.9778 0.9778 197.99 816

3 55 2.4063 0.08750 0.9677 0.9461 191.59 790

4 15 0.6563 0.08750 0.9754 0.9229 186.88 771

5 20 0.9479 0.09479 0.9727 0.8977 181.78 749

6 60 2.8438 0.09479 0.9649 0.8662 175.04 723

7 60 2.8438 0.09479 0.9649 0.8358 169.25 698

8 20 0.9479 0.08750 0.9727 0.8130 164.63 679

9 15 0.6563 0.08750 0.9754 0.7930 160.58 662

10 55 2.4063 0.05000 0.9677 0.7674 155.39 641

11 50 1.2500 0.05000 0.9778 0.7503 151.93 626

Table 4.1 – Tendon Loss Calculations – Friction and Wobble

Figure 4.6 shows a graphical representation of the force along the length of Tendon T2 at full 

jacking force and with losses resulting from friction and wobble. The force at the left end of the 

tendon is equal to the jacking force of 835 kips. The force at the anchorage at End B is 626 

kips.
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Though the radii within the parabolic segment are changing over their lengths, the tendon force 

is assumed to vary linearly over the length of the parabolic segment. This results in the straightline representation shown in figure 4.6. 

It is important to note that the lengths used for computing losses for wobble in this example 

were the horizontal lengths of the parabolic segments. This simplification is made because the 

horizontal projection of the tendon length is not significantly different from the actual tendon 

length for most cast-in-place box girder bridges. When this assumption cannot be made, the 

losses should be computed along the length of the tendon.

Figure 4.6– Tendon Loss Calculations – Friction and Wobble

4.1.2 Elongation

Post-tensioning tendons are stressed to a force chosen by the Engineer and presented in the 

contract drawings. Hydraulic jacks used to stress the tendons are fitted with gauges that relate 

hydraulic pressure to stressing force. Tendon elongations are measured during stressing as a 

secondary method of verifying force in the tendons.

Elongations are computed by considering the average force over a length of tendon: 

(Eqn. 4.16) 
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Where Fi ave is the average force over the length of tendon, li. The total elongation is obtained 

by summing the increments of elongation for each portion of the tendon, based on the average 

of the force at the beginning and end of that portion:

(Eqn. 4.17) 

The table presented in table 4.2 shows the average length of each parabolic segment for 

Tendon T2 of design example 1, the average force over that length (found as the average of the 

force at the beginning and end of the segment) , and the resulting elongation over each 

increment. The total elongation is the sum of the column of incremental elongations.

Point b (ft) Average 

Force (kips)

Incremental 

Elongation (in)

1 50 825.62 4.216

2 55 803.13 4.511

3 15 780.23 1.195

4 20 760.00 1.552

5 60 736.33 4.512

6 60 710.51 4.354

7 20 688.30 1.406

8 15 670.43 1.027

9 55 651.39 3.659

10 50 633.55 3.235

Total Elongation (in) 29.666

Table 4.2– Tendon Elongation

The area below the tendon force diagram in figure 4.7 represents the work done to elongate the 

tendon. This work divided by the tendon area and modulus of elasticity of the prestressing steel 

is equal to the total tendon elongation as demonstrated in figure 4.8.

4.1.3 Anchor Set

When the jacking force is reached at End A, the wedges are made snug and the tendon force 

released. The tendon draws the wedges, or seats them, into the wedge plate. The amount of 

movement that the wedges undergo is referred to wedge or anchor set. Values of anchor set 

vary with post-tensioning system, but typically vary from 1/4 to 3/8 of an inch. Figure 4.7 shows 

a depiction of the wedges before and after anchor set.

As the wedges are seated the tendon is shortened, reducing the tendon force. Often, only a 

portion of the length is affected as the work done in shortening the tendon is less than the work 

to elongate the tendon. Short tendons, however, requiring little elongation to achieve a desired 

force, can have the force affected along their entire length as a result of anchor set.

Figure 4.8 shows the effect of an anchor set of 3/8 inches on the forces along the length of the 

Tendon T2 of design example 1 after stressing at End A. The shaded area in figure 4.8 

represents the work performed during the anchor seating. This area is determined by finding a 

point along the length of the tendon to a location where the anchor set does not impact the 
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Chapter 4 – Prestressing Losses 68 of 389

tendon force (point of zero movement). Working back from the tendon force from that point to 

the anchorage, the loss in tendon force is related to the geometry of the tendon, friction and 

wobble, just as when originally stressed. A typical simplification is to mirror image the force loss 

diagram about a horizontal line passing through the point of zero movement.

Figure 4.7– Anchor Set

Figure 4.8 – Tendon Force Diagram after Anchor Set at End A

The point of zero movement was determined by trial and error to be 90.87’ from the stressing 

end. The convergence on these values was found by equating the shaded area in Figure 4.8, 
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Chapter 4 – Prestressing Losses 69 of 389 

(average changes in force x length along tendon) divided by the tendon cross-sectional area 

and modulus of elasticity, to the anchor set of 3/8”:

(Eqn. 4.18) 

The maximum force in the tendon at the point of zero movement is 796.68 kips. This force 

represents a stress in the tendon of 193.2 ksi, or 71.6 percent of GUTS, which is less that the 

permissible value of 74 percent specified by Table 5.9.3-1 of AASHTO LRFD Article 5.9.3. The 

force at the anchorage at End A is 758.36 kips. This force represents a stress of 183.9 ksi, or 

68.1 percent of GUTS, which is less than the permissible value of 70 percent specified in the 

same AASHTO LRFD Table.

4.1.4 Two-End Stressing

For tendons that are relatively short, or for longer tendons with small vertical draping, single end 

stressing may provide sufficient force throughout the bridge superstructure. Stressing half of 

the tendons from their beginning and half from their end can help produce a more uniform level 

of force along the bridge.

Long tendons that are continuous over multiple spans, where friction losses are high, can be 

stressed at both ends to significantly increase the prestressing force in the bridge. The force in 

Tendon T2 of design example 1, which has been studied in the chapter, can be enhanced by 

this two-end stressing.

After stressing from End A, the force in Tendon T2 was found to be 626 kips (See figure 4.7 and 

4.10). When jacking at End B, the tendon will not begin to elongate until this level of load is 

reached. The jack at End B will pick up load at 626 kips and continue to the final jacking force 

specified by the Engineer, which in this example is chosen to be 835 kips.

From symmetry of the structure and tendon geometry, the force along the length of the tendon 

when stressed from End B at the right would be the mirror image of that for stressing at End A 

as shown in figure 4.6. In this second stressing, however, the force along the tendon length 

would reduce as a result of friction and wobble, to a point where the force in the tendon is equal 

to the force from the first end stressing. No elongation, and therefore no force increase, will 

occur past that point. This is seen in figure 4.9, where the effects of stressing at End B elongate 

only the right half of the tendon. The point of zero movement of the second stressing is at the 

centerline of the main span of the bridge, where the forces from stressing at End A and End B 

both equal 723 kips. 

The shaded area shown in figure 4.9 between the two tendon force diagrams is the work 

performed during the second stressing. This area, divided by the cross-sectional area of the 

tendon and modulus of elasticity is equal to the elongation due to stressing the tendon at End B. 

For the case of Tendon T2 of design example 1, the elongation due to second stage stressing is 

2.3 inches.

The final change in tendon force is a result of the anchor set at End B. By symmetry of the 

tendon and jacking force, the resulting loss due to anchor set is the same as End A. This loss 

and the final tendon force diagram after stressing both ends and anchor set at End B is shown 

in figure 4.10.
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Figure 4.9 – Tendon Force Diagram after Stressing from End B

Figure 4.10 – Final Tendon Force Diagram (After Anchor Set at End B)
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4.1.5 Elastic Shortening (AASHTO LRFD Article 5.9.5.2.3b)

Concrete box girder superstructures are post-tensioned with multiple tendons. When the first 

tendon is stressed, it is pulled to the specified jacking force and seated. The concrete box 

girder shortens as a function of the bridge length and axial stiffness. 

The superstructure is further compressed when the second tendon is stressed, and shortens in 

proportion to the newly applied force. As the superstructure shortens, so does the tendon that 

was stressed first, reducing the force in that tendon. Subsequently stressed tendons 

incrementally reduce the force in all previously stressed tendons, until the last tendon is 

stressed and seated with the specified jacking force.

The difference between the sum of all tendon forces, as they have been affected by each other, 

and the total force, as if all tendons were stressed at the same time, is the instantaneous loss of 

prestress force due to elastic shortening.

Consider a structure that is incrementally prestressed with multiple post-tensioning tendons. 

The tendons are straight with no eccentricity. As the first tendon is stressed, the structure 

shortens an amount equal to:

(Eqn. 4.19) 

Where, Δ1 = Shortening of the superstructure under the force of stressing Tendon 1 (in)

F1 = Stressing force for Tendon 1 (kips)

L = Length of bridge being prestressed (inches)

Ac = Cross-sectional area of the box girder superstructure (in2

) 

Eci = Modulus of Elasticity of the concrete at the time prestressing is applied (ksi)

When the second tendon is stressed, the superstructure shortens more, and the first tendon 

shortens by the same amount. The resulting shortening of the superstructure at this phase of 

construction is:

(Eqn. 4.20) 

Where, Δ2 = Shortening of the superstructure under the force of stressing Tendon 2 (in)

F2 = Stressing force for Tendon 2 (kips)

ΔFp1 = Change in force in Tendon 1 under the action of stressing Tendon 2

Ap1 = Cross-sectional area of Tendon T1 (in2

) 

Eci = Modulus of Elasticity of the concrete at the time prestressing is applied (ksi)

Ep = Modulus of Elasticity of the prestressing steel (ksi)

Equation 4.20 can be rearranged to express the change in stress in Tendon 1 as Tendon 2 is 

stressed:

(Eqn. 4.21) 
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Chapter 4 – Prestressing Losses 72 of 389 

Where, Δfp1 = Change in stress of Tendon 1 while stressing Tendon 2 (ksi)

fcg2 = Stress at the center of gravity of the box girder superstructure as a result of 

stressing Tendon 2 (ksi)

Equations 4.20 and 4.21 were developed for the special case of a concentrically prestressed 

member where the center of gravities of the concrete and tendons coincide. More generally, the 

center of gravity of the post-tensioning is eccentric to the concrete superstructure center of 

gravity, and the stress is different from the axial value. Taking this into consideration, the total 

change in prestressing force in terms of the total stress in the concrete caused by all tendons, 

the expression is that of AASHTO LRFD Equation 5.9.5.2.3b-1:

(Eqn. 4.22) 

Where, ΔfpES = Change in stress of all tendons as a result of elastic shortening (ksi)

N = the number of identical prestressing tendons

fcgp = sum of concrete stress at the center of gravity of the prestressing tendons due 

to the prestressing force after jacking and the self-weight of the member at the 

sections of maximum moment (ksi)

Comments on fcgp: 

• This stress may be computed considering losses in prestressing stress due to friction, early 

portions of steel relaxation, and an estimate of elastic shortening itself.

• Typically, all tendons are stressed prior to grouting. The tendons are then all unbonded and 

fcgp should be calculated as an average stress at the level of the center of gravity of the 

prestressing steel, taken from several locations along the length of the bridge.

4.2 Time-Dependent Losses

Chapter 2 presented time-dependent characteristics for concrete and prestressing steel. The 

equations presented in that Chapter are those presented in the CEB-FIP Model Code (1990). 

The CEB-FIP equations are beneficial for computerized analyses of bridges that use time steps 

to evaluate phased bridge construction. The AASHTO LRFD Specifications provide equations 

for determining lump sum losses in prestressing. These equations are useful for preliminary 

design of complex structures and for the final design of bridges where a phased construction 

analysis is not required.

This section presents the AASHTO LRFD Specification equations for the refined lump sum 

approach for determining long term prestressing force losses. These equation are used chapter 

5 for the preliminary design of the bridge of design example 1. The design example uses a 

computerized analysis that incorporates the CEB-FIP equations.

4.2.1 General (AASHTO LRFD Article 5.9.5.4.1)

Predictions for time-dependent losses presented in the AASHTO LRFD specifications are 

tailored to superstructures made of pretensioned girders with deck slabs made continuous after 

girder placement. The general equation for time-dependent losses is presented in AASHTO 

LRFD Equation 5.9.5.4.1-1:
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(Eqn. 4.23) 

The subscripts for the various changes in prestress stress, Δfp, are:

• id refers to losses that occur between transfer of pretensioning force to the girder and 

placement of the concrete deck.

• df refers to losses that occur after deck placement (until final losses are realized).

• SR, CR and R1 are losses associated with concrete shrinkage, concrete creep and 

prestressing steel relaxation between transfer of pretensioning force to the girder and 

placement of the concrete deck.

• SD, CD and R2 are losses associated with concrete shrinkage, concrete creep and 

prestressing steel relaxation after deck placement.

• SS is a gain in prestressing stress as a result of the shrinkage of the composite deck 

slab

AASHTO LRFD Article 5.9.5.4.5 provides guidance for the application of the general equation 

for post-tensioned non-segmental girders. Recognizing that application of the prestressing and 

engaging of the self weight of the cast-in-place box girder happen at the same time, the part of 

AASHTO LRFD Equation 5.9.5.4.1-1 concerned with the time from transfer to deck placement is 

to be taken as zero. The construction of full cross sections of the box girders at the same time 

also causes the differential effects of the slab shrinkage relative to the webs and bottom slab 

also to be zero. As a result, equation 4.23 can be reduced to:

(Eqn. 4.24) 

The subscript df refers to all losses that occur from initial loading until all time-dependent losses 

are realized. The remainder of this chapter presents the terms of equation 4.24 as applied to 

cast-in-place post-tensioned concrete box girders. Some bridge owners offer simplified 

approaches to determining time-dependent losses. Caltrans, for example, routinely forgoes 

detailed calculations and uses a lump sum loss of 20 ksi.

4.2.2 Concrete Shrinkage (AASHTO LRFD Article 5.9.5.4.3a)

The loss of prestress force resulting from shrinkage of the concrete superstructure is given by 

AASHTO LRFD Equation 5.9.5.4.3a-1 as:

(Eqn. 4.25) 

The strain εbdf, is the shrinkage strain from the time of prestressing to final time as provided by 

AASHTO LRFD Equation 5.4.2.3.3-1:

(Eqn. 4.26) 

The factors in Equation 4.26 are defined by AASHTO LRFD Equations 5.4.2.3.2-2, 5.4.2.3.3-2, 

5.4.2.3.2-4, and 5.4.2.3.2-5:
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(Eqn. 4.27) 

(Eqn. 4.28) 

(Eqn. 4.29) 

(Eqn. 4.30) 

Where, H = relative humidity (%)

ks = factor for the effect of the volume to surface ratio of the component

khs = humidity factor for shrinkage

kf = factor for the effect of concrete strength

ktd = time development factor

t = age of concrete at the end of shrinkage (days)

V/S = volume-to-surface ratio (in.)

The term Kdf is given by AASHTO LRFD Equation 5.9.5.4.3a-2 as:

(Eqn. 4.31) 

Where, epc = eccentricity of the prestressing force with respect to the center of gravity of the 

concrete box girder (in.), positive in typical construction where prestressing 

force is below the centroid of the section

Ac = area of the gross section of the concrete box girder (in.2

) 

Ic = moment of inertia of the gross section of the concrete box girder (in.4

) 

The creep coefficient, ψb, is given by 5.4.2.3.2-1 as:

(Eqn. 4.32) 

Factors ks, kf, and ktd are defined above. Factor khc is defined by AASHTO LRFD Equation 

5.4.2.3.2-3:

(Eqn. 4.33) 

Where, khc = humidity factor for creep

ti = age of concrete at time of load application (days)
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4.2.3 Concrete Creep (AASHTO LRFD Article 5.9.5.4.3b)

The loss of prestress force resulting from the creep of concrete is defined by AASHTO LRFD 

Equation 5.9.5.4.3b-1: 

(Eqn. 4.34) 

Deleting the terms that deal with time of transfer to time of deck placement, this equation can be 

simplified to:

(Eqn. 4.35) 

Where, tf = final age of concrete (days)

4.2.4 Steel Relaxation (AASHTO LRFD Article 5.9.5.4.3c)

The loss of prestress force resulting from the creep of concrete is defined by AASHTO LRFD 

Equation 5.9.5.4.3c-1 for pretensioned conditions, which assigns half of the relaxation from time 

of transfer to deck placement, and half from deck placement to final time:

(Eqn. 4.36) 

This is evaluated by considering AASHTO LRFD Equation 5.9.5.4.2c-1:

(Eqn. 4.37) 

Where, fpt = stress in prestressing strands immediately after transfer, but not less than 

0.55fpy

KL = 30 for low relaxation strands and 7 for other prestressing steels

As an option to evaluating prestress relaxation by the equations above, the AASHTO LRFD 

specifications permits a value of ΔfpR1 equal to 1.2 ksi for low relaxation strand.

Considering relaxation from stressing to final time, the relaxation in cast-in-place box girder 

construction should be equal to two times ΔfpR1, or 2.4 ksi.
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Chapter 5—Preliminary Design

5.1 Introduction

Previous chapters in this manual have introduced cast-in-place box girder construction and 

post-tensioning concepts. This chapter presents a preliminary design process for cast-in-place 

box girder bridge superstructures. The goal of the preliminary design is to select a bridge 

layout, box girder superstructure cross section, and preliminary longitudinal post tensioning 

arrangement in sufficient detail to assure an efficient final design phase. The flow chart shown 

in figure 5.1 outlines the steps of this preliminary design process.

Figure 5.1 – CIP Box Girder Bridge Preliminary Design Flow Chart

The method of presenting this preliminary design process is to discuss influencing factors, and 

then apply them to an example bridge structure. Figure 5.2 shows an elevation view of a three 

span bridge that will be used to develop the preliminary design. 

Figure 5.2 – 3-Span Box Girder Bridge for Preliminary Design
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This bridge is planned as a part of a new highway passing through a relatively flat, developed 

urban area. The bridge is required to cross over an existing four-lane divided highway, with the 

travel lanes separated by 40’ and having required side clear zones of 32’. Figure 5.3 shows the 

cross section of the existing highway lanes to be spanned.

Figure 5.3 – Existing, At-Grade Highway Cross Section

to Be Spanned by Proposed Bridge 

5.2 Establish Bridge Layout

5.2.1 Project Design Criteria

Each bridge design project comes with a set of jurisdictional, technical and site specific criteria 

that need to be considered in establishing the bridge layout. Engineers and planners from 

various disciplines work to meet a wide range of owner requirements for the new facility. The 

most common of these are:

• Nature and function of the facility/geography to be spanned.

• Site topography.

• Required vertical and horizontal clearances.

• Horizontal alignment, profile grade, and superelevation of the proposed bridge.

• Number and widths of lanes and shoulders (highway bridges), dynamic clearance 

envelopes for trains (rail and transit bridges).

• Subsurface and geotechnical investigations.

• Environmental constraints.

• Bridge and Site drainage.

• Limits of Rights-of-Way.

• Utility requirements/conflicts.

• Permitting.

• Maintenance of Traffic.

• Applicable codes and regulations.

Working within the framework of the project design criteria, the bridge engineer begins 

preliminary design by selecting the following:

• Bridge type.

• Pier and abutment locations, resulting in bridge span lengths and overall bridge length.

• Length of bridge between expansion joints.

• Superstructure cross section.

• Pier and abutment types and dimensions.

• Probable foundation types and sizes.
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Bridge type selection studies are often performed before the preliminary design phase. Various 

bridge types are compared using “order of magnitude” estimates, typically based on historic 

bridge cost information. For the purpose of this manual, it is assumed that the preferred bridge 

type is a cast-in-place post-tensioned concrete box girder.

5.2.2 Span Lengths and Layout

In the example project considered in this chapter, the following information has been provided to 

the bridge engineer:

• The bridge will lie on a tangent alignment, crossing the highway with zero skew.

• No permanent new construction is permitted within the clear zones or in the median of 

the underlying highway.

• Bridge depth should be minimized to reduce impacts on developed areas on either end 

of the bridge, height of fill, and length of retaining walls.

• There are no environmental restrictions at the bridge site with regard to the permanent 

bridge and the falsework required to build the bridge.

• The project lies within already procured right-of-way.

• No utility conflicts or future needs have been identified.

• A reduced vertical clearance of 14’-6” is permitted during construction.

• Falsework must span the clear roadways of the underlying highway, with temporary 

supports permitted in the median.

The out-to-out dimension of the clear zones of the highway being crossed is 152’. This length is 

the minimum clear span of the bridge crossing required to clear the underlying highway. With 

consideration for the width of supporting columns, a centerline-to-centerline span of 160’ is 

selected.

A three-span continuous bridge is selected to minimize the depth of the superstructure of the 

example problem of this chapter. For cast-in-place post-tensioned concrete structures, efficient 

end span to main span ratios range from 0.75 to 0.80. Spans as short as 40 percent (requiring 

uplift restraining abutments) and as much as 100 percent of the main span length can be 

accommodated with appropriate post-tensioning design. For this example problem, an end 

span ratio of 0.75 is chosen. Expansion joints will be placed at the abutments.

The resulting span length and pier numbering for the example problem are shown in figure 5.4.

Figure 5.4 –Span Layout for Preliminary Design Example 

Note: The example bridge analyzed in this chapter is supported at the piers and abutments by bearings. 

This facilitates demonstrating the preliminary design of the longitudinal post-tensioning. Often, the 

interior piers are cast monolithically with the superstructure. Chapter 7 presents considerations for 

design when monolithic construction is used. 
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5.3 Cross Section Selection

5.3.1 Superstructure Depth

Past editions of bridge design specifications presented depth to span ratios for various bridge 

types. The current AASHTO LRFD requirements focus on selecting superstructure stiffness to 

meet deflection criteria. AASHTO LRFD Article 2.5.2.6.3 does, however, present optional 

criteria for span-to-depth ratios for use if requested by the Owner.

AASHTO LRFD Table 2.5.2.6.3-1 presents the following traditional minimum depths for constant 

depth cast-in-place prestressed concrete box girder superstructures as a function of span 

length. For simple span girders the traditional minimum depth is:

(Eqn. 5.1) 

(Eqn. 5.2) 

Expressed in terms of span-to-depth ratio, these equations become:

(Eqn. 5.3) 

(Eqn. 5.4) 

Using the traditional span-to-depth ratio for continuous construction, the depth of the structure to 

begin preliminary design would be:

For convenience, the bridge depth carried into preliminary design is 6’-6”.

Note—Other Owner Requirements: Occasionally, an Owner might require an internal minimum height 

within the concrete box girder for easy access of personnel and equipment for inspection and 

maintenance. This can be the case for long bridges with limited points of access, allowing longitudinal 

access along the entire length of the bridge. This can also be the case when the interior of the box girder 

is used to convey utilities such as power, water, and communications.

5.3.2 Superstructure Width

Highway and traffic studies performed for the example project have shown that the new bridge 

will carry three travel lanes in one direction only. The lane widths for the bridge are 12’ and the 

two shoulder widths are 10’ each. The Owner has standard concrete barrier railing that is 1’-9” 

wide. The resulting width of roadway is 59’-6” is shown in figure 5.5.
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Figure 5.5 –Bridge Width and Roadway Features 

5.3.3 Cross Section Member Sizes

This section provides guidance for the initial sizing of the members that comprise the box girder 

cross section. Members discussed are:

• Cantilever Wings.

• Webs.

• Top Slab.

• Bottom Slab.

The last part of this section applies the guidance to the example problem being developed in 

this chapter.

5.3.3.1 Width and Thickness of Cantilever Wing

Side overhangs, or cantilever wings, provide both structural and aesthetic benefits. Structurally, 

some length of overhang provides a development length for top slab reinforcing. Aesthetically, 

the cantilever wing presents a thin fascia and recesses the majority of the superstructure depth. 

Figure 5.6 shows a typical cantilever wing.

Figure 5.6 –Cantilever Wing Dimensions 

Reinforced concrete cantilever wing lengths typically range from approximately 2’ to 8’ in length. 

Shorter lengths are controlled by the minimum length required to develop top slab and web 

reinforcing. Longer lengths can be achieved provided care is taken to assure durability by 
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Chapter 5 – Preliminary Design 81 of 369 

controlling cracking under service level loadings. Transverse post-tensioning should be used to 

assure deck durability for cantilever wings greater than 8’. 

A reinforced concrete cantilever wing typically needs a minimum thickness of 8” to 9” at the wing 

tip beneath the traffic barrier. This thickness is related to the level of crash testing associated 

with the barrier being used. The thickness of the root of the cantilever wing varies with cantilever 

length. The guidance provided in equation 5.5 and 5.6 may be used for initial box girder 

dimensioning.

For cantilever lengths less than 5’ the root thickness can be approximated as:

(Eqn. 5.5) 

For cantilever lengths between 5’ to 8’, the root thickness may be estimated as:

(Eqn. 5.6) 

where: Lc = length of the cantilever wing (ft)

ttip = thickness of the cantilever wing tip (in)

tc = thickness of the cantilever wing root (in)

5.3.3.2 Individual and Total Web Thickness

AASHTO LRFD Article 5.14.1.5.1c provides general criteria for the thickness of individual webs 

for box girder bridges:

“The thickness of webs shall be determined by requirements for shear, torsion, concrete 

cover, and placement of concrete.” 

The commentary to this article provides more definitive guidance:

“For adequate field placement and consolidation of concrete, a minimum web thickness 

of 8” is needed for webs without prestressing ducts; 12” is needed for webs with only 

longitudinal or vertical ducts; and 15” is needed for webs with both longitudinal and 

vertical ducts. For girders over about 8’ in depth, these dimensions should be increased 

to compensate for increased difficulty of concrete placement.”

The great majority of cast-in-place post-tensioned box girder bridges contain only longitudinal 

tendons. As a result, the 12” minimum thickness for individual webs thickness should be used.

Preliminary estimates of overall web thickness (number of webs at the selected thickness) are 

complicated by the various functions the webs play:

• Resisting shear resulting from vertical loads and torsion.

• Relationship between web spacing and top slab thickness.

• Providing sufficient pathways for post-tensioning to achieve desired eccentricities.

AASHTO LRFD does not provide a service limit state verification for shear stresses in webs. 

Approximated shear stresses can, however, help in the initial selection of total web thickness for 
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Chapter 5 – Preliminary Design 82 of 369 

cast-in-place post-tensioned box girder bridges. As a starting point for total web thickness, 

calculate the shear stress from dead loads as:

(Eqn. 5.7) 

Where, pDL = dead load of superstructure (kips/ft)

Lmax = maximum span length in bridge (ft)

h = depth of box girder superstructure (ft)

btotal = total width of web at neutral axis (ft)

Qo = first moment of the area above or below the neutral axis (ft)

The dead load of the superstructure can be estimated by using characteristic average 

thicknesses (cross-sectional area divided by bridge width). Multiplying the average thickness of 

the concrete cross section times the bridge width, times the unit weight of concrete produces 

the self weight of the bridge per foot of span length. The average thickness of cast-in-place box 

girder webs typically ranges from 1’-6” to 1’-9”.

Including weights of barrier railing and wearing surface, the dead load per foot is:

(Eqn. 5.8) 

Where, tave = average superstructure thickness - equal to cross-sectional area divided by

bridge width (ft) 

B = overall bridge width (ft)

pb = weight of barriers (k/ft)

pws = weight of wearing surface (k/ft)

Solving for btotal, and defining an allowable dead load shear stress:

(Eqn. 5.9) 

Limiting the allowable shear stress in Equation 5.9 to between 5√f’c and 7√f’c, typically produces 

a total web thickness that can be adequately reinforced in final design.

Note: AASHTO LRFD specifies limits for principal tension at service limit state only for 

segmental box girder bridges. Principal tension verifications of cast-in-place post-tensioned box 

girders do, however, have great merit.

5.3.3.3 Top Slab Thickness

Minimum thicknesses of decks are either directly or indirectly addressed by several Articles in 

AASHTO LRFD.
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• AASHTO LRFD Article 9.7.1.1 states that, unless approved by the Owner, the minimum 

top slab thickness, excluding grinding and grooving, should be not less than 7”.

• Cover requirements in AASHTO LRFD Article 5.12.3 imply that the minimum thickness 

should be increased by 1⁄2” to 7 1⁄2”, if the bridge deck is exposed to deicing salts, tire 

studs, or chains.

• AASHTO LRFD Article 5.14.1.5.1a provides three criteria for top slab thickness:

− Requirements of AASHTO LRFD Section 9.

− Sufficient to anchor and provide cover for transverse post-tensioning, if used.

− Not less than the clear span between fillets, haunches, or webs divided by 20, 

unless transverse ribs or transverse post-tensioning are provided.

• If the Empirical Design Method is used for the top slab, AASHTO LRFD Article 9.7.2.4 

requires that the ratio of effective length to top slab thickness be between 18.0 and 6.0, 

core depth of the reinforced slab not less than 4”, and effective lengths not greater than 

13.5’.

• AASHTO LRFD Article 5.9.4.2.1 relates allowable compression limits to the slenderness 

ratio of the flanges of concrete box girders. The typical compressive stress limit of 0.6f’c 

is reduced by a factor (φw) when the slenderness ratio as defined in AASHTO LRFD 

Article 5.7.4.7.1 is greater than 15. The minimum reduction factor is 0.75.

• Section 5.3.3.2 describes the relationship between web spacing and top slab thickness. 

Additionally, the commentary of AASHTO LRFD Article 4.6.2.2.1, in presenting load 

distribution equations for “girder lines” in beam-slab bridges, recommends that cell 

width-to-height ratios should be approximately 2:1.

• AASHTO LRFD Article 2.5.2.6.2 provides limits on structure deflections if required by the 

Owner

Considering all of these requirements, the tendency is for top slabs of cast-in-place posttensioned box girders to be on the more robust side of deck thicknesses. Monolithic 

connections to the webs produce higher negative deck moments, which in turn produce 

increased demands on the top-level slab reinforcement. In general, durability of these integral

decks can be improved by increasing cover. However, excessive cover in thin slabs can be 

detrimental, as the larger covers can lead to larger crack widths. Increasing durability is better 

achieved by increasing the overall thickness of the slab while using appropriate concrete 

covers. 

As a rule of thumb for initial box girder dimensioning, the following range of clear span to deck 

thickness ratio may be used:

(Eqn. 5.10) 

Expressing equation 5.10 in terms of the top slab thickness:

(Eqn. 5.11) 
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The thickness of the top slab and clear span between webs or fillets are shown in figure 5.7.

Figure 5.7 –Top Slab Span and Thickness 

Small fillets (3” to 4”), shown in detail A of figure 5.7, are recommended at the top of webs/top 

slab interface. These fillets ease abrupt localized stresses and help facilitate placement and 

consolidation of concrete in the webs.

Using longer slab haunches, as shown in figure 5.8, can help optimize top slab concrete 

quantities and achieve longer slab spans. The haunch thickness at the face of the webs 

effectively accommodates negative flexure, while modifying the slab stiffness to help reduce the 

magnitude of positive moments. The benefits of reduced concrete quantities are sometimes 

offset by increased forming and reinforcing cost required to achieve the haunches.

Figure 5.8 –Top Slab with Haunches 

Transverse post-tensioning is typically required when clear spans in concrete box girder bridges 

are greater than 15’ or when cantilever wing lengths are greater than 8’. The following 

adjustments to slab thicknesses should be considered for initial sizing when transverse posttensioning is used:

• In order to accommodate post tensioning anchorages and local bursting reinforcement, 

use a cantilever wing tip of 9”.

• For shorter cantilever wings, vary depth linearly from tip to root.
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Chapter 5 – Preliminary Design 85 of 369 

• For longer cantilever wings, maintain 9” thickness from tip for about 4’ to 6’, and then 

vary linearly to root.

• Use a cantilever wing root thickness of 9” plus 1⁄2” for each foot of cantilever length. 

• Use a top slab haunch root thickness similar to the cantilever wing root.

• Use top slab interior haunch widths that are approximately 20 percent to 30 percent of 

the clear span between the webs.

• Do not require interior and exterior top slab haunches or fillets to be the same width.

• Use an 8” minimum top slab thickness between haunches. Increase the minimum top 

slab thickness as per equation 5.11.

5.3.3.4 Bottom Slab Thickness

 

AASHTO LRFD Article 5.14.1.5.1b provides criteria for the minimum thickness of bottom slabs:

• Bottom slab minimum thickness of 5.5 inches.

• The distance between fillets or webs of non-prestressed girders divided by 16

• The clear span between fillets, haunches, or webs for prestressed girders divided by 30, 

unless transverse ribs are used

The bottom slab between webs needs to be sufficient to carry its own self weight, superimposed 

loads such as utilities, while effectively participating in general bridge longitudinal and 

transverse behavior. If the bottom slab is to accommodate top and bottom cover and four layers 

of reinforcing (longitudinal and transverse bars in a top and bottom mat), the practical minimum 

thickness is approximately 7”.

Note: Additional thickness may be needed for clearances for longitudinal post-tensioning ducts if 

placed in the bottom slab itself. Haunches at the webs can help provide extra clearance for 

such tendons, allowing for the remainder of the bottom slab to remain at a minimum thickness. 

Local effects of tendons in thin bottom slabs should be evaluated in final design for additional 

reinforcing needs.

5.3.3.5 Member Sizes for Example Problem

In this Section the guidelines for cross section member initial dimensioning are applied to the 

example problem being studied in this chapter.

a. Cantilever Wings—Assume the use of a reinforced concrete deck with cantilever wing 

lengths of 4.75’. This leaves a 50’-0” top slab width between cantilever wings.

b. Total Web Width and Spacing—Use an average cross section thickness of 1’-9”, two 

barrier rails at 0.45 k/ft each, and a 2” overlay at 1.4 k/ft. The approximate dead load per 

foot of span is:

The total width of all webs for a shear stress of 0.158√f’c is:
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Using 0.221√f’c for the shear stress, the total of all web widths would be:

Using a minimum web thickness of each web to be 12”, the limits above would suggest 

either 5 or 4 webs. The centerline spacing of 5 webs would be 49.5/4 = 12.375’. The 

spacing for 4 webs would be 49.5/3 = 16.5’. Use 5 webs to keep the top slab as a 

reinforced concrete member.

c. Top Slab thickness—Using 5 webs of 1’ thickness and 3” fillets, the clear spacing 

between the webs would be 12’-41⁄2” – 1’ – 6” = 10’-101⁄2”. The range of top slab 

thicknesses to start preliminary design would be:

Use a 9” top slab thickness for preliminary design.

d. Bottom Slab Thickness—Use 7” to achieve cover and to fit mats of longitudinal and 

transverse reinforcing.

Note: This Section presented information for the preliminary sizing of a multi-cell concrete box 

girder superstructure. Dimensions for single cell box girders are similar to single cell segmental 

box girder. Information related to the sizing of segmental box girders is presented in LRFD 

Article 5.14.2.3.10.

Applying these initial dimensions with no interior haunches and making adjustments, such as 

sloping the outer webs, produces the preliminary design cross section shown in Figure 5.9.

Figure 5.9 – Example Cross Section Dimensions for Preliminary Design
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5.4 Longitudinal Analysis

5.4.1 Approach

Longitudinal analyses can be performed by any number of computerized solutions specifically 

for post-tensioned structures. General structural analysis programs can also be used with 

attentiveness to developing necessary load cases. Hand calculations like the Method of Joint 

Flexibilities presented in this manual are equally appropriate for longitudinal analysis, especially 

during preliminary design.

Analysis by the Method of Joint Flexibilities begins with the determination of span end rotations 

under the action of the applied loads. These end rotations are then used to determine continuity 

moments at the end of that span. This procedure is performed for all loaded spans, and the 

results summed to complete the analysis of the entire structure. This method is particularly 

helpful in evaluating load effects where the curvature diagram is readily determined, as is the 

case of prestressing primary effects (Fe). Appendix A of this manual presents the generalized 

flexibility-based hand method for solving continuous bridges.

5.4.2 Analysis by Method of Joint Flexibilities

The three-span example bridge being developed in this section is shown schematically in figure 

5.10. The cross section of the superstructure of the box girder is shown in figure 5.9. 

 Figure 5.10 – Model of 3-Span Bridge for Example 1 

5.4.3 Span Properties and Characteristic Flexibilities 

The cross section properties of the box girder superstructure shown in figure 5.9 are: 

The characteristic flexibilities for the 120’ Span are (See appendix A equations A.9 and A.10):

4

2

1

2

3

0

643.7

99.45

2.732

3.768

0.6288

118.1

I ft

A ft

c ft

c ft

Q ft

ρ

=

=

=

=

=

=

( )

120 0.06214

3 3 643.7

L Ea Ec

I = = = =
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Chapter 5 – Preliminary Design 88 of 369 

The characteristic flexibilities for the 160’ Span are:

Note: For numerical ease, the modulus of elasticity of the concrete has been factored out of the 

flexibility coefficients. This will also be done for beam end rotation computations to produce a 

consistent solution.

5.4.4 Analysis Left to Right

Using equations A.28 and A.29 from appendix A:

5.4.5 Analysis Right to Left 

Using equations A.28 and A.29 from appendix A:
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5.4.6 Carry-Over Factors 

For the Superstructure, using equations A.36 and A.37:

5.5 Bending Moments 

Bending moments are computed for the following loadings:

• Dead Load of Components (DC).

• Dead Load of Wearing Surface (DW).

• Live Loads (LL) – Truck and Uniform portions.

• Thermal Gradient (TG).

• Secondary Moments from Prestressing.

Several of these loadings are linearly scaled from the loading of a unit uniform load, which is 

solved first.

5.5.1 Effect of a Unit Uniform Load

The beam end rotations when loaded by a downward acting unit uniform load are:

3

24 i j

L E E

I

ω ω =− =−

( )2 2

23/2

3/32

0.041425

0.08285 0.07101

0.08285 0.06214

b k a

c K =− = − = + +

2/21

1 0.07101 1

0.07101

K = =

12

2/21

0.03107 0.2333

0.06214 .07101

b

c K f =− =− =− + +

23 32

3/32

0.041425 0.2857

0.08285 .06214

b

c K f f = =− =− =− + +

34

4/43

0.03107 0.0

0.06214

b

c K f =− =− = + + ∞

21

1/12

0.03107 0.0

0.06214

b

a K f =− =− = + + ∞

43

3/34

0.03107 0.2333

0.06214 .07101

b

c K f =− =− =− + +
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The beam characteristics of the 160’ center span is:

The moments in Span 1 are:

The end moments in Span 2 are:

The end moments in Span 3 (by symmetry),

The bending moments diagrams for the load in each span and the total of all moments is shown 

in figure 5.11.

 

 

The beam characteristics of the 120’ end spans are:
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Figure 5.11 – Moment Diagram for a Unit Uniform Load 
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5.5.2 Dead Load—DC (Self Weight and Barrier Railing)

The gross area of the box girder superstructure is 99.45 ft2

. The unit weight of concrete for self 

weight computations is 150 lb/ft3

. The resulting weight per foot of superstructure is 14.92 kips/ft. 

The weight of each barrier railing is 0.45 kips/ft. This results in a total dead load weight 15.82 

kips/foot. Rounding up, use 16 kips/ft for finding bending moments. The DC moments are 

found as a linear scaling of the effects of a unit uniform load shown in Figure 5.11.

Figure 5.12 – Moment Diagram for Dead Load (DC) 

5.5.3 Dead Load—DW (Future Wearing Surface)

The bridge is being designed to carry the weight of a 25 psf future overlay equal to 2” of 

concrete. Multiplying by the width of the roadway between barriers, the resulting weight per foot 

is 1.4 kips per foot. The DW moments are found as a linear scaling of the effects of a unit 

uniform load shown in figure 5.11.

Figure 5.13 – Moment Diagram for Dead Load (DW)
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5.5.4 Live Load—LL

Live load bending moments are computed for locations of maximum positive and negative 

moment. The HL-93 notional load contains the design truck and design lane load components. 

These are computed separately and combined later to determine maximum live load effects. 

The moments presented in this section are for a single lane. Total live load effects are 

calculated in section 5.5.4.5.

5.5.4.1 Uniform Load Component

The 0.64 kip/ft uniform lane load component of the AASHTO LRFD notional load can be applied 

in any combination of spans or partial spans to produce maximum and minimum effects. Figure 

5.14 shows the bending moment envelope for the uniform load.

Figure 5.14 – Uniform Live Load Moment Envelope 

5.5.4.2 Truck—Positive Moment in Span 1 or 3

Moments from truck axle loads are computed by determining and summing end rotations of the 

various axles and then solving for continuity moments. Figure 5.15 shows the end rotations for 

a beam subjected to a concentrated load. The rotations are defined in equation 5.12 and 5.13.

Figure 5.15 – Simple Beam Rotations for a Concentrated Load 

(Eqn. 5.12) 

(Eqn. 5.13) 
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The beam end rotation and end moment for the HL-93 design truck located 36’ from the 

Abutment 1 are:

Figure 5.16 shows the bending moment for a single truck with back axle located 36’ from the 

beginning of the bridge.

 

Figure 5.16 – Moment Diagram for HL-93 Design Truck in Span 1 (Positive Bending) 

5.5.4.3 Truck—Positive Moment in Span 2

Figure 5.17 shows the bending moment for a single truck centered on Span 2.

Figure 5.17 – Moment Diagram for HL-93 Design Truck in Span 2 (Positive Bending) 
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5.5.4.4 Truck—Negative Moment over Piers

AASHTO LRFD Article 3.6.1.3.1 calls for 90 percent of two trucks to be applied to produce 

negative moment results in continuous spans. Figure 5.18 shows the moment diagram for two 

HL-93 design trucks located about Pier 2 to generate maximum negative moment.

Figure 5.18 – Moment Diagram for Two HL-93 Design Trucks about Pier 2 (Negative Bending) 

5.5.4.5 Live Load Moment Totals

AASHTO LRFD Article 4.6.2 provides approximate methods for determining live load 

distributions within bridge superstructures. AASHTO LRFD Article 4.6.2.2 provides live load 

distribution equations for beam-slab bridges. One of two approaches may be taken for 

determining live load distribution factors for cast-in-place multi-cell box girders. The first 

approach applies the distribution factors to each “girder line” individually. The effective cross 

section of each girder line consists of the web and the tributary areas of the top and bottom 

slabs. Two designs are then made, one for the interior girders and one for the exterior girders.

The second approach in applying the approximate method of live load distribution is the “wholewidth” method. AASHTO LRFD Article 4.6.2.2.1 (paragraph 9) states:

Cast-in-place multi-cell concrete box girder bridge types may be designed as wholewidth structures. Such cross sections shall be designed for the live load distribution 

factors in Articles 4.6.2.2.2 and 4.6.2.2.3 for interior girders, multiplied by the number of 

girders, i.e., webs.

AASHTO LRFD Table 4.6.2.2.1-1 identifies multi-cell box girder brides as cross section type “d”. 

AASHTO LRFD Table 4.6.2.2.2b-1 provides live load distribution factor equations as a function 

of cross section type. For one design lane loaded, the equation for a type “d” section is:

(Eqn. 5.14)
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The equation for two or more design lanes loaded is:

(Eqn. 5.15) 

where: DF = distribution factor

L = length of span (ft) (60 ≤ L ≤ 240)

Nc = number of cells (Nc ≥3; use Nc=8 if Nc≥8)

S = web spacing (ft) (7.0 ≤ S ≤ 13.0)

For positive moment, negative moment, and shear, L is the length of the span being considered. 

For negative moment between points of contraflexure under uniform loading, L is the average 

length of two adjacent spans.

In the case of the example problem being studied in this chapter, the distribution factors for two 

or more design lanes loaded govern over those for a one design lane loaded. The distribution 

factors are:

The number of design lanes is found from:

(Eqn. 5.16) 

where: NL = number of lanes

Nw = number of webs

For the example problem:
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Chapter 5 – Preliminary Design 97 of 369 

The resulting bending moments for preliminary design, including dynamic load allowance 

(impact) in accordance with AASHTO LRFD Article 3.6.2.1 are:

5.5.5 Thermal Gradient (TG)

Continuous structures are subjected to stress distributions over the depth of the girder as a 

result of thermal gradients. AASHTO LRFD Article 3.12.3 shows the temperature distribution 

which should be applied to the depth of a concrete box girder. Preliminary design is simplified 

by assuming a 20 degree linear variation in temperature with depth for positive gradient, with 

the top slab 20 degrees warmer than the bottom slab. The resulting deformations of a simple 

beam subjected to a linear gradient are shown in figure 5.19.

Figure 5.19 – Simple Beam Subjected to 20°F Positive Linear Gradient 

The change in length of the top slab from the center of no movement (middle of beam) is equal 

to:

(Eqn. 5.17)

The coefficient of thermal expansion for concrete, α, in equation 5.17 is equal to 6x10-6. For 

small changes in length the beam end rotations are equal to this change in length divided by the 

depth of the beam. Considering our solution where E is included in the rotations, 

(Eqn. 5.18)
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For the example bridge being analyzed, the end rotations for the end spans and center span 

are:

The bending moments associated with these rotations are shown in figure 5.20.

Figure 5.20 – Moment Diagram for a 20°F Positive Linear Gradient

5.5.6 Post-Tensioning Secondary Moments

Bending moments for DC, DW, LL, and TG have been computed directly from the applied loads 

or imposed deformations. Secondary moments from prestressing are, however, related to 

tendon geometries and prestressing forces, which are not yet determined. Using parabolic 

segments developed in chapter 3, a tendon profile that represents the center of gravity of all 

tendons can be assumed. The secondary moments can then be determined for a unit 

prestressing force (F=1 kip).

Figure 5.21 shows the cross section of the example bridge studied in this chapter. Detail A, 

detail B, and the position of the strands within the duct have been developed to find probable 

locations of tendons at the high and low points along their profiles. Assuming that three tendons 

per web will be used, the middle tendon profile will represent the center of gravity of all three 

tendons.
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Figure 5.21 – Possible Tendon Locations at Mid-Span at over the Piers 

For the cross section of the example problem, the distances from the neutral axis to the top and 

bottom most fibers of the beam are:

The minimum and maximum eccentricities are:

Using the minimum and maximum eccentricities, the profile of the center of gravity (CG Profile) 

of the post-tensioning can be drawn. Figure 5.22 shows the CG Profile for the end spans. 

Figure 5.23 shows the CG Profile for the middle span. As a guide, the length of the parabolic 

segments at the piers is typically located from 0.10 to 0.15 of the span length. The low point of 

maximum eccentricity in the end spans is typically located from 0.35 to 0.45 of the span length 

measured from the end of the bridge. 

Figure 5.22 – Center of Gravity Profile of Prestressing (End Spans) 
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Figure 5.23 – Center of Gravity Profile of Prestressing (Middle Span)

Conjugate beam loads can be determined from the profiles using the methodology presented in 

chapter 3. Figures 5.24 and figure 5.25 show the conjugate beams and loads for a unit 

prestressing force (F=1).

Figure 5.24 – Conjugate Beam and Loads (End Spans) 

Figure 5.25 –Conjugate Beam and Loads (Main Span) 
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The conjugate beam reactions are equal to the simple beam end rotations. Continuity moments 

are found using these rotations flexibility analysis previously performed. The resulting moments 

for each span and the total secondary prestress moments are shown in figure 5.26.

Figure 5.26 – Secondary Prestressing Moments, M2(F)

5.6 Required Prestressing Force After Losses

In chapter 3 equations for determining post-tensioning requirements to limit tensile stresses in 

the concrete were developed. Equations 3.9 and 3.10 took the form:

(Eqn. 5.19)

1

M Mab F

ρc e

+

≥

+

(Eqn. 5.20)

2

M Mat F

e c ρ

− ≥ −

The total bending moment, M, for checking tensile stresses is that corresponding to Service 

Level III. Using the load factors in AASHTO LRFD Table 3.4.1-1, total bending moment is:

(Eqn. 5.21) 2 0.8 0.5 ( ) M M M M M MF =++ + + DC DW LL I TG +
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Chapter 5 – Preliminary Design 102 of 369

For convenience, equation 5.21 can be expressed as:

(Eqn. 5.22) 

Msum in this equation is the sum of all bending moments except the secondary moments from 

prestressing. Substituting equation 5.22 into equations 5.19 and 5.20:

(Eqn. 5.23) 

(Eqn. 5.24) 

These equations can be rearranged to:

(Eqn. 5.25)

(Eqn. 5.26)

Table 5.1 shows the bending moments determined from the analyses presented earlier in this 

Chapter at three locations in the example bridge. The locations chosen for review are the points 

of maximum positive flexure within the spans and the point of maximum negative bending over 

the piers.

Section DC DW 0.8(LL+I) 0.5TG Msum M2

0.3596 L1 14,896 1303 10,899* 1539 28,637 0.391(F)

0.5 L2 18,848 1649 11,309 4279 36,085 1.087(F)

Pier 2/3 -32,352 -2831 -11,959 -1284** -48,426 1.087(F)

*The maximum moment at 0.4167L (top of Page 24 of 41) was used for this example.

**Negative thermal gradient is -0.3 of the positive moment (LRFD Artivle 3.12.3).

Table 5.1 – Moment Components of Service Level III Loading at Three Locations 

Assuming moderately corrosive conditions, AASHTO LRFD Table 5.9.4.2.2-1 states that the 

allowable stress in the concrete after all losses is 0.19√f’c, where f’c has the units of ksi. This is 

equal to 6√f’c, where f’c has the units of psi. The resulting moments producing allowable 

stresses are:
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Chapter 5 – Preliminary Design 103 of 369

The sign of these moments are consistent with the development of equations 3.9 and 3.10.

Using equations 5.25 and 5.26, the minimum prestressing force requirements at the three 

sections studied, after all losses, are:

. 

5.7 Prestressing Losses and Tendon Sizing for Final Design (Pjack)

The required prestressing force at the center of the middle span is 8,172 kips. Assuming three 

tendons in each of the five webs of the example problem cross section, the final post-tensioning 

force in each of the 15 tendons is 545 kips. The number of prestressing strands required in 

these tendons is a function of the amount of instantaneous and long-term prestressing force 

loss. Losses will be computed in accordance with the information presented in chapter 4.

5.7.1 Losses from Friction, Wobble, and Anchor Set

Using the CG Profile, the forces along the tendon after two-end stressing are computed. The 

assumed values for friction, wobble coefficient, and anchor set are 0.25, 0.0002/ft, and 0.375”, 

respectfully. The geometry required for determining the losses is shown in table 5.2. The 

friction diagram, expressed in terms of stress is shown in figure 5.27. 

Point b (ft) h (ft)

Angular 

Deviation

θ=2h/b (rad)

Angular 

Force 

Coefficient

Cumulative 

Force 

Coefficient

Tendon 

Stress 

(ksi)

1 0 0.0000 0.00000 0.0000 1.0000 202.50 

2 50 2.5000 0.10000 0.9656 0.9656 195.54

3 55 2.9464 0.10714 0.9629 0.9298 188.28

4 15 0.8036 0.10714 0.9707 0.9025 182.76

5 20 0.9375 0.09375 0.9729 0.8781 177.81

6 60 2.8125 0.09375 0.9652 0.8475 171.62

7 60 2.8125 0.09375 0.9652 0.8180 165.65

8 20 0.9375 0.09375 0.9729 0.7959 161.16

9 15 0.8036 0.10714 0.9707 0.7725 156.43

10 55 2.9464 0.10714 0.9629 0.7439 150.63

11 50 2.5000 0.10000 0.9656 0.7183 145.45

Table 5.2 –Data for Friction Diagram for the CG Profile Tendon 
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Chapter 5 – Preliminary Design 104 of 369 

Figure 5.27 – Friction Diagram for the CG Profile Tendon 

5.7.2 Losses from Elastic Shortening

Equation 4.24, repeated here, presented the AASHTO LRFD equation for losses from elastic 

shortening.

(Eqn. 5.27) 

Difficulty arises here, because the value of fcgp, the concrete stress at the center of gravity of the 

prestressing, is computed after jacking and before any long-term losses. This requires an area 

of steel to be multiplied by the tendon stresses in figure 5.29, to produce a prestressing force by 

which concrete stresses can be computed.

This difficulty is overcome by making an initial assumption of all losses equal to 20 ksi. 

Subtracting this from the tendon stress at the center of the middle span (171.62 ksi) produces 

an estimate of final stress in the tendon at this location of 151.62 ksi (56 percent of fpu). The 

estimate of final force per strand in the tendon, using 0.6” diameter strands with an area of 

0.217 in2 is 32.9 kips. Dividing the 550 kips per tendon by the force per strand results in 16.7 

strands required in each tendon. Rounding up to 17 strands per tendon, forces after stressing 

can now be computed as the stresses in figure 5.29 times the tendon area of 0.217(17) = 3.689 

in2

. 
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Chapter 5 – Preliminary Design 105 of 369 

All of the post-tensioning tendons will be grouted within their ducts after installation and 

stressing. During construction, however, tendons are unbonded when elastic shortening 

occurs, assuming all tendons are stressed in one phase of construction.

AASHTO LRFD Article 5.9.5.2.3b states that:

For post-tensioned structures with unbonded tendons, the fcgp value may be calculated as 

the stress at the center of gravity of the prestressing steel averaged along the length of 

the member.

Concrete stresses will be computed at four locations along the bridge and then a weighted 

average will be used for fcgp. The equation used for computing the concrete stress is:

(Eqn. 5.28) 

where: MSW = bending moment from self weight only (ft-kips)

At end of bridge:

At the point of maximum eccentricity in Span 1 and 3

Over Piers 2 and 3

At the center of the middle Span

Using a straight-line average between the stresses at these locations, the average stress in the 

tendon is 106.4 ksf or 0.74 ksi. The loss associated with elastic shortening, using a concrete 

strength at the time of stressing of 4 ksi, is then:

5.7.3 Losses from Concrete Shrinkage

The loss of prestress force resulting from shrinkage of the concrete superstructure is given by 

AASHTO LRFD Equation 5.9.5.4.3a-1 as:
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Chapter 5 – Preliminary Design 106 of 369

(Eqn. 5.29)

The strain εbdf, is the shrinkage strain from the time of prestressing to final time as provided by

AASHTO LRFD Equation 5.4.2.3.3-1:

(Eqn. 5.30)

The factors in Equation 4.25 are defined by AASHTO LRFD Equations 5.4.2.3.2-2, 5.4.2.3.3-2, 

5.4.2.3.2-4, and 5.4.2.3.2-5. Evaluating these for the example problem:

1.45 0.13 1.45 0.13 12.57 .18 1.0) ( ) ( s

V k Use

S

  = − = − =−    

(2.00 0.014 ) (2.00 0.014(75)) 0.95 hs k H =− =− =

'

5 5 1

1 1 4 f

ci

k

f = = = + +

'

10,000 0.996

61 4 61 4(5) 10,000 td

ci

t k

f t

    =   =   = − +   − +  

This results in a shrinkage strain of:

3 (1.0)(0.95)(1.0)(0.996)0.48 10 0.000454 bdf ε − = × =

Evaluation of the transformed section coefficient Kdf requires computing the creep coefficient:

0.118 ( , ) 1.9 b f i s hc f td i t t kk k k t − Ψ =

1.56 0.008 1.56 0.008(75) 0.96 hc k H =− =− =

0.118 ( , ) 1.9(1.0)(0.96)(1.0)(0.996)(28) 1.226 bfi t t − Ψ = =

The transformed section coefficient is then:

2

1

1 1 1 0.7 ( , )

df

p ps c pc

bfi

ci c c

K

E A Ae

Ψtt

EA I

=  

+ ++        
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Chapter 5 – Preliminary Design 107 of 369 

Multiplying the components of equation 5.29, the loss of prestress stress resulting from the 

shrinkage of concrete is:

5.7.4 Losses from Concrete Creep

The loss of stress in the prestressing steel resulting from the creep of concrete is:

5.7.5 Losses from Steel Relaxation

Using the AASHTO LRFD permissible value for low relaxation steel, the loss of prestressing 

stress is resulting from relaxation of the prestressing steel is:

5.7.6 Total of Losses and Tendon Sizing

The total loss of prestressing force after jacking is the sum of elastic shortening, shrinkage, 

creep and relaxation:

The resulting stress in the prestressing steel at the center of the middle span is then:

The resulting force in a 0.6” diameter strand would be 31.9 kips. The number of strands in the 

15 tendons, based on the force requirement at the center of Span 2 of 8,172 kips, would be 

17.2. Therefore, use 18 strand tendons in final design. The jacking stressed assumed in the 

loss calculations was 75 percent of the ultimate strength of the strand. The resulting jacking 

force (Pjack) for each tendon would be 791 kips.

5.8 Service Limit State Stress Verifications

This Section presents stress summaries for important load cases at service limit states. 

Flexural stresses are verified after the tendons are stressed and before grouting, after the 

bridge is open to traffic and before long-term losses, and the bridge in operation after long-term 
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Chapter 5 – Preliminary Design 108 of 369 

losses have occurred. A verification of principal stresses in the webs is made with the bridge in 

operation and after long-term losses have occurred.

5.8.1 Service Flexure—Temporary Stresses (DC and PT Only)

Longitudinal stresses in the concrete superstructure are verified at three locations along the 

bridge. The stresses are being verified after all post-tensioning tendons are stressed, but 

before grouting. The assumption for this example is that there is insufficient bonded mild 

reinforcing in the precompressed tensile zone to permit tension during this phase. Applied 

bending moments are taken from table 5.1. 

Three post-tensioning tendons in each of the five webs of the box girder are stressed. Each of 

the tendons contains 18, 0.6” diameter strands. Tendon forces at the three study sections are 

based on stresses shown in figure 5.27, reduced for elastic shortening losses of 2.6 ksi as 

computed in section 5.7.2.

The allowable concrete stresses in the concrete before losses are presented in AASHTO LRFD 

Article 5.9.4.1. For a 28-day concrete strength of 5 ksi the allowable stresses are:

At the point of maximum eccentricity in Span 1 and 3

Over Piers 2 and 3

At the center of the middle Span
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The concrete stresses in the box girder superstructure are within permissible AASHTO LRFD 

limits. Stresses over Piers 2 and 3 could have been checked at the face of the support.

5.8.2 Service Limit State III Flexure Before Long-Term Losses

Longitudinal stresses in the concrete superstructure are verified at three locations along the 

bridge. The stresses are being verified when the bridge is first open to traffic. The assumption 

for this example is that there is insufficient bonded mild reinforcing in the precompressed tensile 

zone to permit tension during this phase. Applied bending moments are taken from table 5.1. 

Three post-tensioning tendons in each of the five webs of the box girder are stressed. Each of 

the tendons contains 18, 0.6” diameter strands. Tendon forces at the three study sections are 

based on stresses shown in figure 5.27, reduced for elastic shortening losses of 2.6 ksi as 

computed in section 5.7.2.

The allowable concrete stresses in the concrete before losses are presented in AASHTO LRFD 

Article 5.9.4.2. For a 28-day concrete strength of 5 ksi the allowable stresses are:

' 0.6 0.6(1)(5) 3.0 432.0 a wc f f = = = = f ksi ksf (compression)

' 0.19 0.19 5 0.425 61.2 a c f f =− =− =− =− ksi ksf (tension)

At the point of maximum eccentricity in Span 1 and 3

10,819 10,819(2.5)(2.732) 32,867(2.732) 133.5

99.45 643.7 643.7 TOP f =− + = ksf

10,819 10,819(2.5)(3.768) 32,867(3.768) 74.7

99.45 643.7 643.7 BOT f =+ − = ksf

Over Piers 2 and 3

10,556 10,556( 1.25)(2.732) 36,952(2.732) 5.3

99.45 643.7 643.7 TOP f ksf − − =− + =

10,556 10,556( 1.25)(3.768) 36,952(3.768) 245.2

99.45 643.7 643.7 BOT f ksf − − =+ − =

At the center of the middle Span

9,903 9,903(2.5)(2.732) 46,850(2.732) 193.3

99.45 643.7 643.7 TOP f =− + = ksf

9,903 9,903(2.5)(3.768) 46,850(3.768) 29.7

99.45 643.7 643.7 BOT f =+ − = − ksf

The concrete stresses in the box girder superstructure are within permissible AASHTO LRFD 

limits. Stresses over Piers 2 and 3 could have been checked at the face of the support.
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Longitudinal stresses in the concrete superstructure are verified at three locations along the 

bridge. The stresses are being verified when the bridge is first open to traffic. The assumption 

for this example is that there is insufficient bonded mild reinforcing in the precompressed tensile 

zone to permit tension during this phase. Applied bending moments are taken from table 5.1. 

Three post-tensioning tendons in each of the five webs of the box girder are stressed. Each of 

the tendons contains 18, 0.6” diameter strands. Tendon forces at the three study sections are 

based on stresses shown in figure 5.27, reduced by long-term losses of 24.9 ksi as computed in 

section 5.7.6. 

The allowable concrete stresses in the concrete before losses are presented in AASHTO LRFD 

Article 5.9.4.2. For a 28-day concrete strength of 5 ksi the allowable stresses are:

At the point of maximum eccentricity in Span 1 and 3

Over Piers 2 and 3

At the center of the middle Span

The concrete stresses in the box girder superstructure are within permissible AASHTO LRFD

limits. (Stresses checked over Piers 2 and 3 could have been checked at the face of 

the supporting pier.) 

5.8.3 Service Limit State III Flexure After Long-Term Losses
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5.8.4 Principal Tension in Webs after Losses 

AASHTO LRFD does not require principal tension verifications at service limit states for 

concrete box girders other than those built segmentally. These verifications are, however, 

useful for preliminary design, as an indicator whether the webs are sized appropriately. It is 

strongly recommended that principal tension be verified in cast-in-place concrete box girder 

construction when the box girder has only two webs. 

Consider the cross section 0.72h (4.68’) to the left of Pier 2 in Span 2. The shear forces 

acting are: 

16(80 4.68) 1, 205 VDC =−= kips

1.4(80 4.68) 105 VDW = −= kips

(1 ) 53 V kips LL lane =

(1 ) 68 VLL truck = kips

1.33(68) 53 143 VLL I kips + = +=

The live load distribution for shear is found by AASHTO LRFD Table 4.6.2.2.3a-1 for cross 

section type d and two or more design lanes loaded:

0.9 0.1 0.9 0.1

12.25 78 1.157

7.3 12( ) 7.3 12(160)

S d DF

L

      =       = =      

5(1.157) 5.785 N N DF L w =⋅ = =

(0.8)5.785(143) 662 VLL I kips + = =

( )

( ) ( ) 2 2

2 2 0.9375

4.68 0.0219 1.26

20

h

x rad deg b

θ − = = =− =−

VP = − =− 9, 249sin 1.26 203 ( ) kips

∑V = + + +− = 1205 105 662 203 1769 ( ) kips

The stresses acting on an element at the neutral axis are:

9, 249 93.00

99.45 x s = = ksf

0.0 y 

( ) 1,769(118.1) 64.91

( ) 643.7(5) xy

V Q ksf I B

τ = = 
Post-Tensioned Box Girder Design Manual June 2016

The Mohr’s Circle representation of stress at this location is then:

Figure 5.28 – Mohr Circle for Location of Maximum Shear in Middle Span 

The maximum principal tension of -44 ksf is greater than what would be allowable for a 

segmental box girder, but does reflect a level of stress that can be adequately reinforced during 

final design. 

5.9 Optimizing the Post-Tensioning Layout 

Section 5.6 of this chapter showed the determination of the required post-tensioning forces at 

three sections in the example three-span bridge. The three final forces computed were: 

• 4743 kips at 0.4167L of Spans 1 and 3.

• 8256 kips at mid-span of Span 2.

• 7169 kips over Piers 2 and 3.

The large disparity in the required post-tensioning force in the end spans versus the middle 

span indicates that some optimization of the post-tensioning layout is warranted. Decreasing 

the maximum positive eccentricity in Spans 1 and 3 will increase the post-tensioning demand 

Chapter 5 – Preliminary Design 112 of 369

The features of Mohr’s Circle with these stresses are:
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Chapter 5 – Preliminary Design 113 of 369

those spans. In Span 2, however, the effect of reducing the maximum positive eccentricity in 

Spans 1 and 3 will reduce the secondary moments, and the post-tensioning demand in that 

span. 

Figure 5.29 shows a revised center of gravity post-tensioning layout for the end spans of the 

example bridge. The tendons have been raised to 2 feet from the bottom of the bottom slab. 

The resulting maximum positive eccentricity is 1.738’. The maximum eccentricity at the interior 

piers is unchanged.

Figure 5.29 – Revised Center of Gravity Profile of Prestressing (End Spans)

Revised conjugate beam loads are determined from the revised profiles. Figure 5.30 shows the 

updated conjugate beams and loads in the end spans for a unit prestressing force (F=1). The 

conjugate beam loads for the middle span are unchanged.

Figure 5.30 –Revised Conjugate Beam and Loads (End Spans) 

The updated secondary prestress moments are shown in Figure 5.31.
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Figure 5.31 –Revised Secondary Prestressing Moments, M2(F)

The minimum prestressing force requirements at the three sections studied, after all losses, are 

then:

These force requirements are compared to those shown at the top of page 29 of 41. Raising 

the eccentricity in the side spans decreases the overall post-tensioning demand by 510 kips, or 

approximately 6 percent.
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Chapter 6—Substructure Considerations

6.1 Introduction

Chapter 5 presented an approach to the preliminary design of cast-in-place post-tensioned box 

girder bridges. The approach was demonstrated by considering a three-span box girder bridge. 

To focus on superstructure concepts, the three-span example bridge was supported by bearings 

at the piers and abutments. Often, however, the supporting columns are cast integrally with the 

box girder superstructure. When this is the case, the analysis and design of the bridge should 

include the flexural stiffness of the supporting substructure.

This chapter presents bending moments for the same three-span example bridge of chapter 5, 

with the exception that the piers are integral with the superstructure. Figure 6.1 shows an 

elevation view of the example bridge. Figure 6.2 shows a cross section of the bridge supported 

at each pier with two 6’ diameter columns. For this example, each column is supported by a 7’ 

diameter drilled shaft, with an equivalent length to fixity of 10’. The flexibility analysis for this 

structure is provided in appendix A, beginning on page 25 of 33.

Figure 6.1 –Example CIP Box Girder Bridge Elevation

Figure 6.2 –Bridge Cross Section at Piers
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6.2 Bending Moments Caused by Unit Effects

6.2.1 Effect of a Unit Uniform Load

Figure 6.3 – Effect of a Unit Uniform Load 
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6.2.2 Effect of a Unit Lateral Displacement (Side-Sway Correction)

Figure 6.4 – Effect of a Unit Lateral Displacement (Side-sway Correction) 

6.2.3 Effect of a Unit Contraction

Figure 6.5 – Effect of a Unit Contraction 
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6.3 Dead Load—DC (Self Weight and Barrier Railing)

Figure 6.6 – Effect of Dead Load (DC) 

6.4 Dead Load—DW (Future Wearing Surface)

Figure 6.7 – Effect of Dead Load (DW) 
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6.5 Live Load—LL (Lane and Truck Components)

6.5.1 Envelope of Uniform Load Component

Figure 6.8 – Uniform Live Load Moment Envelope 

6.5.2 Truck—Positive Moment in Span 1 or 3

Figure 6.9 – Moment Diagram for HS20 Truck in Span 1 or 3 
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Chapter 6 – Substructure Considerations 120 of 369 

6.5.3 Truck—Positive Moment in Span 2

Figure 6.10 – Moment Diagram for HS20 Truck in Span 2 

6.5.4 Truck—Negative Moment over Piers

Figure 6.11 – Moment Diagram for Two HS20 Trucks about Pier 2 

6.6 Post-Tensioning Secondary Moments—Unit Prestressing Force

Secondary moments from post-tensioning for the framed structure will be different from those of 

a similar bridge supported by bearings. The shortening of the superstructure under the 

prestressing induces bending moments in the columns which are distributed throughout the 

framed structure. The Flexibility Method solution for the frame considers a single rotational 
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Chapter 6 – Substructure Considerations 121 of 369

degree of freedom at each member end. As a result, the effect of the axial shortening needs to 

be added to the flexural results to find the total secondary moments.

The top diagram in figure 6.12 shows the secondary moments considering the rotational 

restraint provided by the columns. The second diagram shows the secondary moments 

resulting from a unit force shortening the bridge. These moments are found by multiplying the 

results for the unit contraction (figure 6.5) times the shortening of the span between the columns 

under the action of a unit force, which is predicted as:

The modulus of elasticity is taken from equation 2.5 for a 28-day strength of 5 ksi and a unit 

weight of 150 lb/ft3

. 

Figure 6.12 – Secondary Prestressing Moments, M2(F)

The component of the secondary moment resulting from axial shortening is slightly 

overestimated in this case as it does not consider the shear forces attracted by the columns. 

This approximation is not the case in a stiffness or flexibility solutions that consider rotational 

and displacement degrees of freedom simultaneously. The approximation could be refined by 

an iterative approach where the column top lateral displacement under the shear force is made 

6 (1)(80) 1.3031 10

(99.45)(617300)

PL

AE

− ∆= = = ×
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Chapter 6 – Substructure Considerations 122 of 369 

equal to the shortening of the span under the unit force, less the column shear force. For this 

example, the refinement reduces the secondary moments from axial shortening by about 4 

percent.

6.7 Thermal Gradient (TG)—20°F Linear

Figure 6.13 – Moment Diagram for a 20°F Positive Linear Gradient 

6.8 Moments Resulting from Temperature Rise and Fall

6.8.1 Temperature Rise—40°F Uniform Rise

Figure 6.14 – Moment Diagram for 40° Temperature Rise 
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6.8.2 Temperature Fall—40°F Uniform Fall

Figure 6.15 – Moment Diagram for 40° Temperature Fall 

6.9 Moments Resulting from Concrete Shrinkage

Bending moments are produced in the rigid frame as the superstructure concrete shrinks and 

displaces the tops of the columns toward the center of stiffness of the structure. These bending 

moments can be estimated by factoring the load case for unit contraction (figure 6.5) by the pier 

top displacements. For the example structure, by inspection of its symmetry, the center of 

stiffness is located at the center of the middle span.

The shrinkage strain can be computed using the CEB-FIP Model Code presented in chapter 2. 

The total shrinkage strain at time t is calculated by:

With,

And,

The values for the beta terms are computed first:
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Assuming day 10,000 to be the day when the concrete shrinkage is assumed to end, and 

considering the outside perimeter and half of the inside perimeter in computing the notional 

thickness (perimeter = 189.83’):

Substituting these values into the equations for concrete strain:

The resulting pier top deflection is:

The resulting bending moment diagram is:

Figure 6.16 – Moment Diagram for Concrete Shrinkage 

The bending moment diagram in figure 6.16 assumes that the same concrete mix is used for the 

superstructure and the substructure. Also, these results assume that the concrete of the 

0.5

1

2

1

( )/ ( ) 350( / ) ( ) /

s s s

o s

tt t

t t

hh t t t β   − − =  

+ −  

0.5

2

(10000 1) /1 (10000 1) 0.8585

350(319.36 /100) (10000 1) /1 β s

  − − =   =

  + −

6 ( ) [160 10(5)(9 42.47 /10)] 10 0.0003977 s cm ε f − =+ − ×=

( ) 0.0003977( 0.8961) 0.0003564 cso s cm RH εε β = = − =− f

( , ) ( ) 0.0003564(0.8585) 0.0003060 cs s cso s s ε εβ tt t t = − =− = −
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superstructure and substructure is poured at the same time (has the same age). Variations in 

these, or other, assumptions will impact the resulting bending moments. For example, the 

bending moments will increase if the columns are poured in advance of the superstructure, 

given that they are made of the same concrete. Oppositely, the bending moments will decrease 

if a lower strength of concrete is used in the columns, reducing their stiffnesses.

6.10 Moments Resulting from Concrete Creep

Cast-in-place post-tensioned concrete bridges will creep under the action of the permanently 

applied loads (typically self weight and post-tensioning). The bending moments determined in 

the elastic structure are compatible with member end rotations. These member end rotations 

will grow as a function of concrete creep, but no redistribution of bending moments will occur, as 

no new end restraints are applied, and the stiffness of the members, though changing with time, 

are relatively the same. Rotations and deflections will increase, but no change in bending 

moments.

Likewise, no creep moments will develop as a result of the continued top of pier displacement 

as a result of concrete creep. Lateral deflections will increase with no increase or decrease of 

bending moments. There will be a lessening of the prestressing force as the span shortens 

which will be opposite in direction to the moments produced during the axial shortening of the 

bridge during stressing.

Creep moments related to member end rotations will develop to reduce deformations induced 

by concrete shrinkage. These moments can be estimated as a percentage of the moments due 

to shrinkage, that percentage being a function of concrete creep characteristics. 

A creep coefficient can be computed using the CEB-FIP Model Code presented in chapter 2. 

The relationship of creep strain to elastic strain is given by:

Where,

The notional creep coefficient is:

With,

And,

0 0 ( )() RH cm f fβ β = f t

1/3 1/3

1 / 1 75 /100 1 1 1.3691

0.46( / ) 0.46(319.36 /100)

o RH

o

RH RH

h h f − − = + = + =

Chapter 6 – Substructure Considerations

0 0.2 0.2

0 1

1 1 ( ) 0.4884

0.1 ( / ) 0.1 (28 /1)

t

t t

β = = = + +

),( )( ),( 0

0 tt

E

t

tt

ci

c

cc o f s

ε =

)(),( 00 0tttt βff c −=
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The development of creep over time is given by:

With,

And,

And,

Solving for the creep coefficient:

The creep moment acts to reduce the moments induced by shrinkage. Consider the total strain 

under a sustained load.

This expression can be rearranged to:

And,
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Expressed in terms of bending moment, the total moment of shrinkage and the creep related to 

shrinkage is:

Solving for the creep moment:

For the example problem considered, the creep moment is estimated by the linear scaling of the 

shrinkage bending moment diagram.

The bending moments resulting from creep of the concrete related to shrinkage are shown in 

figure 6.17.

Figure 6.17 – Moment Diagram for Concrete Creep 

Additional creep bending moments, in addition to those related to concrete shrinkage, can 

be generated in cast-in-place post-tensioned bridges even though the statical scheme 

remains unchanged. These creep moments occur when relative creep characteristics of 

superstructure and substructure concrete are different, whether by being of different mix 

designs or of different casting dates. For example, if the substructure of the example bridge 

were made of the same concrete, but cast at a date earlier than the superstructure, the older 

substructure would resist the creep of the younger superstructure concrete. The moments to 

be considered in calculating this creep moment would be the summation of permanent loads 

(DC and PT). The moments would be estimated by scaling the sum of the permanent loads 

by a relative ratio of remaining creep coefficients of the two concretes. 

Where E’ci is an effective modulus of elasticity equal to the 28-day modulus adjusted by the 

creep coefficient,
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6.11 Bending Moments Summaries 

This Section presents bending moment diagrams for the two example bridges, one on bearings 

and one with integral piers. The moments were determined by Flexibility Analysis. The 

analysis for the bridge on bearings is found in chapter 5, section 5.4.2. The analysis for the 

bridge with integral piers is found in section A.13 of appendix A. 

The “maximum” moments in the side spans are found at the location of maximum load under 

self weight. The location of the truck component of the live loads is the same within the 

spans of both the bridge on bearings and the bridge with integral piers. Though both a 

slightly larger truck component and therefore overall live load moment may be found, the 

values in figure 6.18 were chosen for the comparative exercise in this Chapter. (See section 

5.5.4.5 for the number of design lanes.) 

Load Span 1 (“max”) CL Span 2 CL Pier

Brg Fixed Brg Fixed Brg Fixed

kL 0.3596L1 0.3677L1 0.5L2 0.5L2 0.0L2 0.0L2

DC 14896 15570 18800 17920 -32352 -33280

DW 1303 1362 1649 1568 -2831 -2912

LLu 923 781 1138 898 -1448 -1548

LLt 1475 1381 1657 1408 -1769 -1820

NL 4.544 4.544 4.229 4.229 4.372 4.372

0.8(LL+I) 10487 9516 11306 9374 -11964 -12493

0.5GR 1539 1811 4279 3957 -1284 -1187

M2 0.3159F 0.1796F 0.8785F 1.0736F 0.8785F 1.0736F

Tr 0 1777 0 -2417 0 -2417

Tf 0 -1777 0 2417 0 2417

SH 0 -2265 0 3080 0 3080

CR 0 1427 0 -1940 0 -1940

Table 6.1 – Bending Moments for Bridge on Bearings and Bridge with Fixed Piers 

6.12 Post-Tensioning Force Comparison (after all losses, with thermal effects)

6.12.1 Side Span Positive Bending

For the bridge on bearings:

For the bridge with integral piers:
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For the bridge on bearings:

For the bridge with integral piers:

6.12.3 Negative Bending at Piers

For the bridge on bearings:

For the bridge with integral piers:

6.12.2 Middle Span Positive Bending
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Chapter 7—Longitudinal Analysis & Design

7.1 Introduction

The longitudinal analysis and design of post-tensioned concrete box girder bridge 

superstructures in accordance with AASHTO LRFD specifications culminates in the verification 

of superstructure cross sections at service and strength limit states. The loads for which the 

cross sections are verified under the various limit states are those determined by linear elastic 

analysis methods factored to achieve a desired reliability in design. Included in these elastic 

analyses is the appropriate consideration for the time-dependent characteristics of the 

superstructure concrete and post-tensioning.

Computerized solutions for post-tensioned structures are available from a number of software 

developers. These computer programs are typically based on either two or three-dimensional 

finite element methods using beam/column elements in the elastic range. Built around the finite 

element solutions are material characteristics that, through a series of time updates, allow the 

incorporation of effects such as concrete creep, concrete shrinkage, and relaxation of the posttensioning. Most of these computer programs also allow for the input of post-tensioning 

tendons by geometric definition. The geometric definitions are transformed internally within the 

software into loads applied to the bridge model. Complete packages can include automatic 

generation of AASHTO LRFD live loads and limit state combinations to expedite cross section 

verifications.

Not having specialized software does not leave the engineer without the means to analyze and 

design post-tensioned concrete bridges. Hand calculations like those presented in this Manual, 

or those using general frame analysis software are adequate analysis tools for many bridges. 

More rigor is required in developing these analyses, but often they disclose valuable insights 

into the behavior of the bridge. The effects of post-tensioning are determined by applying 

equivalent loads to the stiffness based solution. The results are the total prestressing effects, 

from which primary effects are subtracted to find the secondary effects. Alternatively, flexibility 

solutions can produce secondary effects directly, which are then added to the primary effects to 

determine the total prestressing effects. Time-dependent effects of the concrete and 

prestressing steel are estimated by code equation and then applied in the form of forces or 

displacements to determine their effect on the bridge. This can be done conservatively by 

applying ultimate values at one time, or applying them incrementally to better estimate the 

diminishing of these effects as time progresses.

This manual utilizes both computer programs specifically developed to facilitate post-tensioned 

concrete bridge design along with supplementary hand calculations to verify limit states. The 

computer programs used are Bridge Designer 2 (BD2) and Bridge Designer 3 (BD3) by 

Interactive Design Systems. BD2, a two-dimensional analysis package, was used in the 

preparation of design example 1 provided in appendix C of this manual. Design example 2, 

provided in appendix D, uses BD3, a three-dimensional solution. The general-purpose finite 

element analysis program LARSA was used in design example 2 for the development of live 

load effects on the two-cell curved structure.

7.2 Modeling Concepts

Stiffness based finite element analyses model complex force-displacement behavior of a 

structure by dividing it into an assemblage of smaller members interconnected at nodes. Nodal 

locations are chosen so that a finite number of independent degrees of freedom will, with 

desired accuracy, capture the displacement characteristics of the structure. Relative 
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Chapter 7 – Longitudinal Analysis & Design 131 of 369

displacements of member connected nodes are established by idealized force-displacement, or 

“stiffness”, characteristics of the smaller finite members. The needed sophistication of stiffness 

characteristics for members in a finite element analysis is related to sophistication of forcedisplacement results needed for subsequent member design. Two-dimensional and threedimensional beam-column elements represented by classic stiffness characteristics are 

elements with appropriate sophistication for the design of most cast-in-place box girder bridges.

7.2.1 Straight Bridges Supported on Bearings

The longitudinal design of straight concrete box girder superstructures can be made using twodimensional analyses supplemented with hand calculations to estimate force effects out of the 

plane of the analysis. Figure 7.1 shows the elevation of a bridge similar to the one presented in 

design example 1 in appendix C. The bridge shown in figure 7.1 is different from the design 

example in that the superstructure is not integral with the piers, but rests on bearings. The 

cross section of this bridge, shown in figure 7.2, is the same as for design example 1.

Figure 7.1 – Example Straight Bridge on Bearings 

Figure 7.2 – Box Girder Superstructure Cross Section

7.2.1.1 Nodes

Figure 7.3 shows one layout for a two-dimensional model representation of the three-span 

bridge in figure 7.1. Nodes are defined by coordinates in an orthogonal coordinate system, 

such as x-coordinates in the longitudinal direction and y-coordinates in the vertical direction.

Longitudinally, nodes were first located at support locations and at even increments along the 

length of the bridge. The number of nodes defined along each span is the decision of the 

analyst, but should be sufficient to allow the analysis software member definitions to accurately 

model bridge behavior. A typical nodal spacing of 10’ was used for the bridge model shown in 

figure 7.3. Additional nodes (3, 15, 17, 22, 35, and 47) were added for ease in using the results 

of the analysis. These nodes coincide with the design cross section for shear (d or h/2 from the 

centerline of pier, depending on the approach in finding nominal capacity). Two other nodes (1 
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Chapter 7 – Longitudinal Analysis & Design 132 of 369 

and 49) were added behind the first and last bearing locations to facilitate the modeling of posttensioning consistent with end anchor details

Nodes shown in figure 7.3 are defined vertically at the center of gravity of the superstructure 

cross section. The vertical location of the nodal coordinates changes with variations in cross 

section such as when a bottom slab is thickened near a pier to control compressive stresses, or 

in a variable depth structure used in longer spans. The vertical locations of the nodes are not 

adjusted for the profile of the bridge, except in the most extreme conditions. There are other 

bridge types, such as cable-stayed bridges, where horizontal forces are sufficiently large as to 

impose significant deck bending because of profile changes.

The analysis model is simply supported at all bearing locations in the vertical direction. A 

horizontal support is provided at Node 2 to assure model stability. Interior piers and bearing 

stiffnesses could be added to this model if the of long-term effects of concrete creep and 

shrinkage and prestressing steel relaxation are desired in the piers. Their effect is typically 

small on the results for the design of the superstructure (see chapter 6). If the bearings are 

flexible, such as for laminated elastomeric bearings, the horizontal stiffness can be input as a 

constraint between pier top and the superstructure. Rotational stiffnesses of these bearings 

should not be accounted for in the model. The very large rotational stiffnesses of these 

bearings rely on large vertical loads to remain active without uplift. The result is an 

inappropriately large attraction of bending moment to the pier if modelled, approaching that of 

integral piers. A rotationally free connection is more appropriate.

Figure 7.3 – Two-Dimensional Analysis Model

7.2.1.2 Elements

Nodes in the two-dimensional analysis model of the straight bridge are connected by idealized 

beam-column elements defined by appropriate stiffness matrices. Figure 7.4 shows a typical 

stiffness matrix for a beam element connecting nodes that have three degrees of displacement 

freedom (vertical, horizontal, and rotational).

Figure 7.4 shows that the information necessary to define the behavior of the beam-column 

element connecting two nodes includes:

• Cross-Sectional Characteristics—Area and Moment of Inertia about the horizontal axis 

passing through the centroid of the cross section. Typically the gross cross section 

characteristics of the concrete girder are considered. There may be instances where the 

cross-sectional area of the ducts is great enough to impact analysis results. When this 

happens, the net cross section should be used for superstructure construction and 

tendon stressing, and the gross properties used for loads applied after the tendons are 

grouted.
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• Modulus of Elasticity—For the purposes of the verification of Service and Strength Limit 

States, cast-in-place post-tensioned bridges are analyzed with fully elastic behavior 

(LRFD C4.5.1). Within these elastic analyses, however, the time-dependent behaviors 

of concrete and prestressing steel must be incorporated (LRFD 4.5.2.2). Timedependent behavior is most typically captured in commercially available software by a 

series of time updates, where changes in concrete dimension are estimated over a time 

step. Typically, the modulus of elasticity is based on the 28-day strength of the concrete 

(see chapter 2).

Figure 7.4 –Typical Element Stiffness Matrix for a Plane Frame Member with 3DOF Nodes

For the cross section shown in figure 7.2, the cross section characteristics are:

Figure 7.5 –Cross Section Properties for the Box Girder shown in Figure 7.2 

End and pier diaphragms are used at the supports to transmit shear forces from the webs to the 

supports and to stiffen the box girder with regard to torsion. End and pier diaphragms are 

typically solid sections of the bridge with small openings to allow for access between the spans 

after construction. It is important to incorporate the weight of these diaphragms into the design 

of the bridge. However, members containing small length diaphragms (up to approximately the 

depth of the box girder superstructure) should not be represented with cross section properties 

including the diaphragms. The typical cross section properties should be used for these 

members. Longer diaphragms could warrant a change in cross section characteristics.
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7.2.1.3 Post-Tensioning

The effects of the post-tensioning tendons are modeled within the software as an equivalent set 

of forces acting on the elements of the bridge. Most post-tensioned bridge analysis software 

packages use a graphical interface so that the tendons can be defined by their desired 

geometry. Internal to the program, the geometry is used to compute fixed end element forces 

which then become the post-tensioning load case within the program. When loaded with the 

equivalent loads, the results are the combination of primary and secondary post-tensioning 

moments. The secondary moments are then determined by subtracting the primary moment, 

from the input tendon geometry, from the total post-tensioning moments.

The effect of time-dependent deformations of the concrete and steel are incorporated through 

time-steps where the long-term deformations are used to change the post-tensioning forces 

along the length of the tendons. Iterative solutions are often used to have the assumed and 

final time-depended deformations converge over each time step. 

General purpose software analysis packages without the capability to define post-tensioning 

tendons geometrically, or to track and update forces with time, can be used for the final design 

of cast-in-place bridge superstructures. An equivalent force approach could be taken with code 

predicted changes in the post-tensioning forces in order to capture appropriately long-term 

behavior. Design Aids 11.1.4 and 11.1.5 in chapter 11 of the Precast/Prestressed Concrete 

Institute Design Handbook, 7th Edition, are good resources for determining equivalent 

prestressing loads.

7.2.2 Straight Bridges with Integral Piers

The superstructure of the straight bridge modeled in section 7.2.1 is the same superstructure 

used in design example 1 in appendix C without integral piers. The cross section of the bridge 

used in design example 1 with integral piers is shown in figure 7.6. The integral pier shown 

consists of twin circular pier columns, each supported by a single drilled shaft foundation.

Figure 7.6 – Cross Section of Design Example 1 Bridge at the Piers 
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Figure 7.7 shows the model for the bridge with integral piers. The layout of nodes and members 

in the superstructure are the same as those for the straight bridge on bearings (section 7.2.1).

Figure 7.7 – Two-Dimensional Analysis Model with Integral Piers

Details of the modeling of the pier and foundation elements are shown in figure 7.8:

Figure 7.8 – Detail of Model at Pier

The model shown in figure 7.7 differs from that of figure 7.3 in three ways:

Pier Elements—The piers are modeled as the combination of the pier shaft properties 

participating in the longitudinal flexure of the bridge superstructure. In this example, the piers 

are modeled with twice the area and twice the inertial of two 6’-diameter pier columns.

Foundation Stiffnesses—In this example problem, the drilled shaft foundations are represented 

by two 7’-diameter concrete drilled shafts in the same way the piers were modeled. An 

equivalent member length to a point of fixity determined by soil-pile interaction was used to 

locate Node 101 relative to Node 102. Depending on the computer program used, the 

foundation could also be modeled with a user defined stiffness matrix determined by a study of 

soil-pile interaction.

Rigid Members—Members from the superstructure center of gravity to the top of the pier have 

been added. This element, modeled as rigid, facilitates the direct output of force and moment 

results at the tops of the piers. The rigid extensions at the end bents are added for dimensional 

consistency with the pier modeling.
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7.2.3 Curved Bridges

LRFD Article 4.6.1.2.3 provides guidelines for including horizontal bridge curvature in the 

analysis and design of concrete girder bridges. Three categories of consideration, related to the 

ratio of span length to horizontal curve radius expressed in terms of span length central angle, 

are presented in the LRFD. The parameters for the evaluation are shown in figure 7.9

Figure 7.9 – Curved Bridge of Design Example 2 

The three categories for consideration of horizontal curvature in LRFD are:

• Central Angle ≤ 12 degrees—A bridge whose largest span length has a central angle not 

greater than 12 degrees (L/R ≤ 0.2) may be designed as straight. No consideration for 

bridge curvature is necessary for the longitudinal design of the concrete box girder 

superstructure.

• 12 degrees < Central Angle ≤ 34 degrees—A bridge whose largest span has a central angle 

greater than 12 degrees (L/R ≤ 0.2) and not greater than 34 degrees (L/R ≤ 0.6) must 

include the horizontal curvature of the bridge in analysis and design. The analysis may be 

comprised of idealized beam-column elements defined between nodes at the center of 

gravity of the cross sections as it follows the horizontal curve. The girder elements may 

themselves be straight between nodes and should include stiffness characteristics that 

relate forces and displacements in 6 degrees of freedom (three translational and three 

rotational). The maximum length of the straight girder elements between nodes shall be 

such that the central angle between adjacent nodes shall not exceed 3.5 degrees (Lnodes/R≤ 

0.06). The assemblage of these elements is sometimes referred to as a spine beam.

• Central Angle > 34 degrees—A bridge whose largest span has a central angle greater than 

34 degrees (L/R ≤ 0.6) shall be analyzed by a proven three-dimensional analysis method 

such as: grillages, folded plates, finite strips, or shell element finite elements.

The bridge presented in design example 2 in appendix D is a three span bridge with the same 

span lengths as the bridge presented in design example 1 in appendix C. Different from design 

example 1, however, is that the design example 2 bridge lies on a horizontal alignment with a 

radius of 600’ over the full length of the bridge. Figure 7.10 shows the bridge of design example 

2. The span lengths of the bridge are 120’, 160’, and 120’. The central angles of the spans are 
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Chapter 7 – Longitudinal Analysis & Design 137 of 369 

11.5 degrees, 15.3 degrees, and 11.5 degrees. The central angle of the middle span is such 

that a curved girder analysis comprised of straight beam elements is appropriate. The model 

used in design example 2 is shown in figure 7.11.

Figure 7.10 – Design Example 2 Bridge 

Figure 7.11 – 3D Model for Bridge in Design Example 2 

The bridge of design example 2 is also different from that of design example 1 in that it is 

supported by three bearings at the piers and abutments. Figure 7.12 shows a cross section of 

the bridge at the pier. Figure 7.13 shows the layout of nodes and members at the piers. Rigid 

elements are used to replicate the rigidity that the superstructure diaphragms add at the pier. 

Other rigid members are used to account for the thickness of the substructure members and 

output convenience.
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Figure 7.12 – Cross Section of Design Example 2 Bridge at the Piers 

Figure 7.13 – 3D Model for Bridge in Design Example 2 at the Piers

Computerized stiffness solutions of three-dimensional spine girders that incorporate bridge 

curvature represent the members with element stiffness matrices similar to those of twodimensional analyses. Two additional cross section characteristics are required to define these 

members: the moment of inertia about the element vertical axis (for transverse flexure), and the 

torsional constant, J, that is used to model the element torsional stiffness. 
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Chapter 7 – Longitudinal Analysis & Design 139 of 369 

Torsional moments applied to closed cross sections such as concrete box girder bridges are, in 

the first order, resisted by a flow of shear around the perimeter of the closed section. Figure 

7.14 shows a single cell box girder subjected to a torsional moment Mt. The shear flow around 

the box girder is defined by the variable “q”, with units force per length.

Figure 7.14 –Torsion in a Single Cell Box Girder 

Appendix B to this manual presents a detailed development of the torsional behavior of closed, 

hollow cross sections. From that development, the shear flow is found to be constant around 

the closed section and related to the torsional moment by:

(Eqn. 7.1)

Where A0 is the area enclosed within a perimeter defined by the median lines of the closed 

section elements (webs, top slab, and bottom slab). The corresponding shear stress in the 

members of the closed section is found by dividing the shear flow by the member thickness:

(Eqn. 7.2) 

The torsional constant for the single cell closed hollow section is:

(Eqn. 7.3) 

The denominator in equation 7.3 is the integral of length along the median perimeter divided by 

the thickness of the members comprising the closed hollow section. For box girder cross 

sections where the members are distinct and linear, equation 7.3 can be written as:
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(Eqn. 7.4) 

Where, the closed loop integral is replaced with a discrete summation.

For cross section comprised of multiple closed hollow cells, torsional moments are resisted by 

shear flow around all of the cells. Figure 7.15 shows the equilibrium of external torsional 

moment and internal shear flow in a two-cell box girder superstructure.

Figure 7.15 –Torsion in a Two-Cell Box Girder

The torsional moment is in equilibrium with the sum of the shear flows multiplied by their 

enclosed medial area:

Evaluation of the shear flow in each cell must include the effects of the shear flow in adjacent 

cells, as influenced by members that are shared in common. Equations B.44 in appendix B 

shows the relationship between cross section rotation of a single box girder cell and shear flow:

Including the effects of shear flow in common members, equation 7.6 can be expressed for 

each cell of the box girder as:
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Compatibility of deformations requires that the rotations of the individual cells are equal to each 

other and are equal to the rotation of the cross section, so that:

(Eqn. 7.9) 

Equations 7.7 and 7.8 can be written as a system of simultaneous equations which include the 

equality of equation 7.9:

(Eqn. 7.10)

(Eqn. 7.11)

Where the shape constants (d) of the expression are:

(Eqns. 7.12)

Solution for the shear flows in equations 7.10 and 7.11 can be made in terms of the cross 

section rotation. The relationship between applied torsional moment and cross section rotation 

is developed in appendix B as:

(Eqn. 7.13)

Equilibrium between applied torsional moment and shear flow in the cells (equation 7.5) can be 

expressed for the two-cell box girder as:

(Eqn. 7.14) 

Equating equations 7.13 and equation 7.14 leads to the calculation of the torsional constant for 

the cross section.

Example: Consider the cross section of design example 1 in appendix C. The 

dimensions of the cross section, enclosed cell areas for torsion, and idealized thin wall 

member thickness and lengths are shown in figure 7.16. 

The system of simultaneous equations that needs to be solved for the four-cell box 

girder is:
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Chapter 7 – Longitudinal Analysis & Design 142 of 369 

Figure 7.16 – Cross Section of Bridge in Example 1, Appendix C 

These equations are modified to discount the influence of cells that are not adjoining:

Using the cross section geometry, and noting symmetry, the shape constants are:

Again noting the symmetry of the section, the system of four simultaneous equations can 

be reduced to two equations: 

 
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Chapter 7 – Longitudinal Analysis & Design 143 of 369 

Or,

From which,

Considering equilibrium, and symmetry:

7.2.4 Other Three-Dimensional Analyses

Bridges containing spans with large central angles (> 34 degrees) are one type of concrete box 

girder superstructure that requires analysis by methods beyond three-dimensional frame (spine 

beam) analyses. There are other concrete box girder bridge types that warrant more rigorous 

three-dimensional analyses, the two most common of these are:

• Bridges with Variable Skew Supports—Bridges, straight or curved, with orientation of 

supports different from right angles to supported cross sections are said to have skewed 

supports. Bridges with skew supports exhibit different distributions of flexural stresses and 

shear forces than those in similar bridges with radial supports. LRFD Article 4.6.2.2 

presents live load distribution factor corrections for bridges with consistently skewed

supports. Special modeling is required to capture flexure and shear behavior when adjacent 

skewed supports are significantly different. 

• Two-Cell Box Girder Bridges—The LRFD live load distribution equations do not address 

bridges with less than three cells. Bridges with two cells require additional analysis to 

determine the distribution of live load effects with the box girder superstructure. These 

bridges may be straight or curved, and with or without skew supports.

The most common methods of analyzing the bridge types described above are by grillage or 

shell finite elements. Grillage methods subdivide the box girder cross section into a series of 

longitudinal and transverse three-dimensional beam elements with characteristics such, that 
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Chapter 7 – Longitudinal Analysis & Design 144 of 369 

when analyzed together, replicate bridge deck behavior with appropriate accuracy. Figure 7.17 

shows the development of nodal locations for a grillage analysis of the superstructure of design 

example 2 in appendix D. Figure 7.18 shows a perspective of a grillage model for that bridge. 

Figure 7.17 – Grillage Model Development for Design Example 2

Figure 7.18 – Grillage Model Design Example 2 

Grillage methods of analysis are relatively easy to develop using general purpose threedimensional analysis software. The quality of the results of the analysis, however, can vary 

significantly depending on the distribution of members in the grillage. Chapter 5 of E.C. 

Hambly’s book “Bridge Deck Behavior” (1998 reprint of the 1991 2nd Edition, ISBN 0-419-17260-
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Chapter 7 – Longitudinal Analysis & Design 145 of 369 

2) is an excellent reference to consult when developing a grillage analysis of a concrete box 

girder bridge.

Finite element methods of analyzing box girder superstructures are typically performed using 

three-dimensional shell elements. For this manual, shell elements are defined as those that 

exhibit in-plane membrane behavior and out-of-plane flexure and shear behavior. Finite 

element methods using shell elements model the continuum of the bridge, reducing the 

influence of mesh choices in grillage methods. That is not to say that they are free from 

inappropriate modeling choices, such as inappropriate element aspect ratios or ineffective 

model discretization. Thus, care must be taken hen developing the model. Many FEM software 

packages have powerful graphical modeling interfaces and/or allow for ease in importing model 

geometry developed in a CAD environment.

Figure 7.19 shows the cross section of the finite element model that was used to find live load 

distribution factors for the bridge in design example 2 in appendix D. The model was generated 

by copying elements along the 600’ horizontal curve radius to produce one foot long elements 

along the centerline of the bridge.

Figure 7.19 – Cross Section of Shell Element FEM Model for Design Example 2 

The reader is encouraged to read NCRP Report 620 “Development of Design Specifications for 

Horizontally Curved Concrete Box-Girder Bridges” (2008) for further information regarding the 

analysis and design of curved box girder bridges.

7.3 Strength Limit Verification—Flexure

Cross sections along the length of the box girder superstructure must satisfy LRFD flexural 

requirements at strength limit states. The number of cross sections verified should sufficiently 

capture the effects of support conditions, individual span lengths within continuous units, 

changes in post-tensioning tendon geometry, and cross section changes, if present. 

The basic equation for factored flexural resistance of a cross section in a box girder bridge is 

presented in LRFD Equation 5.7.3.2.1-1.

(Eqn. 7.15) 

Where, Mr = factored flexural resistance

Mn = nominal flexural resistance

φ = resistance factor as specified in Article 5.5.4.2

Expressing this relationship in terms of the fundamental equation of LRFD (Equation 1.3.2.1-1) 

results in:
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(Eqn. 7.16) 

Where, Mi = bending moment resulting from a force effect or superimposed deformation

γi = load factor

ƞi = load modifier relating to ductility, redundancy, and operational classification

Load factors for the various force effects and superimposed deformations are found in LRFD 

Article 3.4.1. The majority of the load factor information is found in Tables 3.4.1-1, 3.4.1-2, and 

3.4.1-3.

The load modifier (ƞ) is defined in LRFD Equations 1.3.2.1-2 and 1.3.2.1-3. Both of these 

equations contain a form of the following expression:

(Eqn. 7.17) 

Where, ƞD = factor relating to ductility

ƞR = factor relating to redundancy

ƞI = factor relating to importance

For strength limit verifications, each of these three factors vary between 0.95 and 1.05 

depending on conditions described in Articles 1.3.3, 1.3.4, and 1.3.5. For concrete posttensioned box girder bridges designed in accordance with LRFD Specifications and typical 

operational importance, the load modifiers are:

(Eqns. 7.18)

A load modifier related to redundancy (ƞR) equal to 1.0 is valid for both simple span and 

continuous concrete post-tensioned box girder bridges. Though simple span girders do not 

exhibit an external redundancy with regard to longitudinal flexure, there are multiple internal 

load paths to justify increasing load effects.

Equation 7.16 can then be modified to:

(Eqn. 7.19) 

7.3.1 Factored Loads for Longitudinal Flexure

Loads for bridge design specified by the LRFD Specifications are described in Article 3.3.2. 

Those loads that act to produce bending moments in the superstructure of a concrete box girder 

bridge are:

• DC = dead load of structural components and nonstructural components

• DW = dead load of wearing surface and utilities

• LL = vehicular live load

• IM = vehicular dynamic allowance

• PS = secondary moments from post-tensioning

• CR = bending moments caused by concrete creep

• SH = bending moments caused by concrete shrinkage
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• TU = bending moment caused by uniform temperature rise or fall

• TG = bending moments caused by thermal gradient (positive or negative)

Dynamic load allowances are defined in AASHTO LRFD Article 3.6.2. Load effects caused by 

the application of truck and tandem loads are to be increased by a dynamic load allowance 

equal to:

(Eqn. 7.20) 33 1 1.33

100 Dynamic Load Allowance   =+ =    

Moments PS, CR, SH, and TU are defined in Article 3.12 as force effects due to superimposed 

deformations. The secondary post-tensioning moments (PS) are those caused by the 

restraining effect of adjacent spans and integral substructures (see section 6.6). Considering 

construction of complete continuous units in a single phase, the moments caused by concrete 

creep, concrete shrinkage, and uniform thermal changes are induced only when the 

substructure is integral with the superstructure. The shortening/extension of the bridge deck 

displaces the piers, which in turn, induces bending moments in the superstructure (see sections 

6.8, 6.9, and 6.10). For typical bridges on bearings, moments produced by concrete creep, 

concrete shrinkage and uniform temperature change are equal to zero.

The LRFD Specifications define five strength limit state load combinations (Strength I through 

Strength V). Of these five load combinations, those with wind (Strength III and Strength V) are 

not likely to govern with regard to maximum superstructure bending moment. The three 

remaining strength load combinations can be expressed as: 

Strength I: (Eqn. 7.21)

Strength II: (Eqn. 7.22)

Strength IV: (Eqn. 7.23)

For bridges on bearings, these simplify to:

Strength I: (Eqn. 7.24)
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Strength II: (Eqn. 7.25)

Strength IV: (Eqn. 7.26)

The Strength II limit state is intended for special owner-specified design vehicles and/or permit 

vehicles use to evaluate bridges. These vehicles vary with transportation agencies. The 

engineer should consult individual agency requirements and ensure that the newly designed 

bridge adequately carries these vehicles. For this manual, the information presented uses the 

LRFD design notional loading in the Strength I and Strength IV limit states.

Example: Using the three-span bridge in example 1 found in appendix C, whose 

analysis model is shown in figure 7.7, compute factored bending moments at 

selected locations within the bridge.

The bending moment results from analysis by the BD2 computer program in 

units of foot-kips are:

Table 7.1 – Example Bridge 1 Bending Moments (ft-kips)

Using the load factors of equation 7.21 and 7.23 the factored moment 

combinations for Strength Limit States I and IV are:

Table 7.2 – Example Bridge 1 Bending Moments (ft-kips)

Loading 0.42L Span 1

Node 8

1.0L Span 1

Left of

Node 16

0.0L Span 2

Right of

Node 16

0.5L Span 1

Node 25

DC 15,125 -26,585 -29,178 17,761

DW 1,339 -2,353 -2,583 1,572

LL+I 12,306 -11,348 -13,778 11,795

TU 2,165 -5,067 -2,622 2,622

PS 1,964 4,594 9,834 9,799

CR 1,161 2,717 -1,543 -1,543

SH -2,642 -6,181 3,618 3,618

Strength

Case

0.42L Span 1

Node 8

1.0L Span 1

Left of

Node 16

0.0L Span 2

Right of

Node 16

0.5L Span 1

Node 25

I 44,016 -58,023 -53,861 58,386

IV 26,262 -44,811 -37,044 42,185
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7.3.2.1 Strain Compatibility

LRFD Article 5.7.3 provides guidance and requirements for the design of prestressed concrete 

flexural members at strength limit states. Equations leading to the computation of flexural 

resistance for both rectangular sections and t-beam sections are presented in LRFD Article 

5.7.3. This manual presents the more general approach presented in Article 5.7.3.2.5, in which 

strain compatibility throughout the depth of the member is considered for determining internal 

forces under ultimate conditions.

Consider the rectangular prestressed concrete beam shown in figure 7.20. The beam, with 

height of h and width of b, is reinforced with a single layer of prestressing steel with a total area 

of steel equal to Aps, located a distance dp from the extreme compression fiber. The bending 

moment Mn is the nominal resistance of the cross section in flexure. The nominal flexural 

resistance is produced by the internal couple of compression in the concrete and tension in the 

prestressing steel, acting along their lines of action. 

Figure 7.20 – Flexural Resistance by Strain Compatibility 

Figure 7.20 shows a strain diagram over the depth of the member. LRFD Article 5.7.2 provides 

guidelines for establishing the strain diagram:

• The diagram is linear over the depth of the beam.

• The diagram passes through a neutral axis at depth (c) from the extreme compression

fiber.

• The ultimate strain in the concrete (εcu) at the extreme compression fiber is equal to

0.003 in/in.

It is important to note that the prestressing steel only provides resistance in the internal 

equilibrium shown in figure 7.20. This is different from considering the prestressing force as a 

load effect in the summation of stresses at service limit states. The difference is seen in the 

strain in the prestressing steel at nominal flexural resistance. The strain on which the stress in 

the prestressing steel is computed is the sum of the strain at the effective level of stress in the 

prestressing plus the change in strain resulting from strain compatibility. This can be expressed 

as:

(Eqn. 7.27)

7.3.2 Flexural Resistance 
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Where the change in strain is provided by similar triangles:

(Eqn. 7.28)

7.3.2.2 Material Stresses and Internal Forces

Figure 7.20 also shows the stress and resultant forces developed in the cross section at 

nominal resistance. The compressive force in the concrete is found by integrating concrete 

stresses from the neutral axis to the extreme compression fiber. Forces in the prestressing 

steel are found by multiplying the stress in the steel times the area of the reinforcing steel. 

Material stress-strain relationships, such as those presented in chapter 2 of this manual, for 

both concrete and reinforcing steel are required to compute these internal forces. 

To simplify the computation of the resultant compressive force in the concrete, the LRFD 

Specifications permit the use of an equivalent rectangular stress block (Whitney’s Stress Block). 

Details of the rectangular stress block as defined in LRFD Article 5.7.2.2 are shown in figure 

7.21. 

Figure 7.21 – Rectangular Stress Block to Represent Concrete Compression

The actual distribution of concrete stress is replaced by a constant stress equal to 85 percent of 

the 28-day concrete strength. The depth of the block (a), measured from the extreme 

compression fiber, is taken as a percentage of the neutral axis depth (c):

(Eqn. 7.29)

Where, εps = strain in prestressing steel (in/in)

εpe = effective strain in prestressing at time of loading (in/in)

Δεps = change in strain in prestressing as a result of loading (in/in)
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The relationship between the depth of the stress block and the neutral axis depth is a function of 

concrete strength. The parameter β1 is taken as 0.85 for concrete strengths equal to, or less 

than, 4.0 ksi. For greater strengths the parameter is reduced at a rate of 0.05 per 1.0 ksi 

increase in concrete strength above 4.0 ksi, with a lower bound of 0.65. In equation form this is 

expressed as:

(Eqn. 7.30) 

The computation of the resultant tensile force in the prestressing steel also relies on a 

representation of the material’s stress-strain relationship. Figure 7.22 shows the comparison of 

typical stress strain relationships for mild reinforcing and prestressing strand presented in figure 

2.10 of this manual. 

Figure 7.22 – Comparison of Typical Stress-Strain Relationships for Prestressing

Strand and Mild Reinforcing 

Mild reinforcing exhibits a bilinear relationship with a well-defined yield strain. The relationship 

of strains and stresses in the prestressing steel are assumed linear for stresses up to 90 
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percent of the ultimate strength of the strand. Beyond this level of stress the relationship is 

highly nonlinear. 

LRFD Article 5.7.3.1 provides an equation to determine the stress in the prestressing steel at 

nominal resistance for cross sections where it is appropriate to lump the prestressing steel into 

a single level (as shown in figure 7.20) and where the effective stress in prestressing steel is not 

less than 50 percent of specified tensile strength of the prestressing steel. The LRFD 

expression for stress in the prestressing steel under these conditions is:

(Eqn. 7.31)

Where, fps = average stress in prestressing steel at nominal resistance (ksi)

fps = specified tensile strength of prestressing steel (ksi)

c = depth from extreme compression fiber to the neutral axis (in)

dp = depth from extreme compression fiber to the centroid of the steel (in)

The parameter k in equation 7.31 relates the stress in the prestressing steel to the type of 

prestressing steel that is being used. This variation is related to the ratio of the yield stress of 

the prestressing steel to its specified tensile strength:

(Eqn. 7.32)

LRFD Table C5.7.3.1.1-1 provides values for the ratio of stresses in equation 7.32. The great 

majority of prestressing steel used for cast-in-place concrete box girder construction is low 

relaxation steel with a specified tensile strength of 270 ksi. The corresponding values from 

Table C5.7.3.1.1-1 are:

(Eqns. 7.33)

So that for 270 ksi, low relaxation steel,

(Eqn. 7.34)

When the arrangement of prestressing steel is such that the steel cannot be lumped into a 

single layer, another relationship between prestressing steel strain and stress must be used. 

An often used source for estimating the stress in the prestressing steel is found in the 

Precast Concrete Institute Design Manual. Figure 7.23 shows a reproduction of the 

stress-strain relationship presented in Design Aid 11.2.5 of the PCI Design Handbook, 6th

Edition. 

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Up to a strain of 0.0086, the stress in the prestressing steel varies linearly as a function of the 

modulus of elasticity of the steel—in this case 7-wire strand. This level of strain corresponds to 

a stress of approximately 90 percent of the specified strength of the strand for the case of 270 

ksi steel. When the strain in 270 ksi steel is greater than 0.0086 the relationship between stress 

and strain is: 

(Eqn. 7.35)

Figure 7.23 –Stress-Strain Relationships for Prestressing Strand (PCI Design Handbook, 6th Ed.)
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Chapter 7 – Longitudinal Analysis & Design 154 of 369

7.3.2.3 Internal Equilibrium 

With material strains and stresses computed, the internal equilibrium of the cross section 

is considered. For the beam shown in figure 7.20, with a single layer of prestressing 

reinforcing, the horizontal equilibrium is expressed as: 

(Eqn. 7.36)

The condition of a net horizontal force is met through the choosing of the correct neutral axis 

depth. When the condition is met, the flexural resistance of the cross section can be found by 

summing moments about the centroid of the resultant concrete force:

(Eqn. 7.37)

The equation above considers the fundamental case of one layer of prestressing. Posttensioned concrete box girder bridges may have post-tensioning tendons that need to be 

considered at individual levels. In addition, this method of construction also provides for 

longitudinal mild reinforcing in the top and bottom slabs that can provide additional flexural 

resistance. Figure 7.24 shows a cross section with these additional elements.

Figure 7.24 – Flexural Resistance with Multiple Layers of Prestressing Steel and Mild Reinforcing 

Horizontal equilibrium for the cross section in figure 7.24 can be written as:

(Eqn. 7.38)

The nominal resistance can be found by summing moments about the top of the beam: 

(Eqn. 7.39)
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The reduction of stress in the compressive reinforcement is a subtraction from the concrete 

stress block that occupies the same area as the compressive reinforcing. It is accounted for 

with the compressive reinforcing for computational ease. With the flexural resistance computed, 

the conditions of equation 7.19 must be met, expressed here as: 

(Eqn. 7.40)

7.3.3 Resistance Factors (φ) 

Resistance factors (φ), sometimes referred to as capacity reduction factors, for prestressed 

concrete members vary as a function of the controlling behavior of the member with regard to 

specific compression and tension limits. For compression controlled behavior, the resistance 

factor is 0.75. For tension controlled behavior the resistance factor is 1.0. Between code 

specified limitations in strain that define the limits of this behavior, the resistance factor is taken 

as a linear ratio between 0.75 and 1.0 (see figure 7.25) defined in LRFD Equation 5.5.4.2.1-1:

(Eqn. 7.41)

Where, εt = net tensile strain in the extreme tension steel at nominal resistance

εcl = compression-controlled strain limit in the extreme tension steel

εtl = tension-controlled strain limit in the extreme tension steel

Figure 7.25 – Transition of Resistance Factors from

Compression Controlled to Tension Controlled 

The commentary to Article 5.5.4.2 includes a discussion of cross sections reinforced with a 

combination of prestressed reinforcing and mild reinforcing. With regard to resistance factor, a 

cross section is considered as being prestressed when mild reinforcing is used to provide 

additional flexural capacity, provided that the cross section was designed as a prestressed 

section under service limit state requirements. 

Some discernment is required when selecting the resistance factors for prestressed concrete 

members. For the vast majority of superstructure flexure cases for cast-in-place box girder 

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Chapter 7 – Longitudinal Analysis & Design 156 of 369

bridges, the resistance factor will be equal to 1.0. This should not be inferred, however, 

because of the initial level of strain in the prestressing steel. The tensile-controlled 

strain limit of 0.005 would represent a stress in the prestressing steel of 142.5 ksi, using 

a modulus of elasticity of the steel of 28,500 ksi. This is less than the effective stresses in the 

tendons under service limit state conditions. What should be considered is the change in 

prestressing strain in the calculation of flexural resistance. When the change of strain is 

greater than 0.005, the resistance factor of 1.0 is appropriate. Again, this is the case 

for the great majority of prestressed concrete superstructures. 

7.3.4 Limits of Reinforcing 

LRFD Article 5.7.3.3.2 provides minimum reinforcement requirements for prestressed concrete 

members at strength limit states. Two criteria are presented to assure that the flexural 

resistance is greater than the cracking strength of the cross section, avoiding possible brittle 

failures. In equation form, the minimum reinforcing in the cross section must be greater than the 

lesser of: 

(Eqn. 7.42)

Or,

(Eqn. 7.43)

Where Mcr is the cracking moment of the cross section. The cracking strength is determined by 

LRFD Equation 5.7.3.3.2-1. 

(Eqn. 7.44)

This equation accommodates cross sections made of flexural members made composite with a 

concrete deck slab. The equation can be simplified for concrete box girder bridges with 

monolithically cast cross sections (Sc/Snc=1) to: 

(Eqn. 7.45)

Where, γ1 = flexural cracking variability factor (= 1.6 for non-segmental bridges) 

γ2 = prestress variability factor (= 1.1 for bonded tendons)

γ3 = ratio of yield strength to ultimate strength (= 1.00 for prestressed bridges)

S = cross section modulus corresponding to the extreme fiber for which the 

cracking moment is being computed.

fcpe = compressive stress in concrete due to effective prestressing forces 

only(after all losses) at extreme fiber of section where tensile stress is 

caused by external applied loads (ksi)

Mdnc = total unfactored dead load moment acting on the monolithic or 

noncomposite section (kip-in)

Using the applicable values for the gamma coefficients (γ), the resulting equation is then:
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Chapter 7 – Longitudinal Analysis & Design 157 of 369

The modulus of rupture to be used in computing the rupture moment in these equations is found 

in LRFD Article 5.4.2.6. For normal weight concretes with a 28-day strength less than 15.0 ksi, 

the rupture stress is:

(Eqn. 7.47)

Example: Using the three-span bridge in the previous example, find the moment 

controlling the LRFD minimum reinforcement requirements at the Pier 1.

The factored moment at the left of Node 25 in the previous example was:

The stress at the extreme fiber under the action of the prestressing only is 

(See example 1 in appendix C):

The modulus of rupture is:

The resulting cracking moment at this location is:

This moment is compared to 13 percent of the governing factored moment:

As this moment is larger than the cracking moment, the resistance of the 

cross section must be:

Note that change in the sign convention of the example problem to an 

absolute value of the moments for calculating resistance requirements.

(Eqn. 7.46)
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Figure 7.26 shows a flow chart that describes the verification of a prestressed concrete cross 

section at strength limit state for flexure. Two iterative loops are embedded in the flow chart. 

The inner loop is for the appropriate selection of the depth of the neutral axis, which converges 

when horizontal equilibrium of the cross section is reached. The outer loop is for the verification 

that the reduced nominal capacity is greater than the factored loads. Mild reinforcing can be 

considered to add needed flexural capacity. Additional prestressing steel can be added as long 

as service limit state verifications remain valid. If these two approaches do not lead to sufficient 

flexural capacity, more significant changes, such as increased depth of the cross section, may 

be required.

Figure 7.26 – Flow Chart for Verification of Flexure at the Strength Limit State

7.3.5 Procedure
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Example: Using the three-span bridge in the previous example, verify the flexure 

strength limit state at the mid-span of the 160’ central span.

Figure 7.27 shows the cross section of the box girder bridge of this example 

along with the idealized cross section with regard to longitudinal flexure. 

Figure 7.28 shows the location of the 15 (3 per web), 19-strand tendons at 

mid-span.

Figure 7.27 – Idealized Cross Section For Longitudinal Flexure 

Figure 7.28 – Location of Prestressing Reinforcing in Idealized Cross Section

Parameters:

f’c = 5.0 ksi

bts = 714”

bw = 62.8”
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dp = 78”-7.75”-6.5”-1” = 62.75”

Aps = 15(19*0.217) = 61.845 in2

The factored moment at Node 25 in the previous example was:

Determine the moment for minimum reinforcing based on the lesser of the 

cracking moment or 133 percent of the factored moment.

The stress at the extreme fiber under the action of the prestressing only is 

(See example 1 in appendix C):

The modulus of rupture is:

The resulting cracking moment at this location is:

This moment is compared to 133 percent of the factored moment or:

The cracking moment of 52,146 ft-kips is less than the increased factored 

moment and is the moment required for minimum reinforcing. This minimum 

moment is less than the factored moment of 58,386 ft-kips, so the design 

moment is the factored moment.

The depth of the neutral axis from the extreme compression fiber (top of top 

slab at this cross section) is determined by iteration and axial force 

equilibrium checks. After some iteration, assume a neutral axis depth of 6.86 

inches:

The ratio of depth of the rectangular stress block to the neutral axis is:

So that the depth of the rectangular stress block is:
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The change in the prestressing steel strain is:

Assuming an effective strain in the prestressing steel of 63 percent of the 

ultimate strength of the strand, the total strain in the prestressing steel is:

Using equation 7.35 of this Manual for determining stress in the prestressing 

steel:

The force in the prestressing is: 

The force in the rectangular compression block is:

The summation of the forces in the cross section are essentially zero, so the 

nominal flexural resistance can be found by summing moments about the 

center of the compression block:

The level of change in stress in the strands is such that the section is 

compression controlled, so that the resistance factor is 1.0. The reduced 

nominal capacity is such that the strength limit state is satisfied:

Example: Determine the nominal flexural capacity of the cross section of the previous 

example considering mild reinforcing in the bottom slab. Use No. 4 bars top 

and bottom of the bottom slab at a spacing of 12 inches.

The area of mild reinforcing is:
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The yield strain of the mild reinforcing is equal to:

By inspection, the yield strain will be reached for reinforcing located in the 

bottom slab of this example problem. Assuming a neutral axis depth of 7.27”, 

the depth of the rectangular stress block is:

Checking equilibrium in the cross section:

Summing moments about the top of the cross section:

This represents an increase in nominal flexural capacity of 7 percent. A 

similar example could be performed including the top slab longitudinal 

reinforcing. The location of the steel relative to the resulting neutral axis does 

not significantly increase positive flexural capacity for this cross section.
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7.4 Strength Limit Verification—Shear

The verification of the webs of post-tensioned concrete box girder bridge superstructures at 

strength limit states requires the combination of several analyses and design steps. LRFD 

Article 5.8.1.5, repeated below, defines the scope of verifying the webs of curved box girder 

superstructures.

5.8.1.5—Webs of Curved Post-Tensioned Box Girder 

Bridges

Curved post-tensioned box girders having an overall 

clear height, hc, in excess of 4 ft shall be designed for 

the following combined effects before and after losses:

• The combined effects of global shear resulting from 

vertical shear and torsion,

• Transverse web regional bending resulting from 

lateral prestress force, and

• Transverse web bending from vertical loads and 

transverse post-tensioning.

Though Article 5.8.1.5 lists the design components for curved box girder bridges, the webs of all 

box girder bridges must be designed for the first and third bulleted items – global shear and 

transverse flexure.

The transverse analysis and design of concrete box girder bridges (bulleted item 3) is presented 

in chapter 8 of this manual. Chapter 9, section 9.1 of this manual presents information related 

to regional bending in the webs. This section of the manual presents information related to the 

evaluation of the effects of global shear resulting from shear and torsion. 

7.4.1 LRFD Design Procedures for Shear and Torsion

The AASHTO LRFD specifications specifies two design procedures (models) for verifying shear 

at strength limit states. These two models are the Sectional Model, presented in LRFD Article 

5.8.3, and the Strut-and-Tie Model, detailed in LRFD Article 5.6.3. The application of these 

methods is determined by the behavior of the member in the region being verified. The LRFD 

Specifications defines two behavioral regions:

• Flexural Regions (LRFD Article 5.8.1.1)-Locations in reinforced or prestressed concrete 

members where plane sections can be assumed to remain plane under loading. In flexural 

regions the response of the section depends only upon the sectional force effects—moment,

shear, axial load, and torsion—and not on how the force effects are introduced into the 

member.

• Regions near Discontinuities (LRFD Article 5.8.1.2)—Locations where the assumptions of 

plane sections remaining plane cannot be appropriately applied. Examples of these regions 

are locations of abrupt change in cross section, openings, dapped ends, deep beams, and 

corbels. 

The LRFD Specifications, with the limitations presented in paragraph 2 of LRFD Article 5.8.1.1, 

state that either the Sectional Model or the Strut-and-Tie Model can be used to verify shear at 

strength limit states in flexural regions. The commentary presented in LRFD Article C5.8.1.1 

states that design using the Strut-and-Tie Model generally produces designs that are less 
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Chapter 7 – Longitudinal Analysis & Design 164 of 369 

conservative than those designed using the Sectional Model. The LRFD Specifications require 

that the Strut-and-Tie Method be used for the design of regions near discontinuities.

The behavior of post-tensioned concrete box girder bridges, with regard to shear associated 

with the longitudinal superstructure design, are consistent with the LRFD definition of a flexural 

region. As a result, this manual presents information related to the Sectional Model design 

procedure.

7.4.2 General Requirements

Cross sections along the length of the superstructure must satisfy LRFD requirements for shear 

and torsion at strength limit states. LRFD Article 5.8.2.1 provides equations that define the 

resistance with regard to shear and torsion with respect to cross section nominal capacities. 

The resistance with regard to shear is provided in LRFD Equation 5.8.2.1-2:

(Eqn. 7.48) 

Where, Vr = factored shear resistance

 Vn = nominal shear resistance

 φ = resistance factor as specified in Article 5.5.4.2

Extending this equation with regard to applied loads, the resistance with regard to shear must 

be greater than the summation factored load effects:

(Eqn. 7.49) 

Where, Vi = shear resulting from a force effect or superimposed deformations

γi = load factor applicable to force effect or superimposed deformations

An equation similar equation 7.15 is provided for torsion in LRFD Equation 5.8.2.1-1:

(Eqn. 7.50) 

Where, Tr = factored torsional resistance

 Tn = nominal torsional resistance

 φ = resistance factor as specified in Article 5.5.4.2

Again, extending this equation with regard to applied loads, the torsional resistance must be 

greater than the summation of the factored torsional effects:

(Eqn. 7.51)

Where, Ti = torsional moment resulting from a force effect or superimposed deformation

γi = load factor applicable to force effect or superimposed deformation

When the factored torsional moment is small with regard to the torsional moment required to 

crack the cross section, the effects of torsion do not need to be considered in the verification of 

the box girder webs. LRFD Equation 5.8.2.1-3 requires that if the factored torsional moment is 
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Chapter 7 – Longitudinal Analysis & Design 165 of 369 

greater than or equal to one fourth of the torsional cracking moment, the effects of torsion must 

be considered in the design.

Minimum shear and torsion resistance is ensured through the requirement for minimum 

transverse reinforcing in LRFD Article 5.8.2.5.

7.4.3 Sectional Model Nominal Shear Resistance 

Shear design of reinforced and prestressed concrete girders in the AASHTO LRFD 

specifications in flexural regions has long been based on a truss analogy. Figure 7.29 shows a 

typical parallel chord truss for a constant depth beam. The beam is loaded with a concentrated 

vertical load at mid-span. The vertical force is carried to the supports through successive 

inclined compression struts. Vertical tension members lift the vertical component at the bottom 

of a diagonal strut to the top of the adjacent inclined strut. The horizontal component of the 

diagonal strut is held in equilibrium by the compressive forces in top chord members and tensile 

forces in the bottom chord members.

Figure 7.29 – Web Width based on Horizontal Widths 

Using truss analogies for shear design of reinforced concrete beams has its origins in the 

beginning of the last century. Early applications used shear reinforcing to carry the entire shear 

forces predicted in the vertical tension members of the truss. Orientation of the inclined 

compressive struts (θ) for these early solutions was typically 45 degrees from horizontal.

Later testing of concrete beams reinforced using truss analogies revealed shear strengths 

greater than that provided by the reinforcing alone (Vs in LRFD). The additional shear 

resistance was observed to be a somewhat complex combination of behaviors that were a 

function of the magnitude and nature of beam cracking under ultimate loads. Tests of this 

nature led to the empirical development of a shear resisting mechanism associated with the 

girder concrete (Vc in LRFD).

Still later work in the 1980’s advanced the estimation of the concrete contribution from a purely 

empirically based solution to a more analytical approach using the Modified Compression Field

Theory (MCFT). The MCFT methods still have an empirical component, but one based only on 

the nature of reinforced webs to resist in-plane forces. The LRFD Specifications provide both 

empirically based and MCFT based methods for determining the component of shear resistance 

provided by the girder concrete.

The nominal shear resistance of a reinforced or prestressed concrete member with transverse 

reinforcing is presented as the lesser of that predicted by LRFD Equations 5.8.3.3-1 and 

5.8.3.3-2:

(Eqn. 7.52) 
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(Eqn. 7.53)

Where, Vc = shear resistance provided by the cross section concrete

 Vs = shear resistance provided by transverse reinforcing

 Vp = shear resistance provided by the component of the effective prestressing 

force in the direction of the applied shear

 bv = the effective web width (LRFD Article 5.8.2.9)

 dv = the effective shear depth (LRFD Article 5.8.2.9)

The three components of resistance will be discussed in the following sections. This section 

continues by providing more detail with regard to girder dimensions to be used for shear design.

7.4.3.1 Effective Web Width

LRFD Article 5.8.2.9 defines the effective web width as being measured parallel to the neutral 

axis. A literal interpretation of this article is depicted in figure 7.30 for the cross section of 

design example 1. The widths of all webs in this cross section, vertical and inclined, are 12 

inches perpendicular to their axes. A horizontal cut along the neutral axis would cross the 

vertical webs with their 12-inch widths. The horizontal cut through the inclined webs would see 

horizontal widths, bh, equal to 13.42 inches for a web slope of 2:1. The total web width along 

the horizontal cut is then 62.84 inches.

Figure 7.30 – Web Width based on Horizontal Widths 

Unfortunately, the web width computed above is inconsistent with the internal equilibrium of 

shear flow around the cross section with regard to the inclined webs. Consider the cross 

section of the single cell box girder with inclined webs shown in figure 7.31. A shear force, V, is 

applied to the cross section. The inclination of the webs requires that the sum of the shear flow 

in the inclined webs is:

(Eqn. 7.54) 

The vector summation of this force and a horizontal force in the top slab resolves to one half of 

the applied shear force. 
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Figure 7.31 – Shear Flow in Single Cell Box Girder 

The resulting shear stress acting on the web is then:

(Eqn. 7.55) 

Noting that:

(Eqn. 7.56) 

Reduces Equation 7.55 to:

(Eqn. 7.57)

Thus, vertical shear forces are related to web widths perpendicular to their inclined axes. 

Extending these considerations to the cross section shown in figure 7.30, it can be argued that 

the appropriate web width should be 60 inches as opposed to the 62.84 inches of width along a 

horizontal cut through the webs. Using the horizontal width would lead to limit state verifications 

that are approximately 5 percent unconservative. By comparison, the verifications of the single 

cell box girder, using a similar web slope, would be unconservative by 11 percent. The 

engineer should give appropriate consideration to the web width used for each cross section.

In addition to the previous discussion, the effective web width must include reductions in width 

to account for ducts embedded in the webs. Figure 7.32 shows one web of the box girder of 

design example 1. Also shown is the shear stress distribution for a given vertical shear force 

and a detail of the flow of shear around ducts. To accommodate the shear flow concentrations, 

LRFD Article 5.8.2.9 requires a subtraction of one half of the duct diameter for loads applied to 

the bridge when the duct is ungrouted and a subtraction of one fourth of the diameter for loads 

applied after the duct is grouted.
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Figure 7.32 – Shear Stress and Shear Flow Around Ducts 

7.4.3.2 Effective Shear Depth

The effective depth for shear, dv, is required to be at least equal to the distance between the 

centroids of compression in the concrete and tension in the tensile elements. Figure 7.33 

shows one of the webs of the cross section of the box girder of design example 1 at one of the 

interior supports. The centroid of the compression in the concrete was determined as the center 

of gravity of the stress block at nominal loading. The corresponding tension force passes 

through the center of gravity of the post-tensioning tendons.

Figure 7.33 –Effective Depth for Shear Calculations 
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The minimum effective shear is depth can be found as the nominal moment capacity of the 

section divided by either the internal compressive or tensile force:

(Eqn. 7.58) 

The effective shear depth need not be taken less than 90 percent of the depth measured from 

the extreme compression fiber to the centroid of the prestressing steel, de, or 72 percent of the 

overall depth of the box girder.

7.4.4 Shear Resistance from Concrete (Vc) 

The LRFD Specifications provide three different methods for determining the concrete 

contribution of shear resistance in reinforced and prestressed concrete members in flexural 

regions. Method 1 and Method 2 are based on the Modified Compression Field Theory. 

Method 3, included in AASHTO LRFD specifications for many years, is more empirically derived 

from shear testing of prestressed girders. The applicability of the three methods is:

• Method 1 (LRFD Article 5.8.3.1)—for non-prestressed members.

• Method 2 (LRFD Article 5.8.3.4.2 or LRFD Appendix B5)—for both prestressed and nonprestressed members, with or without shear reinforcing.

• Method 3 (LRFD Article 5.8.3.4.3)—for both prestressed and non-prestressed concrete 

members where there is no net axial tensile load and minimum shear reinforcing is provided.

Since the concern of this manual is post-tensioned concrete box girder bridges, this section 

presents the LRFD requirements for Methods 2 and 3. (Method 1, though not a focus of this 

manual, can be viewed as a special case within Method 2.) 

7.4.4.1 Method 2 (Simplified MCFT)

Method 2 for computing the contribution of the concrete to cross section nominal shear 

resistance is based on design procedures derived from the Modified Compression Field Theory. 

Three primary parameters are used to evaluate concrete shear resistance in this method:

θ = angle, from horizontal, of the inclination of the diagonal compressive strut

 β = factor indicating ability of diagonally cracked concrete subjected to in-plane 

shear and normal stresses to transmit tension. 

 εx = longitudinal strain in the web of the member

These three parameters are interdependent in the MCFT, with no direct solution available for 

the wide range of possible girder variables. As a result, the first implementation of MCFT in 

LRFD presented an iterative approach to evaluating these parameters. This iterative approach 

is now provided in LRFD Appendix B5 of LRFD. Beginning in 2008, LRFD offered a simplified, 

non-iterative solution for these parameters. This simplified approach is presented in the 

Manual.

The LRFD application of MCFT is based on a set of assumptions in modeling actual girder 

shear behavior. The top of figure 7.34 depicts a portion of a box girder cross section and 
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Chapter 7 – Longitudinal Analysis & Design 170 of 369 

shows shear stress distributions, longitudinal strains and principal compressive stress 

trajectories. The bottom of figure 7.34 shows the LRFD behavior assumed in design 

procedures using MCFT. Three significant assumptions are:

• The distribution of shear stress over the depth of the section is taken as the value at 

mid-depth of the girder using MCFT methods and the computed longitudinal strain at 

mid-depth. Shear stresses are uniformly distributed over the rectangular area that has a 

height of dv and width of bv, as calculated in the previous sections. 

• The direction of principal compressive stresses remains constant over the height dv. 

• The web is modeled by one biaxial member.

Figure 7.34 –Actual vs. MCFT Girders

In the LRFD approach to MCFT shear design, the sectional forces shown in the bottom right of 

figure 7.34 are placed in equilibrium with forces in an idealized model comprised of compression 

and tension flanges and the web. The forces in these three elements are calculated as follows:

• The sectional bending moment, Mu, is the resisted force in the flanges multiplied by the 

lever arm, dv, between them.

• The sectional axial force, Nu, is resisted by forces in the flanges, one-half assumed to be 

carried by each flange.

• The sectional shear force, Vu, is resisted by the inclined compression strut force, D, 

which is resolved into horizontal and vertical components. The vertical component of the 

force is equal to the factored sectional shear force less the value of any vertical 

component of prestressing acting at the section. The horizontal component is resisted 

by tension in the flanges.
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Figure 7.35 – MCFT Forces and Longitudinal Strain 

The assumed resistance equilibrium for Method 2 is shown above in figure 7.35. Evaluating 

these forces acting at the center of the tensile force, allows the computation of the strain in the 

tensile element, εs, which is needed to evaluate the beta and theta terms. The strain is 

predicted by LRFD Equation 5.8.3.4.2-4:

(Eqn. 7.59) 

Where, fpo = average stress in prestressing steel or 0.7fpu 

An important difference between the simplified and iterative MCFT approaches in Method 2 is 

seen in equation 7.59. The longitudinal strain, εs, at the level of the tensile force is used for the 

simplified approach. The iterative approach presented in LRFD Appendix B5 uses the average 

longitudinal strain in the section. A longitudinal strain of εs/2 is used, which assumes that the 

strain at the level of the center of compression is small.

Equation 7.59 includes another adjustment to produce the simplified version of Method 2. By 

equilibrium, the horizontal component of the compressive strut force divided into the two flanges 

should be:

(Eqn. 7.60) 

The simplified Method 2 fixes the angle θ in this equation such that its cotangent is equal to 2.0 

(θ ≈ 26.6o). This reduces the equation for horizontal components in the flanges to:

(Eqn. 7.61) 

In using equation 7.61 the following considerations should be applied:

• The absolute value of the factored moment, │Mu│, acting with the factored shear, Vu, 

should not be taken less than │Vu – Vp │multiplied by the shear depth, dv. 
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Chapter 7 – Longitudinal Analysis & Design 172 of 369 

• The areas of reinforcing bars and prestressing steel not fully developed at the cross 

section under consideration should be reduced in proportion to their lack of full 

development.

• If the longitudinal strain, es, calculated by equation 7.61 is negative, it can either be 

taken equal to zero or recomputed by equation 7.61 where the denominator is replaced 

with (EsAs + EpAps + EcAc). Ac in this revised denominator is the area of concrete on the 

flexural side of the member. However, εs should not be taken less than -0.4 x 10-3. 

• If the axial tension is large enough to crack the flexural compression face of the section, 

the value calculated by equation 7.61 should be doubled.

• β and θ may be calculated with values of εs larger than that predicted by equation 7.61. 

However, es should not be taken greater than 6.0 x 10-3.

With the sectional forces resolved into the idealized cross section and the longitudinal strain 

computed, the shear resistance contributed by the concrete can be determined. LRFD Article 

5.8.3.3 defines this contribution for the simplified form of Method 2:

(Eqn. 7.62) 

The parameter β, for the simplified approach, is defined in LRFD Article 5.8.3.4.2. When the 

minimum required web (transverse) reinforcing is provided, β is defined as:

(Eqn. 7.63) 

In most cases, the transverse requirements at the critical sections of post-tensioned concrete 

box girders will require more than the minimum shear reinforcing requirements. This reinforcing 

is often used throughout the span to simplify the tying of reinforcing steel. The spacing of this 

reinforcing could be increased where demand is lower, even to that less than the minimum 

requirements. In this case, the LRFD Specifications provides an equation for β when the 

minimum amount of transverse reinforcing is not provided. In all cases, the minimum transverse 

reinforcing spacing requirements must be met. 

Note: It is interesting to be aware of the forms that AASHTO LRFD equations take as a result of 

using consistent units. In the case of equation 7.62 the units are ksi for the concrete strength 

and inches for bv and dv. The parameter β has no units. Historically, the concrete contribution 

to shear resistance was first expressed as a function of the concrete strength in psi. Later, this 

contribution was defined as a multiple of the square root of the concrete strength (again in psi), 

with that multiple typically ranging between a factor of 2 and 4. This multiple is now expressed 

as the β term, with the constant of 0.0316 added for consistent units of ksi. Though more 

cumbersome in appearance, it may be that equation is better understood from the historical 

perspective as:

The concrete strength term in the last expression of this equation is in psi. The engineer is 

encouraged to consider other LRFD equations of similar form in order to retain the historical 

development of the code.
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7.4.4.2 Method 3 (Historical Empirical)

Past editions of the AASHTO Standard Specifications for Bridge Design included an approach 

to shear design based on the nature of girder cracking initiation. Figure 7.36 shows a portion of 

a continuous concrete girder with zones that generally define two types of shear cracks and 

their typical general locations. At the ends of girders and near points of contraflexure girder 

bending is small and web cracking initiates by principle tensile stresses in the webs large 

enough to crack the concrete. In regions of significant flexure girders can crack vertically on the 

tension faces as a result of longitudinal flexural stresses greater than the tensile capacity of the 

concrete. If cracking continues into the webs, the effect of shear stresses in these regions can 

change the direction of these cracks in the webs to be inclined shear cracks.

Figure 7.36 – Types and Locations of Reinforced and Prestressed Girder Cracking

The contribution of the concrete in Method 3 is different depending on the type of cracking likely 

to be found. Two resistance expressions, one for each type of cracking, are defined in LRFD 

Article 5.8.3.4.3. This article requires that concrete resistance for each expression be 

evaluated, and the lesser of the two be used for design.

In regions of web shear cracking, the expression for the shear capacity of the concrete is 

determined from LRFD Equation 5.8.3.4.3-3:

(Eqn. 7.64) 

Where: fpc = compressive stress at the center of gravity of the cross section after all losses 

(ksi)

The first term of equation 7.64 represents, in the form of a shear force, a principle tension in the 

web required to offset the axial compression from the post-tensioning and a conservative 

estimate of the tensile strength of the concrete. This combination of terms was empirically 

derived. The second term of equation 7.64 represents the vertical component of the posttensioning at the section. When equation 7.64 governs the evaluation of the concrete 

resistance, Vp in equation 7.52 (LRFD Equation 5.8.3.3-1) is set equal to zero.

In regions of flexure-shear cracking, the expression for the shear capacity of the concrete is 

determined from LRFD Equation 5.8.3.4.3-1:

(Eqn. 7.65) 
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Where: Vd = shear force at section do to the unfactored combination of DC and DW (kips)

Vi = factored shear force at section due to externally applied loads occurring 

simultaneously with Mmax (kips)

Mcre = moment causing flexural cracking moment at section due to externally 

applied loads (kip-in)

Mmax = moment causing flexural cracking moment at section due to externally 

applied loads (kip-in)

The second and third terms of equation 7.65 represent a shear force consistent with a bending 

moment that causes cracking at the section. The first term of equation 7.65 is an empirically 

derived expression that accounts for an increase concrete resistance developed as the flexure 

crack transforms into a shear crack.

7.4.5 Shear Resistance from Transverse (Web) Reinforcing Steel (Vs) 

The contribution to the nominal shear capacity of a cross section from transverse (web) 

reinforcing is based on the number and orientation of the transverse reinforcing relative to the 

design length of shear crack. Figure 7.37 shows a portion of a girder cracked by shear. The 

reinforcing in this figure is oriented vertically and is spaced along the length of the girder at a 

spacing s. At each bar location there is a resisting force equal to the area of the reinforcing 

times the yield strength. (The area of reinforcing would be the total of all bars in all webs for a 

multi-cell box girder where a spine beam analysis and global forces are used without distinction 

for specific distribution factors to the webs.)

Whether Method 2 (Simplified MCFT) or Method 3 (Historical Empirical) are used to determine 

the contribution of the concrete in resisting shear, the shear depth, dv, multiplied by the 

cotangent of the inclination of the diagonal strut is used to determine the number of bars 

participating to resist shear at the section.

Figure 7.37 – Contribution of Shear Reinforcing to Nominal Shear Resistance 

The general expression for the shear contribution of the transverse reinforcing is determined by 

LRFD Equation 5.8.3.4.3-3:

(Eqn. 7.66) 
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Where: Av = area of transverse reinforcement within distance s (in2) 

fy = minimum yield strength of reinforcing bars (ksi)

dv = effective shear depth, taken as the distance between the resultants of the

tensile and compressive forces due to flexure; it need not be taken less than 

the greater of 0.9de or 0.72h, where de is the distance from the extreme 

compression fiber to the centroid of the tensile force in the tensile 

reinforcement, and where h is the overall depth of the member (in)

s = spacing of reinforcing bars (in)

θ = angle of inclination of diagonal compressive stresses (degrees)

α = angle of inclination of the transverse reinforcing from the longitudinal 

reinforcing

When the transverse reinforcing is placed vertically (α = 90o as shown in figure 7.37), equation 

7.66 simplifles to:

(Eqn. 7.67) 

The angle θ that defines the orientation of the diagonal compressive strut is calculated 

differently in Method 2 and Method 3. When Method 2 is used, the inclination of the 

compressive strut is equal to:

(Eqn. 7.68)

When Method 3 is used to determine concrete resistance, one of two values for the inclination 

of the compressive strut, in terms of its cotangent, must be used. If the resistance determined 

by web cracking (Vcw) is greater than that determined by flexure-shear cracking (Vci), the 

cotangent of the inclination of the compressive strut is taken as:

(Eqn. 7.69) 

When Vci is greater than Vcw the cotangent of the inclination of the compressive strut is taken as:

(Eqn. 7.70)

7.4.6 Shear Resistance from Vertical Component of Effective Prestressing (Vp) 

The shear resistances provided by components of the effective prestressing force in tendons 

are determined from the geometry of the tendons. For the relatively small variations in tendon 

profiles in cast-in-place concrete box girder superstructures, the contribution of the prestressing 

to shear resistance can be taken as: 

(Eqn. 7.71) 

Where: ω = the angle of the tendon at the cross section relative to horizontal (degrees) 
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Depending on the tendon geometry, the component in the direction of the applied shear can 

increase or reduce nominal cross section resistance. Tendon components acting against the 

applied shear force will increase cross section resistance. Tendon components acting with the 

applied shear force will decrease cross section resistance. Remember that Vp is equal to zero 

when using Method 3 for determining the concrete shear contribution.

Figure 7.38 shows a simple span post-tensioned beam with a parabolic tendon profile originally 

presented in figure 3.13.

Figure 7.38 – Simple Span Beam with Parabolically Draped Tendon 

The eccentricity of the tendon along the length of the tendon was found to be:

(Eqn. 7.71) 

The slope of the tendon profile is found by taking the derivative of the equation for eccentricity:

(Eqn. 7.72) 

The maximum vertical component of the prestressing force, also the resistance provided by the 

prestressing, is the maximum shear times the effective prestressing force:

(Eqn. 7.73) 

Figures 7.39 and 7.40 show typical draped tendon profiles typically used for continuous posttensioned box girder bridges. These tendon profiles were previously presented in figures 3.19 

and 20. The profiles of these tendons were developed as parabolic segments with slopes 

compatible with adjacent parabolic segments at their joining. 

Figure 7.39 – Typical Tendon Profile for an End Span of a Continuous Unit 
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Chapter 7 – Longitudinal Analysis & Design 177 of 369

Figure 7.40 – Typical Tendon Profile for an Interior Span of a Continuous Unit 

The definition of the parameters of the parabolic segments, presented in chapter 3, is shown in 

figure 7.41.

Figure 7.41 – Tendon Profile Parabolic Segment 

The elevation of the tendon profile at a horizontal distance from the origin is given as:

(Eqn. 7.74)

The slope of the tendon profile is:

(Eqn. 7.75)

And the angle of the tendon profile at the end of the parabolic segment is:

(Eqn. 7.76)

7.4.7 Longitudinal Reinforcing

Regardless of the method used for determining the contribution of the concrete in resisting 

shear, sufficient tensile capacity in the longitudinal reinforcing is required to appropriately 
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Chapter 7 – Longitudinal Analysis & Design 178 of 369 

develop the shear capacity of the cross section. That is to say, the horizontal component of the 

compression strut in the web has to be resisted by longitudinal tension reinforcement. LRFD 

Article 5.8.3.5 provides the requirements for the tensile force:

(Eqn. 7.77) 

Where: Vs = shear resistance provided by the transverse reinforcing, except that Vs shall 

not be taken greater than Vu /φ 

If this inequality is not met, either the transverse or the longitudinal reinforcement must be 

increased.

7.4.8 Torsion Reinforcing

When torsional forces are large enough to warrant inclusion in design (LRFD Article 5.8.2.1), 

the transverse reinforcing provided in the webs shall be the sum of that required for shear and 

for that required for the concurrent torsion. The nominal torsional resistance is presented in 

LRFD Article 5.8.3.6.2:

(Eqn. 7.78)

 

Where: Ao = area enclosed by the shear flow path around the box girder (in2

) 

At = total area of transverse torsion reinforcing in the exterior web of the box 

girder (in2

) 

The LRFD Specifications also include requirements for the longitudinal reinforcing when torsion 

is included in the design. For box girders, the longitudinal reinforcing for torsion, in addition to 

that required for flexure, is:

(Eqn. 7.79) 

Where: ph = perimeter of the centerline of the transverse reinforcement located in the 

outer most webs and top and bottom slabs (in) 

The longitudinal reinforcing is to be distributed around the outer-most webs and top and bottom 

slab of the box girder.
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Chapter 8—Transverse Analysis

8.1 Introduction

The cross section of a concrete box girder bridge is designed to resist bending moments acting 

transverse to the longitudinal direction of the bridge span. Transverse analyses are preformed 

to evaluate transverse bending moments resulting from both permanent and live loads. 

Permanent loads include self weight, barrier rails, sidewalks, wearing surfaces, and utilities that 

may be attached inside or outside of the box girder superstructure. Live loads are comprised of 

the AASHTO LRFD Design Truck and Design Tandem, located in design lanes arranged to 

produce maximum bending moments at critical sections. Figure 8.1 shows typical box girder 

superstructures, applied loads and likely critical sections. The results of the transverse analysis 

are used to design the transverse reinforcing and post-tensioning (if required) of the cantilever 

wings, top slab, webs, and bottom slab.

Figure 8.1 – Concrete Box Girder Cross Sections and Loads 

8.2 Methods of Analysis

The magnitude and distribution of transverse bending moments due to permanent and live loads 

vary along the length of a span as a function of the deflection behavior of the box girder. Near 

supports, where girder deflections are restrained, localized bending moments remain relatively 

concentrated. Within the span, general longitudinal deflections reduce maximum transverse 

moments as they are distributed along a greater portion of the span.
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AASHTO LRFD provides for three levels of analysis for determining transverse design 

moments:

• Empirical Method—This method, presented in LRFD Article 9.7, presents required top 

slab reinforcing for bridges meeting the criteria of Article 9.7.2.4. No guidance for 

reinforcing or forces in the remainder of the cross section are presented. The use of this 

method of analysis is not recommended for concrete box girders presented in this 

manual.

• Approximate Method—Article 4.6.2 presents an approximate method of analysis in 

where the deck is subdivided into strips perpendicular to the supporting elements 

(webs). Internal forces are determined first by structural analysis of a unit width of 

transverse cross section and then by dividing these force and moment results by the 

width of the strip to determine forces per foot along the superstructure.

• Refined Method—Article 4.6.3 permits the use of refined transverse analyses by 

methods listed in Article 4.4. One typical approach is to use three-dimensional finite 

elements methods with sufficient load cases to generate an envelope of transverse 

design moments. Most often, however, the practicality of detailing and constructing 

superstructures with uniform reinforcing and post-tensioning details typically overrides 

small quantity savings that refined methods offer.

This manual uses the Approximate Method for transverse analysis of the concrete box girder 

cross section, with additional considerations. Multi-cell box girder bridges with three or more 

cells and with web spacing not greater than 15 feet use the LRFD strip method to determine 

transverse moments resulting from concentrated wheel loads. Single-cell or dual-cell box 

girders, and multi-cell box girders with web spacing greater than 15 feet use a refined approach 

(as presented in section 8.5) to account for longitudinal distribution of concentrated wheel loads 

in determining the transverse live load moments. Live load moments are then combined with 

moments resulting from uniformly repeating loads for design.

8.3 Applicable AASHTO LRFD Specifications

AASHTO LRFD Articles governing the design of the concrete box girder cross sections are 

found in four different Sections of the code. The following subsections summarize the most 

relevant of these articles to facilitate transverse analysis and design.

8.3.1 Section 9—Deck and Deck Systems

Section 9 provides general guidance for the design of bridge decks. Requirements related to 

different design limit states are provided in Article 9.5:

9.5.2—Service Limit States

At service limit states, decks and deck systems 

shall be analyzed as fully elastic structures and 

shall be designed and detailed to satisfy the 

provisions of sections 5, 6, 7, and 8. ...
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9.5.3—Fatigue and Fracture Limit State

Fatigue need not be investigated for :

• Concrete decks, ...

9.5.4—Strength Limit States

At strength limit states, decks and deck systems 

may be analyzed as either elastic or inelastic 

structures and shall be designed and detailed to 

satisfy the provisions of sections 5, 6, 7, and 8. ...

9.5.5—Extreme Event Limit States

Decks shall be designed for force effects 

transmitted by traffic and combination railings using 

loads, analysis procedures, and limit states 

specified in section 13. ...

Article 9.6 presents requirements particular to the analysis of bridge decks:

9.6.1—Method of Analysis

Approximate elastic methods of analysis specified 

in Article 4.6.2.1, refined methods specified in 

Article 4.6.3.2, or the empirical design of concrete 

slabs specified in Article 9.7 may be used for 

various limit states as permitted in Article 9.5.

9.6.2—Loading

Loads, load positions, tire contact area, and load 

combinations shall be in accordance with the 

provisions of section 3.

Article 9.7 contains requirements particular to concrete deck slabs:

9.7.1.1—Minimum Depth and Cover

Unless approved by the owner, the depth of a 

concrete deck, excluding any provision for grinding, 

grooving, and sacrificial surface, should not be less 

than 7.0 in.

Minimum cover shall be in accordance with the 

provisions of Article 5.12.3.
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8.3.2 Section 3—Loads

Section 3 presents the various loadings and load combinations for bridge design. Article 

3.6.1.3.3 directs that only the design axles (comprised of wheel loads) of the HL93 and design 

tandem need be used in transverse design:

3.6.1.3.3- Design Loads for Decks, Deck Systems,

and the Top Slabs of Box Culverts

...Where the approximate strip method is used to

analyze decks and top slabs of culverts, force 

effects shall be determined on the following basis:

• Where the slab spans primarily in the 

transverse direction, only the axles of the 

design truck of Article 3.6.1.2.2 or design 

tandem of Article 3.6.1.2.3 shall be applied to 

the deck slab or the top slab of box culverts. ... 

Wheel loads shall be assumed to be equal within 

an axle unit, and amplification of the wheel loads 

due to centrifugal and braking forces need not be 

considered for the design of decks.

The live loads of the Design Truck and Design Tandem are shown in adjacent lanes in figure 

8.2.

Figure 8.2 – AASHTO LRFD Design Truck and Design Tandem 
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Location of the applied wheel loads are in accordance with Article 3.6.1.3:

3.6.1.3—Application of Design Vehicular Live 

Loads

....Both the design lanes and the 10 ft loaded width 

in each lane shall be positioned to produce extreme 

force effects. The design truck or tandem shall be 

positioned transversely such that the center of any 

wheel load is not closer than:

For the design of the deck overhang—1.0 ft from 

the face of the curb or railing, and...

For the design of all other components—2.0 ft from 

the edge of the design lane. ...

Figure 8.3 shows an example loading with two lanes placed at the left side of a box girder cross 

section developed in chapter 5 (also design example 1 in appendix C). In this figure, the left 

most truck is located for design of the overhang. Respecting that, trucks are positioned within 

10’ loaded widths, and that loaded width is located within in the 12’ lane, the left wheel of the 

adjacent truck is positioned no closer than 7’ to the right wheel of truck on the left.

Figure 8.3 – Transverse Truck Placement

When web spacing is small, the application of the wheel loads as uniform pressures, as 

opposed to concentrated loads, can impact the top slab bending moments. The 16 kip HL93 

truck wheel loads or 12.5 kip tandem wheel load may be distributed in accordance with Article 

3.6.1.2.5:

3.6.1.2.5—Tire Contact Area

The tire contact area of a wheel consisting of one

or two tires shall be assumed to be a single 

rectangle, whose width is 20.0 in and whose length 

is 10.0 in...
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Figure 8.4 shows the tire contact area in the transverse direction of a concrete box girder 

bridge. The load width at the center of gravity of the top slab is equal to the 20” transverse tire 

contact dimension, further distributed at 45 degrees on either side over half of the slab 

thickness. The resulting loaded width is 20” plus the slab thickness.

Figure 8.4 – Tire Contact Area in the Transverse Direction

An optional vertical design load for the overhanging slab (cantilever wing) of the box girder is 

provided in Article 3.6.1.3.4. This loading is shown in figure 8.5

3.6.1.3.4—Deck Overhang Load

For the design of deck overhangs with a cantilever, 

not exceeding 6.0 ft from the centerline of the 

exterior girder to the face of a structurally 

continuous concrete railing, the outside row of 

wheel loads may be replaced with a uniformly 

distributed line load of 1.0 klf intensity, located 1.0 ft 

from the face of the railing.

Horizontal loads on the overhang resulting from 

vehicle collisions with barriers shall be in 

accordance with the provisions of section 13.

Figure 8.5 – Alternate Vertical Loading for Overhang Design
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8.3.3 Section 4—Analysis

This manual uses the Approximate Method, or Strip Method, of analysis for the design of the 

transverse cross section of concrete box girder superstructures. Article 4.6.2 presents guidance 

for this analysis approach.

4.6.2.1—Decks

An approximate method of analysis in which the 

deck is subdivided into strips perpendicular to the 

supporting components shall be considered 

acceptable for decks other than:

• Fully filled and partially filled grids for which the 

provisions of Article 4.6.2.1.8 shall apply, and

• Top slabs of segmental concrete box girders for 

which the provisions of 4.6.2.9.4 shall apply.

Where the strip method is used, the extreme 

positive moment in any deck panel between girders 

shall be taken to apply to all positive moment 

regions. Similarly, the extreme negative moment 

over any beam or girder shall be taken to apply to 

all negative moment regions.

Figure 8.6 shows a perspective of the four-cell box girder developed in chapter 5 (design 

example problem 1, appendix C).

Figure 8.6 – Perspective of Multi-cell Box Girder
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Figure 8.7 shows the transverse strip used for computing the longitudinal distribution of 

concentrated wheel loads.

Figure 8.7 – Transverse Strip for Approximate Design Method

Figure 8.8 shows an example loading of the Design Truck in two lanes positioned to the left of 

the roadway width as depicted in cross section view in figure 8.3.

Figure 8.8 – Transverse Strip subjected to two Design Trucks as per Figure 8.3 
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The longitudinal length of the transverse strip is specified in Article 4.6.2.1.3:

4.6.2.1.3—Width of Equivalent Interior Strips

The width of the equivalent strip of a deck may be

taken as specified in Table 4.6.2.1.3-1. ... 

...The equivalent strips for decks that span

primarily in the transverse direction shall not be 

subject to width limits....

Equations for the width of the transverse strip in the longitudinal direction from Table 4.6.2.2.1-1 

for cast-in-place concrete box girder bridges are:

(Eqn. 8.1) W X Overhang = + 45.0 10 ( )

(Eqn. 8.2) W = + 26.0 6.6 ( S Positive Moment)

(Eqn. 8.3) W = + 48.0 3.0 ( S Negative Moment)

Where, S = spacing of supporting components (ft)

X = distance from load to point of support (ft)

W = width of transverse strip in the longitudinal length of transverse strip (in)

Article 4.6.2.1.7 requires that the transverse cross section of the box girder bridge be modeled 

for frame action. 

4.6.2.1.7—Cross-Sectional Frame Action

Where decks are an integral part of box or cellular 

cross sections, flexural and/or torsional stiffnesses 

of supporting components of the cross section, i.e., 

the webs and bottom flange, are likely to cause 

significant force effects in the deck. Those 

components shall be included in the analysis of the 

deck.

If the length of a frame segment is modeled as the 

width of an equivalent strip, provisions of Articles 

4.6.2.1.3, 4.6.2.1.5, and 4.6.2.1.6 may be used.

Article 4.6.2.1.6 presents that the critical sections for negative moments and shear of concrete 

box girders (Section Type “d” in Table 4.6.2.2.1-1) is at the face of the supporting component, 

the webs.

4.6.2.1.6—Calculation of Force Effects

...The design section for negative moments and 

shear forces, where investigated, may be taken as 

follows:
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• For monolithic construction, closed steel boxes, 

closed concrete boxes, open concrete boxes 

without top flanges, and stemmed precast 

beams, i.e., cross sections (b), (c), (d), (e, (f), 

(g), (h), (i), and (j) from Table 4.2.6.2.2.1-1, at 

the face of the supporting component. 

8.3.4 Section 13—Railing

Article 13 presents requirements for the design of overhangs (cantilever wings) with regard to 

vehicle crash loadings on barriers. The intent of the specifications is that overhangs have 

sufficient strength such that the barrier fails first with respect to the design forces. Article 

A13.4.1 presents two design cases to assure this intention is achieved. The third design case 

presented in this Article is the case of normal traffic loading.

A13.4.1—Design Cases

Bridge deck overhangs shall be designed for the

following design cases considered separately:

Design Case 1: the transverse and longitudinal

forces specified in Article A13.2

—Extreme Event Load 

Combination II limit state

Design Case 2: the vertical forces specified in

Article A13.2—Extreme Event

Load Combination II limit state

Design Case 3: the loads, specified in Article 

3.6.1, that occupy the 

overhang—Load Combination 

Strength I limit state

For Design Case 1 and 2, the load factor for dead 

load, γP, shall be taken as 1.0.

Extreme Event Load Combination II for the design of the overhang with regard to vehicle 

collision can be reduced to:

(Eqn. 8.4) 

Two cross sections of the overhang should be verified at this limit state: 1) at the face of the 

barrier and 2) at the root of the cantilevering overhang. The first section is subject to a small 

portion of overhang slab, the weight of the barrier rail, and collision forces. The second section 

includes the remainder of the overhang self weight and vertical wheel loads along the cantilever 

length. Figure 8.9 shows these two critical sections for the cross section of the overhang to be 

designed for the overhang presented for the cross section developed in chapter 5.
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Figure 8.9 – Critical Sections for Overhang Design to Develop Barrier Railing 

Design loads and dimensional considerations for Design Load Cases 1 and 2 specified in Article 

A13.4.1 are presented in Article A13.2. The magnitude of the loads and dimensional 

considerations are a function of the crash testing rating of the barrier railing used on the bridge. 

The superstructures of the design examples in this manual use concrete parapet type barrier 

railing with a TL-4 crash test rating. Design forces for this railing are taken from Table A13.2-1:

Design Forces and Designations TL-4 Crash Test Level

Ft Transverse (kips) 54.0

FL Longitudinal (kips) 18.0

Fv Vertical (kips) Down 18.0

Lt and LL (ft) 3.5

Lv (ft) 18.0

He (min)(in.) 32.0

Minimum H Height of Rail (in.) 32.0

Table 8.1 – Railing Loads for TL-4 Barrier (from AASHTO LRFD Table A13.2-1) 

There are some interesting considerations with regard to the application of Articles A13.4.1 and 

A13.2:

• Design Load Case 2 has implications with regard to Post-and-Beam railings, but will not

govern for overhangs with concrete parapet barriers.

• The longitudinal loads shown in Table A13.2-1 also have implications with regard to

Post-and-Beam railings, but are not meaningful with regard to bridges with concrete

parapet barriers away from railing ends.

• Loads from Table A13.2-1 applied over the specified lengths plus a distribution of 45

degrees over the height of the barrier. This results in a width of application of the forces

in Table A13.2-1 (also table 8.1) of the specified length plus twice the height of the

barrier railing.

Articles A13.4.2 and A13.4.3 present alternate design approaches to Design Case 1 for bridges 

that use concrete parapet and post-and-beam rail. Article A13.4.2 is presented here for the very 

common concrete parapet type of barrier rail.

Article A13.4.2—Decks Supporting Concrete 

Parapet Railings
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For Design Case 1, the deck overhang may be 

designed to provide a flexural resistance, Ms in kipft/ft which, acting coincident with the tensile force T

in kip/ft, specified herein, exceeds Mc of the 

parapet at its base. The axial tensile force, T, may 

be taken as:

(Eqn. 8.5) 

Where,

Rw = parapet resistance specified in Article 

A13.3.1(kips)

Lc = critical length of the yield line failure pattern 

(ft)

H = height of wall (ft)

T = tensile force per unit of deck length (kip/ft)

Design of the deck overhang for the vertical forces 

specified in Design Case 2 shall be based on the 

overhanging portion of the deck.

The parapet resistance, Rw, and critical length Rw, are taken from Article A13.3.1

(Eqn. 8.6)

(Eqn. 8.7) 

Where,

Ft = transverse force specified in Table A13.2-1 assumed to be acting at top of a 

concrete wall (kips)

H = height of wall (ft)

Lc = critical length of yield line failure pattern (ft)

Lt = longitudinal length of distribution of impact force Ft (ft)

Rw = total transverse resistance of the railing (kips)

Mb = additional flexural resistance of beam in addition to Mw, if any, at top of wall (ft-k) 

Mc = flexural resistance of cantilevered walls about an axis parallel to the longitudinal 

axis of the bridge (kip-ft/ft)

Mw = flexural resistance of the wall about its vertical axis (kip-ft/ft)

8.4 Strip Method Analysis for a Multi-Cell Box Girder Superstructure

This section presents the transverse analysis of the four-cell box girder used in the preliminary 

design presented in chapter 5 and in example problem 1 presented in appendix C.
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8.4.1 The Transverse Model

The transverse analysis of the cross section shown in figure 8.9 is a two-dimensional solution 

that can be performed using either a 2D or 3D general-purpose frame analysis program. The 

three-dimensional analysis will typically require additional support conditions out of the model 

plane to provide needed analytical stability.

The analysis performed is a linear elastic analysis utilizing gross section properties of the 

concrete cross section. The top slabs, bottom slabs, webs, and overhangs are modeled as 

beam elements with depths equal to their representative thicknesses. The width of the beam 

elements “out of the plane of the page” is set equal to 1.0’. Node locations are provided at 

intersections of elements, changes in transverse member thickness, changes in transverse 

member orientation, and for convenience at locations of desired output. 

Vertical support of the cross section model is with rigid supports at the bottom of the webs 

(Article 4.6.2.1.6 Paragraph 1). A horizontal support is placed rigidly to restrain the transverse 

movement of the bottom slab. A second horizontal support is provided to restrain the frame 

against side-sway, which better represents the torsional stiffness of the box girder with regard to 

transverse displacements.

Figure 8.10 shows the development of the cross section from transverse dimensions, to 

idealized members to the computer nodal analysis model for the cross section of Design 

Example 1.

Figure 8.10 – Developing the Two-Dimensional Transverse Model
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8.4.2 Transverse Bending Moment Results

Loads that uniformly repeat along the span can be applied directly to the analytical model 

without consideration for longitudinal distribution. These loads include self weight and 

superimposed dead loads such as barrier railing, wearing surfaces, and suspended utilities. 

Figure 8.11 shows the bending moments from the self weight of the cross section. The analysis 

software used developed the applied loads internally based on the member cross section 

dimensions and the unit weight of the concrete. Figure 8.12 shows the effect of barrier railing 

weight 0.45 kips/foot, applied at the center of gravity of the barrier cross section. Figure 8.13 

shows the bending moments resulting from the application of a 25 psf (or 25plf in the unit width 

model) wearing surface.

Small bending moments in webs and bottom slab members are not shown in some of the 

figures of this Section for clarity.

Figure 8.11 – Transverse Self Weight Moments (ft-kip/ft)

Figure 8.12 – Transverse Barrier Railing Moments (ft-kip/ft)

Figure 8.13 – Transverse Wearing Surface Moments (ft-kips/ft)
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Live load moments are determined by positioning both the Design Truck and Tandem wheel 

loads in position to produce maximum transverse moments. Figures 8.14, 8.15 and 8.16 show 

the results of three load cases for the Design Truck. Figure 8.14 shows the position of one truck 

to produce maximum negative transverse bending at the inside face of the outer webs. Figure 

8.15 shows two trucks positioned to produce maximum transverse negative moment at interior 

webs. Figure 8.16 shows the load case producing maximum positive moment in the top slab.

Figure 8.14 – Maximum Negative Design Truck Moment in Outer Web 

Figure 8.15 – Maximum Negative Design Truck Moment at Inner Web 

Figure 8.16 – Maximum Positive Design Truck Moment in Top Slab 
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8.4.3 Transverse Design Moments

Moment results for live load cases need to be magnified for impact and multiple presence 

factors, and then divided by the longitudinal length of the strips, W, based on equations 8.1, 8.2 

and 8.3.

(Eqn. 8.8) 

(Eqn. 8.9) 

(Eqn. 8.10) 

Bending moment results from the transverse analysis of example problem 1 at critical nodes in 

foot-kips/foot are:

Node Type Self Weight Barrier Wearing 

Surface Live Load

3 Overhang (Neg) -1.76 -1.97 -0.15 -32.00

13 Top Slab (Pos) 0.69 0.0 0.15 27.00

15 Top Slab (Neg) -1.06 0.0 -0.24 -36.42

Table 8.2 –Transverse Bending Moment Results from Frame Analysis

Live Load bending moments are magnified by vehicle impact and multi-presence factors, and 

divided over their associated strip width.

The negative live load moment in the overhang at the face of the web is:

(Eqn. 8.11) 

It is interesting to note the magnitude of the overhang moment with regard to the moment 

caused by a 1 kip/foot load acting 1’ from the face of the railing as permitted by Article 3.6.1.3.4. 

This moment would be approximately 1/3 of the moment predicted by the strength method:

(Eqn. 8.12) 

The top slab maximum positive live load moment:

(Eqn. 8.13) 

The negative live load moment in the top slab at the face of the middle web:

(Eqn.8.14) 
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The resulting factored moments for the AASHTO LRFD Strength I case are then:

(Eqn. 8.15) 

(Eqn. 8.16) 

(Eqn. 8.17) 

The three critical cross sections are designed for the amount of required reinforcing per foot 

along the length of the bridge. Reinforcing for the maximum positive and maximum negative 

ultimate moments is used in each slab and over each web. Additional reinforcing may be 

required over the outer webs in order to develop the capacity of the barrier when subjected to 

vehicle collision forces.

8.5 Top Slab Transverse Bending Moment Results for a Single-Cell Box Girder

8.5.1 Introduction

Section 8.2 of this Manual presented limitations on the use of the LRFD Approximate Method for 

certain concrete box girder cross sections. The method can produce unconservative results 

when applied directly to single-cell girders, dual-cell girders, and multi-cell box girders with web 

spacing greater than 15 feet. Using a refined approach to account for longitudinal distribution of 

concentrated wheel loads in determining the transverse live load moments can produce the 

needed conservatism.

The approach taken is the same as specified in Article 4.6.2.9.4 for concrete box girders built 

segmentally. Differences in transverse behavior does not lie in the method of construction, but 

the fact that the majority of box girder bridges built segmentally are constructed with 2 webs 

(single cell) or 3 webs (dual cell), resulting in significant slab spans for most common roadway 

widths.

Figure 8.17 shows the cross section of a single-cell box girder that will be analyzed in this 

Section with regard to transverse flexure. 

Figure 8.17 – Typical Single-Cell Box Girder Cross Section Defined at Mid-Span 
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Figure 8.18 shows a perspective of a span with the cross section under consideration. As 

before, a typical cross section with a unit length is identified in the span for a two-dimensional 

analysis in the plane of the cross section. The extracted typical section for analysis is shown in 

figure 8.19. 

Figure 8.18 – Typical Single-Cell Box Girder Span with Cross Section Defined at Mid-Span 

Figure 8.19 – One Foot Section of Typical Cross Section 

The typical cross section is modeled using beam elements in a general two-dimensional 

structural analysis program. Figure 8.20 shows the transition from typical cross section, to 

idealized beam members, to a node and element layout for a typical analysis model. Special 

transverse modeling considerations include:

a. The model shown in figure 8.20 shows beam elements extending to nodes at the top of 

web/cantilever/top slab intersection and the bottom of web/bottom slab intersection. 

Often, designers will model a portion of these intersecting members as rigid elements, as 

the size of the connection can be significant with regard to member length.
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b. Many precast single-cell box girder cross sections include linear or circular fillets at the 

cantilever/top of web and top slab/top of web connections. The choice of the critical 

sections for design of flexure at these sections is left to the discretion of engineer 

considering the particular geometry of the cross section.

c. Vertical supports are placed under the webs as shown in the bottom sketch of figure 8.20. 

One horizontal support is required for model stability. A second horizontal support has 

been added to restrain side-sway of the two-dimensional model. This support helps 

account for the torsional rigidity of the box girder in the actual three-dimensional structure. 

This second horizontal support can be problematic for some load cases such as 

transverse post-tensioning. Use and placement of the second support is left to the 

discretion of the engineer.

Figure 8.20 – Developing the Two-Dimensional Transverse Model

8.5.2 Analysis for Uniformly Repeating Loads

Transverse bending moments for loads that repeat uniformly along the span can be determined 

directly from the two-dimensional analysis mode. Self weight bending moments can typically 

be generated internally by the analysis software once the unit weight of the concrete has been 
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Chapter 8 – Transverse Analysis 198 of 369 

defined. The per-unit length values of superimposed dead loads are applied directly to the twodimensional analysis model as either concentrated or distributed loads.

The typical cross section shown in figure 8.20 was analyzed for the following loads:

Self Weight: Concrete unit weight = 150 lb/ft3 

Barrier Railing: p = 0.420 kips/ft (8” from edge of the cantilevers) 

Future Wearing Surface: 2” concrete, p = .025 kips/ft/ft

The transverse bending moments resulting from these three load cases are shown in figure 

8.21. 

Figure 8.21 – Transverse Bending Moments for Uniformly Repeating Loads 
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8.5.3 Analysis for Concentrated Wheel Live Loads

Figure 8.22 shows the perspective of a single-cell box girder superstructure loaded with the 

Design Truck portion of the HL-93 notional load. Travel lanes, and the trucks/tandems within 

the lanes, are positioned in number and location in accordance with AASHTO LRFD 

requirements to produce maximum transverse bending moments at critical sections. The 

appropriate multi-presence factor of AASHTO LRFD Article 3.8.1.1.2 should be considered in 

choosing the number of design lanes to apply for a given critical section. Truck/tandem 

locations within the lanes should be in accordance with AASHTO LRFD Article 3.8.1.3.1.

Transverse bending moments resulting from the application of concentrated loads in the span 

are determined in a three step process:

1. Determine maximum transverse bending moments at critical sections of the 

cantilever wing and top slab considering these members as separate threedimensional, fixed-end slab structures. 

2. Use the two-dimensional model to distribute “fixed-end” slab moments around the 

cross section model.

3. Sum the fixed-end moments and redistributed moments to determine live load 

moments for design.

The fixed-end slab structures are typically analyzed using finite element methods or other tools 

such as influence surfaces. When modeled using finite elements, the slab structures are 

supported by full restraints at their intersection with the top of the webs. The longitudinal length 

of the slab structure should be sufficient to not impact transverse bending behavior. Slabs with 

shorter spans may warrant the application of wheel loads as surface loads. The areas over 

which the loads are distributed are the AASHTO LRFD Tire Contact Area (AASHTO LRFD 

Article 3.8.1.2.5) projected to the center of gravity of the top slab using a 45 degree distribution 

in both longitudinal and transverse directions. Bending moments computed in the examples 

included in this Section use influence surfaces and consider only the HL93 Design Truck. 

Figure 8.22 – Truck Loads on a Single-Cell Box Girder Span 
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8.5.4 Live Load Moments in Cantilever Wings.

The maximum fixed-end moment in the cantilever wing is determined by positioning the HL93 

Truck one foot away from the face of the barrier railing in accordance with AASHTO LRFD 

Article 3.8.1.3.1. Figure 8.23 shows this loading on the cross section being analyzed. Figure 

8.24 shows an influence surface for negative moment at the root of the cantilever, along the 

centerline of the influence surface. This figure also shows the location of wheel loads to 

produce the maximum fixed-end moment in the cantilever slab. The unfactored negative 

maximum bending moment found using this cantilever influence surface is 15.2 ft-kips without 

impact. 

Figure 8.23 – Truck Location for Maximum Transverse Bending Moment at Root of Cantilever 

Figure 8.24 – Loaded Influence Surface for the Cantilever Wing.

Transverse bending moments in the other members of the cross section resulting from live load 

in the cantilever are determined by applying the cantilever moment to the cantilever/top 

slab/web intersection in the two-dimensional model. Figure 8.25 shows the distribution of 15.2 

ft-kip cantilever moment around the cross section. Superimposing the cantilever moments with 

the distributed moments produces the final transverse bending moment diagram shown in figure 

8.26.
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Figure 8.25 – Distribution of Cantilever Live Load Moments in the Cross Section 

Figure 8.26 – Final Bending Moments for Live Load in Cantilever 

The shape of the bending moment diagram in the loaded cantilever wing is approximate, as the 

influence surface used only provides the bending moment at the root of the cantilever. Plate 

solutions using finite element methods could be used to produce moments along the loaded 

cantilever.

8.5.5 Negative Live Load Moments in the Top Slab.

Negative live load moments in the top slab require positioning the Design Truck/Tandem for 

maximum negative effect at the end of the isolated top slab structure. In the cross section being 

studied, the maximum negative moment at the left end of the top slab is produced by locating 

one truck in the top slab and adding the effect of one truck in the cantilever wing. Figure 8.27 

shows the locations of these two travel lanes. This loading arrangement can be evaluated as a 

superposition of the moments due to a truck in the cantilever shown in figure 8.24 and moments 

resulting from the single truck located between the webs as shown in figure 8.28 and figure 

8.29.

The fixed-end moments for the top slab plate structure are determined using influence surfaces 

for a doubly fixed plate. Figure 8.28 shows the influence surface for the maximum negative 
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moment at the left end of the slab (10.3 ft-kips). Figure 8.29 shows the inverted influence 

surface used to compute the corresponding moment at the right end of the slab (5.1 ft-kips). 

Figure 8.27 – Truck Location for Maximum Transverse Bending Moment at Middle of Top Slab 

Figure 8.28 – Influence Surface for Maximum Negative Bending at the Left End of the Top Slab. 

Figure 8.29 – Influence Surface for Maximum Negative Bending at the Right End of the Top Slab. 
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The two fixed-end moments can be “released” on the cross section by applying them as 

concentrated moments at the cantilever/top slab/web intersection. Figure 8.30 shows the 

results of the application of the concentrated couples on the two-dimensional analysis model. 

The concentrated couples, though applied as external loads, are actually internal fixed-end 

moments. When applied as external loads with signs opposite to the internally fixed-end 

moments, the moments in the unloaded members (webs and bottom slab) are the final 

moments due to the loading. The moments in the top slab represent the differential moments in 

the loaded member in moving from fixed-end conditions to the actual flexibility of the box girder. 

The final bending moment diagram, shown in figure 8.31, is determined by reducing the fixedend moments by these differential moments.

The total negative moment at the left end of the top slab for Design Trucks arranged in two 

lanes as shown in figure 8.27 are found by summing the maximum moment from the distributed 

cantilever moment (figure 8.26) and the moment resulting from the truck in the top slab (figure 

8.31). The moment at the left end of the top slab is:

(Eqn. 8.18) 

The results of this analysis are valid only at the location of know fixed moments and their 

redistributed differential moments. The shape of the top slab bending moment between the two 

known extremity moments is not an exact representation, but general in nature. Efforts have 

been made in past practice to develop approaches using equivalent forces in equilibrium with 

the extremity moments to determine bending moments within the top slab for a given loading. 

The results of these methods do not prove accurate or necessary for the design of the top slab.

Figure 8.30 – Distribution of Fixed-End Live Load Moments for Maximum Negative Moment Case 

Figure 8.31 – Summed Live Load Moments for the Maximum Negative Moment Case 
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8.5.6 Positive Live Load Moments at Centerline of the Top Slab.

Positive live load moments at the centerline of the top slab are computed in a fashion similar to 

the negative moments at the ends of the slab, with one additional initial step. An influence 

surface for moment at the center of the isolated top slab structure, or a finite element model, is 

first used to determine the arrangement of loads to produce maximum positive moment. Fixedend negative moments are then determined for this load arrangement and distributed around 

the cross section. The final positive moment at the centerline of the slab is the positive moment 

in the fixed-end top slab structure, increased by the release of the end moments.

Figure 8.32 shows the load arrangement for maximum positive bending moment at the center of 

the top slab. Figure 8.33 shows an influence surface for the maximum positive transverse 

bending moment at the center of the top slab for this loading arrangement. The value of this 

bending moment with fixed-end supports is 3.88 ft-kips. The fixed-end bending moments at the 

ends of the top slab structure for the same loading arrangement are 12.0 ft-kips at the left end 

and 11.1 ft-kips at the right end.

Figure 8.32 – Live Load Position for Maximum Positive Bending

These fixed-end moments are released on the cross section as presented in the previous 

Section. Figure 8.34 shows the differential moments, and figure 8.35 shows the summed 

values. The release of the fixed-end moments shifts the bending moment diagram in the top 

slab, increasing the bending moment at the centerline of the top slab. The fixed-end centerline 

moment increases from 3.88 ft-kips/ft to 5.08 ft-kips/ft. Again, the shape of the top slab bending 

moment diagram between known locations is only an approximation.
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Figure 8.33 – Maximum Positive Moment in the Top Slab for Fixed-End Conditions 

Figure 8.34 – Distribution of Fixed-End Live Load Moments for Maximum Negative Moment Case 

Figure 8.35 – Summed Live Load Moments for the Maximum Positive Moment Case 
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8.6 Transverse Post-Tensioning

8.6.1 Transverse Post-Tensioning Tendon Layouts

The cantilever wings and top slab of single-cell box girder superstructures are typically 

prestressed transversely with post-tensioning to offset tensile stresses resulting from permanent 

and live loads. Narrow precast box girders with widths of 16’ or less, often used for single track 

transit systems, may not greatly benefit from transverse post-tensioning. Rails that carry normal 

operating train loads can be positioned adjacent to the webs, limiting top slab bending moments 

to values small enough to not require transverse post-tensioning.

LRFD Article 5.14.2.3.10a provides minimum cantilever wing and top slab thicknesses for 

segmental construction. Cast-in-place box girders with cross sections similar to segmentally 

constructed bridges should follow these provisions. The 15 foot top slab span length limitation 

for requiring transverse post-tensioning should be the total length from face of web to face of 

web (top slab with haunches).

Figure 8.35 shows a typical transverse post-tensioning tendon layout for the typical section 

studied in this section. The transverse tendons are typically comprised of 3 or 4 strands, either 

0.5” or 0.6” diameter each, placed in oval shaped “flat” ducts. The profile of the tendons varies 

to provide needed eccentricity over the webs and at the centerline of the top slab. The 

perspective view in figure 8.35 shows three transverse tendons in a section of a single cell box 

girder superstructure. 

Figure 8.36 – Typical Transverse Tendon Layout.

8.6.2 Required Prestressing Force

The selection of the required prestressing force on a unit length basis is made by summing the 

moments at critical sections, evaluating each section’s prestressing requirement, and choosing 

the greatest required force. AASHTO LRFD Service I is the appropriate limit state for transverse 

prestressing design based on permissible concrete tension (AASHTO LRFD Article 3.4.1). The 

load factors for permanent and live loads at this limit state are equal to 1.0.
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Considering the example presented in this section, the summary of moments at the three 

sections studied are:

Section Root of 

Cantilever

Edge of Top 

Slab CL of Top Slab

Self Weight -8.19 -8.64 1.52

Barrier Railing -4.34 -0.79 -0.43

Wearing Surface -1.52 -1.21 0.34

Live Load -15.2 -12.04 5.08

Multi-Presence 1.20 1.00 1.00

Dynamic Allowance (IM) 1.33 1.33 1.33

Total Service I Moment -38.4 -24.7 8.19

Table 8.3 –Transverse Bending Moment Results from Frame Analysis

The equations governing post-tensioning selection are:

(Eqn. 8.19)

(Eqn. 8.20)

Where: M = Total applied service load moment

Ma = Moment causing allowable tensile stress 

c1 = Distance from neutral axis to extreme top fiber

c2 = Distance from neutral axis to extreme bottom fiber

ρ = Cross section efficiency = 1/3 for a rectangular section

e = Tendon eccentricity considering location of strands within the duct

The moment causing allowable tensile stress (Ma) is equal to the allowable tensile stress (fa) 

multiplied by the section modulus (S=bh2

/6 for a rectangular section), or:

(Eqn. 8.21)

The allowable transverse flexural stress in the top slab of a single-cell box girder is equal to 

0.0948√f’c. Using 6 ksi concrete, the allowable stress would be 0.232 ksi (33.4 ksf). Solving for 

the prestress force requirements:
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Section Root of 

Cantilever

Edge of Top 

Slab CL of Top Slab

Moment -38.4 -24.7 8.19

h/2 = c1=c2 0.87 0.87 0.369 

Efficiency (ρ) 1/3 1/3 1/3

Eccentricity (e) -0.60 -0.60 0.19

Moment Causing fa -18.9 -18.9 4.12

Required PT Force/ft 24.2 8.8 13.0

Table 8.4 –Transverse Bending Moment Results from Frame Analysis

Considering the limited example presented in this section, the governing cross section for 

required prestress force is the root of the cantilever. The prestress force required at this section 

is 24.2 kips/foot.

The final selection of the number and size of the post-tensioning tendons requires a study to 

determine the final force in the tendons. This work must include initial losses caused by friction, 

wobble and anchor set, and long term losses resulting from concrete creep and shrinkage, as 

well as prestressing steel relaxation. Secondary post-tensioning effects must be included when 

present. Several commercially available software packages include modeling of the actual 

geometry of post-tensioning tendons, automatic generation of internal forces due to tendon 

stressing, losses during stressing, and long-term, time-dependent loss calculations. 

Depending on the tendon profile in the top slab, the flexural restraint provided by the webs can 

result in secondary prestressing moments. These moments should be included in the 

summation of service limit state moments when determining the required prestressing force. 

The final value of the secondary moments is a function of the resulting prestress demand, 

thwarting a direct solution and slightly complicating the final prestress force. For the example in 

this section, a concordant tendon profile that produced no secondary moments was used. 

The selection of the required prestress force satisfies just a part of the design requirements for 

the cross section. Other tasks that need to be performed include: 

• Tensile and compressive stress checks at all sections of the cantilever wings and top

slab to verify appropriate service limit states.

• Reinforcing requirement checks in the top slab to verify appropriate strength limit states.

• Transverse bending moments in the webs combined with shear reinforcing requirements

to select final web reinforcing.

8.6.3 Transverse Post-Tensioning Tendon Placement and Stressing

Figures 8.37 and 8.38 show various photographs of the installation and stressing of the 

transverse top slab tendons in the casting yard. Figure 8.37 shows the placement of the 

transverse flat duct within the reinforcing cage after positioning into the casting machine. 

Transverse ducts pass over the longitudinal tendons of this cantilever bridge and are connected 

to the anchorages which are fixed to the cantilever wing bulkhead during the concrete pour. 

The ducts shown are plastic (typically polypropylene) which provide an important layer of 

corrosion protection to the transverse post-tensioning strands. Vertical grout inspection ports 

are located at the top of the anchorages to permit post-grouting inspections within the tendon.
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Figure 8.37 – Transverse Duct Placement in Casting Machine 

The photograph on the left of figure 8.38 shows the stressing of a four strand transverse tendon 

using a mono-strand stressing ram. The photograph of the right is a close-up of the transverse 

tendon anchorage and wedge block after stressing the two central strands. Following stressing 

and verification of tendon elongations by the Resident Engineer and/or Engineer of Record, 

strand “tails” are cut off, permanent grout caps are placed over the anchorages, the tendons are 

grouted, anchorage block-outs are filled with concrete, and protective coatings are applied.

Figure 8.38 – Mono-Strand Stressing of a 4 Strand Tendon (left). 

Anchorage After Stressing 2nd Strand (right). 
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Chapter 9—Other Design Considerations

9.1 Effects of Curved Tendons

Alignments of post-tensioning tendons change direction as they pass through curved ducts cast 

into the surrounding concrete. Figure 9.1 shows two common forms of tendon path deviations 

used in post-tensioned concrete superstructures. The top of figure 9.1 shows a parabolic 

tendon deviation commonly used for vertical deviations in cast-in-place box girder bridges. 

Parabolic deviations exert a uniform load along the horizontal projected length of the tendon 

deviation. This uniform force is a function of tendon force, deviation length and tendon 

eccentricity. The bottom of figure 9.1 shows a tendon deviated along a circular arc. The force 

exerted by the tendon is a radial force proportional to tendon force and inversely proportional to 

the radius of the circular deviation. Circular arc deviations are often used in anchorage blisters 

and at deviations of external tendons. 

Figure 9.1 – Curved Tendon Deviations

Tendons with circular geometry also occur in webs of cast-in-place box girder bridges that curve 

to follow a horizontal circular alignment. Figure 9.2 shows a perspective of a section of curved 

multi-cell box girder superstructure. Radial forces inversely proportional to the radii of a 

particular web act towards the center of curvature as the tendons rise and fall along their vertical 

parabolic profile. The radial forces vary in each web as a function of web radius.

Figure 9.2 – Tendons in Curved Superstructures 
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The horizontal forces per unit length of superstructure load the webs of box girder bridges 

causing transverse bending moments that may increase the required amount of transverse 

reinforcing steel. The lateral tendon forces also cause local stresses in the webs that can lead 

to cover spalling and tendon pull-out. Figure 9.3 shows the deflected shape of a multi-cell box 

girder subjected to lateral loading resulting from tendons following horizontal curvature.

Figure 9.3 – Cross Section of Multi-Cell Box Girder with Lateral Tendon Loads 

This Section addresses the design requirements for the radial effects associated with curved 

tendons in webs. Requirements for this subject are found in LRFD Article 5.10.4.3 which is a 

part of a larger LRFD Article 5.10.4 - Tendon Confinement.

9.1.1 In-Plane and Out-of-Plane Forces

The LRFD Specifications delineate between in-plane and out-of-plane effects when providing 

design requirements for the effects of curved tendons. The direction of in-plane and out-ofplane forces can change along the length of a tendon. Figure 9.4 shows a post-tensioning 

tendon that deviates direction through two circular curves. Curve 1 of the tendon lies in the Y-Z

plane. In-plane forces for the tendon in this location would be forces whose components act in 

the Y and/or Z direction. Out-of-plane forces for this portion of the tendon would be those acting 

in the ±X direction. After exiting Curve 1, the tendon deviates around Curve 2 which lies in the 

X-Y plane. In-plane forces for this curve lie in the XY plane. Out-of-plane forces for this curve 

act in the ±Z direction. 

Figure 9.4 – Tendon Plane of Curvature 
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In-plane force in a region of circular curvature, presented in LRFD Article 5.10.4.3.1a, is shown 

in equation 9.1:

(Eqn. 9.1) 

Where, Fu-in = the in-plane deviation force per unit length of tendon

Pu = the factored tendon force = 1.2 times Pjack (LRFD Article 3.4.3.2)

R = the in-plane radius of curvature of the tendon

Out-of-plane forces acting transverse to in-place forces are the result of the spreading of the 

strands as they bear on the inside face of curved circular ducts. Figure 9.5 shows a section (at 

a profiled tendon high point) of a web of a concrete box girder bridge with a tendon curving to 

produce a downward in-plane curvature. When the tendon is stressed, the strands are drawn to 

the bottom of the duct where they create lateral stresses within the concrete.

Figure 9.5 – In-Plane and Out-of-Plane Tendon Forces 

In-plane force in a region of circular curvature, presented in LRFD Article 5.10.4.3.2, is shown in 

equation 9.2:

(Eqn. 9.2) 

Where, Fu-out = the out-of-plane force per unit length of tendon

Pu = the factored tendon force = 1.2 times Pjack (LRFD Article 3.4.3.2)

R = the in-plane radius of curvature of the tendon

Tendons in cast-in-place concrete box girder bridges are primarily placed to follow a series of 

parabolic curves in the vertical direction. These tendons produce in-plane forces in a plane 

defined by the vertical direction and the longitudinal axis of the bridge. At the same time, these 

tendons can be in webs that are cast to follow a horizontal bridge curvature, producing 

simultaneously acting in-plane forces in a horizontal plane. In theory, the in-plane force effects 

should be verified for both sets of in-plane forces. In reality, the in-plane forces in the vertical 

direction in webs of concrete box girder bridges are resisted by significant concrete as to negate 

concerns. For bridges following horizontal alignments with large radius curves, the in-plane 

forces acting transverse to the webs typically have only the web concrete cover acting to resist 
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Chapter 9 – Other Design Considerations 213 of 369

tendon pullout. Supplemental duct ties are used to resist the in-plane forces for more tightly 

curved bridges.

9.1.2 AASHTO LRFD Design Approach

The requirements of LRFD Article 5.10.4.3 address the design of in-plane and out-of-plane force 

effects of post-tensioning tendons within webs of single-cell and multi-cell concrete box girder 

bridges. This manual presents these requirements in a slightly different arrangement as those 

presented in LRFD Article 5.10.4.3:

• Regional Effects—Flexure of the transverse cross section of the box girder

superstructure, referred to in LRFD Specifications as Regional Bending (5.10.4.3.1d)

• Local Effects—Local flexure and shear behavior at locations of ducts and duct stacks in

the webs. This includes Shear Resistance to Pull-out (5.10.4.3.1b) and Cracking of

Cover Concrete (5.10.4.3.1c)

• Out-of-Plane Force Effects (5.10.4.3.2)

9.1.3 Regional Effects—Transverse (Regional) Bending

Consider the hypothetical case of a tendon following a circular arc within a concrete member 

whose area is concentrated as to be completely coincident with the tendon. Figure 9.6 shows 

this hypothetical case with the tendon shown separated from the concrete for clarity. The 

curved stressed tendon produces an inward radial force. The concrete is placed in compression 

at the tendon anchorages and its curvature induces an outwardly resisting radial load equal and 

opposite to the tendon force. The radial forces negate each other and the curved concrete 

member only shortens under the prestressing force.

Figure 9.6 – Hypothetical Concrete Member Completely Coincident with a Tendon 

Figure 9.7 shows a tendon following a circularly curving plate with a height of hc. In this case, 

the post-tensioning tendon and area of the concrete member being prestressed are not 

coincident. The radial force produced by the post-tensioning tendon is equal to the tendon force 

divided by the radius of the curved member. Circumferential compression is produced in the 

curved plate that is in equilibrium with the prestressing force. This circumferential compression 

is distributed over the height of the plate and produces a distributed pressure in the plate radial 
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Chapter 9 – Other Design Considerations 214 of 369

to the plate curvature. The member will shorten under the action of the prestressing force, but 

as the radial distributed force of the tendon is not coincident with the radial pressure produced in 

the concrete, the plate will bend radially.

Figure 9.7 – Post-Tensioning a Curved Plate 

The curved plate shown in figure 9.7 is free to deflect radially under the applied post-tensioning. 

The top and bottom of the plate will tend to deflect radially outward from the center of curvature, 

while deflections would tend to be inward at the elevation of the post-tensioning tendon. In 

concrete box girder construction the tops and bottoms of webs are restrained by their monolithic 

connections with the top and bottom slabs, producing transverse restraining moments as shown 

in figure 9.8.

Figure 9.8 – Web Flexure Restrained by Top and Bottom Slabs 
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Transverse moments (Regional Bending in LRFD Article 5.10.4.3.1d) must be evaluated and 

combined with other transverse moments and shear requirements to determine the appropriate 

amount and placement of transverse reinforcing. The transverse moments from the in-plane 

radial forces could be determined by either a 2-dimensional analysis of a unit length of 

transverse cross section (as in chapter 8) or a more complex 3-dimensional finite element 

analysis. The LRFD Specifications offers a simplified approach to evaluating these transverse 

moments in the webs of box girder bridges. This approach first considers the web to be simply 

supported at its connection with the top and bottom slab, and loaded at mid-height of the web. 

A continuity factor is then applied to express end moments as a percentage of the maximum 

positive transverse bending moment in the simply supported beam.

Figure 9.9 shows a comparison of the LRFD simplified approach to moments developed by 

more detailed analysis performed on a transverse cross section of a box girder bridge. In the 

detailed analysis, the bending moments can be seen as the superposition of the concentrated 

post-tensioning force and the resisting radial force in the compressed concrete web. Figure 9.9 

shows these component loads and their summation. The LRFD simplified approach is shown at 

the left of figure 9.9. The effects of the radial force produced in the compressed concrete web 

are not considered in the LRFD simplified approach.

Figure 9.9 – Web Transverse (Regional) Bending Moments 

The factored moment in using the LRFD simplified approach is:

(Eqn. 9.3) 

Where, φcont = continuity factor = 0.6 for interior webs and 0.7 for exterior webs

hc = span of the web between the top and bottom slabs, measured along the 

axis of the web

The span of the web for equation 9.3 is shown in figure 9.10. Only components of forces acting 

to produce bending in the direction transverse to the axis of the web needs to be considered. 

Though the height of the inclined outer web is taller than the vertical interior web in figure 9.10 
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Chapter 9 – Other Design Considerations 216 of 369 

by one over the cosine of the web slope, the force transverse to the web axis is the in-plane 

force multiplied by the cosine of the web slope. These two negate each other such that the 

regional bending in the outer web is only different from the inner web by the difference of the 

radii of the two webs.

Figure 9.10 – Web Height for Equation 9.3 

9.1.4 Local Shear and Flexure in Webs

The webs of concrete box girder bridges must be verified with regard to local shear pull-out and 

flexure of cover concrete in the vicinity of ducts and duct banks. Figure 9.11 shows one interior 

web of a post-tensioned box girder bridge that contains three post-tensioning tendons arranged 

in a duct stack. The detail to the right in figure 9.11 is an enlargement of the duct stack. The 

dimensions shown are defined as:

dduct = outside diameter of the post-tensioning tendon duct (in)

dc = cover of the duct on towards the direction of curvature (in)

sduct = clear distance between tendon ducts in the vertical direction (in)

hds = height of the duct stack (in)

tw = web width (in)

Ducts are considered stacked for flexure of the concrete cover when the clear distance between 

ducts (sduct) is less than 1.5 inches. For shear resistance calculations, duct stacks include clear 

spacings greater than the diameter of the ducts.

Figure 9.11 – Parameters for Local Shear and Flexure Design 
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9.1.4.1 Shear Resistance to Pull-out

In-plane tendon forces may cause tendons to pull-out of the webs towards the center of box 

girder curvature unless they are sufficiently embedded in web concrete or tied back to 

transverse reinforcing within the web. Resistance to tendon pull-out is provided by the shear 

capacity of the concrete in the webs on the side of the ducts towards the center of box girder 

curvature. The LRFD Specifications define the shear resistance as:

(Eqn. 9.4) 

In which the nominal capacity is defined by: 

(Eqn. 9.5) 

Where, Vn = nominal shear resistance of two shear planes per unit length (kips/in)

φ = resistance factor for shear = 0.75

f’ci = concrete strength at time of prestressing (ksi)

deff = one-half of the effective length of the failure plane in shear and tension for a 

curved element (in) 

The length of the shear failure plane (deff) varies as a function of the clear spacing (sduct) 

between the tendons in a duct stack. For duct clear spacings less than the diameter of the duct 

(sduct< dduct), the effective length of the failure plane is defined by equation 9.6. The effective 

length of the failure plane for this case is shown in Figure 9.12. The detail to the left in figure 

9.12 shows the special case of zero duct spacing. The detail to the right shows the general 

case for equation 9.6.

(Eqn. 9.6) 

Figure 9.12 – Effective Length of Failure Plane for Equation 9.6 
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Chapter 9 – Other Design Considerations 218 of 369 

When the duct spacing is greater than or equal to the duct diameter, the definition of the 

effective length of the failure plane changes. In this case, the lesser value of equation 9.7 and 

9.8 is used for deff. These two equations are shown graphically in figure 9.13.

(Eqn. 9.7) 

(Eqn. 9.8) 

Figure 9.13 – Effective Length of Failure Plane for Equations 9.7 and 9.8 

Procedure for verifying resistance to shear pull-out:

• Compute the factored in-plane force for all tendons in a web (equation 9.1).

• Compute the effective length of the failure plane for shear by equation 9.6, 9.7 or 9.8.

• Compute the shear resistance using equations 9.3 and 9.4.

• If the factored in-plane force is greater than the shear resistance, provide duct ties and 

stirrup similar to those shown in LRFD Figure C5.10.4.3.1b-1.

• As web radii vary, preform for each web in the cross section.

9.1.4.2 Cracking of Concrete Cover

Excessive in-plane force in a duct stack can cause cracking of the concrete cover. This 

cracking can negatively impact the long-term durability of the web and post-tensioning system. 

The LRFD Specifications requires a calculation of flexural stresses in an idealized beam of 

concrete cover that spans the duct stack as shown in figure 9.14. This calculation is required 

when the clear spacing between ducts is less than 1.5 inches. These flexural stresses are 

combined with flexural stresses from regional bending moments for evaluation.
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Chapter 9 – Other Design Considerations 219 of 369 

Figure 9.14 – Local Bending Moments for Evaluating Cracking of Concrete Cover 

The bending moments in the beam of concrete cover are those of a fixed end beam subjected 

to a uniform load over its length. The magnitude of the uniform load is equal to the summation 

of the factored in-plane forces divided by the duct stack height. The resulting end and mid-point 

bending moments are:

(Eqn. 9.9)

(Eqn. 9.10)

The stresses from the local and regional bending in the web are computed and compared to a 

permissible cracking stress given as:

(Eqn. 9.11) 

Procedure for verifying cracking of concrete cover:

• Compute the factored in-plane force for all tendons in a web (equation 9.1).

• Compute local bending moments from equations 9.9 and 9.10.

• Compute the stresses in the beam of concrete cover.

• Combine stresses resulting from regional bending with those from local bending.

• Compare the stresses to the permissible cracking stress from equation 9.11

• If the stresses are greater than the permissible cracking stress, provide duct ties and 

stirrup similar to those shown in LRFD Figure C5.10.4.3.1b-1 (Caltrans recommends 

duct ties when the bridge horizontal radius is less than 800’).

• As web radii vary, preform for each web in the cross section.
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Chapter 9 – Other Design Considerations 220 of 369 

9.1.5 Out-of-Plane Force Effects

Equation 9.2 defines the out-of-plane force effects resulting from the vertical deviation of posttensioning tendons. For straight bridges, the requirements of equations 9.3 and 9.4 should be 

met with regard to out-of-plane forces, or duct ties should be provided. In bridges with 

horizontal curvature, the out-of-plane force effects should be added to the in-plane force effects 

for the design procedures described in sections 9.1.3 and 9.1.4 of this manual.

9.2 End Anchorage Zones

The anchorages of the post-tensioning tendons for cast-in-place box girder bridges are typically 

located at the end of continuous units of the superstructure. Figure 9.15 shows details 

commonly found at the end of post-tensioned box girder bridge.

Figure 9.15 – Details of End of Post-Tensioned Box Girder Bridge 

The LRFD Article 5.10.9 provides design direction for the development of post-tensioning 

tendon anchorage zones. The concentrated force applied through the anchorage causes 

bursting forces in a local zone directly behind the anchorage. The post-tensioning forces are 

further distributed through a general zone until the beam behavior is achieved. The resistances 
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Chapter 9 – Other Design Considerations 221 of 369 

required in the local and general zones are determined by strut-and-tie modeling. Figure 5.16 

shows a side elevation of one web of a post-tensioned concrete box girder bridge, possible 

strut-and-tie modeling, and end zone reinforcing steel. 

Figure 9.16 – End Zone Design Development 

The specifics of designing anchorage zones are not presented in this manual. The reader is 

directed to FHWA Publication No. FHWA-NHI-07-035 “Load and Resistance Factor Design 

(LRFD) for Highway Superstructures”, Volume 3 “Concrete Bridge Superstructure Design”, 

Section 2.6.4 “Post-Tensioning Anchor Zones” for more detailed design information. Another 

excellent resource is Caltran’s Memo To Designers 11-25 Anchorage Zone Design (October 

2012). This document can be found at: http://www.dot.ca.gov/hq/esc/techpubs/manual/

bridgemanuals/bridge-memo-to-designer/page/Section%2011/11-25.pdf
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Chapter 9 – Other Design Considerations 222 of 369 

9.3 Diaphragms at Supports

Box girder sections at piers, expansion joints and abutments contain diaphragms to stiffen and 

strengthen the typical cross section and to transfer loads from the superstructure to the 

supporting substructure. Diaphragms work to transfer shear forces in the webs to the bridge 

bearings, stiffen the box girder with regard to torsion, and provide a location for anchoring posttensioning tendons. This section develops basic load carrying considerations for diaphragm 

design.

9.3.1 Single-Cell Box Girder Transfer of Vertical Shear to Bearings

Figure 9.17 shows a fundamental arrangement of a vertical web box girder supported by 

bearings on a pier. In this arrangement, the pier is sufficiently wide to allow the bearings to be 

placed directly below the axes of the vertical webs. The diaphragm in this arrangement sees no

force under the action of vertical loads.

Figure 9.17 – Concentric Web/Bearing Orientation

Using the diaphragm to transfer vertical forces, the bearings spacing may be reduced and the 

width of the pier cap greatly reduced. This narrowing of the pier cap provides significant cost 

reduction and can greatly enhance aesthetics. Figure 9.18 shows this bearing configuration for 

the case of the vertical web box girder. Using strut and tie modeling, the horizontal forces 

developed by the eccentricity of the web to the bearing can be computed. Though occurring 

over some depth of the box girder, the transverse reinforcing or post-tensioning used to resist 

the horizontal forces is typically placed near the top of the diaphragm, detailed to fully develop 

the width of the webs.

Figure 9.18 – Eccentric Web/Bearing Orientation
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Chapter 9 – Other Design Considerations 223 of 369 

In addition to horizontal transverse tensions, figure 9.19 shows two other force transfer 

mechanisms that need to be investigated when the line of action of the web is eccentric to the 

bearings. Shear friction at the interface of the web and diaphragm, as shown in the sketch on 

the left in figure 9.19, should be evaluated and reinforced for in accordance with AASHTO LRFD 

requirements. Locally, web reinforcing at the bottom of the web is subjected to a direct tension 

(seen to the right in figure 9.19) that should be included in the web reinforcing selection at the 

piers and abutments.

Figure 9.19 –General Shear Friction and Localized Direct Tension

Further cost savings and improved aesthetics are achieved in box girder construction through 

the use of inclined webs. Figure 9.20 shows the impact on transverse horizontal forces when 

inclined webs are used. The combination of web slope and bearing offset work to increase 

transverse horizontal force in the diaphragm.

Figure 9.20 – Vertical Force Transfer with Inclined Webs

As with the case of the vertical web box girders, the bearing spacing can be further reduced in 

bridges with inclined webs to again reduce substructure costs and improve aesthetics. Figure 

9.21 depicts a box girder with significant eccentricity between the web line of action and 

bearings. In this instance, transverse post-tensioning is used to “lift” the web forces to the top of 

the diaphragm where they are then transferred to the bearings.
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Chapter 9 – Other Design Considerations 224 of 369

Figure 9.21 – Transverse Post-Tensioning in Diaphragms

9.3.2 Single-Cell Box Girder Transfer of Torsion to Bearings

Forces acting on the superstructure eccentric to the center of torsion will produce torsional 

moments in the box girder superstructure (see appendix B for further information). The torsional 

moments are resisted in shear flow around the closed box section. The shear flow and resulting 

shear stress due to torsion are given by:

(Eqn. 9.12) 2

t

o

M

q A =

(Eqn. 9.13) 2

t

i

i oi

q M

t At

t = =

Where: Mt = applied torsional moment (force-length)

Ao = area bounded by the median line of the wall thickness

ti = Thickness of the ith member of the cross section (length)

q = shear flow (force/length)

τi = shear stress in the ith member of the cross section (force/length2

) 

Figure 9.22 shows the concept of shear flow and the limits of Ao. The equations presented 

here, along with those typically used for the torsional stiffness of a box girder superstructure, are 

simplified from more complete expressions that include the effect of the cantilever wings. 

Generally speaking, for most box girder bridges, this behavior is small and may be neglected.

Figure 9.22 – Shear Flow Resulting from Torsional Forces 
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Chapter 9 – Other Design Considerations 225 of 369

Torsional moments along a span are transferred to the substructure at the bearings. The shear 

flow in the top slab caused by the torsional moment reaction produces a horizontal force in the 

top slab as shown in figure 9.22. Diaphragms located at the piers are detailed to resist the 

horizontal force in the top slab, and maintain the integrity of the transverse cross section of the 

superstructure. Figure 9.23 and figure 9.24 show two common configurations of torsion 

resisting diaphragms. The diaphragm in figure 9.23 resists torsion in an “A-shaped” 

configuration, while figure 9.24 resists torsion in a “V-shaped” layout. The tension and 

compression components are evaluated by strut-and-tie models. Tension forces are resisted by 

either mild reinforcing or inclined post tensioning. Compressive forces are verified to establish 

minimum torsion diaphragm concrete dimensions.

Figure 9.23 - A-shaped Torsion Diaphragm 

Figure 9.24 – V-shaped Torsion Diaphragm 

9.3.3 Multi-Cell Box Girder Diaphragms

Diaphragms of multi-cell box girder bridges are often solid sections within the box girder, with 

longitudinal thickness sufficient to accept and develop the reinforcing from integrally cast 

columns. If no access is provided through the diaphragm generally access hatches have to be 

provided in the soffitt of each box cell for each span. Transverse span-to-depth ratios of the 
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Chapter 9 – Other Design Considerations 226 of 369

diaphragms between supporting columns is usually such that a strut-an-tie analysis is 

appropriate for determining reinforcing.

Figure 9.25 shows one possible strut-and-tie layout for the diaphragms of the bridge of design 

example 1 in appendix C. The forces applied to this model are the summation of shear forces in 

the webs. The web forces are applied in halves at the top and bottom of the struts aligned with 

the webs. (This loading pattern is discussed in a 2007 Transportation Research Board article 

entitled “LRFD Design of Integral Bent Caps” by Zayati, Ibrahim and Hida.) In this example the 

diaphragm is simply supported for simplicity of computing the support reactions and member 

forces.

Figure 9.25 – Possible Strut-and-Tie Layout for Diaphragm of Design Example 1 

Figure 9.26 shows another possible strut-and-tie layout for the diaphragms of the bridge of 

design example 1. The alignment of members near the center of compression and tension 

zones in the columns allow for the monolithic connection behavior of the columns to be 

captured.

Figure 9.26 – Strut-and-Tie Layout Considering Monolithic Column Connection 
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Appendix A – Flexibility Method 227 of 369

Appendix A—Analysis of Two-Dimensional

Indeterminate Structures by the Flexibility Method

1. Introduction

Analysis of continuous bridge superstructures is primarily preformed using stiffness based 

computer solutions that solve the following general equation:

(Eqn. A.1) [ ] [ ][ ] F KX =

Where: [F] = vector of forces

[K] = global structure stiffness matrix

[X] = vector of nodal displacements/rotations

The similar general expression in terms of the structure flexibility is: 

(Eqn. A.2) [ ] [ ][ ] X DF =

Where: [D] = global structure flexibility matrix

The development of computerized stiffness solutions, over computerized flexibility solutions, 

grew as a result of several factors. Stiffness solutions lent themselves to more efficient 

programming, an important feature in the early days of computing where CPU costs were very 

expensive. Stiffness methods also fit well with the predominant hand method of solving 

indeterminate structures, e.g. the Hardy Cross Method of Moment Distribution.

Hand solutions for solving indeterminate structures by flexibility methods have also been in use 

for some time. Calperyron produced his Theorem of Three Moments in the 1850’s. However, a 

generalized flexibility method of indeterminate structural analysis, complementary to Moment 

Distribution, was not generally a part of the American engineer’s educational experience.

Flexibility solutions of indeterminate structures did, however, continue to develop outside of the 

U.S. In the 1960’s, Muller and Mathivat summarized a general flexibility method called the

Method of Joint Flexibilities (or the Flexibility Method). 

Both the Method of Moment Distribution and the Flexibility Method rely on the same basic 

assumptions of linear elastic behavior, generalized beam deflection equations, and 

superposition. In practice the two methods are easily understood and have their own 

advantages:

Moment Distribution—Distributing fixed end moments to find final continuity moments.

Flexibility Method—Restraining simple span rotations to find final continuity moments.

The Flexibility Method is a very useful tool when coupled with numerical techniques for finding 

simple beam end rotations, such as the Conjugate-Beam Method or Newmark’s Method of 

Numerical Integration. This is especially true for determining continuity bending moments such 

as secondary moments due to post-tensioning, where simple span rotations can be quickly 

found by equivalent loading or integration of the curvature diagram defined by Fe/EI. 

The Flexibility Method presented in this appendix is that similar to that produced by Muller and 

Mathivat. This method will serve as the basic analysis method for this Manual.
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Appendix A – Flexibility Method 228 of 369 

2. Definition of the Problem

Consider a continuous beam defined by members spanning between supports as shown in 

figure A.1. Nodes are located at the supports and are lettered g through n. Span ij defines 

them member that spans between nodes i and j. Nodes i and j, along with the other nodes of 

the structure, are fixed against displacement but are free to rotate. This one-degree of freedom 

per node indeterminate structure is to be analyzed to by determining member end (continuity) 

moments. 

The top diagram in figure A.1 shows the continuous beam subjected to the load p(x) in Span ij. 

This load produces shear forces and bending moments in the continuous beam as shown in the 

second and third diagrams in figure A.1. The shear forces vary in Span ij because of the loads 

present in that span. The shear forces in the adjacent unloaded spans are constant and only a 

function of making equilibrium within the member end moments. Likewise, the bending 

moments in Span ij are related to the continuity moments and loading within that span, where 

the moments in adjacent unloaded spans are only related to the member end moments.

Figure A.1 – Continuous Beam Load, Shear and Moment Diagrams 

The Flexibility Method is used to solve for continuity moments in each individually loaded span. 

Loaded members are first isolated from the structure and end moments determined based on 

known flexibilities of adjacent restraining spans. The continuity moments are distributed 

throughout the adjacent spans with distribution and carry-over factors. The complete analysis of 

a structure loaded in multiple spans is then solved by summing continuity moments of the 

individual spans.

3. Sign Convention

Positive sign conventions for the Flexibility Method presented in this appendix are shown in 

figure A.2. Positive bending moments cause tensile stresses on the bottom of the beam (beam 

designer’s sign convention). Positive rotations at the ends of the beams are counter-clockwise.
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Figure A.2 – Bending Moment and Rotation Sign Convention 

4. Simple Span Beam Characteristics

Figure A.3 shows a simple span beam subjected to a bending moment, M(x), and resulting end 

rotations ωi and ωj. 

Figure A.3 – Equations for End Rotations of Simple Beams. 

The end rotations of simple span beams can be expressed by Bresse’s formulae:

(Eqn. A.3) 

0

1 ( )

L

i

M L x dx

L EI

ω =− − ∫

(Eqn. A.4) 

0

1 L

j

Mx dx

L EI

ω = ∫

Simple span beam characteristics for the Flexibility Method are defined as the rotations of the 

beam ends under the action of unit moments applied at the beam ends, as shown in figures A.4 

and A.5.

Figure A.4 – Bending Moment Diagram for Unit Moment at Node i. 
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Appendix A – Flexibility Method 230 of 369

Figure A.5 – Bending Moment Diagram for Unit Moment at Node j. 

Applying a unit moment at node i results in the rotations:

(Eqn. A.5)

2

2

0

1( ) L

i

L x dx a

L EI

ω − = − = ∫

(Eqn. A.6) 2

0

1( ) L

j

xL x dx b

L EI

ω − = = ∫

Applying a unit moment at node j results in the rotations:

(Eqn. A.7) 

2

2

0

1 L

j

x dx c

L EI

ω = = ∫

(Eqn. A.8) 2

0

1( ) L

i

xL x dx b

L EI

ω − = − = − ∫

Coefficients a, b, and c are the simple span beam characteristics required in the Flexibility 

Method. The coefficients are always positive and graphically expressed these coefficients are:

Figure A.6 – Simple Span Beam Characteristics 
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Appendix A – Flexibility Method 231 of 369

(Eqn. A.9) 

(Eqn. A.10)

5. Definition of Support Flexibility and Member Flexibility Coefficients

The development of the Flexibility Method begins with isolating the individual spans and defining 

their behavior as a function of the span characteristics and the rotational restraints offered by 

adjacent spans. Consider Span ij which is defined between nodes i and j as shown in figure 

A.7. Span ij is part of an indeterminate planar structure as previously described. Other

members are connected to nodes i and j in this structure are Span ih and Span jm.

Figure A.7 – Span ij in a Continuous Structure.

If Span ij was simply supported it would deflect freely and the rotations of the extremities would 

be the simple span rotations ωi and ωj shown in the second sketch of figure A.7. The adjacent 

connecting members, Span ih and Span jm, restrain these rotations, producing moments Mi

and Mj. The simple span rotations caused by the continuity moments Mi and Mj are αi and αj. 

The final end rotations are the values θi and θj which represent the difference between the 

simple span rotations resulting from loads (ω) and the simple span rotations resulting from 

continuity moments (α).

For a beam with constant inertia: 
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Appendix A – Flexibility Method 232 of 369 

Assuming linear elastic behavior and non-deformable connections between the members, Span 

ij behaves as if each extremity was restrained by a rotational spring simulating the actual 

connecting members, Span ih and Span jm. The isolated Span ij with equivalent end rotational 

springs is shown in figure A.11.

Figure A.8 – Isolating Span ij 

The coefficients Ki/ij and Kj/ij, are called the Support Flexibility Constants, and represent the 

flexibilities that the Span ij feels as a result of its connectivity with the adjacent spans. The 

flexibility of a support is a positive number varying between zero and infinity (0 and ∞). The 

constants are equal to zero when the end of the beam is completely fixed. A simply supported 

beam end has a support flexibility constant of infinity.

The relationship between end rotations and end moments are:

(Eqns. A.11)

The support flexibilities Ki/ij and Kj/ij are related to the flexibility of the adjacent spans connected 

to Span ij at nodes i and j. Figure A.9 shows the unloaded adjacent spans with compatible 

rotations θi and θj, subjected to the end moments mi and mj. Figure A.10 isolates the two 

adjacent members Span hi and Span jm. The coefficients kih/i and kjm/j, are called the member 

flexibility constants of the adjacent member spans. They are defined by the span characteristics 

and the support flexibilities at the far ends of the spans.

Figure A.9 – Compatible rotations of adjacent members
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Appendix A – Flexibility Method 233 of 369 

Figure A.10 – Adjacent Member Flexibilities

The relationship between end rotations and end moments are:

(Eqns. A.12)

 

Setting equations A.11 and A.12 equal to each other we see that:

(Eqns. A.13)

The equality of the support flexibilities with the adjacent member flexibilities, as shown in 

equations A.13, are true for the case of continuous beams with only one span framing into Span 

ij. When several members frame into nodes i and j, the summation of moments result in:

(Eqn. A.14)

(Eqn. A.15)

From these expressions, the general relationships between support and adjacent beam end 

flexibilities are:

(Eqns. A.16)
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6. Relationship between Support Flexibility Constants and Continuity Bending 

Moments

The next step in the development of the Flexibility Method is to express continuity bending 

moments as a result of restraining side spans in relation to the flexibilities that those spans 

provide through Support Flexibility Constants.

By superposition, we define the final rotations in the continuous beam as the summation of the 

simple span rotations under the action of the applied loads and the rotations resulting from the 

continuity moments:

(Eqns. A.17)

Knowing beam characteristics for Span ij (a, b, and c) the rotations of the simple span beam 

resulting from the continuity moments can be written as:

(Eqns. A.18)

Combining equations A.17 and A.18 the final rotations are:

(Eqns. A.19)

Recalling that the final rotations in the continuous beam are also defined to be equal to the

continuity moment times the support flexibility constant (equation A.11), we can write:

(Eqns. A.20)

Solving for the continuity moments Mi and Mj we find:

(Eqns. A.21)
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Appendix A – Flexibility Method 235 of 369

For the special case of a beam fixed against rotations at both ends, Ki/ij and Kj/ij equal zero, 

and:

2

i j

i

c b

M

ac b

ω ω + = −

(Eqns. A.22)

2

i j

j

b a

M

ac b

ω ω + = − −

Comparing equations A.21 and A.22, the elastically restrained span is equivalent to a perfectly 

fixed span having the effective flexibilities of:

/ ' i ij a aK = +

(Eqns. A.23) b b ' =

/ ' j ji c cK = +

For a span perfectly fixed and with constant inertia, the coefficients of flexibility are equal to 

those of equations A.9 and A.10 and equations A.21 can be reduced to:

( ) 2 2 i ij

EI M

L

= + ω ω

(Eqns. A.24)

( ) 2 2 j ij

EI M

L

=− + ω ω

7. Relationships between Support Joint Flexibilities and Member End Flexibilities

Determining the support constants Ki/ij and Kj/ji requires the determination of the member end 

flexibilities kih/i and kjm/j of the members connecting the Span ij to the structure. Consider Span 

hi which is to the left of Span ij. 

Figure A.11 – Member End Flexibility for Span hi 

Apply a moment mi at the i end of Span ih. The resulting end moments in relation to end 

rotations are:
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Appendix A – Flexibility Method 236 of 369 

(Eqns. A.25)

The end rotations in terms of the beam characteristics are:

(Eqns. A.26)

Equating equations A.25 and A.26:

(Eqns. A.27)

Solving for khi/i: 

(Eqn. A.28) 

If we perform the same analysis for Span jm, which is to the right of Span ij, we can solve for 

kjm/j: 

(Eqn. A.29) 

In developing a solution for all spans of the structure, we will require the member end flexibilities 

at either end of each span. For convenience, therefore, we will write the relationships between 

member end flexibility and support flexibility in a more generic form in terms of Span ij: 

(Eqn. A. 30)

(Eqn. A. 31)

Consider two special conditions of known boundary conditions:

a. For a member simple supported at the i end, the flexibility of the support Ki/ij is equal to 

infinity and the member end flexibility reduces to:
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Appendix A – Flexibility Method 237 of 369 

(Eqn. A.32) 

b. When the i end of the member is fixed, the flexibility of the support Ki/ij is equal to zero 

and the member flexibility becomes:

(Eqn. A.33) 

8. Carry Over Factors

In the previous section we developed the member end flexibilities as a function of beam 

characteristics and the support flexibility at the other end of the beam. The top of figure A.12 

shows this modeling for Span ij. The bottom figure shows the moment diagram when there are 

no loads in the span. It is convenient to express the ratio of the moment at the i end of Span ij

to the moment at the j end. For this reason, we define the Carry Over Factor as the ratio of mi

to mj. 

Figure A.12 – Carry Over Factor from j to i

Equating the end rotations at i to those of the restraining side span to the left of i: 

(Eqn. A.34) 

Rearranging and combining like terms:

(Eqn. A.35) 

From this we can define the Carry Over Factor from i to j: 

(Eqn. A.36) 
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Appendix A – Flexibility Method 238 of 369 

Considering Span in the opposite direction, as if it were a member restraining the right end of a 

loaded span, we would find:

(Eqn. A.37) 

Consider two special conditions of known boundary conditions:

a. For a member simple supported at the i end, the flexibility of the support Ki/ij is equal to 

infinity and the Carry Over Factor is equal to zero. This is intuitive, as there is no 

adjacent member connected at i to attract any bending moment.

b. When the both ends of the member are fixed, the flexibility of the supports is equal to 

zero and the Carry Over Factors are:

(Eqns. A.38)

If this span has constant inertia over its length:

(Eqn. A.39) 

9. Continuity Moments Expressed by Carry Over Factors

The Carry Over Factors developed in the previous section can be substituted into the 

relationships for bending moment as a function of beam characteristics and joint support 

flexibilities (equations A.21) to produce the following simplified expressions:

(Eqn. A.40)

For a symmetric span symmetrically loaded, 

(Eqn. A.41) 
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Appendix A – Flexibility Method 239 of 369 

(Eqns. A.42)

For a symmetric span, symmetrically loaded, with symmetrical support flexibilities,

(Eqns. A.43)

(Eqn. A.44) 

For a span hinged at support i, 

(Eqns. A.45)

(Eqn. A.46) 

10. Distribution Coefficients of Moments at Nodes.

In equations 14, 15 and 16 of section 5, we considered the development of the support 

flexibilities when multiple members frame into node i and all work to restrain Span ij. Bending 

moments developed by the restraining members must be distributed between the elements 

which frame together at node i. 

Figure A.13 – Moment Equilibrium at Node i 
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Appendix A – Flexibility Method 240 of 369 

(Eqn. A.47)

(Eqn. A.48)

(Eqn. A.49)

Remembering that stiffnesses are the inverse of flexibility

(Eqn. A.50) 

The distribution of bending moments at nodes is therefore, proportional to the relative stiffness 

of the elements framing at joint i. 

11. Solving the Structure

We have seen that any Span ij is defined by the member characteristics and the Support 

Flexibilities Ki/ij and Kj/ij. The span is first isolated from the structure and analyzed for the 

applied loads to determine continuity moments. The distribution of the continuity moments to

other elements framing into the supports is made in proportion to their relative stiffnesses using 

distribution factors. The moments at the far end of adjacent spans is found through the carryover factors.

For plane frame structures (open systems) the solution begins at a known boundary condition, 

such as and end span whose extremity is free to rotate. The needed coefficients are then 

determined first working left to right from the known boundary condition at the left, and then 

work right to left from the known boundary condition at the right.

In the case of closed systems, such as the cross section of a box girder bridge, the 

determination of the support flexibilities can be made by successive approximations of one 

support and then calculating the others in order. Typically, few analyses around closed 

structure are required to have the assumed value converge to that derived by the analysis. 
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Appendix A – Flexibility Method 241 of 369 

12. Summary of Equations

Support Flexibilities:

(Eqn. A.51) 

(Eqn. A.52) 

Member Flexibilities:

(Eqn. A.53) 

(Eqn. A.54) 

Carry-Over Factors:

(Eqn. A.55) 

(Eqn. A.56) 

End Moments of the Loaded Span:

(Eqn. A.57) 

(Eqn. A.58) 

Distribution Factors:

(Eqn. A.59) 
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Appendix A – Flexibility Method 242 of 369 

13. Example – Flexibility Analysis of a 3 Span Continuous Beam

The three-span example bridge being developed in this Section is shown schematically in figure 

A.14. The cross section of the superstructure of the box girder is shown in figure 5.9 in chapter 

5 of this manual. 

Figure A.14 – Model of 3-Span Bridge for Example 1 

13.1 Span Properties and Characteristic Flexibilities

The cross section properties of the box girder superstructure shown in figure 5.9 in chapter 5 of 

this manual are:

The characteristic flexibilities for the 120’ Span are:

The characteristic flexibilities for the 160’ Span are:
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13.2 Analysis Left to Right

13.3 Analysis Right to Left 

13.4 Carry-Over Factors 

For the Superstructure:

 
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13.5 Effect of a Unit Uniform Load

The beam end rotations when loaded by a downward acting unit uniform load are:

The beam characteristics of the 120’ end spans are:

The beam characteristics of the 160’ center span is:

The moments in Span 1 are:

The end moments in Span 2 are:

The other end moments in Span 2 and 3 are known by symmetry:

The bending moments diagrams for the load in each span and the total of all moments is shown 

in figure 5.11.

3

24 i j

L E E

I

ω ω =− =−

3 120 111.9

24(643.7)

E E ω ω i j =− =− =−

3 160 265.1

24(643.7)

E E ω ω i j =− =− =−

12 M = 0.0 ( ) ( ) 12

21 2

0.2333 111.9 840

.03107

M ft kips b

f ω − =− =− − =− −

( )

( )

32 2 23 3

23

23 32

0.2857 265.1 .2857 (265.1) 1422

1 0.041425 1 .2857 ( 0.2857)

M ft kips b

f ω fω

f f

  − −   − −− =− =−     = − − − −− −    

43 M = 0.0 34 M ft kips =− − 840

23 ( 32 )

3/23

0.041425 0.2857

0.08285 .06214

b by symmetry c K f f =− =− =− = + +

34

4/34

0.03107 0.0

0.06214

b

c K f =− =− = + + ∞ 43

3/34

0.03107 0.2333

0.06214 .07101

b

a K f =− =− =− + +

32 M ft kips =− − 1422
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Figure A.15 – Moment Diagram for a Unit Uniform Load 
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14. Frames Braced Against Side-Sway

The previous analysis considered a three-span beam horizontally restrained by a hinged 

support at the left end of the beam. Now consider a framed structure where the beam in 

integrally connected to supporting columns as shown in figure A.16. The columns, designated 

by the node number of the beam at the top of the column and the letter at the base of the 

column, are fixed at the base.

Figure A.16 – Moment Diagram for a Unit Uniform Load

Figure A.17 shows and individual column, applied pier top forces Q and M, and the resulting 

pier top rotations and deflections. Column characteristic flexibilities are required in order to 

define top of pier lateral and rotational flexibilities.

Figure A.17 – Cantilever Column

The general equations for pier top displacements are:

(Eqns. A.60)

For a moment M applied at the top of the pier:

(Eqns. A.61) 
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For a force Q applied at the top of the pier:

(Eqns. A.62) 

For unit forces and moments the rotations and displacements become the pier flexibilities 

summarized as:

(Eqns. A.63) 

The equations describing the rotation and deflection at the top of the pier are expressed as:

(Eqn. A.64) 

(Eqn. A.65) 

For the case of a framed braced against side-sway, the lateral deflection equals zero:

(Eqn. A.66) 

(Eqn. A.67) 

Substituting equation A.67 into A.64:

(Eqn. A.68) 

Rearranging terms: 

(Eqn. A.69) 

From which we define the pier top rotational flexibility as:

(Eqn. A.70)

We can also compute a carry-over factor from the top to the bottom of the pier by considering 

equilibrium:

(Eqn. A.71)
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Substituting equation A.67 into A.71:

(Eqn. A.72) 

(Eqn. A.73)

For a pier of constant inertia:

(Eqn. A.74) 

With the pier top rotational flexibility and the carry-over factor from the top to the bottom of the 

pier, the framed structure braced against side-sway can be solved.

15. Multiple Pier Elements

The flexibility of the supporting piers can vary with height. Gradual variations, such as linear 

tapers of pier width are accounted for in the development of basic flexibility characteristics 

beginning with equation A.60. Abrupt changes in column properties or where foundation 

elements are modeled with equivalent vertical members, as shown in figure A.18, are treated 

using the following relationships:

(Eqn. A.75) 

(Eqn. A.76) 

(Eqn. A.77) 

Figure A.18 – Column with Multiple Elements
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16. Side-Sway Correction

Figure A.19 shows the lateral displacement for a column rigidly connected to superstructure 

girder.

Figure A.19 – Column in a Rigid Frame

For a unit lateral displacement, the general equations of deflection given in equations A.66 and 

A.67 become:

(Eqn. A.78) 

(Eqn. A.79) 

Where K is the joint flexibility provided by the restraining superstructure:

(Eqn. A.80) 

Solving equation A.78 for Q, substituting the results into equation A.79, and rearranging, the 

moment at the top of the pier from the unit lateral displacement is:

(Eqn. A.81) 

Solving for the shear in the column:

(Eqn. A.82) 

By equilibrium, the moment at the bottom of the column is:

(Eqn. A.83) 

Side-sway corrections are made by restraining shears for the frame braced against side-sway 

and then applying the release of the sway by adding the moments due to side-sway scaled to 

have equity with the base shear.
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17. Example Problem with Side-Sway

The three-span bridge being designed in example 1 is shown schematically in figure A.20. The 

piers consist of two, 6’ diameter columns at supports B and C. Each concrete column is 

supported by a 7’ diameter drilled shaft. The equivalent length of drilled shaft based on the soil 

characteristics at this site has been determined to be 10’. The modulus of elasticity of all 

concrete is assumed constant for this example.

Figure A.20 – Model of 3-Span Bridge for Example 1

17.1 Span Properties and Characteristic Flexibilities

The moment of inertia for the superstructure is:

The characteristic flexibilities for the 120’ Span are:

The characteristic flexibilities for the 160’ Span are:

17.2 Column and Drilled Shaft Characteristic Flexibilities

The gross moment of inertias for the 7’ diameter drilled shafts and the 6’ diameter columns are 

(2 columns at each support):
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The characteristic flexibilities for the 10’ Drilled Shafts are:

The characteristic flexibilities for the 20’ Columns are:

The combined flexibilities for the column/drilled shaft foundations at the top of the rigid 

connection at the center of gravity of the bridge deck are:

The combined pier top flexibilities when sidesway is restrained are:
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17.3 Analysis Left to Right

17.4 Analysis Right to Left 
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17.5 Carry-Over Factors 

For the Superstructure:

For the Substructure:

17.6 Distribution Factors 

Moment at 2 end of member 21:

Moment at 2 end of member 23:
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Moment at 3 end of member 32 (by symmetry):

Moment at 3 end of member 34 (by symmetry):

The distribution of moments at the top of Column 2B to the adjacent spans are:

For a moment at the top of Column 3C (by symmetry):

17.7 End Moments for Unit Uniform Load 

The beam end rotations when loaded by a downward acting unit uniform load are:

The beam characteristics of the 120’ end spans are:

The beam characteristics of the 160’ center span is:
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The moments in Span 1 are:

The end moments in Span 2 are:

The end moments in Span 3 (by Symmetry),

17.8 Distribute End Moments across the Structure 

Carry-Over Factors are applied to compute the moments at the adjacent supports.

Load in Span 1:

Load in Span 2:

 
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Load in Span 3:
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17.9 Effect of a Unit Uniform Load 

Figure A.21 – Moment Diagram for a Unit Uniform Load 
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17.10 Side-Sway Correction 

For a unit lateral displacement of a top of the piers (see appendix A):

From the analysis above:

From which we derive:

And the distribution factors for moments in the piers:

The bending moment and shear force developed at the top of each pier as each pier is 

displaced a unit dimension while the second pier is stationary are:

By equilibrium, the moment at the bottom of the column is (h=33.768’):
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Figure A.22 shows the distribution of the bending moments around the frame for Pier 2B, Pier 

3C, and the sum of the bending moments for a unit side-sway. The base shear created for this 

unit side-sway is:

Side-sway corrections are made by restraining shears for the frame braced against side-sway 

and then applying the release of the sway by adding the moments due to side-sway scaled to 

have equity with the base shear.

Figure A.22 – Moment Diagram for a Unit Lateral Side-Sway 
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Appendix B - Torsion 260 of 369 

Appendix B—Torsion

This Appendix presents information related to the torsional characteristics of box girder bridges. 

A review of torsion in a solid circular bar is first presented to establish basic concepts. This 

information is then extended to the torsional behavior of a thin wall closed shape, specifically, 

the single-cell box girder. Lastly, torsional characteristics of multiple-cell box girders are 

presented.

1. Torsion in a Circular Shaft

The circular shaft is unique with regard to torsion in that the relationship between torsional 

moment and rotation can be determined directly from kinematics, equilibrium, and constitutive 

laws. Consider a circular shaft shown in figure B.1 with length L and outside radius R. Radius r 

denotes the location of point p within the cross section of the bar. The bar is assumed to be 

comprised of a perfectly elastic, homogeneous and isotropic material. The circular bar is 

subjected to an externally applied torsional moment Mt at one end which is resisted by an equal 

and opposite torsional moment at the opposite end. The application and restraint of the 

torsional moments is such that regions of discontinuity in torsional deformations are not created 

at the ends of the bar. The resulting nature of torsional deformation is constant throughout the 

length of the bar.

Figure B.1 – Circular Bar Subjected to Torsional Moment 

Concepts of equilibrium and compatibility of deformations are investigated to determine how a 

series of points within a cross section displace under the action of a torsional moment. Figure 

B.2 presents the circular bar of figure B.1 subdivided into distinct segments of length ∆x. 

Equilibrium requires that when the segments of the beam in figure B.2 are separated, the 

externally applied torsional moment, Mt, must be acting equally and oppositely at either end of 

all segments. 

As the same torsional moment is applied to each segment, each segment will deform in the 

same manner. The kinematics of the individual segments must be such that there is 

compatibility of the shape of radial lines when the segments are reassembled. Figures B.3, B.4, 

and B.5 kinematically consider three different options for the deformations of radial lines. In all 
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cases, the twist on the exterior surface of the bar in these three figures is equal. The plane 

ABCD defines a plane through the center of the bar.

Figure B.2 – Segmented Circular Bar

Figure B.3 shows a non-linear deformation of radial lines in which the rate of displacement 

diminishes with distance from the center of twist. The deformed lines B’C’ and A’D’ in both 

element i and i+1 are consistent for the direction of the applied torsional moments. Inspection 

of the deformed radial lines at the interface of segments i and i+1 reveals that geometric 

compatibility will not allow the reassembly of the two segments when a differential rotation is 

applied to align point E’ to B’ and point H’ to C’.

Figure B.3 – Kinematic Study 1

Figure B.4 shows a second form of non-linear deformation of radial lines in which the rate of 

displacement increases with distance from the center of twist. As with the previous case, the 

radial lines at the interface of segments i and i+1 cannot be re-joined while respecting geometric 

compatibility.
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Figure B.4 – Kinematic Study 2

Figure B.5 presents a third alternative of the deformed shape of radial lines within cross 

sections of the circular bar. In this alternative, the rate of displacement is constant across the 

diameter of the bar. The resulting shape B’C’ and A’D’ is linear. A rigid body rotation of the i+1 

segment allows it to be rejoined to the ith segment.

Figure B.5 – Kinematic Study 3

The linear displacement of radial lines within the cross sections of a circular bar has important 

implications. Figure B.6 shows a circular bar subjected to a torsional moment. The figure 

shows a radial plane abcd and the deformed plane a’b’c’d’. The maximum rotation at the end of 

the beam is denoted f. The previous kinematic study indicates that f varies linearly along the 

length of the bar. A rate of change of twist, θ, can be defined as:

(Eqn. B.1) 
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Figure B.6 – Linear Twisting of the Circular Bar 

Shear strain, γ, is also shown in figure B.6. It is defined as the circumferential displacement of a 

point in a cross section divided by the length of the bar to the cross section under consideration. 

The maximum shear strain, on the outer surface of the bar, at the end of the bar is:

(Eqn. B.2) 

The linear displacement radial lines of the cross section coupled with the assumed material 

characteristics imply a linear distribution of shear stress within the cross section of the bar as 

shown in figure B.7. Equilibrium within the bar requires that, for small rotations, longitudinal 

shear stresses in the bar are also present. Shear stress on an element of the exterior surface of 

the bar is shown in figure B.7. Shear stresses are zero along the center of twist and maximum 

at the exterior radius of the bar.

Figure B.7 –Shear Stresses and Shear Strains in the Circular Bar 
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Continuing with the development of relationships between applied torsional moment and 

resulting rotations, consider a distinct element of length dx of the bar shown in figure B.8.

Figure B.8 – Element of the Circular Bar

Previously defined equations can be rewritten for the element of the bar shown in figure B.8. 

The rate of twist becomes:

(Eqn. B.3) 

And the strain is defined as:

(Eqn. B.4) 

It can also be seen in figure B.8 that the displacement ds over the length dx is equal to:

(Eqn. B. 5) 

Substituting equation B.5 into equation B.4,

(Eqn. B.6)

Substituting equation B.3 into equation B.6,

(Eqn. B.7)

This equation says that the shear strain at any distance from the center of twist is the rate of 

twist multiplied by the radial distance, with a maximum shear strain at the outer surface of the 

circular bar.

Considering now the linear elastic behavior of the bar, the shear stress shown in figure B.7 is 

related to strain by Hooke’s Law:

(Eqn. B. 8) 

Substituting the expression for shear strain in equation B.7 into equation B.8 we find:

(Eqn. B. 9) 
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Consider the equilibrium between the externally applied torsional moment and the resisting 

shear stress in the plane of a cross section within the bar. As shown in figure B.8, call the shear 

stress acting over a small element of surface area dA a force dV. The magnitude of the force is:

(Eqn. B. 10)

The element of torsional moment caused by the force dV is

(Eqn. B. 11) 

Integrating the incremental torsional moments over the cross sectional area yields

(Eqn. B. 12) 

Or,

(Eqn. B.13) 

The integral represents a cross section characteristic known as the torsional constant J: 

(Eqn. B. 14)

For the unique case of the circular bar, the torsional constant equals the polar moment of 

intertia:

(Eqn. B.15)

For a solid circular shaft the polar moment of inertia is:

(Eqn. B.16) 

So, we relate the torsional moment to rotation as follows:

(Eqn. B.17) 

(Eqn. B.18)

(Eqn. B.19)

The shear stress becomes:

(Eqn. B.20)
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The relationships between torsional moment, rotation, and shear stress are consistent with 

those found by a Saint Venant Semi-Inverse solution incorporating Prandtl’s Stress Function to 

more easily address boundary conditions. The fundamental assumptions of Saint Venant 

Torsion are:

• Cross sections of the bar are assumed to rotate as rigid bodies.

• Warping of the cross sections is not restricted, but warping at any point in the cross 

section is constant along the length of the bar.

2. Torsion in a Closed Thin-Walled Shaft

The approach taken for developing torsional characteristics of the circular bar are extended to a 

hollow closed shaft as shown in figure B.9. The length of the shaft is L and the thickness of the 

wall of the shaft is t. The derivation of the torsional constant for the thin-walled shaft is based on 

Saint Venant shear and the assumption that the wall thickness is small compared to the overall 

dimensions of the cross section.

Figure B.9 –Closed Thin-Wall Shaft 

The torsional moment applied to the closed thin-wall shaft will produce shear stress around 

shaft’s the cross section. Figure B.10 shows an element of the shaft shown in figure B.9 with 

length dx. It is convenient to define a system of coordinates where the local x axis is oriented in 

the direction of the global X axis, the s axis is tangential to the centerline of the wall at any 

location within the cross section, and the n axis is normal to the local tangent. The shear 

stresses produced by the torsional moment on the cross section are then labeled τxs. 

It is also convenient, for reasons about to be seen, to introduce the concept of shear flow, q. 

The shear flow is an expression of the shear stress integrated over the thickness of the wall.

(Eqn. B.21) 

Assuming a linear variation in shear stress across the wall thickness, the expression for shear 

flow becomes:
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Appendix B - Torsion 267 of 369 

(Eqn. B.22)

Where the shear stresses in equation B.22 are the shear stresses at the inner face and outer 

face of the wall. The assumption of a thin wall simplifies this equation to an expression where 

the shear stress is assumed constant over the thickness of the wall:

(Eqn. B.23)

In each of these expressions the wall thickness t is assumed to be variable along the s axis.

Figure B.10 includes a segment of the hollow cross section, of width dx and length ds, subjected 

to the incremental forces of shear flow. Tangential shear flow is labeled qxs. Equilibrium of the 

segment requires the presence of incremental shear flow forces in the longitudinal direction, 

labeled qsx. As there is no net longitudinal force applied to the shaft, equilibrium of the segment 

in the longitudinal direction also requires that the incremental shear flow forces qsx must be 

equal and opposite. Satisfaction of these two conditions of equilibrium means, that while shear 

stresses vary as a function of wall thickness at the point where shear stresses are being 

considered, shear flow is constant around the hollow thin-walled cross section. 

Figure B.10 – Segment of Closed, Thin-Wall Shaft 

The shear flow around the cross section must be in equilibrium with the applied torsional 

moment. Figure B.11 shows an end view of the cross section of the hollow thin-walled shaft 

with an applied torsional moment Mt. 
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Figure B.11 – Equilibrium in the Cross Section of the Thin Wall Closed Shaft 

The tangential force caused by the shear flow over ds is:

(Eqn. B. 24) 

The torsional moment caused by this force is:

(Eqn. B. 25)

The dimension h is the perpendicular distance from the line of action of dV to the center of 

rotation. Define a differential area dA as the triangle enclosed by rays from the center of 

rotation to the beginning and end of element ds. The area of the triangle is:

(Eqn. B.26) 

Solving equations B.24 and B.25 for (h)ds and setting them equal to each other produces:

(Eqn. B.27) 

Or,

(Eqn. B. 28) 

The torsional moment is found by integrating this equation around the median line of the wall 

thickness:

(Eqn. B.29) 2 () M q dA q h ds t = =   ∫ ∫
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(Eqn. B. 30)

Where Ao is the area bounded by the median line of the wall thickness. This expression can be 

written into two other convenient forms:

(Eqn. B.31)

(Eqn. B.32) 

Consider now the kinematics of an element on the wall of the thin tube. Figure B.10 also 

includes a sketch of the deformation of an element of the hollow thin-walled shaft. The torsional 

load leads to displacements of the element in the x direction (u) and s direction (v). The shear 

strain in this element is defined as:

(Eqn. B.33)

Where,

(Eqn. B.34)

Applying Hooke’s Law, shear stress divided by the shear modulus can be substituted for strain:

(Eqn. B.35)

This equation can be rearranged to solve for the axial displacement du:

(Eqn. B.36) 

As there is no longitudinally applied load, equilibrium with the applied external forces requires 

that the sum of all longitudinal displacements must be equal to zero, or:

(Eqn. B.37) 

Equation B.36 can be substituted into equation B.37 to give:

(Eqn. B.38) 

Or, 
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For small angles, the tangential displacement dv is also equal to the differential twist multiplied 

by the perpendicular dimension from the center of twist, h:

(Eqn. B. 39)

Inserting equation B.39 into B.38 and rearranging:

(Eqn. B.40) 

Substituting the rate of twist as defined in equation B.3 into this expression yields:

(Eqn. B.41) 

Twice the area enclosed by the centerline of the wall thickness can be substituted for the 

integral on the right side of this expression to produce:

(Eqn. B.42) 

The shear stress in the integral on the left side of this expression can be replaced with the shear 

flow divided by the wall thickness. Multiplying both sides of the resulting equation by the shear 

modules produces:

(Eqn. B.43) 

This equation can be rearranged to produce a relationship between the rate of twist and shear 

flow:

(Eqn. B.44) 

The work developed in this and the previous section resulted in two expressions for the torsional 

moment. equation B.17 and B.30 state:

(Eqns. B.17 and B.30) 

Equating these two equations and solving for the rate of twist produces:

(Eqn. B. 45)

Now equations B.44 and B.45 can be set equal to each other:
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(Eqn. B.46) 

And finally, the torsional constant for the hollow thin-walled shaft can be determined:

(Eqn. B.47)

For concrete box girder sections, the closed loop integral in the denominator of equation B.47 is 

often more conveniently expressed as a discrete summation. The equation for the torsion 

constant then becomes:

(Eqn. B.48) 

Where, si is the length, and ti the thickness of, each of the n individual elements comprising the 

closed cross section.

Example 1: Consider the cross section of a single-cell box girder used in chapter 8, 

Transverse Analysis. The dimensions of the cross section are shown in figure B.12. 

Figure B.13 shows the idealized thin wall members of the box along with their lengths 

and average thicknesses.

Figure B.12 –Example Single Cell Box Girder
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Figure B.13 – Idealized Thin-Walled Members

The coordinates of points 1 through 8 are shown in table B.1. The dimensions of the 

thin wall members and a computation of equation B.48 denominator is shown in table 

B.2.

Point x y 

1 4.75 8.4583

2 10.6326 7.9598

3 7.7545 0.7646

4 4.3306 0.375

5 -4.3306 0.375

6 -7.7545 0.7646

7 -10.6326 7.9598

8 -4.75 8.4583

Table B.1 – Coordinates of Points Defining the Thin-Walled Section 

Element S (ft) t (ft) s/t 

8-1 9.5 0.75 12.667

1-2 5.904 1.245 4.742

2-3 7.749 1.333 5.813

3-4 3.446 1.606 2.146

4-5 8.661 0.75 11.548

5-6 3.446 1.606 2.146

6-7 7.749 1.333 5.813

7-8 5.904 1.245 4.742

Sum 49.617

Table B.2 – Thin-Walled Section Member Dimensions 

The results for the area enclosed by the mid-line of the thin-walled section and the 

summation of member lengths divided by their average thicknesses are:
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Note that the cantilever wings do contribute to the torsional constant for the cross section shown 

in figure B.13. This value of this contribution is, however, small with regard to the contribution of 

the closed cell, and is typically not considered. In an instance where it was considered, thin 

open appendages to a closed cell would add torsional constant with the equation for a thin cross 

section which is:

(Eqn. B.49) 

The total torsional constant would then be:

(Eqn. B.50)

Where m is the number of open appended thin wall sections with depth bk and thickness tk. For 

the example problem above, the average thickness of the cantilever wings is 1.035’ and the 

length of the cantilever is 10.952’. The resulting contribution of the torsional constant is:

This value represents approximately 0.5 percent of the torsional constant of the closed section.

3. Torsion Multi-Cell Box Girder

Torsional moments applied to box girder cross sections comprised of multiple closed cells will 

be resisted by the sum of shear flows around the individual cells. Compatibility of deformation 

of the cross section requires that all cells will rotate the same amount when loaded in torsion. 

This is based on the assumption that the width of the cross section is small with regard to the 

span length. Care should be taken when the ratio of span length to box girder width is less than 

four. In wide bridges, torsional moments tend are often resisted by differential bending in the 

outer webs.
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Figure B.14 shows a typical two-cell box girder superstructure cross section. The applied 

torsional moment, Mt, is resisted by the sum of torsional moments in each of the cells. The 

moments in the cells produce shear flows q1 in Cell 1 and q2 in Cell 2.

Figure B.14 – Two-Cell Box Girder Superstructure

In the previous section the relationship between applied torsional moment and shear flow in a 

single-cell closed shape was presented in equation 30:

(Eqn. B.30) 

For the case of a two-cell structure, this relationship becomes:

(Eqn. B.51) 

Or, for any number of cells:

(Eqn. B.52)

Equation B.44, repeated below, presents an expression for the rate of twist as a function of the 

member dimensions:

(Eqn. B.44) 

 

As the shear flows from adjacent cells will impact the shear flow of a particular cell in members 

that are shared in common, a statically indeterminate solution is required to solve for the

individual shear flows.
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The rotations of the two cells are expressed as the net shear flows:

(Eqn. B.53) 

(Eqn. B.54) 

Where, the open integral portions in the equations are the shear flow in the common members 

multiplied by the lengths of the adjoining sections divided by their thicknesses.

Compatibility of deformations requires that:

(Eqn. B. 55)

This allows us to rewrite equations B.53 and B.54 as:

(Eqn. B. 56)

(Eqn. B. 57) 

Where the shape constants are defined as,

(Eqn. B.58) 

 

(Eqn. B.59) 

And,

(Eqn. B.60) 

(Eqn. B.61) 

The shear flows, q1 and q2, can be solved in terms of the constant Gθ. The torsional constant 

can then be found remembering the relationship between torsional moment and rate of twist 

presented in equation B.17 and repeated here:

(Eqn. B. 17) 
Post-Tensioned Box Girder Design Manual June 2016

Appendix B - Torsion 276 of 369 

Example 2: Determine the torsional constant, J, for the cross section of the two-cell box 

girder shown in figure B.15.

Figure B.15 –Example Two-Cell Box Girder

Evaluating the shape constants:

Equations B.56 and B.57 become
Post-Tensioned Box Girder Design Manual June 2016

Appendix B - Torsion 277 of 369 

From which the shear flows are solved in terms of Gθ:

Considering torsional equilibrium:

Or,

From which the torsional constant is found:

The symmetry of this two-cell box girder cross section would, by inspection, would lead 

to the conclusion that there is no net shear flow in the center web. The result is that the 

torsional constant for this special condition would be the same as if the center web did 

not exist. Computing the torsional constant from equation B.47 verifies this condition. 

Example 3: Consider the cross section of a four-cell box girder shown in figure B.16. 

Figure B.16 also shows the idealized thin wall members of the box along with their 

lengths and average thicknesses.

The system of simultaneous equations that needs to be solved for the four-cell box 

girder is:
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Appendix B - Torsion 278 of 369 

Figure B.16 –Example Four-Cell Box Girder

These equations are modified to discount the influence of cells that are not adjoining:

From the cross section geometry, and noting symmetry:
Post-Tensioned Box Girder Design Manual June 2016

Appendix B - Torsion 279 of 369 

Based on the symmetry of the section, the system of four simultaneous equations 

reduces to two:

Or,

From which,

Considering equilibrium:

Comparing this to the torsional characteristic of the outside perimeter of the bridge only:

This represents a 0.6 percent increase in torsional characteristic.
Post-Tensioned Box Girder Design Manual June 2016

Appendix C – Design Example 1 280 of 369 

Example 1—Multi-Cell Box Girder Bridge

1. Introduction

This example demonstrates the use of the AASHTO LRFD Design 

Specifications in the design of a cast-in-place, post-tensioned concrete, multicell box girder bridge. The design of the supporting substructure is not included 

in this manual.

The bridge considered in this example is shown in figure C.1. The length of the 

three-span bridge from the centerline of the bearings at the abutments is 400’, 

comprised of 120’ side spans and a 160’ center span. The bridge is straight, 

following a tangent alignment, and has no skew. In elevation, the bridge is 

symmetrical in all ways about the centerline of the center span.

Figure C.1 – Elevation of Example 1 Bridge 

Figure C.2 shows a transverse cross section taken through the bridge. The 4-

cell box girder bridge carries three 12’ wide traffic lanes, two 10’ shoulders, and 

two 1’-9” barriers, for a total out-to-out width of 59’-6”. 

Figure C.2 – Cross Section through Example 1 Bridge
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Appendix C – Design Example 1 281 of 369 

Pier 2 and Pier 3 are each comprised of two cylindrical columns with a 

diameter of 6’-0”. The columns are cast monolithic with the superstructure. 

The columns are 20’ in height, measured from the top of foundations to the 

base of the superstructure. The foundations consist of 7’-0” diameter monoshafts supporting each of the columns. For this example, it is assumed that the 

supporting soil provides an equivalent point of fixity 10’ below the top of the 

mono-shafts for all force effects. The distance from the base of the 

superstructure to the point of fixity is 30’.

Summary of Bridge Geometry:

Bridge length between abutment bearings = 400’

Span arrangement = 120’-160’-120’ 

Roadway Width (3-12’ lanes, 2-10’ shoulders) = 56’-0”

Width of Barrier Railing = 1’-9”

Overall Bridge Width = 59’-6”
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Appendix C – Design Example 1 282 of 369 

2. Material Characteristics

2.1. Concrete

The initial and 28-day concrete strengths for the superstructure and 

substructure are:

The unit weight of the concrete is defined below. Note that the unit weight 

used for the self-weight calculations includes an additional weight for the 

reinforcement.

The modulus of elasticity for normal weight concrete, where wc = 0.145 kcf, can 

be taken as:

2.2 Reinforcing Steel

The yield strength and modulus of elasticity are:

2.3 Prestressing Steel

The 0.6 inch diameter low-relaxation prestressing strand is used in this 

example with the following properties:

[3.5.1]

[C5.4.2.4-1] 

[5.4.3] 

[5.4.4] 
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Appendix C – Design Example 1 283 of 369

3. Cross Section Properties

The typical cross section dimensions are shown in figure C.3. The cross 

section dimensions are constant over the length of the bridge:

Figure C.3 – Cross Section Dimensions 

The properties of the typical superstructure cross section are:

A = cross sectional area = 99.45 ft2

Ix = moment of inertia = 643.75 ft4

H = height of section = 6.5 ft

c1 = top of section to centroid = 2.732 ft

c2 = bottom of section to centroid = 3.768 ft

The gross section properties of the substructure elements are used in this 

design example:

Ac = area for two columns = 56.55 ft2

Ixc = moment of inertia for two columns = 127.24 ft4

Ap = area for two monoshafts = 76.97 ft2

Ixp = moment of inertia for two monoshafts = 235.72 ft4
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Appendix C – Design Example 1 284 of 369

∑ηγ f ii i n r Q RR ≤ =

0.95

1 1.0

i DRI

i

DRI

η ηηη

η

ηηη

= ≥

= ≤

1.0

1.0

DRI

i

ηηη

η

= = =

∴ =

4. LRFD General Design Equation

The AASHTO LRFD Design Specification’s general design equation is:

where,

ηi = load modifying factor for ductility, redundancy, and operational 

importance

γ i = load factor

Qi = force effect

f = resistance factor

Rn = nominal resistance

Rr = factored resistance

The load modifier, ηi, takes on two forms. The first is for loads in which a 

maximum value of γ i is appropriate, the second is in which a minimum value of 

γ i is appropriate.

where,

η D = ductility factor

ηR = redundancy factor

ηI = operational importance factor

Since the purpose of this example is to illustrate a conventional design on a 

typical bridge,

Note that the load modifying factors are project and design specific and may 

not equal 1.0 in actual designs.

[1.3.2.1-1] 

[1.3.2.1-2], 

[1.3.2.1-3]

[1.3.3], 

[1.3.4],

[1.3.5]
Post-Tensioned Box Girder Design Manual June 2016

Appendix C – Design Example 1 285 of 369

5. Live Load Distribution Factors

The AASHTO LRFD Design Specifications provide live load distribution tables 

that can be used on structures that meet the requirements of section 4.6.2.2. 

There are two methods of applying the live load distribution factors to 

cast-in-place multi-cell box girders. The first method is to apply the distribution 

factors to each web as an individual girder. According to the AASHTO LRFD 

specifications, the “notional shape” of the web and the tributary areas of the top 

and bottom slabs shall be taken to form a girder. This method would have two 

designs, one for the interior girders and one for the exterior girders.

The second method is the “whole-width” method. A cast-in-place multi-cell box 

girder bridge may be designed as a whole-width structure using the live load 

distribution factors for interior girders multiplied by the number of webs in the 

cross section. This is the method chosen for this design example.

Note that in the development of the distribution factor equations the multiple 

presence factors were considered and included.

Live Load Distribution Factor variables:

L = length of span (ft)*

DF = distribution factor

NL = number of lanes

Nw = number of webs = 5 

Nc = number of cells = 4 

S = web spacing (ft) = 12.25 ft

d = depth of beam (in) = 78 in

* For positive moment, negative moment, and shear, L is the length of the span

for the section considered. For negative moment between points of

contraflexure under uniform loading, L is the average length of two adjacent

spans.

Distribution of live loads for moment in interior beams:

– One design lane loaded

0.45 0.35 1 1 1.75

3.6 c

S DF

L N

    

= +          

0.35 0.45

 2 

12.25 1 1 1.75 0.467

3.6 160 4

DFSpan M ±

     = + =          

0.35 0.45

 1 & 3 

12.25 1 1 1.75 0.517

3.6 120 4

DFSpans M ±

     = + =          

0.35 0.45

 2 & 3 

12.25 1 1 1.75 0.490

3.6 140 4

DFPiers M −

     = + =          

[C3.6.1.1.2] 

[4.6.2.2] 

[c4.6.2.2.1-1] 

[4.6.2.2.2b-1] 
Post-Tensioned Box Girder Design Manual June 2016

Appendix C – Design Example 1 286 of 369

– Two or more design lanes loaded (note these factors control over one lane)

Number of lanes for whole width section

Distribution of live loads for shear in interior beams:

– One design lane loaded

– Two or more design lanes loaded (note these factors control over one lane)

[4.6.2.2.3a-1] 

0.3 0.25 13 1

5.8 c

S DF

N L

     =          

0.3 0.25

 1 & 3 

13 12.25 1 0.909

4 5.8 120

DFSpans M ±

     = =          

0.3 0.25

 2 & 3 

13 12.25 1 0.874

4 5.8 140

DFPiers M −

     = =          

0.3 0.25

 2 

13 12.25 1 0.846

4 5.8 160

DFSpan M ±

     = =          

N N DF L w = ⋅

 1 & 3 5 0.909 4.544 NL Spans M ± =⋅ =

 2 & 3 5 0.874 4.372 NL Piers M − =⋅ =

 2 5 0.846 4.229 NL Span M ± =⋅ =

0.6 0.1

9.5 12

S d DF

L

   =      

0.6 0.1

 1 & 3

12.25 78 0.870

9.5 12 120

DFSpans

   = =       ⋅

0.6 0.1

 2

12.25 78 0.846

9.5 12 160

DFSpan

   = =       ⋅

0.9 0.1

7.3 12

S d DF

L

   =      

0.9 0.1

 1 & 3

12.25 78 1.190

7.3 12 120

DFSpans

   = =       ⋅

0.9 0.1

 2

12.25 78 1.157

7.3 12 160

DFSpan

   = =       ⋅
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Appendix C – Design Example 1 287 of 369

Number of lanes for whole width section

N N DF L w = ⋅

 1 & 3 5 1.190 5.952 NL Spans =⋅ =

 2 5 1.157 5.783 NL Span =⋅ =
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Appendix C – Design Example 1 288 of 369

6. Modeling, Analysis and Results

Figure C.4 – Self Weight and Component Bending Moments 

Figure C.5 – Future Wearing Surface Bending Moments 
Post-Tensioned Box Girder Design Manual June 2016

Appendix C – Design Example 1 289 of 369

Figure C.6 – Concrete Creep Bending Moments 

Figure C.7 – Concrete Shrinkage Bending Moments 
Post-Tensioned Box Girder Design Manual June 2016

Appendix C – Design Example 1 290 of 369

Figure C.8 – Live Load Envelope Bending Moments 

Figure C.9 – Initial Post-Tensioning Bending Moments
Post-Tensioned Box Girder Design Manual June 2016

Appendix C – Design Example 1 291 of 369

Figure C.10 – Bending Moments for Post-Tensioning Losses

Figure C.11 –Final Post-Tensioning Moments 
Post-Tensioned Box Girder Design Manual June 2016

Appendix C – Design Example 1 292 of 369

7. Flexural Design

There are two flexural designs required by the AASHTO LRFD specifications 

for prestressed concrete: flexural designs at the Service Limit State and at the 

Strength Limit State. 

7.1 Service Limit State 

There are two load combinations for the Service Limit State: Service I and 

Service III. The Service I load combination is used to check against maximum 

allowable compression in the superstructure and Service III is used to check 

against maximum allowable tension. 

Load Combination 

Limit State DC DW CR+SH PS TU TG LL+I

Service I 1.0 1.0 1.0 1.0 1.0 0.5 1.0

Service III 1.0 1.0 1.0 1.0 1.0 0.5 0.8

Table C.1 – Service Limit State Load Factors

The maximum allowable compressive stresses are:

• 0.6 f’ci for temporary stresses at the time of tendon stressing.

• 0.45 f’c due to prestressing and permanent loads after all time dependent

effects and losses.

• 0.6 f’c due to prestressing, permanent, and transient loads after all time

dependent effects and losses.

The maximum allowable tensile stresses are:

• 0.24√ f’ci (ksi) for temporary stresses at the time of tendon stressing

(7.5√ f’ci for units in psi).

• 0.019√ f’c (ksi) for stresses after all time dependent effects and losses

(6√ f’c for units in psi).

Temporary initial stress check:

The controlling point for compressive stresses is located in the bottom of the 

section at the down-station face of the diaphragm at Pier 2 (node 249).

[3.4.1] 

[3.4.1-1] 

[5.9.4] 
Post-Tensioned Box Girder Design Manual June 2016

Appendix C – Design Example 1 293 of 369 

Final compression stress check, permanent loads:

The controlling point is located in the bottom of the section at the downstation 

face of the diaphragm at Pier 2 (node 249).

Final compression stress check, all loads:

The controlling point is located in the bottom of the section at the downstation 

face of the diaphragm at Pier 2 (node 249). Use Service I load combination.

Final tensile stress check, all loads:

The controlling point is located in the top of the section at the midspan of 

Span 2 (node 266). Use Service III load combination

The table below contains a summary of the stress checks every ten feet along 

the superstructure.
Post-Tensioned Box Girder Design Manual June 2016

Appendix C – Design Example 1 294 of 369

> -0.6 f'ci > -0.45 f'c

Service I > -0.6 

f'c

Service III < 6 √f'c

Node Distance Bottom Top Bottom Top Bottom Top Bottom Top

Min Min Max Max

(ft) (ksf) (ksf) (ksf) (ksf) (ksf) (ksf) (ksf) (ksf) 

150 0 

151 2.5 -146.4 -87.0 -130.4 -79.0 -137.8 -138.7 -122.0 -62.0

152 10 -138.0 -94.2 -119.1 -88.3 -128.3 -165.8 -93.3 -68.1

154 20 -130.4 -101.3 -109.0 -97.0 -120.9 -193.1 -64.7 -73.1

156 30 -126.5 -105.8 -103.9 -102.4 -119.9 -212.5 -45.2 -74.4

158 40 -126.1 -107.6 -103.7 -103.8 -125.5 -223.2 -33.5 -72.0

160 50 -129.1 -107.1 -108.2 -102.1 -136.3 -226.9 -30.9 -66.2

162 60 -133.3 -105.7 -115.6 -98.1 -149.6 -224.9 -35.3 -58.1

240 70 -136.0 -105.6 -123.4 -94.3 -163.3 -219.2 -44.1 -50.5

242 80 -135.0 -105.8 -129.5 -89.4 -175.6 -209.2 -54.6 -41.5

244 90 -131.6 -106.3 -135.3 -83.3 -187.3 -194.9 -68.3 -31.7

246 100 -127.7 -107.2 -142.4 -76.5 -202.9 -180.0 -83.6 -19.2

248 110 -129.9 -103.3 -156.8 -63.9 -233.7 -161.9 -102.9 3.4

249 117 -156.9 -81.5 -190.7 -37.2 -288.1 -135.6 -135.7 42.1

250 120

251 123 -134.4 -91.2 -137.4 -63.4 -237.9 -150.8 -89.7 7.1

252 130 -99.1 -114.3 -96.6 -90.7 -177.1 -172.3 -55.2 -31.8

254 140 -82.9 -122.8 -70.6 -106.6 -129.2 -195.2 -21.5 -60.4

256 150 -85.7 -119.2 -63.4 -110.4 -112.7 -207.9 -4.5 -69.6

258 160 -87.1 -116.7 -56.6 -113.8 -99.9 -220.4 12.4 -76.4

260 170 -87.4 -115.0 -50.6 -116.8 -90.7 -231.7 27.5 -81.3

262 180 -86.7 -114.1 -45.5 -119.2 -83.3 -240.6 39.8 -85.0

264 190 -85.1 -113.7 -41.2 -120.8 -77.2 -246.3 48.6 -87.7

266 200 -83.2 -113.6 -38.5 -121.3 -72.8 -248.0 52.6 -89.2

336 210 -85.1 -113.7 -41.2 -120.8 -77.2 -246.3 48.6 -87.7

338 220 -86.7 -114.1 -45.5 -119.2 -83.3 -240.6 39.8 -85.0

340 230 -87.4 -115.0 -50.6 -116.7 -90.7 -231.6 27.5 -81.2

342 240 -87.1 -116.7 -56.6 -113.8 -99.9 -220.4 12.4 -76.4

344 250 -85.7 -119.2 -63.4 -110.4 -112.7 -207.9 -4.5 -69.6

346 260 -82.9 -122.8 -70.5 -106.6 -129.1 -195.2 -21.4 -60.4

348 270 -99.1 -114.3 -96.5 -90.7 -177.0 -172.3 -55.1 -31.8

349 277 -134.4 -91.2 -137.4 -63.4 -237.9 -150.8 -89.7 7.1

350 280

351 283 -156.9 -81.5 -190.7 -37.2 -288.1 -135.6 -135.7 42.1

352 290 -129.9 -103.3 -156.7 -63.9 -233.6 -161.9 -102.8 3.4

354 300 -127.7 -107.2 -142.3 -76.5 -202.8 -180.0 -83.5 -19.2

356 310 -131.6 -106.3 -135.3 -83.3 -187.3 -194.9 -68.3 -31.7

358 320 -135.0 -105.8 -129.5 -89.4 -175.6 -209.2 -54.6 -41.5

360 330 -136.0 -105.6 -123.4 -94.2 -163.3 -219.1 -44.1 -50.4

362 340 -133.3 -105.7 -115.7 -98.1 -149.7 -224.9 -35.4 -58.1

440 350 -129.1 -107.1 -108.3 -102.1 -136.4 -226.9 -31.0 -66.2

442 360 -126.1 -107.6 -103.7 -103.7 -125.5 -223.1 -33.5 -71.9

444 370 -126.5 -105.8 -104.0 -102.3 -120.0 -212.4 -45.3 -74.3

446 380 -130.4 -101.3 -109.1 -96.9 -121.0 -193.0 -64.8 -73.0

448 390 -138.0 -94.2 -119.3 -88.1 -128.5 -165.6 -93.5 -67.9

449 397.5 -146.4 -87.0 -130.6 -78.8 -138.0 -138.5 -122.2 -61.8

450 400

Table C.2 – Service Limit State Flexural Verifications
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Appendix C – Design Example 1 295 of 369

[3.4.1] 

[3.4.1-1] 

[1.3.2.1-1], 

[5.7.3.2],

[5.7.3.3.2]

[5.7.3.3.2]

7.2 Strength Limit State

There is one basic load combination used to check the Strength Limit State for 

cast-in-place segmental superstructure: Strength I.

Table C.3 – Strength Limit State Load Factors 

Where, PS represents the secondary moment caused by post-tensioning (note, 

it does not include the primary moment due to post-tensioning)

The factored flexural resistance, Mr, must be greater than the Strength I load 

combination, Mu (ultimate moment), and the minimum reinforcement 

requirement. The minimum reinforcement requirement states that the flexural 

resistance must at least equal to 1.33Mu or the cracking moment, Mcr, 

whichever is less. This can be summarized by the following equation:

7.2.1 Factored Design Moment

The location with the largest demand to capacity ratio, d/c, is the negative 

moment at the downstation face of the diaphragm at Pier 2 (node 249). The 

summation of the Strength I load combination is: 

Minimum reinforcement calculation

Load Combination 

Limit State DC DW CR+SH PS TU TG LL+I

Strength I 1.25 1.5 1.0 1.0 0.5 0.0 1.75
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Appendix C – Design Example 1 296 of 369

[5.7.3.2.2-1]

[5.7.3.2.2]

[5.7.3.1.1-3]

[C5.7.3.1.1-1]

where,

The absolute value of the cracking moment is less than the absolute value of 

1.33Mu, and this value is greater than the absolute value of the factored 

moment, Mu. Thus, the design moment is Mcr. 

7.2.2 Factored Flexural Resistance

The nominal flexural resistance for flanged sections: 

For rectangular sections, bw = b making the second part of the above equation 

drop to zero. Note that the equation is modified from AASHTO LRFD 

neglecting the contributions from mild reinforcement.

The nominal moment calculated at the downstation face of the diaphragm at 

Pier 2 (node 249). The following can be obtained from the section dimensions:

where,

2

3

1.1

7.5 5000 530 76.4

235.6

r

c

f psi psi ksf

S ft

γ =

= = =

=

1

3

1.6

1.0

174.8 cpe f ksf

γ

γ

=

=

=

74099 → =− − M ft kips cr

0.85 ( ) 2 2 2

f

n ps ps p c w f

a a h

M Af d f bb h    

= −+ − − ′        

( )

1

0.85

0.85

ps pu c w f

pu

c w ps

p

Af f bb h

c f f b kA d β

− − ′ =

′ +

2 2 5 3 19 0.217 61.845

270

5

44.5 534

5.2361 62.833

7

0.28

5.03625 60.435

ps

pu

c

w

f

p

A in in

f ksi

f ksi

b ft in

b ft in

h in

k

d ft in

=⋅⋅ ⋅ =

=

′ =

= =

= =

=

=

= =

1 a c = β
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Appendix C – Design Example 1 297 of 369

[5.7.2.2]

[5.7.3.1]

[5.5.4.2]

Solve for c and determine if the section is rectangular or T-section.

If the value of a ≤ hf, then the section is rectangular, otherwise it is considered a 

T-section.

Therefore, this is a T-section.

The average stress in prestressing steel:

The resistance (reduction) factor:

Calculate the flexural resistance, fMn, for negative bending.
Post-Tensioned Box Girder Design Manual June 2016

Appendix C – Design Example 1 298 of 369

Therefore, the flexural resistance is greater than the flexural demand:

The demand to capacity ratio, d/c: 

The above calculations show that the down station face of the diaphragm at 

Pier 2 (node 249) is adequately designed for the flexural demand at that 

location.

The table below contains a summary of the flexural strength design checks at 

typical critical locations along the superstructure.

fM M n cr ≥

75814 74099 ft kips ft kips −> −

d c ≤1.0

74099 0.977 1.0

75814

ft kips

ft kips

− − = < − −
Post-Tensioned Box Girder Design Manual June 2016

Appendix C – Design Example 1 299 of 369

Node 160 249 251 266 349 351 440

Location ~ 0.4 L Pier 2 0.5 L Pier 3 ~ 0.6 L

Distance (ft) 50 117 123 200 277 283 350

DC (ft-kips) 15125 -26585 -29178 17761 -29178 -26585 15125

DW (ft-kips) 1339 -2353 -2583 1572 -2583 -2353 1339

CR (ft-kips) 1161 2717 -1543 -1543 -1543 2717 1161

SH (ft-kips) -2642 -6181 3618 3618 3618 -6181 -2642

PS (ft-kips) 1964 4594 9834 9799 9836 4595 1966

TU

Rise (ft-kips) 2165 5067 -2622 -2622 -2622 5067 2165

Fall (ft-kips) -2165 -5067 2622 2622 2622 -5067 -2165

LL+I

Min (ft-kips) -2264 -11348 -13778 -1530 -13778 -11348 -2264

Max (ft-kips) 12306 14 2438 11795 2438 14 12306

Mu

Min (ft-kips) 16354 -58024 -51237 32445 -51234 -58022 16355

Max (ft-kips) 44017 -33074 -25482 58387 -25479 -33072 44018

1.33Mu

Min (ft-kips) -77171 -68145 -68141 -77169

Max (ft-kips) 58542 77655 58544

Mcr

Min (ft-kips) -74099 -78573 -78573 -74099

Max (ft-kips) 57672 52147 57691

Design 

Moment

Min Mcr 1.33Mu 1.33Mu Mcr

Max Mcr Mu Mcr

Min (ft-kips) -74099 -68145 -68141 -74099

Max (ft-kips) 57672 58387 57691

fMn

Min (ft-kips) -27327 -75814 -75622 -11740 -75622 -75814 -27327

Max (ft-kips) 67604 18917 19105 81151 19105 18917 67604

d/c ratio

Min 0.977 0.901 0.901 0.977

Max 0.853 0.719 0.853

Table C.4 – Flexural Strength Design Verifications
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Appendix C – Design Example 1 300 of 369 

[5.8.2.1]

8. Shear and Torsion Design – Strength Limit State

Like the flexural design, the Strength I Limit State load combination is used for 

the shear and torsion design. It can be found in table 4.2-1 in section 4.

The factored shear resistance must be greater than the Strength I load 

combination, Vu (ultimate shear).

8.1 Torsion Members

According to the AASHTO LRFD specifications, torsion may not be neglected if:

where,

However, by inspection, torsion need not be investigated for this design 

example. Due to the bridge being straight, live loading is the only cause for 

torsion. No dead, miscellaneous, or other loads cause torsion in the 

superstructure. As outlined above in section 2 of this design example, this 

bridge meets the requirements to utilize the live load distribution tables in the 

AASHTO LRFD specifications.

Torsional effects are inherently included in the distribution factor equations. 

The purpose of a live load distribution factor is to capture the maximum 

possible vertical shear load on a web due to live loading. This can be done by 

altering the number of specified live loads and its transverse position on the 

cross section to create the maximum shear effect on the web of interest. 

Torsion affects the vertical shear loads in the webs and is subsequently 

captured by this process.

Therefore, torsion may be neglected in the following calculations of the shear 

design. See design example 2 for a procedure of designing for torsional effects.

8.2 Factored Design Shear

The AASHTO LRFD specifications specifies that the critical section for shear 

design can be located a distance dv from the support or point of discontinuity. 

The highest shear demand and critical section is located just upstation to 

Pier 2, a distance dv (4.68 ft) from the diaphragm (node 252) (see section 8.3 

for calculation of dv).
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Appendix C – Design Example 1 301 of 369

2985.08 → = V kips u

V VV n cs = +

[5.8.3.3]

[5.8.3.4.3]

[5.8.3.4]

The summation of the Strength I load combination is:

where,

VPS = the shear due to secondary post-tensioning effects

8.3 Factored Shear Resistance 

A prestressed concrete beam fails under shear along diagonal cracks. There 

are two types of diagonal cracks that form: Type I and Type II. The first type is 

known as flexure-shear cracks and the second as web-shear cracks. The 

flexure-shear cracks start as a nearly vertical flexural cracks on the tension 

face that later, under a shear demand, spread into diagonal shear cracks 

toward the compression face. Flexure-shear cracks are common in beams with 

low prestressing force near midspan. Web-shear cracks form near the neutral 

axis of the section in the web and spread diagonally. This type of cracking is 

caused by tension in the web called principal tensile stresses and can be found 

near supports.

The AASHTO LRFD specification states the nominal shear resistance to be the 

following:

The shear resistance may be determined using one of three methods outlined in 

the AASHTO LRFD specification. Method 1 is for nonprestressed sections only. 

Method 2 is the general procedure for prestressed and nonprestressed 

members. And Method 3 is the simplified procedure for prestressed and 

nonprestressed members.

This design example uses the Simplified Procedure outlined in the AASHTO 

LRFD specifications. The concrete contribution, Vc, of the nominal shear 

resistance is the minimum of Vci and Vcw, the flexure-shear resistance and 

web-shear resistance respectively.

The nominal shear resistance to combined flexure and shear can be obtained 

from the following equation (units in kips and in):

1143.92

101.25

0

801.89

DC

DW

CR SH PS TU

LL I

V kips

V kips

V VV

V kips

+

+

=

=

= = =

=

V VV c = min , ( ) ci cw

0.02 V f bd V ci c v v o = ′ +
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Appendix C – Design Example 1 302 of 369

[5.8.3.4.3-1]

[5.8.2.9]

[5.8.2.9]

[C5.8.2.9-1]

[5.8.2.9-2]

The above equation can be separated into two parts. The second term, Vo, is 

the shear that occurs at the section when a flexural crack forms. The first term 

is an empirical formula that represents the additional shear force required to 

transform the flexural crack into an inclined shear crack.

The second term is often separated into two parts as follows:

where,

Vd = unfactored dead loads (DC + DW) 

Mcre = the moment required to crack the section above the dead loads

Mmax = the maximum factored moment less the unfactored dead loads

Vi = the factored shear concurrent to the maximum moment less the 

dead load

Therefore, the nominal shear resistance to combined flexure and shear and its 

lower limit can be written (units in kips and in):

The effective web width, bv, is the sum of the web widths less 1⁄4 the diameter of 

the grouted ducts or 1⁄2 the diameter of the ungrouted ducts:

The effective shear depth, dv, can be defined by:

where,
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Appendix C – Design Example 1 303 of 369

1143.92 101.25 1245.17 V V V d DC DW =+ = + = kips

The unfactored dead load, Vd, is:

The factored maximum moment (negative flexure near pier) and the shear that 

acts concurrent to the moment from externally applied loads (Mmax and Vi

respectively) are:

Note that Mmax and Vi contain the secondary post-tensioning effects.

As defined above, Mcre is the moment required, above that caused by dead 

loads, to crack the section. Before looking at the code equation, it is known that 

the moment required to crack a concrete section is:

where:

S = section modulus

fr = modulus of rupture, or the stress at which the concrete cracks

PPT = axial force from post-tensioning (including secondary effects)

A = cross sectional area

Therefore, when the section cracks in flexure, the applied factored moments 

must equal the cracking moment.

Knowing:

where,

MPS = moment from secondary post-tensioning effects

MP = moment primary post-tensioning effects

41624 ( 25746) 15878

889.16

max u d u DC DW

i

M M M M M ft kips

V kips

= − = − =− − − =− − +

=

PT

cr r

P M Sf A

  = +    

1.25 1.5 0.5 1.75 PT

r DC DW CR SH PT TU LL I

P Sf M M M M M M A + +

  + = + + ++ +    

1.25 1.5 0.5 1.75 M M MM M M M u = + + ++ + DC DW CR SH PS + + TU LL I
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Appendix C – Design Example 1 304 of 369 

[5.8.3.4.3-2]

[5.4.2.6]

Therefore, the equation for the cracking moment, Mcre, is written as:

where,

at the top of the section.

For the Vci equation to apply, the following must be true:

The purpose of the Vci equation is to determine the concrete’s resistance to a 

flexural shear crack. If, in this case, the section does not crack in flexure, Vci

does not apply and Vcw controls.
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Appendix C – Design Example 1 305 of 369 

[5.8.3.4.3-3]

[C5.8.2.3]

[C5.8.3.3-1]

[5.5.4.2.1],

[5.8.3.4.3-4]

The nominal shear resistance to excessive principal tension is:

The compressive stress at the centroid of the section, fpc, is:

The vertical component of the prestressing force, Vp, is as follows (positive 

when resisting the applied shear):

Therefore:

section is:

The shear reinforcement required to provide the above shear resistance is:

where,
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Appendix C – Design Example 1 306 of 369

[5.8.2.5-1]

Divided evenly among the 5 webs:

The minimum required reinforcement for the entire cross section is:

Therefore, use #5 bars spaced at 9 inches on each face of each web.

The table below contains a summary of the shear strength design at every 20 

feet. Note that #4 bars at 12 inches is the lowest reinforcement used. This is 

due to engineering judgment.

Node Distance Vu Vc Vs req

Av req

per 

web

Bar 

Area @ 

Av prov

per 

web

(ft) (kips) (kips) (kips) (in2/ft) (in2) (in) (in2/ft) 

150 0 

152 7.18 1946 1434 728 0.288 0.20 12 0.40

154 20 1456 1330 288 0.114 0.20 12 0.40

158 40 726 979 0 0.000 0.20 12 0.40

162 60 -1059 1244 0 0.000 0.20 12 0.40

242 80 -1768 1576 389 0.154 0.20 12 0.40

246 100 -2465 1875 865 0.342 0.20 12 0.40

248 112.32 -2882 1508 1694 0.670 0.31 9 0.83

250 120

252 127.68 2985 1382 1935 0.765 0.31 9 0.83

254 140 2576 1851 1011 0.400 0.20 12 0.40

258 160 1902 1601 512 0.203 0.20 12 0.40

262 180 1229 1286 79 0.031 0.20 12 0.40

266 200 569 1027 0 0.000 0.20 12 0.40

338 220 -1229 1286 79 0.031 0.20 12 0.40

342 240 -1902 1601 512 0.203 0.20 12 0.40

346 260 -2576 1852 1011 0.400 0.20 12 0.40

348 277.32 -2985 1382 1934 0.765 0.31 9 0.83

350 280

352 292.68 2882 1508 1694 0.670 0.31 9 0.83

354 300 2465 1875 864 0.342 0.20 12 0.40

358 320 1768 1576 388 0.154 0.20 12 0.40

362 340 1059 918 0 0.000 0.20 12 0.40

442 360 -726 1161 0 0.000 0.20 12 0.40

446 380 -1456 1330 288 0.114 0.20 12 0.40

448 392.82 -1946 1434 728 0.288 0.20 12 0.40

450 400

Table C.5 – Summary of Shear Design at Strength Limit State 
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Appendix C – Design Example 1 307 of 369 

[5.8.5]

9. Principal Tensile Stresses

The AASHTO-LRFD Design Specifications state that the principal stresses in 

the webs shall be analyzed for all segmental bridges. It does not give direction 

on non-segmental concrete box girder bridges. However, it is a good design 

check and the calculation of principal stresses is included in this example.

The principal tensile stresses shall be calculated using the long-term residual 

axial stress and the maximum shear stress. The Service III Limit State load 

combination is used for both axial and shear stresses. The load combination 

can be found in table 4.1-1 in section 4.

The thermal gradient loading is not considered in the calculation of principal 

stresses. The purpose of thermal gradient is to check longitudinal stresses 

under special thermal effects. Axial stresses resulting from temperature and 

live loading are also not included since they are not long-term loadings. 

The critical section with the maximum principal tensile stress is located just 

upstation to Pier 2, a distance of dv (4.68 ft) from the diaphragm (node 252). 

See section 5 above for the calculation of dv. 

From general mechanics of materials, the shear stress can be taken as:

where,

V = vertical shear in section

= VDC +VDW +VPS +VCR+SH +VTU +0.8·VLL+I

= 1526 kips

Q = first moment of the area = 118.12 ft3

I = section moment of inertia = 643.75 ft4

b = Σ web widths less 1⁄4 duct Ø= 4.53125 ft

9.1 Mohr’s Circle

The Mohr’s circle is a graphical representation of the state of plane stress at a 

given point. The abscissa and ordinate represent the normal and shear stress 

components respectively. Below is a graph of Mohr’s circle and points M, N, 

and C. 
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Appendix C – Design Example 1 308 of 369 

Figure C.12 – Mohr’s Circle

The normal stress at point M is the axial stress on the section at the neutral 

axis. The stresses used 

The Mohr’s circle representation of this state of stress is:

Figure C.13 – State of Stress at Node 252 
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Appendix C – Design Example 1 309 of 369

[5.9.4.2.2-1]

For the case of no vertical compression (sy=0), the radius, R, of Mohr’s circle 

can be expressed as:

The maximum permissible principal tension is 3.5√f’c (psi) (0.110√f’c for units in 

ksi).

The table below contains a summary of the principal tension check at a 

distance dv (4.68 ft) from each diaphragm. These are the critical locations for 

each span in this example. Note that this is not always true for every bridge or 

bridge type.

Node Distance t s s/2 Radius smax √f’c Min Max COG

(ft) (ksf) (ksf) (ksf) (ksf) (ksf) (ksf) 

150 0 

152 7.18 7.1 32.9 -101 -50.5 60.3 9.8 0.96

248 112.32 -56.4 -27.2 -103 -51.3 76.2 24.9 2.45

250 120

252 127.68 33.8 61.8 -94 -46.8 77.5 30.7 3.02

348 277.32 -61.8 -33.8 -94 -46.8 77.5 30.7 3.02

350 285

352 292.68 27.2 56.3 -103 -51.3 76.2 24.9 2.45

448 392.82 -32.9 -7.1 -101 -50.5 60.3 9.8 0.96

450 400

Table C.6 – Principal Tensile Stress Summary 
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Appendix C – Design Example 1 310 of 369 

[5.8.1.5]

[5.12.3]

[4.6.2]

[4.6.2.1.7]

[5.7.3.3.2] 

10. Transverse Design

The transverse design of a multicell concrete box girder bridge may be split into 

four sections: the deck design, overhang design, web design, and bottom slab 

design. Web bending need only be considered for curved post-tensioned box 

girder bridges with a depth of more than 4 feet. Since this example is straight 

and has relatively short bay widths, transverse web bending may be neglected.

The three sections below are designed with mild reinforcement. Therefore, the 

transverse reinforcement is the cover steel, or the outer layer of reinforcement. 

The longitudinal mild reinforcement will be placed inside the transverse 

reinforcement for temperature and shrinkage effects.

The distance, d, from the compression face to the center of the tension 

reinforcement is dependent on the concrete cover. The following are the 

minimum covers used for this example:

Deck surface = 2.5 in

Exterior other than above = 2.0 in

Interior other than above = 1.5 in

10.1 Deck Design

This example uses the Approximate Method outlined in the AASHTO LRFD 

specifications. A cross-sectional frame model one foot in length is used to 

determine the design loads per unit length for the deck. This type of model 

captures the flexural and torsional stiffnesses of the supporting webs and 

bottom slab in a multi-cell box girder.

One or more axles (representing lanes of traffic) are placed on the deck and 

moved transversely to maximize the moments. The minimum distance between 

the wheels of two adjacent axles is four feet. The resulting moments are 

divided by the strip width calculated from AASHTO LRFD Table 4.6.2.1.3-1 to 

obtain the live load moment per unit length of bridge.

10.1.1 Minimum Reinforcing

As shown previously in the Flexural Strength section of this example, the 

design moment is:

And the cracking moment is:
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Appendix C – Design Example 1 311 of 369

[4.6.2.1.3-1]

where,

10.1.2 Negative Moment (max) @ 6 inches from CL of web

Strength I load combination

The moment due to live loading is:

Therefore,

Considering the minimum reinforcement requirement above:

(center web)
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Appendix C – Design Example 1 312 of 369 

[5.7.3]

[4.6.2.1.3-1]

The following formulae are needed to determine the required flexural 

reinforcement:

Manipulation of the above formulae gives the required area of flexural 

reinforcement:

where,

Therefore, use #5 bars at 6 inches. 

10.1.3 Positive Moment @ CL of interior cell

Strength I load combination

The moment due to live loading is:
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Appendix C – Design Example 1 313 of 369 

[A13.4]

[3.4.1-1]

Therefore,

Considering the minimum reinforcement requirement above:

Therefore with d = 7 in: 

For simplicity, use the same reinforcement used in the top: #5 bars at 6 inches

10.2 Overhang (Wing) Design

The deck overhang, or wing, must be designed to resist the maximum of three 

design cases given in Section A13. According to the AASHTO LRFD 

specifications, the flexural resistance, Ms, of the wing shall exceed.

• The transverse vehicular collision forces under Extreme Event II load

combination limit state.

• The vertical vehicular collision force forces under Extreme Event II load

combination limit state.

• The loads that occupy the overhang under the Strength I load

combination limit state.

Load Combination 

Limit State DC DW CR+SH LL+I CT

Extreme Event II 1.0 1.0 1.0 0.5 1.0

Strength I 1.25 1.5 1.25 1.75 0 

Table C.7 – Load Factors for Overhang Design 

This design example does not cover the design of a barrier. Instead, It is 

assumed that a 32” tall barrier of test level TL-4 is used. It has a self-weight of 

0.45 kips/ft and its center of gravity is located 1’-0” from its inside face. The 

three moment capacities (horizontal moment, vertical moment, and additional 

beam moment respectively) needed in the following calculations are given as:

See Figure CA13.3.1-1 in the AASHTO LRFD Design Specifications for a 

depiction of each moment.
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Appendix C – Design Example 1 314 of 369

[A13.4.2]

Figure C.14 – Overhang Design Sections 

The above figure shows the two sections that need to be designed for this 

example. The first section will include design cases 1 and 2. The second 

section will include all three design cases.

10.2.1 Section 1: Overhang design at the face of the barrier

Case 1:

For design case 1, overhang, or wing, of a box girder bridge must be design to 

simultaneously exceed the moment capacity, Mc, at the barrier’s base as well 

as the tensile force, T, from the barrier. In other words, it must be designed to 

resist railing collision forces given in section A13.

Calculate the dead load moments:

where,
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Appendix C – Design Example 1 315 of 369 

[A13.4.2-1]

[A13.3.1-2]

[A13.3.1-1]

Extreme Event II load combination:

Tensile force from the barrier is defined as:

where,

Calculate the factored flexural resistance, fMs. It can be seen from the figure 

and calculations below that the collision tensile force, T, decreases the moment 

resistance of the section.

Figure C.15 – Equilibrium at Strength Limit State 

Take Moment about the compression force, C. 
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Appendix C – Design Example 1 316 of 369

[A13.3.1-1]

and,

Therefore,

Plug a into the moment equation above and solve for the required area of 

reinforcement, As, where:

Assume the bottom reinforcement carries half of the collision tensile force.

Use #7 bars at 6 inches for the top reinforcement and #4 bars at 12 inches for 

the bottom.

Case 2:

According to AASHTO, a vertical downward force of 18 kips is to be distributed 

over 18 ft of the TL-4 barrier. This equates to 1 klf. If we assume this load acts

at the center of gravity of the barrier, the moment it produces on the overhang 

is 1 ft–kips / ft. Note that this moment is less than the effects in design case 1 

and we may skip this design case.

Case 3:

Note that design case 3 does not include any loads that are not included in 

design case 1. Therefore, we may neglect this case at this section.

1 0.85 0.85 c c

s y

C f cb f ab

C Af T

= = ′ ′ β

= −

0.85

s y

c

Af T

a

f b

− = ′

60 5

0.90 12

1.75 9 3 10.11 10.11 2.5 0.5 1 7.11

4.75

11.33 25.53 306.36

y c

u

f ksi f ksi

b in

h in in in d in

T kips M ft kips ft in kips ft

f

= ′ =

= =

= + ⋅ = = − − ⋅=

= = −= −

2 0.98 → = A in ft s

1 2 0.10

2 s

y

T A in ft f f =⋅ =
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Appendix C – Design Example 1 317 of 369

10.2.2 Section 2: Overhang design at the outside face of the web

Case 1:

Figure C.16 – Length of Loaded Areas 

As depicted in the figure above, loads distribute over larger lengths the farther 

the section is from the load application. Assume the distribution length, Lc, 

increases at a 30° angle.
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Appendix C – Design Example 1 318 of 369 

Calculate the dead load moments:

where,

The moment due to 25 psf wearing surface is:

The moment due to live loading is:

Figure C.17 – Live Load on Overhang 
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Appendix C – Design Example 1 319 of 369

[3.6.1.1.2-1]

[3.4.1-1]

where, 

m = multiple presence factor = 1.2

P = AASTHO Truck wheel load = 16 kips

x = distance from load to support = 2 ft

ws = width of equivalent strip (in) = 45 + 10x

Extreme Event II load combination:

Case 2:

Note that design case 2 at this section is similar to section 1 and does not need 

to be investigated.

Case 3:

Strength I load combination:

Note that the collision tensile force is less at section 2 than at section 1, the 

moment is of equivalent magnitude, and the section is much greater in depth. 

Therefore the same reinforcement may be used for the entire overhang.

Verify the area of reinforcement, As, required. Use the same moment equation 

used at Section 1 where:
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Appendix C – Design Example 1 320 of 369

10.3 Bottom Slab Design 

10.3.1 Minimum Reinforcement Cracking Moment

10.3.2 Negative Moment

Considering the minimum reinforcement requirement above:

Using the same formula presented in section 7.1 above, solve for required 

negative area of reinforcement for negative flexure where d = 5 in: 

10.3.3 Positive Moment

Considering the minimum reinforcement requirement above:

Solve for required area of reinforcement for positive flexure where d = 4.5 in: 

10.3.4 Minimum reinforcement for shrinkage and temperature:

where,

2 As ≥ s 0.07in2 ft → ≥ A 0.11in ft
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Appendix C – Design Example 1 321 of 369

Therefore, the minimum required reinforcement for shrinkage and temperature 

controls. Notice that this requires a minimum of # 3 bars spaced at 12 inches. 

This is very light. 

Due to engineering experience and judgment, use #5 bars at 12 inches for the 

top of the bottom slab and #4 bars at 12 inches for the bottom. 
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Appendix D – Design Example 2 322 of 369 

Example 2—Curved Two-Cell Box Girder Bridge

1. Introduction

This Design example is a curved, 3-span concrete box girder with three webs. 

The top slab of the bridge is 47’-6” wide, with a travelled way dimension of 44’-

0”, between the faces of the curbs. The span lengths are 120’-0”, 160’-0”, and 

120’-0” measured along the centerline of the bridge. The radius of curvature of 

the centerline of bridge is 600’-0”. It is assumed that the substructure supports 

are bearings on individual columns beneath each web at the piers and similar 

bearings beneath each web at the abutments. As a result, the substructure 

effectively provides full vertical and torsional fixity at each pier and abutment, 

while allowing longitudinal movement, so that thermal rise and fall, creep, and 

shrinkage shortening do not have an appreciable effect on the superstructure.

The bridge is assumed to require the following concrete covers to the 

reinforcing:

• 2 1⁄2 inches—Top of top slab.

• 2 inches—Other external surfaces.

• 1 1⁄2 inches—Internal surfaces.

It is assumed to be located in a moderate corrosion environment.

The spacing of the three, 1’-0”-thick, vertical webs is 16’-0”. The bottom slab is 

thus 33’-0” wide. Also, the cantilevers of the top slab, beyond the centerline of 

the exterior webs, are 7’-6”. The top slab is thickened at the web locations, and 

is assumed to be transversely post-tensioned, as discussed in section 2.

Figure D.1 shows a perspective of the bridge studied in this design example. 

Figure D.1 – Curved Bridge of Design Example 2 
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Appendix D – Design Example 2 323 of 369 

An elevation view of the bridge for this design example is shown in figure D.2.

Figure D.2 – Elevation of Example 2 Bridge 

Figure D.3 shows a transverse cross section taken through the bridge. The 2-

cell box girder bridge carries two 12’ wide traffic lanes, two 10’, and 1’-9” 

barriers, for a total out-to-out width of 47’-6”. 

Figure D.3 – Cross Section through Example 2 Bridge 

Pier 2 and Pier 3 are each comprised of two cylindrical columns 6’-0” in 

diameter connected by a transverse pier cap. The box girder superstructure is 

supported by bearings placed on the pier cap. The columns are 20’ in height, 

measured from the top of foundations to the base of the superstructure. The 

foundations for this design example consist of 7’-0” diameter mono-shafts. For 

this example, it is assumed that the supporting soil provides an equivalent point 

of fixity 10’ below the top of the mono-shafts for all force effects. The distance 

from the base of the superstructure to the point of fixity is 30’. 
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Appendix D – Design Example 2 324 of 369

2. Material Characteristics

2.1. Concrete

The initial and 28-day concrete strengths for the superstructure and 

substructure are:

The unit weight of the concrete is defined below. Note that the unit weight 

used for the self-weight calculations includes an additional weight for the 

reinforcement.

The modulus of elasticity for normal weight concrete, where wc = 0.145 kcf, can 

be taken as:

2.2 Reinforcing Steel

The yield strength and modulus of elasticity are:

2.3 Prestressing Steel

The 0.6 inch diameter low-relaxation prestressing strand is used in this 

example with the following properties:

[3.5.1]

[C5.4.2.4-1] 

[5.4.3] 

[5.4.4] 
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Appendix D – Design Example 2 325 of 369

3. Cross Section Properties

The typical cross section dimensions are shown in figure D.4. The cross 

section dimensions are constant over the length of the bridge:

Figure D.4 – Cross Section Dimensions 

The properties of the typical superstructure cross section are:

A = cross sectional area = 73.36 ft2

Ix = moment of inertia = 572.7 ft4

H = height of section = 7.0 ft

yct = top of section to centroid = 2.255 ft

ycv = bottom of section to centroid = 4.745 ft

The gross section properties of the substructure elements are used in this 

design example:

Ac = area for two columns = 56.55 ft2

Ixc = moment of inertia for two columns = 127.24 ft4

Ap = area for two monoshafts = 56.55 ft2

Ixp = moment of inertia for two monoshafts = 127.24 ft4
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 326 of 369

∑ηγ f ii i n r Q RR ≤ =

0.95

1 1.0

i DRI

i

DRI

η ηηη

η

ηηη

= ≥

= ≤

1.0

1.0

DRI

i

ηηη

η

= = =

∴ =

4. LRFD General Design Equation

The AASHTO LRFD Design Specification’s general design equation is:

where,

ηi = load modifying factor for ductility, redundancy, and operational 

importance

γ i = load factor

Qi = force effect

f = resistance factor

Rn = nominal resistance

Rr = factored resistance

The load modifier, ηi, takes on two forms. The first is for loads in which a 

maximum value of γ i is appropriate, the second is in which a minimum value of 

γ i is appropriate.

where,

η D = ductility factor

ηR = redundancy factor

ηI = operational importance factor

Since the purpose of this example is to illustrate a conventional design on a 

typical bridge,

Note that the load modifying factors are project and design specific and may 

not equal 1.0 in actual designs.

[1.3.2.1-1] 

[1.3.2.1-2], 

[1.3.2.1-3]

[1.3.3], 

[1.3.4],

[1.3.5]
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 327 of 369

5. Transverse Design

The transverse analysis for this structure was accomplished using the method 

described under LRFD 4.6.2.9.4. This method is a simple way to accurately 

calculate the contribution of the webs and bottom slab to the top slab bending, 

as well as calculate the web and bottom slab bending. For the top slab, which 

is a transversely prestressed member, both service and strength limit states are 

checked. The webs and bottom slab, which are not prestressed transversely, 

are designed at the strength limit state as reinforced members. Because the 

superstructure is symmetrical, the results will be reported for half the top and 

bottom slabs, the center web, and one exterior web.

Figure D.5 shows the transverse model of the bridge, made for the BDII 

program, a program that allows the post tensioning tendons to be input and 

calculates the time-dependent effects of creep and shrinkage. The properties 

of this model assume it is 1’-0” long. The transverse post-tensioning is 1-0.6” 

dia. strand in the 1’-0” long model, which equates to 4-strand tendons spaced 

at 4’-0” along the bridge.

Figure D.5 – Frame Model for Transverse Analysis

Seven live load cases were developed by positioning trucks on the cross 

section to maximize the following moments:

• Negative Moment, Node 3

• Positive Moment, Node 6

• Negative Moment, Node 4

• Negative Moment, Node 8

• Negative Moment, Node 5

• Negative Moment, Node 7

• Negative Moment, Node 2

In this example, all of the maximum moments were created by a single truck on 

the cross section, due to the 1.2 multiple presence factor for a single lane from 

LRFD 3.6.1.1.2. Because a single truck produces the maximum top slab 

moments, the positions of the trucks which maximize the negative moment at 

nodes 3, 4, and 8 also produce maximum bending in the adjacent webs.

Influence surfaces are then used to calculate the Fixed End Moments at the 

end of each slab span for each load case. Also, the maximum moment for a 

beam with fixed ends at nodes 5, 6, and 7 were determined when those were 

the nodes of interest.

[3.6.1.1.2]
Table D.1 – Fixed End Moments for Maximum Flexure at Node 6 

The BDII transverse plane frame model is loaded with the fixed end moments 

at Nodes 4 and 8 in order to determine their redistribution as a result of the 

flexibility of the adjoining top slab. The result of interest due to this load case is 

only the redistributed moment at Node 6, which is +0.6 ft-kips/ft. 

The design positive Live Load Moment at Node 6 is then:

5.1+ 0.6 = 5.7 ft − kips / ft

Similarly, the maximum Live load effects are found on other nodes in the cross 

section and are shown in table D.2. Note that for the top and bottom slabs, 

Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 328 of 369

Once these fixed end moments are calculated for each load case, they are 

input into the BDII model, which calculates the redistributed moments for each 

load case. The redistributed moments are then added to the moments with 

fixed ends to calculate the final live load moments at a node.

The step-by-step process of calculating the positive bending (tension on the 

bottom) live loads at Node 6 will be shown as an example:

The loading producing maximum positive bending at Node 6 is shown in figure 

D.6.

Figure D.6 – Load Location for Maximum Positive Flexure at Node 6

Then, influence surfaces are used to calculate the values of the fixed end 

moments at Nodes 4 and 8 for this loading, as well as the moment with fixed 

ends at Node 6. These moments at Points 4,6,and 8 are shown in the 

following table:

Node FEM

(ft-kips)

4 -14.1

6 5.1

8 6.3
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Appendix D – Design Example 2 329 of 369

positive moments create tension on the bottom. For the webs, positive 

moments create tension on the inside of the exterior webs and the right side of 

the center web. Also, an influence surface was unavailable for Node 2, so a 

plate model of the cantilever overhang was created, and truck wheels were 

placed on the model to give the maximum live load moment at that node.

Node MLL +

(ft-kips) 

MLL – 

(ft-kips)

1 0.0 0.0

2 n/a -3.7

3 n/a -9.5

4 n/a -12.4

5 n/a -3.1

6 5.7 n/a

7 n/a -3.3

8 n/a -12.9

21 0.1 -0.1

22 0.3 -0.5

23 0.1 -0.3

24 0.0 -0.3

25 0.5 -0.4

32 0.5 -1.0

33 4.6 -7.6

34 6.7 -11.0

35 1.2 -1.2

36 6.4 -6.4

37 9.9 -9.9

Table D.2 – Design Live Load Moments 

Once the Live load moments are calculated, they are combined with dead load 

and PT forces in service and strength load groups.

Service Load Groups for Top Slab

Because the top slab is a prestressed member, the Service Limit State that 

must be checked at the time of tendon stressing and after all losses have 

occurred. The Service I load combination is used to check both the allowable 

tension and compression for transverse analysis, using only the truck loads, 

and no lane loads.

The Service I load factors simplify to those shown in table D.3 for this structure.

Table D.3 – Service Limit State Load Factors

Load

Combination DC DW CR + SH PS LL+I

Service I 1.0 1.0 1.0 1.0 1.0

[3.4.1]

[4.6.2.9.4]
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Appendix D – Design Example 2 330 of 369

The maximum allowable compressive stresses are:

• 0.6 f’ci for temporary stresses at the time of tendon stressing.

• 0.45 f’c due to prestressing and permanent loads after all time dependent

effects and losses.

• 0.6 f’c due to prestressing, permanent, and transient loads after all time

dependent effects and losses.

The maximum allowable tensile stresses are:

• 0.24√ f’ci (ksi) for temporary stresses at the time of tendon stressing

(7.5√ f’ci for units in psi).

• 0.0948√ f’c (ksi) for stresses after all time dependent effects and losses

(3.0√ f’c for units in psi).

Temporary Initial Stress Check

The top slab stresses at stressing of PT are shown below in table D.4. With 

6ksi concrete, the allowable stress in compression is -518ksf, and the allowable 

tension stress is +33ksf. The reported stresses are well below these limits.

Node f top

(ksf)

f bot

(ksf)

1 0.0 0.0

2 -56.3 -45.3

3 -61.4 -9.2

4 -78.3 8.4

5 -79.5 -33.9

6 -20.9 -92.6

7 -65.3 -47.8

8 -69.6 -0.4

Table D.4 – Top Slab Stresses at Stressing of PT

Stresses After All Time Dependent Losses

The stresses due to dead load, PT, creep, and shrinkage after all losses are 

shown in table D.5.

[5.9.4]
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Appendix D – Design Example 2 331 of 369

Node f top

(ksf)

f bot

(ksf)

1 0.0 0.0

2 -31.0 -63.4

3 -39.0 -26.7

4 -66.9 2.8

5 -67.4 -36.7

6 -21.4 -82.7

7 -61.1 -42.7

8 -62.2 -2.0

Table D.5 – Top Slab Stresses due to DC, DW, CR, SH after all losses

Combining the stresses from table D.5, above, with those from the Live load 

design moments shown in table D.2 yields the following final stresses, shown in 

table D.6 below:

Node

Max Tension Max Compression

f top

(ksf)

f bot

(ksf)

f top 

(ksf)

f bot

(ksf)

1 0.0 0.0 0.0 0.0

2 8.5 -63.4 -31.0 -102.9

3 9.6 -26.7 -39.0 -75.3

4 -3.5 2.8 -66.9 -60.6

5 -25.6 -36.7 -67.4 -78.5

6 -21.4 -5.8 -98.3 -82.7

7 -16.6 -42.7 -61.1 -87.2

8 3.8 -2.0 -62.2 -68.0

Table D.6 – Top Slab Stresses From Service I (Compression) and 

Service III (Tension) Load Combinations

Strength Limit State for Transverse Analysis

All members must be designed for the Strength Limit State. For the top slab, 

this amounts to a check for the post-tensioning layout determined in the 

Service Limit State design. For the bottom slab, reinforcing will be determined 

based on this work. The required web reinforcing for bending will be 

determined in this stage, but will eventually be combined with that required for 

longitudinal shear and torsion.

The Strength I load factors simplify to those shown in table D.7 for this 

structure.
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 332 of 369

Table D.7 – Strength I Load Factors for Transverse Analysis

The ultimate applied moments are calculated from the output of the BDII 

program (for all except live loads) and the live load moments shown in table 

D.2. Once combined using the above load factors, the ultimate moments for

each node in the cross section are as shown in table D.8

Node Mu + 

(k-ft)

Mu – 

(k-ft)

1 0.0 0.0

2 -4.3 -10.7

3 -8.6 -25.2

4 -1.7 -23.4

5 0.6 -4.8

6 12.4 2.5

7 0.6 -5.2

8 -2.1 -24.7

21 -0.8 -1.1

22 0.4 -1.0

23 1.5 0.8

24 -1.9 -2.4

25 -2.1 -3.7

32 2.7 0.0

33 12.7 -8.6

34 17.9 -13.1

35 2.1 -2.1

36 11.2 -11.2

37 17.3 -17.3

Table D.8 – Ultimate Moments for Transverse Design

The ultimate resistance of the top slab will be calculated separately from the 

resistance of the bottom slab and webs. As mentioned previously, the Strength 

Limit State calculations of the prestressed top slab serve as a check of the 

Service Level design. However, for the webs and bottom slab, reinforcing 

requirements will be determined using this Strength Limit State design.

Top Slab Resistance

The top slab resistance is calculated using LRFD 5.7.3

Load

Combination DC DW CR + SH PS LL+I

Strength I 1.25 1.5 1.0 1.0 1.75

[3.4.1]

[C5.7.3.1.1-1]

[5.7.2.2]
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 333 of 369

Table D.9 presents the ultimate capacity of the top slab. In the table, the 

neutral axis and the stress in the prestressing at ultimate are calculated using 

5.7.3.1.1-4 and 5.7.3.1.1-1, respectively. The ultimate capacity is calculated 

using 5.7.3.2.2-1, without considering mild reinforcing (though it could be 

considered, if necessary).

Table D.9 – Ultimate Capacity of the Top Slab 

The minimum reinforcing provisions of 5.7.3.3.2 should be satisfied, as well. In 

this case, φMn is greater than 1.33·Mu in all cases, thus 5.7.3.3.2 is satisfied. 

The comparison is shown in table D.10.

Node φMn

(k-ft)

1.33•Mu

(k-fft)

1 18.3 0.0

2 -20.1 -14.3

3 -40.2 -33.6

4 -40.2 -31.1

5 -16.0 -6.4

6 23.4 16.5

7 -16.0 -6.9

8 -40.2 -32.8

Table D.10 – φMn vs. 1.33·Mu

Node H 

(in)

dp

(in)

c 

(in)

fps

(ksi) φ φMn

(k-ft)

1 9.00 4.50 1.18 250.1 1.00 18.3

2 9.00 4.86 1.19 251.5 1.00 -20.1

3 13.00 9.02 1.23 259.7 1.00 -40.2

4 13.00 9.02 1.23 259.7 1.00 -40.2

5 8.00 4.00 1.17 247.9 1.00 -16.0

6 8.00 5.56 1.20 253.7 1.00 23.4

7 8.00 4.00 1.17 247.9 1.00 -16.0

8 13.00 9.02 1.23 259.7 1.00 -40.2

  [5.4.2.6]

[5.7.3.3.2]

[5.5.4.2]

[5.7.3.3.22]

[5.7.3]
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 334 of 369

Required Reinforcing in Bottom Slab and Webs

The bottom slab reinforcing is sized for 1.33·Mu to accommodate minimum 

reinforcing requirements. However, since the web reinforcing will ultimately be 

combined with the longitudinal shear reinforcing and regional bending 

reinforcing from post-tensioning (see section 6), the minimum reinforcing 

requirements will be satisfied for the webs. table D.11 shows positive bending 

reinforcement requirements, and table D.12 shows negative bending 

requirements.

Node As/s

(in2

/ft)

d 

(in)

a 

(in) φ

φMn

(+)

(k-ft)

1.33∙ Mu

(k-ft)

Bottom Slab

21 N/A 9.625 0.00 0.9 N/A N/A

22 0.030 4.625 0.03 0.9 0.6 0.5

23 0.095 4.625 0.09 0.9 2.0 2.0

24 N/A 4.625 0.00 0.9 N/A N/A

25 N/A 9.625 0.00 0.9 N/A N/A

Ext. 

Webs

32 0.060 10.125 0.06 0.9 2.7 N/A

33 0.283 10.125 0.28 0.9 12.7 N/A

34 0.400 10.125 0.39 0.9 17.9 N/A

Int. 

Web

35 0.047 10.125 0.05 0.9 2.1 N/A

36 0.248 10.125 0.24 0.9 11.2 N/A

37 0.388 10.125 0.38 0.9 17.3 N/A

Table D.11 – Positive Bending Reinforcement, Bottom Slab and Webs 

Node As/s

(in2

/ft)

d 

(in)

a 

(in) φ φMn (-) 

(k-ft)

1.33∙ Mu

(k-ft)

Bottom Slab

21 0.035 9.625 0.03 0.9 -1.5 -1.5

22 0.070 4.625 0.07 0.9 -1.4 -1.3

23 N/A 4.625 0.00 0.9 N/A N/A

24 0.155 4.625 0.15 0.9 -3.2 -3.2

25 0.115 9.625 0.11 0.9 -5.0 -4.9

Ext. 

Webs

32 0.010 10.125 0.00 0.9 0.0 N/A

33 0.190 10.125 0.19 0.9 -8.6 N/A

34 0.292 10.125 0.29 0.9 -13.1 N/A

Int. 

Web

35 0.047 10.125 0.05 0.9 -2.1 N/A

36 0.248 10.125 0.24 0.9 -11.2 N/A

37 0.388 10.125 0.38 0.9 -17.3 N/A

Table D.12 – Negative Bending Reinforcement, Bottom Slab and Webs 

[5.8.1.5]

[5.7.3]

[5.7.3]
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 335 of 369

The controlling amounts of reinforcing are highlighted in the above tables. For 

positive bending in the bottom slab, the point at the center of the bottom slab 

controls. Likewise, for negative moment in the bottom slab, the locations near 

the center web control. In all cases, the required reinforcing in the bottom slab 

is less than 0.2 in2

/ft, which means #4 bars at 1’-0” spacing will be adequate. 

The web reinforcing for bending is controlled by the reinforcing at the top of the 

web, as expected. The reinforcing required for bending in the center web is 

symmetrical. Also, note that more reinforcing is required on the inside faces 

than is required on the outside faces of the exterior webs. This web reinforcing 

demand will be added to that required for shear and torsion as well as regional 

bending later in the calculations.

Overhang Design

The deck overhang of a box girder bridge must be designed to withstand 

impact on the barrier and to withstand normal (Strength 1) traffic loads, though

not concurrently. There are three design cases given in LRFD A13.4.1. Case 

1 includes lateral load on the barrier which produces moment and tension in 

the overhang. Case 2 is a vertical impact on the barrier, and Case 3 is the 

design accomplished in section 2, with wheel loads on the deck. 

Case 1:

The strength of the deck must simultaneously exceed the moment capacity, 

Mc, of the barrier as well as the tensile force, T, from the barrier. The flexural 

resistance, Ms, of the overhang is designed using the Extreme Event II load 

combination limit state shown below:

Load Combination 

Limit State DC DW CR+SH PS TU TG LL+I

Extreme Event II 1.0 1.0 1.0 1.0 0 0 0.5

Table D.13 – Extreme Event II Load Factors

Typically, barriers are standard designs, and for this problem, a TL-4, 32 inch 

high barrier will be used. The three moment capacities needed in the following 

calculations are given as:

Also given are the weight of the barrier at 0.45kips/ft and its center of gravity, 

1’-0” from its inside face.

For this bridge, the capacity of the slab must be checked at three locations:

• At the inside face of the barrier.

• At the point where the slab bottom surface changes slope (Node 2 in the

transverse model).

[A13.4.1]

[A13.3.1]

[13.6.2]

[A13.4.2]

[3.4.1]
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 336 of 369 

• At the face of the web (Node 3 in the transverse model).

Case 1, Inside Face of Barrier:

1.75ft from the end of the slab:

• PT is at the center of the slab at this location.

• Slab is 9 inches thick.

• Top cover is 2.5 inches to the longitudinal (top layer) of reinforcing, which 

is #5 bars.

Calculate the dead load moments:

Tensile force from the barrier is defined as:

where,

  [A13.4.2-1

[A13.3.1]
Post-Tensioned Box Girder Design Manual June 2016

337 of 369

The reinforcing required to carry the tension will be split between the top and 

bottom layers. The amount required for tension is:

Use #4 at 1’-0” transverse in the bottom = 0.2in2

/ft.

For bending, try #5’s and #4’s alternated, each at 1’-0”, along with the posttensioning (4 strand tendons spaced at 4’-0”). Subtract the amount of 

reinforcing required for tension in the top layer.

Case 1, Node 2:

4.75 ft from the end of the slab:

• PT is 5.07 in above the bottom of the slab at this location.

• Slab is 9 in thick.

• Assume distribution length, Lc, increases at a 30 degree angle.

• Ignore vertical live load during an impact event.

Calculate the dead load moments:

[5.7.3.1]

[5.7.3.2]

MDC 
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 338 of 369

Calculate the impact forces:

For tension, required As per layer is 0.08in2

/ft. The #4’s at 1'-0" standard 

transverse reinforcing satisfies this demand, at 0.2in2

/ft.

Recalling from section 2 the capacity at Node 2 for transverse bending using 

the post-tensioning only,

φMn=Mu for the post-tensioning alone. Thus, the #5’s may be cut off near 

Node 2. Run half of the #5’s 1ft beyond Node 2, and cut the others at Node 2.

Case 1, Face of Web (Node 3):

Impact reinforcing at the face of the web is acceptable by inspection, since it 

was acceptable at Node 2, the #5’s at 1’-0” are continuous, the PT continues to 

move up, and the overhang continues to deepen, all while the applied moment 

continues to spread and decrease. 

Case 2:

Case 2 does not control, since the Fv=18kip vertical load distributed over 

Lv=18ft does not control over a wheel load in the overhang or Case 1.

( )2

0.25 3

0.11

2

DW overlay

DW

M M

ksf ft M ft kips

=

− = =− −

7.35 24.9 16.9

7.35 2 3 tan 30 c

ft M ft kips ft kips ft ft

  =− −   =− − +⋅ °  

3.1 0.11 16.9 20.1 Mu → =− − − =− − ft kips

2 2 tan 30

w

c

R T

L Hx = ++ °

153.8 9.0

8.24 2 2.667 2 3 tan 30

kips kips T

ft ft ft ft = = +⋅ +⋅ ⋅ °

[A13.4.2-1]

[T A13.2-1]

20.1 jM ft kips n =− −
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 339 of 369

Longitudinal Slab and Web Reinforcing

The top and bottom slabs act as one-way members, carrying forces in the 

transverse direction. A summary of the reinforcing to this point is:

• Top slab, top, transverse reinforcing: #4 at 1’-0”, with additional #5 at 1’-0”

in overhang.

• Top slab, top longitudinal reinforcing: #5 at 1’-0” except over the piers,

where additional #5 at 1’-0” are used for negative moment capacity.

• Bottom slab, transverse reinforcing: #4 at 1’-0”, top and bottom.

It remains to calculate the bottom longitudinal reinforcing for the top slab and 

both mats of longitudinal reinforcing for the bottom slab, as well as the 

longitudinal reinforcing in the webs. In each case, the reinforcing is controlled 

by temperature and shrinkage requirements.

For the top slab, bottom mat, using the thickest portion of the slab:

where,

Use #4 bars at 1’-0”, which is 0.2 in2

/ft.

For the bottom slab, the results will be lower. However, use #4 bars at 1’-0”, 

which is 0.2 in2

/ft for the longitudinal reinforcing in each face.

Similarly, for the longitudinal reinforcing in the webs, which are 1’-0” thick,

Use #4 bars at 1’-0”, As=0.2 in2 /ft. 

47.5 570

13

60 y

b ft in

h in

f ksi

= =

=

=

2

0.137 s

in A

ft =

( )

2 2 0.11 0.60

1.30

2

s

s

y

in ft A in ft

bh A

b hf

≤ ≤

≥

+

[5.10.8]

2 2

0.107 0.11 s

in in A

ft ft

= →
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Appendix D – Design Example 2 340 of 369

6. Live Load Distribution

LRFD 4.6.2.2.1 discusses the conditions for calculation of distribution factors. 

It is found that this cross section does not meet the requirements for use of 

4.6.2.2.2, since the number of cells (Nc) is less than 3, and the spacing of the 

of the webs (S) is greater than 13ft. Thus, the number of lanes of live load for 

design is calculated by creating a finite element model of the bridge, as 

discussed in AASHTO LRFD Article 4.6.3. 

The model finite element model uses plates for the top and bottom slabs, rigidly 

connected to beams representing the webs of the box girder. It is loaded with 

live load in positions to create maximum moments and shears for the following 

locations, for both the center and exterior webs:

• End Span—Maximum Positive Bending.

• Center Span—Maximum Positive Bending.

• End Span and Center Span—Maximum Negative Bending.

• End Span—Shear at Abutment.

• End Span—Shear at Pier.

• Center Span—Shear at Pier.

The effects for one, two and three lanes loaded are studied, each with the 

appropriate multiple presence factors applied. In interpreting results for the 

finite element model, it is important to determine how much of the top and 

bottom slabs are attributed to each web by calculating the amount of top and 

bottom slab that creates the same center of gravity for each web as for the 

entire cross section. 

The results show that for bending, 3 lanes loaded creates the maximum 

longitudinal bending in each web for each case, while for shear, 2 lanes loaded 

creates maximum shear in each web for each case.

These results were compared to a single lane loaded on a straight bridge, to 

calculate the number of lanes carried by each web for each case. Table D.14 

shows the results of the bending investigation, and table D.15 shows the 

results for shear.

Center Web Exterior Web

End

Span (+)

Center

Span (+) Negative EndSpan (+)

Center

Span (+)

0.91 0.91 0.93 0.86 0.87

Table D.14 – Number of Lanes per Web - Bending 

[4.6.2.2.1]

[4.6.2.2.2]

[4.6.2.2.3]

[4.6.3]

[3.6.1.1.2-1]
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 341 of 369

Center Web Exterior Web

End Span

@ Abut.

End Span

@ P1

Span 2

@ P1

End Span

@ Abut

End Span

@ P1

Span 2

@ P1

1.30 1.20 1.18 1.24 1.11 1.14

Table D.15 – Number of Lanes per Web - Shear 

All of the results are fairly consistent for bending, with the largest number of 

lanes carried by the center web in negative bending. Therefore, this analysis 

uses 3x0.93=2.79 lanes of live load for bending, including the 0.85 multiple 

presence factor. This is an increase of 9.4 percent over the 3 lanes times 0.85 

multiple presence factor the bridge can carry according to LRFD, where 

3x0.85=2.55 lanes. 

For shear, two different values are chosen, because the variation among the 

cases is greater than it is for bending. At the abutments (and in the ends of the 

end spans), a value of 1.30 lanes per web is used, for a total of 3x1.30=3.9 

lanes. At other locations, a value of 1.20 lanes per web is used, for a total of 

3x1.20=3.6 lanes. 

Note that the bridge can be loaded in the longitudinal model without regard to 

live load torsion, since torsion is implicitly included in the above loadings.

7. Modeling

As discussed in the introduction, this example has supports beneath each web, 

which are considered to fix the bridge for superstructure torsion and transverse 

displacement, but it is free to shorten and lengthen. Thus, the superstructure 

can be separated from the substructure for analysis purposes. 

Once the total number of design lanes is known, the bridge can be modeled as 

a spine beam with straight segments along the curve for all types of loads.

The curvature is within the limits discussed in LRFD 4.6.1, since the largest 

central angle is as follows, and is less than the 34° limit:

Where the span length along the centerline of the box is 160ft, and the radius

of the centerline is 600ft.

For Dead Loads and PT, the bridge is modeled in a three-dimensional program 

t hat allows the actual curved geometry to be modeled, as well as the input of 

the post-tensioning tendons along their paths, and calculates their effects, 

including time-dependent changes. The model is made up of straight

segments, with nodes every 5 ft, beginning at the centerline of 

bearings at the abutments. It has the properties of the gross cross 

section. 

[3.6.1.1.1]

[3.6.1.1.2]

[4.6.3.4]

[4.6.1.2.3]
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 342 of 369

The post-tensioning modeled is three, 19 x 0.6” strand tendons per 

web, draped so that they are low at the midspan and high at the piers. 

At the abutments, the tendons are spread to allow room for the 

anchorages, but otherwise, they are parallel, with 2” clear between the ducts. 

It should be noted that for this type of bridge a time-dependent program is not 

necessary, and LRFD 5.9.5 may be used to estimate the post-tensioning 

losses. 

Live loads, which cannot be run in the time-dependent program, are calculated 

using a separate three-dimensional finite element program, with the same 

spine beam type layout as the one used for dead load. Several cases were 

run, representing the above-calculated numbers of lanes. 

Member forces and stresses were then taken from these two models to 

check Service and Strength Load Groups for flexure and shear. 

8. Flexural Design

There are two flexural designs required by the AASHTO LRFD specifications 

for prestressed concrete: flexural designs at the Service Limit State and at the 

Strength Limit State. 

8.1 Service Limit State 

There are two load combinations for the Service Limit State: Service I and 

Service III. The Service I load combination is used to check against maximum 

allowable compression in the superstructure and Service III is used to check

against maximum allowable tension. Table D.16 shows the Service I and 

Service III load combinations. 

Table D.16 – Service Load Combinations 

The maximum allowable compressive stresses are: 

• 0.6 f’ci for temporary stresses at the time of tendon stressing.

• 0.45 f’c due to prestressing and permanent loads after all time dependent

effects and losses.

• 0.6 f’c due to prestressing, permanent, and transient loads after all time

dependent effects and losses.

The maximum allowable tensile stresses are:

• 0.24√ f’ci (ksi) for temporary stresses at the time of tendon stressing

(7.5√ f’ci for units in psi).

Load

Combination DC DW CR + SH PS TG LL+I

Service I 1.0 1.0 1.0 1.0 .0.5 1.0

Service III 1.0 1.0 1.0 1.0 0.5 0.8

[5.9.5.4.1]

[3.4.1]

[3.4.1-1]

[5.9.4]
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 343 of 369

• 0.019√ f’c (ksi) for stresses after all time dependent effects and losses

(6√ f’c for units in psi).

Temporary Initial Stress Check:

At the time the post-tensioning is stressed, the controlling point for compressive 

stresses is located in the bottom of the section at the centerline of the pier 

(Node 26). 

It should be noted that though there is a diaphragm at this section (which 

changes the section properties), the longitudinal stresses do not typically have 

room to spread significantly through the diaphragm. Thus, the typical section 

properties are used, and the stresses are taken at the centerline of the pier. 

However, they are conservatively taken for the peak moment at this location. 

That peak moment does not exist in reality, since the bearing has a length. 

Thus, if stresses due to the peak moment become critical, that bearing length 

can be accounted for and the design moments reduced somewhat.

At Node 26,

At the time the post-tensioning is stressed, the output from the analysis shows 

there is no tension in the bridge.

Stresses after All Losses:

The controlling tension and compression in a post-tensioned structure is 

generally always after all long term losses have occurred. The relaxation of the 

post-tensioning, as well as shortening due to creep and shrinkage typically 

cause increases in tension at the top of the bridge over the piers, and at the 

bottom of the bridge at midspan. The compression at the bottom of the soffit at 

the piers typically increases over time, as well.

This effect can be seen in table D.17, which summarizes all dead load plus 

post-tensioning effects at the two midspan nodes as well as at the centerline of 

pier, both when the structure is opened to traffic as well as when all losses 

have taken place. The other loads remain constant over time, unless a future 

wearing surface is added, which will increase the differences between the 

stresses at the times shown.

f DC PS btm = + 1.0( )

133.4 0.6 4.0 2.4 345.6 btm f ksf ksi ksi ksf =− >− ⋅ =− =−
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 344 of 369 

Table D.17 – Change in Superstructure Stresses over Time

At the bottom at the centerline of pier (which is a compression point) the 

compression increases over time. At the other locations, which are checked 

against allowable tension, the compression decreases over time.

After all losses have occurred, the controlling point for compression is located 

in the bottom of the section at the center line of the pier (Node 26).

This is the same stress as shown in the above table. 

Final Compression Stress Check, All Loads:

The controlling point is again located at the bottom of the section at the pier 

(Node 26). Use Service I load combination.

Final Tensile Stress Check, All Loads:

The controlling point is located in the bottom of the section at the midspan of 

Span 2 (node 42). Use Service III load combination:

Node Location

Σ (DL + PT)

Open to

Traffic

(ksf)

After All

Losses

(ksf)

12 Bottom, in End Span -88.4 -81.4

26 Top, at Pier -35.9 -28.8

26 Bottom, at Pier -163.8 -164.9

42 Bottom, in Main Span -48.3 -40.9
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 345 of 369

The table below contains a summary of the stress checks every ten feet along 

the superstructure. 

Node Distance

Initial > -

345.6ksf

Permanent > -

518.4ksf Service I > -0.6 f c

Service III > 

6

Bottom

(ksf)

Top

(ksf)

Bottom

(ksf)

Top

(ksf)

Bottom

(ksf)

Top

(ksf)

Bottom

(ksf)

Top

(ksf)

2 0 -116.8 -73.2 -111.1 -64.1 -111.0 -122.3 -45.5 -118.0

4 10 -110.8 -78.7 -96.4 -75.6 -100.3 -150.5 -54.7 -83.5

6 20 -107.3 -82.5 -86.3 -83.9 -94.4 -172.1 -60.6 -57.4

8 30 -106.2 -84.6 -80.6 -89.2 -93.8 -187.3 -63.0 -39.7

10 40 -107.2 -85.2 -79.0 -91.6 -97.3 -196.5 -62.5 -29.8

12 50 -109.8 -84.7 -81.4 -91.2 -104.7 -199.9 -59.2 -27.2

14 60 -111.5 -84.9 -86.2 -89.2 -114.6 -199.1 -54.4 -30.4

16 70 -111 -86.2 -91.6 -86.9 -125.0 -195.3 -49.2 -36.8

18 80 -106.5 -87.9 -95.7 -83.7 -134.1 -188.3 -43.2 -44.9

20 90 -100.9 -90.2 -99.4 -79.6 -142.9 -177.8 -36.1 -55.6

22 110 -95 -92.7 -104.6 -74.4 -153.3 -164.4 -28.0 -69.8

24 100 -110.1 -80.3 -116.9 -64.4 -173.8 -147.4 -13.3 -89.7

26 120 -105.6 -80.8 -164.9 -28.8 -241.0 -113.4 33.0 -135.3

28 130 -75.2 -98.9 -105.8 -66.9 -161.3 -150.1 -16.4 -77.9

30 140 -73.2 -98.5 -79.2 -82.7 -118.8 -172.6 -41.0 -43.5

32 150 -75 -96.3 -69.4 -88.2 -99.5 -186.7 -51.8 -23.8

34 160 -75.7 -94.7 -60.9 -93.0 -88.7 -200.4 -57.9 -4.9

36 170 -75.5 -93.8 -53.6 -97.0 -80.5 -212.1 -62.4 11.2

38 180 -74.5 -93.3 -47.6 -100.0 -74.5 -220.6 -65.4 23.6

40 190 -73.1 -93.1 -43.2 -101.9 -70.0 -225.8 -67.3 31.9

42 200 -73.5 -93.1 -40.9 -102.6 -67.7 -227.5 -68.0 35.4

44 210 -74.9 -93.3 -43.6 -101.9 -70.4 -225.6 -67.3 31.4

46 220 -75.8 -93.8 -48.1 -100.0 -74.8 -220.5 -65.4 23.2

48 230 -75.9 -94.8 -54.0 -97.0 -80.8 -212.1 -62.5 10.8

50 240 -75.1 -96.4 -61.3 -93.1 -88.9 -200.5 -58.0 -5.2

52 250 -73.3 -98.7 -69.7 -88.3 -99.8 -186.9 -52.0 -24.1

54 260 -75.2 -99.2 -79.4 -82.8 -118.7 -172.8 -41.3 -43.6

56 270 -105.6 -81.1 -106.0 -67.1 -161.3 -150.4 -16.7 -77.9

58 280 -110.2 -80.5 -165.0 -29.1 -241.0 -113.7 32.7 -135.4

60 290 -95.1 -92.9 -117.0 -64.7 -173.9 -147.7 -13.6 -89.8

62 300 -101.1 -90.3 -104.7 -74.7 -153.5 -164.6 -28.1 -69.9

64 310 -106.7 -88 -99.6 -79.8 -143.2 -177.9 -36.2 -55.8

66 320 -111.2 -86.3 -95.9 -83.9 -134.5 -188.3 -43.2 -45.3

68 330 -111.6 -84.9 -91.6 -86.9 -125.1 -195.4 -49.2 -36.8

70 340 -109.9 -84.7 -86.3 -89.3 -114.7 -199.2 -54.5 -30.3

72 350 -107.2 -85.2 -81.5 -91.2 -104.7 -199.8 -59.2 -27.4

74 360 -106.2 -84.6 -79.0 -91.6 -97.2 -196.4 -62.5 -29.9

76 370 -107.3 -82.5 -80.6 -89.2 -93.8 -187.2 -63.0 -39.8

78 380 -110.8 -78.8 -86.4 -83.9 -94.4 -172.2 -60.6 -57.4

80 390 -116.7 -73.3 -96.5 -75.6 -99.4 -150.6 -55.2 -83.3

82 400 0 0 -111.3 -64.2 -109.1 -122.4 -46.7 -118.1

Table D.18 – Stress Summaries
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 346 of 369

8.2 Strength Limit State

There is one load combination used to check the strength limit state: Strength I.

Load

Combination DC DW CR + SH PS LL+I

Strength I 1.25 1.5 1.0 1.0 1.75

Table D.19 – Strength Limit State Design Factors 

The factored flexural resistance, Mr, must be greater than the Strength I load 

combination, Mu (ultimate moment), and the minimum reinforcement 

requirement. The minimum reinforcement requirement states that the flexural 

resistance must at least equal 1.33Mu or the cracking moment, Mcr, whichever 

is less. This can be summarized by the following equation (using absolute 

values):

Factored Design Moment

The critical location is the negative moment at the centerline of Pier 2 (Node 

26). The summation of the Strength I load combination is:

Minimum reinforcement calculation

where,

[3.4.1]

[1.3.2.1-1]

[5.7.3.2]

[5.7.3.3.2]

[3.4.1]

[5.7.3.3.2]
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 347 of 369

Therefore, at this location, the design moment is 1.33·Mu. The absolute value 

of 1.33·Mu is less than the absolute value of Mcr, and this value is greater than 

the absolute value of the factored moment, Mu.

Factored Flexural Resistance

Nominal moment calculations will be shown at the critical location. First, the 

nominal moment capacity, Mn, will be calculated using prestressing only. If 

necessary, non-prestressed reinforcing will then be added to increase the 

capacity.

First, LRFD Equation 5.7.3.1.1-4 will be checked, to see if the section is 

rectangular or flanged, as discussed in LRFD 5.7.2.2. The following are known 

from the cross section dimensions, concrete strength, and post-tensioning 

layout at Node 26:

Using these values in equation 5.7.3.1.1-4,

and

which is less than the bottom slab thickness of 7in, so the section can be 

treated as rectangular, and equation 5.7.3.1.1-4 is the correct one to use to 

determine the neutral axis location.

Next, we use equation 5.7.3.1.1-1 to find the stress in the prestressing at 

ultimate, and obtain:

Before the capacity is calculated, find φ.

 

 

[5.7.2.2]

C5.7.3.1.1-1]

[5.5.4.2]
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 348 of 369

where,

Knowing these values, we now use equation 5.7.3.2.2-1 to calculate the 

moment capacity φMn, and obtain:

Comparing φMn to the design moment of 1.33·Mu, we find that the calculated 

capacity is too low. Thus, we will determine if reasonable amount of mild 

reinforcing over the pier can increase the capacity sufficiently.

Try longitudinal #5 bars at 6 inch spacing in the top of the deck. The total 

number of bars is 95. Recalculate the above values, incorporating this 

reinforcing. The values of c, fps, and φ need to be checked before entering 

equation 5.7.3.2.2-1.

And the increased capacity, accounting for mild reinforcing is

Which is greater than the design moment of

The following table contains a summary of the flexural strength design checks 

at typical critical locations along the superstructure.

 
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 349 of 369

Table D.20 – Factored Moments 

9. Web Design

Critical Section:

For shear and torsion design, it is unnecessary to calculate shear and torsion

values closer than dv to supports. LRFD 5.8.2.9 states

Where,

Near the abutments, it is clear that dv=0.72h, since the post-tensioning is not 

near the extremities of the beam. Near the piers, we will calculate dv using the 

properties at the centerline of the pier, for the purposes of determining where 

[5.8.3.2]

[5.8.2.9]

[5.8.2.9-2]

the critical section is located. We know the following values from the ultimate 

moment calculations:

Node 12 26 42

Location ~0.4 L Pier 2 0.5 L

Distance (ft) 50 120 200

DC (ft-kips) 9962.5 -22324.6 12949.7

DW (ft-kips) 1810.7 -4057.5 2353.6

CR (ft-kips) 9.0 21.0 21.1

SH (ft-kips) 0.0 0.0 0.0

PS (ft-kips) 2522.4 6068.9 6082.5

LL+I (ft-kips) 8448.2 -9593.2 9323.9

Mu (ft-kips) 32485.0 -44690.2 42138.1

1.33Mu (ft-kips) 43205.0 -59438.0 56043.6

Mcr (ft-kips) 44047.2 -62507.7 41781.8

Design Moment 1.33Mu 1.33Mu Mu

(ft-kips) 43205.0 -59438.0 42138.1

φMn (ft-kips) 47323.7 -63714.7 55027.4
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 350 of 369

Thus, the critical section could be taken as far away as 5.55 ft from the support. 

However, it is conservative to take the critical section closer to the support 

(since the section properties are constant and shear typically dominates the 

design over torsion), and there is a node at 5.00 ft. So, the critical section will 

be taken at 5.0 ft from the pier and abutments.

Since shear and torsion combine in an exterior web, they must be considered 

together. However, to this point, torsion has only been implicitly considered in 

the calculation of the number of design lanes performed in section 3. In that 

section, loads were placed on the bridge that maximized the shear in the center 

and exterior webs. The interior web controlled, and the number of lanes taken 

by the center web was multiplied by the number of webs to give a conservative 

number of design lanes for the bridge; 3.9 lanes near the abutments and 3.6 

lanes elsewhere. These lanes are placed along the centerline of the bridge in 

the spine model.

LRFD 5.8.2.1 provides a formula that calculates whether torsion needs to be 

considered in the design. If this formula shows that torsion needs to be 

considered, we will calculate the required shear and torsion reinforcing for two 

cases, and use the most conservative design: 

1) Maximum torsion on the bridge with its concurrent shear from a load

case that is derived from the standard LRFD Loading.

2) Maximum shear as calculated in section 3, along with its concurrent

torsion (if torsion needs to be considered for this case).

Determine the number of lanes that produce the maximum torsion for Case 1), 

above. Using the description of lanes and the transverse placement of loads in 

LRFD 3.6.1.1 and 3.6.1.2: 

• Eccentricity for 1 Lane = 17 ft.

• Eccentricity for 2 Lanes = 11 ft.

• Eccentricity for 3 Lanes = 5 ft.

 

 

 

[C5.8.2.1]

[3.6.1.1]
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 351 of 369

The controlling case is found by using number of lanes and multiple presence 

factors: 

Thus, two Lanes controls for maximum torsion.

The finite element spine model is then used to compute the live load shear and 

torsion for two lanes positioned as close to the barrier as possible. The time 

dependent model gives dead load and PT forces and stresses. Once these are 

known, Tu and Tcr can be calculated and LRFD 5.8.2.1 can be entered to find 

whether torsion is critical for Case 1). 

Constants used include the following:

The following table shows the calculation of Tcr and compares it to Tu

according to LRFD 5.8.2.1-3. Tu is calculated according to LRFD 3.4.1. The 

two locations where torsion must be considered, shown in bold, are near the 

piers in the center span.

Node fpc

(ksf)

Tcr

(k-ft)

T DC

(k-ft)

T DW

(k-ft)

T CR

(k-ft)

T PT

(k-ft)

T LL+I

k-ft)

Tu

(k-ft)

0.25φTcr

(k-ft)

3 83.7 27647 555.0 100.9 0.0 225.2 2904 6153 6221

7 85.3 27820 340.9 62.0 -0.1 217.5 2316 4789 6259

11 86.8 27980 11.6 2.1 -0.3 164.4 1594 2971 6296

14 88.0 -28108 -222.3 -40.4 -0.6 93.4 1135 1740 -6324

17 89.0 -28215 -376.0 -68.3 -0.9 -8.2 -1605 -3390 -6348

21 87.2 -28023 -349.2 -63.5 -1.4 -191.6 -2242 -4649 -6305

25 85.1 -27798 -75.4 14.0 -2.1 -414.6 -2811 -5410 -6255

27 83.6 27636 398.3 72.4 2.6 699.8 3218 6940 6218

31 80.9 27343 690.6 125.5 1.9 536.9 2760 6420 6152

35 79.6 27200 596.8 108.5 1.1 373.7 2248 5218 6120

39 78.3 27057 264.0 48.0 0.4 191.4 1692 3554 6088

42 77.4 -26957 -53.9 -9.8 -0.1 42.8 1274 2190 -6065

Table D.21 – Verification of Torsion Considerations 

[5.5.4.2.1]

[5.8.2.9]

[5.8.2.1]

[5.8.2.1-4]

[3.4.1]

[3.6.1.2]
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 352 of 369

Thus, we can develop a load case that requires that torsion be considered for 

at least some locations in the structure. As previously stated, the reinforcing 

for this load case will be compared to that required for Case 2) at node 27, 

where the difference between the applied torsion and the limit is greatest, 

according to the above table.

Shear reinforcing for these two cases will be calculated using the simplified 

procedure of LRFD 5.8.3.4.3.

Case 1), Node 27:

The Governing equation for shear says the capacity, which is made up of the 

strength of concrete plus reinforcing, must be greater than the factored load, or,

And

where

and

Vci relates to the tendency of flexure cracks to become shear cracks, and 

involves the following terms:

• Mmax, which is the ultimate moment under the maximum bending load

case minus the bending due to unfactored dead load. For this load case,

the live load moments are taken from the Case 1) loading, which is two

lanes placed with as much eccentricity as possible.

• Vi, which is the shear concurrent with Mmax.

• Mcre, which is the moment, in addition to unfactored dead load moment,

required to bring the flexural stress up to the code defined cracking

stress, as defined in LRFD Equation 5.8.3.4.3-2.

u

n cs

V V VV

j

=+≥

V VV c = min , ( ) ci cw

max

0.02 0.06 i cre

ci cvv d cvv

V M V f bd V f bd M

= ++ ≥ ′ ′

V f f bd V cw (0.06 0.30 c pc v v p ) =+ + ′

[5.8.3.4.3]

[5.8.3.4.3-1 

[5.8.3.4.3-3]

[5.8.3.3]

[5.8.3.4.3]

[5.8.3.4.3-2]
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 353 of 369

• Vd, which is the unfactored dead load shear.

• bv and dv, which have been previously defined.

Vcw relates to the tendency of the web to crack from principal tension:

• fpc is the compressive stress in the section at the center of gravity.

• Vp is the shear from post-tensioning.

• bv and dv are as previously defined.

Calculate Vci: 

And,

For Vcw: 

And

Thus,

The applied shear, Vu, is calculated from applied dead and live loads for Case 

1):

Node V DC

(k)

V DW

(k)

V CR+SH

(k)

V PT sec

(k)

V LL+I

(k)

Vu

(k)

27 825.3 150.0 0.0 0.0 287.0 1758.8

 

[3.4.1]

Table D.22 – Ultimate Shear at Node 27 
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 354 of 369

The required shear strength from reinforcing is:

Using LRFD Equation 5.8.3.3-4 for Vs, and realizing that the reinforcing will be 

placed at 90° to the longitudinal axis:

To find the required reinforcing, solve the above for Av/s, using known values.

Where

solving,

Since there are 3 webs, and each web has 2 faces, this equates to 

0.373in2

/ft·face.

For Case 1, the reinforcing for torsion must now be calculated. Note that the 

following equation gives the reinforcing required in one outside web: 

From the previous calculations in this section, at Node 27:

 

 

[5.8.3.3-1]

[5.8.3.3-4]

[5.8.3.4.3-4]

[5.8.3.6.2-1]

cotθ =1.7 j = 0.9
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 355 of 369

Now, summing the required reinforcing, the critical exterior web would require:

For Case 1). Note that the reinforcing in the exterior webs would be made 

equal to this minimum, while the interior web could use 0.373 in2

/face, for a 

total of 2.61 in2

/ft in the bridge

Case 2), Node 27:

Case 2) utilizes the load distribution calculated in section 3, with the live load 

placed along the centerline of the bridge, as explained in that section. Thus, 

the only torsion is due to the fact that the bridge is curved.

All permanent load effects are the same for the two cases so only the live load 

varies between the two cases. The table below shows that torsion need not be 

explicitly considered with Case 2).

Node fpc

(ksf)

Tcr

(k-ft)

T DC

(k-ft)

T DW

(k-ft)

T CR

(k-ft)

T PT

(k-ft)

T LL+I

k-ft)

Tu

(k-ft)

0.25φTcr

(k-ft)

3 83.7 27647 555.0 100.9 0.0 225.2 694 2284 6221

7 85.3 27820 340.9 62.0 -0.1 217.5 565 1725 6259

11 86.8 27980 11.6 2.1 -0.3 164.4 288 685 6296

14 88.0 -28108 -222.3 -40.4 -0.6 93.4 -83 -390 -6324

17 89.0 -28215 -376.0 -68.3 -0.9 -8.2 -214 -957 -6348

21 87.2 -28023 -349.2 -63.5 -1.4 -191.6 -290 -1233 -6305

25 85.1 -27798 -75.4 14.0 -2.1 -414.6 -140 -736 -6255

27 83.6 27636 398.3 72.4 2.6 699.8 242 1732 6218

31 80.9 27343 690.6 125.5 1.9 536.9 478 2427 6152

35 79.6 27200 596.8 108.5 1.1 373.7 469 2104 6120

39 78.3 27057 264.0 48.0 0.4 191.4 303 1124 6088

42 77.4 -26957 -53.9 -9.8 -0.1 42.8 145 215 -6065

[5.8.3.6.1]

[3.4.1]

[5.8.2.1]

Table D.23 – Verification of Torsion Considerations 

And we can calculate Av/s for one web as:
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 356 of 369

For Node 27 we find:

Vci, Case 2):

And,

Vcw is the same as for Case 1), and

The applied shear, Vu, is:

Node V DC

(k)

V DW

(k)

VCR+SH

(k)

V PT sec

(k)

V LL+I

(k)

Vu

(k)

27 825.3 150.0 0.0 0.0 510.8 2150.5

Table D.24 – Shear at Node 26 

The required shear strength from reinforcing is:

Using LRFD Equation 5.8.3.3-4 for Vs, and realizing that the reinforcing will be 

placed at 90° to the longitudinal axis:

Solving the above for Av/s, using known values:

 

 

[5.8.3.4.3-1]

[5.8.3.4.3-3]

[3.4.1]

[5.8.3.3-1]

[5.8.3.3-4]
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 357 of 369

Thus, at Node 27, the total amount of shear and torsion reinforcing in the 

bridge is equal to 2.61in2/ft for Case 1), and 3.1in2

/ft for Case 2). Of the two 

nodes that required investigation of torsion under Case 1), the worst case did 

not control over Case 2), and it will be conservative to design the structure 

using the forces developed under section 3, as in Case 2.

9.1 Shear Design

Shear will be investigated at discreet points along the bridge. The symmetry of 

the spans will be considered, and only half of the bridge will be reported. 

Points for investigation are chosen at distances that would be meaningful for 

changes in shear reinforcing, in this case, at about 20 ft.

Dead Load and Post-Tensioning values are from the 3D, time-dependent 

model, and live load is from the spine beam and load cases discussed in

section 3 and Case 2), above. The table below shows the calculation of Vu. 

Note that only shear due to secondary PT moments is considered.

Node Distance

(ft)

V DC

(k)

V DW

(k)

V 

CR

(k)

V PT

sec

(k)

V LL+I

(k)

Vu

(k-ft)

3 5 419.5 76.2 0.2 50.6 443.4 1465.3

7 25 199.4 36.2 0.2 50.6 326.3 925.3

11 45 -20.7 -3.8 0.2 50.6 223.4 410.3

14 60 -185.8 -33.8 0.2 50.6 -241.3 -654.1

17 75 -350.8 -63.8 0.2 50.6 -305.2 -1017.5

21 95 -570.9 -103.8 0.2 50.6 -396.5 -1512.4

25 115 -791.0 -143.8 0.2 50.6 -484.6 -2001.6

27 125 825.3 150.0 0.0 0.0 510.8 2150.5

31 145 605.2 110.0 0.0 0.0 428.7 1671.8

35 165 385.1 70.0 0.0 0.0 344.7 1189.6

39 185 165.1 30 0.0 0.0 262.7 711.0

42 200 0.0 0.0 0.0 0.0 201.2 352.1

Table D.25 – Ultimate Shear Forces

Calculate Vci: 

Using the same methodology and constants as above, the calculation of Vci at 

the nodes is shown in the table below. Note that the absolute values of Mcre, 

Mmax, Vi, and Vd are used in the calculation of Vci. 

Applied Shear:

[3.4.1]

[5.8.3.4.3-1]
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 358 of 369

Node fcpe

(ksf)

Mdnc

(k-ft)

S 

(ft3) 

Mcre

(k-ft)

Mmax 

(k-ft)

Vi

(k)

Vd

(k)

Vci

(k)

3 122.0 2608.0 138.2 23997.2 3784.7 748.3 495.9 5337.2

7 154.8 9939.6 138.2 21205.3 15077.2 517.4 235.8 1060.2

11 166.9 12058.3 138.2 20764.6 20283.6 259.8 -24.3 386.9

14 160.3 10225.4 138.2 21674.7 20570.5 -30.6 -219.3 348.2

17 133.6 5460.7 138.2 22756.7 18048.5 -186.8 -414.4 746.6

21 104.2 -5446.8 200.5 29591.6 -6688.1 -248.1 -674.5 1868.9

25 157.5 -21547.3 200.5 24176.8 -14600.3 -708.7 -934.6 2204.8

27 157.0 -21339.8 200.5 24290.0 -14266.5 779.5 975.3 2399.1

31 107.4 -4411.7 200.5 31273.5 -1720.3 446.7 715.2 8933.2

35 110.2 7318.0 138.2 17661.0 19847.7 525.4 455.1 1019.4

39 145.4 13837.2 138.2 16014.0 25570.1 320.6 195.1 492.5

42 151.7 15303.4 138.2 15411.9 26834.7 -42.1 0.0 290.1

Table D.26 – Concrete Shear Capacity, Vci

Calculate Vcw: 

The calculation of Vcw is performed, again, with the same constants as were 

used for the calculations performed at Node 27, above. Vp is the shear due to 

primary and secondary PT forces, and typically increases the capacity Vcw. 

Table D.27 – Concrete Shear Capacity, Vcw

The final table repeats Vci and Vcw, calculates Vc, and calculates the required 

reinforcing in in2

/ft. Note that at Nodes 7, 11, and 42, the minimum reinforcing 

value controls.

Node Fpc

(ksf)

Vp

(k)

Vcw

(k)

3 83.6 -279.8 913.5

7 85.2 -136.9 777.3

11 86.8 12.2 658.9

14 88.0 171.3 822.9

17 88.9 356.0 1011.5

21 87.2 589.6 1237.9

25 85.1 264.8 904.5

27 83.5 -167.7 801.0

31 80.8 -594.8 1217.1

35 79.6 -372.7 989.9

39 78.3 -157.1 768.8

42 77.4 0.0 608.2

[5.8.3.4.3-3]

[5.8.3.4.3]

[5.8.3.3-4]
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 359 of 369

Node Vci

(k)

Vcw

(k)

Vc

(k) cotθ

Req’d 

Vs 

(k)

Req’d 

Av/s

(in2

/ft)

3 5337.2 913.5 913.5 1.7 714.7 1.4

7 1060.2 777.3 777.3 1.7 250.8 0.5

11 386.9 658.9 386.9 1.0 69.0 0.5

14 348.2 822.9 348.2 1.0 378.5 1.3

17 746.6 1011.5 746.6 1.0 383.9 1.3

21 1868.9 1237.9 1237.9 1.7 442.6 0.8

25 2204.8 904.5 904.5 1.7 1319.6 2.5

27 2399.1 801.0 801.0 1.7 1588.4 3.1

31 8933.2 1217.1 1217.1 1.7 640.5 1.3

35 1019.4 989.9 989.9 1.7 331.9 0.7

39 492.5 768.8 492.5 1.0 297.5 1.0

42 290.1 608.2 290.1 1.0 101.1 0.5

Table D.28 – Required Web Reinforcing for Shear 

9.2 Regional Web Bending 

LRFD 5.8.1.5 states that webs shall be reinforced for vertical shear and torsion, 

regional web bending, and transverse web bending due to vertical loads. The 

above Shear Design section calculated the reinforcing required from shear. 

Section 2 calculated the reinforcing required from web bending. What remains 

is to calculate the reinforcing required from regional web bending.

Regional web bending is bending that places tension and compression on the 

inside and outside faces of the webs, due to the radius of the bridge and, thus, 

the curvature of the tendons. It is partially resisted by the longitudinal 

compression in the web, which bends the web in the opposite direction. 

However, the resistance from the longitudinal compression is not accounted for 

in the following, since it is not mentioned in the AASHTO LRFD Bridge Design 

Specifications.

The regional web bending will be calculated using LRFD 5.10.4.3.1d. The 

values used will be those when the bridge is first opened to traffic, as that is the 

earliest time the web will be subjected to all three effects – shear, transverse 

bending, and regional bending. Thus, the design will be conservative because 

shear and transverse bending are taken when they are most critical (after all 

losses), and regional web bending is taken when it is most critical (at opening 

to traffic).

Observing the maximum force in the post-tensioning tendons at the time the

bridge is opened to traffic, from the time-dependent program output:

[5.8.1.5]

6800kips kips Ppt = = 2266.7

3webs web

Calculating values for the webs, with the radius of the inside web equal to 584 

ft, and the load factor of 1.2,

[3.4.3]
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 360 of 369

For exterior webs,

For the interior web,

No indication of a difference between positive and negative moments is given 

in LRFD, so the assumption is that these values apply to both. For the exterior 

webs, these regional forces work in the same direction, so they put the inside 

face in tension on one web, and the outside in tension on the other. The 

effects of transverse bending are not symmetrical on these webs, so the 

regional bending forces will be used as both positive and negative at all 

locations.

Table D.29 below shows the ultimate moments and required reinforcing for the 

general web bending due to the curved PT tendons:

Node

Positive Moment Negative Moment

Mu (+)

(k-ft)

As/s

(in^2/ft)

Mu (-) 

(k-ft)

As/s

(in^2/ft)

32 3.3 0.072 -3.3 0.072

33 3.3 0.072 -3.3 0.072

34 3.3 0.072 -3.3 0.072

35 0.0 0.000 -3.9 0.085

36 3.9 0.085 0.0 0.000

37 0.0 0.000 -3.9 0.085 

Table D.29 - Regional Web Bending Due to PT Moments 

And Required Reinforcing 

9.3 Total Web Reinforcing

The reinforcing for shear, transverse bending due to vertical loads, and regional 

web bending due to PT will now be combined to find the total reinforcing 

required in the webs. The shear reinforcing requirements vary along the 

bridge, and the bending reinforcing requirements vary from web to web and 

vertically along each web, as well as from face to face in each web. Also, the 

live loading that produces maximum shear in a given web will not 

produce maximum bending. The converse is also true, as a single truck 

 

[5.10.4.3.1a]

[5.10.4.3.1d]

[5.7.3.1]

[5.7.3.2]
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 361 of 369

produces maximum web bending, but that loading produces shears that 

are approximately one-third of the maximum shear design case. The only 

constant amount is for the regional bending from PT lateral forces, 

because LRFD 5.10.4.3.1d presents only one formula. As the tendons travel 

up and down the webs, this value would also change. Thus, there are 

many variables to accommodate in calculating web reinforcing. 

It remains to the engineer to select web reinforcing that is efficient, not only in 

material price but also to place. Too much variation in shear reinforcing 

(from web to web, top to bottom or face to face) presents opportunities for 

incorrect placement. Placing the same reinforcing in each web is logical, as is 

using the same reinforcing in each face of each web, because a single Ushaped bar can be used for reinforcing, in this case. This will also provide the 

same reinforcing from top to bottom in each web. 

As discussed above, the exact amount of reinforcing used at each location 

would be very cumbersome to calculate, as each bending case would have a 

corresponding shear reinforcing requirement, and vice-versa. Thus, it is simple 

to use a method that relates only to the maximums required. The following 

combination formula has proven to simplify the calculations and be 

conservative. 

Where,

• V = shear reinforcing required at a given location

• B = maximum transverse bending reinforcement at a given face and

height

• P = regional web bending from PT loads

• A = area of reinforcing to use

The design zones along the bridge are simplest to quantify if the locations and 

shear reinforcing requirements from Section 6.1 are recalled. In the table 

below, Node 2 is the abutment, Node 26 is the pier, and Node 42 is the 

centerline of Span 2. Note that the values shown are for the entire bridge width 

(three webs, with two faces each):

[5.8.1.5]

A V BP V BP V B P = + + ++ + + max 0.5 ,0.5 ,0.7( ) ( )
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 362 of 369

Node Distance

(ft)

Req’d 

Av/s

(in2

/ft)

3 5 1.4

7 25 0.5

11 45 0.5

14 60 1.3

17 75 1.3

21 95 0.8

25 115 2.5

27 125 3.1

31 145 1.3

35 165 0.7

39 185 1.0

42 200 0.5

Table D.30 – Shear Reinforcing Requirements 

Using the values from the above table, the design regions chosen are 

expressed in the following table:

Design

Region

Begin

Distance

End

Distance

Shear

Reinforcing

1 B/B 25 1.4

2 25 45 0.5

3 45 75 1.3

4 75 120 2.5

5 120 145 3.1

6 145 185 1.3

7 185 M/S 1.0

Table D.31 – Shear Reinforcing Design Regions 

In the above table, B/B refers to Begin Bridge and M/S refers to Midspan of

Span 2.

For each region above, the transverse bending and regional web bending must

be added, in accordance with the combination formula.

The table below recalls and summarizes the reinforcing requirements for 

positive and negative bending in the webs due to transverse bending and 

regional bending from PT, found in sections 2 and 6.2. The node 

numbers refer to the transverse analysis model. Note that the reinforcing 

shown is for one face of a web. 
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 363 of 369

Table D.32 – Web Bending Reinforcing Requirements 

As an example, the reinforcing in Region 1 would be calculated as in the 

following, for each face of each web:

Then, the combination formula yields, at the exterior webs, at the top of the 

web (Node 34), for positive bending:

The maximum of:

So, at this location, the design would require 0.589in2

/ft on each of the six 

faces.

Likewise, at the interior web, at the top of the web(Node 37), for negative 

bending:

Node

Transverse Regional

Positive

As/s

(in2

/ft)

Negative

As/s

(in2/ft)

Positive

As/s

(in2/ft)

Negative

As/s

(in2/ft)

Ext.

Webs

32 0.060 0.010 0.072 0.072

33 0.283 0.190 0.072 0.072

34 0.400 0.292 0.072 0.072

Int.

Webs

35 0.047 0.047 0.000 0.085

36 0.248 0.248 0.085 0.000

37 0.388 0.388 0.000 0.085
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 364 of 369

And the design value would be 0.590in2

/ft. Since these two values are in the 

same design region, 0.590in2

/ft would control and would become the design 

value for that region, if there were no other locations which required more 

reinforcing.

Similarly, using the combination formula to calculate the reinforcing 

requirements for both positive and negative moment at each point in exterior 

and interior webs for each design region leads to the following web reinforcing

Design

Region Required

Distance Use

Provided

Reinforcing

(in2

/ft)

1 0.590 #5@6” 0.620

2 0.515 #5@7” 0.531

3 0.581 #5@6” 0.620

4 0.696 #5@5” 0.744

5 0.796 #6@6” 0.880

6 0.581 #5@6” 0.620

7 0.556 #5@6” 0.620

Table D.33 – Final Web Reinforcing at Each Face of Each Web 

9.4 Principal Tension

The LRFD Design Specifications state that the principal stresses in the webs 

shall be analyzed for all segmental bridges. It does not give direction on nonsegmental concrete box girder bridges. However, it is a good design check and 

the calculation of principal stresses is included in this example, because such a 

check will help prevent web cracking at the service level.

The principal tensile stresses are calculated using the long-term residual axial 

stress and the maximum shear stress. The Service III Limit State load 

combination is used for both axial and shear stresses.

[5.7.3.1]

[5.7.3.2]

[5.8.5]
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 365 of 369

The critical section with the maximum principal tensile stress is located just 

upstation of Pier 2, a distance of dv (approximately 5.0 ft) from the diaphragm 

(Node 27). See section 5 above for the calculation of dv. In this case, as 

discussed in section 6, torsion need not be included. However, the principal 

tension will be calculated at the critical location including torsion, as an 

example of how to calculate the effects of torsion. 

Shear Stress from Vertical Shear 

From general mechanics of materials, the shear stress can be taken as: 

where, 

V = vertical shear in section

= VDC +VDW +VPS +VCR+SH +0.8•VLL+I

= 1216.3 kips

Q = first moment of the area = 96.0 ft3

I = section moment of inertia = 572.7 ft4

b = web width less 1⁄4 duct Ø = 0.906 ft

nw = number of webs = 3 

This calculation assumes that the shear effects all webs equally, and is 

conservative due to the fact that the load distribution in section 3 calculated the 

largest load to one web (the center one, in this case) then applied this value to 

all the webs to get the total load on the bridge.

Shear Stress from Torsion

According to section 6, torsion need not be included in this design. 

Calculations are presented here as an example only. 

In a box girder, St. Venant Torsion (the dominant effect) produces shear in the 

exterior webs. In one exterior web it adds to the vertical shear, and in the other 

it subtracts from it. On the interior webs, the shears from this torsion cancel. 

Typically, when torsion is included, it is calculated in the critical web and the 

required reinforcing is used in all webs for simplicity of placement.

Mechanics of materials gives the shear flow around a box as:

where, T = Torsion at the section

= TDC +TDW +TPS +TCR+SH +0.8•TLL+I

= 1366.5 ft-kips

w

VQ

Ibn

t =

2 o

T

q A =

[C5.8.2.1]

3

4

1216.3 96.0 75.0

572.7 0.906 3

kips ft ksf ft ft

t ⋅ → = = ⋅ ⋅
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 366 of 369

Ao = as defined in LRFD 5.8.2.1 = 203.2ft2 

This value has units of force per unit length, and represents the force acting 

along the web. To calculate the shear stress in the web, simply divide by the 

effective thickness of the web, bv: 

In this case, the shear stress due to torsion, 3.7ksf, is about 5 percent of the 

shear stress due to vertical shear, 75.0 ksf. 

Mohr’s Circle

Using Mohr’s circle, the principal tension can be calculated at the neutral axis:

The normal stress at Node 27 is:

From Mohr’s circle, the radius, R, and then the principal stress, sp, can be 

calculated with and without torsion.

With torsion,

Without torsion,

2

1366.5 3.36

2 203.2 T

ft kips kips q ft ft− → = = ⋅

3.36

3.7

0.906

T

T

v

kips

q ft ksf b ft

t = = =

83.5 DC DW PS CR SH ss s s s ksf − = + ++ = +

( ) ( )

2

2 2 2 41.75 78.7 89.1

2

R T ksf s

t t

  − = ++ = + =    

41.75 89.1 47.4

2 p R ksf s

s → = += + = −

41.75 85.8 44.1

2 p R ksf s

s → = += + = −

( ) ( )

2

2 2 2 41.75 75.0 85.8

2

R T ksf s

t t

  − = ++ = + =    
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 367 of 369

At this location, both with and without torsion included, the principal tension is 

greater than the allowable. In order to reduce the principal tension, several 

options are available:

• Increase the axial stress by adding post-tensioning.

• Increase the thickness of the webs locally.

• Increase the strength of the concrete.

Of the above solutions, locally thickening the webs is the simplest for a cast-inplace structure. The webs can be tapered from 1 ft thick at 10 ft from the pier 

to 1’-6” at the pier. Thus, at 5 ft from the pier, they will be 1’-3” thick, with 

bv=1.156 ft.

The revised section properties at Nodes 27 and 25, with 1’-3” thick webs are:

A = 96.0 ft3

Q = 96.0 ft3

I = 572.7 ft4

B = 0.906 ft

Recalculating the principal tension with the revised section at the critical 

location shows that the principal tension is now below the required value. Note 

that for such a modest, localized web thickening, it is unnecessary to 

recalculate the remainder of the design values.

The table below contains a summary of the principal tension check at the same 

locations that were used for shear. Torsion is not explicitly included.

Table D.34 – Results of Principal Tension Check 

Node bv t θ Radius θp √f’c

(ft) (ksf) (ksf) (ksf) (ksf) (psi)

3 0.906 35.2 -83.6 54.7 12.9 1.15

7 0.906 22.2 -85.2 48.1 5.4 0.49

11 0.906 10.3 -86.8 44.6 1.2 0.11

14 0.906 14.9 -88.0 46.4 2.4 0.22

17 0.906 18.7 -88.9 48.2 3.8 0.34

21 0.906 24.8 -87.2 50.1 6.6 0.59

25 1.156 51.1 -83.1 65.9 24.3 2.22

27 1.156 58.8 -79.2 70.9 31.3 2.85

31 0.906 28.6 -80.8 49.5 9.1 0.81

35 0.906 22.1 -79.6 45.5 5.7 0.51

39 0.906 15.3 -78.3 42.0 2.9 0.26

42 0.906 9.9 -77.4 39.9 1.3 0.11

However, the maximum allowable principal tension is:
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 368 of 369

9.5 Longitudinal Shear Reinforcing

The longitudinal reinforcing required for shear is calculated using values 

already known from the calculations. The table below shows that the 

prestressing alone is sufficient.

Node Mu

(k-ft)

dv

(ft)

Vu

(k)

Vp

(k)

Vs

(k) Cotθ

Required

Apsfps

(k)

Provided

Apsfps

(k)

3 6393 5.04 1465.3 279.8 714.7 1.7 2964 9741

7 25017 5.04 925.3 136.9 250.8 1.7 6285 9790

21 12135 5.04 1512.4 589.6 442.6 1.7 3922 9640

25 36148 5.04 2001.6 264.8 1319.6 1.7 9412 9748

27 35606 5.04 2150.5 167.7 1588.4 1.7 9506 9748

31 6132 5.04 1671.8 594.8 640.5 1.7 2807 9649

Table D.35 – Verification of Longitudinal Shear Reinforcing

9.6 Duct Pull-Out

Ducts are assumed to be supported in the center of a web. Thus, the cover to 

the ducts, dc, in these calculations is 3.75 inches. It is also assumed that the 

tendons are stressed to 75 percent of their capacity, when the concrete 

strength is 4500 psi. The center-to-center spacing of the ducts is 6.5 inches, 

and the clear space between ducts is 2 inches. 

Out-of-plane forces must also be added to Fu-in. The minimum radius on a 

tendon occurs on the lowest tendon as it crosses the pier, and the average 

radius in that region is 162.2 ft. The out-of-plane force

Post-tensioning ducts in curved webs apply lateral pressure to the webs. In 

addition to the regional web bending discussed earlier, local forces must also 

be addressed.

[5.8.3.5]

[5.10.4.3]

[5.10.4.3.1b]

[5.10.4.3.1a]

[5.10.4.3.2]
Post-Tensioned Box Girder Design Manual June 2016

Appendix D – Design Example 2 369 of 369

Thus, since Fu-TOTAL is less than φVn, no ties are necessary.

The 2 inch clear duct spacing must also be checked against the in-plane radius 

of the vertical duct curvature, to insure that stressing a tendon does not cause 

a shear failure that crushes an adjacent duct. Using the previous value of 

R=162.2 ft,

Thus, the 2 inch clear spacing between ducts will not require additional ties.

u

u out

P F

π R − =

( )

2

1.2 0.75 3 19 0.217 270

0.48

162.2 12 u out

in

str ksi kips str F in in ft ft

π −

⋅⋅ ⋅

= =

⋅ ⋅

0.428 0.48 0.91 u TOTAL u in u out

kips kips kips F FF

in in in − −− =+ = + =

4.5 2 3.125

4 eff

in d in =+ = in

0.75 0.15 3.125 4.5 0.75 n

kips V in ksi

in j =⋅⋅ ⋅ =

( )

2

1.2 0.75 19 0.217 270

0.51

162.2 12

u in

in

str ksi kips str F in in ft ft

−

⋅ ⋅

= =

⋅

[5.10.4.3.1d]