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Three-dimensional nonlinear seismic soil-abutment-foundation-structure interaction analysis of skewed bridges

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Three-dimensional nonlinear seismic soil-abutment-foundation-structure interaction analysis of skewed bridges

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Content THREE-DIMENSIONAL NONLINEAR SEISMIC SOIL-
ABUTMENT-FOUNDATION-STRUCTURE INTERACTION
ANALYSIS OF SKEWED BRIDGES

by

Anooshirvan Shamsabadi

A Dissertation Presented to the
FACULTY OF THE GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(CIVIL ENGINEERING)

May 2007

Copyright 2007 Anooshirvan Shamsabadi

ii
DEDICATION

To the memory of my parents for all their supports and sacrifices.

iii
ACKNOWLEDGMENTS

The author would like to express his sincere and deepest thanks to his
advisor, Professor Geoffrey R. Martin, for many hours he has spent with me for his
research project and review of this dissertation.
The author is very grateful to the rest of his committee members: Professor
Jean-Pierre Bardet, Professor Vincent Lee, and Professor Frank Crosetti.
I would like to thank Dr. Liping Yan of Arroya Geotechnical for the
friendship, consultations, and review and very helpful suggestions for this
dissertation.
In addition, I am very thankful to Mr. Po Lam of Earth Mechanics, Inc. for
his intuition and insightful suggestions in implementation and practical application
of this dissertation for bridge abutment design under seismic loading.
The comments and suggestions of my academic and professional colleagues
are also greatly appreciated.
My utmost gratitude goes to my family members for their love, patience and
understanding.
.

iv
TABLE OF CONTENTS

DEDICATION ............................................................................................................. ii

ACKNOWLEDGMENTS.........................................................................................iii

LIST OF TABLES....................................................................................................viii

LIST OF FIGURES ................................................................................................... ix

ABSTRACT.............................................................................................................xxiii

CHAPTER ONE: INTRODUCTION ................................................................. 1
1.1 Background ............................................................................................ 1
1.2 Scope and Objectives ............................................................................. 4
1.3 Organization of Dissertation .................................................................. 8

CHAPTER TWO: LITERATURE REVIEW .................................................. 11
2.1 Introduction..........................................................................................11
2.2 Abutment Behavior during a Seismic Event ........................................ 26
2.3 Caltrans Abutment Design Criteria ...................................................... 28

CHAPTER THREE: ABUTMENT LSH MODEL.......................................... 36
3.1 Introduction..........................................................................................36
3.2 Types of Bridge Abutments.................................................................. 37
3.3 Seismic Behavior of Seat-Type Abutments ......................................... 40
3.4 Seismic Behavior of Monolithic Abutments ........................................ 41
3.5 Force-Displacement Capacity of Bridge Abutment ............................. 41
3.6 Mechanism of the Abutment Backfill Failure...................................... 44
3.7 Abutment Backfill Constitutive Model ................................................ 49
3.8 Nonlinear Hyperbolic Model................................................................ 52
3.9 Nonlinear Abutment Backfill Capacity................................................ 58
3.10 Verification of LSH Model ............................................................... 68
3.11 Full-Scaled Abutment Experiments .................................................. 68
3.11.1 UCLA Abutment Experiment..................................................... 68
3.11.2 UCD Abutment Experiment....................................................... 72
3.12 Full-Scaled Pile Cap Experiment...................................................... 72
3.12.1 Clean Sand..................................................................................73
3.12.2 Silty Sand...................................................................................73
3.12.3 Fine-grained Gravel....................................................................77
3.12.4 Coarse-grained Gravel................................................................77

v
3.13 RPI Centrifuge Experiment in Nevada Sand .................................... 77
3.13.1 Seat-type abutment.....................................................................80
3.13.2 Pile Cap......................................................................................80
3.14 3.14 Small-Scaled Experiment of Wall in Loose Sand..................... 81
3.15 Recommended Abutment Force-deformation Relationship.............. 81
3.15.1 Application of HFD Model for other Height.............................. 90
3.15.2 Development of the Height Factors............................................ 93
3.15.3 HFD Height Validation .............................................................. 96
3.16 Conclusions.....................................................................................100

CHAPTER FOUR: ABUTMENT CONTINUUM FINITE ELEMENT
MODEL............................................................................ 102
4.1 Introduction........................................................................................102
4.2 Method of Analysis ............................................................................ 103
4.2.1 Presumptive Values..................................................................103
4.2.2 Closed-Form Solution..............................................................106
4.2.3 Simplified Solution...................................................................107
4.2.4 Numerical Solution...................................................................107
4.3 Constitutive Models for Bridge Structures......................................... 108
4.4 Constitutive Model for Abutment Backfill ........................................ 111
4.4.1 Primary Loading Stiffness........................................................ 113
4.4.2 Unloading/Reloading Stiffness.................................................115
4.4.3 Oedometer Stiffness.................................................................115
4.4.4 Dilatancy..................................................................................117
4.5 Verification of the Constitutive Model .............................................. 118
4.6 2-D Finite Element Model.................................................................. 123
4.6.1 Interface Elements....................................................................125
4.6.2 2-D Finite Element Simulations for Various Backfill.............. 126
4.6.3 UCLA-CT Full-Scaled Abutment Experiment......................... 127
4.6.4 Selection of Parameters Used in the Finite Element Model..... 130
4.6.5 UCLA 2-D Abutment Finite Element Model........................... 131
4.6.6 Sequence of the Events in the Model ....................................... 133
4.7 UCD Abutment Field Experiment...................................................... 136
4.7.1 UCD Abutment 2-D Finite Element Model ............................. 140
4.7.2 BYU Pile Cap Experiments 2-D Finite Element Model .......... 144
4.7.3 Pile Cap Backfill Parameters.................................................... 145
4.8 Backfill Behavior Using 3-D Finite Element Model.......................... 148
4.8.1 3-D Finite Element Simulation of UCLA Abutment Test........ 151
4.8.2 Testing Sequence..................................................................... 153
4.9 3-D Finite Element Simulation of UCD Abutment Test.................... 158
4.9.1 Sequence of the Events............................................................. 158
4.10 3-D Finite Element Simulation of BYU Pile Cap Experiments...... 163
4.10.1 3-D Finite Element Simulations Using Half Model................. 165
4.10.2 Testing Sequence......................................................................166

vi
4.11 Comparisons of Various Models Versus Experimental Data ......... 171
4.12 3-D Finite Element Model for Skewed Abutment .......................... 178
4.13 Summary.........................................................................................185

CHAPTER FIVE: NONLINEAR SEISMIC RESPONSE OF
SKEWED BRIDGES.................................................. 187
5.1 Introduction........................................................................................187
5.2 Impact of Ground Motion Characteristics.......................................... 190
5.3 Selection of the Ground Motions ....................................................... 192
5.4 Model’s Boundary Conditions ........................................................... 209
5.4.1 Abutment Soil-Structure Interaction ........................................ 209
5.4.2 Abutment Gaps.........................................................................210
5.4.3 Abutment Backwall-Backfill Longitudinal Response.............. 210
5.4.4 Abutment Shear Key ................................................................ 212
5.4.5 Validation of Abutment Model ................................................ 214
5.4.6 Bridge Deck Model .................................................................. 217
5.4.7 Bridge Columns........................................................................219
5.5 Bridge Models....................................................................................220
5.6 Analytical Study.................................................................................226
5.6.1 Bridges Types...........................................................................226
5.6.2 Single-Span Bridge...................................................................227
5.6.3 Two-Span Bridge.....................................................................228
5.6.4 Three-Span Bridge with a Single-Column Bent....................... 229
5.6.5 Two-Span Bridge with Two-Column Bent .............................. 229
5.6.6 Three-Span Bridge with Tow-Column Bents........................... 230
5.7 Longitudinal and Transverse Abutment Response............................. 240
5.7.1 Effects of Skew Angles on the Bridge Abutment .................... 240
5.8 Response Due to Spectra-Compatible Time History Motions ........... 248
5.8.1 Abutment Response..................................................................248
5.8.2 Rotation Due to Deck Flexibility ............................................. 251
5.8.3 Rotation Due to Column Rigidity ............................................ 251
5.9 Analytical Results of the Single-Column Two-Span Bridge ............. 256
5.9.1 Abutment Behavior in the Longitudinal Direction................... 264
5.9.2 Abutment Behavior in the Transverse Direction...................... 266
5.10 Results of the Analysis for All Bridge Models............................... 271
5.10.1 Rotational Response.................................................................271
5.11 Conclusions.....................................................................................296

CHAPTER SIX: CASE STUDY OF AN INSTRUMENTED SKEWED
BRIDGE............................................................................... 298
6.1 Introduction........................................................................................298
6.2 Bridge Description.............................................................................300

vii
6.3 Seismic Instrumentation and Input Ground Motion........................... 304
6.4 Geotechnical Information...................................................................309
6.5 Idealized Soil Profile and Properties.................................................. 312
6.6 Global Bridge Model.......................................................................... 317
6.7 Global Bridge Modeling Using Direct Method.................................. 319
6.7.1 Pile Foundations.......................................................................325
6.7.2 Abutment Soil-Structure Interaction ........................................ 325
6.7.3 Global Bridge Displacement Response.................................... 330
6.7.4 Abutment Force and Displacement Response.......................... 343
6.8 Substructuring Technique...................................................................349
6.8.1 Procedures To Develop Stiffness Matrix ................................ 353
6.9 Bridge Response Using Substructuring Technique............................ 354
6.10 Conclusions.....................................................................................364

CHAPTER SEVEN: CONCLUSIONS AND RECOMMENDATIONS ..... 365
7.1 Summary and Conclusions................................................................. 365
7.2 Recommendations..............................................................................367
7.3 Further Research.................................................................................368

REFERENCES ........................................................................................... 371

viii
LIST OF TABLES

Table 3.1 Typical Values of ε
50
..................................................................... 56
Table 3.2 Mobilized Earth Pressure Coefficients Using Three Methods ..... 66
Table 3.3 Parameters for HFD Model........................................................... 89
Table 3.4 Suggested HFD Parameters for Abutment Backfills .................... 89
Table 3.5 Height factors to calculate abutment force-displacement............. 96
Table 4.1 Parameters of the HS model........................................................ 115
Table 4.2 Input Parameters Used in the UCLA-Caltrans PLAXIS
Model .......................................................................................... 130
Table 4.3 Input Parameters for UCD PLAXIS Model................................ 140
Table 4.4 Input Parameters for BYU Pile Cap Analyses ............................ 145
Table 5.1 Selected Earthquake Records...................................................... 193
Table 5.2 Input Ground Motion Characteristics ......................................... 194
Table 6.1 Characteristics of the Input Motions of the Painter Street
Bridge.......................................................................................... 304
Table 6.2 Idealized Soil Parameters for Painter Street Bridge Site ............ 316
Table 6.3 Idealized Stiffness Parameters to Develop p-y Springs for
Painter Street Bridge Site............................................................ 316
Table 6.4 Bent Foundation Stiffness Matrix (kips, inch and radian).......... 354

ix

LIST OF FIGURES

Figure 2.1: Linear Elastic Finite Element Model ............................................ 12
Figure 2.2: Idealized Spline Bridge Model ..................................................... 14
Figure 2.3: 3-D Continuum (Sweet and Morill, 1993) .................................... 19
Figure 2.4: 3-D Continuum (Maroney,1995) .................................................. 19
Figure 2.5: Passive Earth Pressure Coefficient (Caquot and Kerisel 1948).... 23
Figure 2.6: Stress-Strain Relationship (Shamsabadi et al., 2005) ................. 26
Figure 2.7: Backwall Shear Failure and Mobilized Passive Wedge................ 27
Figure 2.8: UCD Test Setup (Maroney,1995) ................................................. 29
Figure 2.9: Estimated Bilinear Abutment Stiffness......................................... 30
Figure 2.10: Effective Abutment Stiffness (SDC, 2004)................................... 34
Figure 2.11: Effective Abutment Area (SDC, 2004)......................................... 34
Figure 2.12: Effective Abutment Width for Skewed Bridges (SDC, 2004)...... 35
Figure 2.13: Abutment Seat Width Requirements (SDC, 2004) ....................... 35
Figure 3.1: Seat-Type Abutment and Foundation System .............................. 38
Figure 3.2: Monolithic Abutment and Foundation System ............................. 39
Figure 3.3: Example of Mobilized Passive Wedges........................................ 40
Figure 3.4: Mobilization of Passive Resistance............................................... 45
Figure 3.5: Mobilized Wedges During Seismic Event ................................... 48
Figure 3.6: Various Types of Elstoplastic Soils Behavior (Bardet, 1997) ...... 50
Figure 3.7: Stress-Strain Relationship for Typical Abutment Backfill ........... 51
Figure 3.8: Hyperbolic Model ......................................................................... 53

x
Figure 3.9: Modified Hyperbolic Stress-Strain Relationship .......................... 55
Figure 3.10: Mobilized Passive Wedge (Shamsabadi et al., 2005) .................. 58
Figure 3.11: Associated Mohr Circle and Soil Strain........................................ 61
Figure 3.12: Flowchart of LSH Procedure ........................................................ 63
Figure 3.13: Simple Wedge with Planar Failure ............................................... 64
Figure 3.14: Mobilized Earth Pressure Coefficients Using Three Methods ..... 67
Figure 3.15: LSH Model Failure Wedges Versus Experimental Failure
Wedges (UCLA Test) ................................................................... 70
Figure 3.16: LSH Prediction of UCLA Abutment Test..................................... 71
Figure 3.17: LSH Prediction of UCD-CT Abutment test.................................. 74
Figure 3.18: LSH Prediction of BYU Pile Cap Test in Clean Sand.................. 75
Figure 3.19: LSH Prediction of BYU Pile Cap Test in Silty Sand................... 76
Figure 3.20: LSH Prediction of BYU Pile Cap Test in Fine Gravel ................. 78
Figure 3.21: LSH Prediction of BYU Pile Cap in Test Coarse Gravel ............ 79
Figure 3.22: LSH Prediction of RPI Abutment Test in Dense Sand ................ 82
Figure 3.23: LSH Prediction of RPI Pile Cap Test in Nevada Sand ................. 83
Figure 3.24: LSH Prediction of Wall Test in Loose Sand............................... 84
Figure 3.25: Hyperbolic Force-Displacement Formulation .............................. 85
Figure 3.26: UCLA Abutment Experiment Compared with Various
Models........................................................................................... 88
Figure 3.27: UCD Abutment Experiment Compared with Various Models ..... 92
Figure 3.28: Backbone Curves for Various Abutment Height (Silty Sand)...... 94
Figure 3.29: Backbone Curves for Various Abutment Height (Clay)............... 95
Figure 3.30: Comparisons of LSH and HFD Models for Cohesive Backfill..... 97
Figure 3.31: Comparisons of LSH and HFD Models for Silty Sand................. 98

xi
Figure 3.32: HFD Validation Using BYU Silty Sand Backfill ......................... 99
Figure 4.1: Abutment Force-Deformation Based on Presumptive Value...... 104
Figure 4.2: Closed-Form Solution ................................................................. 107
Figure 4.3: Hysteretic Behavior of Confined Concrete................................. 110
Figure 4.4: Behavior of Steel......................................................................... 110
Figure 4.5: Hardening Soil (HS) Model ........................................................ 114
Figure 4.6: HS Model in Stress Space........................................................... 116
Figure 4.7: PLAXIS Dilatancy Model........................................................... 118
Figure 4.8: Triaxial: (a) Specimen; (b) Finite Element Model...................... 120
Figure 4.9: Triaxial Test of the Cohesionless Backfill.................................. 121
Figure 4.10: Triaxial Test of the Abutment Backfill for Cohesive Soil.......... 122
Figure 4.11: 2-D Finite Element Mesh with Backwall Interface .................... 124
Figure 4.12: Mohr-Coulomb Interface Stress-Strain Relationship.................. 126
Figure 4.13: Seat-type Abutment and Foundation System.............................. 128
Figure 4.14: UCLA-CT Full-Scaled Abutment Test with Sandy Backfill ...... 129
Figure 4.15: UCLA-CT Abutment Backfill Triaxial Test............................... 131
Figure 4.16: UCLA-CT Triaxial Test of the Abutment Backfill..................... 132
Figure 4.17: Limits of 2-D Finite Element Model........................................... 133
Figure 4.18: 2-D Finite Element Deformed Shape for UCLA Abutment
Test.............................................................................................. 134
Figure 4.19: 2-D Finite Element Displacement Vectors for UCLA
Abutment Test............................................................................. 134
Figure 4.20: 2-D Finite Element Displacement Contours for UCLA
Abutment Test............................................................................. 134
Figure 4.21: 2-D Finite Element Failure Surfaces for UCLA Abutment
Test.............................................................................................. 135

xii
Figure 4.22: 2-D Finite Element Incremental Strains for UCLA Abutment
Test.............................................................................................. 135
Figure 4.23: 2-D Cyclic UCLA Abutment Test versus Finite Element
Model Predictions ....................................................................... 137
Figure 4.24: UCD Abutment Test Setup ......................................................... 138
Figure 4.25: UCD 3-D Abutment Field Experiment ....................................... 138
Figure 4.26: Failure Mechanism of the UCD Abutment Test ......................... 139
Figure 4.27: 2-D Finite Element Deformed Shape for UCD Abutment
Test.............................................................................................. 141
Figure 4.28: 2-D Finite Element Displacement Vectors for UCD
Abutment Test............................................................................. 141
Figure 4.29: 2-D Finite Element Displacement Contours for UCD
Abutment Test............................................................................. 141
Figure 4.30: 2-D Finite Element Failure Surfaces for UCD Abutment Test... 142
Figure 4.31: 2-D Finite Element Incremental Strains for UCD Abutment
Test.............................................................................................. 142
Figure 4.32: UCD Abutment Test versus Finite Element Model Prediction... 143
Figure 4.33: Finite Element Model for BYU Clean Sand Backfill ................. 144
Figure 4.34: 2-D Finite Element Deformed Mesh for BYU Clean Sand
Backfill........................................................................................ 146
Figure 4.35: 2-D Finite Element Displacement Vector Field for BYU
Clean Sand Backfill .................................................................... 146
Figure 4.36: 2-D Finite Element Displacement Contours for BYU Clean
Sand Backfill............................................................................... 147
Figure 4.37: 2-D Finite Element Mohr-Coulomb Failure Surface for BYU
Clean Sand Backfill .................................................................... 147
Figure 4.38: 2-D Finite Element Incremental Strains for BYU Clean Sand
Backfill........................................................................................ 147

xiii
Figure 4.39: BYU Pile Cap Test versus Finite Element Model Prediction
For Clean Sand Backfill.............................................................. 150
Figure 4.40: 3-D Solid Element....................................................................... 151
Figure 4.41: Section Through the Full 3-D Finite Element Model of the
UCLA Abutment Experiment ..................................................... 152
Figure 4.42: Full 3-D Finite Element Model of the Deformed Shape for
UCLA Abutment Experiment ..................................................... 154
Figure 4.43: Section Through Gypsum Columns for the Full 3-D Finite
Element Model for UCLA Abutment Experiment...................... 154
Figure 4.44: Full 3-D Finite Element Model of the Displacement Vectors
for UCLA Abutment Experiment................................................ 155
Figure 4.45: Full 3-D Finite Element Model of the Displacement
Contours of UCLA Abutment Experiment ................................. 155
Figure 4.46: Full 3-D Finite Element Model of the Plastic Points Through
Gypsum Columns for UCLA Abutment Experiment.................. 156
Figure 4.47: Full 3-D Finite Element Model of the Incremental Strains
Through Gypsum Columns for UCLA Abutment Experiment... 156
Figure 4.48: Measured Experimental Data Versus Finite Element
Prediction for UCLA Abutment Experiment .............................. 157
Figure 4.49: Finite Element Model of the of the Deformed Shape for
UCD Abutment Experiment........................................................ 158
Figure 4.50: Full 3-D Finite Element Model of the Displacement Vectors
for UCD Abutment Experiment.................................................. 160
Figure 4.51: Full 3-D Finite Element Model of the Displacement
Contours for the UCD Abutment Experiment ............................ 160
Figure 4.52: Full 3-D Finite Element Model of Mohr-Coulomb Plastic
Points for UCD Abutment Experiment....................................... 161
Figure 4.53: Full 3-D Finite Element Model of the Incremental Strains for
UCD Abutment Experiment........................................................ 161
Figure 4.54: Measured Experimental Data Versus Finite Element
Prediction for UCD Abutment Experiment................................. 162

xiv
Figure 4.55: Schematic of a Typical 3-D Wedge Failures and Cracking
Patterns of BYU Pile Cap Experiments ...................................... 163
Figure 4.56: Full 3-D Finite Element Model Displacement Contours of
BYU Pile Cap Experiment.......................................................... 164
Figure 4.57: 3-D Finite Element Using Half Model of BYU Pile Cap
Experiment .................................................................................. 165
Figure 4.58: 3-D Finite Element Model of the Displacement Contours for
BYU Pile Cap Experiment.......................................................... 167
Figure 4.59: 3-D Finite Element Model of the Displacement Contours for
BYU Pile Cap Experiment.......................................................... 168
Figure 4.60: Full 3-D Finite Element Model of Mohr-Coulomb Plastic
Points for BYU Pile Cap Experiment ......................................... 168
Figure 4.61: Full 3-D Finite Element Model of Mohr-Coulomb
Incremental Strains for BYU Pile Cap Experiment .................... 169
Figure 4.62: Measured Experimental Data Versus Finite Element
Prediction for BYU Silty Sand Backfill Experiment ................. 170
Figure 4.63: Measure Force-Deformation of the UCLA Abutment Test
Versus Predictions by Various Methods..................................... 172
Figure 4.64: Measure Force-Deformation of the UCD Abutment Test
Versus Predictions by Various Methods..................................... 173
Figure 4.65: Measure Force-Deformation of the BYU Pile Cap Test
Versus Predictions by Various Methods for Silty Sand
Backfill........................................................................................ 174
Figure 4.66: Measure Force-Deformation of the BYU Pile Cap Test
Versus Predictions by Various Methods for Clean Sand
Backfill........................................................................................ 175
Figure 4.67: Measure Force-Deformation of the BYU Pile Cap Test
Versus Predictions by Various Methods for Fine Gravel
Backfill........................................................................................ 176
Figure 4.68: Measure Force-Deformation of the BYU Pile Cap Test
Versus Predictions by Various Methods for Coarse Gravel
Backfill........................................................................................ 177

xv
Figure 4.69: Clockwise Deck Rotation During a Seismic Event .................... 178
Figure 4.70: Non-uniform Passive wedge behind skewed abutment .............. 179
Figure 4.71: Passive Soil Wedge Plunging in Between Wingwalls ................ 180
Figure 4.72: Full 3-D Finite Element Model of the Passive Wedge
Formation Behind Skewed Abutment......................................... 181
Figure 4.73: Nonlinear Normal and Tangential Components of the
Abutment-Backfill Resistance for a 30 Skew angle ................... 183
Figure 4.74: Impact of Skew Angles on Nonlinear Abutment Force-
Deformation Relationship ........................................................... 184
Figure 5.1: Components of Bridge System.................................................... 189
Figure 5.2: Measured Shaking Table Hysteretic Response of Bridge
Column (Phan et al., 2005) ......................................................... 191
Figure 5.3: Input Ground Motion 1 in Longitudinal Direction ..................... 195
Figure 5.4: Input Ground Motion 1 in Transverse Direction ........................ 196
Figure 5.5: Input Ground Motion 2 in Longitudinal Direction ..................... 197
Figure 5.6: Input Ground Motion 2 in Transverse Direction ........................ 198
Figure 5.7: Input Ground Motion 3 in Longitudinal Direction ..................... 199
Figure 5.8: Input Ground Motion 3 in the Transverse Direction .................. 200
Figure 5.9: Input Ground Motion 4 in the Longitudinal Direction ............... 201
Figure 5.10: Input Ground Motion 4 in the Transverse Direction .................. 202
Figure 5.11: Input Ground Motion 5 in the Longitudinal Direction ............... 203
Figure 5.12: Input Ground Motion 5 in the Transverse Direction .................. 204
Figure 5.13: Input Ground Motion 6 in the Longitudinal Direction ............... 205
Figure 5.14: Input Ground Motion 6 in the transverse Direction.................... 206
Figure 5.15: Input Ground Motion 7 in the Longitudinal Direction ............... 207
Figure 5.16: Input Ground Motion 7 in the Transverse Direction .................. 208

xvi
Figure 5.17: Bridge Monolithic Abutment Model........................................... 211
Figure 5.18: Abutment Shear Key Experiment (Bozorgzadeh et al., 2005).... 212
Figure 5.19: Generic Model of the Exterior Shear Key Backbone Curve...... 213
Figure 5.20: Displacement Cycles in the UCD Abutment Test (Romstad
et al., 1995).................................................................................. 215
Figure 5.21: Measured Load-Deformation in the UCD Abutment Test
(Romstad et al., 1995) ................................................................. 215
Figure 5.22: Simulated Longitudinal Response of the UCD Abutment
Test.............................................................................................. 216
Figure 5.23: Longitudinal and Transverse Backbone Curves Used in the
Bridge Global Models................................................................. 216
Figure 5.24: Typical Bridge Model ................................................................. 218
Figure 5.25: Column Moment-Curvature Relationship................................... 219
Figure 5.26: Single-Span Bridge ..................................................................... 221
Figure 5.27: Single-Column Two-Span Bridge............................................... 222
Figure 5.28: Single-Column Three-Span Bridge............................................. 223
Figure 5.29: Tow-Columns Two-Spans Bridge............................................... 224
Figure 5.30: Tow-Columns Three-Spans Bent Bridge.................................... 225
Figure 5.31: Transverse and Longitudinal Sections of the Single-Span
Bridge.......................................................................................... 231
Figure 5.32: Transverse Section of the Two-Span Single-Column Bridge..... 232
Figure 5.33: Analytical Models of the Two-Span Single-Column Bridge...... 233
Figure 5.34: Analytical Models of the Three-Span Single-Column Bridge.... 234
Figure 5.35: Transverse Section of the Two-Span Two-Column Bridge........ 235
Figure 5.36: Analytical Models of the Two-Span Two-Column Bridge......... 236
Figure 5.37: Transverse Section of the Three-Span Two-Column Bridge...... 237

xvii
Figure 5.38: Analytical Models of the Three -Span Two-Column Bridge...... 238
Figure 5.39: Columns Longitudinal and Transverse Moment Curvatures ...... 239
Figure 5.40: Single Span Bridge With 0o and 45o Skew Angles.................... 241
Figure 5.41: Recorded Renaldi Longitudinal Motion ..................................... 244
Figure 5.42: Recorded Renaldi Transverse Motion......................................... 245
Figure 5.43: Variation of Normal Abutment Impact Forces For a Single
Span-Bridge With 0o Skew Angle.............................................. 246
Figure 5.44: Variation of Normal Abutment Impact Forces For a Single
Span-Bridge With 45o Skew Angle During the First 4
Seconds of Shaking..................................................................... 247
Figure 5.45: Variation of Normal Abutment Impact Forces For a Single
Span-Bridge With 45o Skew Angle Between 2 Seconds to
2.6 Seconds of Shaking............................................................... 248
Figure 5.46: Longitudinal Hysteretic Behavior of the Single-Span Bridge
Abutment With 0o Skew Angle .................................................. 250
Figure 5.47: Longitudinal Hysteretic Behavior of the Single-Span Bridge
Abutment With 45o Skew Angle ................................................ 252
Figure 5.48: Transverse Hysteretic Behavior of the Single-Span Bridge
Abutment With 45o Skew Angle ................................................ 253
Figure 5.49: Average Rotation Due to Deck Flexibility of the Single-Span
Bridge With 45o Skew Angle ..................................................... 254
Figure 5.50: Average Rotation Due to Presence of the Number of Spans
With 45o Skew Angle ................................................................. 255
Figure 5.51: Transverse Displacement Due Motion 1 For a Two-Span
Single-Column Bridge ................................................................ 257
Figure 5.52: Transverse Displacement Due Motion 2 For a Two-Span
Single-Column Bridge ................................................................ 258
Figure 5.53: Transverse Displacement Due Motion 3 For a Two-Span
Single-Column Bridge ................................................................ 259

xviii
Figure 5.54: Transverse Displacement Due Motion 4 For a Two-Span
Single-Column Bridge ................................................................ 260
Figure 5.55: Transverse Displacement Due Motion 5 For a Two-Span
Single-Column Bridge ................................................................ 261
Figure 5.56: Transverse Displacement Due Motion 6 For a Two-Span
Single-Column Bridge ................................................................ 262
Figure 5.57: Transverse Displacement Due Motion 7 For a Two-Span
Single-Column Bridge ................................................................ 263
Figure 5.58: Hysteretic Abutment Backfill Response For a Two-Span
Single-Column Bridge with 0o Skew Angle............................... 264
Figure 5.59: Hysteretic Abutment Backfill Response For a Two-Span
Single-Column Bridge with 25o Skew Angle............................. 265
Figure 5.60: Hysteretic Abutment Backfill Response For a Two-Span
Single-Column Bridge with 60o Skew Angle............................. 265
Figure 5.61: Shear Key Failure Due to Deck Rotation 1994 Northridge
Earthquake .................................................................................. 268
Figure 5.62: Hysteretic Abutment Shear Key Response at the Acute
Corners For a Two-Span Single-Column Bridge with 0o
Skew Angle ................................................................................. 269
Figure 5.63: Hysteretic Abutment Shear Key Response at the Obtuse
Corners For a Two-Span Single-Column Bridge with 45o
Skew Angle ................................................................................. 270
Figure 5.64: Rotation Time History of Single Span Bridge With 25o
Skew Angle (Length =102’) ....................................................... 274
Figure 5.65: Rotation Time History of Single Span Bridge With 45o
Skew Angle (Length =102’) ....................................................... 275
Figure 5.66: Rotation Time History of Single Span Bridge With 60o
Skew Angle (Length =102’) ....................................................... 276
Figure 5.67: Rotation Time History of Single Span Bridge With 25o
Skew Angle (Length =204’) ....................................................... 277
Figure 5.68: Rotation Time History of Single Span Bridge With 45o
Skew Angle (Length =204’) ....................................................... 278

xix
Figure 5.69: Rotation Time History of Single Span Bridge With 60o
Skew Angle (Length =204’) ....................................................... 279
Figure 5.70: Rotation Time History of Two- Span Single-column Bridge
With 25o Skew Angle ................................................................ 280
Figure 5.71: Rotation Time History of Two- Span Single-column Bridge
With 45o Skew Angle ................................................................ 281
Figure 5.72: Rotation Time History of Two- Span Single-column Bridge
With 60o Skew Angle ................................................................ 282
Figure 5.73: Rotation Time History of Three- Span Single-column
Bridge With 25o Skew Angle .................................................... 283
Figure 5.74: Rotation Time History of Three- Span Single-column
Bridge With 45o Skew Angle .................................................... 284
Figure 5.75: Rotation Time History of Three- Span Single-column
Bridge With 60o Skew Angle .................................................... 285
Figure 5.76: Rotation Time History of Two- Span Two-column Bridge
With 25o Skew Angle ................................................................ 286
Figure 5.77: Rotation Time History of Two-Span Two-column Bridge
With 45o Skew Angle ................................................................ 287
Figure 5.78: Rotation Time History of Two-Span Two-column Bridge
With 60o Skew Angle ................................................................ 288
Figure 5.79: Rotation Time History of Three-Span Two-column Bridge
With 25o Skew Angle ................................................................ 289
Figure 5.80: Rotation Time History of Three-Span Two-column Bridge
With 45o Skew Angle ................................................................ 290
Figure 5.81: Rotation Time History of Three-Span Two-column Bridge
With 60o Skew Angle ................................................................ 291
Figure 5.82: Maximum Deck Rotations .......................................................... 292
Figure 5.83: Average Residual Deck Rotations .............................................. 293
Figure 5.84: Two-Span Bridge Maximum Residual Deck Rotations.............. 294
Figure 5.85: Three-Span Bridge Maximum Residual Deck Rotations............ 295

xx
Figure 6.1: 3-D View of the Painter Street Bridge........................................ 301
Figure 6.2: Elevation and Plan Views of the Painter Street Bridge ............. 302
Figure 6.3: Cross Section of Superstructure and Pile Foundation of the
Painter Street Bridge ................................................................... 303
Figure 6.4: Seismic Instrumentation at the Painter Street Bridge ................. 305
Figure 6.5: Input Motion in Longitudinal Direction (Channel 12) ............... 306
Figure 6.6: Input Motion in Vertical Direction (Channel 13) ....................... 307
Figure 6.7: Input Motion in Transverse Direction (Channel 14)................... 308
Figure 6.8: Layout of Abutment Backfill Borings at the Painter Street
Bridge Site................................................................................... 310
Figure 6.9: Geometry and Idealized Soil Profile at the Painter Street
Bridge Site................................................................................... 313
Figure 6.10: Total System Versus Substructure System ................................. 318
Figure 6.11: Painter Street Bridge Model for Direct Method ......................... 320
Figure 6.12: Bent and columns Sections for Painter Street Bridge................. 322
Figure 6.13: Moment Curvature Relationship for Painter Street Bridge......... 323
Figure 6.14: Shear Key Capacities at West Abutment for Painter Street
Bridge.......................................................................................... 324
Figure 6.15: Bridge Pile Foundation Model.................................................... 328
Figure 6.16: Longitudinal Abutment-Soil Loading-Unloading Curves .......... 329
Figure 6.17: Bridge Monolithic Abutment Model........................................... 329
Figure 6.18: Recorded Response Versus Analytical Response of the
Direct Method (Channel 18) ....................................................... 331
Figure 6.19: Recorded Response Versus Analytical Response of the
Direct Method (Channel 11) ....................................................... 332
Figure 6.20: Recorded Response Versus Analytical Response of the
Direct Method (Channel 17) ....................................................... 333

xxi
Figure 6.21: Recorded Response Versus Analytical Response of the
Direct Method (Channel 4) ......................................................... 334
Figure 6.22: Recorded Response Versus Analytical Response of the
Direct Method (Channel 3) ......................................................... 335
Figure 6.23: Recorded Response Versus Analytical Response of the
Direct Method (Channel 9) ......................................................... 336
Figure 6.24: Recorded Response Versus Analytical Response of the
Direct Method (Channel 5) ......................................................... 337
Figure 6.25: Recorded Response Versus Analytical Response of the
Direct Method (Channel 8) ......................................................... 338
Figure 6.26: Recorded Response Versus Analytical Response of the
Direct Method (Channel 2) ......................................................... 339
Figure 6.27: Channel 1 Transverse at Column base of the Direct Method ..... 340
Figure 6.28: Comparisons of the Northwest and Southeast Corners
Displacement Response in the Transverse Directions Using
Direct Method ............................................................................. 341
Figure 6.29: Deck Rotational Response Using Direct method........................ 342
Figure 6.30: Force-Displacement Response at The West Abutments Using
Direct method.............................................................................. 344
Figure 6.31: Force-Displacement Response at The East Abutments Using
Direct method.............................................................................. 345
Figure 6.32: Displacement response at the West Abutment Seat Using
Direct method.............................................................................. 346
Figure 6.33: Force-Displacement Capacity of Shear Key at West
Abutment Using Direct method .................................................. 347
Figure 6.34: Force-Displacement Capacity of Shear Key at West
Abutment Using Direct method .................................................. 348
Figure 6.35: Schematic of soil-pile interaction ............................................... 349
Figure 6.36: Dynamic Impedance Versus frequency ...................................... 352
Figure 6.37: Substructuring Approach For Abutments and Bent.................... 355

xxii
Figure 6.38: Soil-Pile Interaction .................................................................... 356
Figure 6.39: Recorded Response Versus Analytical Response Using
Substructure Method (Channel 1)............................................... 357
Figure 6.40: Recorded Response Versus Analytical Response Using
Substructure Method (Channel 3)............................................... 358
Figure 6.41: Recorded Response Versus Analytical Response Using
Substructure Method (Channel 4)............................................... 359
Figure 6.42: Recorded Response Versus Analytical Response Using
Substructure Method (Channel 7)............................................... 360
Figure 6.43: Recorded Response Versus Analytical Response Using
Substructure Method (Channel 9)............................................... 361
Figure 6.44: Recorded Response Versus Analytical Response Using
Substructure Method (Channel 5)............................................... 362
Figure 6.45: Recorded Response Versus Analytical Response Using
Substructure Method (Channel 11)............................................. 363
Figure 7.1: Abutment Seat Width Requirements........................................... 368

xxiii
ABSTRACT

The purpose of this thesis is to investigate the nonlinear global seismic soil-
abutment-foundation-structure interaction behavior of typical highway skewed-
bridge structures subjected to near-fault ground motions with high velocity pulses.
Three-dimensional nonlinear finite element models of typical bridges with
various skew angles were developed. The bridge deck was modeled using shell
elements referred as “shell models” and beam elements referred as “spline models”.
The validity of the spline models was established

by comparing results obtained from
shell models. There is a very good agreement between the shell and the spline
models. The bridge columns were modeled as beam elements with cracked sectional
properties. The abutment-backfill and the transverse shear keys were simulated using
nonlinear springs. The structural models were excited using seven sets of bilateral
ground motions with the near fault effects.
The limit-equilibrium methods using mobilized Logarithmic-Spiral failure
surfaces coupled with a modified Hyperbolic soil stress-strain behavior referred here
as the “LSH” model is employed to capture the nonlinear abutment-backfill force-
displacement relationship. The validity of the LSH model was established using
experimental data and nonlinear continuum finite element models. The predicted
results obtained using the LSH model is in good agreement with the experimental
force-displacement capacity and the finite element model.

xxiv
A nonlinear Hyperbolic Force-Deformation relationship referred here as the
“HFD” model is developed as a powerful and effective tool for practicing bridge
engineers to develop nonlinear abutment backbone curves for typical abutment
backfill.
Case study based on the recorded response of a skewed-two-span reinforced
concrete box girder under strong shaking was performed. The bridge system was
subjected to the three-component recorded free-field earthquake motions. The
resulting dynamic response of the bridge model was found to be in good agreement
with most of the motions recorded at various locations of the bridge. This validates
the practical application and the methodology developed in this dissertation for
evaluating the seismic response of other skewed bridges that is realistic, repeatable
and reliable.

1
CHAPTER ONE
INTRODUCTION

1.1 Background
Earthquake records with near-source ground motion characteristics, such as
those of the 1994 Northridge earthquake (U.S.A.), the 1995 Kobe earthquake
(Japan), the 1999 Izmit and Duzce earthquake (Turkey), and 1999 Chi-Chi
earthquake (Taiwan), have increased the awareness of importance of nonlinear
seismic analyses employing soil-foundation-structure-interaction on bridge
structures. It has been long recognized that the short-span highway bridges and in
particular skewed-bridges are highly influenced by characteristics of bridge
abutments during a strong seismic excitation. During the 1971 San Fernando
earthquake (magnitude 6.7), many bridges in particular bridge structures with high
skew angles such as Northbound Truck Route Undercrossing and Roxford Street
Undercrossing, concrete box-girder bridges both built in 1969, resulted in significant
amount of rotation and translation of the superstructure as shown in Figure 1.1.
Northbound Truck Route Undercrossing was a curved-three-span bridge with
high skew angle. As a result of in-plane rotation and longitudinal translation the
abutments end diaphragm and the wingwalls sheared off approximately at the bottom
of the deck elevation. The end diaphragm abutments were pinned at the bottom and
supported on spread footings, allowing it to immediately engage the soil as a result
of the earthquake movement. Roxford Street Undercrossing was a two-parallel-

2
single-span bridge with integral abutments supported on pile foundation. During the
earthquake, the bridge moved about three feet transversely, shearing off all eight
wingwall connections to the abutments. The fill around the abutment settled several
feet, exposing the abutment piles which were tilted in the transverse direction.

(a) Northbound Truck Route

(b) Roxford Street
Figure 1.1: Abutment Failure During 1971 San Fernando Earthquake

The research conducted by Chen and Penzin (1977) on the effect of
poundings between the bridge deck and the abutments during the 1971 San Fernando
earthquake has motivated challenges and the needs to further study and understand
the seismic behavior of skewed bridges.
During the 1994 Northridge earthquake, Route 14/I5 Separation and
Overhead (Bridge #53-1960F) collapsed primarily as consequence of high horizontal
ground motion and subsequent rotational response of the bridge. This ten-span

3
curved box-girder bridge connected westbound traffic on SR14 to southbound I-5.
The bridge was constructed as five frames separated by expansion hinges.
Prestressed concrete box girders were used in the frames at each end of the bridge
and in the central frame. The east wingwall at Abutment 1 was severely damaged,
presumably by impact of the bridge deck. The bridge deck lost seat support and
moved about 5 feet north of the abutment face as shown in Figure 1.2.

Figure 1.2: Abutment Shear Key Failure Due to Deck Rotation
During a seismic event the bridge deck undergoes significant amount of
rotations about the vertical axis due to the effect of bilateral seismic excitations. In

4
particular when the center of the mass and the center of stiffness of the global bridge
system are not coincident, the inertia loading on the bridge tends to create torsional
bridge response about the vertical axis. As a consequence of superstructure rotations
about the vertical axis, excessive transverse movement can result in unseating of the
superstructure and pounding on the abutment wall.
Global seismic behavior of a skewed bridge is affected by a number of
factors, including bridge skew angle, bridge width, deck flexibility, number of spans,
the ratio of spans length to bridge length, number of columns per bent, columns
ductility, soil-abutment-superstructure interaction, abutment shear keys, abutment
bearing pads soil-bent foundation-structure interaction and characteristics of the
seismic source.
In this dissertation, the bridge elements considered include the nonlinear
abutment-backfill, bridge deck, bent cap and a pier column. The bridge deck is
modeled using full 3-dimensional (3-D) shell elements to account for the realistic
flexibility of the superstructure. The computational efficiency is achieved in the
three-dimensional finite element model by converting the shell model to three-
dimensional stick models for practicing bridge engineers.
1.2 Scope and Objectives
Significant difficulties have been encountered for design of short span
skewed bridges as observations from the past earthquakes suggest strong coupling
behavior between longitudinal and transverse movements. In-plane rotation of the
bridge superstructure about the vertical axis is a contributing factor leading to some

5
of bridge damages and current design practice does not explicitly address how to
handle such a mechanism. The lateral response of the abutment-embankment during
strong shaking is highly nonlinear. Therefore, it is not suitable to represent the lateral
stiffness of the abutment-backfill with an elastic element during strong shaking. In
most cases, bridge engineers ignore the contributions of the abutment resistance in
seismic design of bridge structures due to complexity of the problem.
In recent decades, however, Performance-Based Earthquake Engineering
(PBEE) has been identified as a quantitative means for design of the bridge
structures to provide life safety for the public. PBEE involves the design of ductile
bridge structures that will resist earthquake loads in a predicable manner. Therefore,
new bridge designs with ductile columns will impose large displacements on
abutments and this may cause more damage to the abutments than the damage levels
that have been observed in the past earthquakes. Proper evaluation and design of the
bridge abutment reduces the columns displacement demand during earthquake
shaking, this leads to a more efficient and economical design. Modeling assumptions
made for nonlinear abutment-embankment stiffness as well as hysteretic and
geometrical damping of the abutment can have significant effects on the global
seismic response characteristics of the short span skewed-bridges. Large-scaled field
experiments, observation following the major seismic event and system
identification techniques on the instrument bridges have identified that:
(1) The abutment-soil-structure-interaction plays a significant role in the global
response of this class of bridges.

6
(2) Stiffness and strength of the abutment-backfill depends on the level of shaking
and exhibits significant degradation at the large deformation.
(3) The abutment-backfill undergoes well into inelastic range and dissipate
significant amount of energy through the soil hysteretic action as a result of
abutment damage during strong seismic event.
(4) The abutment-backfill not only can provide significant lateral resistance but also
is a good source of energy dissipation at large deformation due to nonlinear
hysteretic behavior of the abutment-soil system.
The overall objective of the research presented in this dissertation is to
evaluate and understand the global seismic response of actual skewed-bridges
employing soil-abutment-foundation-structure interaction models subjected to
seismic bilateral excitations with near-source response-spectra-compatible ground
motion characteristics. The abutment-backfill enclosed in between the abutment
wingwalls provides significant resistance to the bridge deck motions during a
seismic event for single-span, two-span and three-span bridges, but becomes less
effective as the number of spans and columns increased. However, for multi-span
bridges with continuous deck and small diameter column-pile-extension the bridge
engineers can transfer the lateral seismic load to the bridge abutment. Therefore, in
the present research the global response of continuous single-span, two-spans-single-
column bent, two-spans-two-column bent, three-spans-single-column bents and
three-spans-two-column bents reinforced bridge structures with various skewed
angles is investigated. In most cases, the pier columns are supported by a pinned

7
connection at the base of the column, and thus the need to explicitly model the pier
foundations is not required. Since the primary objective of this research is to
understand the effect of the abutment-soil-structure interaction on the global
behavior of the bridge structures for the first part of the dissertation only this class of
bridge structure is selected. The second part of the dissertation is focused on the
skewed-bridge models including the pile foundations for the bents and the
abutments. The findings from this research can be used to improve the current bridge
design practice for seismic design of skewed bridges. To achieve this objective the
following tasks were undertaken:
• A practical and simplified design tool was developed and calibrated with all
available experimental data to predict the nonlinear force displacement capacity
of the abutment backfill.
• Full three-dimensional nonlinear finite element models were developed to
simulate the skew abutment-backfill nonlinear behavior and to understand the
mechanism of the problem.
• A nonlinear closed form hyperbolic force-deformation relationship which takes
the backfill stiffness and ultimate capacity of the backfill into account is
developed as a powerful and effective tool for practicing bridge engineers to
calculate a realistic non-linear load versus displacement relationship for
abutment-backfill without tedious finite element analysis.
• Using the nonlinear abutment springs developed above, three-dimensional
nonlinear finite element models for wide ranges of bilateral ground motions and

8
bridges with various skewed angles were carried out to quantify the global
seismic response for this class of bridge structures.
• Parametric studies on all bridge models were carried out to better understand the
mechanics of skewed bridge behavior. The parameters included nonlinear wide
ranges of skew angle, bridge width, span length, number of columns per bent,
number of actual earthquakes recorded motions and response spectra-compatible
time history ground motions. All the motions have near-source ground motion
characteristics with high velocity pulses.
• Case study based on the recorded response of a skewed-two-span reinforced
concrete box girder under strong shaking was performed to validate the modeling
techniques developed in this dissertation. The bridge system was subjected to the
three-component recorded free-field earthquake motions at the bridge site.
1.3 Organization of Dissertation
Chapter 2 presents a comprehensive review of seismic behavior of skew
bridges and the issues regarding nonlinear longitudinal abutment-backfill force-
deformation, nonlinear transverse abutment-shear-keys force-deformation and their
implementations in the current Caltrans seismic design criteria for the global bridge
behavior analysis.
Chapter 3 describes the proposed plane-strain two-dimensional (2-D) model
to evaluate abutment stiffness. The limit-equilibrium methods using mobilized
Logarithmic-Spiral failure surfaces coupled with a modified Hyperbolic soil stress-
strain behavior (referred here as the “LSH” model) is employed to capture the

9
nonlinear abutment-backfill force-displacement relationship. A nonlinear closed-
form hyperbolic force-deformation relationship which takes the backfill stiffness and
ultimate capacity of the backfill into account is developed to be a powerful and
effective tool for practicing bridge engineers.
Chapter 4 uses 2-D and 3-D finite element models to validate the LSH model.
The computer program Plaxis (Brinkgreve, 2006) was used to perform finite-element
analyses to evaluate the development of passive resistance of the bridge abutment-
backfill based on experimental data. The constitutive Hardening Soil (HS) model
(Schanz et al., 1999) available in Plaxis was used to model the nonlinear abutment
backfill behavior. This material model defines the soil stiffness moduli that reduce
with strain according to a hyperbolic relationship modified from the well-known
Duncan-Chang hyperbolic model (Duncan et al., 1970).
Chapter 5 discusses the three-dimensional dynamic behavior of bridges with
wide ranges of skew angles, implementation of the nonlinear abutment-structure
interaction into the global bridge model, parametric studies and discussion regarding
the impact of ground motions with high velocity pulses.
Chapter 6 deals with validation of the modeling technique applied in this
dissertation to investigate the behavior of an instrumented Painter Street Overpass in
Humboldt County, California. A three-dimensional nonlinear finite-element model
of bridge was developed. Direct method and substructure method of analysis were
considered. The direct model includes the superstructure, the nonlinear abutment
springs, and pile foundations with full coupling between structure and foundation

10
soils. For the substructure model, the pile foundations are represented by a
condensed foundation matrices. Realistic geotechnical soil properties obtained from
a geotechnical field exploration were used to represent the nonlinear soil support
provided by the abutment backfill and the pile foundations. The bridge system was
subjected to the three-component Cape Mendocino/Petrolia 1992 free-field
earthquake motions. The model was calibrated and verified using the recorded data
from the Painter Street Bridge.
Chapter 7 provides a critical discussion of the research results and the
conclusion.

11
CHAPTER TWO
LITERATURE REVIEW

2.1 Introduction
The literature review presented herein focuses on analytical research as well
as experimental studies on the bridge abutment-backfill behavior and the response of
instrumented ordinary box girder bridges during a seismic event.
The dynamic behavior of the skew bridges following the 1971 San Fernando
earthquake (magnitude-6.7) has received considerable attention and has motivated
challenges and the needs to further study and understand the seismic behavior of
skewed bridges. Chen and Penzin (1977) studied the effect of seismic soil-
foundation-structure interaction on the global behavior of skew bridges using a finite
element model. Their model included linear elastic beam to represent the bridge deck
and the bridge columns, linear springs were used to represent the foundation
flexibility. A three-dimensional linear continuum finite element was used to
represent the backfill and the abutment wall as shown in Figure 2.1. The elastic
perfectly-plastic Mohr-Coulomb yield criterion was used to represent the nonlinear
abutment-backfill interface interaction. They concluded that the foundation
flexibility and in particular the poundings between the bridge deck and the
abutments have significant influence on the global response of the bridge structures
and should be included in the bridge model.

12

6H
ABUTMENT WALL
FRICTION ELEMENT
7H
6H
ABUTMENT WALL
FRICTION ELEMENT
7H
(a) Plan
H H

(b) Elevation
Figure 2.1: Linear Elastic Finite Element Model

13
Traditional bridge design practice evaluates dynamic performance of skewed
bridges using spline models. The spline model is a collection of beam elements with
cross section properties adjusted from geometric data as shown in Figure 2.2.
Researchers have used both the simple spline models and the detail
continuum finite element models. Maragakis and Jennings (1987) used a rigid beam
element to model the bridge deck.
Wilson and Tan (1990) used elastic spline model to analyze the seismic
response of the Meloland Road Overcrossing (MRO). The MRO is a single column
bent concrete box-girder bridge with a monolithic abutment located near El Centro
in a high seismic region in the southern California. They carried out time history
analysis and high damping values were used to match the computed response of the
model to the recorded response of the structure. They concluded that, due to the
softening effect of the abutment backfill, the frequency of the vibration of the
abutment system decreased during the strong portion of the shaking and increased
near initial value after the strong shaking decreased.
Werner et al. (1994) also applied system identification technique for the
MRO to identify parameters for implementation of a simple elastic spline model.
They concluded that high modal damping ratios and low abutment stiffness were
necessary to replicate the recorded response of the bridge. They attribute this
observation to evidence of nonlinear behavior of the abutment-backfill system during
the seismic event.

14

Figure 2.2: Idealized Spline Bridge Model

15
Their evaluation also indicated that the wingwalls behave as a flexible plate
rather than a rigid retaining wall, and do not appear to have the capacity to mobilized
resistance of the soil between the wing walls in the transverse direction. However,
shearing resistance of the soil along the abutment end walls may carry some fraction
of the seismic load.
Wakefield et al. (1991) used beam elements to model concrete box girder
bridge deck, supporting columns, and the bent cap. McCallen and Romstad (1994)
simulated the bent cap using a beam element and the bridge deck was modeled by
flexible beam along the bridge length and a series of transverse rigid bars.
Tirasit, Kazuhiko and Kaeashima (2005) used spline model to investigate the
torsional response of skewed bridge columns during a seismic event. Watanabe and
Kaeashima (2004) used spline model to study the effect of the seismic cable
restrainers for the retrofit of skewed bridges.
Sweet and Morril (1993) and MaCallen and Romstad (1994) developed a
large three-dimensional finite element model including large volume of soil to
include abutment embankment and surrounding soil around the pile foundations for
the Painter Street Overcrossing to back calculate response of the structure during the
past earthquake.
The lateral response of the bridge abutments has been investigated through
theoretical models, half scale load tests on abutments, small-amplitude field
vibration tests, centrifuge tests, and analyses of recorded motions of actual bridges
during earthquakes. Wilson and Tan (1990), Levine and Scott (1989), and Wilson

16
(1988) proposed theoretical models for determining abutment stiffness based on the
soil properties and abutment dimensions. However, these models do not include the
significant effects of nonlinear soil behavior (Shamsabadi et al., 2005, 1998;
Siddaharthan et al., 1995). Martin and Yan (1998) conducted research on load-
deformation characteristics of bridge abutments under cyclic loading. In their
research, they used a large-strain finite difference computer program, FLAC yielding
highly nonlinear abutment response.
Several researchers have attempted to determine abutment stiffness and/or
vibration properties from field vibration tests on highway bridges (Crouse et al.,
1987; Gates and Smith, 1982; Douglas et al., 1990; Ventura et al., 1995). However,
such small amplitude tests lead to results that are not useful in design for intense
earthquake motions, because the stiffness of abutment depends on level of shaking.
Recognizing this limitation of small-amplitude tests, several investigations to
estimate abutment stiffness from motions of bridges recorded during earthquakes
have been reported (Maroney et al., 1990; McCallen and Romstad, 1994; Werner et
al., 1994; Goel and Chopra, 1997). Maroney et al. (1994) described the results of a
half scale load test on a monolithic abutment tested to failure.
It is evident that from the abutment backfill experimental studies (Romstad et
al. 1995), system identifications techniques (Wilson et al., 1990; Werner et al., 1994;
Goel et al., 1997) and theoretical studies (Shamsabadi et al., 2005; Martin et al.,
1999; and Sidharthan et al.. 1997) that the behavior of the bridge abutment-backfill
system is increasing nonlinear as displacement increases. Experiments conducted by

17
Thomson and Lutenegger (1998), Fang and Ishibashi (1994), Sherif et al. (1992),
Row (1954), and Terzaghi (1936) show that both the deformation mode and
magnitude of the deformation affect the magnitude and distribution of the earth
pressure. Results from the full-scale cyclic tests (Duncan and Mokwa, 2001; Rollins
and Sparks, 2002; and Rollins and Cole, 2005) to measure a lateral resistance of a
pile cap with a various backfill soils suggest that the force-displacement relationship
of the backfill is highly nonlinear and it is a function of backfill properties, formation
of the gap between the backfill and the pile cap at each loading cycle and the level of
pile cap displacement. Gadre (1997) performed centrifuge tests on model pile cap
and seat type abutment in dry Nevada sand. The most prominent feature in the
mobilized passive force response was the large reduction of the backfill stiffness as a
function of displacement after unloading or reloading occurs. There was a large
drop-off on the load due to the formation of the gap between the backfill and the
structure under cyclic loading. Similar to the Rollins’ pile cap test, upon gap closure
and structure contact with the backfill the load started to build up. The experimental
results also indicate that limiting equilibrium analysis using a logarithmic spiral
failure surface may be more appropriate to compute the ultimate passive earth
pressure.
Several researchers have attempted to capture seismic response of the
instrumented short span highway bridges (i.e., Painter Street Overpass, a skewed-
bridge and Meloland Overpass, a non-skewed-bridge) using field vibration test
records or earthquake seismic records and using spline models or more complicated

18
finite element models. Very high modal damping and discrete elastic abutment
stiffness values were selected as they lead to a good match between the elastic
earthquake response of the models and the motions recorded during the earthquake.
These may not be valid assumptions because the global system will behave in a
nonlinear manner during strong shaking (Goel and Chopra, 1997). Zhang and Markis
(2001) adopted a substructure approach where the kinematic motions and linear
elastic foundations and abutments stiffness were computed separately and
subsequently incorporated in the dynamic model for the Meloland and Painter Street
Models. Substructuring system is a matrix reduction technique that has been used
efficiently for seismic analysis linear systems. However, as mentioned above the
bridge abutment is highly nonlinear and substructuring technique is not suitable to
represent bridge abutments.
Sweet and Morill (1993) presented a three-dimensional continuum finite
element analyses which included large volume of, surrounding embankment soil and
the structure system for the Painter Street Overcrossing as sown in Figure 2.3.
However, the soil properties were based on numerous assumptions that are not valid
based on a recent investigation conducted at the bridge site. Furthermore, the
analysis approach is not suited for typical bridge design applications.

19
Figure 2.3: 3-D Continuum (Sweet and Morill, 1993)

MaCallen and Romstad (1994) also developed a large three-dimensional
finite element model for the Painter Street as sown in Figure 2.4.

Figure 2.4 : 3-D Continuum (Maroney,1995)

These types of simulations require the use of rigorous three-dimensional
finite element models. The three-dimensional finite element computational is very
expensive and time consuming. The models require the use of appropriate boundary

20
conditions, advanced nonlinear constitute models with a robust interface element
between the abutment-backfill, abutment backwall, bridge deck and the pile
foundations and surrounding soils. For these types of large models if coarse mesh is
selected the level of accuracy will be lost. Therefore, these types of models with high
complexity are unrealistic and should not be used in engineering practice. As an
alternative to the three-dimensional finite element models, simple and practical
nonlinear soil springs connected to the beam elements representing the pile
foundations. Nonlinear springs should be used to connect abutment-backfill to bridge
deck.
The impact of the bridge abutment on the overall response of a bridge
structure depends on many factors including bridge displacement, abutment skew
angle, and backfill strength and stress-strain properties. The nonlinear force
displacement capacity of the bridge abutment in a seismic event is developed mainly
from the mobilized passive pressure behind the abutment wall. From the time of the
French scientist Coulomb (1776) to the present, the analysis of lateral earth pressure
evaluation has been of prime interest to civil engineers. Literature on the subject of
earth pressure computation is rather abundant; however, most of it involves certain
simplifying assumptions. Coulomb (1776) presented a theory to evaluate the lateral
earth pressure against retaining structures based on the concept of limiting
equilibrium for cohesionless backfill. He assumed that the slip surface is a plane and
passes through the heel of the wall at a certain angle. The friction between the wall
and the adjacent soil is taken into account.

21
Rankine (1857) considered the equilibrium of a soil element within a semi-
finite soil mass bounded by a plane surface. Rankine's theory is the same as
Coulomb's theory except in his assumption no shearing stress exist along the wall,
and therefore, the friction between the wall and the soil does not exist. However,
frictional forces do exist specially during earthquake the behavior of the soil is
governed by compression forces induced by the abutment tending to push into the
soil. Thus the basic assumption used by Rankine is not valid. Tschbotarioff (1951),
in his book Soil Mechanics Foundation and Earth Structure, quotes Terzaghi, “The
fundamental assumptions of Rankine in earth pressure theory are incompatible with
the known relation between stress and strain in soils, including sand. Therefore, the
use of this theory should be discontinued.”
All of the methods now in common use are based on these theories. The
limiting equilibrium approach originating with Coulomb is statically indeterminate
along the straight line failure plane. However, it is statically determinate along a
curved failure plane. Developments since 1920, largely due to the influence of
Terzaghi, have led to a better understanding of the limitations and appropriate
applications of the classical earth pressure theories. Many experiments have been
conducted to prove the validity of the wedge theory and it has been found that the
sliding surface is not a plane, but a combination of a curve and straight line.
Furthermore these experiments have shown that the classical earth pressure theories
for cohesionless soil lead to quite accurate results for backfill of clean dry sand for

22
low wall-backfill friction angle. However, realistically most structural backfill
materials possess cohesion and a structure soil interface friction angle.
Kerry (1936) developed the friction circle method using trial and error
procedures. In this method the lower portion of the slip surface is assumed to be
curved in the form of arc of a circle which joins continuously to the upper portion of
the slip surface which is a straight line.
Caquot and Kerisel (1948) provided tables for passive earth pressure
coefficients only for cohesionless soil using the arc of an ellipse for the failure plane
as shown in Figure 2.5.
Sokolovski (1960) developed the Method of Characteristic by using finite
difference technique. Rosenfard and Chen (1972) applied plasticity theory and their
results compared very well with those of Sokolovski. Lambe and Whitman (1969)
reported that Sokolovski’s method is an alternative which may be used to evaluate
soil passive resistance. However this method gives the same solution as that
proposed by Terzaghi (1943) using a trial curved failure surface.
Janbu (1957) proposed the use of method of slices and bearing capacity
calculation techniques for evaluating passive earth pressure coefficients.

23

Figure 2.5: Passive Earth Pressure Coefficient (Caquot and Kerisel 1948)

24
Rowe and Peaker (1965) conducted several laboratory tests on rigid walls
and concluded that the peak values of the soil friction angle φ and the wall friction
angle δ should not be used in theoretical expressions for earth pressure computations
without a reduction factor.
Narain, Saran and Nandakunran (1969) have conducted several laboratory
tests with a rough wall having a vertical face and horizontal backfill. Passive
pressure at various depths of the wall was measured when the wall was subjected to
rotation about its bottom, its top and horizontal translation. They concluded that
none of the more common theories including Terzaghi’s were found to give the
passive earth resistance or the size and shape of the rupture wedge correctly.
Shields and Tolunay (1973) used simplified Bishop’s method of slices and
logarithmic failure surface to predict passive earth pressure coefficients. They
assumed no interslice shear force developed along the slice boundary. This indicates
that there is only a large shear force at the first slice (face of the wall) compare to
other slices. They concluded that for dense sand (φ = 40
o
) the method of slices gives
earth pressure coefficient Kp values which are in better agreement with experimental
values than other theories. For loose sand (φ =34
o
) the other theoretical values of Kp
are closer to the experimental values, but the slice method values are lower and,
therefore, more conservative. However, recent experimental test results conducted by
various universities and institutions indicated that using simplified Bishop’s method
of slices under predict the passive earth pressure capacity for dense sand since the
effect of vertical interslice forces are being ignored.

25
In the derivation of the classical earth pressure formulation, no wall
movement was specified. Dubrova (1963) recognized this fact and assumed that the
backfill material mobilizes its strength to a certain extent that is in proportion to the
corresponding wall movement and that the strength varies along the height of the
wall. The extensive experimental work conducted by James and Bransby (1970,
1971) remarkably showed that wall movement is a function of backfill shear strain
and mobilized shear strength.
Prakash and Rafnsson (1991) used a simplified method to estimate the
backfill stiffness as a function of three variables which are displacement, at-rest earth
force and ultimate earth forces. Because stiffness by definition is the ratio of change
in force to change in displacement, they used the following relationship to estimate
the passive stiffness of the backfill.
K= (Fp - Fo)/ dp (2.1)
Rahardjo, Fredlund, and Fan (1995) used the method of slices and two-
dimensional finite element analysis to compute lateral earth pressure for dry sand.
They concluded that the interslice shear forces occur as a result of soil displacement,
the magnitude of mobilized wall-soil interface friction angle δ and the shear forces
dissipate from the maximum at wall to zero at some distance away from the wall and
the selection of proper interslice force function is essential for an accurate passive
earth pressure computation.
Martin, Yan, and Lam (1997) conducted advanced theoretical studies at the
University of Southern California using the two-dimensional explicit finite

26
difference computer program FLAC (ITASCA, 1995) for improved seismic design of
bridge abutment. They used a simple elastoplastic constitutive modeling in a
numerical analysis to evaluate passive earth pressure capacity of several abutment-
soil combinations. Based on this study it can also be concluded that all of the
acceptable common theories for passive earth pressure computation over predict the
bridge abutment capacity for competent embankment approach fills.
Shamsabadi et al. (2005) developed nonlinear abutment force-deformation
based on the exponential empirical stress-strain relationship (Norris, 1998, 1979)
shown in Figure 2.6.
SL e
(3.707 SL)
100
(m + q)
50
=
−∗ ≥
=
+
⎡
⎣
⎢
⎤
⎦
⎥
≥

e
λε
ε
ε
ε
ε ε
for
for
80%
02
80%
ln( . )
ε
800
ε
50
ε
100
SL
1.0
0.8
0.5
λ=≤
≥
⎧
⎨
⎪
⎩
⎪
Linear variation for 0.5 < SL < 0.8
3.19 for SL 0.5
2.14 for SL 0.8
SL e
(3.707 SL)
100
(m + q)
50
=
−∗ ≥
=
+
⎡
⎣
⎢
⎤
⎦
⎥
≥

e
λε
ε
ε
ε
ε ε
for
for
80%
02
80%
ln( . )
ε
800
ε
50
ε
100
SL
1.0
0.8
0.5
SL e
(3.707 SL)
100
(m + q)
50
=
−∗ ≥
=
+
⎡
⎣
⎢
⎤
⎦
⎥
≥

e
λε
ε
ε
ε
ε ε
for
for
80%
02
80%
ln( . )
ε
800
ε
50
ε
100
SL
1.0
0.8
0.5
λ=≤
≥
⎧
⎨
⎪
⎩
⎪
Linear variation for 0.5 < SL < 0.8
3.19 for SL 0.5
2.14 for SL 0.8

Figure 2.6: Stress-Strain Relationship (Shamsabadi et al., 2005)
2.2 Abutment Behavior During a Seismic Event
Reconnaissance reports after a number of earthquakes around the world all
indicated that approach embankment failure resulted in many bridge closures.
During a seismic event, depending on its magnitude, the abutment structural

27
capacity, and the pile-abutment connection details different types of damage will
occur. For instance, the backwall shear failure and formation of the mobilized
passive wedge (shown in Figure 2.7) are the result of the bridge cyclic displacement
of the bridge deck in the longitudinal direction.

Figure 2.7: Backwall Shear Failure and Mobilized Passive Wedge

The top of the abutment is pushed back by the impact from the
superstructure. Since there is no longer sufficient lateral embankment resistance, as a
result of abutment displacement and rotation the bottom of the abutments moves
forward, pushing out the slope paving and imposing high flexural as well as shear
demand on the abutment piles. During the June 2001 Atica earthquake in Peru, the

28
north abutment of the Puente Los Banos bridge (a three-span continuous RC box
girder structure supported on two-column bents and seat-type abutments)
experienced significant displacement and rotation.
2.3 Caltrans Abutment Design Criteria
Field experiment from the full scale abutment testing conducted at the
University of California, Davis forms the basis for development of abutment springs
in longitudinal direction to simulate bridge-abutment-backfill interaction during a
seismic event. The experimental set up is shown in Figure 2.8. The wall height and
width were was 5.5 feet and 10 feet, respectively. The soil used to construct the
embankment in this abutment test was Yolo Loam-compacted clay with an undrained
shear strength of about 2 ksf and unit weight of 120 pcf. The abutment-embankment
interface friction angle (δ) was estimated to be 22 degrees. The horizontal force
versus displacement curve measured from the test is shown in Figure 2.9 indicating
highly nonlinear behavior. The measure ultimate abutment backfill capacity was
approximately 312 kips. The abutment experienced over 6 inches of longitudinal
displacement and 2 degrees of rotation.

29

Figure 2.8: UCD Test Setup (Maroney,1995)

30

Figure 2.9 : Estimated Bilinear Abutment Stiffness

31
The nonlinear force-deformation relationship and the idealized bilinear curve per
Caltrans recommendation is also shown in Figure 2.9. This idealization resulted
ultimate abutment pressure of 5 ksf and an average stiffness of 20 kips per inch per
foot of abutment. For the longitudinal abutment response currently per Caltrans
Seismic Design Criteria (SDC, 2004), acceptable design procedure for the seismic
design of the bridge abutment is the idealized bilinear force deformation relationship
shown in Figures 2.9 and 2.10.
Adjustment height factor based on the height of the UC Davis abutment is
considered according to the following equations for other wall heights.
The initial stiffness and the maximum passive capacity provide by the
abutment backfill are adjusted proportional to the abutment backwall height as
shown in Equations (2.2) and (2.3).

*
W
*
K
abut
K
i
= α (2.2)

*
5.0
*
F
abut
A
e
= α (2.3)
Where
α =
⎛
⎝
⎜
⎞
⎠
⎟
h
55 .
(2.4)

W is the width of the backwall and A
e
is the effective width of the backwall.
Regarding the seat-type abutment shown in Figure 2.11b, for the linear
elastic demand model effective abutment stiffness, K
eff
that accounts for expansion
gaps, and incorporates a realistic value for the abutment backfill response is used.

32
For diaphragm abutments shown in Figure 2.11a, the entire diaphragm above
and below the soffit is typically designed to engage the backfill immediately when
the bridge is displaced longitudinally. Therefore, the effective abutment area is equal
to the entire area of the diaphragm. If the diaphragm has not been designed to resist
the passive earth pressure exerted by the abutment backfill, the effective abutment
area is limited to the portion of the diaphragm above the soffit of the deck.

A
e
h
bw
W
bw
A
e
h
dia
W
dia
=
=
Saet Abutment
Diaphragm Abutment
(2.5)

h
dia
= h
d
d
d
i
i
i
a
a
a
* = Effective height if the diaphragm is not designed for full soil pressure
(see Figure 2.11: Effective Abutment Area).
h
dia
= h
dia
** = Effective height if the diaphragm is designed for full soil pressure (see
Figure 2.11: Effective Abutment Area ).
Per current Caltrans SDC, typically abutment shear keys are expected to
transmit the lateral shear forces generated by small earthquakes and service loads.
Determining the earthquake force demand on shear keys is difficult. The forces
generated with elastic demand assessment models should not be used to size the
abutment shear keys. Per SDC 2004, shear key capacity for seat abutments shall be
limited to the smaller of the following:
⎪
⎩
⎪
⎨
⎧
×
×
≤
∑
sup
3 . 0
75 .
dl
pile
sk
P
V
F (2.6)

33
abutment the at reaction load dead Axial
capacity pile lateral the of Sum
sup
=
=
∑
dl
pile
P
V

For abutments supported on spread footings the shear keys are only designed
to 0.3P. Wide bridges may require internal shear keys to insure adequate lateral
resistance is available for service load and moderate earthquakes. Internal shear keys
should be avoided whenever possible because of maintenance problems associated
with premature failure caused by binding due to the superstructure rotation or
shortening. Sufficient abutment seat width shall be available to accommodate the
anticipated thermal movement, prestress shortening, creep, shrinkage, and the
relative longitudinal earthquake displacement. Per SDC 2004, the seat width normal
to the centerline of bearing shall be calculated by equation 2.7 but not less than 30
inches as shown in Figure 2.13.
N
Aps crsh temp eq
≥+
+
++
⎛
⎝
⎜
⎞
⎠
⎟ ∆∆ ∆ ∆
/
+4 (2.7)
Where
N
A
= Abutment seat width normal to the centerline of bearing
∆
p/s
= Displavement attributed to pre-stree shortening
∆
cr+sh
= Displacement attributed to creep and shrinkage
∆
tepm
= Displacement attributed to thermal expansion and contraction
D
eq
= The largest relative earthquake displacement between the
superstructure and the abutment calculated by global or stand-alone analysis

34

P
bw
∆
gap
K
eff
Deflection
Force
K
abut
∆
eff

Figure 2.10: Effective Abutment Stiffness (SDC, 2004)

**
w
dia
h
dia
*
h
dia
**
w
dia
h
dia
*
h
dia
w
dia
h
dia
*
h
dia

(a) Diaphragm Abutment

w
bw
h
bw
w
bw
h
bw

(b) Seat Abutment

Figure 2.11: Effective Abutment Area (SDC, 2004)

35

w
abut
w
abut

Figure 2.12: Effective Abutment Width for Skewed Bridges (SDC, 2004)

4"
∆
eq
∆
p/s
+
∆
cr+sh
+
∆
temp
30 in Seat
≥
N
A
C Brg.
L
4"
∆
eq
∆
p/s
+
∆
cr+sh
+
∆
temp
30 in Seat
≥
N
A
C Brg.
L

Figure 2.13: Abutment Seat Width Requirements (SDC, 2004)

36
CHAPTER THREE
ABUTMENT LSH MODEL

3.1 Introduction
Current seismic design of bridges is based on a displacement performance
philosophy. This type of bridge design necessitates that geotechnical engineers
predict the resistance of the abutment backfill soils, which is inherently nonlinear
with respect to the displacement between soil backfill and the bridge structure.
Usually bridge engineers ignore the contributions of the abutment resistance in
seismic design of bridge structures due to complexity of the abutment soil-structure
interaction.
The objective of this chapter is to apply a limit-equilibrium method using
mobilized Logarithmic-Spiral failure surfaces coupled with a modified Hyperbolic
soil stress-strain behavior (the “LSH” model) to capture the nonlinear abutment
force-displacement relationship. The LSH model is developed to estimate abutment
nonlinear force-displacement capacity as a function of wall displacement and soil
backfill properties. The predicted results obtained using the LSH model are
compared with the results obtained from a total of nine experiments conducted on
various typical structure backfills. The predicted capacities calculated from the LSH
model are in good agreement with the measured capacities from the experiments.
The LSH model can also be expressed directly as a function of average soil stiffness

37
and ultimate soil capacity that can be used as a powerful and effective tool for
performance-based bridge design.
3.2 Types of Bridge Abutments
A bridge abutment consists of stem walls to support the bridge deck. The
footing to support the stem and the wingwalls attached at the end of each abutment
to retain the abutment-backfill in between the wingwalls. Abutments are basically
classified into two types: (1) seat-type-abutments, and (2) monolithic abutments.
Seat-type-abutments are located at or near the top of approach fills, with a
backwall depth sufficient to accommodate the structure depth. The seat-type
abutment is constructed separately from the bridge deck. The bridge deck rests on
the abutment seat through bearings pads as shown in Figure 3.1.
Monolithic abutments are cast integrally with the superstructure and are
supported on either spread footings or pile foundations as shown in Figure 3.2. For
monolithic abutments, the entire backwall is typically designed to engage the
backfill immediately when the bridge is displaced longitudinally. If the backwall has
not been designed to resist the passive earth pressure exerted by the abutment
backfill, the effective abutment height is limited to the depth of the bridge deck.
Figure 3.3 shows an example of the abutment damage occurred in the June
2001 Attica earthquake in Peru. The north abutment of the Puente Los Banos Bridge
(a three-span continuous RC box girder structure supported on two-column bents and
seat-type abutments) experienced significant displacement and rotation.

38

Figure 3.1: Seat-Type Abutment and Foundation System

39

Figure 3.2 : Monolithic Abutment and Foundation System

40

Figure 3.3: Example of Mobilized Passive Wedges

The top of the abutment was pushed back by the impact from the
superstructure. Since there was no sufficient lateral embankment resistance as a
result of abutment displacement and rotation, the bottom of the abutments moves
forward, pushing out the slope paving and imposing high flexural as well as shear
demand on the abutment piles.
3.3 Seismic Behavior of Seat-Type Abutments
During a seismic event, the bridge moves laterally and collides with the
abutment backwall in between the wingwalls. The backwall is designed to break off
as a result of seismic force F in order to protect the foundation from inelastic action

41
as a result of backwall displacement. This type of abutment allows the bridge
engineers to control the amount bridge deck forces and or displacement that are to be
transferred to the abutment backfill.
The abutment force-deformation capacity is provided by the passive
resistance of the abutment backfill. A typical highway bridge is wide and has a
moderate back wall height, often 5 to 6 feet. The earth pressure problem is then a
plane strain problem and 2-D simulations may be sufficient to simulate the
abutment-backfill response.
3.4 Seismic Behavior of Monolithic Abutments
Contrary to the seat-type abutment, the monolithic abutment has the potential
for heavy damage during a major seismic event. There is no relative displacement
allowed between the superstructure and abutment. All the superstructure forces at the
bridge ends are transferred to the abutment backwall and then to the abutment
backfill and foundations. The lateral force-deformation capacity of the monolithic
abutment is a function of abutment backfill properties as well as foundation and
structural capacity of the abutment backwall.
3.5 Force-Displacement Capacity of Bridge Abutment
Bridges are one of the most crucial parts of the transportation network which
have been struck by earthquakes in the past. It is generally recognized that when the
bridge deck moves laterally towards the abutment during a seismic event, the bridge
structure applies a lateral compressive force to the abutment which mobilizes passive
resistance in the soil backfill and results in permanent soil displacement. When the

42
bridge moves away from the abutment, a gap can form between the bridge deck and
the abutment backfill.
When bridges are subjected to small earthquake-induced lateral forces, they
generally remain in the elastic range. When subjected to strong earthquake shaking,
however, the dynamic response of the bridge becomes nonlinear and is largely
dependent on the nonlinear soil-structure interaction effects between the abutments
and the backfill soils. The nonlinear force-displacement-capacity of the bridge
abutment in a seismic event is developed mainly from the mobilized passive pressure
behind the abutment backwall. Proper modeling of the abutment-backfill system is
therefore critical and the assumptions made for the nonlinear stiffness as well as the
hysteretic of the abutment have been shown to have a profound effect on the global
seismic response and performance of the bridge (Shamsabadi et al., 2005;
Shamsabadi et al., 2007; Faraji et al., 2001; El-Gamal and Siddharthan, 1998).
There are many bridges with seat-type abutments in which the bridge deck is
supported by the abutments on bearings and the columns are supported by a pinned
connection at the base of the column to the pile caps or spread footings. The
performance of these bridges during seismic shaking is profoundly affected by the
interaction between the backfill soil and the abutment structure which involves
relative displacement and soil stress-strain behavior.
Earth pressure theories are developed based on different assumptions and
employ various methods to predict lateral soil-abutment capacity. As a result, the
capacities they predict can vary drastically from each other. The distribution and the

43
magnitude of the lateral soil-abutment resistance are highly dependent on the
abutment displacement. In spite of this, classical earth pressure theories give no
information about the wall movement. In order to calculate the wall passive pressure
as a function of wall movement, advanced analytical model such as finite element
(Shamsabadi et al., 2006) or finite difference (Martin et al., 1997) models should be
used. The analyses using these types of models are very expensive and time
consuming. They are also complex because they require the use of appropriate
boundary conditions, advanced nonlinear constitute models with a robust interface
element between the abutment-backfill and the abutment backwall. Therefore, they
are not feasible to be used in day- to-day bridge design.
The purpose of this chapter is to present a new model to predict the realistic
nonlinear lateral force-displacement capacity of a regular bridge abutment as a
function of common backfill properties and structural configurations. The basic
framework of the formulation is based on the mobilized Logarithmic Spiral (LS)
failure coupled with modified Hyperbolic (H) abutment-backfill stress-strain
behavior.
The LSH relationship which was developed from evaluation of a large
number of experimental test data can used to calculate the backfill capacity. The
LSH model can be used by geotechnical and bridge engineers to calculate the
backfill capacity as a function of soil stress-strain and strength characteristics, and
therefore it requires an understanding of soil mechanics.

44
3.6 Mechanism of the Abutment Backfill Failure
In the derivation of the classical earth pressure formulation, no wall
movement was specified. Dubrova (1963) recognized this fact and assumed that the
backfill material mobilizes its strength to a certain extent that is in proportion to the
corresponding wall movement and that the strength varies along the height of the
wall. The extensive experimental work conducted by James and Bransby (1970,
1971) remarkably showed that wall movement is a function of backfill shear strain
and mobilized shear strength. Therefore, when an abutment wall is loaded
monotonically by a horizontal force F, the wall is resisted by the mobilized passive
resistance of the abutment backfill as a function of relative displacement ∆ between
the wall and the backfill. For intermediate levels of displacement (∆<∆
ult
),

the shear
strength of the backfill can not be fully mobilized and therefore, the final passive
wedge can not be formed behind the abutment-wall. It is assumed that at each level
of displacement, ∆, a mobilized passive wedge is formed and as a result, an
intermediate passive resistance force F is developed. When the displacement
becomes large enough (∆ = ∆
ult
), the shear strength of the backfill will be fully
mobilized and the ultimate passive backfill capacity F
ult
develops as shown in Figure
3.4.

45
F
ult
∆
ult
1
2
3
i
ult
∆
F
F ult
i
h
i
h
ult
i
3
2
1
F
ult
∆
ult
1
2
3
i
ult
∆
F
F ult
i
h
i
h
ult
i
3
2
1

(a) Force-Displacement Relationship

() σ
1
i
ε
1
2
3
i
ult
σ
τ
φ
1
() σ
1
i
σ
3
σ
3
σ
3
φ
i
φ
3
φ
2
φ
ult
φ
ult
c
ult
() σ
3 i
σ
() σ
1
i
ε
1
2
3
i
ult
σ
τ
φ
1
() σ
1
i
σ
3
σ
3
σ
3
φ
i
φ
3
φ
2
φ
ult
φ
ult
c
ult
() σ
3 i
σ

(b) Stress-Strain Relationship

Figure 3.4: Mobilization of Passive Resistance

46
The formation of these wedges is primarily a function of the stress-strain
behavior of the backfill. The intermediate levels of displacement are associated with
intermediate states of strain (ε) and stress (τ). The ultimate wedge forms at
maximum level of strain (ε
ult
). This stress-strain relationship will be used to calculate
the “mobilized” shear strain levels (γ
i
) and shear strength parameters of the backfill
soil in order to predict the nonlinear passive force as a function of wall-soil
displacement.
The abutment-backfill failure mechanism has been observed to occur for both
wall rotations about the toe as well as translations during major seismic events (Kosa
et al., 2001), small-scaled laboratory experiments (Maciejeweski et al., 2004;
Bransby, 1971) ) and large-scaled field experiments (UCLA, 2006; Rollins et al.,
2006).
Bransby (1970, 1971) observed in rigid wall load tests that the failure
surfaces in dense sand progress from the top down for both failure modes.
Maciejeweski and Jarzebowski (2004) made similar observations in load tests of a
rigid wall pushed into silty sand for translation.
Rollins (2006) showed the traces of multiple passive wedges in cracking
patterns behind a pile cap progressing with increasing distance from the pile cap.
As part of UCLA-Caltrans research program, a full-scale cyclic load tests
was conducted to develop abutment nonlinear force-deformation relationship for a
typical backfill (2006). Before the test, 3-inch diameter vertical holes were drilled
along the longitudinal centerline of the abutment into the abutment backfill and filled

47
with brittle gypsum columns to map the failure mechanism of the abutment backfill
placed behind the backwall as shown in Figure 3.5a. After the completion of the test,
a longitudinal trench was excavated and the failure mechanism of the backfill was
carefully investigated by mapping the deformation and cracks of the gypsum
columns. The pattern of the cracks developed in the gypsum columns illustrates the
development of successive plane-strain failure surfaces that mobilize as a function of
lateral displacement and backfill properties. The deformed wedges started to develop
within the upper soil layer and progress deeper down and away from the backwall.
Observations and post earthquake investigations have also indicated that
during a major seismic event mobilized passive wedges will form within the
abutment backfill. The effect of an actual earthquake pushing a bridge deck into the
abutment-backfill is shown in Figure 3.5b. This is an example of the mobilized
passive wedge formation when a bridge superstructure has been pushed into the
abutment-backfill due to longitudinal seismic excitation. The surface cracks were
developed in the roadway pavement behind the northern (77-feet wide, near-normal
5
o
skew) abutment of the Shiwei Bridge in Taiwan during the Chi-Chi earthquake
(Kosa et al., 2001).
The LSH model is developed to simulate such a failure mechanism as a
function of abutment height and abutment backfill strength and stress-strain
properties. Similar failure mechanism has been observed using two-dimensional and
three-dimensional nonlinear finite element models. The finite element models are
presented in the next chapter.

48
70”
A
B
C
D
E
70”
A
B
C
D
E

(a) UCLA Abutment Field Experiment

(b) Shiwei Bridge after Chi-Chi earthquake (Kosa et al., 2001)

Figure 3.5: Mobilized Wedges During Seismic Event

49
3.7 Abutment Backfill Constitutive Model
When soils are subjected to stress changes in the laboratory and in the field,
they deform in complicated ways, which can be represented in terms of stress-strain
relationship (Bardet, 1997). Stress-strain characteristics of the soils are extremely
complex, highly nonlinear and inelastic. Bardet (1983) has shown several types of
stress-strain behavior that have been observed for various soils as shown in Figure
3.6. In all cases, the soil undergoes both elastic and plastic deformation. The yield
stress σ* marks the transition between elastic and plastic soils behavior.
Determination of σ* is not always trivial and may be subjected to interpretations
(Bardet, 1997). This above statement is also true based on all the small-scaled and
full-scaled abutment experimental nonlinear force-deformation backbone curves.
This is the main reason that the current Caltrans Seismic Design Criteria (SDC,
2006) suggests an average stiffness rather than the initial stiffness should be used to
calculate a bilinear force-deformation relationship to model the nonlinear behavior
of the backfill as a set of independent horizontal springs based on large-scaled
abutment and pile cap field experiments.
The realistic constitutive model must be able to distinguish between the
elastic and plastic deformation of the soil behavior up to and beyond the failure.
As part of Caltrans seismic research program, Earth Mechanic Inc. (EMI)
conducted State-wide field exploration and cyclic triaxial tests on the in-situ soil
samples to characterize strength properties and stress-strain behavior of abutment
backfills.

50

Figure 3.6: Various Types of Elstoplastic Soils Behavior (Bardet, 1997)

The results of the EMI’s laboratory cyclic triaxial experiments indicate that
the stress-strain behavior of the backfill is highly plastic and nonlinear from the very
early stages of the loading as shown in Figure 3.7. The decreasing stiffness and
simultaneously irreversible plastic strain were present for all the backfills. Typical
abutment-backfill triaxial experimental stress-strain data and the triaxial finite element
simulations will be presented in the next chapter.

51
ε
Stress
Strain
σ
ε
Stress
Strain
σ

Figure 3.7: Stress-Strain Relationship for Typical Abutment Backfill

Constitutive modeling of the backfill soil mass behavior is an essential
component for prediction of abutment force-displacement relationships. It requires
an understanding of the shear strength parameters and stress-strain characteristics of
the abutment-backfill material. The backfill material model is described by a set of
equations that define a nonlinear relationship between the stress and strain. Norris
(1998, 1979) developed the basic idea for the formulation of the intermediate
mobilized passive wedge coupled with an exponential stress-strain relationship based
on triaxial test data to capture the behavior of pile foundations subjected to lateral
loading. Shamsabadi et al. (2005) used this relationship to calculate the nonlinear
passive force-displacement capacity for cohesionless (c = 0) and purely cohesive
(φ = 0) abutment backfills.

52
Kondner et al. (1963) have shown that the stress-strain behavior of various
soil types can be approximated by a hyperbolic relationship. Duncan and Chang
(1970) developed a widely used nonlinear material model which employs this
hyperbolic relationship to capture the soil stress-strain behavior. In this chapter, the
hyperbolic stress-strain relationship is modified to develop the mobilized backfill
shear strength parameters (φ−c) as a function of strain. The modified hyperbolic
model consists of one expression compared to the exponential model which consists
of three equations. The hyperbolic stress-strain model is found to have better quality
and to be easier in matching load test data compared to the exponential relationship.
3.8 Nonlinear Hyperbolic Model
The basic idea for the formulation of the intermediate mobilized passive
wedge formation is based on the hyperbolic relationship between the vertical strain ε
and deviatoric stress (σ
1
-σ
3
) during triaxial loading. The hyperbolic model described
by Duncan and Chang (1970) is shown in Figure 3.8 and defined as follows:
()
()
σσ
ε
ε
σσ
13
13
1
−=
+
−
i
i
o
i
ult
E

(3.1)

In the hyperbolic relationship, the deviatoric stresses increase towards an asymptotic
value of (σ
1
-σ
3
)
i
and a stress level of (σ
1
-σ
3
)
f
must be defined at which the soil is
postulated to fail. The failure ratio R
f
is introduced that specifies the stress level at
failure as follows:

53

() ) ) σσ σ
φφ
13 3
2 −= + −
⎛
⎝
⎜
⎞
⎠
⎟
++
i
i
i
i
c tan (45 1 tan(45
2o o
22
() σ
1
i
σ
3
() σ
1
i
σ
3
σ
τ
σ
3
c
ult
φ
i
() σ
1
i
() ) ) σσ σ
φφ
13 3
2 −= + −
⎛
⎝
⎜
⎞
⎠
⎟
++
i
i
i
i
c tan (45 1 tan(45
2o o
22
() σ
1
i
σ
3
() σ
1
i
σ
3
σ
τ
σ
3
c
ult
φ
i
() σ
1
i
() σ
1
i
σ
3
() σ
1
i
σ
3
() σ
1
i
σ
3
() σ
1
i
σ
3
σ
τ
σ
3
c
ult
φ
i
() σ
1
i

(a) Stresses

ε
1
Asymptote
Assumed
Failure
1.0
0.5
SL
0.0
0.0
E
o
ε
50
ε
f
SL
E
R
i
i
f
o
fi
()
()
ε
ε
σσ
ε
=
−
+
13
1
2 13
() σσ −
f
() σ σ
13
−
f
() σ σ
13
−
ult
() σ σ
13
−
ε
1
Asymptote
Assumed
Failure
1.0
0.5
SL
0.0
0.0
E
o
ε
50
ε
f
SL
E
R
i
i
f
o
fi
()
()
ε
ε
σσ
ε
=
−
+
13
1
2 13
() σσ −
f
() σ σ
13
−
f
() σ σ
13
−
ult
() σ σ
13
−

(b) Stress-Strain Relationship

Figure 3.8: Hyperbolic Model

54

R
f
f
ult
=
−
−
()
()
σ σ
σσ
13
13
(3.2)
The hyperbolic relationship then can be expressed as:
()
()
σσ
ε
ε
σσ
13
13
1
−=
+
−
i
i
o
fi
f
E
R
(3.3)
By normalizing the stresses to the stress at failure, the hyperbolic relationship can be
expressed in terms of the deviatoric stress ratio SL(ε
i
):
SL
E
R
R
i
i
f
i
f
o
fi
i
ofi
()
()
()
()
ε
σ σ
σσ
ε
σσ
ε
ε
εε
=
−
−
=
−
+
=
+
13
13
13
(3.4)
The modulus E
o
is determined from the slope at the departure of the nonlinear stress-
strain curve and is typically difficult to obtain due to the nonlinear nature of curve as
was discussed earlier. For practical purposes, the secant modulus E
50
associated with
ε
50
is used instead (Schanz et al., 1999), where ε
50
is the strain at which 50% of the
failure strength is achieved as illustrated in Figure 3.8.
Figure 3.8 shows three following boundary conditions that the hyperbolic
relationship should satisfy:
()
() .
() .
ISL at
II SL at
III SL at
i
i
if
==
==
==
⎫
⎬
⎪
⎭
⎪
ε
εε
εε
0
05
10
50
(3.5)
Eq. (3.4) satisfies boundary condition (I). It is found that Eq. (3.4) satisfies boundary
condition (II) only if the R
f
factor is set to 1 and the strain ε
50
=ε
ο
. However, in that

55
case the equation does not converge as shown in Figure 3.8 and as a result, the
hyperbolic function must be modified as shown in Figure 3.9. The hyperbolic
equation in a more general form is as follows:
SL
AB
i
i
i
() ε
ε
ε
=
+
(3.6)

ε
50
ε
f
0
05 .
10 .
SL
SL
A B
i
i
i
() ε
ε
ε
=
+
E
o
E
50
00 .
1
2 13
() σσ −
f
() σ σ
13
−
f
1
1
() σ σ
13
−
ε
50
ε
f
0
05 .
10 .
SL
SL
A B
i
i
i
() ε
ε
ε
=
+
E
o
E
50
00 .
1
2 13
() σσ −
f
() σ σ
13
−
f
1
1
() σ σ
13
−

Figure 3.9 Modified Hyperbolic Stress-Strain Relationship
A and B are constants that can be found by applying the same boundary conditions in
Eq. (3.5) shown in Figure 3.9. The resulting constants are:
A
f
f
=
−
ε ε
εε
50
50
and B
f
f
=
−
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
εε
εε
2
50
50
(3.7)
The modified hyperbolic relationship of Eq. (3.6) satisfying the boundary conditions
then becomes

56
SL
i
i
f
f
f
f
i
() ε
ε
εε
εε
εε
εε
ε
=
−
+
−
−
⎛
⎝
⎜
⎞
⎠
⎟
50
50
50
50
2
(3.8)
This modified relationship was used in subsequent LSH applications
presented later and requires parameters ε
50
and ε
f
to be defined. The ε
50
can be
determined from geotechnical laboratory testing of a sample from the backfill. The
sample needs to be tested to failure and the stress-strain curve is recorded. When
there is a lack of laboratory test data, typical values of ε
50
are recommended to be
selected from Table 3.1, which was developed by matching LSH model fittings with
field data and from correlation with published values. For sand, ε
50
is a function of
relative density (D
r
), grain size distribution (coefficient uniformity C
u
), grain shape,
and confinement (σ
3
). For clay, ε
50
is a function of undrained shear strength (S
u
) and
plasticity index (PI).
Table 3.1 Typical Values of ε
50
ε
50
Predominant Soil Type
Range
Presumptive
Value
Gravel 0.001 to 0.005
Clean Sand (0-12% Fines*) 0.002 to 0.003
Silty Sands (12-50% Fines*) 0.003 to 0.005
0.0035
Silt 0.005 to 0.007
Clay 0.0075
0.007
Note: *) “Fines” is the percentage by weight of soil grain sizes smaller than 0.075 mm.
From the evaluation of 144 triaxial tests (Norris, 1979), it was back
calculated that the strain at failure ε
f
is approximately 31 times larger than ε
50
:

57
ε ε
f
≅⋅ 31
50
(3.9)
From comparison of this relationship with the original relationship of Eq.
(3.4), it is found that
ε
ε
f
f
R
=
−
50
1
(3.10)
From Eq. (3.9) and Eq. (3.10), the corresponding R
f
value would be approximately
0.97. Typical R
f
values between 0.94 and 0.98 were obtained for various soil types
from backfitting of load test data presented in the next section.
By substituting Eq. (3.10) into Eq. (3.8), the modified hyperbolic stress-strain
relationship can be expressed in terms of R
f
and ε
50
:
SL
RR
i
i
ff
i
()
()
ε
ε
ε
ε
=
+
−
50
2
1
(3.11)
Based on the concepts of triaxial tests, the stress level SL can be also expressed in
terms of shear strength as the ratio of incremental deviatoric stress (σ
1
-σ
3
)
i
to
deviatoric stress at failure (σ
1
-σ
3
)
f
for the stress level SL =1 as shown in Figures 3.8
and 3.9:
( )
()
SL c
c
c
ii
i
f
ii i
()
(45 1 tan(45
tan (45 1 tan(45
2o o
2o o
φ
σσ
σσ
σφ φ
σφ φ
,
()
()
tan ) )
))
=
−
−
=
+− + +
+− + +

13
13
3
1
2
1
2
3
1
2
1
2
2
2
(3.12)
where
c
c
i
i
=
tan
tan
φ
φ
(3.13)

58
Eq. (3.12) can be also expressed in terms of Rankine earth pressure coefficients:
SL K
KcK
KcK
pi
pi i pi
pp
()
1)
1
=
−+
−+

σ
σ
3
3
2
2
(
()
(3.14)
where K
pi
is the intermediate mobilized passive earth pressure coefficient develops as
a function of soil strain ε
i
during triaxial tests, and K
p
is the ultimate passive earth
pressure coefficient of the soil at failure.
3.9 Nonlinear Abutment Backfill Capacity
Figure 3.10 shows the geometry and forces acting on a mobilized
logarithmic-spiral failure surface (Shamsabadi et al., 2005).
j
Slice
α
ij
Z
ij
Z
ij+1
∆Z
ij
∆y
ij
Mobilized failure
surface
O
1 j n
r
i1
r
ij
α
i1
α
ij
α
iw
θ
ij
h
i
j
Slice
α
ij
Z
ij
Z
ij+1
∆Z
ij
∆y
ij
Mobilized failure
surface
O
1 j n
r
i1
r
ij
α
i1
α
ij
α
iw
θ
ij
h
i

(a) Geometry
C
ij
1
j n
α
ij
φ
ij
+
j
F
iv
F
ih
T
(i+1)j
E
(i+1)j
W
ij
δ
(i+1)j
δ
iw
E
i1
R
ij
α
ij
90
o
φ
ij
−
δ
ij
E
ij
T
ij
C
ij
1
j n
α
ij
φ
ij
+
α
ij
α
ij
φ
ij
φ
ij
+
j
F
iv
F
ih
T
(i+1)j
E
(i+1)j
W
ij
δ
(i+1)j
δ
iw
E
i1
R
ij
α
ij
α
ij
90
o
φ
ij
−
90
o
φ
ij
−
φ
ij
φ
ij
−
δ
ij
E
ij
T
ij

(b) Forces
Figure 3.10 Mobilized Passive Wedge (Shamsabadi et al., 2005)

59
Where
i = subscript denoting a quantity associated with intermediate mobilized
failure surface i,
j = subscript denoting a quantity associated with the slice j,
W
ij
= intermediate mobilized total weight of slice,
L
ij
= intermediate mobilized length of failure plane of slice,
α
ij
= intermediate mobilized inclination of the failure plane at the slice base
(with respect to horizontal),
φ
ij
= intermediate mobilized soil interface friction angle,
R
ij
= intermediate mobilized resultant friction force at mid point of slice
failure surface,
C
ij
= intermediate mobilized resultant cohesion force along failure surface of
a slice,
δ
ij
= intermediate mobilized friction angle between slices (with respect to
horizontal).
E
ij
= intermediate mobilized force of slice,
h
i
= intermediate mobilized wall height Figure 3.9, and
δ
iw
= intermediate mobilized wall-soil interface friction angle.
The mobilized logarithmic-spiral failure surface is defined as follows:

60
rr
tan
ij i1
ij
=

e
ij
θ φ
(3.14)
where
θ
δ
ij
=
−
−
1
21
1
tan
2Ktan
K
ij

K
AA
A
=
+
12
3

A
h
ij
ij
ij
ij 1
2 =+ 1+ sin
c
2
φφ sin

A
c
z
c
z
c
z
ij ij
ij
ij
ij
ij
ij
ij
ij
ij 2
22 2
2441 =++ + − (cos ) (tan ) tan [ ( ) tan ] φφ
γ
δ
γ
φ

A
ij ij 3
2
4 =+ cos tan
2
φδ

Here, K is a ratio of horizontal to vertical stresses in the slice (Shamsabadi et al.,
2005). For the mobilized logarithmic-spiral failure surface shown in Figure 3.9, the
horizontal component ∆E
ij
resulting from the interslice forces E
ij
and E
(i+1)j
acting at
the sides of slice j can be expressed as (Shamsabadi et al., 2005):
[ ]
∆E
Wtan c L sin
1- tan tan
ij
ij ij ij ij ij ij ij ij ij
ij ij ij
=
++ + +
+
() tan( )cos
()
αφ α α φ α
δα φ
(3.16)
Summation of the ∆E
ij
forces yields the mobilized horizontal passive capacity F
ih

associated with the mobilized failure surface i and mobilized displacement:
F=
E
tan tan( + ) 1-
ih
ij
j
n
iw iw i
∆
=
∑
1
δα φ
(3.17)

61
where
α θ α
iw i i
= +
11

as shown in Figure 3.10. The local horizontal displacement of slice j as shown in
Figure 3.11 associated with the mobilized failure surface i is as follows (Shamsabadi
et al., 2005):
γ
ij
max
ij
22
2 =
γ
α sin
ε
v
pole
ε
σ
=
∆
h
E
()
γε−ε ε ν
ν max
22
1
==
+
2
γ
i
2
γ
i
2
ν σ
νε
∆
h
E
=− ε
ε
2α
ij
α
ij
γ
max
2
γ
2
γ
ij
max
ij
22
2 =
γ
α sin
ε
v
pole
ε
σ
=
∆
h
E
()
γε−ε ε ν
ν max
22
1
==
+
2
γ
i
2
γ
i
2
ν σ
νε
∆
h
E
=− ε
ε
2α
ij
α
ij
γ
max
2
γ
2

Figure 3.11: Associated Mohr Circle and Soil Strain

∆∆ ∆ yz
2
z
1
2
12
ij ij
ij
ij ij ij
== +
γ
να ε()sin
(3.18)
where
υ = Poisson’s ratio of the soil,
ε
ij
= axial strain in the slice, and
γ
ij
= shear strain in the slice.

62
The Mohr circle associated with failure surface i (see Figure 3.11) demonstrates the
relationship between the normal strain ε and shear stain γ/2 in the soil (Shamsabadi
et al., 2005).
The displacement of the entire mobilized logarithmic-spiral failure surface is
then obtained by summation of the displacements ∆y
ij
of all slices (Shamsabadi et
al., 2005):
y= y
iij
j
n
∆
=
∑
1
(3.19)
The entire LSH procedure to develop the nonlinear abutment force-displacement
curve using the modified hyperbolic soil model is illustrated in Figure 3.12.
For the simple wedge with planar failure surface shown in Figure 3.13, the
mobilized horizontal passive force F
ih
associated with each stress level SL(ε
i
) can be
calculated as follows:
( )
F
ih
ii i ii i i i i
iw i i

Wtan c L sin
1- tan tan
=
++ + +
+
() tan( )cos
()
αφ α α φ α
δαφ
(3.20)

63

Estimate ε
50
from
laboratory test,
Table 3.1, or literature
Estimate ε
f
from lab tests,
Eq. (3.9), Eq. (3.10)
using R
f
or literature
R
f
backfitted from
laboratory test,
or 0.97
END
Calculate SL(ε
i
)
using Eq. (3.8) or (3.11)
Select strain level ε
i
(0 < ε
i
< ε
f
)
Determine mobilized soil
shear strengths φ
i
and c
i
using Eqs. (3.12) & (3.13)
START
Step 1. Estimate Model Parameters
Step 2. Develop
Stress-Strain Curve
Calculate Force F
ih
using Eq. (3.17)
Calculate displacement y
i
using Eqs. (3.18) & (3.19)
Stress level SL=1?
Step 3. Develop
Force-Displacement
Curve
No
Yes
Estimate ε
50
from
laboratory test,
Table 3.1, or literature
Estimate ε
f
from lab tests,
Eq. (3.9), Eq. (3.10)
using R
f
or literature
R
f
backfitted from
laboratory test,
or 0.97
END
Calculate SL(ε
i
)
using Eq. (3.8) or (3.11)
Select strain level ε
i
(0 < ε
i
< ε
f
)
Determine mobilized soil
shear strengths φ
i
and c
i
using Eqs. (3.12) & (3.13)
START
Step 1. Estimate Model Parameters
Step 2. Develop
Stress-Strain Curve
Calculate Force F
ih
using Eq. (3.17)
Calculate displacement y
i
using Eqs. (3.18) & (3.19)
Stress level SL=1?
Step 3. Develop
Force-Displacement
Curve
No
Yes
Figure 3.12: Flowchart of LSH Procedure

64

1
2
3
i
ult
h
i
h
ult
1
2
3
i
ult
h
i
h
ult

(a) Mobilized Failure Surfaces

() α φ
ii
+
F
i
F
ih
W
i
F
iv
R
i
C
i
α
i
δ
iw
() 90
o
i
− φ
() α φ
ii
+
F
i
F
ih
W
i
F
iv
R
i
C
i
α
i
α
i
δ
iw
δ
iw
() 90
o
i
− φ

(b) Forces acting on the simple wedge

Figure 3.13: Simple Wedge with Planar Failure
For a given wall interface friction angle δ
iw
and a given mobilized failure
surface angle α
i
, Eq. (3.20) yields the exact same value as Coulomb’s equation. If
the wall interface friction angle δ
iw
=0 and the mobilized failure surface angle α
i
=45
o
-
φ/2, then Eq. (3.20) yields the exact same value per Rankine’s equation. Therefore,
once the mobilized backfill properties as a function of stress-strain is determined,
both Rankine’s and Coulomb’s equations can be used to determine the nonlinear
passive earth pressure as a function of wall displacement. The mobilized
displacement for the planar failure surface i can be calculated as follows:

65
∆y
2
1
2
12
ii ii i
hh == +
γ
να ε()sin
(3.21)
where
h
i
= the intermediate mobilized wall height, and
α
i
= inclination of intermediate mobilized failure surface of a planar
Rankine’s or Coulomb’s wedge (with respect to horizontal).
The mobilized soil strength parameters coupled with soil hyperbolic stress-
strain relationship (the “LSH procedure”) can be used with any earth pressure theory.
Table 3.2 compares the capacities obtained using the LSH procedure with two limit-
equilibrium methods for a 10-foot high wall. The material properties and LSH model
parameters used are also given in Table 3.2. The resulting coefficients are shown in
Table 3.2 and plotted in Figure 3.14.
It can be seen that (1) Coulomb’s nonlinear earth pressure coefficient is very
stiff, (2) Coulomb ultimate earth pressure coefficient is more that 5 times larger than
Rankine’s coefficient and about 2 times larger than the Log Spiral earth pressure
coefficient, and (3) Coulomb reaches the ultimate failure at lower displacement
because the mobilized wedge is larger. There is a large spread of the coefficients at
larger displacement. For displacement performance-based design, it is important to
select the earth pressure theory that is most suitable to a particular application in
terms of acceptable displacement criteria.

66
Table 3.2 Mobilized Earth Pressure Coefficients Using Three Methods
Earth Pressure Coefficient, K
pi

SL
ε
i
Mobilized φ
i
Mobilized δ
i

Rankine LSH Coulomb
0.000
8
0.0000 0.00 0.0 0.00 0.00 0.00
0.027 0.0001 2.61 0.80 1.10 1.13 1.13
0.233 0.0011 17.17 6.98 1.84 2.32 2.45
0.468 0.0031 27.21 14.05 2.69 4.16 4.95
0.645 0.0061 32.48 19.34 3.32 5.91 8.04
0.762 0.0101 35.33 22.86 3.74 7.27 11.07
0.839 0.0151 36.99 25.18 4.02 8.26 13.68
0.891 0.0211 38.02 26.74 4.21 8.96 15.82
0.927 0.0281 38.70 27.82 4.34 9.47 17.51
0.953 0.0361 39.17 28.59 4.43 9.85 18.84
0.972 0.0451 39.51 29.15 4.50 10.13 19.89
0.986 0.0551 39.76 29.58 4.55 10.35 20.74
0.997 0.0661 39.95 29.91 4.59 10.51 21.41
1.000 0.0781 40.00 30.00 4.60 10.56 21.59
Notes: Wall height: h
ult
= 10 ft
Soil parameters: φ = 40
o
, δ = 30
o

LSH model parameters: ν = 0.35, R
f
= 0.95, ε
50
= 0.0035

67
Rankine
LSH Model
Coulomb
0.0
5.0
10.0
15.0
20.0
25.0
K
ph
0.00.5 1.01.5 2.0
∆/H (%)

Figure 3.14 : Mobilized Earth Pressure Coefficients Using Three Methods

68
3.10 Verification of LSH Model
The passive resistance of the backfill has been studied by various researchers
in lateral centrifuge experiments, small-scale laboratory experiments and full-scale
field experiments on walls and pile caps subjected to monotonic and cyclic lateral
loadings. Analytical models have been developed to better understand the nonlinear
response of the bridge abutment as a function of displacement (Shamsabadi et al.,
2005; Martin et al., 1996; Siddharthan et al., 1997).
For the present study, the nonlinear backfill force-displacement capacity
curves from selected tests on various soil types are summarized. The results are
compared with the LSH model predictions.
3.11 Full-Scaled Abutment Experiments
As part of the Caltrans seismic research program, full-scale abutment field
experiments were conducted at University of California, Los Angeles (2006) and
University of California, Davis (Romstad et al., 1995; Maroney et al., 1994). A brief
description of the abutment experiments and the LSH prediction is described herein.
3.11.1 UCLA Abutment Experiment
A full-scale cyclic load test was performed by UCLA research team on a
15 feet by 3 feet abutment wall with a height of 5.5 feet having a silty sand backfill.
The purpose of the test was to simulate the seat-type abutment shown in Figure 3.1.
The backfill was placed in layers and compacted to over 95% Modified Proctor
density behind the wall and was extended about 3 times the backwall height in the
longitudinal direction. The backwall was pushed horizontally in between the

69
abutment wingwalls without any vertical movement. The abutment wingwalls were
constructed using smooth plywood. Plastic sheeting was placed at the interior face of
the plywood to minimize the friction along the wingwalls in order to simulate a plane
strain condition.
After the completion of the test, a longitudinal trench was excavated and the
failure mechanism of the backfill was carefully investigated by mapping the
mobilized deformed passive wedges along the abutment backfill. The deformed
wedges started to develop within the upper soil layer and progress deeper down and
away from the abutment backwall. The final failure surface extended from the
bottom of the abutment backwall and intersected the backfill surface at about 3 times
the height of the backwall. The mobilized deformed wedges and the final failure
surface which were mapped after the completion of the experiment is shown in
Figure 3.15a. The mobilized passive wedges predicted by the LSH model are shown
in Figure 3.15b. The final logarithmic spiral failure surface predicted by the LSH
model intersected the abutment backfill at about 16 feet behind the abutment
backwall, which remarkably matches the experimental failure surface mapped in the
field. The backfill properties, the LSH parameters and the measured data and
predicted capacity curves are shown in Figure 3.16. The backfill properties are based
on the triaxial test results conducted by EMI. The curve predicted by the LSH model
is in good agreement with the experimental data.

70

70”
A
B
C
D
E
70”
A
B
C
D
E

(a)
0
1
2
3
4
5
0 5 10 15
(b)
Figure 3.15: LSH Model Failure Wedges Versus Experimental Failure Wedges
(UCLA Test)

71

0
100
200
300
400
500
Passive Capacity (Kips)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Displacement (inches)
Measured
Predicted

Figure 3.16: LSH Prediction of UCLA Abutment Test

72
3.11.2 UCD Abutment Experiment
A full-scale test was conducted on a monolithic abutment in University of
California, Davis by applying cyclic longitudinal loading to ultimate failure
(Romstad et al., 1995, Maroney et al., 1994). The backfill consisted of compacted
Yolo Loam clay. The soil properties and the measured force-displacement response
of the abutment wall are shown in Figure 3.17. The predicted force-displacement
response using the LSH model and the LSH model parameters are also shown in
Figure 3.17. The predicted abutment wall capacity using the LSH model is in good
agreement with the experimental data. UCD abutment experiment will be explained
in more details in the next chapter as part of the advanced analytical modeling.
3.12 Full-Scaled Pile Cap Experiment
A series of full-scale static load tests were performed by Rollins and Cole
(2006) on a 17-foot by a 10-foott pile cap with a height of 3.67 feet. The pile cap was
placed on a 3 by 4 group of 12-inch diameter steel pipe piles driven in saturated low-
plasticity silts and clays. The passive resistance of the backfill against the side of pile
cap was determined to be about 40% of the total load resistance. The tests were
designed to differentiate the passive resistance of the pile cap in four different types
of backfill: clean sand with small amount of silt, silty sand, fine-grained gravel, and
coarse-grained gravel. The sand was compacted to approximately 95% of modified
Proctor density per ASTM D-1557 (ASTM, 2003). Adjustment factors were applied
to the structure width in the LSH model to account for the three-dimensional wedge
observed in the field. The factors were based on measurements of observed surface

73
cracking patterns in Rollins’ field tests (2006) and are similar to Ovesen-Brinch
Hansen correction factors based on field tests (Ovesen, 1964).
3.12.1 Clean Sand
The clean sand (SP) backfill was fine to coarse-grained, poorly-graded with
less than 2% fines and no gravel. The soil strength properties, the LSH parameters
and measured capacity of the pile cap are shown in Figure 3.18. For the LSH model,
a small apparent cohesion of 80 psf due to the presence of silt was applied. Rollins et
al. (2006) found that the maximum width of the 3-dimensional passive wedge
observed from cracking patterns at the ground surface was 40% larger than the width
of the pile cap. To account for three-dimensional effects, the pile cap width was
increased by 1.4 in the LSH model. The nonlinear response of the pile cap predicted
by the LSH model is in good agreement with the measured data.
3.12.2 Silty Sand
The silty sand (SM) backfill had a maximum particle size of 12.5 mm with
approximately 90% passing the No. 40 sieve and 45% non-plastic fines. The C
u
and
C
c
coefficients were 14.8 and 2.8, respectively. The pile cap width was increased by
1.2 in the LSH model for three-dimensional effects based on findings by Rollins et
al. (2006) from the field data. Figure 3.19 shows the key soil properties, the LSH
model parameters and the measured and predicted pile cap capacity. The computed
curve obtained by the LSH model is close to the measured data.

74

0
50
100
150
200
250
300
350
Passive Capacity (Kips)
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Displacement (inches)
Measured
Predicted
Figure 3.17: LSH Prediction of UCD-CT Abutment test

δ
o

20.0
ε
50

0.007
c lb/ft
2

2000.0
φ
o

0
R
f

0.96
Soil Strength Parameters
γ lb/ft
3

120.0
LSH Model Parameters
ν
0.45

75
0
100
200
300
Passive Capacity (Kips)
0.0 0.5 1.0 1.5 2.0 2.5
Displacement (inches)
Measured
Predicted

Figure 3.18: LSH Prediction of BYU Pile Cap Test in Clean Sand

δ
o

30
ε
50

0.002
c lb/ft
2

80
φ
o

39
R
f

0.98
Soil Strength Parameters
γ lb/ft
3

117
LSH Model Parameters
ν
0.30

76
0
50
100
150
200
250
300
350
Passive Capacity (Kips)
0.0 0.5 1.0 1.5 2.0 2.5
Displacement (inches)
Measured
Predicted

Figure 3.19: LSH Prediction of BYU Pile Cap Test in Silty Sand

δ
o

21
ε
50

0.003
c lb/ft
2

630
φ
o

27
R
f

0.97
Soil Strength Parameters
γ lb/ft
3

122
LSH Model Parameters
ν
0.35

77
3.12.3 Fine-grained Gravel
The fine-grained gravel backfill was poorly graded and had a maximum
particle size of 19 mm with approximately 50% gravel, 30% sand, and 20% fines.
The soil properties, the LSH model parameters, and the measured nonlinear pile cap
capacity are shown in Figure 3.20. The pile cap width was increased by 1.4 for three-
dimensional effects based on Rollins et al. (2006) field data. The LSH prediction
matched the measured load-displacement curve well as shown in Figure 3.20.
3.12.4 Coarse-grained Gravel
The coarse-grained gravel backfill was poorly graded and had a maximum
particle size of 100 mm with approximately 36% gravel, 15% sand, and 20% non-
plastic fines. The soil properties, the LSH model parameters and the measured pile
cap resistance are shown in Figure 3.21. The pile cap width was again increased by
1.4 for three-dimensional effects based on Rollins et al. (2006) field data. The
predicted capacity from the LSH model captures the rising trend of the measured
data.
3.13 RPI Centrifuge Experiment in Nevada Sand
Nonlinear abutment and pile cap backfill behavior under large amplitude
displacement has been investigated using centrifuge experiments. Scaled-centrifuge
abutment and pile cap experiments conducted at RPI has also provided a unique
opportunity to validate the LSH model.

78
0
50
100
150
200
Passive Capacity (Kips)
0.0 0.5 1.0 1.5 2.0
Displacement (inches)
Measured
Predicted

Figure 3.20 : LSH Prediction of BYU Pile Cap Test in Fine Gravel

δ
o

26
ε
50

0.0015
c lb/ft
2

82
φ
o

34
R
f

0.98
Soil Strength Parameters
γ lb/ft
3

132
LSH Model Parameters
ν
0.3

79
0
100
200
300
400
500
Passive Capacity (Kips)
0.0 0.5 1.0 1.5 2.0 2.5
Displacement (inches)
Measured
Predicted

Figure 3.21: LSH Prediction of BYU Pile Cap in Test Coarse Gravel

δ
o

30
ε
50

0.005
c lb/ft
2

0
φ
o

40
R
f

0.95
Soil Strength Parameters
γ lb/ft
3

148
LSH Model Parameters
ν
0.3

80
3.13.1 Seat-Type Abutment
Gadre and Dobry (1998) conducted a number of cyclic load tests in the
centrifuge on a seat-type bridge abutment prototype using dry dense Nevada sand.
The prototype represented a bridge abutment 5 feet high and 18.77 feet long. A
three-dimensional correction factor of 1.25 was applied to the abutment width
(Gadre, 1997). The soil properties, the LSH model parameters and the measured
force-displacement capacity of the abutment wall are shown in Figure 3.22. The sand
was glued to the back face of the wall to simulate a rough concrete surface (δ =
φ = 39
ο
). The predicted curve using the LSH model is in good agreement with the
experimental data.
3.13.2 Pile Cap
Gadre and Dobry (1998) also conducted a number of cyclic load tests in the
centrifuge on a pile cap prototype model embedded in dry dense Nevada sand. The
soil properties are shown in Figure 3.23. The prototype represented a pile cap with a
width of 3.74 feet and a height of 2.76 feet. An equivalent surcharge load of about 12
inches was also imposed on the pile cap. Since the contribution of the side and the
base friction of the pile cap were subtracted from the total measured resistance, no
correction for three-dimensional effects per Ovesen and Stromann (1972) was
applied in the model. The sand was again glued to the back face of the Pile Cap (δ =
φ = 39
ο
). The soil properties, the LSH model parameters and the measured force-
displacement curve for the pile cap are shown in Figure 3.23. The predicted curve
using the LSH model is in good agreement with the experimental data.

81
3.14 3.14 Small-Scaled Experiment of Wall in Loose Sand
Fang et al. (1994) conducted a small-scale laboratory test on a rigid vertical
retaining wall about 1.64 feet high by 3.28 feet wide. The wall was backfilled with
dry loose sand and subjected to slow monotonic, lateral loading. The soil properties
and LSH model parameters are given in Figure 3.24. The figure shows the horizontal
force-displacement response from strain gage measurements at four wall locations.
The curve predicted by the LSH model reasonably matches the experimental results.
3.15 Recommended Abutment Force-deformation Relationship
The relationship used in Eq. (3.8) and Eq. (3.11) was expressed in terms of
stress and strain. In seismic bridge design practice, however, abutment soil capacity
is typically based on an average soil stiffness K and the maximum abutment force
F
ult
developed at a maximum displacement y
max
. All three quantities are typically
provided by the geotechnical engineer. This abutment force-displacement
relationship (“backbone curve”) is shown in Figure 3.25 with the average stiffness
defined as:
K
F
y
ult
ave
=
1
2
(3.22)

82
0
100
200
300
400
Passive Capacity (Kips)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Displacement (inches)
Measured
Predicted

Figure 3.22: LSH Prediction of RPI Abutment Test in Dense Sand

δ
o

39
ε
50

0.0035
c lb/ft
2

0
φ
o

39
R
f

0.95
Soil Strength Parameters
γ lb/ft
3

103
LSH Model Parameters
ν
0.35

83
0
5
10
15
20
25
30
Passive Capacity (Kips)
0.00.5 1.01.5 2.02.5 3.03.5 4.04.5 5.0
Displacement (inches)
Measured
Predicted

Figure 3.23: LSH Prediction of RPI Pile Cap Test in Nevada Sand

δ
o

39
ε
50

0.007
c lb/ft
2

0
φ
o

39
R
f

0.96
Soil Strength Parameters
γ lb/ft
3

103
LSH Model Parameters
ν
0.35

84
Experimental Set 1
Experimental Set 2
Experimental Set 3
Experimental Set 4
LSH Model
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Passive Capacity (Kips)
0.00.5 1.01.5 2.02.5 3.03.5 4.0
Displacement (inches)

Figure 3.24: LSH Prediction of Wall Test in Loose Sand

δ
o

19.2
ε
50

0.014
c lb/ft
2

0
φ
o

30.9
R
f

0.96
Soil Strength Parameters
γ lb/ft
3

98.6
LSH Model Parameters
ν
0.45

85
0
K
y
max
y
ave
F
ult
2
F
ult
y
0
K
y
max
y
ave
F
ult
2
F
ult
y

Figure 3.25: Hyperbolic Force-Displacement Formulation
The force-displacement relationship in general hyperbolic form is:
Fy
y
ABy
() =
+
(3.23)
The constants A and B can be found by applying the following boundary conditions
as shown Figure 3.25:
()
()
()
max
IFL aty
II FL F at y y
III FL F at y y
i
ult i ave
ult i
= =
==
==
⎫
⎬
⎪
⎭
⎪
00
1
2
(3.24)
The resulting constants are:
A
y
Ky F
ult
=
−
max
max
2
and B
Ky F
FKy F
ult
ult ult
=
−
−
2
2
()
()
max
max
(3.25)

86
Therefore, the hyperbolic force-displacement (“HFD”) relationship is:
Fy
y
y
Ky F
Ky F
FKy F
y
ult
ult
ult ult
()
()
()
max
max
max
max
=
−
+
−
− 2
2
2
(3.26)
which can also be expressed as
Fy
FKy F y
Fy Ky F y
ult ult
ult ult
()
()
()
max
max max
=
−
+
−
2
2
(3.27)
Current Caltrans Seismic Design Criteria (SDC, 2006) uses a bilinear force-
deformation relationship to model the nonlinear behavior of the bridge abutment. It
is based on an average abutment-backfill capacity of 5 kips per square foot of
effective abutment backwall area and an average abutment backwall stiffness of 20
K/in per foot of wall length obtained from UCD’s field experiment (Maroney, 1994)
of a 5.5-ft high wall in cohesive soil. Eq. (3.28) is used to adjust for various
abutment heights:
KKinftf
FksfA
e
f
Abut
ult
= ⋅
=⋅ ⋅
20
5
//
(3.28)
where
f
H
ft
=
55 .

H is the wall height, and ft is the height adjustment factor.

87
The average stiffness (K) based on the UCD data shown in Figure 3.27 is
about 25 K/in/ft and the ultimate capacity (F
ult
) is about 5.5 K/ft
2
at approximately
6.6 inches of displacement.
Based on the recent UCLA abutment field experiment (2006) performed on a
5.5-ft high wall on compacted silty sand, the average abutment backwall stiffness is
about double that of UCD abutment experiment. However, the measured ultimate
capacity F
ult
is about the same as the UCD abutment experiment at approximately
3.3 inches of displacement as shown in Figure 3.26.
Table 3.3 summarizes the average stiffness, and the predicted maximum
capacity and displacement for each of the nine case studies presented in the prior
section. The y
max
/H ratio was based on the maximum passive force. For any of the
cases studies, the predicted HFD curves using Eq. (3.27) based on the stiffness and
maximum capacity and displacement parameters shown in Table 3.3 is nearly the
same as predicted using the LSH approach.
For a typical concrete highway bridge when no geotechnical data is available,
the following presumptive HFD parameters shown in Table 3.4 can be used to
develop the nonlinear force-displacement curve for engineered abutment backfill
based on the full-scale experimental test results and the author’s experience.

88
UCLA Test Data
HFD Model
Bilinear
LSH Model
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
Passive Capacity (Kips/ft)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
Displacement (inches)

Figure 3.26 : UCLA Abutment Experiment Compared with Various Models

89
Table 3.3 Parameters for HFD Model
Case Study (Ref.) F
ult

(Kips)

y
max
(in)
y
max
/H K
(K/in/ft)

Silty Sand (UCLA, 2006) 455.0 3.3 0.05 54
Clean Sand (BYU, 2005) 245.0 2.2 0.05 51
Silty Sand (BYU, 2005) 414.0 2.2 0.05 53
Fine Gravel (BYU, 2005) 175.0 1.8 0.04 51
Coarse Gravel (BYU, 2005) 453.0 2.6 0.06 46
Sand (BYU, 2002) 345.0 1.30 0.03 42
Sand/Abutment (RPI, 1998) 343.0 4.3 0.06 17
Sand/Pile cap (RPI, 1998) 28.30 4.3 0.10 7.5
Clay (UCD, 1994) 312.0 6.6 0.10 25
Sand (Fang, 1994) 2.20 4.3 0.20 2.4

Table 3.4 Suggested HFD Parameters for Abutment Backfills
Abutment Backfill
Type
Pressure
(ksf)

Ave. Soil Stiffness
(K/in/ft)

y
max
/ H
Granular* 5.5 50 0.05
Cohesive* 5.5 25 0.10
Note: * Compacted to at least 95% relative compaction per ASTM D-1557
Abutment Backwall Height = 5.5 ft

The constants A and B in Eq. (3.25) can be found by substituting the values
give in Table 3.4 as follows.
For a granular backfill:
Hft = 55 .
F
ult
ft ksf k ft = • = 55 55 3025 .. . /
KKinft
Abut
= 50 / /

90
Solve for A and B then:
A
B
=
=
0 011
0 030
.
.

Substitute A and B into Eq. (3.23), then
Fy
y
y
yF
ult
()
..
( =
+ 011 03
in inches, in K per ft of wall) (3.29)
For a cohesive backfill
Hft = 55 .
F
ult
ft ksf k ft = • = 55 55 3025 .. . /
KKinft
Abut
= 25 / /
Solve for A and B then
A
B
=
=
0 022
0 030
.
.

Substitute A and B into Eq. (3.23), then
Fy
y
y
yF
ult
()
..
( =
+ 022 03
in inches, in K per ft of wall) (3.30)
Comparisons of the experimental data, LSH model, bilinear model and HFD
model using Eq. (3.29) and Eq. (3.30) are shown in Figure 3.26 and Figure 3.27. The
results of the LSH model and HFD model are remarkably close.
3.15.1 Application of HFD Model for other Height
The experimental data and the LSH model were used to develop Eq. (3.29)
and Eq. (3.30) to calculate nonlinear force-deformation relationship for a 5.5 feet
high abutment-backwall. In order to develop nonlinear abutment force-deformation

91
relationship for other wall height, Eq. (3.29) and Eq. (3.30) are multiplied by the
height adjustment factors as shown in Eq. (3.31) and Eq. (3.32) for granular and
cohesive backfills, respectively.
For the granular backfill:
Fy
fy
y
yF
s
ult
()
..
( =
+
⋅
011 03
in inches, in K per ft of wall) (3.31)
For the cohesive backfill:

Fy
fy
y
yF
c
()
..
( =
+
⋅
022 03
in inches, in K per ft of wall). (3.32)
The height factors f
s
and f
c
are defined as follows
f
s
H
=
⎛
⎝
⎜
⎞
⎠
⎟
55
15
.
.
. (3.33)
f
c
H
=
⎛
⎝
⎜
⎞
⎠
⎟
55 .
. (3.34)
Substitute Eq.(3.33) and Eq.(3.34) into Eq.(3.31) and Eq.(3.32), then following
force-displacement relationship per foot of abutment-backwall is recommended for
granular backfill:
Fy
y
y
Hy F ()
..
.
( =
+ 14 38
15
in inches, in kips per ft of wall) (3.35a)
Fy
y
y
H
y
y
H
y
y
H ()
.
.
.
.
.
.
.
.
.
=
+
=
+
=
+
14
1
38
14
15
7143
1 2 714
15
789
13
15
(3.35b)

92
UCD Test Data
HFD Model
Bilinear
LSH Model
0.0
4.0
8.0
12.0
16.0
20.0
24.0
28.0
32.0
Passive Capacity (Kips/ft)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
Displacement (inches)

Figure 3.27 : UCD Abutment Experiment Compared with Various Models

93
For all practical purposes Eq.(3.35b) can be simplified to Eq.(3.36):
Fy
y
y
Hy F ()
.
( ≈
+
8
13
15
in inches, in kips per ft of wall) (3.36)
For the cohesive backfill, the force-displacement relationship is given in Eq.(3.37).
Fy
y
y
Hy F ()
..
( =
+ 12 16
in inches, in kips per ft of wall) (3.37a)
Fy
y
y
H
y
y
H
y
y
H ()
.
.
.
.
..
=
+
=
+
=
+
12
1
16
12
833
1133
8
1128
(3.37b)
Eq.(3.37) can be simplified to Eq.(3.38)
Fy
y
y
Hy F ()
.
( ≈
+
8
113
in inches, in kips per ft of wall) (3.38)
3.15.2 Development of the Height Factors.
To develop the height adjustment factors, the following steps were followed:
• Developed nonlinear force-deformation relationship for various abutment
heights using the LSH model, as shown in Figures 3.28 and 3.29.
• Normalized ultimate capacity of various wall heights to the ultimate capacity of
the 5.5-feet wall as shown in Table 3.5:
f
r
F
ult
F
ult
H
=
55 .
(3.39)
where F
Hult
is the ultimate calculate capacity for various abutment height and
F
5.5ult
is the ultimate capacity of the experimental abutment height.
Use Eq.(3.33) and Eq.(3.34) the value of f
c
and f
s
listed in Table 3.5.

94
0.0
6.0
12.0
18.0
24.0
30.0
36.0
42.0
48.0
54.0
60.0
Passive Capacity (Kips/ft)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Displacement (inches)
3 ft
4 ft
5 ft
5.5 ft
6 ft
7 ft
8 ft

Figure 3.28: Backbone Curves for Various Abutment Height (Silty Sand)

95
0.0
6.0
12.0
18.0
24.0
30.0
36.0
42.0
48.0
Passive Capacity (Kips/ft)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0
Displacement (inches)
3 ft
4 ft
5 ft
5.5 ft
6 ft
7 ft
8 ft

Figure 3.29: Backbone Curves for Various Abutment Height (Clay)

96
Table 3.5 Height factors to calculate abutment force-displacement
Height H (feet) 3 4 5 5.5 6 7 8
Granular Backfill
f
r

0.41 0.61 0.86 1.0 1.15 1.47
1.83
f
s

0.41 0.62 0.87 1.0 1.14 1.44
1.75
Cohesive Backfill
f
r

0.53 0.72 0.90 1.0 1.10 1.29
1.49
f
c
0.55 0.73 0.91 1.0 1.09 1.27 1.45

Figures 3.30 and 3.31 show comparisons of the nonlinear abutment force-
deformation relationship predicted using the LSH model and the HFD model with
Eq.(3.35) and Eq.(3.37).
3.15.3 HFD Height Validation
Experimental result of the full-scale static load test performed by Rollins and
Cole (2006) at Brigham Young University (BYU) using a silty sand backfill
discussed in section 3.12.2 was selected to examine the validity of Eq.(3.36). The
Ovesen-Brinch Hansen 3-D correction factor (Ovesen, 1964) of 1.2 was applied to
the pile cap width in the Eq.(3.36) to account for the three-dimensional wedge effect
similar to the LSH model. Substituting the pile cap dimensions:
Height ft = 367 .
;
Width ft = 17
;
312 D Factor = .
and the adjustment factor into Eq.(3.36), we obtain
Fy
y
y
yF () ( . )
.
.( =
+
⎡
⎣
⎢
⎤
⎦
⎥
17
8
13
367
15
12 in inches, in kips ) (3.36)
The force-displacement relationship using Eq.(3.36) is shown in Figure 3.32.

97
0.0
6.0
12.0
18.0
24.0
30.0
36.0
42.0
48.0
54.0
60.0
Passive Capacity (Kips/ft)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Displacement (inches)
5 ft
6 ft
7 ft
8 ft
HFD
LSH

Figure 3.30: Comparisons of LSH and HFD Models for Cohesive Backfill

98
0.0
6.0
12.0
18.0
24.0
30.0
36.0
42.0
48.0
Passive Capacity (Kips/ft)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
Displacement (inches)
5 ft
6 ft
7 ft
8 ft
HFD
LSH

Figure 3.31: Comparisons of LSH and HFD Models for Silty Sand

99
BYU Test Data
LSH Model
HFD Model
0.0
50.0
100.0
150.0
200.0
250.0
300.0
350.0
Passive Capacity (Kips/ft)
0.00.5 1.01.5 2.02.5
Displacement (inches)

Figure 3.32: HFD Validation Using BYU Silty Sand Backfill

100
3.16 Conclusions
The LSH model was developed to provide a simplified means to estimate
nonlinear abutment backfill force-displacement capacity. The model employs a full-
length logarithmic-spiral failure wedge mechanism coupled with a modified
hyperbolic stress-strain relationship. The model has seven model parameters, five of
which characterize the soil properties (total unit weight γ, shear strength parameters
φ and c, Poisson’s ratio ν, and a parameter ε
50
relating to the low-strain Young’s
modulus), which may be determined from field or laboratory testing or estimated.
The ε
50
parameter can be determined from testing of soil samples in a geotechnical
laboratory to failure and recording of the stress-strain curve. When there is a lack of
laboratory test data, the presumptive values of ε
50
(0.0035 for granular soils and
0.007 for cohesive soils) are recommended. The sixth parameter δ is the structure-
soil interface friction angle and experimental values are given in the literature. The
last model parameter R
f
describes the theoretical ultimate capacity of the soil in the
hyperbolic stress-strain relationship and typical values are between 0.94 and 0.98 for
all soil types considered. When no capacity data is available for calibrating the LSH
model, it is found that a value of 0.97 is reasonable.
The LSH model is applicable to all soil types that can be reasonably
characterized by these seven parameters. The validity of the LSH model is
established by comparison with experimental nonlinear force-deformation results
from full-scale tests, centrifuge model tests and small-scale laboratory tests of
abutments and pile caps in a variety of structure backfills.

101
A modified hyperbolic force-displacement (HFD) equation was developed,
which requires only three parameters: average soil stiffness K, and the ultimate
passive capacity F
ult
, and the maximum displacement y
max
at which F
ult
is mobilized.
The first parameter can be readily determined using presumptive soil stiffnesses,
such as used in Caltrans seismic design of bridge abutments (SDC, 2004). The latter
two parameters can be estimated from available experimental test data (such as
Caltrans SDC, 2006) or by the geotechnical engineer using a selected earth pressure
theory depending on the application. Using the HFD parameters derived directly
from the nine experimental test data, the HFD model is found to match all test data
well (nearly the same as the LSH predicted curves). Nonlinear closed-form solutions
are to develop the nonlinear force-displacement curve for compacted abutment
backfills when no geotechnical data is available. The LSH and HFD models are
practical and versatile tools that can be used by structural and geotechnical engineers
in seismic analysis for bridge design.

102
CHAPTER FOUR
ABUTMENT CONTINUUM FINITE ELEMENT MODEL

4.1 Introduction
The finite element method (FEM) is the most powerful numerical technique
to solve soil-abutment-structure interaction problems for seismic analysis of bridge
structures. The FEM can provide insight into the behavior and cause of abutment-
backfill failure during a seismic event. Before using FEM, at least a preliminary
solution should be obtained using previously available solutions or experimental
data. In common practice, bridge engineers use presumptive value to calculate a
bilinear force-deformation relationship to model the nonlinear behavior of the
abutment backfill.
Depending on the complexity of the bridge abutment system, there is more
than one method of analysis to develop nonlinear abutment-backfill springs. The
purpose of this chapter is to carry out a displacement finite element analysis to
capture the nonlinear force-deformation capacity of the abutment backfill and to
investigate the mechanism of the backfill failure in particular the bridge abutments
with high skew angles. PLAXIS, a finite element code that is capable of performing
both two-dimensional and three-dimensional analysis is used to calculate backfill
backbone curves for both skewed and nonskewed abutments. The results of the finite
element analysis is compared with results from full-scale experiments, presumptive
values, a closed-form solution and a simplified solution. It is assumed that the

103
abutment backwall is loaded very gradually such that it remains in a state of static
equilibrium, for which the static action of the backwall and reaction of the backfill
equilibrate each other and time has no influence on the results.
4.2 Method of Analysis
For exact theoretical solution of bridge abutment backfill behavior, the
requirements for equilibrium, compatibility, and constitutive stress-strain
relationship with the proper boundary conditions must all be satisfied. Nonlinear
force-deformation relationship for seismic design and analysis of bridge abutment-
backfill interaction can be calculated using the following methodologies: (1)
Presumptive values, (2) closed-form solution, (3) simplified solution and (4)
numerical solution.
4.2.1 Presumptive Values
In bridge engineering practice, it is common to use a presumptive value to
calculate abutment bilinear force-deformation relationship for seismic design and
analysis of bridge structures. Bilinear force-deformation relationship in longitudinal
direction of the bridge superstructure is developed using Eq.(4.1) (SDC, 2004).
KK f
FA
e
f
Abut Exp
Ult Exp
=
=
*
**
σ
(4.1)
The allowable presumptive value is calibrated based on full-scale abutment field
experiments. Bilinear abutment force-deformation relationship assumes linear
elasticity for all stress states below the yield point. Once idealized backfill passive

104
capacity is reached, the ultimate abutment force-capacity remains constant with
increasing displacement as shown in Figure 4.1
F
F
Ult
y
K
Abut
∆h
abut
F
F
Ult
y
K
Abut
∆h
abut

Figure 4.1: Abutment Force-Deformation Based on Presumptive Value

The parameters in Eq.(4.1) and Figure 4.1 are defined as follows:
K
abut
is the average abutment stiffness in the longitudinal direction,
F
ult
is the ultimate allowable passive resistance provided by the abutment
backfill,
σ
exp
is the ultimate allowable stress resistance provided by the abutment
backfill,
f is the width and height adjustment factor,
h
abut
is the height of abutment backwall,

105
∆ is a factor as a function of backfill properties to define the yield point, and
h
exp
is the height of field experiment abutment backwall height,
The current Federal Highway Administration’s Seismic Retrofit Manual for
Highway Structures (MCEER, 2006) suggests for integral or monolithic abutments,
an initial secant stiffness (K
abut
) shown in Eq.(4.2) may be used to calculate bilinear
abutment backbone curve as shown in Figure 4.1.
K
F
ult
h
Abut
=
002 .
(4.2)
where
∆ = 0.02h which defines the yield point shown in Figure 4.1.
For the seat-type abutments, the expansion gap should be included in the calculation
of the secant stiffness as follows:
K
F
ult
hD
g
Abut
=
+ (. ) 002
(4.3)
where
D
g
is the gap width.
Current Caltrans Seismic Design Criteria (SDC, 2006) suggests presumptive
value for the abutment stiffness (K
abut
) of 20 k/in per foot of wall and a presumptive
value of 5 kips per square foot of abutment backwall should be used to calculate the
yield plateau of the bilinear abutment force-deformation relationship. Per current
Caltrans criteria, the adjustment factor f used in Eq.(4.1) should be set equal to the

106
ratio h
abut
/5.5 (5.5 is wall height in feet based on the UCD abutment test). The
abutment stiffness should be modified to account for the expansion gap.
4.2.2 Closed-Form Solution
Simple nonlinear closed-form solution is the ultimate goal for bridge
engineers to develop discrete abutment springs for various backfill. In order to get a
solution for more realistic abutment-backfill behavior, good engineering judgment
and approximations must be introduced. As shown in the previous chapter, for the
engineered abutment backfill it is possible to establish a realistic closed form
solution as shown in Figure 4.2 to develop an abutment-backfill nonlinear force-
deformation relationship.
As discussed in Chapter 3, based on experimental data and parametric
studies using the LSH model, the hyperbolic force-displacement (HFD) relationships
per foot of abutment-backwall for cohesionless and cohesive backfills, respectively,
are expressed below:
Fy
y
y
Hy F ()
.
( =
+
8
13
15
in inches, in kips per ft of wall) (4.4a)
Fy
y
y
Hy F ()
.
( =
+
8
113
in inches, in kips per ft of wall) (4.4b)

107

y
max
0
F
y
A By
ult
=
+
K
abut
1
2
F
ult
F
ult
1
F
y
50
y
max
0
F
y
A By
ult
=
+
K
abut
1
2
F
ult
F
ult
1
F
y
50

Figure 4.2: Closed-Form Solution
4.2.3 Simplified Solution
As shown in the previous chapter, the LSH model can capture the abutment
behavior fairly accurately. The method can be used by both geotechnical and bridge
engineers to calculate the nonlinear abutment backbone curves based on strength and
stress-strain behavior of the backfill. The requirements of equilibrium, stress-strain
and deformation compatibility are all satisfied.
4.2.4 Numerical Solution
The most common numerical method used in the geotechnical engineering to
solve complicated soil-structure interaction problems is the finite element method.
Unlike the classical analysis methods (e.g., limit equilibrium, limit analysis, etc), it is

108
capable of satisfying all four basis requirements for a complete theoretical solution.
The requirements of limiting equilibrium, compatibility of displacement, material
constitutive behavior, boundary conditions and stage constructions are all satisfied.
The geometry of the abutment-backfill system, loading conditions, nonlinear
material properties and boundary are absolute necessary to be accurately modeled.
The accuracy of displacement finite element method is dependent on the realistic
stress-strain characteristics of the abutment backfill. The intent of this section is to:
• Develop a finite element model to validate the constitutive soil model based on
realistic triaxial test data for existing bridge abutment-backfill;
• Develop 2-D finite element models to simulate full-scaled abutment and pile cap
experiments conducted at UCLA, UCD and BYU;
• Develop 3-D finite element models to simulate full-scaled experiments and
investigate the mechanism of the skewed abutment failure; and
• Compare the results of the experimental data versus closed-form solution,
simplified solution and the 2-D and 3-D finite element solutions.
4.3 Constitutive Models for Bridge Structures
Bridge structures considered herein are constructed from reinforced concrete.
Steel reinforcement is to provide tensile capacity and concrete is to provide
compressive capacity. A brief description of the constitutive models for the
reinforced concrete is given below. Extensive experimental and analytical research
has resulted in significant advances and the development of reinforced accurate

109
concrete constitutive relationships. However, in the bridge community, there is not a
well define stress-strain relationship for the bridge abutment backfill.
Figure 4.3 shows the constitutive stress-strain model including unloading
and reloading branches for a confined concrete section which is used in the analysis
to determine the capacity of the ductile concrete members. The envelope for the
model is the monotonic stress-strain curve based on Mander et al. (1988) confined
concrete model. Where (f’
cc
) is expected concrete compressive strength, (ε
cc
) is the
concrete confined compressive strain and (ε
cu
) is the concrete ultimate compressive
strain.
The tensile stress-strain relationship for typical reinforcing steel used in
bridge structures is shown in Figure 4.4. The steel stress-strain relationship exhibits
an initial linear elastic portion up to point A, a yield plateau AB, and a strain
hardening range in which the stress increases with strain. The yield point is defined
by the expected yield stress of the steel f
ys
. The length of the yield plateau is a
function of the steel strength and bar size. The strain-hardening portion is nonlinear
and is terminated at the ultimate tensile strain ε
su
.

110
ε
cc
ε
cu
cc
Stress
Strain
E
c
f’
ε
cc
ε
cu
cc
Stress
Strain
E
c
f’

Figure 4.3: Hysteretic Behavior of Confined Concrete

f
y
ε
y
ε
sh
ε
s
ε
su
f
s
f
u
Stress
Strain
E
s
AB
C
D
f
y
ε
y
ε
sh
ε
s
ε
su
f
s
f
u
Stress
Strain
E
s
AB
C
D

Figure 4.4: Behavior of Steel

111
4.4 Constitutive Model for Abutment Backfill
A detailed description of the abutment backfill stress-strain relationship was
given in the previous chapter. Based on the data of triaxial tests conducted on state-
wide abutment-backfill using both in-situ and remolded samples, the decreasing
stiffness and simultaneously irreversible plastic strain were characteristics of all the
abutment backfill. Advanced constitutive modeling such as bounding surface
plasticity model (Bardet, 1986) has been developed to simulate the nonlinear soil
behavior. When these constitutive models are implemented in finite element
computer programs, they can be used to solve difficult geotechnical engineering
problems (Bardet, 1997). For the seismic analysis of bridge abutment, a constitutive
model should be selected such that it would be possible to obtain the model
parameter values in a simple manner from conventional geotechnical field
exploration and laboratory test data.
The purpose of this section is to use a finite element model which not only
can simulate the stress-strain behavior of the abutment backfill up to and beyond the
failure, but also can capture the nonlinear force-deformation of the abutment-
backfill. The application of finite element method for abutment-backfill interaction
requires not only knowledge of the fundamentals of the method, but also
understanding of the material properties used for abutment backfill. A hardening soil
model, Hardening Soil (HS), which is an advanced double stiffness model available in
the finite element code, PLAXIS, is selected for the simulation of the nonlinear
abutment backfill behavior. The model captures the abutment backfill behavior in a

112
tractable manner on the basis of only two stiffness parameters and is very much
appreciated by the practicing geotechnical engineers due to its simplicity. The stiffness
moduli are stress-dependent, which are reduced with the strain according to a hyperbolic
relationship. This model represents an updated version of the well-known Duncan-
Chang hyperbolic model (Duncan et al., 1980), however, it supersedes the Duncan-
Chang model by: (1) using theory of plasticity rather than elasticity, (2) including soil
dialatancy, and (3) introducing a yield cap.
The full explanation and derivation of this model is described by Schantz (1999)
and Brinkgreve (2006). A brief description of the model formulation and parameters
are given below. More detailed explanation of nonlinear soil behavior and application
of plasticity theory to soil behavior are described by Bardet (1983).
The basic idea for the formulation of the HS model is the Hyperbolic
relationship between the deviatoric stress (q
i
) and the vertical strain (ε
i
) in primary
triaxial loading as shown in the following equation.
q
E
R
q
i
i
fi
f
=
+
ε
ε
1
2
50
(4.5)
where
() qc
f
=−
−
cot
sin
sin
φσ
φ
φ
3
2
1
(4.6)

The value of q
f
is the ultimate value of the deviatoric stress which can be
express as
() qR
ff
=− σ σ
13
(4.7)

113
There are two types of hardening, namely shear hardening and compression
hardening in the HS model. The shear hardening controls the irreversible shear strains
and the shear yield surface of the HS model. The compression hardening controls the
irreversible plastic straining of the HS model due to compressional loading. The
limiting states of stress are described by means of conventional Mohr-Coulomb
parameters (soil friction angle, φ; the cohesion, c; and the dilatancy angle, ψ).
The nonlinear stress-strain behavior in loading is represented by the hyperbolic
function as shown in Figure 4.5. The HS model enables a realistic description of the
stiffness. The model is consisted of three stiffnesses: (1) primary loading stiffness, (2)
unloading/reloading stiffness and (3) oedometer stiffness.
4.4.1 Primary Loading Stiffness
The primary loading stiffness (E
50
) is an average secant modulus and is the
confining stress dependent. The value of the parameter E
50
is calculated using Eq.(4.8).
EE
a
ap
ref
ref
m
50 50
3
=
+
+
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
σ
(4.8)
where
φ tan / c a =

The shear hardening is used to model irreversible strains due to primary
deviatoric loading. The triaxial modulus controls the shear yield surface of the HS
model. The parameters used in the HS model are summarized in Table 4.1.

114
ε
1
0.0
E
50
ε
50
ε
f
1
2
q
f
q
f
q
E
ur
05 .
10 .
SL
ε
1
0.0
E
50
ε
50
ε
f
1
2
q
f
q
f
q
E
ur
05 .
10 .
SL
() σ
3
i
() σ
3
i
() σ
1
i
() σ
1
i
ε
1
0.0
E
50
ε
50
ε
f
1
2
q
f
q
f
q
E
ur
05 .
10 .
SL
ε
1
0.0
E
50
ε
50
ε
f
1
2
q
f
q
f
q
E
ur
05 .
10 .
SL
() σ
3
i
() σ
3
i
() σ
1
i
() σ
1
i
() σ
3
i
() σ
3
i
() σ
1
i
() σ
1
i

(a) Triaxial Test

0.0
E
oed
ref
ε
1
p
ref
σ
1
1
σ
1
0.0
E
oed
ref
ε
1
p
ref
σ
1
1
0.0
E
oed
ref
ε
1
p
ref
σ
1
1
σ
1
σ
1

(b) Oedometer Test

Figure 4.5: Hardening Soil (HS) Model

115
Table 4.1 Parameters of the HS model
Parameters Expression Description
E
ref
50

Primary loading reference modulus in drained triaxial test
E
oed
ref

Reference modulus for primary loading in oedometer test
E
eur
ref

Unloading/reloading reference modulus in drained triaxial test
M Modulus exponent for stress dependency
Deformation

ν
ur
Poisson’s ratio for loading/unloading
C Effective cohesion at failure
φ Effective friction angle at failure Strength
ψ Dilatancy angle at failure

4.4.2 Unloading/Reloading Stiffness
For unloading and reloading stress path, a much stiffer linear response in
unloading is described by the parameter, E
ur
. The value of the parameter E
ur
is
calculated using Eq.(4.9).

EE
a
ap
ref
ur ur
ref
m
=
+
+
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
σ
3
(4.9)
4.4.3 Oedometer Stiffness
The oedometer modulus (E
oed
) controls the cap yield surface. It is a compression
hardening which is used to model irreversible plastic straining due to primary
compression in oedometer loading and isotropic loading as shown in Figure 4.5. The
value of the parameter E
oed
is calculated using Eq.(4.10).

116
EE
a
ap
ref
oed oed
ref
m
=
+
+
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
σ
1
(4.10)

The parameter m used in Eq. (4.8), Eq. (4.9) and Eq. (4.10) controls variation of
stiffness with confining pressure. The value of the parameter m for sand varies from
0.40 to 0.70; for highly overconsolidated clays it varies from 0.50 to 0.90; and for
normally consolidated clays it is about 1.0 (PLAXIS, 2006 Manual).
The plastic behavior is defined through the yield/failure surfaces. In contrast to
the elastic perfectly-plastic model, the yield surface of the HS is not fixed in principal
stress space, but will expand due to plastic straining as shown in Figure 4.6 .

CAP
Shear yield surface
σ
1
σ
2
σ
3
CAP
Shear yield surface
σ
1
σ
2
σ
3

Figure 4.6: HS Model in Stress Space

117
4.4.4 Dilatancy
Dilatancy Ψ is defined as the volume change associated with the application
of shear stresses. An increase in volume, or expansion, is known as positive dilation,
while a decrease in volume, or contraction, is known as negative dilation. The stress-
dilatancy theory is able to explain qualitatively and quantitatively how sandy soils
dilate when subjected to shearing stresses (Bardet, 1983). The amount of dilatancy
that an abutment-backfill can experience is dependent on particle interlocking, which
relates to the fabric of the material. Dilatancy can be estimated from the volumetric
strain versus axial strain curve (Figure 4.7) of a material subjected to shearing with
the following expression as stated by Bolton (1986) for plane-strain conditions and
later derived by Schanz and Vermeer (1996) for triaxial test conditions.
sin
sin sin
sin sin
ψ
φ φ
φφ
m
mcv
mcv
for e e
cv
=
−
−
<
1

(4.11)
where:
sin
sin sin
sin sin
φ
φ ψ
φψ
cv
m
m
=
−
− 1

(4.12)
sinψ
m
for e e
cv
=≥ 0

(4.13)
φ
m
is the mobilized friction angle, and
φ
cv
is the critical friction angle at the maximum void ratio.
The void ratio is related to the volumetric strain, ε
v
by the relationship given
in Eq.(4.14):

118
εε
vo v
e
e
o
−=
+
−
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
ln
1
1
(4.14)
where an increment of ε
v
is negative for dilatancy.
ε
v
dilation
ε
y
1-sin ψ
2sin ψ
contraction
e
max
ε
v
dilation
ε
y
1-sin ψ
2sin ψ
contraction
e
max
Figure 4.7: PLAXIS Dilatancy Model
The initial void ratio, e
o
, is the in-situ void ratio of the abutment backfill. The
maximum void ratio, ε
cv
is the void ratio of the material in a state of critical void
(critical state). Critical state is an asymptotic state eventually reached during loading
characterized by volume change, no stress change and infinite deviatoric stain: “It is
as material has melted under stress.” (Bardet, 1983).
4.5 Verification of the Constitutive Model
A seismic soil-structure interaction analysis of the bridge abutment generally
requires knowledge of both the shear strength and nonlinear stress-strain behavior of
the abutment backfill. In particular for the deformation analysis of the abutment,
description of the stress-strain behavior is necessary, as the overall performance of

119
the abutment is defined mostly within the range of nonlinear stress-strain behavior of
the abutment backfill. To describe the abutment nonlinear behavior using a finite
element model, the modified hyperbolic stress-strain relationship in the PLAXIS
code is proposed herein. In this section, the verification of the constitutive soil model
selected for the abutment backfill is utilized using triaxial test data conducted by
Earth Mechanics In. (EMI, 2005) to characterize representative bridge abutment
backfill properties as part of a Caltrans seismic research program. Finite element
analysis of the triaxial tests was performed and compared with the measured stress-
strain response for the Painter Street Overcrossing abutment backfill. In the finite
element analysis, the PLAXIS HS model was used to simulate the triaxial tests.
Figure 4.8 shows the test specimen and axisymmetric finite element analysis of the
triaxial test.
The left hand side and the bottom of the model are axes of symmetry. At
these boundaries, the displacements normal to the boundary are fixed and tangential
displacements are free for move. Two representatives of the in-situ samples from the
bridge abutment backfill are selected to validate the constitutive model used in the
finite element analysis: one predominantly cohesive soil and one predominantly
cohesionless soil. The triaxial test specimens were run at three confining pressures
and three deviatoric stresses to obtain cyclic stress-strain relationships, friction angle
and cohesion values of the abutment backfill at the failure.

120

(a)
h
∆h
Axial Symmetry
σ
1
σ
3
h
∆h
Axial Symmetry
σ
1
σ
3

(b)
Figure 4.8: Triaxial: (a) Specimen; (b) Finite Element Model
The cyclic stress-strain relationships measured and predicted from the finite
element analysis are plotted in Figure 4.10 and Figure 4.11 showing three unloading
and reloading cycles. The plots include the three associated Mohr circles, the shear
strength parameters at the failure the nonlinear stress-strain behavior of the backfill.
Figure 4.9 and Figure 4.10 demonstrate that the measured and calculated stress-strain
curves show a good match for both triaxial tests. Therefore, the HS model available
in PLAXIS is capable of capturing the nonlinear stress-strain behavior of both
cohesive and cohesionless abutment backfills.

121
0.0 2.0 4.0 6.0 8.0
Normal Stress (ksf)
0
2
4
6
Shear Stress(ksf)
φ = 43.3
o
c = 0.13 psf

(a) Mohr Circles

0.0 2.0 4.0 6.0 8.0
Axial Strain ε (%)
0
2
4
6
8
Devetoric Stress q(ksf)
FEM
Experiment

(b) Stress-strain Behavior

Figure 4.9: Triaxial Test of the Cohesionless Backfill

122
0.0 4.0 8.0 12.0 16.0
Normal Stress (ksf)
0
2
4
6
8
10
Shear Stress(ksf)
φ = 11.5
o
c = 3.90 psf

(a) Mohr Circles
0.0 10.0 20.0 30.0
Axial Strain ε (%)
0
4
8
12
Devetoric Stress q(ksf)
FEM
Experiment

(b) Stress-strain Behavior

Figure 4.10: Triaxial Test of the Abutment Backfill for Cohesive Soil

123
4.6 2-D Finite Element Model
A serious of two-dimensional (2-D) displacement-controlled finite element
(FE) simulations using PLAXIS have been carried out to calculate the nonlinear
backbone curves for full-scaled abutments and pile cap experiments. Detail
description of these experiments were presented in the previous chapter. The finite
element mesh used in the simulations is shown in Figure 4.11. For each simulation, a
geometry model was first created. The geometry model is a representation of the real
problem and consists of points, lines and clusters (elements), which includes a
representative abutment backfill, construction stages and loading. The model was
sufficiently large so that the boundaries did not influence the results. The points were
used to position the point of fixities and local refinement of the finite element mesh.
The lines were used to define the physical boundaries of the geometry and the
backfill for staged construction. Clusters are the areas that are fully enclosed by
lines. The finite element model was generated based on the composition of the
clusters and lines in the geometry model.
During the generation of the mesh, the clusters were divided into 15-noded
triangular elements. The distribution of nodes over the elements are shown in Figure
4.11. During the finite element calculation, displacements are calculated at the
nodes. In contrast to displacements, stresses are calculated at individual Gaussian
integration points (stress points) rather than at the nodes.

124
-Nodes
-Interface
Interface Element
Triangular Element
H
6H
-Nodes
-Interface
-Nodes
-Interface
Interface Element
Triangular Element
H
6H
Figure 4.11: 2-D Finite Element Mesh with Backwall Interface

The right vertical boundary was placed over a distance of 6H, the left
boundary was placed 1H away from the abutment backwall and the bottom boundary
was placed 1H below the bottom of the backwall, which were similar to those used
by Martin et al. (1997). Numerical simulation of the experiments involved two steps
of analysis: the first step was to establish the geostatic state by applying gravity load
and the second step was to push the wall by prescribing either cyclic displacements
or monotonic displacement until the backfill failed and the passive wedge was
formed.

125
4.6.1 Interface Elements
The abutment backwall-backfill interaction is modeled with the interface
elements. Figure 4.11 shows how the abutment backwall interface elements are
connected to backfill elements. When using 15-node soil elements, the
corresponding interface elements are defined by five pairs of nodes. In Figure 4.11
the interface elements are shown to have a finite thickness, but in the finite element
formulation the coordinates of each node pair are identical, which means that the
element has a zero thickness.
A bilinear model is used to describe the behavior of interfaces for modeling
backfill-abutment interaction. The Coulomb criterion is used to distinguish between
elastic behavior (where small displacements can occur within the interface) and
plastic interface behavior (slip).
For the interface to remain elastic, the shear stress τ is given by:
τ σ φ <+
n
c tan

(4.15)
For plastic behavior, shear stress τ is given by:
τ σ φ =+
n
c tan

(4.16)
where φ

and c

are the friction angle and cohesion of the interface and σ
n
and τ are the
normal stress and shear stress acting in the interface. The strength properties of
interfaces are linked to the strength properties of the abutment-backfill using a

126
strength reduction factor (R
inter
). The interface properties are calculated using the
following relationships:
cR c
er soil
=
int

(4.17)
τ φ φ = ≤ R
er soil soil int
tan tan

(4.18)
Figure 4.12 shows stress-strength and stress-strain relationship of the
classical Mohr-Coulomb model applied to the interface elements. Once the specified
shear stress is reached, the shear stress is assumed to remain constant with increasing
slip.

ε
E
τ
σ
τ
φ
()
τ φ = + Rc
er
soil
int
tan
c
ε
E
τ
σ
τ
φ
()
τ φ = + Rc
er
soil
int
tan
c c

Figure 4.12: Mohr-Coulomb Interface Stress-Strain Relationship
4.6.2 2-D Finite Element Simulations for Various Backfill
The previous chapter has dealt with LSH model for full-scale abutment
backwall, full-scale pile cap, small-scale laboratory for retaining wall and centrifuge
experiments for abutment backwall and pile cap. This section presents simulations

127
using finite element models only for the full-scale experiments, which include
UCLA and UCD abutment test and BYU pile cap experiments.
4.6.3 UCLA-CT Full-Scaled Abutment Experiment
A detailed description of the abutment experiment was given in Chapter 3. A
brief description of the test related to the analytical model is given herein. As shown
in Figure 4.13a, the rigid abutment-backwall was constructed to be 15 feet wide but
the backfill was constructed to be 16 feet wide. Figure 4.13b shows a typical seat
type bridge abutment, indicating how the bridge may move in the longitudinal
direction and collide with the abutment backwall during a seismic event. Per
Caltrans Seismic Design Criteria (SDC, 2004), the backwall is designed to break
away when the bridge deck jolts against it during a seismic event, pushing the
backwall into the backfill and forming a passive wedge between the wingwalls.
Therefore, the width of the abutment-backwall was constructed to be less than the
width of the backfill to simulate the real seat-type abutment. Figure 4.14a shows the
schematics of the abutment field experiment with the three-dimensional mobilized
passive wedges bounded by a logarithmic-spiral type failure surface within the
abutment backfill in-between the wingwalls.
Figure 4.14b shows observed crack patterns in the brittle gypsum columns
within the abutment backfill. The patterns of the cracks developed in the gypsum
columns illustrates the development of successive failure surfaces that mobilize as a
function of lateral displacement and backfill properties.

128

(a) Field Experiment

(b) Plunging Action
Figure 4.13: Seat-type Abutment and Foundation System

129
5.5 ft
15 ft
17 ft
16 ft
A
A
5.5 ft
15 ft
17 ft
16 ft
A
A

(a) Schematic of Mobilized Failure Surfaces

(b) Section A-A Mapped Mobilized Final Failure Surface
Figure 4.14: UCLA-CT Full-Scaled Abutment Test with Sandy Backfill

130
The deformed wedges started to develop within the upper soil layer and
progress deeper down and away from the abutment backwall. This failure
mechanism was observed using the LSH model and field observation which was
described in Chapter 3.
The failure surface started from the bottom of the abutment backwall and
extended upward with a log spiral shape intercepting the backfill surface at about 3
times the height of the backwall. The height of the rupture zone of the gypsum
columns along the logarithmic failure surface indicates that the final failure was not
a distinct line but it manifested a shear band, a zone of intense shearing to form the
log spiral failure surface. The numerical simulations and formations of the shear
band in geomaterials have been well demonstrated by Bardet (1992) and are beyond
the scope of this dissertation.
4.6.4 Selection of Parameters Used in the Finite Element Model
The cyclic triaxial laboratory test was conducted on the remolded samples to
obtain the stiffness and strength properties of the abutment backfill as shown in
Figure 4.15. The plot of the stress-strain behavior and the associated Mohr circles are
shown in Figure 4.16. The backfill strength values and the stiffness parameters
obtained from the laboratory test are listed in Table 4.2.
Table 4.2 Input Parameters Used in the UCLA-Caltrans PLAXIS Model
Backfill Type
γ
[lb/ft
3
]
ϕ
Friction
c
[psf]
ψ
Dilatancy
R
inter
Wall
interface
R
f

E
50
ref
[psf]
E
ur
ref
[psf]
Sand/UCLA 130 40
0
300 0
0
0.70 0.97 1.4E5 2.8E5

131

Figure 4.15: UCLA-CT Abutment Backfill Triaxial Test
4.6.5 UCLA 2-D Abutment Finite Element Model
The finite element model used in the simulation of the UCLA-CT abutment
experiment is shown in Figure 4.17. The height of the abutment backwall is 5.5 feet.
The right vertical boundary of the model was placed a distance of about 28 feet from
the face of the backwall and the left vertical is placed about 5 feet from the wall face.
The bottom boundary is set about 5.5 feet below the bottom of the backwall. Control
horizontal cyclic displacements were applied at the face of the backwall, while
displacement of the bottom and vertical boundaries were constrained in both
horizontal and vertical directions.

132
0.0 4.0 8.0 12.0 16.0 20.0
Normal Stress (ksf)
0
4
8
12
16
Shear Stress(ksf)
φ = 40
o
c = 300 psf
φ

(a) Stress-strain Behavior
01 23 4
Axial Strain ε (%)
0
4
8
12
16
Deviatoric Stress q(ksf)
Experiment
E
50
E
ur
E
50
E
ur
E
50
= 1400
E
ur
= 2800

(b) Mohr Circles
Figure 4.16: UCLA-CT Triaxial Test of the Abutment Backfill

133
5.5’
33’
5.5’
33’

Figure 4.17: Limits of 2-D Finite Element Model
4.6.6 Sequence of the Events in the Model
The analysis was performed in steps to simulate sequence of the real events
during the field experiment. The backfill was placed and compacted behind the wall
and was extended more than 3 times the abutment wall height behind the backwall
laterally. No backfill was placed at the exterior sides of abutment wingwalls. The
computations were performed using the following steps:
(1) Starting with a level ground the initial stresses were calculated.
(2) Excavation was performed by deactivating 5.5 feet of clusters of elements in
the front of the abutment face.
(3) Both cyclic displacement history and monotonic displacement were applied to
the abutment vertical face.
The deformed mesh, displacement vectors and contours of the FE model at the final
stage of the backfill failure are shown in Figure 4.18 through Figure 4.20,
respectively.

134
Before
After
Before
After

Figure 4.18: 2-D Finite Element Deformed Shape for UCLA Abutment Test

Figure 4.19: 2-D Finite Element Displacement Vectors for UCLA Abutment Test

Figure 4.20: 2-D Finite Element Displacement Contours for UCLA Abutment Test

135
The formation of the plastic points is shown in Figure 4.21. The plastic points
are stress points in plastic state, displayed on undeformed model geometry. The
plastic points shown by open squares indicate the stresses that lie on the surface of
the Coulomb failure envelope within the shear zone. The incremental strain of the
backfill shown in Figure 4.22 is also a very good indication of the most critical
failure surface and formation of the shear zone. The failure surface predicted by the
analytical model remarkably resembles the experimental failure surface which was
mapped in the field.

Figure 4.21: 2-D Finite Element Failure Surfaces for UCLA Abutment Test

Figure 4.22: 2-D Finite Element Incremental Strains for UCLA Abutment Test

136
The load-displacement curves predicted by the finite element model applying
cyclic displacements and monotonic displacement versus experimental data are
shown in Figure 4.23. There is good agreement between the experimental data and
the model prediction. The backbone curve predicted by applying monotonic
displacement is slightly higher than the backbone curve predicted by the cyclic
displacement at higher displacement. For all practical purposes, the difference
between the two models is insignificant.
4.7 UCD Abutment Field Experiment
The simulation using the displacement controlled finite element model for the
UCD-Caltrans abutment experiment conducted by Maroney et al. (1994) is presented
herein. Figure 4.24 illustrates the abutment test setup. This is an example of the
monolithic abutment which was described in detail in Chapter 3.
Before the test, 3-inch diameter vertical holes were drilled along the
longitudinal centerline of the abutment into the abutment backfill and filled with
liquid styrofoam columns as shown in Figure 4.25 and Figure 4.26. After the
completion of the test, a longitudinal trench was excavated and the failure
mechanism of the backfill was carefully investigated by mapping the deformation of
the styrofoam columns. The longitudinal failure features exposed by trenching
behind the abutment wall is shown in Figure 4.26. The center of the deformed shape
of the styrofoam columns were mapped in the field (Romstad et al., 1995) as shown
in Figure 4.26b.

137
UCLA Test Data
2D Cyclic
2D Monotonic
0
50
100
150
200
250
300
350
400
450
500
550
Passive Capacity (kips)
0.00.5 1.01.5 2.02.5 3.03.5 4.04.5 5.0
Displacement (inches)

Figure 4.23 : 2-D Cyclic UCLA Abutment Test versus Finite Element Model Predictions

138

Figure 4.24: UCD Abutment Test Setup

Figure 4.25: UCD 3-D Abutment Field Experiment

139

Exposed
Plane
Exposed
Plane
AA
9.67 feet
10.0 feet
Exposed
Plane
Exposed
Plane
Exposed
Plane
Exposed
Plane
AA
9.67 feet
10.0 feet

(a) Plan
ABUTMENT BACKWALL
Center of Deformed Regions
of Styrofoam Columns
Styrofoam Columns
5.5’
Wingwall
Idealized Failure Surface
ABUTMENT BACKWALL
Center of Deformed Regions
of Styrofoam Columns
Styrofoam Columns
5.5’
Wingwall
Idealized Failure Surface

(b) A-A Elevation
Figure 4.26: Failure Mechanism of the UCD Abutment Test

140
The approximate idealized failure surface is shown through the center of
deformed regions of the styrofoam columns. Maroney (1995) reported that the
failure surface extended from the bottom of the abutment backwall at initially a zero
slope and upward to the embankment surface with increasing slope and intercepting
embankment surface at near by twice the height of the backwall.
4.7.1 UCD Abutment 2-D Finite Element Model
The geometry and the boundary conditions of the finite element model used
in the simulation is the same as shown in Figure 4.17. The backfill strength values
and the stiffness parameters are listed in Table 4.3.
Table 4.3 Input Parameters for UCD PLAXIS Model
Backfill Type
γ
(lb/ft
3
)
ϕ
Friction
c
(psf)
ψ
Dilatancy
R
inter
Wall
interface
R
f

E
50
ref
(psf)
E
ur
ref
(psf)
Clay/UCD 120 0
0
2000 0
0
0.70 0.95 3.0E5 9.0E5

The deformed mesh of the finite element model at the final stage of the
backfill failure is shown in Figure 4.27. The analytical model indicates that the
failure surface extended from the bottom of the abutment backwall at initially a zero
slope and intersected the abutment backfill at surface near twice the height of the
backwall. The displacement vectors of the deformed model and are shown in Figure
4.28. The displacement contours are shown in Figure 4.29. The Mohr-Coulomb
failure surface of the model is shown in Figure 4.30 and the incremental strain is
shown in Figure 4.31.

141
Before
After
Before
After

Figure 4.27: 2-D Finite Element Deformed Shape for UCD Abutment Test

Figure 4.28: 2-D Finite Element Displacement Vectors for UCD Abutment Test

Figure 4.29: 2-D Finite Element Displacement Contours for UCD Abutment Test

142

Figure 4.30: 2-D Finite Element Failure Surfaces for UCD Abutment Test

Figure 4.31: 2-D Finite Element Incremental Strains for UCD Abutment Test
The failure mechanism of the backfill is similar to the UCLA experiment
which was discussed in the previous section. However, the extent of the failure
surface within the backfill is quite different. The UCLA abutment experiment
intersected the backfill surface at about three times the height of the backwall due to
presence of the cohesionless backfill. As shown in Figure 4.21 and Figure 4.30, it
appears that there are more mobilized failure surfaces present in the cohesionless
abutment backfill than in the cohesive backfill. The plastic points are mostly
concentrated in a single shear zones for the cohesive backfill. The load-displacement
curve of the model and the experimental data are shown in Figure 4.32. There is a
good agreement between the experimental data and the model prediction.

143
0
50
100
150
200
250
300
350
Passive Capacity (Kips)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
Displacement (inches)
2D FEM
Data

Figure 4.32: UCD Abutment Test versus Finite Element Model Prediction

144
4.7.2 BYU Pile Cap Experiments 2-D Finite Element Model
The geometry of all the finite element models for all BYU pile cap
experiment are identical. Therefore, only a typical finite element model for one case
is presented in this section. The load-displacement curves calculated by LSH, HFD,
2-D and 3-D finite element models for all cases will be presented later in this
chapter.
The 2-D finite element model of the BYU pile cap with clean sand backfill is
shown in Figure 4.33. The height of the wall is 3.67 feet. The right vertical
boundary of the model is place about 20 feet from the wall face and the left vertical
is placed about 4 feet from the wall face. The bottom boundary is set about 4 feet
below the bottom of the wall. A controlled horizontal displacement was applied at
the face of the wall, while the bottom and vertical boundaries were constrained in
both horizontal and vertical directions.

Figure 4.33: Finite Element Model for BYU Clean Sand Backfill

145
4.7.3 Pile Cap Backfill Parameters
The backfill strength values and the stiffness parameters used for the
Hardening Soil model available in PLAXIS are shown in Table 4.4 Because the
stress-strain measurements for the BYU experiments were not available, the stiffness
parameters used in the finite element models for the backfill were back-calculated. It
is important to mention that the purpose of the validation was not to obtained a
perfect match between the measured structural-backfill force-displacement and the
results from the finite element model, but rather to establish a set of reasonable
stiffness parameters to be used by the practicing engineers. Perfect match could have
been obtained if strength and stiffness variation were utilized for every separate test.
The recommended stiffness parameters listed in Table 4.4 provides a good estimate
for practicing geotechnical and bridge engineers to develop nonlinear force-
displacement relationships of the bridge abutment-backfill using the PLAXIS HS
model.
Table 4.4 Input Parameters for BYU Pile Cap Analyses
Backfill Type
γ
(lb/ft
3
)
ϕ
Friction
c
(psf)
ψ
Dilatanc
y
R
inter
Wall
interface
R
f

E
50
ref
(psf)
E
ur
ref
(psf)
Clean Sand 117 39
o
80 9
o
0.70 0.97 2.0E6 4.0E6
Fine Gravel 132 34
o
80 4
o
0.70 0.97 2.0E6 4.0E6
Coarse Gravel 148 40
o
250 10
o
0.70 0.97 2.0E6 4.0E6
Silty Sand 122 27
o
648 0
o
0.70 0.97 1.0E6 2.0E6

The strength parameters used in the Hardening Soil model are identical to the
parameters used in the “LSH” model presented in the previous chapter. The
Hyperbola cut-off parameter, Rf, was set equal to 0.97. This gives a better match

146
with the measured nonlinear force-displacement than the default value of 0.90
proposed in the PLAXIS User Manual (2006). The deformed mesh, displacement
vector, displacement contours, Mohr-Coulomb plastic points and the incremental
strains of the model are shown in Figure 4.34 through Figure 4.38. The Mohr-
Coulomb failure surface of the model is also shown in Figure 4.38.

Figure 4.34: 2-D Finite Element Deformed Mesh for BYU Clean Sand Backfill

Figure 4.35: 2-D Finite Element Displacement Vector Field for BYU Clean Sand
Backfill

147

Figure 4.36: 2-D Finite Element Displacement Contours for BYU Clean Sand
Backfill

Figure 4.37: 2-D Finite Element Mohr-Coulomb Failure Surface for BYU Clean
Sand Backfill

Figure 4.38: 2-D Finite Element Incremental Strains for BYU Clean Sand Backfill

148
The failure surface remarkably resembles the logarithmic spiral failure
surface which was mapped in the field (Rollins et al., 2006). The Mohr-Coulomb
plastic points and the incremental shear strain contours illustrating the progressive
logarithmic-spiral shear surfaces similar to the LSH model and the experimental
data.
As shown in Figure 4.36 through Figure 4.38, the failure surface initiated
from the bottom of the abutment backwall at a negative slope. This was referred to as
“take off angle” using the LSH model in the previous chapter. The failure surface
then extended upward to the backfill free surface with increasing slope, intercepting
backfill surface at about three times the height of the backwall similar to the UCLA
abutment experiment. The finite element model demonstrates a good match between
the simulated deformed shapes of the passive wedges and the slip surfaces mapped
in the field (Rollins et al., 2006).
The load-displacement curve of the model and the experimental data are
shown in Figure 4.39. The calculated load-displacement curve was multiply by the
same adjustment factor explained in the previous chapter to account for the 3-D
effect. There is a good agreement between the experimental data and the model
prediction only after applying an adjustment factors which were discussed in the
previous chapter.
4.8 Backfill Behavior Using 3-D Finite Element Model
In this section, a three-dimensional continuum finite element model is
presented to investigate the abutment-backfill behavior. In the previous section, it

149
was shown that a two-dimensional plane-strain finite element model is capable of
modeling the behavior of the bridge abutment. However, a three-dimensional finite
element model must be used to simulate the soil-structure behavior of bridge
abutments with high skewed angles due to significant out-off-plane rotation during a
seismic event since axisymmetric condition does not exist anymore. Using a three-
dimensional finite element model, bridge engineers are able to develop nonlinear
abutment force-deformation and understand the mechanism of the abutment failure.
At the present time, there is no experimental data available for skewed abutments.
Therefore, full-three-dimensional finite element models of the abutments and pile
cap experiments mentioned in the previous sections are used to calibrate the three-
dimensional finite element models for skewed abutments. A realistic bridge
abutment with typical bridge abutment backfill is used to examine the mechanism of
the skewed-abutment-backfill behavior which will be presented later in this chapter.

150

Experiment
2D FEM Model
0
50
100
150
200
250
300
Passive Capacity (Kips/ft)
0.0 0.5 1.0 1.5 2.0 2.5
Displacement (inches)

Figure 4.39: BYU Pile Cap Test versus Finite Element Model Prediction For Clean Sand Backfill

151
4.8.1 3-D Finite Element Simulation of UCLA Abutment Test
Simulation using a full 3-D finite element model for the UCLA Abutment
field experiment is presented herein. The abutment backfill is modeled using a 15-
node solid elements. These elements are composed of 6-node triangular at each face
as is shown in Figure 4.40.
x
Nodes
Stress Points
1
2
3
7
8
9
4
5
6
13
14
15
11
12
10
x
x
x
x
x
x
X
Y
Z
x
Nodes
Stress Points
1
2
3
7
8
9
4
5
6
13
14
15
11
12
10
x
x
x
x
x
x
X
Y
Z
X
Y
Z

Figure 4.40: 3-D Solid Element

The 3-D passive wedge with gypsum columns and section A-A of the 3-D
finite element model are shown in Figure 4.41. Low strength properties were
assigned to the gypsum columns. The left, right and bottom boundaries of the 3-D
finite element model are the same as the 2-D model. The 2-D finite element model

152
was extruded 16 feet in the z direction. The constitutive Hardening Soil (HS) model
was used to simulate the abutment-backfill stress-strain behavior.

(a) Schematic of UCLA 3D Abutment Passive Wedge Formation
8.0’
X
Y Z
8.0’ 8.0’ 8.0’
X
Y Z
X
Y Z

(b) Section A-A of the Backfill
Figure 4.41: Section Through the Full 3-D Finite Element Model of the UCLA
Abutment Experiment

153
4.8.2 Testing Sequence
Similar to the 2-D analysis, the analysis was performed in steps to simulate the
sequence of actual events that occurred during the field experiments.
The width of the abutment backwall was set to be 15 feet and the backfill width
was set to 16 feet to simulate the field experiment and the backfill behind the shear
keys in the longitudinal direction. A uniform displacement of 3.5 inches was applied
only to the 15 feet of abutment-backfill. Zero displacement was applied to the
remaining 6-inch of backfill at each side of the backwall as shown in Figure 4.42 and
Figure 4.43. The computations were performed using the following steps:
(1) Starting with a level ground, the initial stresses were calculated.
(2) Excavation was performed by deactivating entire 16 feet of clusters of elements
in the front of the abutment face.
(3) A 3.5 inches of monotonic uniform horizontal displacement was applied at the
5.5-foot by 15-foot of vertical abutment-backfill.
(4) Zero displacement was applied at the vertical backfill face at each side of the
abutment-backwall as shown in Figure 4.43 and Figure 4.44.
The deformed mesh, displacement vector, displacement contours, Mohr-Coulomb
plastic points and the incremental strains of the model are shown in Figure 4.43
through Figure 4.47. The calculated force-displacement relationship of the 3-D finite
element model is shown in Figure 4.48. The results are very close to the 2-D finite
element model shown in Figure 4.23 and the LSH model presented in the previous

154
chapter, demonstrating that the force-displacement relationship of the bridge
abutment can be simulated using a simple 2-D plane-strain analysis.
16.0’
X
Y Z
5.5’
6.0” Thick
16.0’
X
Y Z
X
Y Z
5.5’
6.0” Thick

Figure 4.42: Full 3-D Finite Element Model of the Deformed Shape for UCLA
Abutment Experiment

Figure 4.43: Section Through Gypsum Columns for the Full 3-D Finite Element
Model for UCLA Abutment Experiment

155

Figure 4.44: Full 3-D Finite Element Model of the Displacement Vectors for UCLA
Abutment Experiment

Figure 4.45: Full 3-D Finite Element Model of the Displacement Contours of UCLA
Abutment Experiment

156

Figure 4.46: Full 3-D Finite Element Model of the Plastic Points Through Gypsum
Columns for UCLA Abutment Experiment

Figure 4.47: Full 3-D Finite Element Model of the Incremental Strains Through
Gypsum Columns for UCLA Abutment Experiment

157
UCLA Test Data
3D FEM Model
0
50
100
150
200
250
300
350
400
450
500
550
Passive Capacity (Kips)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
Displacement (inches)

Figure 4.48:: Measured Experimental Data Versus Finite Element Prediction for UCLA Abutment Experiment

158
4.9 3-D Finite Element Simulation of UCD Abutment Test
Full 3-D finite element model simulation for the UCD Abutment field
experiment is presented herein. The dimensions and 3-D deformed mesh of the finite
element model is shown in Figure 4.49. The left, right and bottom boundaries of the
3-D finite element model are the same as the 2-D model. The 2-D finite element
model was extruded 10 feet in the z direction. The constitutive Hardening Soil (HS)
model was used to model the nonlinear abutment backfill behavior.

33.0 ft
10.0ft
5.5 ft
33.0 ft
10.0ft
5.5 ft

Figure 4.49: Finite Element Model of the of the Deformed Shape for UCD Abutment
Experiment

4.9.1 Sequence of the Events
The analysis was performed in steps to simulate sequence of the real events
during the field experiments, where the backfill was placed and compacted behind

159
the wall and was extended some distance behind the back wall laterally. The width
of the backfill and the abutment wall were identical, therefore there was no 3-D
effect. The computations were performed using the following steps:
(1) Starting with a level ground the initial stresses were calculated.
(2) Excavation was performed by deactivating clusters of elements in the front of
the abutment face.
(3) The entire 5.5 feet of vertical abutment face was pushed up to backfill failure
using a uniform displacement.
The deformed mesh, displacement vector and counters, Mohr-Coulomb plastic
points and the incremental strains of the model are shown in Figure 4.49 through
Figure 4.53. The three-dimensional finite element analysis also indicates that the
failure surface does not extend more that twice the height of the backwall.
The calculated force-displacement relationship predicted by the model is
shown in Figure 4.54. There is a good agreement between the experimental data and
the model prediction. The results are very close to the 2-D finite element model and
the LSH model. This shows that the force-displacement relationship of the bridge
abutment can be simulated using a simple 2-D plane strain analysis.

160

Figure 4.50: Full 3-D Finite Element Model of the Displacement Vectors for UCD
Abutment Experiment

Figure 4.51: Full 3-D Finite Element Model of the Displacement Contours for the
UCD Abutment Experiment

161

Figure 4.52: Full 3-D Finite Element Model of Mohr-Coulomb Plastic Points for UCD
Abutment Experiment

Figure 4.53: Full 3-D Finite Element Model of the Incremental Strains for UCD
Abutment Experiment

162
0
50
100
150
200
250
300
350
Passive Capacity (kips)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
Displacement (inches)
3D FEM
Data

Figure 4.54: Measured Experimental Data Versus Finite Element Prediction for UCD Abutment Experiment

163
4.10 3-D Finite Element Simulation of BYU Pile Cap Experiments
The pile cap experiments conducted by Kyle et al. (2006) were described in
the previous chapter. Cracking pattern of the pile cap backfill for each experiment
was measured. The surface cracking for all experiments were extended beyond the
edges of the pile caps in the longitudinal as well as transverse direction. Figure 4.55
shows the schematics of a typical BYU pile cap field experiment with the three-
dimensional passive wedges bounded by a logarithmic-spiral type failure surface,
based on field measurements of observed cracking patterns and wedge deformations
by Rollins and Cole (2006).

Figure 4.55: Schematic of a Typical 3-D Wedge Failures and Cracking Patterns
of BYU Pile Cap Experiments

164
The 3-D finite element model simulations were performed using both full-
wall-width model and half-wall-width model due to symmetry of the problem as
shown in Figure 4.56 and Figure 4.57. The deformed finite element mesh of the full
model with displacement contours is also shown in Figure 4.57 to illustrate the
capability of the model and to examine the mechanism of the failure mode. The
backfill material was extended 11.5 feet in the lateral directions at each side of the
wall to account for the three-dimensional effect of the backfill. A 2.5-inch uniform
monotonic displacement was applied at across the wall (3.67 feet by 17 feet). A
zero displacement was applied at the vertical face of the soil each side of the wall
(3.67 feet’ by 11.5 feet).

Figure 4.56: Full 3-D Finite Element Model Displacement Contours of BYU
Pile Cap Experiment

165
8.5 ft
11.5 ft
20.0 ft
3.67 ft
Plane of Symmetry
7.35 ft
8.5 ft
11.5 ft
20.0 ft
3.67 ft
Plane of Symmetry
7.35 ft

Figure 4.57: 3-D Finite Element Using Half Model of BYU Pile Cap Experiment

The testing sequence for the analytical model will be presented in Section
4.10.2. Since the full three-dimensional finite element model is relatively large and
computationally expensive, only half models were developed for all BYU pile cap
experiments. The result of the half models and the full models were identical. The
three-dimensional finite element model demonstrates a good match between the
simulated deformed shapes of the passive wedges and the slip surfaces mapped in
the field (Rollins et al., 2006).
4.10.1 3-D Finite Element Simulations Using Half Model
The 3-D finite element model simulations were performed using half-wall-
width model due to symmetry of the problem as shown in Figure 4.57. The
constitutive Hardening Soil (HS) model was used to simulate the nonlinear pile cap
backfill stress-strain behavior.

166
4.10.2 Testing Sequence
The analysis was performed in steps to simulate sequence of actual events
during the field experiments, where the backfill was placed and compacted behind
the pile cap and was extended some distance behind the pile cap laterally. No
backfill was placed at the sides of the pile cap, therefore the simulations were
performed only with a vertical face of the backfill. The 2-D finite element model was
extruded in the z direction as shown in Figure 4.57. The computations were
performed using the following steps:
(1) Starting with a level ground the initial stresses were calculated.
(2) Excavation was performed by deactivating clusters of elements in the entire
front face of the pile cap and the vertical face in the lateral direction.
(3) A 2.5-inch uniform displacement was applied to the 8.5 feet of the pile cap. A
zero displacement was applied to the remaining 11.5 feet of the model shown
in Figure 4.57.
The extent of the deformed mesh and the displacement contours in the longitudinal
and transverse direction of the pile cap backfill is shown in Figure 4.57. This
illustrates the realistic three-dimensional simulation of the backfill as shown in
Figure 4.55. The computed load-displacement components using the “LSH” model
and 2-D finite element model were multiplied by an adjustment factor α varying
between 1.2 and 1.4 to account for the three-dimensional effect of the mobilized
passive wedge in the backfill. However, the force-deformation relationships
calculated using 3-D finite element model were compared against experimental data

167
without any adjustment factor. Figure 4.56 and Figure 4.55 show the formation of
the three-dimensional passive wedge using half-wall-width. Figure 4.58 through
Figure 4.61 show the total displacement contours, displacement vectors Mohr-
Column plastic points and incremental shear strains reflecting the location of the
failure surface.

Figure 4.58: 3-D Finite Element Model of the Displacement Contours for BYU
Pile Cap Experiment

168

Figure 4.59: 3-D Finite Element Model of the Displacement Contours for BYU
Pile Cap Experiment

Figure 4.60: Full 3-D Finite Element Model of Mohr-Coulomb Plastic Points for
BYU Pile Cap Experiment

169

Figure 4.61: Full 3-D Finite Element Model of Mohr-Coulomb Incremental
Strains for BYU Pile Cap Experiment

The load-displacement curve of the model and the experimental data for the pile cap
with the silty sand backfill is shown in Figure 4.62. In contrast to the 2-D model, the
calculated coordinates of the load- displacement curve are the direct results of the
three-dimensional finite element without any adjustment factor. There is a good
agreement between the experimental data and the model prediction.

170

0
50
100
150
200
250
300
350
Passive Capacity (Kips)
0.0 0.5 1.0 1.5 2.0 2.5
Displacement (inches)
3D FEM
Data

Figure 4.62: Measured Experimental Data Versus Finite Element Prediction for BYU Silty Sand Backfill Experiment

171
4.11 Comparisons of Various Models Versus Experimental Data
Numerical results are compared with the measurements by means of set of
load-displacement curves that show backfill capacity as a function of displacement.
The comparisons of the load- displacement curves resulted from the closed-form
solution (HFD model), simplified solution (LSH model) and the numerical solution
(2-D and 3-D finite element models) for all the full-scale abutments and pile cap
experiments are presented herein. The comparisons of the abutment backbone curves
predicted by all methods presented are nearly the same for all practical purposes.
Therefore, typical bridge abutments behavior is a 2-D plane-strain problem.
However, for the pile caps, the coordinate of the backbones developed by the LSH or
2-D finite element model were multiplied by an adjustment factor to account for the
three-dimensional effect. The pile cap backbone curves predicted using the 3-D finite
element model need not to be multiplied by an adjustment factor. The comparisons
the analytical models versus the experimental data are shown in Figure 4.63 through
Figure 4.68.

172
0
50
100
150
200
250
300
350
400
450
500
550
Passive Capacity (kips)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
Displacement (inches)
HFD
LSH
2D FEM
3D FEM
Data

Figure 4.63: Measure Force-Deformation of the UCLA Abutment Test Versus Predictions by Various Methods

173
0
50
100
150
200
250
300
350
Passive Capacity (kips)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0
Displacement (inches)
HFD
LSH
2D FEM
3D FEM
Data

Figure 4.64: Measure Force-Deformation of the UCD Abutment Test Versus Predictions by Various Methods

174

0
50
100
150
200
250
300
350
Passive Capacity (kips)
0.00.5 1.01.5 2.02.5
Displacement (inches)
HFD
LSH
2D FEM
3D FEM
Data

Figure 4.65: Measure Force-Deformation of the BYU Pile Cap Test Versus Predictions by Various Methods for Silty Sand
Backfill

175
0
50
100
150
200
250
300
Passive Capacity (kips)
0.00.5 1.01.5 2.02.5
Displacement (inches)
HFD
LSH
2D FEM
3D FEM
Data

Figure 4.66 : Measure Force-Deformation of the BYU Pile Cap Test Versus Predictions by Various Methods for Clean
Sand Backfill

176
0
50
100
150
200
Passive Capacity (kips)
0.00.5 1.01.5 2.02.5
Displacement (inches)
HFD
LSH
2D FEM
3D FEM
Data

Figure 4.67: Measure Force-Deformation of the BYU Pile Cap Test Versus Predictions by Various Methods for Fine Gravel
Backfill

177
0
50
100
150
200
250
300
350
400
450
500
Passive Capacity (kips)
0.00.5 1.01.5 2.02.5
Displacement (inches)
HFD
LSH
2D FEM
3D FEM
Data

Figure 4.68: Measure Force-Deformation of the BYU Pile Cap Test Versus Predictions by Various Methods for Coarse
Gravel Backfill

178
4.12 3-D Finite Element Model for Skewed Abutment
During seismic events, the bridge deck experiences significant rotational
motions about the its vertical axis. As a result, the deck first collides with the
abutment–backfill system. The collision continues for some times and then the
clockwise rotation of the deck as shown in Figure 4.69 results in the separation of
the deck from the abutment.

Figure 4.69: Clockwise Deck Rotation During a Seismic Event

The photograph of Figure 4.70 shows manifestation of the asymmetric
passive wedge and ground heave in the west half of the roadway for more than 2
feet, adjacent to the obtuse corner of the bridge deck. This river bridge crosses the
Chelongpu reverse-thrust Fault that generated slip movements causing incremental
collapse of the southern two deck spans and driving the northern span into the
skewed abutment backfill a distance of 7.25 feet.
Inspection of existing skewed abutments after recent earthquakes indicates
that the passive wedges that form behind the skewed walls tend to be asymmetric
along the abutment backwall due to deck rotation as shown in Figure 4.71. Such

179
behavior was observed at the northern abutment of the skewed Wushi highway
bridge in Taiwan that was severely damaged during the recent Chi-Chi earthquake as
shown in Figure 4.70.

Figure 4.70: Non-uniform Passive wedge behind skewed abutment

A set of 3-D finite-element analyses using PLAXIS was performed to
evaluate the development of passive resistance behind a 75-feet wide abutment with
a 5.5-feet high backwall of varying skew angles. First, the soil in the front of the
abutment backwall was excavated and then the abutment backwall was loaded

180
monotonically using a displacement control normal to the abutment backfill to
simulate the non-skewed abutment failure mechanism as shown in Figure 4.71. It
was assumed that the bridge deck will be pushed between the wingwalls in a
plunging mode during a seismic event. The wingwalls were kept in stationary in the
analytical model. The PLAXIS HS model was used to simulate the nonlinear
abutment backfill. Linear elastic material properties were assigned to the abutment
wingwalls. The formation of the mobilized passive wedge and the displacement
contours are shown in Figure 4.71.

Figure 4.71: Passive Soil Wedge Plunging in Between Wingwalls

The same 3-D displacement control finite element model was used to
investigate the failure mechanism of the skewed-abutment with various skewed

181
angles. Figure 4.72 shows the shape of the fully formed 3-D passive wedge formed
between the wingwalls of the abutment with 45
o
skew.

(a) Deformed Mesh and Displacement Contours

(b) Total Displacement Contours

Figure 4.72: Full 3-D Finite Element Model of the Passive Wedge Formation
Behind Skewed Abutment
As a result of deck clockwise rotation, the abutment backwalls tend to be
pushed primarily in the obtuse corners of the deck, causing asymmetric passive
wedges to form behind the abutment backwall. In skewed abutments, the non-

182
uniform loading of the abutment backwall can result in a reduced mobilized soil
capacity as compared to ordinary non-skewed abutments. The ground heave at the
far half of the wall width (see Figure 4.72) illustrates the overstress and breakdown
of the passive wedge, resulting in the reduction of soil resistance.
Due to in-plane motions and induced pounding forces (compression) of the
bridge deck, the abutment-backfill response consists of normal and tangential
passive resistance. Figure 4.73 shows example of the nonlinear tangential and
normal components of 30
o
skewed-abutment-backfill backbone curves. Therefore,
for the global seismic analysis of the skew bridges both tangential and normal
components of the abutment backfill should be considered. The tangential
component of passive resistance about one third of the normal component.
The normal components of the abutment passive resistance for various skew
angles are shown in Figure 4.74. The results indicate that the mobilized passive
capacity might decrease as a function of skew angles at large displacement levels.
At very high-skew abutments, the passive capacity can decrease significantly,
which is a result of separation of the deck at the acute corners and “disintegration” of
the passive wedge after significant plastic ground deformation and heave has
occurred near the obtuse corners of the deck. These findings raise the possibility that
a skew abutment may develop a considerably reduced soil resistance in comparison
to a similar normal abutment, affecting overall bridge response.

183
0
300
600
900
1200
1500
1800
Passive Capacity (Kips)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0
Displacement (inches)
Tangential
Normal

Figure 4.73: Nonlinear Normal and Tangential Components of the Abutment-Backfill Resistance for a 30 Skew angle

184

Figure 4.74: Impact of Skew Angles on Nonlinear Abutment Force-Deformation Relationship

185
4.13 Summary
The full-scale field experiments conducted on the abutments backwall and
pile caps provide the opportunity to compare actual recorded measurements with the
results from Hyperbolic Force-Displacement (HFD model) relationship, the limiting
equilibrium using Log Spiral combined with modified Hyperbolic backfill stress-
strain relationship (LSH model), 2-D and 3-D finite element model. The 3-D finite
element models were calibrated against the experimental data and was used to
investigate the failure mechanism and the response of the skew abutments.
From comparison of field observations after a seismic event and computer
simulations, it is clear that the skewed abutment tend to develop an asymmetric
passive soil wedge that is less wide and generates less soil resistance than the normal
passive wedge behind a non-skewed abutment. The size (width) and capacity of this
passive wedge depends on abutment width and skew angle. These factors affect the
interaction of the bridge deck with the abutment. Soil resistance does not increase
with increasing skew angle as could be expected from a combination of passive
resistance normal to the wall and additional soil traction developed along the back
face of the abutment wall. Current analyses indicate that the width and total
resistance of the mobilized passive wedge is maximum for zero skew and decreases
as the magnitude of skew angles is increased.
It is assumed that the abutment wingwalls are expected to yield or fail during
a seismic event because they is designed to retain the abutment backfills in sloped
ground and not to provide passive resistance in the transverse directions. The

186
transverse capacity of the abutment is provided by the abutment shear keys which
will be described in the following chapter.

187
CHAPTER FIVE
NONLINEAR SEISMIC RESPONSE OF SKEWED BRIDGES

5.1 Introduction
Global seismic behavior of skewed bridges is affected by a number of factors,
including bridge skew angle, deck width, deck flexibility, number of spans, number
of columns per bent, column ductility, soil-abutment-superstructure interaction,
abutment shear keys, soil-bent foundation-structure interaction, abutment bearing
pads, and characteristics of the seismic source. The objective of this chapter is to
evaluate the dynamic nonlinear soil-abutment-structure interaction (SASI) behavior
of typical straight, concrete box girder highway bridges at several skew angles
subjected to a suit of ground motions with the near fault effect. Skewed bridges tend
to rotate during a seismic event, which can cause excessive transverse movement and
unseating of the superstructure and pounding to the abutment backwall.
A number of SASI studies have been performed on skewed bridges.
Traditional bridge design practice evaluates dynamic performance of skewed bridges
using two dimensional stick models with lumped springs to represent the abutment
structure and foundation. However, when a bridge has a skew abutment, the
longitudinal bridge response is affected by transverse loading due to the coupling
nature of the two horizontal directions. Full three-dimensional bridge models which
include bridge deck, bent caps, and ductile columns, seat-type abutments with

188
abutment-backfill and abutment shear keys are developed to perform nonlinear
dynamic time history analysis for various typical bridge structures.
Figure 5.1 shows various components of a bridge system and the modeling
assumptions used in this chapter. The bridge deck is model either using shell
elements (referred to as shell model) and or beam elements (referred to as spline
model). The cracked moment of inertia obtained from the moment-curvature analysis
of the column cross section is used to model the bridge columns. Nonlinear frame
elements with the moment-curvature properties are used to model the top of columns
to allow plastic hinge formations during a major seismic event. Pinned connections
are assumed at the base of the columns. Abutment-soil interaction was modeled by
an expansion gaps and nonlinear normal springs skewed to the principal bridge axis.
Abutment shear keys were modeled using nonlinear springs and an expansion gaps
in the skewed transverse direction. Nonlinear response-spectra-compatible time
history analyses were performed using seven sets of ground motions with two lateral
components incorporating near-fault effects. The analyses show that the
superstructure undergoes significant rotations about the vertical axis, this result in
permanent lateral deck offset at the abutments.

189

Figure 5.1: Components of Bridge System

190
5.2 Impact of Ground Motion Characteristics
The effects of rupture directivity on near-fault ground motions have been
recognized by engineering seismologists for several decades. The propagation of
fault rupture toward a site at a velocity close to the shear wave velocity causes most
of the seismic energy from the rupture to arrive in a single large long-period pulse of
motion that occurs at the beginning of the record (Somerville et al., 1997). Current
seismic design of ordinary bridges is based on the response spectrum approach.
However, the response spectrum does not provide an adequate characterization of the
ground motions with the near-fault effect. Current trends for seismic response of
bridge structures have embraced the concept of performance based design. The
validity of performance-based design depends on realistic specification of ground
motion inputs, realistic models of the bridge structure and realistic boundary
conditions of the bridge model. Therefore, time history input ground motions should
be used instead of a response spectrum to adequately characterize the nonlinear
response of the bridge models due to near-fault ground motions.
Ground motions with an asymmetrical and high amplitude velocity pulse
characteristic have the tendency of producing a biased, one-sided response of the
bridge structures. Asymmetrical impulsive loading generates large displacements in
one direction leading to a significant residual displacement. As part of the FHW
seismic research program, the effect of near-fault motions on bridge columns was
studied using shake table tests at the University of Nevada, Reno (Phan et al., 2005).
The asymmetry in the directivity pulse generated an asymmetric response in column

191
specimens. The measured results revealed the important role of ground motion
characteristics on the bridge column hysteretic behavior. The most unique aspect of
the measured response was the presence of high residual displacements. The
measured force displacement hysteretic behavior of the column specimen is shown
in Figure 5.2. The one-sided high velocity pulse of the input ground motion caused
the hysteretic response of the column to be biased in one direction.

-4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0
Displacement (inches)
-30
-20
-10
0
10
20
30
Force (Kips)
UNR Column Test

Figure 5.2: Measured Shaking Table Hysteretic Response of Bridge Column (Phan et
al., 2005)

In this chapter, in addition to the seven response-spectra-compatible time
history ground motions, the two-components recorded Renaldi ground motion was
used to investigate the response of skew bridges in particular the abutment response

192
due to the asymmetrical impulse loading of the ground motions with near-fault
effect.
5.3 Selection of the Ground Motions
Many toll bridge designs require a minimum of three sets of spectrum-
compatible time history ground motions. However, three sets of time histories are
still not sufficient to cover aleatory variability in ground motion parameters; as a
result, envelop of the results is used for the design of all toll bridges. If one wishes to
use average structural response (rather than envelop), the number of time histories
needs to be significantly more than three. The requirement of number of time
histories has been discussed in several literatures and in many project meetings for
Caltrans Toll Bridge projects. The consensus has been to employ seven spectrum-
compatible time histories to be eligible for averaging. Same number or more has
been recommended in UBC 1997 and IBC 2000.
For this research project, it is more important to obtain statistically stable
mean values in order to make reasonable conclusions of bridge abutment response
under general conditions. Since the conclusion must be applicable for a wide range
of seismological considerations including near fault directivity effects, seven
spectrum-compatible time histories with the near-fault effects are used.
The time histories were developed to match the Caltrans standard SDC curve
having a Magnitude 8 with a peak ground acceleration of 0.7 g on Soil Type D (stiff
soil with shear wave velocity 600< Vs.<1200 ft/s). All seven sets of start-up motions
are modified to represent time history motions that are response spectrum compatible

193
with the target SDC curve. All the time histories have been baseline corrected. The
earthquake records with high velocity pulses selected for this research are listed in
Table 5.1
Table 5.1 Selected Earthquake Records
No. Startup Motion
1 1979 Imperial Valley, Array 7
2 1994 Northridge, Slymar Record
3 1992 Landers, Lucerne Record
4 1994 Northridge, Renaldi Record
5 1989 Loma Prieta, Los Gatos Record
6 1995 Kobe, Takatori Record
7 1992 Turkey Erzincan Record

Since the purpose of the research is to investigate the in plane motion of the
bridge deck, the vertical components of the ground motions were ignored. The two
components were applied in the longitudinal and transverse directions of the bridge.
All input motions exhibit high-velocity pulses; the components with the largest
velocity pulses were applied in the longitudinal direction. The peak accelerations,
velocities and displacements including the time of the velocity pulses for both
components are listed in Table 5.2. The largest velocity pulses occurred between 4.9
to 12.9 seconds. Figure 5.3 to Figure 5.16 show the response spectra compatible
acceleration, velocity and displacement time histories of all seven input motions
applied in the longitudinal and transverse bridge directions of the bridge models.

194
Table 5.2 Input Ground Motion Characteristics
Bridge Longitudinal Direction Bridge Transverse Direction
Motion
No
PGA PGV PGD t
p
PGA PGV PGD t
p

1 0.7 56 38 8.3 0.7 51 40 8.8
2 0.7 68 21 7.9 0.7 47 30 10.6
3 0.7 53 23 12.6 0.7 43 23 11.0
4 0.7 79 23 4.0 0.7 52 25 7.3
5 0.7 51 36 9.9 0.7 51 27 9.9
6 0.7 44 32 7.7 0.7 43 28 10.5
7 0.7 58 36 4.9 0.7 39 39 6.0
Note:
PGA- Peak Ground Acceleration (g)
PGV-Peak Ground Velocity (in/sec)
PGD-Peak Ground Displacement (in)
t
p
-Time of the peak (sec)

195

0 1020 3040
Time (sec)
-60
-30
0
30
60
Velocity (in/sec)
0 1020 3040
Time (sec)
-1.0
-0.5
0.0
0.5
1.0
Acceleration (g)
(a)
(b)
(c)
0 1020 3040
Time (sec)
-40
-20
0
20
40
Displacement (inches)
02 468 10
Period (sec)
0.0
0.5
1.0
1.5
2.0
Spectral Acceleration (g)
0
20
40
60
80
Relative Displacement (inches)
Pseudo Acceleration
Target
Relative Displacement
(d)

Figure 5.3: Input Ground Motion 1 in Longitudinal Direction

196

0 10203040
Time (sec)
-60
-30
0
30
60
Velocity (in/sec)
0 10203040
Time (sec)
-1.0
-0.5
0.0
0.5
1.0
Acceleration (g)
(a)
(b)
(c)
0 10203040
Time (sec)
-40
-20
0
20
40
Displacement (inches)
02 4 6 8 10
Period (sec)
0.0
0.5
1.0
1.5
2.0
Spectral Acceleration (g)
0
20
40
60
80
Relative Displacement (inches)
Pseudo Acceleration
Target
Relative Displacement
(d)

Figure 5.4: Input Ground Motion 1 in Transverse Direction

197

0 10 2030 4050 60 70
Time (sec)
-60
-30
0
30
60
Velocity (in/sec)
0 10 2030 4050 60 70
Time (sec)
-0.5
0.0
0.5
-0.25
0.25
Acceleration (g)
(a)
(b)
(c)
0 10 2030 4050 60 70
Time (sec)
-20
0
20
Displacement (inches)
02 4 6 8 10
Period (sec)
0.0
0.5
1.0
1.5
2.0
Spectral Acceleration (g)
0
20
40
60
80
Relative Displacement (inches)
Pseudo Acceleration
Target
Relative Displacement
(d)

Figure 5.5: Input Ground Motion 2 in Longitudinal Direction

198

0 10 2030 4050 60 70
Time (sec)
-60
-30
0
30
60
Velocity (in/sec)
0 10 2030 4050 60 70
Time (sec)
-0.5
0.0
0.5
-0.25
0.25
Acceleration (g)
(a)
(b)
(c)
0 10 2030 4050 60 70
Time (sec)
-30
-20
-10
0
10
20
30
Displacement (inches)
02 4 6 8 10
Period (sec)
0.0
0.5
1.0
1.5
2.0
Spectral Acceleration (g)
0
20
40
60
80
Relative Displacement (inches)
Pseudo Acceleration
Target
Relative Displacement
(d)

Figure 5.6: Input Ground Motion 2 in Transverse Direction

199
0 10 20 304050
Time (sec)
-60
-30
0
30
60
Velocity (in/sec)
0 10 20 304050
Time (sec)
-0.5
0.0
0.5
-0.25
0.25
Acceleration (g)
(a)
(b)
(c)
0 10 20 304050
Time (sec)
-30
-20
-10
0
10
20
30
Displacement (inches)
02 4 6 8 10
Period (sec)
0.0
0.5
1.0
1.5
2.0
Spectral Acceleration (g)
0
20
40
60
80
Relative Displacement (inches)
Pseudo Acceleration
Target
Relative Displacement
(d)

Figure 5.7: Input Ground Motion 3 in Longitudinal Direction

200
0 10 20 304050
Time (sec)
-60
-30
0
30
60
Velocity (in/sec)
0 10 20 304050
Time (sec)
-0.5
0.0
0.5
-0.25
0.25
Acceleration (g)
(a)
(b)
(c)
0 10 20 304050
Time (sec)
-30
-20
-10
0
10
20
30
Displacement (inches)
02 4 6 8 10
Period (sec)
0.0
0.5
1.0
1.5
2.0
Spectral Acceleration (g)
0
20
40
60
80
Relative Displacement (inches)
Pseudo Acceleration
Target
Relative Displacement
(d)

Figure 5.8: Input Ground Motion 3 in the Transverse Direction

201
010 20 30
Time (sec)
-40
0
40
80
Velocity (in/sec)
010 20 30
Time (sec)
-0.5
0.0
0.5
-0.25
0.25
Acceleration (g)
(a)
(b)
(c)
010 20 30
Time (sec)
-30
-20
-10
0
10
20
30
Displacement (inches)
02 4 6 8 10
Period (sec)
0.0
0.5
1.0
1.5
2.0
Spectral Acceleration (g)
0
20
40
60
80
Relative Displacement (inches)
Pseudo Acceleration
Target
Relative Displacement
(d)

Figure 5.9: Input Ground Motion 4 in the Longitudinal Direction

202

010 20 30
Time (sec)
-40
-20
0
20
40
60
Velocity (in/sec)
010 20 30
Time (sec)
-0.5
0.0
0.5
-0.25
0.25
Acceleration (g)
(a)
(b)
(c)
010 20 30
Time (sec)
-30
-20
-10
0
10
20
30
Displacement (inches)
02 4 6 8 10
Period (sec)
0.0
0.5
1.0
1.5
2.0
Spectral Acceleration (g)
0
20
40
60
80
Relative Displacement (inches)
Pseudo Acceleration
Target
Relative Displacement
(d)

Figure 5.10: Input Ground Motion 4 in the Transverse Direction

203

0 10203040
Time (sec)
-60
-40
-20
0
20
40
Velocity (in/sec)
0 10203040
Time (sec)
-0.5
0.0
0.5
-0.25
0.25
Acceleration (g)
(a)
(b)
(c)
0 10203040
Time (sec)
-20
-10
0
10
20
30
40
Displacement (inches)
02 4 6 8 10
Period (sec)
0.0
0.5
1.0
1.5
2.0
Spectral Acceleration (g)
0
20
40
60
80
Relative Displacement (inches)
Pseudo Acceleration
Target
Relative Displacement
(d)

Figure 5.11: Input Ground Motion 5 in the Longitudinal Direction

204

0 10203040
Time (sec)
-60
-40
-20
0
20
40
Velocity (in/sec)
0 10203040
Time (sec)
-0.5
0.0
0.5
-0.25
0.25
Acceleration (g)
(a)
(b)
(c)
0 10203040
Time (sec)
-20
-10
0
10
20
30
40
Displacement (inches)
02 4 6 8 10
Period (sec)
0.0
0.5
1.0
1.5
2.0
Spectral Acceleration (g)
0
20
40
60
80
Relative Displacement (inches)
Pseudo Acceleration
Target
Relative Displacement
(d)

Figure 5.12: Input Ground Motion 5 in the Transverse Direction

205

0 10 20 304050
Time (sec)
-60
-40
-20
0
20
40
60
Velocity (in/sec)
0 10 20 304050
Time (sec)
-0.5
0.0
0.5
-0.25
0.25
Acceleration (g)
(a)
(b)
(c)
0 10 20 304050
Time (sec)
-40
-30
-20
-10
0
10
20
30
Displacement (inches)
02 4 6 8 10
Period (sec)
0.0
0.5
1.0
1.5
2.0
Spectral Acceleration (g)
0
20
40
60
80
Relative Displacement (inches)
Pseudo Acceleration
Target
Relative Displacement
(d)

Figure 5.13: Input Ground Motion 6 in the Longitudinal Direction

206

0 10 20 304050
Time (sec)
-60
-40
-20
0
20
40
60
Velocity (in/sec)
0 10 20 304050
Time (sec)
-0.5
0.0
0.5
-0.25
0.25
Acceleration (g)
(a)
(b)
(c)
0 10 20 304050
Time (sec)
-30
-20
-10
0
10
20
30
Displacement (inches)
02 4 6 8 10
Period (sec)
0.0
0.5
1.0
1.5
2.0
Spectral Acceleration (g)
0
20
40
60
80
Relative Displacement (inches)
Pseudo Acceleration
Target
Relative Displacement
(d)

Figure 5.14: Input Ground Motion 6 in the transverse Direction

207

0 10 20 304050
Time (sec)
-80
-40
0
40
80
Velocity (in/sec)
0 10 20 304050
Time (sec)
-0.5
0.0
0.5
-0.25
0.25
Acceleration (g)
(a)
(b)
(c)
0 10 20 304050
Time (sec)
-40
-30
-20
-10
0
10
20
30
40
Displacement (inches)
02 4 6 8 10
Period (sec)
0.0
0.5
1.0
1.5
2.0
Spectral Acceleration (g)
0
20
40
60
80
Relative Displacement (inches)
Pseudo Acceleration
Target
Relative Displacement
(d)

Figure 5.15: Input Ground Motion 7 in the Longitudinal Direction

208

0 10 20 304050
Time (sec)
-40
-20
0
20
40
Velocity (in/sec)
0 10 20 304050
Time (sec)
-0.5
0.0
0.5
-0.25
0.25
Acceleration (g)
(a)
(b)
(c)
0 10 20 304050
Time (sec)
-40
-30
-20
-10
0
10
20
30
40
Displacement (inches)
02 4 6 8 10
Period (sec)
0.0
0.5
1.0
1.5
2.0
Spectral Acceleration (g)
0
20
40
60
80
Relative Displacement (inches)
Pseudo Acceleration
Target
Relative Displacement
(d)

Figure 5.16: Input Ground Motion 7 in the Transverse Direction

209
5.4 Model’s Boundary Conditions
Three-dimensional analytical finite element models are developed as a
computational representation of various bridge structures. In order for the analytical
models to accurately predict bridge-structure response, realistic boundary conditions
are essential part of models. For the single-span bridge structures, the dynamic
behavior of the structural system is completely dominated by the boundary
conditions at the beginning and end of the analytical model. These boundary
conditions for a single-span bridge are the nonlinear abutment-backfill and nonlinear
abutment shear keys in the longitudinal and transverse directions respectively. For
the multi-span bridge structures, in addition to boundary conditions at the beginning
and end of the bridge system, nonlinear boundary conditions of the deck-column-
soil-structure-foundation must be modeled accurately. For the dynamic analysis of
bridge systems considered in this chapter, the beginning and ending boundary
conditions (bridge abutments) are modeled as a set of nonlinear springs in both
transverse and longitudinal directions. Pinned connections are used at the base of
the columns. Descriptions of the boundary conditions for various bridge models are
given later in this chapter.
5.4.1 Abutment Soil-Structure Interaction
The longitudinal abutment-backfill-structure interaction model is shown in
Figure 5.17. The dynamic interaction between the deck, abutment backwall in the
longitudinal direction and the abutment shear keys in the transverse direction were
modeled by, gap elements between the bridge deck and the abutment backfill, gap

210
elements between the bridge deck and the abutment shear keys, nonlinear springs in
the longitudinal direction representing the abutment backfill and nonlinear springs in
the transverse direction representing the abutment shear keys. The bridge abutments
are constrained in the vertical direction, while free to move in the horizontal
longitudinal and transverse direction. However, once the relative motion of the
bridge deck exhausts the abutment gap the bridge deck starts pounding on the
abutment backfill.
5.4.2 Abutment Gaps
At each abutment, a one-inch structural expansion gap exists between the
bridge deck and the abutment-backwall in the longitudinal direction as well as
between the bridge deck and the abutment shear key in the transverse direction. Two
methods were used to simulate the expansion gaps, (1) the gap elements in series
with nonlinear backfill link elements, (2) only one nonlinear link element which
includes the abutments expansion gaps as part of the nonlinear abutment-backfill
backbone curve as shown in Figure 5.17 and Figure 5.19. The second nonlinear
abutment model is more efficient and computationally less expensive.
5.4.3 Abutment Backwall-Backfill Longitudinal Response
The nonlinear spring represents the near-field load-deformation behavior at
the longitudinal abutment-embankment soil interface. The hysteretic behavior of the
backbone curve is modeled using the multi-linear plasticity model with the tension

211
side of the curve set to zero. The behavior is essentially that of a gap element in
series with a compressive plastic spring s shown in Figure 5.17.

Deck
Stem
Backfill
Backwall
Footing
Weak Link
Deck
Backfill
Nonlinear Spring
Expansion Gap
Backwall
d
F
Gap
Deck
Stem
Backfill
Backwall
Footing
Weak Link
Deck
Backfill
Nonlinear Spring
Expansion Gap
Backwall
d
F
Gap

Figure 5.17: Bridge Monolithic Abutment Model

Upon load reversal, the spring unloads elastically until zero force is reached,
with net permanent deformation present. Further loading in the tension direction acts
as an open gap, with no force exhibited. Reloading in the compressive direction
remains at zero force until the gap is closed at a deformation equal to the permanent
plastic deformation. The spring loads elastically until the backbone curve is reached,
then follows the backbone with increasing plastic deformation. A series of two
nonlinear longitudinal abutment springs was used at the top and bottom of each
girder.

212
5.4.4 Abutment Shear Key
The abutment shear keys are designed to support the bridge deck in the
transverse direction and to act as a fuse in order to protect the abutment piles failure
during a seismic event. As part of Caltrans seismic research program a large-scale
experiment was conducted at USCD to investigate the seismic behavior of the
abutment exterior shear key. The experimental nonlinear force deformation of the
abutment exterior shear key is in shown in Figure 5.18 (Bozorgzadeh et al., 2005).

Figure 5.18: Abutment Shear Key Experiment (Bozorgzadeh et al., 2005).

In the present study the abutment transverse shear key behavior was
simulated using a multi-linear plasticity model that addresses the structural capacity
of the shear key as well as contribution of passive resistance of the abutment

213
embankment in the transverse direction as a function of relative displacement
between bridge deck and abutment. The model included the 1-inch expansion gap
between shear key and deck, and the limiting passive capacity of the embankment
soil in the transverse direction. The generic nonlinear force-deformation relationship
of the shear key backbone curve based on the USCD experiment used in the present
study is shown in Figure 5.19.

F
d
1
d
2
d
3
d
4
d
res
Gap
d
F
d
1
d
2
d
3
d
4
d
res
d
res
Gap
d

Figure 5.19: Generic Model of the Exterior Shear Key Backbone Curve

The curve has one inch gap, a nonlinear ascending branch from d
1
to d
2
, a
yield plateau from d
2
to d
3
at the combined ultimate force deformation capacity of the
shear key and transverse-abutment-soil capacity, a descending branch from d
3
to d
4
and residual branch d
res.

214
5.4.5 Validation of Abutment Model
Nonlinear link element available in SAP2000 computer program was used to
model the abutment backfill and the shear keys. The UCD abutment test which was
subjected to a series of longitudinal displacement cycles as shown in Figure 5.20.
Figure 5.21 was used to validate the cyclic behavior of the nonlinear link element. The
link element was model using the nonlinear backbone curve predicted using LSH
model. The link element was subjected to the experimental UCD cyclic loading shown
in Figure 5.20. Figure 5.22 shows the simulated load-displacement response of the
abutment backfill using the cyclic displacement history of the test specimen. The
simulated results from SAP2000 agree reasonably well with the measured results for the
displacement controlled longitudinal abutment field experiment. Therefore, link
elements available in SAP2000 are capable of simulating the dynamic behavior of the
bridge abutment subjected to earthquake loading. The longitudinal and transverse
abutment backbone curves including one inch expansion gap used in all the global
models presented in this chapter are shown in Figure 5.23.

215
0 2 4 6 8 10 12 14 16 18 20 22 24 26
Time
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
Lateral Displacement (inches)

Figure 5.20: Displacement Cycles in the UCD Abutment Test (Romstad et al.,
1995)

Figure 5.21: Measured Load-Deformation in the UCD Abutment Test (Romstad
et al., 1995)

216
-4.0 -2.0 0.0 2.0 4.0 6.0 8.0
Displacement (inches)
0
100
200
300
400
Force (Kips)
UCD Experiment
Sap2000 Cyclic
LSH BBC

Figure 5.22: Simulated Longitudinal Response of the UCD Abutment Test

0.0 2.0 4.0 6.0
Displacement (inches)
0
50
100
150
200
250
Force (Kips)
Gap
Backfill
Shear Key

Figure 5.23: Longitudinal and Transverse Backbone Curves Used in the Bridge
Global Models

217
5.4.6 Bridge Deck Model
Since the bridge decks are extremely stiff and strong in compression with
supporting columns and abutments, the bridge response during a major seismic event
is primarily dominated by the inelastic deformation of the columns and bridge
abutment backfill. The deck will remain elastic and, therefore, can be modeled by
linear elastic elements. Both shell elements and frame elements with superstructure
properties can be used to model the bridge deck. Shell and frame elements have their
advantages and disadvantages. Since the global seismic behavior of skewed bridges
is affected not only by skew angle but also by the deck width, the deck flexibility
and the deck mass distribution, shell elements are more appropriate to represent the
bridge deck. However, the shell elements are computationally expensive. In this
respect, first the bridge deck is modeled using full three-dimensional shell elements
(shell model) to account for the realistic properties of the bridge deck. On the other
hand the bridge deck modeled with beam elements (spline model) can represent
effective stiffness characterization very well but it is not a good representation of
bridge width and the mass distribution of the bridge deck. Therefore, first the models
were developed using shell elements to represent the bridge deck model. Figure 5.24
shows the 3D view for the shell and spline models of a single-span bridge structure.
The seismic response of shell and spline models was found to be comparable. The
simpler spline models were much less computationally intensive than the shell
models.

218

(a) Bridge Geometry

(b) Shell Model

(c) Spline Model

Figure 5.24: Typical Bridge Model

219
5.4.7 Bridge Columns
Per Caltrans SDC, moment-curvature of the column cross sections for a range
of strain values were calculated to develop an idealized moment-curvature
relationship and plastic moment capacity of the column as shown in Figure 5.25.

Figure 5.25: Column Moment-Curvature Relationship

The calculated plastic moment was used at the top of the column to simulate the
plastic hinge action during a seismic event. Pinned connections were used at the base
of the column. SAP2000 was used to develop the moment-curvature relationship of
the column cross-section.

220
5.5 Bridge Models
In the present research, the longitudinal and transverse response of
continuous reinforced concrete box girder bridge structures of 1, 2 and 3 spans as
shown in Figure 5.26 to Figure 5.30 are investigated. All bridge models are
subjected to the same seismic ground motions with high velocity pulses. The bridge
decks were supported by bearing pads on seat-type abutments and rigidly connected
to reinforced concrete columns. The following classes of bridge structures are
investigated.
1. Bridge No. 1 - A typical single-span bridge,
2. Bridge No. 2 - A two-span with a single-column center bent,
3. Bridge No. 3 - A two-span bridge with a dual-column at the center bent,
4. Bridge No. 4 - A three-span bridge with single-column bents, and
5. Bridge No. 5 - A three-span bridge on two bents supported by two
columns per bent based.
In order to determine a meaning full seismic analysis of the above bridge
configurations, typical existing bridge structures with well defined traffic lanes,
shoulders as well as span length were selected. These bridges were based on actual
bridge plans from a single-span Meloland Road Bridge in Imperial County, a two-
span LaVeta Avenue Bridge and a three-span Redhill Avenue Bridge both located in
Orange County. Three-dimensional finite-element models of various bridge
structures were developed using SAP2000 (2006). Descriptions of bridge
components used in the three-dimensional models are given bellow.

221

102’ 102’

Figure 5.26: Single-Span Bridge

222

102’ 102’ 102’ 102’

Figure 5.27: Single-Column Two-Span Bridge

223
148’ 256’ 138’ 148’ 256’ 138’

Figure 5.28: Single-Column Three-Span Bridge

224
155’ 145’ 155’ 145’

Figure 5.29: Tow-Columns Two-Spans Bridge

225
148’ 256’ 138’ 148’ 256’ 138’

Figure 5.30: Tow-Columns Three-Spans Bent Bridge

226
5.6 Analytical Study
The investigation is limited to single-span, two-span and three-span ordinary
bridge structures. The following are some of the criteria used in selection of bridges
for this study:
Bridge alignment has no horizontal or vertical curves
• Original models started with no skew angle
• Bridge models with 25, 45 and 60 degree skew angle were developed
from the original models
• The bridge is constructed very recently with modern design and
construction techniques
Three-dimensional finite-element models of various bridge structures were
developed using both shell and frame elements with appropriate superstructure
properties. Frame elements with moment curvature and cracked sectional properties
were used to model bridge columns. Nonlinear springs were used to model the
bridge abutments. A total of five typical bridge configurations were modeled. The
bridge structures selected for this study are described in the following sections.
5.6.1 Bridges Types
The investigation was limited to single-span, two-span and three-span
ordinary standard bridges. The bridge alignments have no horizontal or vertical
curves, but the skew angles vary from 0 to 60 degrees. Full 3D nonlinear finite
element model of the soil-abutment-structure interaction (SASI) for all bridge
structures were developed. The purpose of the first three analytical models is to

227
study the impact of the skew angles on the dynamic behavior of ordinary bridge
structures. The Meloland Road Overpass, located near El Centro in southern
California was selected for the first three analytical models.
The Meloland Road Overpass is a very simple non-skewed two-span cast-in-
place reinforced concrete box girder bridge structure with monolithic abutments. The
bridge was strongly shaken by the October 1979 Imperial Valley Earthquake of
magnitude 6.4. However, even bridge was shaken by such strong earthquake
virtually no damage was observed in the bridge. Since the purpose of this chapter is
to investigate the nonlinear SASI, it was assumed that all bridge structures have set-
type abutment for the analytical models.
5.6.2 Single-Span Bridge
The longitudinal and transverse sections of the model are shown in Figure
5.31. It is assumed that the deck is supported at the two-ends on abutment bearing
pads resting on the set-type abutment with one-inch expansion gap. The length and
width of the bridge is assumed approximately 102 feet long and 34 feet wide
respectively. The depth of the superstructure is 5.5 feet. The bridge model is
constrained at each abutment in the vertical direction but free to move in the
horizontal direction. In the analyses, the ground motion is prescribed at the end of
the nonlinear abutment springs. Since there is no bridge column, during a seismic
event upon the abutments gap closure the bridge deck will impose time varying
pounding forces and displacement directly to the bridge abutment in the transverse

228
and longitudinal directions. Three-dimensional shell and spline finite element
models of the bridge are shown in Figure 5.24.
5.6.3 Two-Span Bridge
Only the deck cross section and the column cross section of this model was
developed based on the Meloland Road Overpass. However, for the analytical model
a two-span cast-in-place reinforced concrete box girder bridge with a continuous
deck which is supported by bearing pads on the seat-type abutments with one-inch
expansion gap was considered. The bridge deck is rigidly connected to a single
reinforced concrete column at the bent cap as shown in Figure 5.27.
The geometry of the model was exactly the same as that of Meloland Road
Overpass. The model bridge is approximately 204 feet long and 34 feet wide. The
depth of the superstructure is about 5.5 feet. The bridge column is 35 feet tall and 5
feet in diameter as shown in Figure 5.32. The bridge deck is constrained in all
degrees of freedom at the bent cap but it is constrained at each abutment only in the
vertical. In the analyses, the ground motion is prescribed at the bottom of the column
and the end of the nonlinear abutment springs. First the three-dimensional finite-
element models of bridge deck was developed using shell elements. The shell model
was converted to the spline model.
The frame element with moment curvature and cracked sectional properties
was used to model bridge column. Plastic hinge was used at the top and pinned
connections at the base of the column. The plastic hinge was calculated using an
idealized bilinear moment-curvature relationship taking bridge axial load as well as

229
confinement effect into account. Cross section and idealized moment-curvature of
the column is shown in Figure 5.33.
5.6.4 Three-Span Bridge with a Single-Column Bent
The third bridge considered in the analysis is a continuous three-span cast-in-
place reinforced concrete box girder bridge which is supported by bearing pads on
the seat-type abutments with one-inch expansion gap and rigidly connected to a
single reinforced concrete column at the bent caps. The bridge has a total length of
642 feet as shown in Figure 5.28. The cross section of the deck, the cross section of
the columns and the height of the columns are identical to the two-span bridge
structure shown in Figure 5.32. The analytical models of the bridge are shown in
Figure 5.34.
5.6.5 Two-Span Bridge with Two-Column Bent
The analytical model was developed based on the As-Built plan for the
LaVeta Avenue Overcrossing bridge structure. The existing bridge structure is
approximately 286 feet long and 75.5 feet wide. The deck is a 6.25 feet thick box
girder supported on two-column bents, each 25.5 feet high. The span lengths are 155
and 145 feet, respectively. The bridge deck is continuous and is rigidly connected to
the reinforced concrete columns at the bent cap and is supported by bearing pads on
the seat-type abutments with one-inch expansion gap. There is pin connection at the
bottom of the flared columns to the pile cap. The longitudinal and transverse

230
sections of the model are shown in Figure 5.35. Cross section and idealized moment-
curvature of the column is shown in Figure 5.36.
5.6.6 Three-Span Bridge with Tow-Column Bents
The analytical model is based on Redhill Avenue Over Crossing. The bridge
site is located in the southern end of the Los Angeles physiographic basin, at the San
Diego freeway (I-405) adjacent to the John Wayne Airport. The Red Hill Avenue
Over Crossing is a three-span haunch cast-in-place prestressed reinforced concreted
box girder bridge supported on dual-column bents and seat type abutments with a 25
degree skewed angle. The average height of the columns is approximately 40 feet.
The bridge is approximately 642 feet long and 72 feet wide with spans measuring
148, 256, and 138 feet. The depth of the deck varies from minimum of 6.5 feet (at
each abutment and at the midpoint of each span) to a maximum of 13 feet (at the
bent caps). There is pin connection at the bottom of the flared columns to the pile
cap. The reinforced concrete columns with interlocking spiral reinforcements were
used to construct the bridge columns. The longitudinal and transverse sections of the
model are shown in Figure 5.37. The analytical shell and spline model are shown in
Figure 5.38. The interlocking spirals provide confinement to enhance ductility of the
columns. The idealized moment-curvature of the columns in the longitudinal and
transverse direction of the bridge alignment are shown in Figure 5.39.

231
(a) Section S-S
(b) Section T-T
Figure 5.31: Transverse and Longitudinal Sections of the Single-Span Bridge

232

(a) 3D View

(b) Section s-s

Figure 5.32: Transverse Section of the Two-Span Single-Column Bridge

233

(a) 3D Shell Model

(b) 3D Spline Model

0.000 0.002 0.004 0.006 0.008
Curveture ( φ)
0
4000
8000
12000
Moment (k-ft)
M- φ
Idealized-M- φ

Figure 5.33: Analytical Models of the Two-Span Single-Column Bridge

234

(a) 3D View

(b) Shell Model

(c) Spline Model

Figure 5.34: Analytical Models of the Three-Span Single-Column Bridge

235

(a) 3D View

(c) Section S-S

Figure 5.35 Transverse Section of the Two-Span Two-Column Bridge

236
(a) Shell Model
(b) Spline Model

0.000 0.002 0.004 0.006 0.008 0.010
Curveture ( φ)
0
4000
8000
12000
16000
20000
Moment (k-ft)
M- φ
Idealized-M- φ

Figure 5.36: Analytical Models of the Two-Span Two-Column Bridge

237

(a) 3D View

(b) Section S-S

Figure 5.37: T Transverse Section of the Three-Span Two-Column Bridge

238

(a) 3-D Shell Model

(b) 3-D Spline Model

Figure 5.38: Analytical Models of the Three -Span Two-Column Bridge

239
0.000 0.002 0.004 0.006 0.008
Curveture ( φ)
0
10000
20000
30000
40000
50000
60000
70000
Moment (k-ft)
M- φ
Idealized-M- φ

(a) Longitudinal

0.000 0.001 0.002 0.003 0.004 0.005
Curveture ( φ)
0
20000
40000
60000
80000
100000
Moment (k-ft)
M- φ
Idealized-M- φ

(b) Transverse

Figure 5.39: Columns Longitudinal and Transverse Moment Curvatures

240
5.7 Longitudinal and Transverse Abutment Response
In order to understand the impact of the ground motions on the bridge
abutment the soil-abutment interaction of a single span bridge structure is first
investigated without the complexity of the bridge columns.
5.7.1 Effects of Skew Angles on the Bridge Abutment
As indicated earlier, the ground motions with an asymmetrical and high
amplitude velocity pulse characteristics have the tendency of producing a biased,
one-sided response of the bridge structures. Asymmetrical impulsive loading
generates large displacements in one direction leading to a significant rotation and
residual displacement on the bridges with skew-abutment. To evaluate the
interaction behavior of a skewed bridge deck with a skewed abutment, a global
three-dimensional nonlinear dynamic model of a concrete box-girder bridge shown
in Figure 5.40 was performed using a zero skew angle and a 45 degree skew angle.
The interaction between abutments and backfill was modeled by two rows of
four nonlinear soil springs denoted A, B, C and D (Figure 5.40) at each abutment,
oriented normal to the backwall. Each spring was modeled by a nonlinear plasticity
link element using the coordinate of the backbone curve shown in Figure 5.23. The
model was excited by the two-horizontal-component earthquake motion with high-
velocity pulses from the Renaldi record of the 1994 Northridge earthquake.

241

(a) Zero Skew Angle

(b) 45 Degree Skew Angle

Figure 5.40: Single Span Bridge With 0
o
and 45
o
Skew Angles

Two components of the recorded motion at the Renaldi station are shown in
Figure 5.41 and Figure 5.42. The southwest direction motion has peak velocity pulse
amplitude of 62 in/s in one direction and peak velocity pulse amplitude of 31 inch/s
in the other direction. The northwest direction motion has peak velocity pulse
amplitude of 25 inch/s in one direction and peak velocity pulse amplitude of 22
inch/s in the other direction. The recorded motion in the southwest direction has

242
biased (residual) acceleration, velocity, and displacement in one direction. Therefore,
both components must be considered when conducting shake table experiments or
performing nonlinear analytical models.
Figure 5.43 shows the abutment-backfill response due to deformation and
compressive pounding force time histories of the bridge deck along the abutment-
backwall. Despite the presence of the biased velocity pulse in the longitudinal
direction of the model, the normal passive forces are distributed uniformly along the
width of the abutment backwall. The abutment backfill provided resistance during
the entire shaking without any significant bridge rotation. The first impact between
the abutment-backfill and the bridge deck took place at about 2 seconds from the
beginning of the excitation and continued pounding on the abutment-backfill up to
about 11 seconds of the earthquake duration. The results of the analysis indicate that
the bridge deck stopped pounding on springs A, B and C after about 9 second of the
earthquake. However, the acute corner of the bridge deck continued pounding spring
D up to 11 seconds of the shaking. This is an indication of slight counterclock wise
rotation of the bridge deck. The residual displacement of the abutment-backfill is
about 2.5 inches including the 1-inch expansion gap (net displacement about 1.5
inches).
Variation of normal passive forces and abutment deformation during the
shaking of the single-span bridge with a 45 degree skew angle are shown in Figure
5.44. The first impact between the abutment-backfill and the bridge deck took place
at about 2.17 seconds from the beginning of the excitation and continued pounding

243
on the abutment-backfill only up to about 2.6 seconds of the earthquake duration.
The dynamic response of the abutment-backfill indicates that, the superstructure
underwent significant clockwise rotations about the vertical axis and was
permanently displaced from its original position by approximately 20 inches of the
end of ground shaking. Figure 5.44 also illustrates the variation of abutment impact
forces as a function of time across the abutment backwall. The passive forces were
developed in each of the four normal springs from acute corner (NW) to obtuse (SW)
corner at the abutment during initial shaking of the earthquake. It is interesting to
note that the normal passive forces are distributed non-uniformly along the width of
the abutment walls due to bridge deck rotation. It can also be observed that this non-
uniform loading of the abutment backwalls resulted in a smaller magnitude of total
soil resisting force in the acute corners of the deck as compared to non-skewed
abutments. As the bridge deck is pushed into the abutment during ground shaking,
the abutment backwall generates asymmetric passive soil resistances that cause the
ends of the deck to “bounce” off the abutment seat in the bridge transverse direction,
resulting in deck rotation. The width and capacity of this passive wedge depends on
abutment (embankment) width, skew angle and ground motion characteristics.
Therefore, for the bridge structures with high skew angles ground motions with high
velocity pulse plays significant which can not be capture using response spectra
analysis.

244
0 5 10 15 20
Time (sec)
-60
-40
-20
0
20
40
60
Velocity (in/sec)
0 5 10 15 20
Time (sec)
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
Acceleration (g)
(a)
(b)
(c)
0 5 10 15 20
Time(sec)
-10
-5
0
5
10
Displacement (inches)

Figure 5.41: Recorded Renaldi Longitudinal Motion

245
0 5 10 15 20
Time (sec)
-60
-40
-20
0
20
40
60
Velocity (in/sec)
0 5 10 15 20
Time (sec)
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
Acceleration (g)
(a)
(b)
(c)
0 5 10 15 20
Time (sec)
-10
-5
0
5
10
Displacement (inches)

Figure 5.42: Recorded Renaldi Transverse Motion

246
0 2 4 6 8 10 12 14 16 18 20
Time (sec)
-150
-100
-50
0
50
100
150
Force (Kips)
A_U1
B_U1
C_U1
D_U1
B_F1
C_F1
D_F1
A_F1
-10
-5
0
5
10
Displacement (inches)

(a)

2 468 10 12
Time (sec)
-150
-100
-50
0
50
100
150
Force (Kips)
A
B
C
D

(b)

Figure 5.43: Variation of Normal Abutment Impact Forces For a Single Span-
Bridge With 0
o
Skew Angle

247

0 2468 10 12 14 16 18 20
Time (sec)
-100
-75
-50
-25
0
25
50
75
100
Force (Kips)
A_U1
B_U1
C_U1
D_U1
B_F1
C_F1
D_F1
A_F1
-60
-45
-30
-15
0
15
30
45
60
Displacement (inches)
24
Time (sec)
-100
-75
-50
-25
0
25
50
75
100
Force (Kips)
-60
-45
-30
-15
0
15
30
45
60
Displacement (inches)

Figure 5.44: Variation of Normal Abutment Impact Forces For a Single Span-
Bridge With 45
o
Skew Angle During the First 4 Seconds of Shaking

248
2.0 2.2 2.4 2.6
Time (sec)
0
20
40
60
80
100
Force (Kips)
A
B
C
D

Figure 5.45: Variation of Normal Abutment Impact Forces For a Single Span-Bridge
With 45
o
Skew Angle Between 2 Seconds to 2.6 Seconds of Shaking

5.8 Response Due to Spectra-Compatible Time History Motions
The global response and the abutment response of various bridge
configurations using seven sets of the Response-Spectra-Compatible Time History
Ground Motions with high velocity pulses are discussed in this section.
5.8.1 Abutment Response
Figure 5.46 shows the result of the abutment hysteretic behavior for a single-
span bridge structure subjected to motion number 3 and number 5 motions. It can
observe that the ultimate abutment passive force developed in the abutment-backfill
is almost the same for both input ground motions. However, the loop of the

249
abutment force-displacement curve and loading and unloading features of the
hysteretic response differ from one input ground motion to another, which reflects
the influenced of different dynamic characteristics of input ground motions on the
bridge abutment.
Figure 5.47 shows the hysteretic behavior of the abutment backfill at the
obtuse corner (south west) and acute corner (south east) of the 45 degree skewed-
bridge abutment due to input motion number 3 and number 7. From Figure 5.47 it
can be observed that in addition to the characteristics of the ground motion the
abutment-backfill hysteretic force-displacement response differs for the skewed
abutments due to clockwise rotation versus non-skewed abutment. As the result of
bridge rotation, the obtuse corners of the bridge deck have been pushed into the
abutment backfill and the passive force is fully mobilized, while the acute corners
have moved away from the abutment backfill and the passive wedge has been
mobilized only partially. More discussion of the abutment-bridge interaction will be
presented later on in this chapter.
Figure 5.48 shows example of the hysteretic abutment shear keys force-
displacement response at the obtuse corner and acute corner of the 45 degree
skewed-bridge abutment due to input motion number 3. The shear keys at the acute
corners are observed to approach their ultimate soil-structural capacities. This result
suggests that these keys at the acute corners of the bridge deck failed and the deck
become unseated due to deck rotation and lateral movement.

250

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0
Displacement (inches)
0
40
80
120
160
Force (Kips)
Motion #3
West Abut.
East Abut.

(a)

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0
Displacement (inches)
0
40
80
120
160
Force (Kips)
Motion #7
West Abut.
East Abut.

(b)
Figure 5.46: Longitudinal Hysteretic Behavior of the Single-Span Bridge
Abutment With 0
o
Skew Angle

251
5.8.2 Rotation Due to Deck Flexibility
Global seismic behavior of skewed bridges is dominated not only by the
abutment springs, number of spans and skew angles but also by the deck flexibility.
The more the bridge deck is flexible the lees rotation occurs during a seismic event.
In order to investigate the impact of the deck flexibility, a 102-feet long single-span
bridge and a 204-feet single-span bridge with a 45 degree skew angle were exited
using seven sets of response-spectra-compatible time history input motions. Figure
5.49 shows the average rotational response of both bridges subjected to seven sets of
response-spectra-compatible time history ground motion. Since the 204-feet long
bridge is more flexible it has experienced less rotation than the 102-feet long bridge.
Bridge rotation will be discussed in more detail later in this chapter.
5.8.3 Rotation Due to Column Rigidity
In order to investigate the impact of the bridge column a two-span bridge
(204-feet long) and a single-span bridge (102-feet long) were excited using seven
sets of time history input motions. The average rotational response of both bridges
subjected to seven sets of response-spectra-compatible time history ground motion is
shown in Figure 5.50. Even though both bridges have the same span length, the
presence of the column causes the rotational response of the two-span bridge to drop
more than half.

252
-8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0
Displacement (inches)
0
40
80
120
160
Force (Kips)
Motion #3
West Abut.
East Abut.

(a)

-8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0
Displacement (inches)
0
40
80
120
160
Force (Kips)
Motion #7
West Abut.
East Abut.

(b)
Figure 5.47: Longitudinal Hysteretic Behavior of the Single-Span Bridge
Abutment With 45
o
Skew Angle

253
0.0 2.0 4.0 6.0 8.0 10.0
Displacement (inches)
0
40
80
120
160
200
Force (Kips)
Motion #3
Acute Shear Key
Backbone

(a)

0.0 2.0 4.0 6.0 8.0 10.0
Displacement (inches)
0
40
80
120
160
200
Force (Kips)
Motion #3
Obtuse Shear Key
Backbone

(b)
Figure 5.48: Transverse Hysteretic Behavior of the Single-Span Bridge Abutment
With 45
o
Skew Angle

254

0.06
0.04
0.02
0
-0.02
Rotatio (rad)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
Average
Span Len = 204 ft
Span Len = 102 ft

Figure 5.49: Average Rotation Due to Deck Flexibility of the Single-Span Bridge With 45
o
Skew Angle

255

0.06
0.04
0.02
0
Rotatio (rad)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
Average
2-Spans
1-Span

Figure 5.50: Average Rotation Due to Presence of the Number of Spans With 45
o
Skew Angle

256
5.9 Analytical Results of the Single-Column Two-Span Bridge
In order to investigate a realistic global behavior of skew bridges, first the
global response of a simple two-span bridge will be discussed. Figure 5.51 through
Figure 5.57 show examples of the transverse displacement time history response at
the acute corners (northwest and southeast) of the bridge deck for all seven input
motions used in the analysis. The abutment interaction in the longitudinal direction
was modeled as a set of 4 nonlinear springs shown in Figure 5.23. The bridge
column was modeled using a frame element with a bi-linear moment curvature
shown in Figure 5.33. The response time histories demonstrate that the bridge decks
experienced significant amounts of transverse displacement and rotation about the
vertical axis during seismic ground shaking when the abutments are skewed, whereas
the non-skewed bridge showed little or no rotation. Since the two span lengths are
the same for all models, the center of mass and the center of rigidity of the bridge
system coincides at the top of the bent.
The transverse displacement results of the analyses show that the bridge
dynamic behavior is dependent on the characteristics of the input ground motions.
The permanent residual displacements of the acute deck corners at end of shaking.
The magnitude of permanent residual displacements (deck rotation) varies among all
seven input motions and all three skew angles.

257
-75
-50
-25
0
25
50
75
Displacement (inches)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
60
o
Skew, NW
45
o
Skew, NW
25
o
Skew, NW
0
o
Skew, NW
60
o
Skew, SE
45
o
Skew, SE
25
o
Skew, SE
0
o
Skew, SE

Figure 5.51: Transverse Displacement Due Motion 1 For a Two-Span Single-Column Bridge

258
-75
-50
-25
0
25
50
75
Displacement (inches)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
60
o
Skew, NW
45
o
Skew, NW
25
o
Skew, NW
0
o
Skew, NW
60
o
Skew, SE
45
o
Skew, SE
25
o
Skew, SE
0
o
Skew, SE

Figure 5.52 : Transverse Displacement Due Motion 2 For a Two-Span Single-Column Bridge

259
-75
-50
-25
0
25
50
75
Displacement (inches)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
60
o
Skew, NW
45
o
Skew, NW
25
o
Skew, NW
0
o
Skew, NW
60
o
Skew, SE
45
o
Skew, SE
25
o
Skew, SE
0
o
Skew, SE

Figure 5.53: Transverse Displacement Due Motion 3 For a Two-Span Single-Column Bridge

260
-75
-50
-25
0
25
50
75
Displacement (inches)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
60
o
Skew, NW
45
o
Skew, NW
25
o
Skew, NW
0
o
Skew, NW
60
o
Skew, SE
45
o
Skew, SE
25
o
Skew, SE
0
o
Skew, SE

Figure 5.54: Transverse Displacement Due Motion 4 For a Two-Span Single-Column Bridge

261
-75
-50
-25
0
25
50
75
Displacement (inches)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
60
o
Skew, NW
45
o
Skew, NW
25
o
Skew, NW
0
o
Skew, NW
60
o
Skew, SE
45
o
Skew, SE
25
o
Skew, SE
0
o
Skew, SE

Figure 5.55: Transverse Displacement Due Motion 5 For a Two-Span Single-Column Bridge

262
-75
-50
-25
0
25
50
75
Displacement (inches)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
60
o
Skew, NW
45
o
Skew, NW
25
o
Skew, NW
0
o
Skew, NW
60
o
Skew, SE
45
o
Skew, SE
25
o
Skew, SE
0
o
Skew, SE

Figure 5.56: Transverse Displacement Due Motion 6 For a Two-Span Single-Column Bridge

263
-75
-50
-25
0
25
50
75
Displacement (inches)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
60
o
Skew, NW
45
o
Skew, NW
25
o
Skew, NW
0
o
Skew, NW
60
o
Skew, SE
45
o
Skew, SE
25
o
Skew, SE
0
o
Skew, SE

Figure 5.57: Transverse Displacement Due Motion 7 For a Two-Span Single-Column Bridge

264
5.9.1 Abutment Behavior in the Longitudinal Direction
Figure 5.58 through Figure 5.60 show the hysteretic behavior of the abutment
backfill at the acute and obtuse corners west abutment (for 0, 25 and 60-degree
skew) due to set number 3 input motion. Figure 5.58 shows the backbone curve and
the hysterical behavior of non-skewed abutment –backfill. It can be observed that the
backfill has been fully engaged during the seismic loading and the ultimate abutment
force has been fully mobilized and is uniformly distributed along the backwall.

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0
Displacement (inches)
0
40
80
120
160
Force (Kips)
NW Corner
SW Corner
Backbone Curve

Figure 5.58: Hysteretic Abutment Backfill Response For a Two-Span Single-
Column Bridge with 0
o
Skew Angle

265

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0
Displacement (inches)
0
40
80
120
160
Force (Kips)
NW Corner
SW Corner

Figure 5.59: Hysteretic Abutment Backfill Response For a Two-Span Single-
Column Bridge with 25
o
Skew Angle

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0
Displacement (inches)
0
40
80
120
Force (Kips)
NW Corner
SW Corner

Figure 5.60 Hysteretic Abutment Backfill Response For a Two-Span Single-
Column Bridge with 60
o
Skew Angle

266
However, Figure 5.59 and Figure 5.60 demonstrate that the abutment
hysteretic force-displacement response differ for the skewed abutments due to
clockwise rotation. As the result of bridge rotation similar to the single-span bridge
structures, the obtuse corners of the bridge deck have been pushed into the abutment
backfill and the passive force is fully mobilized, while the acute corners have moved
away from the abutment backfill and the passive wedge has been mobilized only
partially. It was observed that as the skew angle increased the abutment participation
becomes less at both the acute and obtuse corners. This behavior reflects the
influence of skew angle on abutment participation during a major seismic event.
The results of the analytical model revealed the important role of ground
motion characteristics on the bridge abutment hysteretic behavior. The most unique
aspect of the calculated response was the presence of high residual rotation of the
bridge deck. Figure 5.59 and Figure 5.60 show that the combination of skew angles
and the presence of one-sided high velocity pulse in the input ground motion caused
the hysteretic response of the abutment to be biased in one direction similar to the
shake table text conducted at University of Neveda Reno (Phan et at., 2005) as
shown Figure 5.59 and Figure 5.60.
5.9.2 Abutment Behavior in the Transverse Direction
Reconnaissance reports after earthquakes such as the 1994 Northridge
Earthquake indicated many bridge abutment shear keys failure as a result of deck
rotations and lateral movements. Figure 5.61 shows an example of the shear key
failure due to deck rotation at the east side of abutment 1 during the 1994 Northridge

267
earthquake. The shear key on the west side of the abutment did not experience any
significant damage. An analytical example of the hysteretic behavior of the
transverse abutment shear keys of bridge model with 45 degree skew angle is shown
in Figure 5.62 and Figure 5.63 for the acute and obtuse corners deck. The abutment
shear keys were modeled using a nonlinear plasticity link element. Figure 5.23
shows the expansion gap and the nonlinear shear key backbone curve used in the
model. As shown in Figure 5.62 the shear keys located on the acute corners are
observed to approach their ultimate structural capacities during the shaking. This
result suggests that these keys could potentially fail when subjected to stronger
shaking, which could allow the bridge to rotate and cause the deck to become
unseated from the abutment due to lateral movement. The first impact between the
abutment-shear keys and the bridge deck took place after the gap closure. Hysteretic
behavior of the shear keys indicates that there is less damage at the obtuse corners
than the acute corners due to deck rotation. As shown in Figure 5.63 the
deformations of the shear keys on the obtuse corners are in the order of couple of
inches including the 1 inch expansion gap opening while the deformations on the
acute corners are in excess of several inches.

268
Left Corner
Shear Key Failure
Right Corner
No Shear Key Failure
Left Corner
Shear Key Failure
Right Corner
No Shear Key Failure

Figure 5.61: Shear Key Failure Due to Deck Rotation 1994 Northridge
Earthquake

269
0.0 2.0 4.0 6.0 8.0 10.0
Displacement (inches)
0
40
80
120
160
200
Force (Kips)
Motion #3
Acute Shear Key
Backbone

(a)

0.0 2.0 4.0 6.0 8.0 10.0
Displacement (inches)
0
40
80
120
160
200
Force (Kips)
Motion #3
Acute Shear Key
Backbone

(b)
Figure 5.62: Hysteretic Abutment Shear Key Response at the Acute Corners
For a Two-Span Single-Column Bridge with 0
o
Skew Angle

270
0.0 2.0 4.0 6.0 8.0 10.0
Displacement (inches)
0
40
80
120
160
200
Force (Kips)
Motion #3
Obtuse (SW) Shear Key
Backbone

(a)

0.0 2.0 4.0 6.0 8.0 10.0
Displacement (inches)
0
40
80
120
160
200
Force (Kips)
Motion #3
Obtuse (NE) Shear Key
Backbone

(b)
Figure 5.63 Hysteretic Abutment Shear Key Response at the Obtuse Corners
For a Two-Span Single-Column Bridge with 45
o
Skew Angle

271
5.10 Results of the Analysis for All Bridge Models
In the previous sections various components and global behavior of a single-
span and a two-span-single-column bent was investigated the mechanism of the
problem. The purpose of the flowing sections is to investigate global response for all
five bridge configurations due to rotation of various skew angles using seven sets of
ground motions.
5.10.1 Rotational Response
Figure 5.64 to Figure 5.81 show that the deck rotation for the 25, 45 and 60-
degree skew built up during the initial peak cycles of shaking of all seven ground
motions. The time histories for the non-skewed configurations are not shown since
the rotations were insignificant. The results of analysis indicate that once this large
rotation had occurred, the deck did not return to its original position for all three
skewed configurations. In contrast, the non-skewed bridge deck experienced neither
significant rotation nor permanent transverse displacement despite the large velocity
pulses in the input motions. From these analyses, the following observations are
made for all bridge structures in this chapter:
• The bridge dynamic behavior is dependent on the characteristics of the input
ground motions. The magnitude of permanent rotation varies among all seven
input motions, particularly for Bridges No. 1 to 3, and all bridge skew
configurations (Figure 5.64 to Figure 5.72).

272
• The bridges experienced significant amounts of rotation about the vertical
axis during seismic ground shaking when the abutments are skewed, whereas
the non-skewed bridge showed little or no rotation.
• The decks experienced significant amount of rotation which builds up during
initial peak cycles shortly after the velocity pulses occurred (about 3 to 13
seconds). None of the decks returned to their original position. Bridge No. 1
(single-span without column) and No. 2 (with one single-column bent)
experienced the largest magnitudes of rotation.
• The bridge decks then rotated back in a reverse direction by a small amount.
The amount of this rotation was largest for Bridges No. 1 and 2. Bridges No.
3 to 5, which consist of more rigid bridge structure systems, showed little or
no significant reverse rotation.
• Subsequent deck rotations were small for Bridges No. 1 and 2. The response
is observed to undergo more and more oscillation for the more rigid Bridges
No. 3 to 5 (with multiple-column bents and more spans).

Figure 5.82 and Figure 5.85 show the three-dimensional histogram depicting
maximum and the average residual deck rotations developed from the seven
response time histories motions for a single-span, two-span-single-column bent and
three-span-single-column bent bridge models (Bridge No. 1, Bridge No. 2 and
Bridge No. 3) for three skew configurations. It is also observed that there is a clear
trend between the magnitude of deck rotation and skew angle among all seven input

273
motions. As the number of spans increases the deck rotation decreases due to
presence of the bridge columns.
Figure 5.84 is a three-dimensional histogram depicting maximum deck
rotations due to the seven input ground motions for a two-span-single-column bent
and two-span-two-column bent bridge models (Bridge No. 2, and Bridge No. 4) for
three skew configurations. The results of the analysis indicate that as the number of
column per bent increase the deck rotation decreased due to column rigidity.
Figure 5.85 shows the results of the maximum rotation for a three-span-
single-column bent and three-span-two-column bent bridge models (Bridge No. 3,
and Bridge No. 5). The results of the analysis indicate that as the number of column
per bent increase the deck rotation decreased due to column rigidity. However, the
magnitude of the deck rotations for all three skew angles is similar (25, 45, and 60
degree skew). As a mater of fact the magnitude of the deck rotation for a 45 degree
skew angle is slightly higher than the deck rotation for the 60 degree skew angle.

274
0.08
0.06
0.04
0.02
0
-0.02
Rotatio (rad)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
Motions
1
2
3
4
5
6
7
Average

Figure 5.64 :Rotation Time History of Single Span Bridge With 25
o
Skew Angle (Length =102’)

275

0.12
0.08
0.04
0
-0.04
Rotatio (rad)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
Motions
1
2
3
4
5
6
7
Average

Figure 5.65: Rotation Time History of Single Span Bridge With 45
o
Skew Angle (Length =102’)

276

0.12
0.08
0.04
0
-0.04
Rotatio (rad)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
Motions
1
2
3
4
5
6
7
Average

Figure 5.66: Rotation Time History of Single Span Bridge With 60
o
Skew Angle (Length =102’)

277
0.05
0.04
0.03
0.02
0.01
0
-0.01
Rotatio (rad)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
Motions
1
2
3
4
5
6
7
Average

Figure 5.67: Rotation Time History of Single Span Bridge With 25
o
Skew Angle (Length =204’)

278
0.05
0.04
0.03
0.02
0.01
0
-0.01
Rotatio (rad)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
Motions
1
2
3
4
5
6
7
Average

Figure 5.68: Rotation Time History of Single Span Bridge With 45
o
Skew Angle (Length =204’)

279

0.05
0.04
0.03
0.02
0.01
0
-0.01
Rotatio (rad)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
Motions
1
2
3
4
5
6
7
Average

Figure 5.69: Rotation Time History of Single Span Bridge With 60
o
Skew Angle (Length =204’)

280

0.03
0.02
0.01
0
-0.01
Rotatio (rad)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
Motions
1
2
3
4
5
6
7
Average

Figure 5.70: Rotation Time History of Two- Span Single-column Bridge With 25
o
Skew Angle

281

0.05
0.04
0.03
0.02
0.01
0
-0.01
Rotatio (rad)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
Motions
1
2
3
4
5
6
7
Average

Figure 5.71: Rotation Time History of Two- Span Single-column Bridge With 45
o
Skew Angle

282
0.05
0.04
0.03
0.02
0.01
0
-0.01
Rotatio (rad)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
Motions
1
2
3
4
5
6
7
Average

Figure 5.72 : Rotation Time History of Two- Span Single-column Bridge With 60
o
Skew Angle

283
0.01
0.008
0.006
0.004
0.002
0
-0.002
Rotatio (rad)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
Motions
1
2
3
4
5
6
7
Average

Figure 5.73: Rotation Time History of Three- Span Single-column Bridge With 25
o
Skew Angle

284

0.012
0.008
0.004
0
-0.004
Rotatio (rad)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
Motions
1
2
3
4
5
6
7
Average

Figure 5.74: Rotation Time History of Three- Span Single-column Bridge With 45
o
Skew Angle

285

0.01
0.008
0.006
0.004
0.002
0
-0.002
Rotatio (rad)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
Motions
1
2
3
4
5
6
7
Average

Figure 5.75: Rotation Time History of Three- Span Single-column Bridge With 60
o
Skew Angle

286

0.01
0.008
0.006
0.004
0.002
0
-0.002
Rotatio (rad)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
Motions
1
2
3
4
5
6
7
Average

Figure 5.76: Rotation Time History of Two- Span Two-column Bridge With 25
o
Skew Angle

287

0.016
0.012
0.008
0.004
0
-0.004
Rotatio (rad)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
Motions
1
2
3
4
5
6
7
Average

Figure 5.77: Rotation Time History of Two-Span Two-column Bridge With 45
o
Skew Angle

288

0.02
0.016
0.012
0.008
0.004
0
-0.004
Rotatio (rad)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
Motions
1
2
3
4
5
6
7
Average

Figure 5.78: Rotation Time History of Two-Span Two-column Bridge With 60
o
Skew Angle

289
0.006
0.004
0.002
0
-0.002
Rotatio (rad)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
Motions
1
2
3
4
5
6
7
Average

Figure 5.79: Rotation Time History of Three-Span Two-column Bridge With 25
o
Skew Angle

290

0.008
0.006
0.004
0.002
0
-0.002
Rotatio (rad)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
Motions
1
2
3
4
5
6
7
Average

Figure 5.80: Rotation Time History of Three-Span Two-column Bridge With 45
o
Skew Angle

291
0.008
0.006
0.004
0.002
0
-0.002
Rotatio (rad)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
Motions
1
2
3
4
5
6
7
Average

Figure 5.81: Rotation Time History of Three-Span Two-column Bridge With 60
o
Skew Angle

292

Single-Span Two-Spans-Single-
Column Bent
Single-Spans-Two-
Columns Bent

Figure 5.82: Maximum Deck Rotations

293

Single-Span
Two-Spans-Single-
Column Bent
Single-Spans-Two-
Columns Bent

Figure 5.83: Average Residual Deck Rotations

294

Two-Spans-Single-Column Bent
Two-Spans-Two-Column Bent

Figure 5.84: Two-Span Bridge Maximum Residual Deck Rotations

295

Three-Spans-Single-Column Bent Three-Spans-Two-Column Bent

Figure 5.85: Three-Span Bridge Maximum Residual Deck Rotations

296
5.11 Conclusions
Three-dimensional models of a various bridge configurations were developed
incorporating 0
o
, 25
o
, 45
o
and 60
o
skew angles. Nonlinear force-deformation
relationship was used to model the interaction between bridge deck and the abutment
backfill. The abutment shear keys were modeled using combination of the UCSD
experimental backbone cures and contribution of the soil capacity in the transverse
direction of the abutment. Nonlinear time-history analyses using a total of seven sets
of two-component response-spectra-compatible time history input motions were
performed to evaluate the seismic response of the abutments and the superstructure
as a function of skew angle. All motions exhibited velocity pulses to characteristic of
near-field effects.
From the results, it was found that the skewed-bridge decks undergo
significant rotations about the vertical axis during seismic ground events and are
permanently displaced from the original location by the end of shaking. The non-
skewed bridge decks did not experience significant rotation or permanent transverse
displacement despite the large velocity pulses in the input motions. Depending on
the intensity of the velocity pulses, this may cause the deck to become unseated at
the abutments. The deck rotation is due to development of an non-uniform passive
soil wedge behind the abutment wall that results in asymmetric soil reactions
between the acute and obtuse corner of the wall. The width and capacity of this
passive wedge depends on a number of factors, particularly abutment (embankment)
width, skew angle and ground motion characteristics. Such behavior has been

297
observed at existing skew abutments during the recent earthquakes. Most of the
permanent rotation (transverse deck offset) was observed to build up during the
initial peak cycles of shaking by all seven ground motions.

298
CHAPTER SIX
CASE STUDY OF AN INSTRUMENTED SKEWED BRIDGE

6.1 Introduction
Nonlinear soil-abutment-structure interaction analysis of typical ordinary
highway bridge structures subjected to seismic ground motions with high velocity
pulses was discussed in the previous chapter. The purpose of this chapter is to
investigate the seismic Soil-Abutment-Foundation-Structure Interaction (SAFSI)
analysis of ordinary highway bridge structures. The numerical methods used for
SAFSI of highway bridge structures can be classified into direct approach and
substructure approach. In the direct approach, nonlinear soil and foundation
behaviors are explicitly included in the global model to account for geotechnical and
structural behavior of foundations. The substructure approach divides the system into
two subsystems, a superstructure that includes the bridge columns, bridge deck and
bridge abutments and a substructure that includes the foundations and the
surrounding soil media.
Many bridges are supported on pile foundations that may penetrate multiple soil
layers with varying stiffness and shear strength properties. For a deep pile foundation,
ground motion excitation with depth is felt along the pile length. Hence, seismic loading
criteria must account for the variation in ground motion with depth, rather than merely
adopting surface motion as the basis for the earthquake design. The effects of depth-
varying ground motion can rigorously be addressed by modeling the global bridge

299
including representation of each individual pile, with distributed soil springs, extending
from ground surface to the pile tip. The depth-varying ground motions from a free-field
site response analysis can be used to excite the soil-pile structure system; whereby, the
depth varying soil-pile properties and ground motions are rigorously taken into account.
Such an analysis is referred as a complete system or direct approach. The advantage of
employing the complete system approach includes possible implementation of non-
linear soil-foundation supports arising from either material or geometric non-linearity.
Alternatively, a substructure system can be used to reduce the size of the
problem in the complete structural model. This approach establishes the structural
model without the complete foundation element. The foundation substructure is
modeled by a linear stiffness matrix representing the entire soil-pile system and a set of
kinematic ground motion representing effective shaking arising from the depth-varying
motions acting along the piles. The substructuring technique to compute the stiffness
matrix and kinematic motion involves modeling each individual pile to a convenient
interface with the superstructure. Then static condensation is used to derive the
condensed foundation substructure stiffness and the effective ground excitation
(kinematic motion) transmitted to the superstructure.
While the detailed structural bridge models consisting of a complete foundation
system are frequently used in high profile projects where plenty of resources are
available, this kind of bridge model becomes economically not feasible for many
ordinary bridges where resources are limited. The substructure foundation system serves

300
as an alternative to the complete foundation system for the structure bridge model and
yet provides reasonably accurate representation.
The objective of this chapter is to compare the results of the direct approach
with those of the substructure approach through a case study of an instrumented
bridge structure.
6.2 Bridge Description
The Painter Street Overpass is located in Rio Dell, California. The structure
is a two-span cast-in-place prestressed reinforced concreted box-girder bridge
supported on two-column bents and integral abutments, as shown in Figure 6.1 and
Figure 6.2. The bridge is approximately 265 feet long and 52 feet wide with spans
measuring 146 and 119 feet with a 39
o
skew angle, as shown in Figure 6.2. The
depth of the deck is 5.67 feet. The average height of the columns is approximately 24
feet and each is supported on 4 x 5 driven 45-ton concrete piles, as shown in Figure
6.3. The average height of the monolithic abutment backwall is approximately 12
feet. The west abutment backwall rests on a neoprene bearing strip lubricated with
grease to allow thermal movement between the abutment wall and the backfill. There
is a 1-inch gap between the abutment wall and the abutment backfill. The west
abutment is supported on a single row of 16 concrete piles. The east abutment
backwall is monolithic, e.g., the wall is cast to the deck and the pile cap and it is
supported on a single row of 14-ton driven concrete piles. As a result, the west
abutment is more flexible than the east abutment.

301

(a) General View

39
O
265’
146’
119’
39
O
265’
146’
119’

(b) Structural components
Figure 6.1: 3-D View of the Painter Street Bridge

302
265’
146’ 119’
265’
146’ 119’
265’
146’ 119’

(a) Elevation

Skew = 39
o
52
’
A
A
67’
143’
116’ 7
’
Skew = 39
o
52
’
A
A
67’
143’
116’ 7
’
52
’
A
A
A
A
67’
143’
116’ 7
’

(b) Plan
Figure 6.2: Elevation and Plan Views of the Painter Street Bridge

303
2’-6”
5’-6”
5’-8”
BENT CAP
52’
1:2
5’-8”
24’
Pile Foundation
2’-6”
5’-6”
5’-8”
BENT CAP
52’
1:2
5’-8”
24’
Pile Foundation

Figure 6.3: Cross Section of Superstructure and Pile Foundation of the Painter Street Bridge

304

6.3 Seismic Instrumentation and Input Ground Motion
The Painter Street Overpass is seismically instrumented and has been shaken
by several earthquakes for which records are available. The largest earthquake that
shook the structure is the 6.9 magnitude 1992 Cape Mendocino/Petrolia earthquake.
The location of the instruments that measured the free-field (Channels 12, 13 and 14)
motions and structure accelerations (all other channels) are shown in Figure 6.4.
For performing dynamic analyses with a time-domain approach, the unscaled
recorded free-field acceleration-time histories of the Cape Mendocino/Petrolia
earthquake were used as input motions to excite the bridge. The strongest component
of the input motion was in the bridge transverse direction. The acceleration, velocity
and displacement time histories of the recorded and the input motions are shown in
Figure 6.5, Figure 6.6 and Figure 6.7. The peak ground accelerations, velocities and
displacements of the three input motions are summarized in Table 6.1.
Table 6.1 Characteristics of the Input Motions of the Painter Street Bridge
Bridge Direction Peak Acceleration
(g)
Peak Velocity
(in/sec)
Peak Displacement
(in)
Transverse (N-S) 0.54 18.30 2.06
Longitudinal (E-W) 0.38 14.61 2.97
Vertical 0.20 3.85 0.84

305

(a) Seismic Instrumentations
39
O
265’
146’
119’
39
O
265’
146’
119’

(b) 3-D Structural Configuration
Figure 6.4: Seismic Instrumentation at the Painter Street Bridge

306
0 10 20 304050 60
Time (sec)
-15
-10
-5
0
5
10
15
Velocity (in/sec)
0 10 20 304050 60
Time (sec)
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
Acceleration (g)
(a)
(b)
(c)
0 10 20 304050 60
Time (sec)
-5
-4
-3
-2
-1
0
1
2
3
4
5
Displacement (inches)

Figure 6.5: Input Motion in Longitudinal Direction (Channel 12)

307
0 102030 40 50 60
Time (sec)
-4
-2
0
2
4
Velocity (in/sec)
0 102030 40 50 60
Time (sec)
-0.2
-0.1
0.0
0.1
0.2
Acceleration (g)
(a)
(b)
(c)
0 102030 40 50 60
Time (sec)
-1.0
-0.5
0.0
0.5
1.0
Displacement (inches)

Figure 6.6: Input Motion in Vertical Direction (Channel 13)

308
0 102030 40 5060
Time (sec)
-20
-15
-10
-5
0
5
10
15
20
Velocity (in/sec)
0 102030 40 5060
Time (sec)
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
Acceleration (g)
(a)
(b)
(c)
0 102030 40 5060
Time (sec)
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
Displacement (inches)

Figure 6.7: Input Motion in Transverse Direction (Channel 14)

309
6.4 Geotechnical Information
The As-built Plans from 1973 for Painter Street Bridge include a Log of Test
Boring Sheet which shows a soil boring that was drilled at Bent 2 to a depth of 55
feet. The log displays soil type and strength descriptions and Standard Penetration
Test (SPT) blowcounts at 5-ft depth intervals. As shown in Figure 6.8, the recent site
investigation (EMI, 2004) consisted of two soil borings, two in-hole pressuremeter
tests, and two seismic cone penetrometer soundings in the eastbound lane behind
both abutments near the approach slabs, a small distance behind the abutment walls.
The depth of interest in that study was in the upper few feet of the abutment
backwalls where the backwall engages (pushes into) the backfill soils and the
passive wedge are expected to mobilize. Two eight-inch diameter shallow dry auger
borings were drilled through the embankment fills at each abutment down to 12.5
and 14 feet depth below existing road grade. No groundwater was encountered in
these borings. Geotechnical soil sampling consisted of relatively “undisturbed” ring
samples collected using a Modified California sampler and disturbed bag samples
collected with a SPT sampler at 2.5 and 5-feet depth intervals. The Drive sampler is
a split-barrel sampler (2.5-inch ID) with a tapered cutting tip and lined with a series
of brass rings. The standard split-spoon sampler has a 1.4-inch ID and 2-inch OD. At
each depth, the samplers were alternated and driven either 18 inches into virgin soil
or until refusal using a 140-lb hammer free-falling from a height of 30 inches.

310
39
O
265’
146’
119’
39
O
265’
146’
119’

Figure 6.8: Layout of Abutment Backfill Borings at the Painter Street Bridge Site

311
The blowcounts for the last 12 inches of penetration were recorded on the
boring logs. Selected soil samples were tested in a soils laboratory to determine
backfill properties such as grain size distribution and soil strength tests. Large bulk
samples of near-surface soils were collected to determine for properties such as
gravel and silts contents, plasticity and expansion potential.
Two pressuremeter tests were performed near each soil boring to determine
in-situ soil resistance. Two probe holes were drilled adjacent to the soil borings and a
70-mm-diameter mechanical RocTest Texam Pressuremeter probe placed tight in
virgin soil using an NX drill bit at 5 feet and 7.5 feet depth below existing road
grade. The test consists of inflating a rubber membrane enclosed in a flexible metal
shield with water to exert lateral pressure on the backfill soil with cycles of
unloading and reloading. The test measures the in-situ nonlinear lateral stress-strain
behavior and stiffness (modulus) of the backfill.
The two conventional CPT soundings (denoted as CPT-1 and CPT-2) were
conducted next to the borings to allow correlation with the soil type and measured
soil strength with the shear wave velocity measurements. The soundings used a
standard cone with 10 cm
2
area using a 25-ton truck with on-board data acquisition.
During penetration of the cone, the cone tip resistance and sleeve friction is
continually measured. This data was correlated to soil behavior type, undrained shear
strength and friction angle. Shear wave and compression wave velocities were
measured every 2.5 feet depth using the seismometer built into the cone. Shear

312
waves were generated by impacting a wooden beam that is coupled to the ground
surface by the truck dead weight. The sledgehammer impact triggers the electronic
recording of the wave arrival time measured by the seismic cone in the ground.
6.5 Idealized Soil Profile and Properties
Figure 6.9 shows an idealized soil profile with design properties based on
the available geotechnical data and the bridge foundation configuration. The strata
are as follows:
• The fill embankments at the Abutments consist of compacted dense to very
dense, damp, fine to coarse sand with some silt and gravel up to 1inch size
down to about 6.5 feet depth below road grade. Sand equivalent (SE) is a
measure of the amount of fines and coarse fractions in predominantly granular
samples and a large bulk sample tested as 43, satisfying Caltrans’ standard
abutment backfill specification of 20.
• The natural soils below consist of stiff to very stiff low-plasticity clayey silt to
plastic silty clay with traces of fine to medium sand, subrounded gravel up to ½
inch in size, and organic material down to about El. 102 feet.
• A layer of medium dense sands and silts down to about El. 87 feet
• A dense to very dense sand with silt and gravel that serves as a bearing stratum
for the pile foundations down to at least El. 64 feet explored.

313

Figure 6.9: Geometry and Idealized Soil Profile at the Painter Street Bridge Site

314
High groundwater was encountered in 1973 and then was lowered below the
bottom of the existing roadcut. Current groundwater levels are not known, but are
assumed at the bottom of the present roadway base fills. Figure 6.9 also shows
graphs of uncorrected field SPT blowcounts, CPT measurements and shear wave
velocities. Non-standard blowcounts were converted to SPT-equivalent blowcounts.
The SPT value is widely used in geotechnical design practice to correlate soil
strengths and other parameters. Table 6.2 and Table 6.3 summarize the properties of
the existing backfills and natural soils for use in our computer model. These values
were derived as follows:
• Field soil densities were measured in the laboratory directly from carefully
preserved ring samples per ASTM D-2937 and D-2216 methods. The
maximum density and optimum moisture content was determined on shallow
bulk samples using the Modified Proctor Test in a 4-inch diameter mold
(ASTM D-1557) using four trial points to develop a dry density versus
moisture curve.
• Gradation of sandy soils was determined by standard sieve tests per Caltrans
Test Methods 202/203. The grain size curves show the presence of fines and
coarse fractions.
• Plasticity of cohesive samples was determined from laboratory Atterberg limits
(plasticity index, liquid limit and plastic limits) using Caltrans Test Method
204. The plastic limit determines whether a soil is a silt or clay, and the degree
of plasticity (e.g., low or high).

315
• Shear Strength Parameters (total-stress friction angles and cohesion strengths)
were determined from load-displacement curves and Mohr circles obtained
from cyclic unconsolidated undrained (UU) triaxial tests at failure and residual
(large displacement) values from direct shear tests. The triaxial tests were
performed on extruded ring samples per ASTM D-2850. Sand samples were
screened through a #4 sieve (to remove large gravel) and placed in a rubber
membrane. Specimens were then consolidated and tested in a Geomatic triaxial
apparatus to failure under unconsolidated undrained conditions at three
different confining pressures and three unloading and reloading cycles. Direct
shear tests were conducted using ASTM-D3080 on ring samples of sandy soils
subjected to three vertical pressures near the overburden pressure.
• Shear wave velocities (SV) were measured using the seismic cone soundings.
Caltrans BDS indicates a presumptive shear wave velocity of 800 feet per
second for compacted backfills.
• Soil Moduli were determined from the pressuremeter and triaxial tests from
stress-strain curves for initial monotonic loading (Young’s modulus), secant
modulus E
50
(50% of the failure stress), and the modulus for unloading and
reloading. These moduli are representative of the strain range (up to 25%)
expected in the subsequent engineering application and are much lower than
very-small-strain moduli calculated from shear wave velocity.

316

Table 6.2 Idealized Soil Parameters for Painter Street Bridge Site

Soil Properties
Unit Soil Type (USCS Symbol)
γ' (pcf) φ (deg) c (psf) v
s
(fps)
1 Compacted Sandy Fill (SP, GP) 130 38 50 670
2 Stiff Silt and Clay (ML/CL) 128 11 3,300 1,000
3 Medium dense Sand (SP) 57 34 0 NA
4 Dense Sand with Gravel (SP) 63 36 0 NA
Notes: γ' = Effective Unit Weight, φ = Friction Angle ‘ c = Cohesion,
v
s
= Shear wave velocity

Table 6.3 Idealized Stiffness Parameters to Develop p-y Springs for Painter Street
Bridge Site

p-y Curve Parameters Soil Stiffness
Unit Soil Type
K (pci) ε
50
E
s
(ksf) E
50
(ksf) E
r
(ksf)
1 Compacted Sandy Fill 60 - NA NA NA
2 Stiff Silt and Clay) - .005 90 110 300
3 Medium dense Sand 60 - NA NA NA
4 Dense Sand with Gravel 80 - NA NA NA
Notes:
ε
50
= Strain Parameter for p-y curve, J = Empirical Coefficient for p-y curve
k = Modulus of subgrade reaction
E
50
= Stiffness at 50% of Ultimate Stress, E
r
= Unloading/ Reloading modulus

317
6.6 Global Bridge Model
The skewed bridge abutment foundations and the surrounding soils constitute
a strongly-coupled system. The complete soil-abutment-foundation-structure
interaction of the Painter Street bridge system, is separated into two substructures
separated by a foundation interface: (1) superstructure (2) pile foundation and
surrounding soils. Two approaches, namely the direct method and the substructure
method are used for analyzing the global bridge response as shown Figure 6.10.
In the direct approach all structural components, all foundation components
and all the soil support springs are explicitly included in the bridge model. A more
feasible alternative is to use substructuring concept to reduce the structuring
modeling and analysis to a manageable seize. The concept of substructuring can be
applied either to a single pile or a pile group foundation. On the other hand, the
substructure system simplifies the foundation to a reduced degree-of-freedom system
by using a substructuring technique. The choice between substructuring of the
individual pile or the entire pile group depends on the application. For example, if
plastic hinge behavior of the pile is to be considered in the global bridge model,
substructuring of individual piles would be used in which case individual pile head
loads (shear and moment) can be obtained directly from the global analysis. If
substructuring of the entire pile group is adopted, the individual pile loads would be
available only after a back-substitution process is carried out on the foundation
substructure, common known as pushover analysis.

318

Figure 6.10: Total System Versus Substructure System

319
For the direct method the foundation system, non-linear soil springs were
developed using the site specific geotechnical data. The soil springs were not only
non-linear but also inelastic upon unloading to allow for a hysteretic behavior. When
the substructuring technique was used to simplify the foundation in the global bridge
model, the pile group was represented by a 6x6 linear stiffness matrix. Since soil
springs were nonlinear, the first step towards developing the foundation stiffness
matrix involved linearization of the non-linear soil springs by performing a lateral
pushover analysis on a single pile to a representative displacement level expected
during the design earthquake. Once the non-linear soil springs were linearized on the
basis of the lateral pushover analysis, the problem became analogous to beams on
elastic springs, and the method of substructuring was used to obtain a condensed
stiffness matrix.
6.7 Global Bridge Modeling Using Direct Method
For the complete detailed fully-coupled three-dimensional nonlinear dynamic
finite-element (FE) model considering soil-abutment-foundation-structure interaction
(SAFSI) bridge was developed.using the site specific geotechnical data. The global
bridge structure and foundation was modeled using the computer program SAP2000
as shown Figure 6.11 to perform nonlinear time history analysis to obtain the “exact”
response of the bridge as a function of time. The gravity load was applied to the
bridge system prior to the time history analysis. The displacements due to gravity
load were removed from the calculated displacement.

320

Figure 6.11: Painter Street Bridge Model for Direct Method

321
The bridge deck and the abutments wall were modeled as shell elements with
applicable structural properties. The support provided by the west abutment was
modeled using a friction isolator to simulate the neoprene pad and to decouple the
superstructure and abutment backwall from the pile cap. The isolator was fixed in the
vertical direction only. The support provided by the east abutment was fixed to the
pile cap. At the west abutment, lateral sliding friction of the endwall on the concrete
pile cap was incorporated in the model. Using a frictional coefficient of 0.45 and
assuming one-half of each the span dead weight contributes to the weight at each
abutment, a total linear frictional stiffness of 420 kips per inch of displacement was
estimated following Section 14 of the Caltrans Bridge Design Specifications (2000).
This was distributed using a total of 14 lateral springs applied between the bottom
the wall and the pile heads. The structural behavior of the piles and columns was
modeled using frame elements. The transverse section of the bridge structure and the
cross section of the column used in the analysis are shown in Figure 6.12. SAP2000
computer program was used to develop the moment-curvature relationship of the
column cross-section as shown in Figure 6.13.

322

39
O
265’
146’ 119’
S
S
39
O
265’
146’ 119’
S
S

3D View

AA
52’
Section A-A
24’
5.67’
0.62’
1’
.46’
3” cl.
#4 Spiral
36 #11
5.0’
5.0’
3” cl.
#4 Spiral
36 #11
5.0’
5.0’
2
1

Section S-S

Figure 6.12: Bent and columns Sections for Painter Street Bridge

323
0.000 0.002 0.004 0.006 0.008
Curveture (φ)
0
4000
8000
12000
Moment (k-ft)
M-φ
Idealized-M-φ
I
cr
M
E
y
=
φ
(φ
y
,M)

Figure 6.13: Moment Curvature Relationship for Painter Street Bridge

The pile cap at the west abutment has one longitudinal and two transverse
shear keys. When the bottom of the abutment endwall pushes toward the keys, the
pile cap and underlying foundation are engaged. A nonlinear plasticity model was
used to simulate the abutment shear key behavior as observed during a prior
Caltrans-UCSD field experiment (Bozorgzadeh et al. 2003). The nonlinear backbone
curve was scaled to produce the structural shear key capacity of the abutment as a
function of displacement between bridge deck and abutment pile cap. In addition, the
curve was offset to incorporate the 1-inch expansion gap. At the tail end of the curve,
a fourth segment was added to account for the tangential component of the
abutment-backfill passive capacity due to deck rotation and the passive capacity

324
contribution of the exterior embankment soil. The longitudinal abutment-backfill
was modeled by a series of nonlinear link elements distributed along each abutment
backwall. One transverse key was applied at the north end and one at the south end
of each abutment. The nonlinear backbone curves for each shear key (four in the
transverse direction) and soil springs (twenty one in the normal direction) are shown
in Figure 6.14.

01234 5 6
Displacement (inches)
0
100
200
300
400
500
600
700
800
900
1000
Force to Be Finalized (kips)
Total Abutment
Longitudidal Spring
Shear Key @ Each Corner
Transverse Spring
Figure 6.14: Shear Key Capacities at West Abutment for Painter Street Bridge

325
The nonlinear backbone curves at each abutment was divided into 21
nonlinear discrete backbone curves longitudinal direction perpendicular to the skew
angle to represent the backfill distributions at the abutment stem walls.
6.7.1 Pile Foundations
The pile foundations were modeled as beam elements with nonlinear springs
to represent interaction between the piles with the surrounding soil as shown in
Figure 6.15. Lateral pile-soil support curves were generated according to API
criteria (API, 1993) using the parameters given in Table 6.3. For the abutment piles,
p-y curves were generated at depths of 1, 2, 3, 5, 10, 15, 20, and 30 feet from pile top
and at the pile tip at 37.5 ft. For the bent piles, p-y curves were calculated at depths
of 1, 2, 3, 5, 10, 15, 20 feet, below pile top and pile tip at 25 ft. The p-y soil
resistance springs are omni-directional, e.g. they can be thought of as “rotating”
around the pile axis into the direction of the soil resistance during seismic loading. In
the pile foundation model, two nonlinear horizontal springs (one in the bridge
longitudinal and one in the bridge transverse direction) were applied along each pile
and at each of the above depths.
6.7.2 Abutment Soil-Structure Interaction
The dynamic interaction between the deck, abutment wall and the
embankment soil in the direction perpendicular to the abutment wall was modeled by
a gap element between the bridge deck and the abutment backfill and a nonlinear
spring. The dynamic deck-abutment interaction along the skew-angle in the

326
transverse direction was model by a gapping element, nonlinear shear key and
nonlinear soil springs. The detailed model to the abutment is shown in Figure 6.17.
Gaping at the west abutment, a 1-inch structural expansion gap exists
between the structural and the soil backfill. Gapping elements were incorporated to
simulate these gaps between the soil and the bridge deck in the longitudinal and
transverse directions. The initial opening was set to one inch. The east abutment is
monolithic abutment and in direct contact with backfill. Gap elements with zero
initial opening were incorporated to disallow tension in the backfill.
The nonlinear spring represents the near-field load-deformation behavior at
the longitudinal abutment-embankment soil interface. A separate continuum finite-
element model was developed using PLAXIS (2005) with a strain-hardening-soil
model to develop this load-deformation relationship (backbone curve) as shown
schematically in Figure 6.17. This model incorporates the combined response of the
two soil layers behind the abutment backwall. The hysteretic behavior of the
backbone curve is modeled using the multi-linear plasticity model with the tension
side of the curve set to zero. The behavior is essentially that of a gap element in
series with a compressive plastic spring at the west abutment and only compression
spring at the east abutment. Upon load reversal, the spring unloads elastically until
zero force is reached, with net permanent deformation present. Further loading in the
tension direction acts as an open gap, with no force exhibited. Reloading in the
compressive direction remains at zero force until the gap is closed at a deformation

327
equal to the permanent plastic deformation. The spring loads elastically until the
backbone curve is reached, then follows the backbone with increasing plastic
deformation as shown in Figure 6.16.
The soil mass contribution from the passive wedge is attached at each end of
nonlinear spring as shown in Figure 6.17. McCallen and Romstad (1994) showed
that dynamic behavior of a bridge system is relatively insensitive to variation of
abutment-embankment soil mass when the soil is undergoing small strains. However,
the system dynamics becomes more sensitive to the soil mass when the abutment-
embankment soil becomes softer as would be expected due to soil softening during
strong shaking. In fact, for any soil-structure interaction problem there are two
mechanisms affect energy dissipation during seismic loading: the strain-dependent
energy dissipation mechanism (material damping) associated with nonlinear
hysteretic behavior of the soil and radiation damping (geometric damping) associated
with dissipation of elastic wave energy away from the bridge. When proper
nonlinear inelastic abutment springs are implemented allowing for hysteretic
behavior, the strain-dependent energy dissipation mechanism is automatically
simulated, accounting for material damping in the abutment. Due to highly nonlinear
nature of the abutment-backfill in the vicinity of the abutment the contribution of the
damping associated with radiation damping is insignificant and was not considered
in this study.

328

Figure 6.15: Bridge Pile Foundation Model

329
-4 -3 -2 -1 0 1 2 3
Displacement (inches)
0
10
20
30
40
50
Resistance Force (Kips)
East Abutment
Weat Abutment
Backbone Curve

Figure 6.16: Longitudinal Abutment-Soil Loading-Unloading Curves

Near-Field
Bridge Nonlinear Spring
Soil Mass
Neoprene
Bridge
CLAY
SAND
d
F
pad
Near-Field
Bridge Nonlinear Spring
Soil Mass
Neoprene
Bridge
CLAY
SAND
d
F
d
F
d
F
pad

Figure 6.17: Bridge Monolithic Abutment Model

330
6.7.3 Global Bridge Displacement Response
Figures 6.18 through 6.27 show a comparison of the calculated displacement
response of the analytical model with those obtained from the acceleration records
for major points and directions on the bridge. The results of the model match the
records remarkably well, particularly the longitudinal response at both abutments. It
should be noted that the model used did not formally include wingwalls and thus did
not allow comparisons with recorded wingwall motions (Channels 15 through 20).
Since the west span of the bridge is larger than the east span and the bridge
deck has much more flexibility than at the west abutment, the center of rigidity of
the bridge system is closer to the east abutment. As a result, the bridge is expected
to rotate about its vertical axis during seismic shaking. The fact that recorded
transverse peak ground accelerations at the west abutment was about 1.5 times
higher than at the east abutment indicates that the bridge must have undergone
such rotation during the earthquake. Figure 6.28 presents the transverse
displacement-time histories results from the analytical model at opposing far
corners of the bridge deck (shown as “NW” and “SE” in Figure 6.4). Figure 6.29
presents the deck rotation time histories results from the analytical model. From
these comparisons, the superstructure underwent significant rotations during the
earthquake and returned to its original position by the end of ground shaking.
Inspection of the concrete barrier rails and curbs and the pavement striping during
EMI’s field investigation (EMI, 2004) did not show any permanent lateral offset at
either abutment.

331
-3
-2
-1
0
1
2
3
4
Displacement (inches)
010 20 30
Time (sec)
Recorded (Channel 18)
Model (Channel 18)

Figure 6.18: Recorded Response Versus Analytical Response of the Direct Method (Channel 18)

332
-3
-2
-1
0
1
2
3
4
Displacement (inches)
010 20 30 40 50 60
Time (sec)
Recorded (Channel 11)
Model (Channel 11)

Figure 6.19: Recorded Response Versus Analytical Response of the Direct Method (Channel 11)

333
-3
-2
-1
0
1
2
3
4
Displacement (inches)
010 20 30 40 50 60
Time (sec)
Recorded (Channel 7)
Model (Channel 7)

Figure 6.20: Recorded Response Versus Analytical Response of the Direct Method (Channel 17)

334
-3
-2
-1
0
1
2
3
4
Displacement (inches)
010 20 30 40 50 60
Time (sec)
Recorded (Channel 4)
Model (Channel 4)

Figure 6.21: Recorded Response Versus Analytical Response of the Direct Method (Channel 4)

335
-2
-1
0
1
2
Displacement (inches)
010 20 30 40 50 60
Time (sec)
Recorded (Channel 3)
Model (Channel 3)

Figure 6.22: Recorded Response Versus Analytical Response of the Direct Method (Channel 3)

336
-3
-2
-1
0
1
2
3
Displacement (inches)
010 20 30 40 50 60
Time (sec)
Recorded (Channel 9)
Model (Channel 9)

Figure 6.23: Recorded Response Versus Analytical Response of the Direct Method (Channel 9)

337
-2
-1
0
1
Displacement (inches)
010 20 30 40 50 60
Time (sec)
Recorded (Channel 5)
Model (Channel 5)

Figure 6.24: Recorded Response Versus Analytical Response of the Direct Method (Channel 5)

338
-2
-1
0
1
Displacement (inches)
010 20 30 40 50 60
Time (sec)
Recorded (Channel 8)
Model (Channel 8)

Figure 6.25: Recorded Response Versus Analytical Response of the Direct Method (Channel 8)

339
-2
-1
0
1
Displacement (inches)
010 20 30 40 50 60
Time (sec)
Recorded (Channel 2)
Model (Channel 2)

Figure 6.26: Recorded Response Versus Analytical Response of the Direct Method (Channel 2)

340
-3
-2
-1
0
1
2
3
4
Displacement (inches)
010 20 30 40 50 60
Time (sec)
Record (Channel 1)
Model (Column Base)

Figure 6.27: Channel 1 Transverse at Column base of the Direct Method

341
-3
-2
-1
0
1
2
3
4
Displacement (inches)
010 20
Time (sec)
NW Corner Deck Transverse
SE Corner Deck Transverse

Figure 6.28: Comparisons of the Northwest and Southeast Corners Displacement Response in the Transverse Directions
Using Direct Method

342
0.0012
0.0008
0.0004
0
-0.0004
-0.0008
Rotatio (rad)
0 5 10 15 20 25 30 35 40 45 50 55 60
Time (sec)
Deck Rotation ( Rad)

Figure 6.29: Deck Rotational Response Using Direct method

343
6.7.4 Abutment Force and Displacement Response
Figure 6.30 and Figure 6.31 show the average force-displacement response of
the abutments. At the West Abutment, the backfill participates to resist seismic
forces after 4.5 seconds of shaking following closure of the 1-inch expansion gap. At
the East Abutment (where there is no gap), the backfill begins to resist lateral forces
at the instance the bridge moves eastward.
Figure 6.32 shows the transverse displacement response between the
abutment (pile cap) and the bridge deck. The difference between these two time
histories corresponds to transverse offset of the deck relative to the seat of up to
about 2 inch and illustrates the bridge deck rotation in time. The hysteretic behavior
of the north and south transverse shear keys at the west abutment are shown in
Figure 6.33 and Figure 6.34. The two shear keys are observed to approach their
ultimate structural capacities. This result suggests that these keys could potentially
fail when subjected to stronger shaking, which could allow the bridge to rotate and
cause the deck to become unseated at the east abutment due to lateral movement.

344
(a)
02 468 10 12
Time (sec)
-50
-40
-30
-20
-10
0
10
20
30
40
50
Force (Kips)
Force
Displacement
-3
-2
-1
0
1
2
3
Displacement (inches)

(a)
(b)
01 23
Displacement (inches)
0
10
20
30
40
50
Force (Kips)
Model
Backbone

(a)
Figure 6.30: Force-Displacement Response at The West Abutments Using Direct
method

345
(a)
02 468 10 12
Time (sec)
-50
-40
-30
-20
-10
0
10
20
30
40
50
Force (Kips)
Force
Displacement
-3
-2
-1
0
1
2
3
Displacement (inches)

(a)
(b)
01 23
Displacement (inches)
0
10
20
30
40
50
Force (Kips)
Model
Backbone

(b)
Figure 6.31: Force-Displacement Response at The East Abutments Using Direct
method

346
-3
-2
-1
0
1
2
3
4
Displacement (inches)
010 20
Time (sec)
Recorded (West Abutment)
Model ( Abutment Pilecap)

Figure 6.32: Displacement response at the West Abutment Seat Using Direct method

347
(a)
02 46 8 10 12
Time (sec)
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
Force (Kips)
Force
Displacement
-3
-2
-1
0
1
2
3
Displacement (inches)

(a)
(b)
0123 4 5 6
Displacement towards South (inches)
0
100
200
300
400
500
600
700
Force (Kips)
Model
Backbone

(b)
Figure 6.33: Force-Displacement Capacity of Shear Key at West Abutment Using
Direct method

348
(a)
02 4 6 8 10 12
Time (sec)
-600
-500
-400
-300
-200
-100
0
100
200
300
400
500
600
Force (Kips)
Force
Displacement
-3
-2
-1
0
1
2
3
Displacement (inches)

(a)
(b)
65 432 1 0
Displacement towards North (inches)
0
100
200
300
400
500
600
700
Force (Kips)
Model
Backbone

(b)
Figure 6.34: Force-Displacement Capacity of Shear Key at West Abutment Using
Direct method

349
6.8 Substructuring Technique
The concept of the substructuring system and foundation impedance can be
explained using equation of motion in the frequency domain. Several researches
have developed numerous procedures to determine dynamic impedance of pile
foundation. The purpose of this section is to explain why substructuring technique is
the most efficient way to represent the bridge foundation for seismic global analysis
of the bridge structure. Figure 6.35 shows a linear pile foundation system of mass M,
Lateral stiffness K and damping C subjected to a horizontal ground excitation force.

Dashpot C
Spring K
Mass M
Ground
Motion
Pile Response X
MX t CX t KX t M U t
g
( ) () () ()
•• • ••
++ =−
U
g
••
(
Dashpot C
Spring K
Mass M
Ground
Motion
Pile Response X
MX t CX t KX t M U t
g
( ) () () ()
•• • ••
++ =−
U
g
••
(
Figure 6.35: Schematic of soil-pile interaction

At each instant of time t, the equation of motion from the free body shown in
Figure 6.35 can be expressed in a general form as shown in Eq. (6.1).
MX t CX t KX t M U t
g
() () () ()
•• • ••
++ =− (6.1)
where

350
Xi X
Xi X
••
•
=
=
22
ω
ω
(6.2)
Substituting Eq. (6.2) into Eq. (6.1), then
Mi Xt Ci Xt KX t MU t
g
22
ωω () () () () ++ =−
••
(6.3)
In which M, C and K are mass, damping and stiffness matrix respectively, of
a single degree freedom of the structure, (X) is the total displacement vector of the
system, and U
g
is the acceleration vector of the free-field ground motion.
Let
−=
••
MU t F t
g
() ()

Where F(t) is the free field excitation force due to ground acceleration
therefore, Eq. (6.2) can be expressed as follow:
Mi Xt Ci Xt KX t F t
22
ωω () () () () ++ =
(6.4)
Then
Ft
K M Ci
Xt
()
()
−+
=
ωω
2
(6.5)
where

351
[] [ ]{} []{ }
KM Ci K iC −+ =+ ωω ω
2

Let
KM K −= ω
2

K is the static stiffness,
K
is the dynamic stiffness.

Ki C + ω
is the dynamic impedance.
The dynamic impedance is a complex stiffness which is consisted of real part
and imaginary part. The real part is the dynamic stiffness K is a function of static
stiffness K and the circular frequencyω . The imaginary part of the stiffness is a
function of frequency and viscous damping C and the circular frequencyω .
Figure 6.36 shows a general plot of the dynamic impedance versus frequency
of the system. It can be seen that as the frequency increases the dynamic stiffness
decreases, however, in the frequency range of interest for bridge structure the
dynamic stiffness is practically the same as the static stiffness. That is the main
reason substructuring technique using static condensation can be used to develop 6x6
foundation stiffness matrix.

352
Frequency
Range For
Bridges
KM − ω
2
K
ω
Frequency
Range For
Bridges
KM − ω
2
K
ω

Figure 6.36: Dynamic Impedance Versus frequency

The substructuring technique using static condensation is used to develop the
foundation stiffness matrices for the each pile group at the bent and at the each
abutment as shown below.
K
Pile
KK
K
KK
KK
K
KK
=
−
−
⎡
⎣
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
⎥
⎥
⎥
11 16
22
33 34
43 44
55
61 66
000 0
00000
00 0 0
00 0 0
00 0 0 0
000 0

where:
k
11
, k
22
, k
33
are corresponding to translational degrees of freedom
k
44
, k
55
, k
66
are corresponding to rotational degrees of freedom
k
16
, k
34
, k
61
k
43
are corresponding to cross coupling degrees of freedom

353
As shown in Figure 6.37 each column is supported on 20 concrete piles and
the west and the east abutments are supported on a single row of 16 and 14 concrete
piles respectively. The pile group supporting each individual column and the
surrounding soil are represented by a 6x6 stiffness matrix and individual pile and the
surrounding soil supporting the bridge abutments are represented by a 6x6 matrix.
6.8.1 Procedures To Develop Stiffness Matrix
The soil-foundation-structure interaction of a bridge system depending on the
soil profile and structural integrity of the piles and pile cap system could be highly
nonlinear. In order to represent the foundation nonlinearity using a linear 6x6
stiffness matrix, equivalent linear foundation stiffness matrices were developed
based on the average recorded displacement obtained from channel 1,2 and 3 located
close to the bottom of the column at bent number 2. The piles were pushed at the top
for the given displacement. The soils reactions along the soil profile were divided by
the deformation along the pile length in order to calculate the linear subgrade
reactions. The subgrade reactions were multiplied by each tributary length in
between the nodal points to calculate the linear springs at each nodal point along the
pile length. To model the soil pile interaction effect, the stiffness of the soil
surrounding each pile were modeled using lumped generalized spring elements
attached to pile at discrete nodal points located along the centerline of the pile. The
linear translational spring elements, namely longitudinal and transverse springs
acting parallel and perpendicular to the bridge centerline, respectively, were used at

354
each pile nodal point for modeling the lateral soil stiffness and a linear vertical
spring was attached at the tip of each pile for modeling vertical pile stiffness. The
combination of the linear springs at the nodal points and the stiffness of the piles
were condensed to a full 6x6 matrix at the pile caps. Using the above procedures as
an example the full 6x6 matrix for the pile group at the bent is given in Table 6.4.
Table 6.4 Bent Foundation Stiffness Matrix(kips, inch and radian)
3.21E+03 0 0 0 0 0
0 3.21E+03 0 0 0 0
0 0 3.03E+03 0 0 0
0 0 0 7.87E+7 0 0
0 0 0 0 4.92E+7 0
0 0 0 0 0 1.35E+7
6.9 Bridge Response Using Substructuring Technique
Figure 6.39 through Figure 6.45 shows comparisons of the calculated
displacement time history response versus recorded response of the bridge
superstructure using a substructure technique. The comparison between the recorded
motions and the computed motions using substructure approach is remarkably close.

355
Springs
K
33
K
66
K
55
K
22
K
11
K
44
K
33
K
66
K
55
K
22
K
11
K
44
Springs
K
33
K
66
K
55
K
22
K
11
K
44
K
33
K
66
K
55
K
22
K
11
K
44

Figure 6.37: Substructuring Approach For Abutments and Bent

356

Pile depth
Soils Reaction Deformation
Subgrade Reaction
L
Pile depth
Soils Reaction
Pile depth
Soils Reaction Deformation Deformation
Subgrade Reaction
L
Subgrade Reaction
L

Figure 6.38: Soil-Pile Interaction

357
-3
-2
-1
0
1
2
3
4
Displacement (inches)
010 20 30 40 50 60
Time (sec)
Recorded (Channel 1)
Model (Column Base)

Figure 6.39: Recorded Response Versus Analytical Response Using Substructure Method (Channel 1)

358
-3
-2
-1
0
1
2
3
Displacement (inches)
010 20 30 40 50 60
Time (sec)
Recorded (Channel 3)
Model (Column Base)

Figure 6.40: Recorded Response Versus Analytical Response Using Substructure Method (Channel 3)

359
-3
-2
-1
0
1
2
3
4
Displacement (inches)
010 20 30 40 50 60
Time (sec)
Recorded (Channel 4)
Model (Channel 4)

Figure 6.41: Recorded Response Versus Analytical Response Using Substructure Method (Channel 4)

360
-3
-2
-1
0
1
2
3
Displacement (inches)
010 20 30 40 50 60
Time (sec)
Recorded (Channel 7)
Model (Channel 7)

Figure 6.42: Recorded Response Versus Analytical Response Using Substructure Method (Channel 7)

361
-3
-2
-1
0
1
2
3
Displacement (inches)
010 20 30 40 50 60
Time (sec)
Recorded (Channel 9)
Model (Channel 9)

Figure 6.43: Recorded Response Versus Analytical Response Using Substructure Method (Channel 9)

362
-1
0
1
Displacement (inches)
010 20 30 40 50 60
Time (sec)
Recorded (Channel 5)
Model (Channel 5)

Figure 6.44: Recorded Response Versus Analytical Response Using Substructure Method (Channel 5)

363
-2
-1
0
1
2
3
Displacement (inches)
010 20 30 40 50 60
Time (sec)
Recorded (Channel 11)
Model (Channel 11)

Figure 6.45: Recorded Response Versus Analytical Response Using Substructure Method (Channel 11)

364
6.10 Conclusions
A global, three-dimensional finite-element structural model of Painter Street
Overpass Bridge which is seismically instrumented was developed to investigate
using two different approaches. The remarkable match achieved between models and
the recorded motions is due to additional steps taken to realistically estimate passive
soil and shear key capacities at the abutment walls. The quality of the structural
bridge evaluations depends directly on the magnitude of soil properties and
capacities. The soil-abutment-foundation-structure interaction model parameters
were based on actual engineering soil properties. The abutment-backfill interaction
was modeled separately (using PLAXIS) and this requires an understanding of earth
pressure theories; the pile-soil interaction was developed using pile design criteria
which are based on pile load tests. The comparison between the recorded motions
and the computed motions is also very favorable for the both foundation modeling
approaches. The study led to conclusions that both the complete foundation system
and the substructure foundation system can offer reasonable solutions. The results
are very encouraging for the geotechnical engineers as significant time saving can be
realized if the foundation substructure can be simplified in seismic response studies.
While the complete foundation system is the most desirable modeling approach, it
may not be required for simple ordinary bridges. Reasonably accurate solutions can
be obtained by the substructure foundation system.

365
CHAPTER SEVEN
CONCLUSIONS AND RECOMMENDATIONS

7.1 Summary and Conclusions
The purpose of this dissertation is to investigate the nonlinear global seismic
soil-abutment-foundation-structure interaction behavior of typical highway skewed-
bridge structures subjected to near-fault ground motions with high velocity pulses.
Three-dimensional nonlinear finite element models of typical box girder
bridges for various skew angles were developed. The bridge decks were modeled
using both three-dimensional shell elements and frame elements. Moment-curvature
relationships were developed to model the nonlinear behavior of the reinforced
concrete columns. Nonlinear link elements were used to model the abutment-backfill
and the expansion gaps in the longitudinal directions and the shear keys and the
expansion gaps in the transverse directions. The structural models were excited using
seven sets of bilateral response-spectra-compatible time history ground motions with
the near fault effects.
A practical and simplified design tool was developed and calibrated with all
available experimental data to predict the nonlinear force displacement capacity of
the abutment backfill. In addition advanced three-dimensional nonlinear finite
element models were developed to simulate the skew abutment-backfill nonlinear
behavior and to understand the mechanism of the problem. A nonlinear closed form
hyperbolic force-deformation relationship which takes the backfill stiffness and

366
ultimate capacity of the backfill into account is developed as a powerful and
effective tool for practicing bridge engineers. Parametric studies were carried out to
better understand the mechanics of skewed bridge behavior. The parameters included
nonlinear wide ranges of skew angle, bridge width, span length, number of columns
per bent, number of actual earthquakes recorded motions and response spectra-
compatible time history ground motions. All the motions have near-source ground
motion characteristics with high velocity pulses. Case study based on the recorded
response of a skewed-two-span reinforced concrete box girder under strong shaking
was performed to validate the modeling techniques developed in this dissertation.
Chapter 3 describes the limit-equilibrium method using mobilized
Logarithmic-Spiral failure surfaces coupled with a modified Hyperbolic soil stress-
strain behavior referred here as the “LSH” model to capture the nonlinear abutment-
backfill force-displacement relationship. A nonlinear closed form hyperbolic force-
deformation relationship referred here as the “HFD” model is developed as a
powerful and effective tool for practicing bridge engineers.
Chapter 4 presents the applications of the two- and three-dimensional finite-
element analysis to investigate the mechanism of the abutment-backfill behavior in
particular bridge abutments with high skew angles. Three-dimensional finite-element
analysis indicates that the total resistance of the mobilized passive wedge is
maximum for the zero skew angle and decreases as the magnitude of skew angles are
increased.

367
Chapter 5 discusses the mechanics and the behavior of the three-dimensional
dynamic behavior of bridges with wide ranges of skew angles, implementation of the
nonlinear abutment-structure interaction into the bridge global model, parametric
studies and discussion regarding the impact ground motions with high velocity
pulses. It was concluded that seismic response of the bridge structure depends on
many factors such as nonlinear abutment springs, column ductility, skew angles and
characteristics of the ground motions.
Chapter 6 presents a three-dimensional nonlinear finite-element model of an
instrumented bridge structure for the validation of the modeling technique applied in
this dissertation. Both direct model and substructure model were implemented. The
direct model included the superstructure, nonlinear columns, nonlinear bridge
abutments and pile foundations with full coupling between structure and foundation
soils. For the substructure model the pile foundations were represented by condensed
stiffness matrices. The bridge system was subjected to the three-component
recorded free-field earthquake motions. The modeling techniques were calibrated
and verified using the recorded response of the bridge structures at various locations.
7.2 Recommendations
The simple HFD relationship shown in Figure 7.1 is recommend to develop
longitudinal nonlinear abutment springs for seismic design of typical highway bridge
when no geotechnical data is available. Eq. (7.1) and Eq. (7.2) may be used to
develop abutment springs for the cohesionless and cohesive backfill respectively.

368

0
K
y
max
y
ave
F
ult
2
F
ult
y
0
K
y
max
y
ave
F
ult
2
F
ult
y
Figure 7.1 Abutment Seat Width Requirements

Fy
y
y
Hy F ()
.
( =
+
8
13
15
in inches, in kips per ft of wall) (7.1)
Fy
y
y
Hy F ()
.
( =
+
8
113
in inches, in kips per ft of wall) (7.2)
The expansion gaps may be model as part of the abutment backbone curve for
computational efficiency.

7.3 Further Research
Although the results of the analysis illustrated that with proper bridge
boundary conditions and suitable bridge components parameters, it is possible to
capture the global seismic behavior of the bridge structures, additional research
studies and verification testing is recommended. Additional research includes
nonlinear dynamic analyses and centrifuge and shaking table tests of skewed bridge

369
structures systems, under earthquake excitations with high velocity pulses. Shaking
table tests of reasonably scaled complete systems, and centrifuge tests of complete
systems are of particular priority. Regarding the analytical research, more parametric
studies should be carried on various components of the bridge structures as listed
below to analyze:
(1) The contribution of the far field geometric damping (which was neglected in
this investigation) along with the nonlinear force-deformation relationship of
the bridge abutment.
(2) The variation of nonlinear abutment backbone curves in both transverse and
longitudinal directions.
(3) Modeling of a realistic representation of the elastomeric pads.
(4) Variations of span length and depth of the superstructure, number of columns
per bent and columns ductility.
(5) The seismic response of bridges with modern column-shaft construction.
(6) Switching the ground motion in the transverse and longitudinal direction.

Regarding the experimental research the following studies is recommended to
validate the analytical models and to develop guidelines and procedures for the
seismic performance based-design of bridge structures:
(1) Conduct large-scale field experiments in the longitudinal direction of the
abutment-embankment bridge approaches constructed with mechanically
stabilized

earth (MSE) walls to develop nonlinear abutment backbone curve.

370
(2) Conduct large-scale field experiments on a skew abutment to develop nonlinear
abutment backbone curves for skewed abutments.
(3) Conduct shaking table tests of reasonably scaled complete systems using 2-
component motions to investigate the global behavior of the bridge-abutment-
structure interaction system.
(4) Conduct geotechnical-structural centrifuge tests including a bridge soil-
abutment-pile-foundation-structure with various bridge geometries and skew
angles using 2-component ground motions with and without velocity pulses to
investigate the global behavior of the bridge system.

371
REFERENCES

AASHTO (1992). Standard Specifications for Highway Bridges, Fifteenth
Edition, Washington, D.C.

Applied Technology Council (1996), Seismic Evaluation and Retrofit of Concrete
Buildings, ATC-40, Report No. SSC 96-01, November.

Ashour, M., Norris, G., and Pilling, P. (1998). Lateral Loading of a Pile in
Layered Soil Using the Strain Wedge Model. Journal of Geotechnical and
Geoenvironmental Engineering, ASCE, Vol. 124, No. 4, pp. 303-315.

American Society for Testing and Materials (ASTM) (2003). “Annual Book of
Standards,” Soil and Rock; Dimension Stone; Geosynthetics, Vol. 04.08.

Bardet, J. P. (1997). Experimental Soil Mechanics, Prentice Hall, Upper Saddle
River, New Jersey 07458.

Bardet, J. P., and S.M. Mortazavi, (1987). “simulation of shear band formation in
overconsolidated clay” Proc. 2d Int. Conf. Const, Laws for Engr.
Mat.,Tuscon, 805-812.

Bhatia, S. K., and Bakeer, R. M. (1989). “Use of the finite element method in
modeling a static earth pressure problem.” International Journal for Numerical
and Analytical Methods in Geomechanics, 13, 207-213.

Bozorgzadeh, A., Megally, S., Restrepo, J., and Ashford, S., 2005. Seismic
Response and Capacity Evaluation of Sacrificial Exterior Shear Keys of
Bridge Abutments, Dept. of Structural Engineering, UC San Diego.

Bransby, P. L., and Smith, I. A. A. (1975). “Side friction in model retaining-wall
experiments.” Journal of the Geotechnical Engineering Division, Proc. of
ASCE, 101(GT7), July, 615-632.

Caltrans (1989). Bridge Design Aids, Division of Structures, October, 14-1-14-7.

Caltrans Seismic Design Criteria (2004).

Caquot, A., and Kerisel, J. (1948). Tables for the Calculation of Passive Pressure,
Active Pressure and Bearing Capacity of Foundations, Gauthier-Villars, Paris.

372
Chen, W. F., and Mizuno, E. (1990). Nonlinear Analysis in Soil Mechanics,
Developments in Geotechnical Engineering Vol. 53, Elsevier.

Chu, S.-C., and Su, J.-J. (1994). "A method for passive earth pressure
computation on sands." Proc. of the Eighth International Conference on
Computer Methods and Advances in Geomechanics, Morgantown, West
Virginia, USA, 22-28 May 1994, Siriwardane and Zaman (eds), Balkema,
Rotterdam, Vol. III, 2441-2445.

Clough, G. W., and Duncan, J. M. (1991). “Earth pressures.” Foundation
Engineering Handbook, 2nd Edition, Fang (ed), van Nosstrand Reinhold, New
York.

Craig, R. F. (1992). Soil Mechanics, Chapter 6, 4th Edition, Chapman & Hall.

Crouse, C. B., and Hushmand, B. (1987). “Estimate of bridge foundation
stiffnesses from forced vibration data." Proc. of the Third International
Conference on Soil Dynamics and Earthquake Engineering, Princeton
University, June.

Crouse, C. B., Hushmand, B., and Martin, G. R. (1987). “Dynamic soil-structure
interaction of a single-span bridge.” Earthquake Engineering and Structural
Dynamics, 15, 711-729.

Crouse, C. B., Hushmand, B., Liang, G., Martin, G. R., and Wood, J. (1985).
“Dynamic response of bridge - abutment-backfill systems.” Proc. Joint U.S.-
New Zealand Workshop on Seismic Resistance of Highway Bridges, San
Diego, CA, May 8-10.

Cutillas, A. M., Lera, M. S. C., and Alarcon, E. (1994). "Dynamic soil-structure
interaction in bridge abutments." Proc. of the Eighth International Conference
on Computer Methods and Advances in Geomechanics, Morgantown, West
Virginia, USA, 22-28 May 1994, Siriwardane and Zaman (eds), Balkema,
Rotterdam, 3, 905-910.

Das, B. M. (1996). Principles of Geotechnical Engineering, 3rd Edition, PWS-
KENT Publishing Company, Boston.

Desai, C. S., and Christian, J. T. (1977). Numerical Methods in Geomechanics,
McGraw-Hill, New York.

373
Desai, C. S., and Siriwardane, H. J. (1984). Constitutive Laws for Engineering
Materials with Emphasis on Geologic Materials, Prentice-Hall, Inc.,
Englewood Cliffs, NJ 07632.

Duncan, J. M., Byrne, P., Wong, K. S., and Mabry, P. (1980). Strength, Stress-
Strain and Bulk Modulus Parameters for Finite Element Analyses of Stresses
and Movements in Soil Masses, Report No. UCB/GT/80-01, August,
Department of Civil Engineering, University of California, Berkeley,
California.

Duncan, J. M., and Chang, C. Y. (1970). "Non-linear analysis of stress and strain
in soils." Journal of Soil Mechanics and Foundations Division, ASCE, 96
(SM5), 1629-1653.

Dubrova, G. A. (1963). Interaction of Soil and Structures. Izd. Rechnoy
Transport, Moscow, U.S.S.R.

Earth Mechanics, 2005. Field Investigations Report for Abutment Backfill
Characterization, Response Assessment of Bridge Systems – Seismic, Live,
and Long Term Loads, Task 3, RTA Contract No. 59, Report to Dept. of
Structural Engineering, University of San Diego, La Jolla, January 18.

El-Gamal, M. E.-S. (1992). Optimum Design for Bridge Abutments Subjected to
Strong Seismic Excitation, M.S. Degree Thesis, University of Nevada, Reno.

Fang, Y.-S., Chen, T.-J., and Wu, B.-F. (1994). "Passive earth pressures with
various wall movements." Journal of Geotechnical Engineering, ASCE,
120(8), 1307-1323.

Faraji, S. Ting, J.M., Crovo, D.S. and Ernst, H. (2001). “Nonlinear analysis of
integral bridges; finite-element model.” Journal of Geotechnical and
Geoenvironmental Engineering., ASCE, Vol. 127, No. 5, 454-461.

Gadre, A. (1997). Lateral Response of Pile-Cap Foundation Systems and Seat-
Type Bridge Abutments in Dry Sand, Ph. D. Thesis, Rensselaer Polytechnic
Institute, August.

Goel, R. K., and Chopra, A. K. (1995). “Evaluation of abutment stiffness in a
short bridge overcrossing during strong ground shaking.” Proc. of the Fourth
U. S. Conference on Lifeline Earthquake Engineering, Technical Council on
Lifeline Earthquake Engineering Monograph No. 6, August.

374
Griffiths, D. V. (1990). “Failure criteria interpretation based on Mohr-Coulomb
friction.” Journal of Geotechnical Engineering, ASCE, 116(6), 986-999.

Griffiths, D. V., and J. H. Prevost, J. H. (1990). “Stress-strain curve generation
from simple triaxial parameters.” International Journal for Numerical and
Analytical Methods in Geomechanics, 14, 587-594.

Griggs, M. N. (1992). Experimental and Analytical Investigation of Laterally
Loaded Piles, M.S. Thesis, Department of Civil and Environmental
Engineering, University of California, Davis.

Hardin, B. O., and Drnevich, V. P. (1972). "Shear modulus and damping in soils:
design equations and curves." Journal of Soil Mechanics and Foundations
Division, ASCE, 98 (SM7), 667-692.

Hausmann, M. R. (1990). Engineering Principles of Ground Modification,
McGraw-Hill Publishing Company.

Hill, R. (1950). The Mathematical Theory of Plasticity, Oxford University Press,
Ely House, London.

James, R.G. and Bransby, P.L. (1971). Experimental and Theoretical
Investigations of a Passive Earth Pressure Problem, Geotechnique, Vol. 20,
No. 1, pp. 17-36, London, England.

Janbu, N. (1957). Earth Pressure and Bearing Capacity Calculations by
Generalized Procedure of Slices. Proc., 4th Int. Conf. on Soil Mechanics and
Foundation Eng., Vol. 2, pp. 207-213.

Joyner, W. B. (1975). "A method for calculating of nonlinear seismic response in
two dimensions." Bulletin of the Seismological Society of America, October,
65(5), 1337-1357.

Joyner, W. B., and Chen, A. T. F. (1975). "Calculation of nonlinear ground
response in earthquakes." Bulletin of the Seismological Society of America,
October, 65(5), 1315-1337.

Kondner, R. L. (1963). "Hyperbolic stress-strain response: cohesive soils."
Journal of Soil Mechanics and Foundations Division, ASCE, 89 (SM1), 115-
143.

Kumar, J., and Subba Rao, K. S. (1997). "Passive pressure coefficients, critical
failure surface and its kinematic admissibility." Geotechnique, 47(1), 185-192.

375

Lam, I. P., Martin, G. R., and Imbsen, R. (1991). “Modeling bridge foundations
for seismic design and retrofitting.” The Third Bridge Engineering
Conference, Denver, Colorado, March 10-13.

Lambe, T. W., and Whitman, R. V. (1969). Soil Mechanics, John Wiley & Sons,
New York.

Larkin, T. J., and Marsh, E. J. (1992). "Two dimensional nonlinear site response
analysis." Proc. Pacific Conference on Earthquake Engineering, 3, 217-227,
Auckland, November.

Levine, M. B., and Scott, R. F. (1989). "Dynamic response verification of
simplified bridge-foundation model." Journal of Geotechnical Engineering,
ASCE, 115(2), February, 246-260.

Mace, N., and Martin, G. R. (1996). “Ground remediation - aspects of the theory
and approach to compaction grouting.” Proc. of the Sixth Japan-Us Workshop
on Earthquake Resistant Design of Lifeline Facilities and Countermeasures
Against Soil Liquefaction, Tokyo, Japan.

Maciejeweski, J. and Jarzebowski, A. (2004), Application of Kinematically
Admissable Solutions to Passive Earth Pressure Problems, ASCE Journal of
Geomechanics, June, Vol. 127.

Malvern, L. E. (1969). "Introduction.” Mechanics of a Continuous Medium,
Prentice H all, Englewood Cliffs, New Jersey.

Maragakis, E. (1985). A Model for the Rigid Body Motions of Skew Bridges,
Report No. EERL 85-02, Earthquake Engineering Research Laboratory,
Caltech, Pasadena.

Maragakis, E., Thornton, G., Saiidi, M., and Siddharthan, R. (1989). “A simple
non-linear model for the investigation of the effects of the gap closure at the
abutment joints of short Bridges.” Earthquake Engineering and Structural
Dynamics, 10, 1163-1178.

Maroney, B., and Chai, Y. H. (1994). “Bridge abutment stiffness and strength
under earthquake loadings.” Proc. of the Second International Workshop on
the Seismic Design of Bridges, Queenstown, New Zealand, August 9-12.

376
Maroney, B., Kutter, B., Romstad, K., Chai, Y. H., and Vanderbilt, E.(1994).
“Interpretation of large scale bridge abutment test results.” Proc. of the Third
Caltrans Seismic Research Workshop, Sacramento, June 27-29.

Martin, G. R., Lam, I. P., Yan, L.-P., Kapuskar, M., and Law, H. (1996). “Bridge
abutments - modelling for seismic response analysis.” Proc. of the 4th
Caltrans Seismic Research Workshop, Sacramento, July 9-11.

Martin, G. R., Yan, L.-P., and Lam, I. P. (1997). Development and
Implementation of Improved Seismic Design and Retrofit Procedures for
Bridge Abutments, Final Report on a Research Project Funded by the
California Department of Transportation, September.

Matsuo, M., Kenmochi, S., and Yagi, H. (1978). “Experimental study on earth
pressure of retaining wall by field tests.” Soils and Foundations, 18 (3), 27-
41.

Meyerhof, G. G. (1963). “Some recent research on the bearing capacity of
foundations.” Canadian Geotechnical Journal, 1 (1), 16-26.

Mokwa, R. L. and Duncan, J. M. (2001). Experimental Evaluation of Lateral-
Load Resistance of Pilecaps. Journal of Geotechnical and Geoenv. Eng.,
ASCE, Vol. 127, No. 2, pp. 185-192.

Naval Facilities Engineering Command (NAVFAC, 1986). Foundation & Earth
Structures, Design Manual 7.02, U.S. Government Printing Office.

Norris, G.M. (1977). The drained shear strength of uniform quartz sand as related
to particle size and natural variation in particle shape and surface roughness.
Ph.D. Thesis, University of California, Berkeley, CA.

Norris, G.M. (1998). Lateral Loading of a Pile in Layered Soil Using the Strain
Wedge Model, Journal of Geotechnical and Geoenvironmental Eng., ASCE,
Vol. 124, No. 4, pp. 303-315.

Ovesen, N.K., 1964, Anchor Slabs, Calculation Methods, and Model Tests.
Danish Geotechnical Institute, Bulletin No. 16, Copenhagen.

Ovesen, N.K., and Stromann, H. (1972). Design Methods for Vertical Anchor
Slabs in Sand. ASCE, Proc., Spec. Conf. Perf. Earth and Earth-Supp. Struct.,
Vol. 2.1, pp. 1481-1500.

377
Pastor, M., Zienkiewicz, O. C., and Chan A. H. C. (1990). “Generalized plasticity
and the modelling of soil behaviour.” International Journal for Numerical and
Analytical Methods in Geomechanics, 14, 151-190.

Prevost, J. H. (1985). "A simple plasticity theory for frictional cohesionless soils."
Soil Dynamics and Earthquake Engineering, 4(1), 9-17.

Prevost, J. H., and Popescu, R. (1996). " Constitutive relations for soil materials."
Electronic Journal of Geotechnical Engineering, October.

Richart, F. E., Jr., Hall, J. R., Jr., and Woods, R. D. (1970). Vibrations of Soils
and Foundations, Prentice-Hall, Englewood Cliffs, NJ.

Rollins, K.M., and Cole, R.T. (2005). Cyclic Lateral Load Behavior of a Pile Cap
and Backfill. Accepted for Publication, Journal of Geotechnical and
Geoenvironmental Engineering.

Rollins, K.M., and Sparks, A. (2002). Lateral Resistance of Full-Scale Pilecap
with Gravel Backfill. ASCE, Journal of Geotechnical and Geoenv. Eng., Vol.
128, No. 9, pp. 711-723.

Romstad, K., Kutter, B., Maroney, B., Vanderbilt, E., Griggs, M., and Chai, Y. H.
(1995). Experimental Measurements of Bridge Abutment Behavior. Dept. of
Civil and Environmental Eng., University of California, Davis, Report No.
UCD-STR-95-1, Sept.

Romstad, K., Kutter, B., Maroney, B., Vanderbilt, E., Griggs, M., and Chai, Y. H.
(1995). Experimental Measurements of Bridge Abutment Behavior,
Department of Civil and Environmental Engineering, University of California,
Davis, Report No. UCD-STR-95-1, September.

Rowe, P. W., and Peaker, K. (1965). Passive Earth Pressure Measurements.
Geotechnique, Vol. 15, No. 1, pp. 22-52, London, England.

Scott, R. F. (1985). "Plasticity and constitutive relations in soil mechanics."
Journal of Geotechnical Engineering, ASCE, 111(5), 563-605.

Seed, H. B., and Idriss, I. M. (1970). Soil Moduli and Damping Factors for
Dynamic Response Analysis, Report No. EERC 70-10, University of
California, Berkley.

SAP 2000 (2005). Integrated Finite Element Analysis and Design of Structures.
by Computers and Structures, Inc., Berkeley, CA.

378

Schanz, T., Vermeer, P.A., and Bonnier, P.G. (1999). Formulation and
Verification of the Hardening-Soil Model. in Brinkgreve, R.B.J., Beyond
2000 in Computation. Geotechnics, Balkema, pp. 281-290.

Shamsabadi, A., Rollins, K., and Kapuskar, M., 2007. “Nonlinear Soil-Abutment-
Bridge Structure Interaction for Seismic Performance-Based Design,” Journal
of Geotechnical and Geoenvironmental Engineering, ASCE (under
publication).

Shamsabadi, A., Law, L., and Martin, G. (2006) Comparison of direct method
versus substructure method for seismic analysis of a skewed bridge,”
Proceedings, 5th National Seismic Conference for Bridges and Highways, San
Francisco, Sept.

Shamsabadi, A., Nordal, S. (2006) “Modeling passive earth pressures on bridge
abutments for nonlinear seismic soil-structure interaction using PLAXIS.”
PLAXIS Bulletin issue 20.

Shamsabadi, A., Law, L., and Martin, G. (2006) Comparison of direct method
versus substructure method for seismic analysis of a skewed bridge,”
Proceedings, 5th National Seismic Conference for Bridges and Highways, San
Francisco, Sept.

Shamsabadi, A., Kapuskar, M., and Zand, A., 2006. “Three-Dimensional
Nonlinear Finite-Element Soil-Abutment Structure Interaction Model for
Skewed Bridges,” Proceedings, 5th National Seismic Conference for Bridges
and Highways, San Francisco, Sept.

Shamsabadi, A., and Kapuskar, M. (2006), “Practical Nonlinear Abutment Model
for Seismic Soil-Structure Interaction Analysis,” 4th Int’l Workshop on
Seismic Design and Retrofit of Transportation Facilities, San Francisco,
March 13-14.

Shamsabadi, A., Ashour, M., and Norris, G. (2005). Bridge Abutment Nonlinear
Force-Displacement-Capacity Prediction for Seismic Design. Journal of
Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 131, No. 2, pp.
151-161.

Shamsabadi, A., Yan, L., and Martin, G. (2004). Three Dimensional Nonlinear
Seismic Soil-Foundation-Structure Interaction Analysis of a Skewed Bridge
Considering Near Fault Effects. Proceedings, US-Turkey Soil-Structural
Interaction Workshop, Oct.

379

Shamsabadi, A., Yan, L., and Law, H. K. (2002a). Nonlinear Abutment-Backfill-
Structure Interaction Behavior for Seismic Design of Bridges. Proceedings,
3rd International Workshop on Performance-Based Seismic Design and
Retrofit of Transportation Facilities, Tokyo, Japan, July 9-11.

Shamsabadi, A., Yan, L., Martin, G. R., Lam, I. P., Tsai, C.-F., and Law, H. K.
(2002b). Effects of Abutment Modeling on Seismic Response of Ordinary
Standard Bridges. Proceedings, Transportation Research Board Road
Builders' Clinic, March.

Shen, J., and Lopez, M. (1997). "Seismic performance of integral abutment
bridges." Seismic Analysis and Design for Soil-Pile-Structure Interactions,
Geotechnical Special Publication No. 70, ASCE, 111-125.

Shields, D. H., and Tolunay, A. Z. (1973). “Passive pressure coefficients by
method of slices.” Journal of the Soil Mechanics and Foundation Division,
Proc. of ASCE, 99(SM12), December, 1043-1053.

Siddharthan, R. El-Gamal, M., and Maragakis, E. A. (1995). “Influence of free-
field strains on nonlinear lateral abutment stiffnesses." Proc. of the Seventh
Canadian Conference on Earthquake engineering, Montreal, June.

Stewart, J., Tacirogla, E., and Wallace, J. (2006). “Full Scale Large Deflection
Testing of Foundation Support Systems for Highway Bridges.” UCLA Draft
Report on a Research Project Funded by the California Department of
Transportation, December.

Taylor, P. W., Bartlett, P. E., and Wiessing, P. R. (1981). “Foundation rocking
under earthquake loading.” Proc. of the Tenth International Conference on
Soil Mechanics and Foundation Engineering, 3, 313-322.

Terzaghi, K. (1943). Theoretical Soil Mechanics, John Wiley & Sons, Inc., New
York, N.Y.

Welsh, A. K. (1967). “Discussion on dynamic relaxation.” Proc. of Institute of
Civil Engineers. 37, 723-750.

Wiessing, P. R. (1979). Foundation Rocking on Sand, M. E. Thesis, University of
Auckland, School of Engineering Report No. 203, November.

380
Wilkins, M. L. (1964). “Fundamental methods in hydrodynamics.” Methods in
Computational Physics, Alder et al. Eds., New York, Academic Press, 3, 211-
263.

Wilson, J. C. (1988). “Stiffness of non-skew monolithic bridge abutments for
seismic analysis.” Earthquake Engineering and Structural Dynamics, 16, 867-
883.

Wilson, J. C., and Tan, B. S. (1990a). “Bridge abutments: formulation of simple
model for earthquake response analysis..” Journal of Engineering Mechanics,
ASCE, 116(8), 1828-1837.

Wilson, J. C., and Tan, B. S. (1990b). “Bridge abutments: assessing their
influence on earthquake response of Meloland Road Overpass.” Journal of
Engineering Mechanics, ASCE, 116(8), 1838-1856.

Wong, K. S., and Broms, B. B. (1994). “Analysis of retaining walls using the
hyperbolic model.” Soil-Structure Interaction: Numerical Analysis and
Modeling, Chapter 17, Bull, J. W. (editor), E & FN SPON.

Abstract (if available)

Abstract The purpose of this thesis is to investigate the nonlinear global seismic soil-abutment-foundation-structure interaction behavior of typical highway skewed-bridge structures subjected to near-fault ground motions with high velocity pulses.

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Asset Metadata

Creator Shamsabadi, Anooshirvan (author)

Core Title Three-dimensional nonlinear seismic soil-abutment-foundation-structure interaction analysis of skewed bridges

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Civil Engineering (Geotechnical Engineering)

Publication Date 02/15/2007

Defense Date 12/15/2006

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag 3D Plaxis,abutment ex,abutment finite element model,bridge abutment,finite element application in geotechnical engineering,hardening soil model,HFD model,log spiral,LSH model,nonlinear load deformation,OAI-PMH Harvest,passive earth pressure,plaxis 3D modeling,SAP2000,skew bridges,skewed abutment,soil-foundation -- structure interaction

Language English

Advisor Martin, Geoffrey R. (committee chair), Bardet, Jean-Pierre (committee member), Corsetti, Frank A. (committee member), Lee, Vincent W. (committee member)

Creator Email anoosh_shams@yahoo.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m271

Unique identifier UC1222631

Identifier etd-Shamsabadi-20070215 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-169791 (legacy record id),usctheses-m271 (legacy record id)

Legacy Identifier etd-Shamsabadi-20070215.pdf

Dmrecord 169791

Document Type Dissertation

Rights Shamsabadi, Anooshirvan

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu