Examination of Bridge Performance through the Extension of Simulation Modeling
and Structural Identification to Large Populations of Structures
A Thesis
Submitted to the Faculty
Of
Drexel University
By
David Robert Masceri Jr.
in partial fulfillment of the
requirements for the degree
of
Doctor of Philosophy
September 2015
© Copyright MMXV
David R. Masceri Jr. All Rights Reserved.
ii
Dedication
For Mom and Dad…
iii
Acknowledgement s
This research was funded by contributions from Drexel University, the National Institute of
Standards and Technology Research Innovation Program, The National Cooperative Research
Program, the National Science Foundation CAREER Award program, the Federal Highway
Administration Long Term Bridge Performance Program, and Pennoni Associates, Inc.
I would like to thank my dissertation committee: A.E. Aktan, I. Bartoli, A.W. Lau, and Y.
Shifferaw for their comments, criticisms, and advice. My advisor, Frank Moon, was a great
mentor throughout my graduate studies at Drexel University and I would like to extend my
sincere gratitude to him for giving me this opportunity.
I want to thank my good friends John Braley and Nick Romano for their support and assistance
with this research, without which this project would not have been completed. I’d also like to
thank John DeVitis for his close support in the lab, at work, and in the field and for being my
friend and co-conspirator from the very beginnings of this journey.
iv
Table of C onte nts
LIST OF TABLES............................................................................................................................................. xii
LIST OF FIGURES .......................................................................................................................................... xv
1. INTRODUCTION.......................................................................................................................................... 1
1.1
Guiding Need for Research ......................................................................................................... 1
1.2
Research Objectives ...................................................................................................................... 4
1.3
Research Scope .............................................................................................................................. 5
1.4
1.3.1
AAHSTO Design Code and Structural Analysis Model.................................................. 5
1.3.2
Effect of Local and Global Structural Abnormalities on Model Updating Behaviors 7
Summary of Thesis Chapters....................................................................................................... 8
1.4.1
Chapter 2: Literature Review ............................................................................................ 8
1.4.2
Chapter 3: Study Design for the Investigation of Inherent Bias in the AASHTO
Single Line-Girder Model ..................................................................................................... 9
1.4.3
Chapter 4: Automated Member Sizing of for Steel Multi-Girder Bridges ................ 9
1.4.4
Chapter 5: Automated Finite Element Model Creation............................................... 9
1.4.5
Chapter 6: Automated Finite Element Model Analysis and Simulation ................. 10
1.4.6
Chapter 7: Investigation of Inherent Bias in the AASHTO Single Line-Girder
Model for Steel Multi-Girder Bridges............................................................................... 10
1.4.7
Chapter 8: Finite Element Model Calibration ............................................................. 11
1.4.8
Chapter 9: Case Studies for Rapid Model Calibration Using Unknown and Known
Parameters ............................................................................................................................. 11
1.4.9
Chapter 10: Conclusions and Further Work ................................................................. 11
2. LITERATURE REVIEW............................................................................................................................. 12
v
2.1
Visual Inspection and Connection to Safety ........................................................................... 12
2.2
Comparison of Single Liner Girder Model with Finite Element Models .......................... 13
2.3
Previous Usage of FEM Models ............................................................................................... 15
2.4
Population Assessment and Research ...................................................................................... 16
2.4.1
2.5
Evaluation of Bias/Non-uniformity in Design Code .................................................... 17
Support Movement ..................................................................................................................... 18
2.5.1
Vertical Support Movement ............................................................................................... 18
2.5.2
Horizontal (Lateral) and Rotational Support Movements ............................................ 21
3. STUDY DESIGN FOR THE INVESTIGATION OF INHERENT BIAS IN THE AASHTO
SINGLE LINE-GIRDER MODEL....................................................................................................... 25
3.1
Summary of SLG Bias Investigation ........................................................................................ 25
3.2
Parametric Study Design ............................................................................................................ 27
3.2.1
Input Parameters of Interest .............................................................................................. 28
3.2.2
Sensitivity Study Results...................................................................................................... 33
3.2.3
Summary................................................................................................................................ 45
3.2.4
Sampling Method Overview .............................................................................................. 47
3.2.5
Notes on Parameters ........................................................................................................... 53
4. AUTOMATED MEMBER SIZING OF FOR STEEL MULTI-GIRDER BRIDGES ................... 55
4.1
Introduction ................................................................................................................................. 55
4.2
Historical Development of Bridge Girder Design and Rating Methods............................ 56
4.3
Development of Automated Member-Sizing ......................................................................... 57
4.3.1
Girder Sizing Algorithm to Satisfy Capacity and Prescriptive Requirements ............ 59
4.3.2
Single Line-Girder Dead and Live Load Demand Calculation .................................... 62
4.3.3
Allowable Stress Design Criteria ....................................................................................... 66
4.3.4
Load and Resistance Factor Design .................................................................................. 77
vi
4.4
Evaluation of Automated Member Sizing for the Study of Single Line-Girder Model
Bias................................................................................................................................................. 89
4.5
4.4.1
Allowable Stress Design...................................................................................................... 89
4.4.2
Load and Resistance Factor Design .................................................................................. 91
Notes on LRFD Girder Sizing for the Study of Bias and Tolerable Support Settlement94
5. AUTOMATED FINITE ELEMENT MODEL CREATION ............................................................ 98
5.1
Overview....................................................................................................................................... 98
5.2
Model Form ................................................................................................................................. 99
5.3
5.4
5.2.1
Girders ................................................................................................................................. 100
5.2.2
Diaphragms ......................................................................................................................... 100
5.2.3
Deck ..................................................................................................................................... 104
5.2.4
Sidewalk ............................................................................................................................... 104
5.2.5
Barriers................................................................................................................................. 104
5.2.6
Boundary Conditions ........................................................................................................ 105
5.2.7
Non-structural Mass .......................................................................................................... 106
5.2.8
Composite Action .............................................................................................................. 106
Model Creation .......................................................................................................................... 107
5.3.1
Strand7 API ........................................................................................................................ 107
5.3.2
Node Placement Algorithm ............................................................................................. 108
5.3.3
Beam Element Placement................................................................................................. 111
5.3.4
Continuity Element Placement ........................................................................................ 111
5.3.5
Shell Placement Algorithm ............................................................................................... 112
Property Assignment ................................................................................................................ 112
5.4.1
Shell Elements .................................................................................................................... 112
5.4.2
Beam Elements .................................................................................................................. 113
vii
5.4.3
5.5
Composite Action Elements ............................................................................................ 119
Verification of Automated Finite Element Modeling for the Study of Bias in the Single
Line-Girder Model .................................................................................................................... 124
5.5.1
Common Modeling Approaches For Multi-Girder Bridges ....................................... 125
5.5.2
Effects of Modeling Choice on Performance of Shell and Element Level Model
Types .................................................................................................................................... 128
5.5.3
Comparison of Results Convergence Agreement Between Shell Element and
Element-Level Model Types ............................................................................................ 141
5.5.4
Investigation of Automated Analysis and Results Extraction Methods ................... 142
5.5.5
Summary and Conclusions of the Composite Beam Modeling Study ...................... 145
5.5.6
Final Investigation into Model Form Using Benchmark Full-Bridge Models ......... 146
6. AUTOMATED FINITE ELEMENT ANALYSIS AND SIMULATION........................................152
6.1
Introduction ............................................................................................................................... 152
6.2
Load Application ....................................................................................................................... 153
6.3
6.4
6.2.1
Dead Load........................................................................................................................... 154
6.2.2
Live Load............................................................................................................................. 156
6.2.3
Support Movement ............................................................................................................ 164
Results Extraction ..................................................................................................................... 167
6.3.1
Response Locations of Interest ....................................................................................... 167
6.3.2
Live Load and Dead Load Results Extraction Steps ................................................... 171
Computation of Rating Factors .............................................................................................. 172
6.4.1
Member Response ............................................................................................................. 173
6.4.2
Load Rating Factors .......................................................................................................... 175
6.4.3
Tolerable Support Settlement .......................................................................................... 176
7. INVESTIGATION OF INHERENT BIAS IN THE AASHTO SINGLE LINE-GIRDER
viii
MODEL FOR STEEL MULTI-GIRDER BRIDGES.......................................................................178
7.1
Introduction ............................................................................................................................... 178
7.2
Sample Population Evaluation ................................................................................................ 178
7.3
7.2.1
Results Convergence ......................................................................................................... 183
7.2.2
Linear Regression Analysis ............................................................................................... 192
Population-Based Comparison of Single Line-Girder and Finite Element Model
Demands for Simply Supported Structures .......................................................................... 193
7.3.1
Single Line Girder Ratings................................................................................................ 193
7.3.2
Finite Element Rating Controlling Girder – Nominal Diaphragm Stiffness ........... 197
7.3.3
Finite Element Ratings – Nominal Diaphragm Stiffness ............................................ 201
7.3.4
Effect of Diaphragm Stiffness on FE Rating Factors ................................................. 219
7.3.5
Effect of Deck Thickness on FE Rating Factors ......................................................... 237
7.3.6
Bivariate Analysis of Ratio of FE and SLG Rating Factors – Nominal Diaphragm
Stiffness and Inclusion of Infinite Fatigue Life Design Criteria ................................ 241
7.3.7
Moment Demands – Nominal Diaphragm Stiffness and Inclusion of Infinite
Fatigue Life Design Criteria ............................................................................................. 249
7.3.8
Bivariate Analysis of Ratio of FE and SLG Moment Demands – Nominal
Diaphragm Stiffness and Inclusion of Infinite Fatigue Life Design Criteria ........... 254
7.4
Population-Based Investigation of Single Line-Girder Bias for Two-Span Continuous
Structures .................................................................................................................................... 266
7.4.1
Single Line Girder Ratings................................................................................................ 266
7.4.2
Finite Element Ratings ...................................................................................................... 270
7.4.3
Tolerable Support Movement .......................................................................................... 278
8. FINITE ELEMENT MODEL CALIBRATION ................................................................................. 292
8.1
Overview..................................................................................................................................... 292
ix
8.2
Single Model Optimization Methods for Updating Parameters ........................................ 294
8.2.1
8.3
Gradient-based Least Squares Minimization ................................................................. 295
Parameter Configuration .......................................................................................................... 302
8.3.1
Unknown Global Parameters........................................................................................... 302
8.3.2
Mass Redistribution ........................................................................................................... 305
8.4
Graphical User Interface for Parameter Estimation............................................................ 308
9. CASE STUDIES FOR RAPID MODEL CALIBRATION USING UNKNOWN AND KNOWN
PARAMETERS........................................................................................................................................ 324
9.1
Overview..................................................................................................................................... 324
9.2
Mossy Creek Bridge .................................................................................................................. 325
9.3
FE Model .................................................................................................................................... 327
9.4
Simulation of Experimental Data ........................................................................................... 329
9.5
Case 1: Global Loss of Composite Action ............................................................................ 331
9.6
9.5.1
Initial Model-Experiment Comparison .......................................................................... 331
9.5.2
Mass Redistribution as Model Fitness Check for a Priori Model................................ 338
9.5.3
Parameter Estimation ........................................................................................................ 341
9.5.4
Mass Redistribution as Model Fitness Check for Updated Model ............................ 344
Case 2: Local Loss of Composite Action along Two Girders ........................................... 346
9.6.1
Initial Model-Experiment Comparison .......................................................................... 347
9.6.2
Mass Redistribution as Model Fitness Check for a Priori Model................................ 355
9.6.3
Parameter Estimation ........................................................................................................ 356
9.6.4
Mass Redistribution as Model Fitness Check for Updated Model ............................ 358
10. CONCLUSIONS AND FUTURE WORK........................................................................................... 360
10.1
Summary of Research Objectives and Scope........................................................................ 360
10.2
Conclusions ................................................................................................................................ 361
x
10.2.1 Objective 1: Development of Automated Design, Modeling, and Simulation Tool
361
10.2.2 Objective 2: Establish the Bias, Trends, and Variability in Performance Due to the
LRFD Design Model and Common Design Assumptions ......................................... 364
10.2.3 Objective 3: Examine Resiliency for Extraneous Demands due to Inherent
Conservatism in Bridge Design Practice ........................................................................ 371
10.2.4 Objective 4: Development of a Streamlined Parameter Estimation Tool ................ 372
10.3
Future Work ............................................................................................................................... 373
LIST OF REFERENCES.............................................................................................................................. 374
APPENDIX A. SUPPORT SETTLEMENT SENSITIVITY................................................................. 379
A.1
Total Composite Section Stress:
Shear Deformation Off – Barrier and Sidewalk
Stiffness On ................................................................................................................................ 379
A.2
Total Composite Section Stress:
Shear Deformation Off – Barrier and Sidewalk
Stiffness Off ............................................................................................................................... 383
A.3
Deck Stress: Shear Deformation Off – Barrier and Sidewalk Stiffness On ................... 387
A.4
Deck Stress: Shear Deformation Off – Barrier and Sidewalk Stiffness Off ................... 392
A.5
Vertical Reaction at the Support: Shear Deformation Off – Barrier and Sidewalk
Stiffness On ................................................................................................................................ 396
A.6
Vertical Reaction at the Support: Shear Deformation Off – Barrier and Sidewalk
Stiffness Off ............................................................................................................................... 401
APPENDIX B. SUPPLEMENTAL MATERIAL TO THE INVESTIGATION OF BIAS IN THE
AASHTO SINGLE LINE-GIRDER MODEL .................................................................................. 406
B.1
Finite Element Rating Controlling Girder ............................................................................ 406
B.1.1
B.2
Service II Limit State Including Out of Plane Bending ............................................... 406
Finite Element Ratings – Nominal Diaphragm Stiffness ................................................... 408
xi
B.2.1
B.3
B.4
B.5
B.6
Service II Limit State Including Out of Plane Bending ............................................... 408
Finite Element Ratings – 10x Nominal Diaphragm Stiffness ............................................ 412
B.3.1
Strength I Limit State ........................................................................................................ 412
B.3.2
Strength I Limit State without Infinite Fatigue Life Design Criteria......................... 416
B.3.3
Service II Limit State ......................................................................................................... 420
B.3.4
Service II Limit State without Infinite Fatigue Life Design Criteria ......................... 424
Finite Element Ratings – 30x Nominal Diaphragm Stiffness ............................................ 428
B.4.1
Strength I Limit State ........................................................................................................ 428
B.4.2
Strength I Limit State without Infinite Fatigue Life Design Criteria......................... 432
B.4.3
Service II Limit State ......................................................................................................... 436
B.4.4
Service II Limit State without Infinite Fatigue Life Design Criteria ......................... 440
Bivariate Analysis of Ratio of FE and SLG Rating Factors ............................................... 444
B.5.1
Strength I Limit State ........................................................................................................ 444
B.5.2
Service II Limit State ......................................................................................................... 448
B.5.3
Service II Limit State with the Inclusion of Out of Plane Bending Moment .......... 451
Bivariate Analysis of Ratio of FE and SLG Moment Demands........................................ 458
B.6.1
Dead Load........................................................................................................................... 458
B.6.2
Superimposed Dead Load Moment Demand ............................................................... 461
B.6.3
Live Load Moment Demand for Interior Girders........................................................ 462
B.6.4
Live Load Moment Demand for Exterior Girders ...................................................... 464
VITA .................................................................................................................................................................. 467
xii
List of Tables
1. Input Parameters for Sensitivity Study.................................................................................................. 28
2. Bridge Configuration Parameters .......................................................................................................... 46
3. LRFD Sizing Constraints ........................................................................................................................ 77
4. Design Criteria for Simply Supported Structures................................................................................ 78
5. Additional Steel Girder Sizing Criteria for Continuous-Span Structures ........................................ 80
6. Distribution Factors Calculated with Section Dimensions................................................................ 82
7. Distribution Factors Calculated with the Lever Rule ......................................................................... 82
8. Fatigue Limit State ................................................................................................................................... 83
9. Benchmark Design Structure Comparison........................................................................................... 90
10. Deck and Sidewalk Concrete Material Property Assignment.......................................................... 113
11. Girder Steel Material Property Assignment ....................................................................................... 114
12. Girder Steel Section Property Assignment......................................................................................... 114
13. Diaphragm Steel Material Property Assignment ............................................................................... 115
14. Diaphragm Channel Section Property Assignment .......................................................................... 116
15. Diaphragm Angle Section Property Assignment .............................................................................. 117
16. Barrier Concrete Material Property Assignment ............................................................................... 118
17. Barrier Rectangular Section Property Assignment ............................................................................ 118
18. Composite Action Element Steel Material Property Assignment .................................................. 119
19. Composite Section Property Assignment........................................................................................... 120
20. Sensitivity Study for Use of Moment of Inertia in Beam Elements as Composite Action Links
Using Two-Beam Model ....................................................................................................................... 122
xiii
21. Sensitivity Study for Use of Shear Area Adjustment in Beam Elements as Composite Action
Links Using Two-Beam Model ............................................................................................................ 123
22. Sensitivity Study for Use of Deck Modulus in Beam Elements as Composite Action Links Using
Two-Beam Model ................................................................................................................................... 124
23. Summary of Demands and Reponses Used in Benchmark Study.................................................. 129
24. Benchmark Model Details..................................................................................................................... 129
25. Element Sizes for Discretization Study .............................................................................................. 132
26. Dead Load Stage Parameter Modifications ........................................................................................ 155
27. Live Load Application Types ............................................................................................................... 157
28. Load Rating Lanes .................................................................................................................................. 158
29. Demand and Truck Load Locations ................................................................................................... 160
30. Support Movement Types .................................................................................................................... 165
31. Response Types and Locations of Interest ........................................................................................ 168
32. Support Movement Locations and Resultant Response Locations of Interest............................ 169
33. Load Factor Calculation Steps.............................................................................................................. 173
34. Load and Resistance Factor Rating Limit States ............................................................................... 176
35. Effective Flexibility Ratio for Bridges Suites 1 and 4 ....................................................................... 224
36. Population Statistics for Effect of Effective Flexibility Ratio on Strength I Rating Factor....... 236
37. Population Statistics for Effect of Effective Flexibility Ratio on Service II Rating Factor ....... 236
38. Two-Span Continuous Rating Factor Population Statistics ............................................................ 266
39. Population Statistics for Tolerable Support Settlement ................................................................... 278
40. A Priori Parameter Values ..................................................................................................................... 330
41. Fixed Bearing Degrees of Freedom..................................................................................................... 330
42. Expansion Bearings Degrees of Freedom .......................................................................................... 330
43. A Priori Model-Experiment Comparison for Global Loss of Composite Action ....................... 332
xiv
44. Mass Zone Multipliers at End of Redistribution for Global Loss of Composite Action .......... 340
45. Model-Experiment Comparison after Initial Mass Redistribution for Global Loss of Composite
Action ....................................................................................................................................................... 341
46. A Priori and Converged Parameter Values ......................................................................................... 342
47. Model Experiment Comparison .......................................................................................................... 342
48. Model Experiment Comparison of Mass Redistribution Solution after Parameter Estimation for
Global Loss of Composite Action....................................................................................................... 345
49. A Priori Model Experiment Comparison for Local Loss of Composite Action .......................... 348
50. Final Mass Redistribution Coefficients for Local Loss of Composite Action ............................. 355
51. Model-Experiment Comparison of Local Loss of Composite Action to Initial Mass Updating356
52. A Priori and Converged Parameter Values for Local Loss of Composite Action ....................... 357
53. Model Experiment Comparison for Local Loss of Composite Action ......................................... 357
54. Model-Experiment Comparison for Local Loss of Composite Action for Mass Zone Updating
................................................................................................................................................................... 359
xv
List of Figure s
1. The M28.9 Bridge ..................................................................................................................................... 14
2. Comparison of Predicted FE Model and Measured Deflection ....................................................... 15
3. Schematic of horizontal and rotational deformations due to settlement and rotation of
foundations at (a) piers and (b) abutment ............................................................................................ 24
4. Schematic Illustrating the Support Movement Considered for the Preliminary Parametric Study
..................................................................................................................................................................... 28
5. Plan View of Skewed Median Model with Shell Element View On ................................................ 29
6. Plan View of Skewed Median Model with Beam Elements .............................................................. 30
7. Isometric View of Skewed Median Model ........................................................................................... 30
8. Isometric View of Skewed Median Model with Vertical Settlement at the Near Abutment and
Contour Shading of Total Fiber Stress in the Beams ......................................................................... 31
9. Plan View of Straight Median Model with Shell Element View On ................................................ 31
10. Plan View of Straight Median Model with Beam Elements Shown................................................. 32
11. Isometric View of Straight Median Model with Beam Elements Shown........................................ 32
12. Isometric View of Straight Median Model with Deflection. Contours Show Deck Shell Stress
and Beam Element Total Fiber Stress. ................................................................................................. 33
13. Effect of Girder Spacing on Total Composite Section Stress .......................................................... 34
14. Effect of Span Length on Total Composite Section Stress Normalized by Span Lngth ............. 35
15. Effect of Span Length on Total Composite Section Stress ............................................................... 36
16. Effect of Skew Angle on Total Composite Section Stress ................................................................ 37
17. Effect of Span Length to Beam Depth Ratio on Total Composite Section Stress ....................... 37
18. Effect of Girder Spacing on Deck Stress ............................................................................................. 38
19. Effect of Span Length on Deck Stress Normalized by Span Length .............................................. 39
20. Effect of Span Length on Deck Stress ................................................................................................. 39
xvi
21. Effect of Skew on Deck Stress............................................................................................................... 40
22. Effect of Span length to Beam Depth Ratio on Deck Stress............................................................ 40
23. Effect of Girder Spacing on Vertical Reaction at the Support ......................................................... 41
24. Effect of Span Length on Vertical Reaction at the Support Normalized by Span Length .......... 42
25. Effect of Span Length on Vertical Reaction at the Support ............................................................. 43
26. Effect of Skew Angle on Vertical Reaction at the Support ............................................................... 44
27. Effect of Span Length to Beam Depth Ratio on Vertical Reaction at the Support ...................... 44
28. Basic Study Workflow ............................................................................................................................. 49
29. Detailed Study Workflow ........................................................................................................................ 50
30. Sampling Methodology for Tolerable Support Study......................................................................... 51
31. Latin Square............................................................................................................................................... 52
32. Overall Girder Design Process .............................................................................................................. 58
33. Girder Sizing Algorithm .......................................................................................................................... 60
34. Fixed End Forces ..................................................................................................................................... 64
35. Simply Supported Girder Design Process ............................................................................................ 79
36. Continuous Span Girder Design Process ............................................................................................. 81
37. Influence of Addition of Fatigue I Limit State on Flange Area for Interior Girders for Ten
Sample Bridges.......................................................................................................................................... 83
38. Influence of Fatigue Limit State on Interior Girder Strength I Rating Factors for Ten Sample
Bridges ........................................................................................................................................................ 84
39. LRFD Section 6.10.6 ............................................................................................................................... 85
40. LRFD Section 6.10.7 ............................................................................................................................... 86
41. LRFD Section 6.10.8 ............................................................................................................................... 87
42. LRFD Appendix A................................................................................................................................... 88
43. FE Model Creation Overview ................................................................................................................ 99
xvii
44. 3D Element Level FE Model ............................................................................................................... 100
45. Channel Section Diaphragms ............................................................................................................... 101
46. Cross-Bracing Diaphragms ................................................................................................................... 101
47. Chevron-Bracing Diaphragms.............................................................................................................. 102
48. Cross-Bracing Diaphragm Connectivity ............................................................................................. 102
49. Skew Bridge with Parallel Diaphragms (applicable to bridges with skew angles less than 20o) 103
50. Straight-Skew Bridge with Normal Contiguous Diaphragms ......................................................... 103
51. Skew Bridge with Normal Non-contiguous Diaphragms ................................................................ 104
52. Illustration of “Alignment” and “Longitudinal” Special Boundary Condition Cases. ................ 106
53. Deck Node Placement ........................................................................................................................... 109
54. Deck Node Placement ........................................................................................................................... 110
55. Overhang Deck Node Placement ........................................................................................................ 110
56. Two-Beam Model with Live Load Combination Applied ............................................................... 121
57. Effect of Composite Action Beam Moment of Inertia on Live Load Moment and Live Load
Distribution ............................................................................................................................................. 121
58. 3D Geometric Element-Level Model ................................................................................................. 127
59. Element-level Model Continuity and Boundary Conditions ........................................................... 130
60. Shell Element Model Continuity and Boundary Conditions ........................................................... 131
61. Discretization Levels of Single-Girder Element-level Model ......................................................... 133
62. Discretization Levels of Two-Girder Element-level Model ............................................................ 134
63. Discretization Levels of Single Girder Shell Element Model.......................................................... 135
64. Discretization Levels for Two-Girder Shell Element Model .......................................................... 136
65. Effect of Shear Deformation Calculation on Shear Force Convergence as a Function of Beam
Depth to Element Length Ratio Subject to Vertical Abutment Settlement in the Two-Girder
Model........................................................................................................................................................ 137
xviii
66. Effect of Shear Deformation Calculation on Moment Convergence as a Function of Beam
Depth to Element Length Ratio Subject to Vertical Abutment Settlement in the Two-Girder
Model........................................................................................................................................................ 138
67. Effect of Shear Deformation Calculation on Axial Force Convergence as a Function of Beam
Depth to Element Length Ratio Subject to Vertical Abutment Settlement in the Two-Girder
Model........................................................................................................................................................ 138
68. Effect of Shear Deformation Calculation on Shear Force Convergence as a Function of Beam
Depth to Element Length Ratio Subject to Point Load at Mid-Span in the Two-Girder Model
................................................................................................................................................................... 139
69. Effect of Shear Deformation Calculation on Moment Convergence as a Function of Beam
Depth to Element Length Ratio Subject to Point Load at Mid-Span in the Two-Girder Model
................................................................................................................................................................... 140
70. Effect of Shear Deformation Calculation on Axial Force Convergence as a Function of Beam
Depth to Element Length Ratio Subject to Point Load at Mid-Span ........................................... 140
71. Effect of Element Size on Deck Stress Under a Vertical Settlement in a Two-Girder Shell
Element Model ....................................................................................................................................... 142
72. Stress Contour in the Principle XX Direction in a Shell Element Model Due to Point Load at
Mid-Span. (a) Deck (b) Beam Web ................................................................................................... 144
73. Computation Time as a Function of Element Discretization Level .............................................. 145
74. Restrained Boundary Degrees of Freedom ........................................................................................ 146
75. Steel W-Shape (I-Beam) ........................................................................................................................ 147
76. Typical Multi-Girder Bridge Cross Section ........................................................................................ 147
77. Percent Change in Response to Support Settlement with Decreasing Mesh Size ....................... 149
78. Percent Change in Response to Dead Load with Decreasing Mesh Size ..................................... 149
79. Percent Change in Response to Point Load with Decreasing Mesh Size ..................................... 150
xix
80. Transverse Lane Positions .................................................................................................................... 158
81. Truck Positions for Simply Supported Bridges ................................................................................. 161
82. Truck positions for Two-span Continuous Bridges ......................................................................... 162
83. Truck Point Loads on FE Model Shell Element Faces.................................................................... 163
84. Lane Point Loads on FE Model Shell Element Vertex Nodes....................................................... 163
85. Simulated Load Combination. Actual Load Combinations are Calculated Using Superimposed
Results ...................................................................................................................................................... 164
86. “Clockwise” Transverse Rotation Support Movement .................................................................... 166
87. “Counter-Clockwise” Transverse Rotation Support Movement .................................................... 166
88. Response Locations of Interest for Support Movements Occurring at the Abutment. ............. 170
89. Response Locations of Interest for Support Movements Occurring at the Pier. ........................ 171
90. Composite Section Stress Superposition ............................................................................................ 174
91. Distribution of Sample Space for Continuous Parameters for Bridge Suite 1 ............................. 179
92. Distribution of Sample Space for Continuous Parameters for Bridge Suite 2 ............................. 180
93. Distribution of Sample Space for Continuous Parameters for Bridge Suite 3 ............................. 181
94. Girder Design Time ............................................................................................................................... 182
95. Number of Required Design Iterations to Achieve Solution.......................................................... 182
96. Suite 1, 2, and 3 Empirical Cumulative Distribution Function of Ratio of FE to SLG LRFR
Strength I Rating with Barrier Stiffness Off ...................................................................................... 184
97. Suite 1, 2, and 3 Empirical Cumulative Distribution Function of Ratio of FE to SLG LRFR
Strength I Rating with Barrier Stiffness On ....................................................................................... 185
98. Suite 1, 2, and 3 Empirical Cumulative Distribution Function of Ratio of FE to SLG LRFR
Service II Rating with Barrier Stiffness Off ....................................................................................... 186
99. Suite 1, 2, and 3 Empirical Cumulative Distribution Function of Ratio of FE to SLG LRFR
Service II Rating with Barrier Stiffness On ........................................................................................ 187
xx
100.Quantile-Quantile Plot of Empirical Cumulative Distribution Function of Ratio of FE to SLG
LRFR Strength I Rating with Barrier Stiffness On with Residual Error Bars .............................. 189
101.Quantile-Quantile Plot of Empirical Cumulative Distribution Function of Ratio of FE to SLG
LRFR Strength I Rating with Barrier Stiffness off with Residual Error Bars .............................. 190
102.Quantile-Quantile Plot of Empirical Cumulative Distribution Function of Ratio of FE to SLG
LRFR Service II Rating with Barrier Stiffness On with Residual Error Bars .............................. 191
103.Quantile-Quantile Plot of Empirical Cumulative Distribution Function of Ratio of FE to SLG
LRFR Service II Rating with Barrier Stiffness off with Residual Error Bars ............................... 192
104.Frequency of Single Line-Girder LRFR Strength I Rating.............................................................. 194
105.Frequency of Single Line-Girder LRFR Strength I Rating without Consideration of Infinite
Fatigue Life Design Criteria .................................................................................................................. 195
106.Frequency of Single Line-Girder LRFR Service II Rating............................................................... 196
107.Frequency of Single Line-Girder LRFR Service II Rating............................................................... 197
108.Frequency of Finite Element LRFR Strength I Rating Controlling Girder .................................. 198
109.Frequency of Finite Element LRFR Strength I Rating Controlling Girder Order from Center
Girder ....................................................................................................................................................... 199
110.Frequency of Finite Element LRFR Service II Rating Controlling Girder Order from Center
Girder without Inclusion of Out of Plane Moment ......................................................................... 200
111.Frequency of Finite Element LRFR Service II Rating Controlling Girder Order from Center
Girder without Inclusion of Out of Plane Moment ......................................................................... 201
112.Frequency of Finite Element LRFR Strength I Rating .................................................................... 203
113.Frequency of Ratio of LRFR Strength I Finite Element Rating to Single Line-Girder Rating . 204
114.Frequency of Ratio of LRFR Strength I Finite Element Rating to Single Line-Girder Rating for
Interior Girders ....................................................................................................................................... 205
xxi
115.Frequency of Ratio of LRFR Strength I Finite Element Rating to Single Line-Girder Rating for
Exterior Girders...................................................................................................................................... 206
116.Frequency of Finite Element LRFR Strength I Rating without Consideration of Infinite Fatigue
Life Design Criteria ................................................................................................................................ 207
117.Frequency of Ratio of LRFR Strength I Finite Element Rating to Single Line-Girder Rating
without Consideration of Infinite Fatigue Life Design Criteria...................................................... 208
118.Frequency of Ratio of LRFR Interior Girder Strength I Finite Element Rating to Single LineGirder Rating without Consideration of Infinite Fatigue Life Design Criteria ............................ 209
119.Frequency of Ratio of LRFR Exterior Girder Strength I Finite Element Rating to Single LineGirder Rating without Consideration of Infinite Fatigue Life Design Criteria ............................ 210
120.Frequency of Finite Element LRFR Service II Rating ..................................................................... 211
121.Frequency of Ratio of LRFR Service II Finite Element Rating to Single Line-Girder Rating .. 212
122.Frequency of Ratio of LRFR Service II Finite Element Rating to Single Line-Girder Rating for
Interior Girders ....................................................................................................................................... 213
123.Frequency of Ratio of LRFR Service II Finite Element Rating to Single Line-Girder Rating for
Exterior Girders...................................................................................................................................... 214
124.Frequency of Ratio of LRFR Service II Finite Element Rating to Finite Element Rating
Including Out of Plane Bending .......................................................................................................... 215
125.Frequency of Finite Element LRFR Service II Rating without Consideration of Infinite Fatigue
Life Design Criteria ................................................................................................................................ 216
126.Frequency of Ratio of LRFR Service II Finite Element Rating to Single Line-Girder Rating
without Consideration of Infinite Fatigue Life Design Criteria...................................................... 217
127.Frequency of Ratio of LRFR Interior Girder Service II Finite Element Rating to Single LineGirder Rating without Consideration of Infinite Fatigue Life Design Criteria ............................ 218
xxii
128.Frequency of Ratio of LRFR Exterior Girder Service II Finite Element Rating to Single LineGirder Rating without Consideration of Infinite Fatigue Life Design Criteria ............................ 219
129.Simplified Model for Diaphragm Flexibility Contribution .............................................................. 221
130.Ratio of FE to SLG Strength I Rating Factor as a Function of the Ratio of the Minimum
Theoretical Distribution Factor to the Maximum Live Load Moment Distribution Factor with
Nominal Diaphragm Stiffness .............................................................................................................. 225
131.Ratio of FE to SLG Strength I Rating Factor as a Function of the Ratio of the Minimum
Theoretical Distribution Factor to the Maximum Live Load Moment Distribution Factor with
30X Nominal Diaphragm Stiffness ..................................................................................................... 226
132.Percent Change in Effective Flexibility Ratio as a Function of Change in Diaphragm Stiffness227
133.Percent Change in Ratio of FE to SLG Rating Factor as a Function of Diaphragm Stiffness for
Bridge #83 ............................................................................................................................................... 228
134.Percent Change in Ratio of FE to SLG Rating Factor as a Function of Diaphragm Stiffness for
Bridge #61 ............................................................................................................................................... 229
135.Percent Change in FE to SLG Rating Factor as a Function of Effective Flexibility Ratio for
Bridge #83 ............................................................................................................................................... 229
136.Percent Change in FE to SLG Rating Factor Ratio as a Function of Effective Flexibility Ratio
for Bridge #61 ........................................................................................................................................ 230
137.Percent Change in Interior Girder Strength I Finite Element Rating Factor as a Function of
Percent Increase in Effective Flexibility Ratio................................................................................... 232
138.Percent Change in Exterior Girder Strength I Finite Element Rating Factor as a Function of
Percent Increase in Effective Flexibility Ratio................................................................................... 233
139.Percent Change in Interior Girder Service II Finite Element Rating Factor as a Function of
Percent Increase in Effective Flexibility Ratio................................................................................... 234
xxiii
140.Percent Change in Exterior Girder Service II Finite Element Rating Factor as a Function of
Percent Increase in Effective Flexibility Ratio................................................................................... 235
141.Ratio of LRFR Strength I FE Rating to SLG Rating as a Function of Length............................ 243
142.Ratio of LRFR Strength I FE Rating to SLG Rating as a Function of Skew Ratio .................... 244
143.Ratio of LRFR Strength I FE Exterior Girder Rating to SLG Rating as a Function of Exterior
Girder Distribution Factor.................................................................................................................... 245
144.Ratio of LRFR Service II FE Rating to SLG Rating as a Function of Length ............................ 246
145.Ratio of LRFR Service II FE Rating to SLG Rating as a Function of Skew................................ 247
146.Ratio of LRFR Service II FE Rating to SLG Rating as a Function of Skew Ratio ..................... 248
147.Ratio of LRFR Service II FE Exterior Girder Rating to SLG Rating as a Function of Exterior
Girder Distribution Factor.................................................................................................................... 249
148.Frequency of Ratio of Predicted SLG Dead Load Moment Demand to Maximum FE Dead Load
Moment Demand ................................................................................................................................... 250
149.Frequency of Ratio of Predicted SLG Superimposed Dead Load Moment Demand to Maximum
FE Superimposed Dead Load Moment Demand ............................................................................. 251
150.Frequency of Ratio of Predicted SLG Live Load Moment Demand to Maximum FE Live Load
Moment Demand for Interior Girders ............................................................................................... 253
151.Frequency of Ratio of Predicted SLG Live Load Moment Demand to Maximum FE Live Load
Moment Demand for Exterior ............................................................................................................. 254
152.Ratio of Maximum FE Dead Load Demand to SLG Dead Load Demand as a Function of Span
Length ...................................................................................................................................................... 255
153.Ratio of Maximum FE Dead Load Demand to SLG Dead Load Demand as a Function of
Girder Spacing ........................................................................................................................................ 256
154.Ratio of Maximum FE Superimposed Dead Load Demand to SLG Superimposed Dead Load
Demand as a Function of Width ......................................................................................................... 257
xxiv
155.Ratio of Maximum FE Superimposed Dead Load Demand to SLG Superimposed Dead Load
Demand as a Function of Girder Spacing.......................................................................................... 258
156.Ratio of Maximum FE Superimposed Dead Load Demand to SLG Superimposed Dead Load
Demand as a Function of Skew Ratio................................................................................................. 259
157.Ratio of Maximum Interior Girder FE Live Load Demand to SLG Live Load Demand as a
Function of Length ................................................................................................................................ 260
158.Ratio of Maximum Interior Girder FE Live Load Demand to SLG Live Load Demand as a
Function of Skew Ratio ......................................................................................................................... 261
159.Ratio of Maximum Interior Girder FE Live Load Demand to SLG Live Load Demand as a
Function of Interior Live Load Distribution Factor ........................................................................ 262
160.Ratio of Maximum Exterior Girder FE Live Load Demand to SLG Live Load Demand as a
Function of Length ................................................................................................................................ 263
161.Ratio of Maximum Exterior Girder FE Live Load Demand to SLG Live Load Demand as a
Function of Skew Ratio ......................................................................................................................... 264
162.Ratio of Maximum Exterior Girder FE Live Load Demand to SLG Live Load Demand as a
Function of Exterior Girder Live Load Distribution Factor .......................................................... 265
163.Frequency of Single Line-Girder Rating of Positive Moment Region for Strength I Limit State
................................................................................................................................................................... 267
164.Frequency of Single Line-Girder Rating of Negative Moment Region for Strength I Limit State
................................................................................................................................................................... 268
165.Frequency of Single Line-Girder Rating of Positive Moment Region for Service II Limit State269
166.Frequency of Single Line-Girder Rating of Negative Moment Region for Service II Limit State
................................................................................................................................................................... 270
167.Frequency of Finite Element Rating of Positive Moment Region for Strength I Limit State ... 271
168.Frequency of Finite Element Rating of Negative Moment Region for Strength I Limit State . 272
xxv
169.Frequency of Ratio of Finite Element Rating to Single Line-Girder Rating of Positive Moment
Region for Strength I Limit State ........................................................................................................ 273
170.Frequency of Ratio of Finite Element Rating to Single Line-Girder Rating of Negative Moment
Region for Strength I Limit State ........................................................................................................ 274
171.Frequency of Finite Element Rating of Positive Moment Region for Service II Limit State .... 275
172.Frequency of Finite Element Rating of Negative Moment Region for Service II Limit State .. 276
173.Frequency of Ratio of Finite Element Rating to Single Line-Girder Rating of Positive Moment
Region for Service II Limit State ......................................................................................................... 277
174.Frequency of Ratio of Finite Element Rating to Single Line-Girder Rating of Negative Moment
Region for Service II Limit State ......................................................................................................... 278
175.Frequency of Strength I Tolerable Support Movement Under Transverse Rotation of Abutment
(in Inches) – Bending Response Over Pier ........................................................................................ 280
176.Frequency of Strength I Tolerable Support Movement Under Transverse Rotation of Pier (in
Inches) – Bending Response at Mid-Span.......................................................................................... 281
177.Frequency of Strength I Tolerable Support Movement Under Vertical Translation of Abutment
(in Inches) – Bending Response Over Pier ........................................................................................ 282
178.Frequency of Strength I Tolerable Support Movement Under Vertical Translation of Pier (in
Inches) – Bending Response at Mid-Span.......................................................................................... 283
179.Frequency of Strength I Tolerable Support Movement Under Transverse Rotation of Abutment
(in Inches) – Shear Response at Pier ................................................................................................... 284
180.Frequency of Strength I Tolerable Support Movement Under Transverse Rotation of Pier (in
Inches) – Shear Response at Abutment.............................................................................................. 285
181.Frequency of Strength I Tolerable Support Movement Under Vertical Translation of Abutment
(in Inches) – Shear Response at Pier ................................................................................................... 286
xxvi
182.Frequency of Strength I Tolerable Support Movement Under Vertical Translation of Pier (in
Inches) – Shear Response at Abutment.............................................................................................. 287
183.Frequency of Service II Tolerable Support Movement Under Transverse Rotation of Abutment
(in Inches) – Bending Response at Pier .............................................................................................. 288
184.Frequency of Service II Tolerable Support Movement Under Transverse Rotation of Pier (in
Inches) – Bending Response at Mid-Span.......................................................................................... 289
185.Frequency of Service II Tolerable Support Movement Under Vertical Translation of Abutment
(in Inches) – Bending Response at Pier .............................................................................................. 290
186.Frequency of Service II Tolerable Support Movement Under Translation of Pier (in Inches) –
Bending Response at Mid-Span ........................................................................................................... 291
187.Schematic of Iterative Parameter Identification Process ................................................................. 294
188.MAC Matrix Plot .................................................................................................................................... 299
189.Development of MAC Matrix Plot with Model Updating .............................................................. 301
190.Symmetric Lateral Redistribution of Deck Mass............................................................................... 307
191.Symmetric Longitudinal Redistribution of Deck Mass .................................................................... 307
192.Asymmetric Lateral Redistribution of Deck Mass ............................................................................ 308
193.Parameter Edit GUI Window – Parameter Group Number .......................................................... 309
194.Parameter Edit GUI Window – Parameter a Priori Value ............................................................... 310
195.Parameter Edit GUI Window – Update Logic Box ......................................................................... 310
196.Parameter Edit GUI Window – Starting, Minimum, Maximum Alpha Values and Alpha Scale310
197.Parameter Edit GUI Window – Starting, Minimum, Maximum Parameter Value...................... 311
198.Parameter Sensitivity GUI Window - Deck Stiffness Sensitivity with a Linear Alpha Scale ..... 312
199.Parameter Sensitivity GUI Window - Deck Stiffness Sensitivity with a Linear Alpha Scale ..... 313
200.Parameter Sensitivity GUI Window - Deck Stiffness Sensitivity with a Linear Alpha Scale ..... 314
201.Parameter Sensitivity GUI Window - Deck Stiffness Sensitivity with a Linear Alpha Scale ..... 315
xxvii
202.Parameter Sensitivity GUI Window - Deck Stiffness Sensitivity with a Linear Alpha Scale ..... 316
203.Parameter Sensitivity GUI Window - Deck Stiffness Sensitivity with a Linear Alpha Scale ..... 317
204.Parameter Sensitivity GUI Window – Composite Action Sensitivity with a Logarithmic Alpha
Scale .......................................................................................................................................................... 318
205.Parameter Sensitivity GUI Window – Boundary Rotational Spring Sensitivity with a Logarithmic
Alpha Scale .............................................................................................................................................. 319
206.Model-Experimental Comparison GUI Window ............................................................................. 322
207.Parameter Estimation GUI Window .................................................................................................. 323
208.Topside of Mossy Creek Bridge ........................................................................................................... 326
209.Underside of Mossy Creek Bridge ....................................................................................................... 327
210.3D FE Model of Mossy Creek Bridge ................................................................................................ 328
211.3D FE Model of Mossy Creek Bridge Shown without Deck Shell Elements.............................. 328
212.Location of “Experimental” Nodes and Girder Numbers.............................................................. 329
213.A Priori MAC Matrix for Global Loss of Composite Action.......................................................... 332
214.A Priori Mode 1....................................................................................................................................... 333
215.“Experimental” Mode 1 ........................................................................................................................ 333
216.A Priori Mode 2....................................................................................................................................... 333
217.“Experimental” Mode 2 ........................................................................................................................ 333
218.A Priori Mode 3....................................................................................................................................... 334
219.“Experimental” Mode 3 ........................................................................................................................ 334
220.A Priori Mode 5....................................................................................................................................... 334
221.“Experimental” Mode 4 ........................................................................................................................ 334
222.A Priori Mode 10..................................................................................................................................... 335
223.“Experimental” Mode 5 ........................................................................................................................ 335
224.A Priori Mode 6....................................................................................................................................... 335
xxviii
225.“Experimental” Mode 6 ........................................................................................................................ 335
226.A Priori Mode 4....................................................................................................................................... 336
227.“Experimental” Mode 7 ........................................................................................................................ 336
228.A Priori Mode 8....................................................................................................................................... 336
229.“Experimental” Mode 8 ........................................................................................................................ 336
230.A Priori Mode 11..................................................................................................................................... 337
231.“Experimental” Mode 9 ........................................................................................................................ 337
232.A Priori Mode 7....................................................................................................................................... 337
233.“Experimental” Mode 10 ...................................................................................................................... 337
234.5 Lateral Mass Zones ............................................................................................................................. 338
235.5 Longitudinal Mass Zones................................................................................................................... 338
236.3x3 Grid Mass Zones ............................................................................................................................ 339
237.MAC Matrix Plot at 5 Iterations .......................................................................................................... 343
238.MAC Matrix Plot at Parameter Convergence (10 Iterations).......................................................... 343
239.Final MAC Matrix Plot for Mass Redistribution Convergence with 5 Lateral Zones for Global
Loss of Composite Action .................................................................................................................... 345
240.Final MAC Matrix Plot Mass Redistribution Convergence with 3x3 Grid Zones for Global Loss
of Composite Action ............................................................................................................................. 346
241.Isometric View of 3D FE Model of Mossy Creek Bridge Indicating Two Girders with Total Loss
of Composite Action ............................................................................................................................. 347
242.A Priori Mac Matrix Plot for Local Loss of Composite Action ...................................................... 348
243.A Priori Mode 1....................................................................................................................................... 349
244.“Experimental” Mode 1 ........................................................................................................................ 349
245.A Priori Mode 2....................................................................................................................................... 349
246.“Experimental” Mode 2 ........................................................................................................................ 349
xxix
247.A Priori Mode 3....................................................................................................................................... 350
248.“Experimental” Mode 3 ........................................................................................................................ 350
249.A Priori Mode 5....................................................................................................................................... 350
250.“Experimental” Mode 4 ........................................................................................................................ 350
251.A Priori Mode 6....................................................................................................................................... 351
252.“Experimental” Mode 5 ........................................................................................................................ 351
253.A Priori Mode 4....................................................................................................................................... 351
254.“Experimental” Mode 6 ........................................................................................................................ 351
255.A Priori Mode 11..................................................................................................................................... 352
256.“Experimental” Mode 7 ........................................................................................................................ 352
257.A Priori Mode 10..................................................................................................................................... 352
258.“Experimental” Mode 8 ........................................................................................................................ 352
259.A Priori Mode 7....................................................................................................................................... 353
260.“Experimental” Mode 9 ........................................................................................................................ 353
261.A Priori Mode 9....................................................................................................................................... 353
262.“Experimental” Mode 10 ...................................................................................................................... 353
263.Example of Software GUI for Model-Experimental Comparison with Local Loss of Composite
Action ....................................................................................................................................................... 354
264.MAC Matrix Plot at Parameter Convergence for Local Loss of Composite Action .................. 358
265.MAC Matrix Plot at Mass Redistribution Convergence with 5 Lateral Zones for Local Loss of
Composite Action .................................................................................................................................. 359
266.Effect of Deck Strength on Total Composite Section Stress ......................................................... 379
267.Effect of Deck Thickness on Total Composite Section Stress....................................................... 380
268.Effect of Girder Spacing on Total Composite Section Stress ........................................................ 380
269.Effect of Span Length on Total Composite Section Stress ............................................................. 381
xxx
270.Effect of Span Length Normalized by Length on Total Composite Section Stress ................... 381
271.Effect of Skew Angle on Total Composite Section Stress .............................................................. 382
272.Effect of Span Length to Girder Depth Ratio on Total Composite Section Stress.................... 382
273.Effect of Deck Strength on Total Composite Section Stress ......................................................... 383
274.Effect of Deck Thickness on Total Composite Section Stress....................................................... 384
275.Effect of Girder Spacing on Total Composite Section Stress ........................................................ 384
276.Effect of Span Length Normalized by Length on Total Composite Section Stress ................... 385
277.Effect of Span Length on Total Composite Section Stress ............................................................. 385
278.Effect of Skew Angle on Total Composite Section Stress .............................................................. 386
279.Effect of Span Length to Girder Depth Ratio on Total Composite Section Stress.................... 386
280.Effect of Deck Strength on Deck Stress ............................................................................................ 387
281.Effect of Deck Thickness on Deck Stress ......................................................................................... 388
282.Effect of Girder Spacing on Deck Stress ........................................................................................... 388
283.Effect of Span Length on Deck Stress ............................................................................................... 389
284.Effect of Span Length on Deck Stress ............................................................................................... 390
285.Effect of Skew Angle on Deck Stress ................................................................................................. 390
286.Effect of Span Length to Girder Depth Ratio on Deck Stress ...................................................... 391
287.Effect of Deck Strength on Deck Stress ............................................................................................ 392
288.Effect of Deck Thickness on Deck Stress ......................................................................................... 393
289.Effect of Girder Spacing on Deck Stress ........................................................................................... 393
290.Effect of Span Length on Deck Stress ............................................................................................... 394
291.Effect of Span Length Normalized by Length on Deck Stress ...................................................... 395
292.Effect of Deck Strength on Vertical Support Reaction ................................................................... 396
293.Effect of Deck Thickness on Vertical Support Reaction ................................................................ 397
294.Effect of Girder Spacing on Vertical Support Reaction .................................................................. 397
xxxi
295.Effect of Span Length on Vertical Support Reaction ...................................................................... 398
296.Effect of Span Length Normalized by Length on Vertical Support Reaction ............................. 399
297.Effect of Skew Angle on Vertical Support Reaction ........................................................................ 399
298.Effect of Span Length to Girder Depth Ratio on Vertical Support Reaction ............................. 400
299.Effect of Deck Strength on Vertical Support Reaction ................................................................... 401
300.Effect of Deck Thickness on Vertical Support Reaction ................................................................ 402
301.Effect of Girder Spacing on Vertical Support Reaction .................................................................. 402
302.Effect of Span Length on Vertical Support Reaction ...................................................................... 403
303.Effect of Span Length Normalized by Length on Vertical Support Reaction ............................. 404
304.Effect of Skew Angle on Vertical Support Reaction ........................................................................ 404
305.Effect of Span Length to Girder Depth Ratio on Vertical Support Reaction ............................. 405
306.Frequency of Finite Element LRFR Service II Rating Controlling Girder with Inclusion of Out
of Plane Moment .................................................................................................................................... 406
307.Frequency of Finite Element LRFR Service II Rating Controlling Girder Order from Center
Girder with Inclusion of Out of Plane Moment ............................................................................... 407
308.Frequency of FE LRFR Service II Rating Including Out of Plane Bending................................ 408
309.Frequency of Ratio of Service II FE Rating Factor to SLG Rating Factor Including Out of Plane
Bending .................................................................................................................................................... 409
310.Frequency of Ratio of Service II FE Rating Factor to SLG Rating Factor Including Out of Plane
Bending for Interior Girders ................................................................................................................ 410
311.Frequency of Ratio of Service II FE Rating Factor to SLG Rating Factor for Exterior Girders
Including Out of Plane Bending .......................................................................................................... 411
312.Frequency of Finite Element LRFR Strength I Rating without Consideration of Infinite Fatigue
Life Design Criteria ................................................................................................................................ 412
xxxii
313.Frequency of Ratio of LRFR Strength I Finite Element Rating to Single Line-Girder Rating
without Consideration of Infinite Fatigue Life Design Criteria...................................................... 413
314.Frequency of Ratio of LRFR Interior Girder Strength I Finite Element Rating to Single LineGirder Rating without Consideration of Infinite Fatigue Life Design Criteria ............................ 414
315.Frequency of Ratio of LRFR Exterior Girder Strength I Finite Element Rating to Single LineGirder Rating without Consideration of Infinite Fatigue Life Design Criteria ............................ 415
316.Frequency of Finite Element LRFR Strength I Rating .................................................................... 416
317.Frequency of Ratio of LRFR Strength I Finite Element Rating to Single Line-Girder Rating . 417
318.Frequency of Ratio of LRFR Strength I Finite Element Rating to Single Line-Girder Rating for
Interior Girders ....................................................................................................................................... 418
319.Frequency of Ratio of LRFR Strength I Finite Element Rating to Single Line-Girder Rating for
Exterior Girders...................................................................................................................................... 419
320.Frequency of Finite Element LRFR Service II Rating ..................................................................... 420
321.Frequency of Ratio of LRFR Service II Finite Element Rating to Single Line-Girder Rating .. 421
322.Frequency of Ratio of LRFR Service II Finite Element Rating to Single Line-Girder Rating for
Interior Girders ....................................................................................................................................... 422
323.Frequency of Ratio of LRFR Service II Finite Element Rating to Single Line-Girder Rating for
Exterior Girders...................................................................................................................................... 423
324.Frequency of Finite Element LRFR Service II Rating ..................................................................... 424
325.Frequency of Ratio of LRFR Service II Finite Element Rating to Single Line-Girder Rating .. 425
326.Frequency of Ratio of LRFR Service II Finite Element Rating to Single Line-Girder Rating for
Interior Girders ....................................................................................................................................... 426
327.Frequency of Ratio of LRFR Service II Finite Element Rating to Single Line-Girder Rating for
Exterior Girders...................................................................................................................................... 427
xxxiii
328.Frequency of Finite Element LRFR Strength I Rating without Consideration of Infinite Fatigue
Life Design Criteria ................................................................................................................................ 428
329.Frequency of Ratio of LRFR Strength I Finite Element Rating to Single Line-Girder Rating
without Consideration of Infinite Fatigue Life Design Criteria...................................................... 429
330.Frequency of Ratio of LRFR Interior Girder Strength I Finite Element Rating to Single LineGirder Rating without Consideration of Infinite Fatigue Life Design Criteria ............................ 430
331.Frequency of Ratio of LRFR Exterior Girder Strength I Finite Element Rating to Single LineGirder Rating without Consideration of Infinite Fatigue Life Design Criteria ............................ 431
332.Frequency of Finite Element LRFR Strength I Rating .................................................................... 432
333.Frequency of Ratio of LRFR Strength I Finite Element Rating to Single Line-Girder Rating . 433
334.Frequency of Ratio of LRFR Strength I Finite Element Rating to Single Line-Girder Rating for
Interior Girders ....................................................................................................................................... 434
335.Frequency of Ratio of LRFR Strength I Finite Element Rating to Single Line-Girder Rating for
Exterior Girders...................................................................................................................................... 435
336.Frequency of Finite Element LRFR Service II Rating ..................................................................... 436
337.Frequency of Ratio of LRFR Service II Finite Element Rating to Single Line-Girder Rating .. 437
338.Frequency of Ratio of LRFR Service II Finite Element Rating to Single Line-Girder Rating for
Interior Girders ....................................................................................................................................... 438
339.Frequency of Ratio of LRFR Service II Finite Element Rating to Single Line-Girder Rating for
Exterior Girders...................................................................................................................................... 439
340.Frequency of Finite Element LRFR Service II Rating ..................................................................... 440
341.Frequency of Ratio of LRFR Service II Finite Element Rating to Single Line-Girder Rating .. 441
342.Frequency of Ratio of LRFR Service II Finite Element Rating to Single Line-Girder Rating for
Interior Girders ....................................................................................................................................... 442
xxxiv
343.Frequency of Ratio of LRFR Service II Finite Element Rating to Single Line-Girder Rating for
Exterior Girders...................................................................................................................................... 443
344.Ratio of LRFR Strength I FE Rating to SLG Rating as a Function of Width ............................. 444
345.Ratio of LRFR Strength I FE Rating to SLG Rating as a Function of Girder Spacing ............. 445
346.Ratio of LRFR Strength I FE Rating to SLG Rating as a Function of Span Length to Girder
Depth Ratio ............................................................................................................................................. 446
347.Ratio of LRFR Strength I FE Interior Girder Rating to SLG Interior Girder Rating as a Function
of Interior Girder Distribution Factor ................................................................................................ 447
348.Ratio of LRFR Service II FE Rating to SLG Rating as a Function of Width .............................. 448
349.Ratio of LRFR Service II FE Rating to SLG Rating as a Function of Girder Spacing .............. 449
350.Ratio of LRFR Service II FE Rating to SLG Rating as a Function of Span Length to Girder
Depth Ratio ............................................................................................................................................. 449
351.Ratio of LRFR Service II FE Interior Girder Rating to SLG Rating as a Function of Interior
Girder Distribution Factor.................................................................................................................... 450
352.Ratio of LRFR Service II FE Rating with the Inclusion of Out of Plane Bending to SLG Rating
as a Function of Length ........................................................................................................................ 451
353.Ratio of LRFR Service II FE Rating with the Inclusion of Out of Plane Bending to SLG Rating
as a Function of Width .......................................................................................................................... 451
354.Ratio of LRFR Service II FE Rating with the Inclusion of Out of Plane Bending to SLG Rating
as a Function of Skew ............................................................................................................................ 452
355.Ratio of LRFR Service II FE Rating with the Inclusion of Out of Plane Bending to SLG Rating
as a Function of Girder Spacing .......................................................................................................... 453
356.Ratio of LRFR Service II FE Rating with the Inclusion of Out of Plane Bending to SLG Rating
as a Function of Span Length to Girder Depth Ratio...................................................................... 454
xxxv
357.Ratio of LRFR Service II FE Rating with the Inclusion of Out of Plane Bending to SLG Rating
as a Function of Span Length to Girder Depth Ratio...................................................................... 455
358.Ratio of LRFR Service II FE Interior Girder Rating with the Inclusion of Out of Plane Bending
to SLG Rating as a Function of Interior Girder Distribution Factor ............................................ 456
359.Ratio of LRFR Service II FE Exterior Girder Rating with the Inclusion of Out of Plane Bending
to SLG Rating as a Function of Exterior Girder Distribution Factor........................................... 457
360.Ratio of Maximum FE Dead Load Demand to SLG Dead Load Demand as a Function of Width458
361.Ratio of Maximum FE Dead Load Demand to SLG Dead Load Demand as a Function of the
Ratio of Span Length to Girder Depth............................................................................................... 459
362.Ratio of Maximum FE Dead Load Demand to SLG Dead Load Demand as a Function of Span
Length to Girder Depth Ratio ............................................................................................................. 459
363.Ratio of Maximum FE Dead Load Demand to SLG Dead Load Demand as a Function of Skew
Ratio.......................................................................................................................................................... 460
364.Ratio of Maximum FE Superimposed Dead Load Demand to SLG Superimposed Dead Load
Demand as a Function of Skew ........................................................................................................... 461
365.Ratio of Maximum FE Superimposed Dead Load Demand to SLG Superimposed Dead Load
Demand as a Function of Span Length to Girder Depth Ratio ..................................................... 461
366.Ratio of Maximum FE Superimposed Dead Load Demand to SLG Superimposed Dead Load
Demand as a Function of Skew Ratio................................................................................................. 462
367.Ratio of Maximum FE Live Load Demand to SLG Live Load Demand as a Function of Width
................................................................................................................................................................... 462
368.Ratio of Maximum FE Live Load Demand to SLG Live Load Demand as a Function of Girder
Spacing ..................................................................................................................................................... 463
369.Ratio of Maximum FE Live Load Demand to SLG Live Load Demand as a Function of Span
Length to Girder Depth Ratio ............................................................................................................. 463
xxxvi
370.Ratio of Maximum FE Live Load Demand to SLG Live Load Demand as a Function of Width
................................................................................................................................................................... 464
371.Ratio of Maximum FE Live Load Demand to SLG Live Load Demand as a Function of Skew
................................................................................................................................................................... 465
372.Ratio of Maximum FE Live Load Demand to SLG Live Load Demand as a Function of Girder
Spacing ..................................................................................................................................................... 465
373.Ratio of Maximum FE Live Load Demand to SLG Live Load Demand as a Function of Span
Length to Girder Depth Ratio ............................................................................................................. 466
1
Abstract
Examination of Bridge Performance through the Extension of Simulation Modeling and
Structural Identification to Large Populations of Structures
David R. Masceri Jr.
The long-term strength and serviceability of common multi-girder bridges in the United States
has been the subject of considerable inquiry in the modern era, in part due to the limited
resources allocated to the preservation of large populations of bridges throughout the U.S. that
are approaching the end of their originally envisioned design lives. While, the conservatism that
has served the civil engineering profession well for over two centuries is still appropriate for new
design, in the case of aging infrastructures it has proven ill-equipped with a resulting track
record of “crying wolf.”
Current methods of population-scale evaluation are primarily qualitative and thus struggle to
effectively support proper prioritization for preservation or replacement of the large numbers of
bridges built during the infrastructure expansions of the 20th Century. The disparity between
what is predicted through current methods of evaluation and what has been shown by refined
quantitative testing indicates that concerns over safety are largely unfounded and hence provides
little evidence for the need to drastically modify current design methodologies; therefore research
in this area must concentrate on strategies for understanding this safety bias and the factors that
influence its behavior on a quantifiable level so it may be used as factional information by
infrastructure stakeholders.
The overarching aim of the research reported herein is to establish a framework whereby realistic
simulations and structural identification may be brought to bear on furthering the understanding
of performance of large populations of bridges.
The completed objectives outlined in this
dissertation include: (1) Develop and validate an automated steel girder design/modeling tool
2
capable of developing realistic estimates of the structural characteristics/responses for broad
populations of bridges.
(2)
Using the tool developed in (1), establish the extent to which
common design assumptions can result in deterministic trends of structural characteristics within
populations of bridges. (3) Using the tool developed in (1), examine how the current practice of
bridge design (inclusive of the conservatism introduced through common assumptions) may
produce bridges that are capable of meeting demands that were not explicitly considered during
member sizing. (4) Develop and validate a streamlined parameter identification tool capable of
reliably improving the representative nature of simulation models through the use of field
measurements.
Key conclusions from this research include: (1) Design decisions such as diaphragm type and
girder spacing that are made based on arbitrary criteria can have significant influence over the
actual properties and reserve capacity of highway bridges. (2) Bias implicit in conventional
design processes provides reserve capacity that is critical to accommodating limit states not
explicitly considered during design.
(3) When incorporating field measurements within
structural assessment, it is crucial to perform model updating. The non-uniqueness associated
with this inverse problem can be reduced through the updating and interpretation of both global
and spatially varying deterministic parameters.
1
1. Introduction
1.1
Guiding Need for Research
The long-term strength and serviceability of common multi-girder bridges in the United States
has been the subject of considerable inquiry in the modern era. This interest has arisen, in part,
due to the limited resources allocated to maintain and preserve the large populations of bridges
throughout the U.S. that are approaching the end of their originally envisioned design lives.
Although it is hard to argue that funding in this area is sufficient, there exists a potentially larger
challenge associated with transforming the bridge engineering profession from one focused on
new design to one equipped to address the full life-cycle management of bridges.
The reality is that the conservatism that has served the profession well for over two centuries is
no longer good enough for this task. The complex challenges posed by aging infrastructures with
their engineering, economic and political dimensions cannot be met by antiquated approaches
developed for a different problem and a different time. While conservatism is appropriate for
new design, in the case of aging infrastructures this approach has proven ill-equipped and has
resulted in a track record of “crying wolf”. Bridge owners faced with budget shortfalls and asked
to make difficult trade-offs have been given a list of over 70,000 “structurally deficient” bridges
with little guidance or basis for prioritization.
Even when performance problems are well
defined and characterized, the conservatism that pervades the profession does not permit
accurate and reliable diagnosis of root causes and thus struggles to develop effective intervention
strategies.
Current methods of population-scale evaluation are primarily qualitative and thus struggle to
effectively support proper prioritization for preservation or replacement of the large numbers of
bridges built during the infrastructure expansions of the 20th Century. The disparity between
2
what is predicted through current methods of evaluation and what has been shown by refined
quantitative testing indicates that concerns over safety are largely unfounded. The uncertainty
inherent in this relationship also provides little evidence for the need to drastically modify
current design methodologies. Further research into new means of large-scale infrastructure
evaluation and management can ameliorate much of the lack of knowledge about the current
system.
The majority of bridges are subjected only to qualitative investigation such as visual inspection,
and the quantitative analysis of these bridges that follows inspection is frequently based upon
that subjective foundation. Visual inspection methods, however, require relatively little time and
cost. The great number of bridges in this country as well as limited budgets mandate cheap,
regular review. Compounding the problem are the simplifying assumptions made by designers
during analysis: the methods that enable the efficient and uniform design of large numbers of
bridges may mask many structural phenomena. A small number of bridges are investigated with
more refined methods, however sophisticated bridge analysis is costly and time-consuming.
Furthermore, in-depth analysis of bridges is myopic: the results from these tests are not used to
supplement overall bridge population management.
The problem may be divided into two distinct but interdependent parts that exist on different
scales: this first part of the problem is long term, low stochasticity, and underlies the second,
which can be frequently short term and random. A metaphor may be the comparison of ocean
currents and local turbulence. A snapshot of a bridge’s performance in time may not help
anyone discern the causes or outcomes of that state, because local turbulence and global currents
are overlapping. The relative dominance of turbulence or current over the current state may be
unknown as well.
In the search for ocean currents it is important to examine simplifying assumptions, rules or
thumb, or other codified heuristics that have implicitly influenced the design of large numbers of
3
bridges.
Perhaps the most obvious such source is the simplified structural analysis model
(commonly referred to as the single-line-girder model) that has been employed to drive bridge
designs since the 1930s. This simplified model encompasses a set of implicit assumptions made
by designers and design code that ultimately govern the structural characteristics of bridges that
may, in turn, influence (in both positive and negative ways) numerous performances through the
service life of the structure.
On the other end of the spectrum, local turbulence within a bridge population is developed due
to circumstances that are unique to specific structures and thus appear random when examined
in the context of broad populations.
An example of this type of influence would be poor
construction quality and/or fabrication errors that may be outliers, but may govern the
performance of certain bridges.
The measurement methods found in the state of practice are coarse enough that measuring the
state vector is impossible.
Refined investigations can describe the state vector however
prediction of future states is unlikely because it cannot discern what is current and what is
turbulence. There is no context for the refined analysis to relate the present state to what may
have brought it there or where it may take it next. Previous research of existing structures has
either looked at the current from a far, coarse perspective, or it has looked closely, but has been
unable to decouple current and turbulence.
Given the random nature of local turbulence, it is difficult to study as it would require detailed
field studies of large populations of bridges. This is currently being undertaken by the U.S.
Federal Highway Administration (FHWA) through the Long-Term Bridge Performance (LTBP)
Program. In contrast however, understanding the potential ocean currents that exist within the
U.S. bridge population is much easier to study due to its more deterministic nature. Achieving
an understanding of these influences is necessary to properly interpret the results from the LTBP
4
Program and ultimately to gain a sound understanding of bridge performance and its multifaceted causes.
1.2
Research Objectives
The overarching aim of the research reported herein is to establish a framework whereby realistic
simulations and structural identification may be brought to bear on furthering the understanding
of performance of large populations of bridges.
More specifically, the following research
objectives were defined and adopted to guide this research effort:
1.
Develop and validate an automated design/modeling tool capable of developing
realistic
estimates
of
the
structural
characteristics/responses
for
broad
populations of bridges. This tool should be capable of (a) sizing members as per
the current AASHTO LRFD Bridge Design Specifications for different bridge
configurations, (b) constructing 3D FE models of common bridge types as per
best practices approaches, (c) simulating a wide range of demands (including
dead load, superimposed dead load, live load, etc.) as per current design
practice, and (d) automating the response extraction process for the various
considered demands.
2.
Using the tool developed in (1), establish the extent to which common design
assumptions can result in deterministic trends of structural characteristics within
populations of bridges. The specific design assumptions selected for this study
include (a) the use of distribution factors to estimate the transverse distribution
of live load and (b) the equal distribution of superimposed dead load across all
girders.
5
3.
Using the tool developed in (1), examine how the current practice of bridge
design (inclusive of the conservatism introduced through common assumptions)
may produce bridges that are capable of meeting demands that were not
explicitly considered during member sizing. The demand selected for this study
was differential vertical and rotational support movement within continuous
bridges.
4.
Develop and validate a streamlined parameter identification tool capable of
reliably improving the representative nature of simulation models through the
use of field measurements. To permit the reliable implementation to populations
of bridges, this tool must provide the user with the ability to quickly and
effectively identify and diagnose error sources that may compromise the model
updating process and distort the representative nature of the model.
1.3
Research Scope
1.3.1
AAHSTO Design Code and Structural Analysis Model
The AASHTO structural design codes focuses heavily on the distinct and opposing definitions of
global loads and local capacities. Global demands rely on system-level force magnitudes, spatial
distribution of responses, probabilities of load combinations and resultant load factors, while
local capacities are concerned with individual members, material stress limits, and member
proportioning criteria for stability or ductility concerns. To permit comparison for the various
limit states it is necessary to reconcile these two scales, that is, to estimate the member- and/or
material-level responses (e.g. moments and stresses) caused by the global demands (e.g. truck
weight and configuration) prescribed by the code. This translation from global to local demands
is a primary goal of structural analysis. Although the specific manner in which this translation is
6
carried out can profoundly impact design, the AASHTO specifications are far less prescriptive
with structural analysis than global demands and local responses.
The modeling approach included in the bridge design specifications explicitly permits the
simulation of common bridge types using a single beam known as the single line-girder (SLG)
model. To enable this simplification the specifications provide distribution factor equations
which allow the designer to estimate the percentage of the global demands that should be used in
the analysis and member-sizing of each individual girder.
The initial distribution factor
equations relied solely on girder spacing to predict load sharing while further evolution of the
design method led to the current distribution factor expressions that consider effects such as span
length and skew. The general approach to relying on simple expressions to account for the load
sharing, however, has remained consistent for over nearly a century.
While much work has gone into tuning load and resistance factors to provide both a uniform and
specified structural reliability, all such work has implicitly assumed that the structural analysis
that remains critical to reconciling the scales of demands and capacities within the design process
is accurate for a broad range of bridge types and configurations. The guiding hypothesis for the
research described in this thesis is that such an assumption is incorrect, and that the SLG model
introduces biases within the bridge population. Further, by documenting and quantifying these
biases this research will (1) help bridge owners to better understand their bridge inventories and
allocate resources to preservation and renewal activities, and (2) provide a quantitative basis for
improving design approaches that ensure a level of uniform safety factor for robustness and
resilience against demands not foreseen in the design phase.
7
1.3.2
Effect of Local and Global Structural Abnormalities on Model Updating
Behaviors
Model-experiment correlation is used to improve the predictive ability of finite element bridge
models. A common misconception is that any calibrated model developed through structural
identification (StID) is more accurate than its a priori precursor and practitioners of StID find the
validity of a calibrated model inherent while ignoring concerns about the validity of the
calibration process itself. Many of the concerns about updated models and the validity of StID
has been addressed by further development of advanced parameter estimation algorithms
The trend towards complexity has many unintended consequences. Advanced algorithms that
use global optimization searches and Bayesian probability seem to produce more objective results
by removing the engineer from the parameter estimation process, yet they may instead front load
the subjective choice in the creation of the updating methodology. Furthermore, better results are
assumed due to the existence of complexity itself, meanwhile the underlying phenomena that
lead to results are masked. An additional byproduct of this is that, due to time commitments,
engineers are overly reliant on their results lest they must repeat the process.
Simple deterministic model updating schemes are easy to implement and have a low time cost.
Many users of these methods do not have sufficient understanding of the requirements for their
successful use or their limitations such as non-unique solutions for a given parameter set, local
minimums as solutions, and choosing proper parameter step sizes and starting values. Their
transparent behavior also has the benefit of keeping the engineer “in the loop.” This oversight
allows an engineer to use heuristics in judging the value of parameter updating results; in fact,
parameter updating behaviors may be used themselves as indicators of structural phenomena.
8
The goal of this study is to revisit these more fundamental model calibration methods and study
their behavior in response to unknown characteristics or defects commonly encountered in
bridges. In order to investigate how uncertainty affects deterministic, gradient-based parameter
estimation algorithms, a series of case control studies will be performed using 3D geometric
element-based FE models. In the first stage of the study, FE models with distorted parameters
will be substituted for an in situ structure and analytical results will be used to simulate dynamic
testing data – specifically, natural frequencies and mode shapes. These distorted parameters will
simulate the presence of unknown global or local characteristics such as the stiffness of the
concrete bridge deck or the loss of composite action along a single girder. The response of
unknown global parameters and spatially varying known parameters during an iterative model
updating process will be investigated.
1.4
Summary of Thesis Chapters
1.4.1
Chapter 2: Literature Review
Presented in this chapter is an overview of current bridge performance evaluation methods
including the use of visual inspection and structural identification. Common analysis models for
the prediction of live and dead load demands are compared as well as an investigation into
previous usage of model forms for the prediction of population-wide performance. Experimental
and meta-data studies on bridge population performance are also present.
Following is a
presentation of known research on the bias and variability of historic and current design codes
and load models. Also presented is an overview of the literature on tolerable support settlement.
9
1.4.2
Chapter 3: Study Design for the Investigation of Inherent Bias in the
AASHTO Single Line-Girder Model
This chapter summarizes the investigation of single line-girder model bias and the design of a
parametric study for this research. Presented is the selection of input parameters, the sensitivity
of performance indices of interest – AASHTO LRFR rating factors and tolerable support
displacements - to input parameters, an overview of sampling methods used for the study, and
various errata related to the modifications of other products of this research in the preparation for
the parametric study.
1.4.3
Chapter 4: Automated Member Sizing of for Steel Multi-Girder Bridges
This chapter discusses the historical development of bridge girder design and rating methods
beyond what is presented in the literature review. The development of an automated girder
sizing algorithm is presented; this includes an overview of the algorithm in order to satisfy
capacity and prescriptive design requirements, the method for calculation of single line-girder
demands, and design criteria used to size steel girders according to AASHTO Allowable Stress
Design and Load and Resistance Factor Design methods.
Also presented is the validation of the
automated member sizing process. Additional notes on design heuristics, design algorithm
modifications, and other criteria specific to the study of single line-girder model bias and
variability are included at the end of the chapter.
1.4.4
Chapter 5: Automated Finite Element Model Creation
Detailed in this chapter is an overview of the method for the production of three-dimensional
finite element models in a guided or automated fashion. Included is a discussion of general
model form, element type, continuity conditions, and boundary conditions. The method for
10
accessing the application programming interface of a finite element solver with common
scripting languages is discussed along with and the model creation algorithm for the placement
of nodes, elements, and property assignment.
Also discussed is a verification of use of
automatically created finite element models for mass simulation of bridges and their use in
research on the bias of the AASHTO single line-girder design model.
1.4.5
Chapter 6: Automated Finite Element Model Analysis and Simulation
This chapter presents methods for the automated analysis and load rating of finite element
models developed using software described in the preceding chapters. Load application, results
extraction, and the computation of AAHSTO Load and Resistance Factor Rating Factors and
resultant tolerable support movements are detailed for simply supported and two-span
continuous steel multi-girder bridges
1.4.6
Chapter 7: Investigation of Inherent Bias in the AASHTO Single LineGirder Model for Steel Multi-Girder Bridges
Presented in this chapter are the preliminary findings of research into the effect of bias and
variability in the AASHTO single line-girder structural analysis model. First discussed are the
qualifications for sample population convergence and acceptance. Following is a presentation of
the variability and bias of the single line-girder model for simply supported structures; this
section contains a discussion of the population single line-girder ratings, finite element model
ratings, controlling girders, effects of diaphragm stiffness on load rating, and the ratio of finite
element ratings and dead and live load moment demands to those predicted with the single linegirder model. An overview of results from the investigation into rating and tolerable support
movement of two-span continuous structures is found at the end of the chapter.
11
1.4.7
Chapter 8: Finite Element Model Calibration
A discussion of the software tools developed to assist in rapid model parameter estimation for
structures with dynamic experimental data is contained within this chapter. First presented are
details for the specific optimization algorithm used for parameter estimation as well as the
method for interfacing finite element models developed as part of this research with the
parameter estimation tool.
Discussed at the end of the chapter is the graphical user interface
developed for rapid structural identification, including software tools for parameter editing,
parameter sensitivity studies, model/experimental data comparison, and parameter estimation.
1.4.8
Chapter 9: Case Studies for Rapid Model Calibration Using Unknown and
Known Parameters
The findings from two case studies on parameter estimation on a simply supported steel multigirder bridge are discussed. A single structure is modified to simulated global or local damage
and used to develop simulated experimental data. Globally distributed parameter estimation for
unknown parameters is used to update the model as well as locally varying mass distribution.
The response of global parameters to simulated structural anomalies as well as mass
redistribution to determine model fitness and anomalous experimental behavior are presented.
1.4.9
Chapter 10: Conclusions and Further Work
This chapter presents conclusions from this thesis as well as recommendations for future work.
12
2. Literature Review
Presented in this chapter is an overview of current bridge performance
evaluation methods including the use of visual inspection and structural
identification. Common analysis models for the prediction of live and
dead load demands are compared as well as an investigation into
previous usage of model forms for the prediction of population-wide
performance. Experimental and meta-data studies on bridge population
performance are also present. Following is a presentation of known
research on the bias and variability of historic and current design codes
and load models. Also presented is an overview of the literature on
tolerable support settlement.
2.1
Visual Inspection and Connection to Safety
Currently, the routine evaluation method for existing highway bridges consists primarily of
visual inspection coupled with structural analysis of the simplified line-girder model. Following
the collapse in 1967 of the Silver Bridge over the Ohio, the federal government mandated regular
inspection of all public road bridges, as well as the maintenance of an inventory of these bridges.
The National Bridge Inventory (NBI) database contains geometry and condition information of
each bridge, and has been the primary source of management data for decades. Inspections are
mandated once every two years for most bridges, and annually for a subset of bridges that
includes fracture critical structures, among others. Section loss, scour, concrete spall, and bearing
alignment, among other visually identifiable defects, are recorded and then used to influence
rating calculations and maintenance recommendations.
Evaluation in most cases is still
performed using the line-girder model. Bridge rating may also be performed utilizing a finite
element model, however this is not required in most cases. An a priori FEM model or “tuned”
FEM model may be used, depending on engineering judgment.
The pertinent issue with visual inspection is that there is a lack of correlation between structural
safety and appearance. While it is certain that deteriorated members, section loss, and deck
cracking contribute to lower load carrying ability in bridges, the magnitude of performance loss
13
is unknown. This is partly due to the uncertainty via bias in structural design; while some
bridges with large safety margins may still reside well above operating strength levels despite
considerable deterioration, others may be significantly affected.
Studies have indicated that the results from traditional visual inspection techniques are subjective
and unreliable.
The Federal Highway Administration demonstrated that only 68% of the
Condition Ratings will vary within one rating point of the average, and 95% will vary within two
points.
(Moore et al. 2001)
Other studies have attempted to utilize finer-resolution visual
inspection records, such as PONTIS, for performance evaluation with limited results; the
conclusion from one such study marked that “most often some conservative assumptions will be
necessary” and that “visual inspection data will never be a substitute for…NDE inspection.”
(Estes and Frangopol 2003)
2.2
Comparison of Single Liner Girder Model with Finite Element Models
Current practice in bridge design is heavily dependent upon single line girder (SLG) models,
where bridges are analyzed as an “equivalent” single girder through making assumptions related
to transverse distribution of forces based on various parameters (such as span length girder
spacing, etc.). While SLG models have proved conservative for design related to live load and
dead load actions, comparisons to field tests show such models significantly underestimate
stiffness and thus would be nonconservative for certain support movement-induced actions. To
illustrate the disparity, consider the MP28.9 Bridge (Figure 3.1) that the PI recently load-tested.
This multi-girder steel bridge was composed of six plate girders with spans of 145 ft. and a 44o
skew.
14
Figure 2.1. The M28.9 Bridge
Figure 2.1 shows a comparison between the simulated deflections of the bridge using 3D
element-level FE model (see Chapter 5) for the details of this modeling approach) and the
deflections measured during a load test. Specifically, this plot shows the mid-span deflection of
each girder (i.e. the displacements across a transverse section of the bridge at mid-span) due to a
single 65 kip tri-axle truck located over the edge girder at mid-span. As shown by this figure, the
FE model captures the transverse deformed shape quite well, and generally over-predicts the
measured displacements by 10% to 20%. Through a detailed model calibration these differences
were traced primarily to the barrier stiffness and the stiffness of the asphalt overlay (the
temperature at the time of test was below 40o F), which were not included in the model shown in
the comparison.
15
Figure 2.2. Comparison of Predicted FE Model and Measured Deflection
An analysis was also carried out using a single-line girder model, and produced displacement
predictions over two times larger than the displacements actually measured (110% to 150%
error). For this particular bridge, the single-line girder model had stiffness consistent with a
deflection limitation of L/2100, while the 3D FE model had a stiffness consistent with an L/5000
deflection limitation.
This significant under-prediction of stiffness by the single-line girder
modeling approach is not uncommon and in most cases it is conservative (Hevener 2003, Eom
and Nowak 2001). For example, in the case of live load demands, the decreased stiffness of the
single-line girder model adds an implicit level of conservatism during the design phase.
2.3
Previous Usage of FEM Models
The uniformity of design codes have also been studied in comparison with results from in situ
testing; this performance difference has frequently been further compared to the predicted
performance of both a priori and calibrated FEM models. Load testing has also demonstrated that
girder distribution factors are consistently lower than those predicted by the AASHTO code.
Analysis has shown that LRFD and AASHTO Standard distribution factors are conservative
compared to FEM models. (Hevener 2003) In some cases, the AASHTO codes have predicted
load distributions 50 to 80% greater than those predicted by FEM analysis, and that FEM analysis
16
predictions are consistently closer to actual bridge performance than the AASHTO codes. (Eom
and Nowak 2001) Catbas et al. demonstrated the variability of bridge rating when comparing
line-girder analysis with a priori and calibrated FEM models. (Catbas et al. 2001)
FEM models have been used in parametric studies to determine the variability or reliability of
line-girder bridge designs. In one such study, LRFD and ASD design standards were used to
design a small number of bridges using the line girder method, 3D FEM models were created for
these designs and the performance of these designs under simulated truck load were compared.
(Baber and Simons 2007) Hevener developed FEM models for bridges tested in separate studies
and found reasonable agreement between measured and simulated deflection. The same study
demonstrated a procedure for creating a parametric study using FEM models based on
hypothetical line-girder designs, simulating truck loads on these FEM models, and developing a
live-load distribution factor equation for line-girder design and analysis. (Hevener 2003)
2.4
Population Assessment and Research
Population performance has also been investigated using a mixture of visual inspection, nondestructive evaluation, and structural identification of a sample set of bridges. A sample set of
bridges were used in a prior study to inform the predictive analysis of a population of bridges in
Pennsylvania. This study used the similarities in design, construction, and material of concrete
T-beam to extrapolate structural identification information from a set of tested bridges to the
larger state-wide population. (Catbas et al. 2001) The Long Term Bridge Performance Program
managed by the Federal Highway Administration is currently seeking to further the knowledge
of overall bridge performance using a targeted sample set, however that project’s focus has been
primarily on material degradation and maintenance, environmental, and rehabilitative effects.
(Friedland et al. 2007)
17
2.4.1
Evaluation of Bias/Non-uniformity in Design Code
The uniformity of the AASHTO design codes has been studied previously using various state
and federal bridge databases along with the traditional line-girder model analysis method. The
National Cooperative Highway Research Program’s Report 700 compared LFR and LRFR rating
factors for 1,500 separate bridges using the AASHTOWare VIRTIS software and Wyoming DOT
BRASS Girder software. VIRTIS, unlike the NBI, includes element-level data; however this study
still utilized only the line-girder method to compute rating factors. Tabsh and Nowak have
shown that the “reliability of bridges designed according to AASHTO Specifications (AASHTO
1998) vary depending on span and type of material.” (Tabsh and Nowak 1991) Tabsh has shown
that “beam designs based on the current AASHTO’s LFD and LRFD methods result in nonuniform safety for different span lengths, section sizes, and beam spacing” by utilizing Monte
Carlo methods to simulate variable loading requirements as well as material properties of
composite steel beams.
(Tabsh 1996)
He has further shown that distribution factors are
conservative for bridges with large girder spacing. (Tabsh 1996) The AASHTO ASD standard
has been shown to be conservative for short span lengths with smaller girder spacing and longer
spans with large girder spacing. The AASHTO LRFD standard is also conservative, however this
has shown, in contrast to the study by Tabsh, to be constant despite changes in span length or
girder spacing. (Eom and Nowak 2001) NCHRP project 20-07/task 122 study of a small sample
of bridges reported that LRFR average about 7% higher than LFR for design-load inventory
ratings (FHWA 2005).
Bridge foundations and geotechnical features should be designed so that their deformations or
differential movements will not cause structural damage to the bridge or any of its auxiliary
features.
Uneven displacements of bridge abutments and pier supports can deteriorate the
quality of the ride, public safety, aesthetics, and structural integrity of a bridge. These types of
movements often lead to expensive maintenance and repairs. Therefore, geotechnical limit states
with consideration of bridge structures are related to foundation deformations. Foundation
18
deformations within the service limit states can be categorized into vertical, horizontal, and
rotational movements.
The following sections provide background information and design
criteria regarding these limit states as well as issues related to construction sequencing.
2.5
Support Movement
2.5.1
Vertical Support Movement
Depending on the type of superstructure, the connection between the superstructure and
substructure, and the span lengths and widths, the magnitudes of differential settlement that can
cause damage to the bridge can vary significantly. In a continuous span bridge, differential
settlements induce bending moments and shear in the superstructure and can potentially cause
structural damage. They can also cause damage to a simple span bridge, although to a lesser
extent. With simple span bridges the major concern is with ride quality and aesthetics. Without
continuity over the supports, the change in slope of the riding surface near the supports may be
more severe than those in a continuous span. It has been found in a number of studies (Grant et
al., 1974 and Skempton and MacDonald, 1956) that the extent of damage of structures caused by
differential settlement is roughly proportional to the angular distortion. The angular distortion is
the normalized measure of differential settlement, including the distance over which the
settlement occurs. For bridge structures, the two points to evaluate the differential settlement are
commonly the distance between adjacent supports.
Currently, the only definitive guidance related to the effect of foundation deformations on bridge
structures is based on a report by the FHWA (1985).
From an evaluation of 314 bridges
nationwide, FHWA (1985) arrived at the following conclusions:
19
The results of this study have shown that, depending on type of spans, length and stiffness of
spans, and the type of construction material, many highway bridges can tolerate significant
magnitudes of total and differential vertical settlement without becoming seriously overstressed,
sustaining serious structural damage, or suffering impaired riding quality. In particular, it was
found that a longitudinal angular distortion (differential settlement/span length) of 0.004 would
most likely be tolerable for continuous bridges of both steel and concrete, while a value of angular
distortion of 0.008 would be a more suitable limit for simply supported bridges.
Another study (NCHRP 1983) states:
In summary, it is very clear that the tolerable settlement criteria currently used by most
transportation agencies are extremely conservative and are needlessly restricting the use of spread
footings for bridge foundations on many soils. Angular distortions of 1/250 of the span length
and differential vertical movements of 2 to 4 inches (50 to 100 mm), depending on span length,
appear to be acceptable, assuming that approach slabs or other provisions are made to minimize
the effects of any differential movements between abutments and approach embankments. Finally,
horizontal movements in excess of 2 in. (50 mm) appear likely to cause structural distress. The
potential for horizontal movements of abutments and piers should be considered more carefully
than is done in current practice.
Based on the above studies, AASHTO LRFD C10.5.5.2 indicates that angular distortions between
adjacent foundations greater than 0.008 rad. in simple spans and 0.004 rad. in continuous spans
should not be permitted in settlement criteria.
This same article states that “other angular
distortion limits may be appropriate after consideration of cost of mitigation through larger
foundations, realignment or surcharge, rideability, aesthetics, and safety.”
In a survey performed for SHRP 2 (2011) regarding the allowable movement of new structures, it
was found that a majority of agencies are not following the guidance on tolerable movement
provided in the AASTHO LRFD Specification. Agencies differed in their criteria for tolerable
20
movement, with some on a case-by-case bases while others had general quantitative
requirements. An example of the use of more stringent criteria can be found in the Pennsylvania
Department of Transportation’s (PennDOT) Structures Design Manual (2012), which states:
The allowable settlement for shallow footings supporting bridge structures shall be based on the
angular distortion (δ'/l) between adjacent support units (i.e., between piers or piers and
abutments) where δ' and l are the differential settlement and span between adjacent units,
respectively. In addition, the maximum net settlement of a footing shall not exceed 1 inch. The
dimensionless ratio δ'/l shall be limited to 0.0025 and 0.0015 for simple and continuous span
bridges, respectively.
Another example of the use of more stringent criteria is from Chapter 10 of the Arizona
Department of Transportation (ADOT) Bridge Design Guidelines (ADOT 2009), which states the
following:
The bridge designer should limit the total settlement of a foundation per 100 ft. span to 0.5 in.
Linear interpolation should be used for other span lengths. Higher total settlement limits may be
used when the superstructure is adequately designed for such settlements. The designer shall also
check other factors such as rideability and aesthetics. Any total settlement that is higher than 2.5
in, per 100 ft. span, must be approved by the ADOT Bridge Group.
From a structural perspective, bridges can handle more movement that traditionally allowed.
There are no technical reasons for agencies to set such arbitrary limitations to the criteria found in
AASHTO LRFD C10.5.5.2. There are practical limits to limiting deformation based on other
structures associated with a bridge, e.g., utilities, approach slabs, wing walls, drainage grades,
etc. It is understood that the differential movement limitations provided in AASHTO LRFD
should be considered in conjunction with the movement tolerances of all bridge facilities.
Comprehensive guidance in design, however, is currently lacking.
21
2.5.2
Horizontal (Lateral) and Rotational Support Movements
According to Moulton et al. (1985) both the frequency and magnitude of vertical movements are
often substantially greater than horizontal movements, but horizontal movements tend to be
more damaging to bridge superstructures.
Herein the word “horizontal” is considered
synonymous with “lateral” (i.e. in the out-of-plane direction of substructures; longitudinal to the
superstructure). Tolerance of the superstructure to horizontal movement depends greatly on the
bridge seat or joint widths, bearing type(s), structure type, and load distribution effects. In the
ideal case, such deformations are accommodated by movement systems and thus do not deform
or result in forces within the superstructure. As a result, the tolerances built into the movement
systems define the degree to which they may isolate the superstructure from lateral and
rotational support movements. If exceeding the isolation capability of movement systems is
defined as intolerable, then simple, rigid-body geometric models are sufficient to compute the
associated limits (since the stiffness characteristics of the superstructure are not engaged).
Moulton et al. (1985) found that horizontal movements less than 1 in. were almost always
reported as being tolerable, while horizontal movements greater than 2 in. were quite likely to be
considered to be intolerable. Based on this observation, Moulton et al. (1985) recommended that
horizontal movements be limited to 1.5 in. The data presented by Moulton et al. (1985) show that
horizontal movements tended to be more damaging when they are accompanied by vertical
movements than when they were not. This is likely because when horizontal movements are
combined with vertical movements, they tend to create rotational demands which have different
implications for various superstructure elements, e.g., simple shear deformations in elastomeric
bearing pads, rotational considerations for pot bearings, cracking within tall (slender)
substructure elements, etc.
For foundations, regardless of whether they are shallow (e.g., spread footings) or deep (e.g.,
driven piles or drilled shafts), horizontal and rotational deformations can occur because of either
22
lateral loads or lateral squeeze of the foundation soil. The following two sections provide of
details related to these two mechanisms.
2.5.2.1
Horizontal and Rotational Deformations Due to Lateral Loads
Assuming that adequate drainage features are in-place and functioning satisfactorily, the primary
source of lateral loads at abutments is earth fill and any surcharges behind the abutment. If
appropriate drainage is not provided, then additional lateral loads can occur due to the build-up
of hydrostatic pressures and frost action. Assuming that the abutment walls are free to displace
laterally and the foundation soils are competent, the minimum movement that can be anticipated
for design is the movement required to mobilize the active earth pressure.
Such lateral
movements can occur by sliding at the base of the spread footing, rotation of pile/shaft caps,
and/or by rotation of the abutment stem wall. In any case, the primary concern is the horizontal
movement and rotation at the superstructure level.
Generally granular fills are used at abutment locations. For these types of materials, the typical
horizontal movements that can be anticipated are in the range of 0.001 to 0.004 times the height of
the abutment wall. Thus, for example, if the abutment is 20-ft tall, horizontal movements in the
range of ¼ in. to 1 in. may be anticipated. In a general construction sequence, the earth fill
behind the abutment is substantially complete prior to placement of the superstructure. In this
case, the horizontal movement at the superstructure level is virtually complete and should not be
of concern assuming that the vertical joint between the end of the superstructure and the
abutment back-wall was designed properly to accommodate the movement. However, the lateral
movements caused by lateral loads due to surcharges, such as live loads and thermal effects,
experienced by the abutment after the placement of the superstructure should be considered in
the design of the bridge structure.
23
At pier locations, the primary source of lateral loading is from thermal effects, braking forces and
forces due to unequal spans if any exist on either side of the pier. Assuming that the pier
substructure has sufficient structural resistance, these lateral loads are primarily resisted by
sliding resistance at the base of the spread footing or the structural resistance at the connection of
the cap with the underlying deep foundations. Where the foundation soils are weak in shear
strength (e.g., fine-grained clayey soils) the interface shear strength may be small which increases
the potential for sliding. Once the interface shear strength is overcome by the horizontal forces,
large sliding movements can occur.
2.5.2.2
Horizontal and Rotational Deformations Due to Lateral Squeeze
In addition to lateral forces, lateral squeeze is often a source of horizontal and rotational
substructure movements (Samtani and Nowatzki, 2006; Samtani et al. 2010).
Figure 2.3 shows
schematics of such movements at pier and abutment locations. The lateral squeeze phenomenon
is due to an unbalanced load at the surface of the relatively soft soil with the depth of significant
influence (DOSI) of the foundation subsurface stresses. The lateral squeeze behavior may be: (a)
short-term undrained deformation that results in horizontal deformation from a local bearing
resistance type of failure, or (b) long-term drained, creep-type deformation. Creep refers to the
slow deformation of soils under sustained loads. In addition to rigid-body deformation of the
substructures, the flexibility of the substructures themselves can act to amplify the resulting
support movements experienced by the superstructure.
24
Fill
Figure 2.3. Schematic of horizontal and rotational deformations due to settlement and
rotation of foundations at (a) piers and (b) abutment
25
3. Study Design for t he Investigation of Inherent Bias i n the
AASHTO Single Li ne-Girder Model
This chapter summarizes the investigation of single line-girder model
bias and the design of a parametric study for this research. Presented is
the selection of input parameters, the sensitivity of performance indices
of interest – AASHTO LRFR rating factors and tolerable support
displacements - to input parameters, an overview of sampling methods
used for the study, and various errata related to the modifications of
other products of this research in the preparation for the parametric
study.
3.1
Summary of SLG Bias Investigation
As noted in Chapter 2, research suggests that the demands found in in situ multi-girder bridges
are usually less than that predicted by the single line girder model and that this difference
between the predicted and experimentally determined loads is highly variable. Also noted is
previous research that has shown a priori 3D geometric finite element models of bridges to
predict demands closer to those found in actual structures than predictions based on the SLG
model. This study utilizes the ability of the RAMPS software, described in Chapters 4, 5, and 6,
to rapidly design, and then construct and analyze FE models of large numbers of steel multigirder bridges to investigate the bias introduced by the SLG model (compared to an element-level
FE model that explicitly simulates transverse stiffness mechanisms).
Rating factors were examined in order to better understand the magnitude and variability of the
inherent conservatism imposed by the SLG model. In addition, to illustrate the value of this
conservatism in accommodating demands that were not explicitly considered during design, this
research examined the level of support movement – which is not considered in current design
approaches – that may be tolerated because of the SLG model bias.
A group of common geometric constraints, continuity conditions, and bearing conditions were
identified as being the most impactful on global structural performance for simply supported and
26
two-span continuous steel multi-girder bridges. After these final set of parameters (and their
ranges) were identified, it was necessary to sample them to develop a representative “bridge
suite” (or sample of bridges) to allow identify the impact of these factors on rating factor and
tolerable support movement. This representative bridge suite was used to simulate in situ global
structural responses to dead load, live load, and support movement.
Steel girders were sized using the AASHTO LRFD code and the SLG demand model for the given
inputs:
1.
Length
2.
Width
3.
Skew
4.
Girder Spacing
5.
Span length to girder depth ratio
FE models were created using the design and then analyzed for dead load, live load, and support
movement demands with the following continuity and boundary conditions:
1.
Concrete barrier stiffness on and pinned bearings
2.
Concrete barrier stiffness off and pinned bearings
The responses of these bridges to dead and live load as well as support movement were
compared to those predicted using the AASHTO LRFD and SLG models and studied to identify
sources of bias and variability.
27
3.2
Parametric Study Design
In order to properly plan for the multivariate study of bias, a smaller, single degree-of-freedom
study was performed using the current bridge design, model development, and analysis
software. Specifically, a set of bridge parameters were varied for a benchmark structure that
consisted of a two-span continuous bridge. For each set of parameters, appropriate girder sizes
were selected using the AASHTO specifications and an element-level FE model was created for
each design (consistent with the modeling approach described in Chapter 5). These FE models
were then used in two phases following phases: Phase I examines the bias – or “extra”
conservatism” – inherent in the use of the SLG model in LRFD design; Phase II examines how
such extra conservatism is important in the accommodation of demands not explicitly considered
or foreseen in design – i.e. support movement. Response values (such as various stresses and
reaction forces) were extracted from the simulation results and compared to the bridge
parameters to identify levels of sensitivity. The goal of this portion of the study was to determine
whether the performance indices of interest were sensitive to the chosen parameters and to verify
that the parameter ranges of interest were adequate for the larger parametric study.
In order to simplify this process, only one type of demand input was studied: a 1 in. vertical
settlement applied to one abutment of the two-span continuous benchmark structure (Figure 3.1).
Three responses to this input were studied, namely, longitudinal stresses in the girders (termed
total fiber stress), tensile stresses in the deck, and the reactions at the supports. Given the
benchmark structure selected (2-span continuous bridge), the maximum value for each of the
three responses will be located over the interior support. Therefore, each of these responses were
extracted above the interior support at three locations: at the exterior (or fascia) girder, at the 1st
interior girder, and at the center interior girder.
28
Figure 3.1. Schematic Illustrating the Support Movement Considered for the Preliminary
Parametric Study
3.2.1
Input Parameters of Interest
The study carried out under this task varied a series of parameters over a specific range one at a
time while all other parameters were held constant at their “median” value. Table 3.1 provides
the parameters included in this study, their ranges, and their median values.
Table 3.1. Input Parameters for Sensitivity Study
Constant Parameters
Varied Parameters
Parameter
Values
Median Value
Span Length
40 ft to 160 ft
100 ft
Bridge Width
36 ft. to 90 ft.
60 ft
Girder Spacing
5 to 12 ft
Skew Angle
0o to 60o
7.5 ft
0o and 20o degree skews were
both used as median values.
Span to Depth Ratio
Stiffness of non-structural
components (barriers and
sidewalks)
“Primary” Bridge Types
Design Method
Superstructure Continuity
Shear deformation of members
Overhang
Material Properties (elastic
modulus of concrete)
Deck Thickness
Sidewalk Dimensions
Barrier Dimensions
L/20, L/22, L/25,
L/28, L/30
Assumed fully
active or ignored
L/25
N/A
Not Varied
Not Varied
Not Varied
Not varied
Dependent on other
parameter
Steel
Allowable Stress Design
2-span continuous
Off
Not Varied
4000 ksi
Not Varied
Not Varied
Not Varied
8 inches
10 in high x 48 in wide
27 in high x 12 in wide
½ of girder spacing
29
A few parameters, such as bridge type, design methodology, and superstructure continuity were
not varied as they were either (1) deemed to be essential to the larger multi degree-of-freedom
study and therefore sensitivity studies were unnecessary, or (2) not currently feasible due to the
present state of development of the design and model building software.
In addition, the
Allowable Stress Design method was employed as the incorporation of the LRFD approach was
not developed at that stage of the research.
Additional parameters that were held constant include the overhang dimension of the deck and
the barrier and sidewalk dimensions. The overhang for each structure was kept to one half of the
girder spacing. The sidewalks were modeled as 10 in. high by 48 in. wide while the barriers were
modeled as 27 in. high by 12 in. wide.
These values may be modified in later studies.
Overhangs, sidewalks, and barriers dimensions were chosen as a generic average that would
provide approximate loading effects to the exterior girders under dead load as well as stiffness
(when required) during dead, live, and settlement loading. To provide a more comprehensive
study, two median values of skew were employed. Specifically, the median values of skew were
taken as both zero degrees and 20 degrees, which results in two “parallel” single degree-offreedom studies. Figures Figure 3.2 though Figure 3.9 show the two “median” models.
Figure 3.2. Plan View of Skewed Median Model with Shell Element View On
30
Figure 3.3. Plan View of Skewed Median Model with Beam Elements
Figure 3.4. Isometric View of Skewed Median Model
31
Figure 3.5. Isometric View of Skewed Median Model with Vertical Settlement at the Near
Abutment and Contour Shading of Total Fiber Stress in the Beams
Figure 3.6. Plan View of Straight Median Model with Shell Element View On
32
Figure 3.7. Plan View of Straight Median Model with Beam Elements Shown
Figure 3.8. Isometric View of Straight Median Model with Beam Elements Shown
33
Figure 3.9. Isometric View of Straight Median Model with Deflection. Contours Show
Deck Shell Stress and Beam Element Total Fiber Stress.
3.2.2
Sensitivity Study Results
Included in this chapter is a subset of the results obtained from the single degree-of-freedom
sensitivity studies, which were selected to illustrate the key findings. The figures below show
both 0o and 20o skew bridges on the same graph. All graphs shown below are for FE models with
non-structural element stiffness turned on and shear deformation of all beam elements ignored.
The complete set of results from this study can be found in Appendix A.
3.2.2.1
Total Composite Section Stress
Total composite section stress was calculated for the exterior, 1st interior, and centerline girder
over the central pier of a number of 2-span continuous bridges. Figure 3.10 through Figure 3.14
detail the relationship between a subset of the parameters of interest and the total stress in the
composite section. This stress was taken at the extreme bottom fiber of a two-node beam element
directly over the center support. The other parameters not shown did not exhibit a strong or
34
unexpected relationship with total composite section stress; these results are included in
Appendix A.
Figure 3.10 illustrates the relationship between girder spacing and total composite section stress
over the center pier due to a 1 inch settlement at an exterior abutment. Stresses for most girders
follow a slightly quadratic trend, however the exterior girder from the 0° skew bridge and 2nd
girder from the 20° skew bridge diverge from the apparent trend. Although the cause of this
apparent anomalous behavior is still in question, based on the interpretation conducted thus far,
it appears to be due to the fact that girder spacing, bridge width, and overhang are all coupled
parameters. Regardless, the influence of these parameters appears quite small compared with
span length and skew angle (See Figure 3.12 and Figure 3.13).
Figure 3.10. Effect of Girder Spacing on Total Composite Section Stress
35
Figure 3.11 and Figure 3.12 show the influence of span length on total fiber stress both
normalized by span length and non-normalized, respectively. As apparent from these figures,
the relationship between girder stress and span length is nonlinear. In addition, span length has
the largest influence over girder stress due to support settlement, with stress varying over 800%
for the 160 ft. to 40 ft. bridge spans examined.
Figure 3.11. Effect of Span Length on Total Composite Section Stress Normalized by
Span Length
36
Figure 3.12. Effect of Span Length on Total Composite Section Stress
Figure 3.13 shows the influence of skew on girder stress due to a support settlement, and
indicates that the exterior girder stresses are quite sensitive to skew angle, while the interior
girder stresses are not. Further, the influence of girder depth to span ratio is shown in Figure
3.14. Based on this figure it appears the relationship between girder depth to span ratio and
stress is linear in nature.
37
Figure 3.13. Effect of Skew Angle on Total Composite Section Stress
Figure 3.14. Effect of Span Length to Beam Depth Ratio on Total Composite Section
Stress
38
3.2.2.2
Deck Stress
The influence of a number of parameters on the deck stress directly above the exterior, 1st
interior, and centerline girder over the central pier of the benchmark structure was examined.
Figure 3.15 through Figure 3.19 detail the relationship between a subset of the parameters of
interest and the stress in the topmost fiber of the deck at the centerline of each girder. A complete
set of results can be found in Appendix A.
The deck stresses shown in these plots were
calculated using the procedure outlined in Chapter 6 of this thesis. As apparent from these
figures, the same trends observed for girder stresses were also observed in the case of deck
stresses.
Figure 3.15. Effect of Girder Spacing on Deck Stress
39
Figure 3.16. Effect of Span Length on Deck Stress Normalized by Span Length
Figure 3.17. Effect of Span Length on Deck Stress
40
Figure 3.18. Effect of Skew on Deck Stress
Figure 3.19. Effect of Span length to Beam Depth Ratio on Deck Stress
41
3.2.2.3
Vertical Reaction at the Support
The influence of various parameters on the vertical reaction of the supports at the exterior, 1st
interior, and centerline girders over the central pier of the benchmark structure was examined.
Figure 3.20 and Figure 3.24 show the relationship between a subset of the parameters of interest
and these boundary reactions. A full set of results can be found in Appendix A.
As apparent from these plots, similar trends observed for both total fiber stress in the girders and
deck stresses were also observed in the case of boundary reactions. Some notable exceptions
include more consistent trends related to girder spacing, opposite trends between exterior and
interior reactions relative to skew, and a nonlinear relationship with girder depth to span length
ratio.
Figure 3.20. 1. Effect of Girder Spacing on Vertical Reaction at the Support
42
Figure 3.21. Effect of Span Length on Vertical Reaction at the Support Normalized by
Span Length
43
Figure 3.22. Effect of Span Length on Vertical Reaction at the Support
44
Figure 3.23. Effect of Skew Angle on Vertical Reaction at the Support
Figure 3.24. Effect of Span Length to Beam Depth Ratio on Vertical Reaction at the Support
45
3.2.3
Summary
The goals of this task were (1) to determine whether the performance indices of interest were
sensitive to the chosen parameters and (2) to verify that the parameter ranges of interest were
adequate for the bias study. To satisfy these objectives a series of single degree of freedom
parametric studies were carried out on a set of two-span continuous, steel multi-girder
benchmark bridges. Based on the results of this study, the following conclusions were drawn.
1.
The parameters that exert the most significant influence on the responses due to
support movements are Span Length, Skew Angle, and Girder Depth-to-Span
Ratio. While Span Length and Skew Angle had nonlinear influences over all
responses examined, Girder Depth to Span showed a linear influence over
stresses and slightly nonlinear influence over support reactions.
2.
Parameters such as Girder Spacing, Bridge Width, and Barrier/Sidewalk
Participation have significantly less influence on the responses induced by
vertical support settlement.
3.
While several parameters showed relatively small influence over responses to
support movement, such parameters have been shown (through past studies by
the investigators) to have influence over both live and dead load demand
calculations. As such, in order to reliably identify live load rating factor bias and
tolerable support movements for the limit states outlined in this this chapter,
such parameters must be included in the larger study.
4.
Live load and dead are sensitive to the parameters selected for this study, and as
these parameters are also integral to calculations of AASHTO LRFR rating
factors, these parameters are appropriate for use in the bias study.
46
5.
The results of this initial study indicated that the initially proposed parameters
and bounds are appropriate for the bias study. Table 3.2 provides a summary of
these parameters.
Table 3.2. Bridge Configuration Parameters
Parameter
“Primary” Bridge Types
Discrete
Superstructure Continuity
Superstructure-toSubstructure Continuity
Girder Spacing
Stiffness of non-structural
components (barriers and
sidewalks)
Steel Multi-Girder
Simple, 2-span
Continuous
Fixed-expansion
bearings and integral
abutments (settlement
only)
Allows the investigation of the
influence of different support
restraints
5 ft. to 12 ft.
These nominal girder spacings will
be adjusted (rounded) based on
bridge width to define the number
of girders
Assumed fully active
or ignored
Since these items increase the
superstructure stiffness, ignoring
them is not necessarily
conservative in the case of support
movements
20 ft. to 160 ft.
Bridge Width
36 ft. to 72 ft.
Span to Depth Ratio
Material Properties (elastic
modulus of concrete)
Notes
This bridge type was selected as it
represents the most likely bridges
to be designed/ constructed in the
future
These levels of superstructure
continuity allow for the
investigation of both positive- and
negative- dominant bending
designs
Span Length
Skew Angle
Continuous
Bounds, Limits
0o to 60o
L/20 to L/30
3,500 ksi to 8,000 ksi
Typical span-length bounds for
multi-girder bridges
Approximately 2 to 4 lanes
Larger skew angles will require
using advanced analysis methods
These represent typical bounds on
girder depth and also govern the
relationship between girder
strength and stiffness
Elastic modulus of concrete has
significant influence on
superstructure and deck stiffness
and has a high variability
(compared with steel)
47
Stiffness of non-structural
components (barriers and
sidewalks)
Span Length
20 ft. to 160 ft.
Bridge Width
36 ft. to 72 ft.
Continuous
Skew Angle
3.2.4
Assumed fully
active or ignored
0o to 60o
Span to Depth Ratio
L/20 to L/30
Material Properties
(elastic modulus of
concrete)
3,500 ksi to 8,000
ksi
Since these items increase the
superstructure stiffness,
ignoring them is not necessarily
conservative in the case of
support movements
Typical span-length bounds for
multi-girder bridges
Approximately 2 to 4 lanes
Larger skew angles will require
using advanced analysis
methods
These represent typical bounds
on girder depth and also govern
the relationship between girder
strength and stiffness
Elastic modulus of concrete has
significant influence on
superstructure and deck
stiffness and has a high
variability (compared with
steel)
Sampling Method Overview
The parametric study was designed using commonly accepted design of experiments sampling
methods. A combination of sampling methods were utilized to sample from a multivariate
parameter space.
parameters.
A set of independent random samples were created from the sampled
Following sampling, the input parameters were used to define global bridge
geometry and girder configurations for each independent random sample set. These parameter
sets were used for both simply supported and continuous bridges, with the only difference being
that the two-span continuous bridges essentially has a “copy” of the simply supported span
attached to it. These geometries were used to develop girder designs and then build FE models
using the girder designs. The FE models were analyzed and results extracted. The results were
compared for agreement between the two sample sets. If the results showed disagreement the
48
parameters were sampled again and that set of results were compared to the combination of the
first two samples. This process was repeated until the results showed convergence. Figure 3.25
presents a workflow diagram of the study, including the main phases of “Preliminary Activities,”
“Data Generation,” and “Data Analysis.” This flow chart is a representation of the key action
items within the research plan. Preliminary Activities and Data Generation are discussed in this
chapter. Data Analysis will be discussed in Chapter 7 along with the results from the main study
of bias. Details regarding the general functionality of the software for design, model creation,
and analysis may be found in Chapters 4, 5, and 6.
In order to carry out the planned parametric study, it was necessary to sample the parameters of
interest to develop a representative sample of bridges to fully examine the impact of total and
differential support movements. The goal of the sample procedure is to effectively and efficiently
cover the parameter space, and to satisfy this objective a hybrid sampling method that utilizes a
statistical sampling approach known as Latin Hypercube Sampling (LHS) for a set of continuous
parameters, and a Design of Experiments (DoE) approach for different set of discrete parameters.
As shown in Figure 3.25 through Figure 3.27, the proposed sampling methodology for rating
factor analysis and tolerable support settlement analysis uncouples the discrete parameters from
the continuous parameters to allow for finer sampling of the later, as the discrete parameters
assume such a small number of values that a full-factorial approach (i.e. every possible
combination of these parameters) is feasible. Due to the large number of values that continuous
parameters may assume, such a simple sampling approach cannot efficiently cover their space.
In these cases, random sampling approaches are generally used. For the task at hand, the LHS
sampling approach was used. This process was done for both simply-supported and 2-span
continuous structures. For the study of load rating only one discrete parameter – the stiffness of
barriers – was used while for support settlement, a second discrete parameter, boundary
condition fixity, was added. This second parameter either frees or restricts the rotation about all
three principle degrees of freedom to simulate the bearing fixity of either pinned/expansion
49
bearings or integral abutments, respectively. In the case of tolerable support settlement, the total
number of discrete parameter combinations is 22, or 4.
50
Figure 3.25. Basic Study Workflow
51
Figure 3.26. Detailed Study Workflow
52
LHS Sampling of
DOE Sampling of
Continuous Parameters
Discrete Parameters
Divide Continuous
Generate a
Parameters into n
full-factorial sample
intervals
(i.e. all possible
combinations of
Draw a random sample
discrete parameters)
from each interval for
each parameter
2 Support Types
Randomly combine
x 2 Barriers/Sidewalk
samples from each
4 Samples of Discrete
parameter to generate n
Parameters
sets of continuous
parameters
Pair each sample of continuous parameters with each
sample of discrete parameters
Total number of samples within Bridge Suite = 4*n
Figure 3.27. Sampling Methodology for Tolerable Support Study
3.2.4.1
DoE Sampling
Factorial experiments, or DoE sampling, are methods by which parameters from with discrete
values are sampled by forming experimental units of possible combinations of each parameter.
Full factorial experiments create an experimental unit for each possible combination; fractional
53
factorial designs sample from a subset of combinations in order to produce the maximum
amount of information.
3.2.4.2
Latin Hypercube Sampling
Latin hypercube sampling (LHS) is a random sampling method for generating a sample of
possible combinations of parameter values. Latin hypercube sampling works on the principle of
the Latin square, where each sample position is in a unique row and column (Figure 3.28). The
Latin hypercube works in this same manner but within any multidimensional space. Parameters
do not have to be discrete, but may be broken up into bins.
A modification of the Latin
hypercube is orthogonal sampling. Orthogonal sampling usually divides up each parameter into
bins based on probability, so that each bin has the same probability density.
Figure 3.28. Latin Square
Although conventional Monte Carlo (MC) sampling is also possible for the random sampling
process, the LHS method generally covers a multi-dimensional continuous parameter space with
54
fewer samples. This increased efficiency is primarily due to the stratification of the parameter
space that does not permit the clustering of samples that MC approaches are susceptible to
(Smith and Saitta, 2008). LHS divides each parameter distribution into n number of bins with
equal probability density.
For this study each parameter was assumed to have a uniform
probability distribution, therefore all bins were of equal size. Parameters are randomly sampled
from a combination of bins as well as a random point within each bin, and each bin is sampled
from only once. A Matlab function was written to develop the preliminary sample space used for
the load rating and support settlement studies. This function uses LHS to sample five continuous
parameters: span length, skew, exterior-to-exterior girder width, girder spacing, and span-todepth ratio. Each of the parameters is given a range of values to sample based on predetermined
upper and lower bounds.
3.2.4.3
Sample Set Convergence
As with any sampling approach, it is critical that a criteria and strategy be defined to ensure that
the sample is representative of the population from which it is drawn.
To allow for this
convergence check, it is anticipated that the original sampling runs will generate 100 samples
from the continuous parameters, which will produce either a simply supported or two-span
bridge configuration. If the two independent samples produce results that are significantly
different, then each population will be increased in increments of 100 until convergence is
realized.
3.2.5
Notes on Parameters
3.2.5.1
Girder Spacing and Width
Combining the random sample of both width and girder spacing one value must be “corrected”
as the number of girders must be an integer value. Given its relative insensitivity, it was decided
55
to adjust the exterior-to-exterior girder width to the closest multiple of the girder spacing.
Specifically, the exterior-to-exterior width value obtained using LHS is adjusted by first dividing
this value by the girder spacing and rounding to the nearest whole number. This number gives
the total number of girder spaces, which is then multiplied by the actual girder spacing to obtain
the
adjusted
exterior-to-exterior
girder
width.
56
4. Automated Me mbe r Sizing of for Steel Multi-Girder Bridges
This chapter discusses the historical development of bridge girder design
and rating methods beyond what is presented in the literature review.
The development of an automated girder sizing algorithm is presented;
this includes an overview of the algorithm in order to satisfy capacity and
prescriptive design requirements, the method for calculation of single
line-girder demands, and design criteria used to size steel girders
according to AASHTO Allowable Stress Design and Load and Resistance
Factor Design methods.
Also presented is the validation of the
automated member sizing process.
Additional notes on design
heuristics, design algorithm modifications, and other criteria specific to
the study of single line-girder model bias and variability are included at
the end of the chapter.
4.1
Introduction
Automated girder design for American Association of State Highway Transportation Officials
(AASHTO) Allowable Stress Design (ASD) and Load and Resistance Factor Design (LRFD) is
performed according to the AASHTO Standard Specifications (AASHTO 2002) and the AASHTO
LRFD Bridge Design Specifications (AASHTO 2014), respectively. The girder design process aims
to replicate the industry practice of using the AASHTO code along with a single line-girder
analysis model to appropriately size girders for composite steel multi-girder bridges. A set of
girder sizing algorithms were developed in the Matlab programming environment to find a steel
section that can meet the design criteria for simply-supported and two-span continuous bridges.
The ASD algorithm identifies both a rolled American Institute of Steel Construction (AISC, Date)
wide-flange rolled section (W-Shape) or welded plate-girder section that satisfies the AASHTO
requirements for live and dead load for simply-supported structures.
The LRFD algorithm
identifies an appropriate welded plate-girder section that satisfies the AASHTO requirements for
live and dead load for simply-supported and two-span continuous structures.
57
4.2
Historical Development of Bridge Girder Design and Rating Methods
The majority of steel multi-girder bridges in the United States have been designed using the
coded specifications set forth by the AASHTO in the Standard Specifications for Highway Bridges
and the LRFD Bridge Design Specifications. There are three design codes historically used in bridge
design: Allowable Stress Design (ASD), Load Factor Design (LFD), and Load Factor Resistance
Design (LRFD). ASD design, developed in the early 1930’s, was the first nationally utilized
bridge design code in the United States (ref). It was revised extensively up until the 1940’s, and
from then on was largely consistent in both its specifications as well as effects on design. LFD
was introduced in the 1970’s; however it retained many of the same specifications and loading
mechanisms of the ASD specifications. LRFD was not introduced until 1994, though many states
did not adopt this design code until the 21st century. As of October 2007, the LRFD design
specifications were used throughout the U.S. as the sole criteria for new bridge design (FHWA
2006). These design codes were developed to ensure a minimum level of safety (and in some
cases serviceability) for all structures designed according to their specifications.
The LRFD AASTHO bridge design specifications are assumed to produce uniform safety and
performance across all designs, regardless of structure type. The demand and capacity model
utilized by AASHTO – known as a “line-girder” model – simplifies the complex geometry of a
bridge by analyzing it as a single beam. The design method outlined in the Standard Specifications
for Highway Bridges – which contains provisions for ASD and LFD – and the LRFD Bridge Design
Specifications provide a level of conservatism that makes this simplification possible.
ASD
provides a safety factor for strength-based performance by allowing only a fraction of the total
yield stress of steel to be reached for regular loading. LFD provides a safety factor for various
performance states through load factor coefficients while LRFD adds to the LFD model with
modifications to the total demand and resistance of a structure based on probabilistic models.
“The notional highway loading HL-93 for LRFD and LRFR was developed to provide a more
58
uniform safety factor for structures over various lengths, be more inclusive of AASHTO and State
legal loads, and to include legacy exclusion trucks” (FHWA 2005). The NCHRP project 2007/task of a small sample of bridges reported that LRFR ratings average about 7% higher than
LFR ratings for design-load inventory and that LRFD provides a less conservative design than
ASD (FHWA 2005).
4.3
Development of Automated Member-Sizing
The goal of the automated member sizing tool developed as part of this research is to emulate the
traditional single line-girder sizing process through a combination of the AASHTO code, AISC
manuals, design heuristics, and design algorithms developed in Matlab. The process is outlined
in Figure 4.1. The first stage of the design process requires input of material properties and
structure geometry. The second stage requires the selection of diaphragm type, configuration,
and section dimensions. This process may be handled automatically by the software. The third
stage includes the girder sizing process. Part of this third stage requires the input of design code,
design load, girder spacing, number of girders, overhang, and span length-to-girder depth ratio.
In the case of continuous spans, the software permits the user to opt for the inclusion of a
negative moment region cover-plate. The software also permits distinct interior and exterior
girder sections or constraining them to the same section. The girder sizing algorithm outputs
section sizes and computes section properties.
59
Figure 4.1. Overall Girder Design Process
60
4.3.1
Girder Sizing Algorithm to Satisfy Capacity and Prescriptive Requirements
Member sizing is based on the single-line girder (SLG) method of structural analysis as defined
by the AASHTO LRFD Bridge Design Specifications. Figure 4.2. Girder Sizing Algorithm the
flow of the algorithm. The software is capable of building single span models with a rolled or
plate girder section, as well as continuous models of two or more spans with a plate girder
section. Rolled sections are used for single spans when the section meets all AASHTO LRFD
Specifications. If a rolled section does not satisfy all requirements for the single span bridge, or if
the bridge is multiple-span continuous, the software builds the model with a plate girder section.
The plate girder section is a doubly symmetric I-shape section. The optimization algorithm is
given a set total girder height – the height of the web plus both flange thicknesses – and returns
the steel section with the smallest area that may still satisfy all AASHTO ASD or LRFD demand
and section proportion requirements.
61
Figure 4.2. Girder Sizing Algorithm
62
Plate girder sections are sized using a built-in optimization algorithm in Matlab called fmincon.
The fmincon algorithm is a nonlinear solver that seeks to find the scalar minimum of a function
using a set of user-supplied constraints such that:
𝑐𝑐(𝑥𝑥) ≤ 0
⎧
⎪
⎪ 𝑐𝑐𝑐𝑐𝑐𝑐(𝑥𝑥) = 0
𝑚𝑚𝑚𝑚𝑚𝑚 𝑓𝑓(𝑥𝑥) 𝑠𝑠𝑠𝑠𝑠𝑠ℎ 𝑡𝑡ℎ𝑎𝑎𝑎𝑎
𝐴𝐴 ∙ 𝑥𝑥 ≤ 𝑏𝑏
𝑥𝑥
⎨
⎪
⎪𝐴𝐴𝐴𝐴𝐴𝐴 ∙ 𝑥𝑥 = 𝑏𝑏𝑏𝑏𝑏𝑏
⎩ 𝑙𝑙𝑙𝑙 ≤ 𝑥𝑥 ≤ 𝑢𝑢𝑢𝑢
4.1
Only c(x) and lb and ub are used as constraints for girder size optimization. x is a vector of the
design variables solved for using fmincon, and is made up of the values of flange thickness, flange
width, and web thickness. The x vector also includes cover-plate thickness for continuous span
structures when that design option is selected. Web height is the difference between the total
girder depth and twice the flange thickness. C is a function that contains all pertinent ASD or
LRFD design criteria (see Section 4.3.4).
fmincon uses the same “Trust Region Reflective”
algorithm that is fully described in Chapter 8.
The scalar, or “objective” function, within the member-sizing algorithm is the area of the steel
section. In the same manner that a typical designer may attempt to find the most economical
section that still passes all constraints set by AASHTO LRFD Specifications, the fmincon algorithm
attempts to find the combination of variables—plate girder dimensions—that pass all constraints
while minimizing the area (which is taken as a surrogate for economy). Using fmincon and the
proper sizing constraints, the steel girder cross-sections can be sized such that it is the least
conservative section possible that still passes all the requirements of the AAHSTO LRFD
Specifications.
That is, the section is sized right on the margin of capacity and demands
(considering all of the applicable limit states).
63
4.3.2
Single Line-Girder Dead and Live Load Demand Calculation
For dead load demands, member actions of the single-line girder are obtained by first applying a
unit distributed load and then calculating the resulting member actions (moments and shears).
Using the principle of superposition, the dead load demand is obtained by scaling those actions
by the actual distributed dead load calculated for the structure. For live load, single-lane member
actions are obtained by stepping point loads (representing the axle loads of the design trucks)
across the entire length of the bridge together with distributed lane loads (when applicable) as
per the AASHTO LRFD Specifications. Single-lane member actions (moments and shears) are
then calculated for each combination and scaled by the dynamic impact factor. All of the singlelane member actions are combined based on the applicable load combinations and the resulting
envelopes are scaled using the applicable distribution factors
4.3.2.1
Single Line-Girder Finite Element Approximation Method
Single line-girder responses are developed using numerical finite element approximation
methods. A beam element is divided into foot-long sections and a global stiffness matrix is
developed by the superposition of each individual beam elements stiffness matrix into a single
matrix. Each node in the analysis beam has two degrees of freedom, vertical translation and
rotation. The 4x4 beam element stiffness matrix utilized ignores shear deformation is provides in
Equation 4.2.
12
6𝐿𝐿
𝐸𝐸𝐸𝐸 6𝐿𝐿
4𝐿𝐿2
𝑘𝑘 = 3 �
𝐿𝐿 −12 −6𝐿𝐿
6𝐿𝐿
2𝐿𝐿2
Or in the general form (Equation 4.3):
−12 6𝐿𝐿
−6𝐿𝐿 2𝐿𝐿2
�
12 −6𝐿𝐿
−6𝐿𝐿 4𝐿𝐿2
4.2
64
𝑘𝑘11
𝐸𝐸𝐸𝐸 𝑘𝑘
𝑘𝑘𝑖𝑖𝑖𝑖 = 3 � 21
𝐿𝐿 𝑘𝑘31
𝑘𝑘41
𝑘𝑘12
𝑘𝑘22
𝑘𝑘32
𝑘𝑘42
𝑘𝑘13
𝑘𝑘23
𝑘𝑘33
𝑘𝑘43
𝑘𝑘14
𝑘𝑘24
�
𝑘𝑘34
𝑘𝑘44
4.3
The global element stiffness matrix, K, can be formed by superimposing the beam element
stiffness matrices in the following manner (Equation 4.4):
(1)
⎡𝑘𝑘11
⎢ (1)
⎢𝑘𝑘21
⎢
(1)
𝐸𝐸𝐸𝐸 ⎢𝑘𝑘31
𝐾𝐾 = 3 ⎢
𝐿𝐿 ⎢ (1)
𝑘𝑘
⎢ 41
⎢
⎢ 0
⎢
⎣ 0
(1)
𝑘𝑘12
(1)
𝑘𝑘22
(1)
(1)
(1)
𝑘𝑘13
(1)
𝑘𝑘23
(1)
𝑘𝑘14
(2)
𝑘𝑘32
𝑘𝑘33 + 𝑘𝑘11
0
𝑘𝑘31
(1)
𝑘𝑘42
0
(1)
𝑘𝑘43
+
(2)
𝑘𝑘21
(2)
(2)
𝑘𝑘41
(1)
𝑘𝑘24
(1)
0
(2)
𝑘𝑘34 + 𝑘𝑘12
(1)
𝑘𝑘44
+
(2)
𝑘𝑘22
(2)
𝑘𝑘32
(2)
𝑘𝑘42
0
(2)
𝑘𝑘13
(2)
𝑘𝑘23
(2)
𝑘𝑘33
(2)
𝑘𝑘43
0 ⎤
⎥
0 ⎥
⎥
(2)
𝑘𝑘14 ⎥
⎥
(2) ⎥
𝑘𝑘24
⎥
(2) ⎥
𝑘𝑘34 ⎥
⎥
(2)
𝑘𝑘44 ⎦
4.4
The global stiffness matrix is assembled then the rows and columns corresponding to the fixed
degrees of freedom (DOFs) are removed. The external force vector is then assembled. Truck
loads and concentrated point loads (in the case of ASD) are applied only at each node and
therefore may be inserted directly into the global force vector; if they were applied between
nodes on each beam element the resultant fixed end forces would need to first be solved for.
Lane loads and dead load are considered distributed loads and are accounted for in the force
vector by first calculating the fixed end moments and shears as in Figure 4.3). With a beam
discretization level of one foot sections treating distributed loads as resultant point loads as every
65
node would produce minimal error, however the software was developed to maximize utility for
different mesh sizes.
Figure 4.3. Fixed End Forces
The global displacement vector is solved for each global load condition by:
𝐷𝐷 = 𝐾𝐾 −1 𝐹𝐹
4.5
Where D is the global displacement vector, K is the global stiffness matrix, the F is the global
force vector. These displacements are then used to determine the internal nodal force vector, f,
for each local beam element n using the following:
𝑓𝑓𝑛𝑛 = 𝑘𝑘𝑛𝑛 𝑑𝑑𝑛𝑛
With the local force vector
4.6
66
𝑓𝑓𝑛𝑛 =
𝑀𝑀𝑖𝑖
⎧ 𝑉𝑉 ⎫
𝑖𝑖
⎨𝑀𝑀𝑗𝑗 ⎬
⎩ 𝑉𝑉𝑗𝑗 ⎭𝑛𝑛
4.7
And the local displacement vector
𝜃𝜃𝑖𝑖
𝑣𝑣𝑖𝑖
𝑑𝑑𝑛𝑛 = �𝜃𝜃 �
𝑗𝑗
𝑣𝑣𝑗𝑗 𝑛𝑛
4.8
Where Mi is nodal moment at the i end and Vj is the nodal shear force at the j end. The local force
vectors are superimposed by summing the i and j force quantities from each n+1 and n beam
element, respectively.
4.3.2.2
Load Application
Dead load is calculated using the FE SLG approximation with a unit dead load. This unit dead
load may be multiplied by the changing dead load of steel as beam dimensions are iteratively
updated by the girder sizing algorithm.
Live load application follows the requirements outlined in the AASHTO design codes. Truck
loads are placed at each node along the beam starting with the front axle of a truck at the first
node along the beam and ending with the rear axle of a truck at the last node along the beam.
Single and dual trucks are used. The distance between the rear axle of the leading truck and the
front axle of the following truck is set at 50 ft. for simply-supported spans or in the case of
continuous spans set at the minimum of 50 ft. or 0.8 times the length minus 28 ft. (0.8L – 28 ft.),
67
where L is span length. The second spacing option places the center axle of each truck in the dual
truck configuration approximately 0.6L away from the center piers. Impact factors are applied to
truck and point loads when inserted into the force vector.
Lane loads are applied at each span
independently. The displacements and forces from each individually loaded span are combined
to produce the greatest moment and shear.
H-10, H-15, HS-15, H-20, HS-25 truck loads are varied using three rear axle spacing of 14 ft., 22 ft.,
and 28 ft.. The alternate military load places two 25 kip axles 4 ft. apart. Point loads are applied
in combination with lane loads when applicable. HL-93 is varied using three rear axle spacing of
14 ft., 22 ft., and 28 ft.. The design tandem places two 25 kip axle loads 4 ft. apart.
4.3.3
Allowable Stress Design Criteria
The following tables outlines the ASD criteria used for girder design. Also in this section are
outlines of rolled girder and welded plate girder sizing algorithms.
4.3.3.1
Distribution Factors
Distribution factors determine the portion of a wheel line that is taken up by a single girder and
simplifies the transverse load distribution phenomena (Equation 1.9):
𝐷𝐷𝐷𝐷 =
𝑆𝑆
5.5
4.9
Where S is the girder spacing in feet. The total live load on each girder is the portion of a design
truck that is specified using these distribution factors (AASHTO Standard Specification Section
3.23).
68
4.3.3.2
Effective Deck Width
Effective width of the composite section is determined for both short- and long- term composite
action. The deck width for each girder is taken as the minimum of the following:
•
The span length divided by 4
•
The girder spacing
•
The deck thickness multiplied by 12
The effective width of concrete is determined using the ratio of the moduli of elasticity of steel (E
= 29000 ksi) to those of normal weight concrete (145 pcf) with varying design strength as follows:
f’c =
Ultimate compressive strength of
concrete
n=
Ratio of modulus of elasticity of
steel to that of concrete. The value
of n is given for the following:
f’c =
2,000 – 2,300
n=
11
2,400 – 2,800
10
2,900 – 3,500
9
3,600 - 4,500
8
4,600 – 5,900
7
6,000 or more
6
The effects of creep are accounted for in the long-term composite action where dead loads act on
the composite section. In the case of this study, these are assumed to result from sidewalk and
barrier dead load. The stresses due to these loads are computed with both n- and the n- value
69
multiplied by 3, whichever produces greater stresses (AASHTO Standard Specification Section
10.38).
4.3.3.3
Steel Strength
Unless explicitly reported, the steel strength is determined by the year of construction, as
indicated by the AASHTO Manual for Bridge Evaluation guide (AASHTO Manual for Bridge
Evaluation Section 6B.5.2.1) for cases when steel grade is unknown. For structures built before
1963, it is specified that steel strength should be assumed to be equal to 33 ksi. For structures
built 1963 and after, steel grade should be assumed to be equal to 36 ksi.
4.3.3.4
Concrete Strength
For unknown concrete types, the AASHTO Allowable Stress Rating guide specifies that concrete
strength for structures built before 1959 should be assumed to be 2500 psi. All other structures
should be assumed to have 3000 psi concrete (AASHTO Manual for Bridge Evaluation Section
6B.5.2.4).
4.3.3.5
Design Truck
May be any of the standard AASHTO design truck loads for ASD. Both H and HS trucks with 10,
15, 20, and 25 designations are available. Alternate military loading (otherwise known as a
“design tandem” is also used in the design process and compared to standard truck loads by
default. HL-93 trucks are listed in the NBI records, however the use of this design load implies
an LRFD-based design, and therefore structures with this design load were not studied in this
research.
70
4.3.3.6
Number of Design Lanes
The number of design lanes is an integer value equal to the number of whole 12 ft. design lanes
that can fit in the clear roadway width between non-mountable curbs. The 10 in. sidewalk height
implies a non-mountable curb. This is determined according to AASHTO Standard Specification
Section 3.6.
4.3.3.7
Live Load Reduction Factor
The reduction in live load intensity is derived from a lower probability of all design lanes being
loaded simultaneously and may be determined by the following, where the following
percentages are the maximum load (AASHTO Standard Specification Section 3.12).
4.3.3.8
One or two lanes:
100 percent
Three lanes:
90 percent
Four or more lanes:
75 percent
Live Load Impact Factor
The live load impact factor is determined by the length of the structure using the following:
𝐼𝐼 =
50
(𝐿𝐿 + 125)
Where I has a maximum value of 30 percent. This value is added to unity and multiplied by live
load (known as live load plus impact, or LL + I) (AASHTO Standard Specification Section 3.8).
71
4.3.3.9
Rolled Girder Design
The following lists the steps the design algorithm takes to choose the best girder size.
1.
Rolled section is chosen from list
Rolled sections are designed according to the moment of inertia method, as specified in
AASHTO Standard Specification Section 10.38. The list of rolled wide flange sections is
chosen from the AISC Steel Construction Manual. All Wide-Flange sections were used in
this study. Future studies will only include Wide-Flange sections that are appropriate for
use as beams.
2.
Check beam and composite section depth criteria
Minimum depth criteria for beams and composite sections are based on the Standard
Specifications:
•
Beam depth is preferably not be less than L/30 for composite sections and L/25
for non-composite sections
•
Composite section depth is preferably not less than L/25
For the design algorithm for composite sections, the total beam depth must be greater
than or equal to L/30 and the depth of the section plus the deck thickness must be greater
than or equal to L/25. For non-composite sections, the total beam depth must be greater
than or equal to L/30. The parametric determinism for beam depth to span length ratio is
not used in the case of rolled section (AASHTO Standard Specification Section 10.5).
3.
Get associated diaphragm section
Load and aspect ratio requirements for diaphragms are determined in accordance with
AASHTO Standard Specifications Section 10.20 and 10.21. All cross bracing is designed
72
to withstand a lateral wind force of 50 pounds per square foot on the exterior face of the
girders. The code also notes that no cross-bracing angle member shall be less than 3 x 2
½ inches, however this requirement was omitted from the presented study and will be
added in future work.
The requirement for member slenderness ratio, KL/r, for a secondary member given as a
maximum of 140. This requirement is used in any diaphragms designed with angle
sections (AASHTO Standard Specification Section 10.5).
Diaphragm configuration is determined according to beam depth.
The diaphragm
configuration determines whether the transverse bracing elements are constructed out of
a single channel section or of multiple angle sections.
The software allows the
specification of whether the angles are in one of two common configurations: “Cross”
(‘X’) or “Chevron” (‘K’) bracing types. For this study, only “Cross” configurations are
used when the diaphragms are built of angle sections.
A channel section is selected when the total beam depth is less than or equal to 30 inches.
This is derived from the Standard Specifications requirement that the depth of the beam
section used as a diaphragm shall not be less than 1/3 of the girder depth, and preferably
not less than ½ the girder depth. The maximum channel section depth in the ASIC Steel
Construction Manual is listed as 15 inches as thus leads to using a channel section as a
diaphragm only when beam sections are less than 30 inches deep. For beams greater
than 30 inches deep, angles section in the “Cross” configuration are used (AASHTO
Standard Specification Section 10.38).
4.
Deflection due to live load plus impact factor is calculated
Loading due to live and dead loads is determined based on Sections 3.3, 3.4, 3.5, 3.6, and
3.7.
73
Live load plus impact factor is determined according to the ASD standard specifications
and is explained in the previous section on parameter configuration.
The maximum allowable deflection for a structure is given in the standard specifications
as L/800 (AASHTO Standard Specification Section 3 and 10.6).
5.
Moment and shear stresses due to dead load, superimposed dead load, and live load
are determined
Moment and shear stresses are calculated according to mechanics of materials
approaches. Live load stresses are modified by an impact factor, distribution factors, and
a live load reduction factor (also known as multi-presence factor) as indicated in the
Standard Specifications.
Dead load is that load consisting of all steel and the concrete deck and is assumed to be
carried only by the steel stringers.
Superimposed dead load is that consisting of the sidewalks and barriers and is carried by
the long-term composite section (if the design of the bridge is composite). This loading is
assumed to be carried equally across all members.
Live load is carried by the short-term composite section (if applicable) and consists of
three loading scenarios:
A. AASHTO ASD truck load multiplied by the distribution factors and impact
factors.
B. AASHTO ASD lane load multiplied by the distribution factors and impact
factors.
C. AASHTO ASD truck load and lane loads in the following equation:
𝐿𝐿𝐿𝐿 × 𝐼𝐼𝐼𝐼 × # 𝑜𝑜𝑜𝑜 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 × 𝐿𝐿𝐿𝐿 𝑅𝑅𝑅𝑅𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹
𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁 𝑜𝑜𝑜𝑜 𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺𝐺
74
Where:
LL = live load
Im = Impact factor + 1
The maximum stressed produced from any of the loading scenarios is used for design
(AASHTO Standard Specification Section 3.23).
Allowable stresses are determined according to Standard Specifications Table 10.32.1A: The
allowable stress in any member that is unsupported or partially supported is given as
0.55Fy, where Fy is the strength of steel.
6.
Beam is deemed a candidate or rejected
The current beam is deemed a candidate if stresses are less than the allowable limit.
7.
Current beam section candidate is compared to last candidate, and more efficient
section is kept
Beam efficiency is determined by area. The lightest section that may meet the allowable
stress requirements under loading is chosen.
8.
Iterate
Procedure repeats until all rolled sections are examined and selects the most efficient
candidate. Future developments in the software will likely streamline this process by
only considering likely sections that meet some basic criteria.
75
8.
Final demands and responses computed
For the most efficient section identified, the live load deflections, diaphragm
requirements, and section forces and stresses are calculated.
4.3.3.10
1.
Welded Plate Girder Design
Choose initial plate girder dimensions
Initial plate girder dimensions are chosen following two rules: (1) The depth of the beam
web is chosen by the depth to span ratio initialization parameter given to the program.
This value is rounded to the nearest inch in order to better replicate fabrication realities.
(2) The initial value for the flange width is chosen to be 25% of the web depth. The initial
flange thickness is chosen to be 5% of the web depth. The initial web thickness is chosen
to be 1.25% of the web depth. The flange width, flange thickness, and web thickness
values are rounded to the nearest ½, ¼, and 1/8 inch, respectively.
In addition, flange width, flange thickness, and web thickness are given minimum values
of 6, 1, and ¼ inch, respectively; and maximum values of the girder spacing divided by 3,
10, and 20, respectively. These lower and upper bounds are rarely reached by the design
algorithm, however, and don’t accurately reflect the final values achieved. These bounds
were chosen in order to give the algorithm enough “room” to determine the response
space while still giving tight enough bounds for computational efficiency.
2.
Check beam and composite section depth criteria
Depth criteria are calculated as noted in the above section describing rolled section
design excepting that the web depth is chosen in the manner described above.
3.
Check plate dimension proportion criteria
76
Plate dimension proportion criteria are checked in accordance with Standard
Specification Section 10.34:
•
The compression flange width shall preferably not be less than 0.2 times the web
depth. The design algorithm makes this a requirement.
•
The compression flange thickness shall preferably be no less than 1.5 the web
thickness. This is made a requirement in the design algorithm.
•
4.
The width to thickness ratio of the flanges shall not exceed 24.
Deflection due to live load is calculated
Deflection due to live load is calculated in the manner described for rolled sections.
5.
Get associated diaphragm section
Diaphragm criteria are calculated as noted in the above section describing rolled section
design.
6.
Moment and shear stresses due to dead load, superimposed dead load, and live load
are determined
Live load deflection, diaphragm, moment and shear stress criteria are calculated as noted
in the above section describing rolled section design.
7.
Compression flange check
Compression flange dimension proportion is checked against calculated compressive
bending stress and proportion requirements (AASHTO Standard Specification Section
10.34).
77
8.
Web plate thickness calculation
The required web plate thickness for beams without longitudinal stiffeners is calculated
considering bending stresses and assuming no longitudinal stiffeners (AASHTO Standard
Specification Section 10.34).
9.
Transverse stiffener requirement due to average unit-shearing stress is calculated in
the gross section of the web plate is calculated
Web plate gross section for beams without transverse stiffeners is checked according to
AASHTO Standard Specification Section 10.34.
10. If section is a composite design, neutral axis location is checked
Neutral axis location is checked according to Section 10.38. While not a requirement,
Standard Specifications notes that it is preferable to locate the neutral axis of composite
sections within the steel and not in the concrete (AASHTO Standard Specification Section
10.38).
11. Moment and shear criteria due to ASD Service Loads I and IA are calculated
Moment and shear criteria due to service loads are calculated according to Section 3.22 for
the Service Load method (AASHTO Standard Specification Section 3.22).
12. Optimization algorithm adjusts plate dimensions according to calculations 2 through
11
Algorithm is run using the Matlab function fmincon, which is a nonlinear minimization
algorithm with constraints. The function to be minimized is the area of the beam, or
weight.
13. Iterate operations 2 through 12 until optimal design is achieved
78
13. Final deflection, diaphragm requirements, and section forces and stresses are
calculated for chosen section
4.3.4
Load and Resistance Factor Design
The following tables outlines the LRFD criteria used for welded plate girder design as well as the
girder sizing algorithm. The design process includes specifications for Strength I and Service II
limit states as well as considerations for infinite fatigue life.
4.3.4.1
LRFD Sizing Constraints
The sizing constraints imposed by the LRFD Specifications are provided in Table 4.1.
Table 4.1. LRFD Sizing Constraints
Sizing Constraints
AASHTO LRFD Section
Ductility
6.10.7.3
Flange Proportioning Limits
6.10.2.2
Web Proportioning Limits
6.10.2.1
Strength I Flexure
6.10.6 for Positive & 6.10.8 for Negative
Service II Flexure
6.10.4
Shear
6.10.9
Fatigue I
6.6.1.2
79
4.3.4.2
Simply Supported Structure Design
Design criteria for simply supported structures is found in Table 4.2. The design process is
outlined in Figure 4.4. Fatigue life is also used in the design process and is discussed fully in
Section 4.3.4.3
Table 4.2. Design Criteria for Simply Supported Structures
5) Depth Criteria (2.5.2.6.3)
5) Service Limit (6.10.4)
a) 𝐿𝐿 ∗ 0.033 ≤ 𝐷𝐷𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏
a) For Compact:
b) 𝐿𝐿 ∗ 0.040 ≤ 𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
6) Ductility (6.10.7.3)
a) 𝐷𝐷𝑝𝑝 ≤ 0.42 ∗ 𝐷𝐷𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
7) Web Thickness (6.7.3)
a) 𝑡𝑡𝑤𝑤 ≥ 0.3125"
8) Section Proportions (6.10.2)
a)
b)
c)
𝐷𝐷𝑤𝑤
𝑡𝑡𝑤𝑤
𝑏𝑏𝑓𝑓
2𝑡𝑡𝑓𝑓
≤ 150
≤ 12
𝑏𝑏𝑓𝑓 ≥
𝐷𝐷𝑤𝑤
6
d) 𝑡𝑡𝑓𝑓 ≥ 1.1 ∗ 𝑡𝑡𝑤𝑤
0.1 ≤
𝐼𝐼𝑦𝑦𝑦𝑦
≤ 10
𝐼𝐼𝑦𝑦𝑦𝑦
i.
𝑓𝑓𝑐𝑐, 𝑡𝑡 ≤ 0.95 ∗ 𝐹𝐹𝑦𝑦
b) For Non-Compact:
i.
Does not control.
(C6.10.4.2.2)
6) Strength Limit (6.10.6)
a) For Compact:
i.
ii.
iii.
2𝐷𝐷𝑐𝑐
𝑡𝑡𝑤𝑤
≤ 3.76�
𝑀𝑀𝑢𝑢 ≤ 𝑀𝑀𝑛𝑛
𝐸𝐸
𝐹𝐹𝑦𝑦
𝑉𝑉𝑢𝑢 ≤ 𝑉𝑉𝑛𝑛
b) For Non-compact:
i.
ii.
iii.
2𝐷𝐷𝑐𝑐
𝑡𝑡𝑤𝑤
≥ 3.76�
𝑓𝑓𝑐𝑐, 𝑡𝑡 ≤ 𝐹𝐹𝑛𝑛
𝑉𝑉𝑢𝑢 ≤ 𝑉𝑉𝑛𝑛
𝐸𝐸
𝐹𝐹𝑦𝑦
80
Figure 4.4. Simply Supported Girder Design Process
4.3.4.1
Continuous Structure Design
While the process is similar between simple and multiple-span continuous bridges, one key
consideration with continuous bridges is the negative moment region over the pier(s). In the
negative moment region, the cross-section is considered non-composite with the deck. As this
region generally governs the design for multiple-span continuous bridges, the automated
member sizing software is capable of including cover plates to reinforce this region or to design a
plate girder that remains constant throughout all spans. Sizing with constant cross-section girder
introduces built-in conservatism in the positive moment region for multiple-span continuous
models due to the increased material needed in the negative moment region in order to satisfy
the AASHTO LRFD Specifications. For this reason, the least conservative path was chosen, to
81
size the cross section with a cover plate in the negative moment region, allowing both the
positive moment region and negative moment region cross-sections to be sized without any
arbitrary, additional conservatism. Fatigue life specifications, however, tend to add capacity in
the positive moment region of continuous structures. Due to the symmetric geometry of the
girders sized in this research, this tends to add excess capacity to negative moment regions. The
refined software allows a cover plate to be included over the negative moment region if needed.
The associated requirements for a cross-section in the negative moment region are presented in
Table 4.3.
The FE approximation of moment demands accommodate beams with changing cross-sections
using an iterative process. The approach takes the basic design algorithm and “wraps” it in
another layer of iteration. The process begins by approximating moment demands assuming a
negative moment region with an EI twice that of the positive moment region. The software
chooses the appropriate girder dimensions using the regular algorithm then reruns the FE single
line-girder demand approximation using the updated EI values for the two steel sections. The
updated moment demand envelope is compared to that of the previous iteration at the same
points of interest. If there is less than 5% difference in moment at all points of interest, the latest
girder dimensions will be accepted.. If there is a greater than 5% difference, the algorithm will rerun the FE approximation for moment demand and restart the girder sizing process. The flow
chart below shows the girder design and moment demand calculation synthesis.
Table 4.3. Additional Steel Girder Sizing Criteria for Continuous-Span Structures
9) Service Limit (6.10.4)
a) 𝑓𝑓𝑐𝑐, 𝑡𝑡 ≤ 0.95 ∗ 𝐹𝐹𝑦𝑦
b) 𝑓𝑓𝑐𝑐 ≤ 𝐹𝐹𝑐𝑐𝑐𝑐𝑐𝑐
10) Strength Limit (6.10.8)
a)
2𝐷𝐷𝑐𝑐
𝑡𝑡𝑤𝑤
≤ 3.76�
b) 𝑓𝑓𝑐𝑐 ≤ 𝐹𝐹𝑛𝑛
c)
𝑓𝑓𝑡𝑡 ≤ 𝐹𝐹𝑦𝑦
𝐸𝐸
𝐹𝐹𝑦𝑦
82
Figure 4.5. Continuous Span Girder Design Process
4.3.4.2
Distribution Factors
Distribution factors are referenced in Figure 4.8 through Figure 4.11 and are calculated using the
expressions shown in Table 4.4 and Table 4.5. The maximum distribution factor is used for
83
design.
Distribution factors are updated for each iteration of the algorithm and adjusted
according to new section sizes.
Table 4.4. Distribution Factors Calculated with Section Dimensions
One Design Lane Loaded
Two Design Lanes Loaded
0.1
𝐾𝐾𝑔𝑔
𝑆𝑆 0.4 𝑆𝑆 0.3
𝐷𝐷𝐷𝐷1 = 0.06 + � � � � �
�
14
𝐿𝐿
12 𝐿𝐿 𝑡𝑡𝑠𝑠3
0.1
𝐾𝐾𝑔𝑔
𝑆𝑆 0.6 𝑆𝑆 0.2
𝐷𝐷𝐷𝐷2 = 0.075 + � � � � �
�
9.5
𝐿𝐿
12 𝐿𝐿 𝑡𝑡𝑠𝑠3
Table 4.5. Distribution Factors Calculated with the Lever Rule
4.3.4.3
One Design Lane Loaded
Two Design Lanes Loaded
𝐷𝐷𝐷𝐷1 = 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅
𝐷𝐷𝐷𝐷2 = 𝑒𝑒 ∗ 𝐷𝐷𝐷𝐷𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖
Fatigue Limit State Design Criteria
While this limit state is not directly related to the Strength I and Service II limit states used for
rating of structures it needs to be considered during the design of notional bridges as it can have
an impact on member sizes (especially in the positive moment region). The fatigue limit state
limits the difference in live load stress that a member can see due to vehicular loads. The aim of
the limit state is to reduce the stress at certain fatigue prone details. The design algorithm
chooses the diaphragm connection detail as the detail in question.
Fatigue limit states are
applicable to girder sections undergoing net tension, i.e. the bottom flange in the positive
moment region of a beam and the top flange in the negative moment region of a beam. The
fatigue limit state used for the design process is the infinite fatigue life limit state (Table 4.6). The
result of this limit state is an increase in average flange area in the positive moment region, and,
in turn, the reduction of the size of (or even the need for) cover-plates in the negative moment
region. As noted previously, the flanges on all beams are sized equally, therefore increasing the
84
tension flange thickness in the positive moment region due to fatigue will increase the tension
flange thickness in the negative moment region. For some structures, Fatigue I is the controlling
limit state and additional capacity is introduced for Strength and Service Limit States. Figure 4.6
and Figure 4.7 illustrate the change in flange area and Strength I rating factors due to the
inclusion of Fatigue Limit State I for two-span continuous structures for a sample of ten bridges.
Table 4.6. Fatigue Limit State
Fatigue Limit (6.6.1)
For Load-Induced Fatigue:
𝛾𝛾(∆𝑓𝑓) ≤ (∆𝐹𝐹) 𝑇𝑇𝑇𝑇
Where γ is the fatigue load factor and ∆𝑓𝑓 is the net change in tensile stress and (∆𝐹𝐹) 𝑇𝑇𝑇𝑇 the fatigue
resistance of the member.
Figure 4.6. Influence of Addition of Fatigue I Limit State on Flange Area for Interior
Girders for Ten Sample Bridges
85
Figure 4.7. Influence of Fatigue Limit State on Interior Girder Strength I Rating Factors
for Ten Sample Bridges
4.3.4.4
Diaphragm Sizing
The sizing criteria for diaphragms are the same as those utilized in ASD design and are detailed
in Section 4.3.3.9
4.3.4.5
of this chapter.
LRFD Design Code Algorithm Flow Chart
The following flow charts depict the step-by-step process taken by the software and reference the
software sub-functions (Figure 4.8 through Figure 4.11). These are provided in a similar fashion
to the flow charts provided in AASHTO Appendix C6 and cover the design paths for the LRFD
design code Sections 6.10.6, 6.10.7., 6.10.8., and A.
86
Figure 4.8. LRFD Section 6.10.6
87
Figure 4.9. LRFD Section 6.10.7
88
Figure 4.10. LRFD Section 6.10.8
89
Figure 4.11. LRFD Appendix A
90
4.4
Evaluation of Automated Member Sizing for the Study of Single Line-Girder
Model Bias
To ensure the relevancy of the overall study, it is imperative that the designs produced in this
task are representative of the ones produced by designers following the AASHTO LRFD
Specifications and common conventions based on heuristics (e.g. sizing increments for webs,
flanges, etc.). Given the size of the sample needed to investigate all of the potentially influential
parameters identified, automation of the member sizing processes was required.
The ASD
method was first developed in code in order to assess the ability of a software algorithm to
properly size a girder based on the AASHTO code. The success of this process led to the
development of an algorithm to design girders based on the LRFD method.
4.4.1
Allowable Stress Design
The algorithm described above was first used to size girders for three New Jersey bridges that
have been previously studied by Drexel University in order to evaluate the accuracy of the
member sizing approach. The availability of design drawings enables the comparison of the
properties of the actual sections with those produced by the software. For this study, composite
design and an L/25 beam depth was assumed. The girder spacing was not varied, but instead the
actual girder spacing of each structure was used.
By deterministically choosing the girder
spacing, a large source of uncertainty was eliminated and this permits the differences between
the automated and actual member sizes to be evaluated. The shortest bridge, Pennsauken Creek,
was designed with a rolled section while the two longer bridges, MP 28.9 and US202/23, were
designed with welded plate girder sections.
91
Table 4.7. Benchmark Design Structure Comparison
Pennsauken Creek
22
38
31
5761
7450 (29)
10.7x104
428
477 (11)
3,880
3,810 (2)
2,900
3,100 (7)
1.5x104
2.0x104
(31)
20.7x105
31.3x104
(51)
15.9x104
20.0x104
(26)
646
659 (2)
4,800
4,520 (6)
3,440
3,620 (5)
Actual
106
128
66
Automated
(% diff)
71 (33)
116 (9)
91 (38)
17.4x104
(63)
US202/NJ23
Automated
(% diff)
22 (-)
41 (8)
33 (6)
Actual
Flange Area (in2)
Total Area (in2)
Girder Depth (in)
Girder Moment of
Inertia (in4)
Girder Section
Modulus (in3)
Composite
Moment of Inertia
(in4)
Composite
Section Modulus
(in3)
NJ TPK (MP28.9)
Actual
101
123
65
9.4x104
Automated
(% diff)
88 (13)
112 (9)
72 (11)
11.1x104
(18)
Examining Table 4.7 the following observations can be made that illustrate both the nature and
accuracy of automated member sizing.
•
Automating the member sizing process will generally produce very similar results for the
governing limit state, but may vary for other limit states. For the examples shown, the
governing limit state was related to strength and thus the composite section moduli had
relatively low discrepancies (less than 10%). Conversely, the discrepancies in moments
of inertia were relatively large, since the deflection criteria did not govern the design.
•
The somewhat arbitrary selection of girder span-to-depth ratio has significant influence
over the relative levels of strength (section moduli) and stiffness (moment of inertia). For
these designs a span-to-depth ratio of L/25 was assumed when the actual ratios for the
NJTPK and US202 bridges were closer to L/35 (which is unusually shallow) and L/30,
respectively. The result of assuming a deeper girder was that the automated design
approach produced lighter (less total area) and stiffer girders (but with the same section
moduli).
92
•
In addition to girder depth, incremental sizing rules are also somewhat arbitrary and
influence the balance between conservatism and efficiency.
For example, if flange
dimensions/girder widths were treated as continuous variables, a truly “optimum” and
unique solution to the design problem is possible.
In reality however, plate
sizes/thicknesses are typically defined in increments and thus minimum dimensions are
rounded up resulting in additional strength/stiffness. These arbitrary sizing rules (and
the additional strength and stiffness they produce) vary with regard to region, owner,
and even time period, and thus is not something an automated software can recreate
accurately.
4.4.2
Load and Resistance Factor Design
Development and validation of automated member-sizing based on AASHTO LRFD
Specifications for steel simply-supported and multi-girder bridges was begun after the initial
investigation into automated ASD method showed that an algorithm could size girders
appropriately to both satisfy the AASHTO code requirements as well as result in member
dimensions that were consistent with what is found in actual structures. The critical path in the
development of the LRFD algorithm development was the production of accurate “designs” that
produce representative results. The approach for the automated member sizing is intended to
parallel current practice to the fullest possible extent while neglecting conservative practices that
are not codified or explicitly required by the AASHTO LRFD Specifications. Member sizing is
based on replication of the process of single-line girder method of structural analysis and
constraints defined in the AASHTO LRFD Specifications.
In an effort to validate the member-sizing software, researchers from the University of Delaware
acted as independent partners to provide a “peer review” of the model design philosophy and
assumptions utilized in the development of the software. A “one-to-one” approach was used in
93
this validation effort. Several designs were conducted by hand, and key parameters of these
designs were compared to the members sized using the software. A design spreadsheet based on
the single line-girder design method was developed using Microsoft Excel. The spreadsheet
requires cross section dimensions, load cases, and bridge orientation information as inputs, and
calculates all relevant section properties as well as the flexural and shear strength of the girder
following the AASHTO LRFD Specifications.
Flexural strength checks include: local buckling resistance, lateral torsional buckling resistance,
and tension flange yielding resistance for both composite (positive moment region) and noncomposite sections (negative moment region). The shear strength calculations only include the
resistance of an unstiffened web. This was done in an attempt to parallel the model design
approach of the automated modeling software, which does not include transverse or longitudinal
shear stiffeners.
In addition to the strength checks, all proportional limits were checked. To size the girders using
the spreadsheet, calculated moment and shear capacities were checked against the factored load
cases to determine the most efficient girder cross section. In this case, efficiency is determined by
minimizing the cross sectional area. Several design examples were manually generated utilizing
the design spreadsheet in order to compare results and validate the automated design approach.
“One-to-One” Validation Checks:
•
Flange Area
•
Web Area
•
Girder Depth
•
Girder Moment of Inertia
•
Girder Section Modulus
•
Girder Capacity (Lateral Torsional, Local Buckling Resistance
Calculations)
94
•
Composite Moment of Inertia
•
Composite Section Modulus
•
Composite Section Capacity (Plastic Neutral Axis and Plastic
Moment Calculations)
•
Single Line Dead Load Computations
•
Single Line Live Load Computations
The “one-to-one” validation approach was very effective, but in order to validate all parts of the
possible design paths a line by line analysis was also performed. This analysis was used to
ensure that there were no typos and all appropriate equations were being calculated properly.
Using Microsoft Excel, the research personnel from the University of Delaware developed a
design spreadsheet based on the single line girder design method. The spreadsheet requires
cross section dimensions, load cases, and bridge orientation information as inputs, and calculates
all relevant section properties as well as the flexural and shear strength of the girder following
the AASHTO LRFD Specifications. Flexural strength checks include: local buckling resistance,
lateral torsional buckling resistance, and tension flange yielding resistance for both composite
(positive moment region) and non-composite sections (negative moment region). The shear
strength calculations only include the resistance of an unstiffened web. This was done in an
attempt to parallel the approach of the automated modeling software, which does not include
transverse or longitudinal shear stiffeners. In addition to the strength criteria, all proportional
limits were checked. To size the girders using the spreadsheet, calculated moment and shear
capacities were checked against the factored load cases to determine the most efficient girder
cross section.
In this case, efficiency is determined by minimizing the cross sectional area.
Several design examples were manually generated utilizing the design spreadsheet in order to
compare results and validate the automated member sizing approach.
95
To further validate the process, the flow of the software was compared to the flow charts
provided in AASHTO LRFD Specs Section C.6. It was apparent that the automated membersizing software calculates and evaluates all necessary constraints, with a work flow similar to that
represented in the flow charts provided by AASHTO.
4.5
Notes on LRFD Girder Sizing for the Study of Bias and Tolerable Support
Settlement
For this research a number of design choices were made to simplify the girder sizing process and
promote uniformity of bias study results. Rolled sections were not used for this study to prevent
the influence of rounding sections. Continuous structures were designed with a coverplate
however there was no restriction on the minimum thickness of the coverplate sections over the
negative moment region. Interior and exterior girder were designed separately and used for
their respective location to reduce any inherent conservatism in designing a girder that would
satisfy the unique demands requirements of both interior and exterior girders.
In depth
discussion follows.
4.5.1.1
Rounding Plate Dimensions
Plate girder sections are dimensioned using a non-linear least-squares minimization algorithm
built-in to Matlab called fmincon that adjusts a set of dimensional parameters in order to
minimize a scalar objective function. In the same manner that a typical designer may attempt to
find the most economical section that still passes all constraints set by the AASHTO LRFD
Specifications, the fmincon algorithm attempts to find the combination of variables—plate girder
dimensions—that pass all constraints while minimizing the area (minimizing the objective
function). The dimensional parameters in this case are flange width, flange thickness, web
thickness, and for multiple span continuous bridges, the thickness of the cover plate in the
96
negative moment region.
By utilizing a minimization algorithm with the proper sizing
constraints, the steel member cross-section can be sized such that it is the least conservative
section possible that still passes all AAHSTO LRFD Specifications: the section is sized right on the
margin of capacity and demands.
More simply stated, the steel cross-section sized by the
algorithm has no excess capacity for the controlling demand. Strength I, Service II, Shear, and
Fatigue limit states for steel were used for the design of the girders. If Strength I was the
controlling demand – the demand that the algorithmic sizing process attributed zero excess
capacity to, the Strength I Inventory/Operating ratings for all members dimensioned using this
process would be found to be 1.0 and 1.3, respectively.
By using a section with no excess capacity, the results of the population study for total and
differential support movements will provide the most conservative estimate of allowable
settlement. Rounding girder plate sizes provides additional conservativism that, while common
in practice due to the steel fabrication process, was not explicitly required by the AASHTO LRFD
Specifications.
4.5.1.2
Continuous Span Bridges
The automated member sizing software is capable of including cover plates to reinforce the
negative moment region or designing a plate girder that remains constant throughout all spans.
A negative moment region cover-plate may be included in continuous designs as a result of a
number of factors that contribute to this region controlling designs: first, concrete bridge decks
are considered to contribute negligible capacity in the negative moment region as the concrete is
in tension; second, the resistance of steel reinforcing bars that would exist in a bridge design are
neglected for the entire design process for this project; third, the AASHTO truck load
combinations provided in LRFD result in large negative moments over interior supports. It was
concluded that with a constant cross-section girder (no coverplates) there would be built-in
conservatism in the positive moment region for multiple-span continuous models due to the
97
increased material needed in the negative moment region in order to satisfy the AASHTO LRFD
Specifications. Consequently the design process was changed to design all continuous span
bridges with negative moment region cover-plates.
A related issue that arose due to the decision to move to non-uniform cross-sections for
continuous bridges is that the determination of the demand for the continuous single-line girder
models implicitly assumed a constant cross-section. That is, when the girder section over the pier
is stiffened with a cover plate the increased stiffness will act to attract more moment, and this
additional moment is currently not being considered in the design of the negative moment
region. The design of continuous bridges was modified to include an iterative process to ensure
that the elastic distribution of forces is consistent with the selected member sizes. The influence
of this phenomenon (and the current errors it produces) is discussed further in this chapter.
4.5.1.3
Exterior and Interior Girders
The differing design requirements for interior and exterior girders were also investigated in the
initial stages of this research. The original strategy employed when designing bridges using the
ASD code was to size a member capable of satisfying both interior and exterior girder
requirements; however this approach resulted in excess capacity in whichever girder had the
lesser load demands. For example, take the case of an exterior girder that may have a lower live
load distribution factor than an interior girder of the same structure while having the same
capacity; the interior girder would have a resulting line-girder load rating of exactly 1.0 while the
exterior girder would rate at 1.35. The design methodology was subsequently changed to size
exterior and interior girders independently and resulted in both exterior and interior girders
rating at 1.0.
98
4.5.1.4
Infinite Fatigue Life Design Criteria
The overall effect of fatigue life in girder design will be studied as part of this research. Although
the main body of research will concentrate on those populations that have been designed with
infinite fatigue life considerations, as sub-population will be designed and analyzed to determine
whether the inclusion of fatigue alters the relationships between the independent design input
parameters and FE ratings and the ratio of FE ratings to SLG ratings. Fatigue considerations
dramatically increase the ratio of longitudinal stiffness of composite girder section to the
transverse structural stiffness due to deck and diaphragms. The ratio of longitudinal global
stiffness to transverse global stiffness increases when fatigue limit states are included in design,
however this relationship has not been quantified and will be considered as part of this research.
4.5.1.5
Rationally Sized Diaphragms
The AASHTO LRFD design code specifications for diaphragm sizing result in minimal
diaphragm sizing that does not agree with the stiffness of elements found in the field. The design
requirements for both LRFD and ASD design are detailed in Section 4.3.3.9
of this chapter.
The effect of the choice to design diaphragms for this minimum constraint and to not include
rationally-sized diaphragms to avoid the effects on conservatism will be studied by re-rating two
bridges suites with diaphragms with 10 times and 30 times the stiffness and comparing these
ratings to the original bridge suite ratings.
This increase will be studied for bridges with
diaphragm configurations using chevron or cross-bracing.
Channel sections are limited in
diaphragm sizing due to the lower limit on diaphragm depth. These sections used in the bridge
suites for FE ratings are already close to the rationally sized diaphragms found in practice and
their stiffness will not be increased. Both bridges designed with and without infinite fatigue life
considerations
will
be
investigated
for
this
effect.
99
5. Automated Finite Element Model Creation
Detailed in this chapter is an overview of the method for the production
of three-dimensional finite element models in a guided or automated
fashion. Included is a discussion of general model form, element type,
continuity conditions, and boundary conditions.
The method for
accessing the application programming interface of a finite element
solver with common scripting languages is discussed along with and the
model creation algorithm for the placement of nodes, elements, and
property assignment.
Also discussed is a verification of use of
automatically created finite element models for mass simulation of
bridges and their use in research on the bias of the AASHTO single linegirder design model.
5.1
Overview
This feature in the software provides assistance to the user in the semi-automatic creation of finite
element (FE) models of multi-girder bridges. Given the somewhat regular details of structural
design and symmetric geometries of common highway bridges, features such as roadway
geometry, girder type and spacing, cross-bracing configuration, and bearing type may be entered
by the user to create a 3D geometric element-level FE model in a matter of minutes. Normally,
model creation involves a process over hours and involves the element-by-element creation and
manipulation by a human user via a graphical user interface (GUI).
The software system developed as part of this research automates the placement of nodes,
elements, links, and boundary conditions, as well as the application of section dimensions,
properties, and material. The software, written in the Matlab scripting language, may be run
either through a GUI developed through this study or through a scripted command set. The GUI
interface allows the creation of a single FE model while the scripting interface is may be used for
the creation of sets of models that can be utilized in parametric population studies. Both methods
for model creation interface with the finite element software package, Strand7 (Strand7 2014), for
FE model creation through the application programming interface (API).
100
5.2
Model Form
The modeling process begins with the initialization of the Maltab-Strand7 API link. Next, a
global grid based on user-specified mesh parameters is defined. This global grid is used as the
basis for the creation of model nodes in the 3D space. Elements are created by connecting nodes
with two-node beam elements for girders, diaphragms, concrete traffic and safety barriers, and
girder-deck composite action links; three- and four-node shell elements are created for deck and
sidewalk; rigid links are created to enforce planar girder sections, link girder elements to
composite action elements and diaphragm elements, and link girder elements to boundary nodes.
Continuity and boundary conditions are set by applying fixity, translational stiffness, and
rotational stiffness to boundary nodes.
Element properties are then defined using section
property and material property assignment. Figure 5.1 illustrates the model creation process and
Figure 5.2 illustrates the model form.
Figure 5.1. FE Model Creation Overview
101
Beam
Shell
Rigid Link
Beam
Figure 5.2. 3D Element Level FE Model
5.2.1
Girders
Girders are two-node beam elements that may either be a standard AISC wide flange section (Wshape) or a doubly-symmetric I-shape with defined web height and thickness, and flange width
and thickness. In addition, the model may use separate girder section properties for exterior and
interior girders. Models of continuous span bridges may have different girder properties in the
positive and negative moment regions. Rigid links are used to enforce compatibility between the
top and bottom surfaces of the girder flanges. Rigid links connect the girder centroid to the top
and bottom flange nodes. These nodes in turn are connect to the deck through another set of
adjustable stiffness links or are used as boundary nodes.
5.2.2
Diaphragms
Diaphragms are two-node beam elements that may either be a standard AISC channel section (Cshape) or angle section (L-shape) in the case of cross-brace- or chevron-type diaphragms.
102
5.2.2.1
Diaphragm Type
The diaphragm type determines both the section type and in the case of angle-sections the
connection configuration. Diaphragms may be a channel section (Figure 5.3), a cross-braced
angle section (Figure 5.4), or a chevron-braced angle section (Figure 5.5). Channel sections are
connected directly to girder centroid nodes. Cross- and chevron-bracing are connected to girders
by the top and bottom flange girder nodes. These nodes connect to the girder beam element by a
rigid continuity link between the top and bottom flange nodes and the girder centroid node
(Figure 5.6).
Figure 5.3. Channel Section Diaphragms
Figure 5.4. Cross-Bracing Diaphragms
103
Figure 5.5. Chevron-Bracing Diaphragms
Figure 5.6. Cross-Bracing Diaphragm Connectivity
5.2.2.2
Diaphragm Direction
The diaphragm direction determines whether the bracing elements are oriented parallel to the
skew angle (if any) or normal to the bridge girders. In the case of a structure with no skew, this
option does not have any effect, as the bracing elements are oriented both parallel to the skew
(which is equal to zero) and normal to the girders (Figure 5.7).
The AASHTO specifications indicate that on any structure with a support skew angle greater
than 20 degrees, the diaphragms shall be normal to the girders (AASHTO Standard Specifications
104
Sec. 10.20). The software allows for normal diaphragms that are located in contiguous rows
across all girder bays (Figure 5.8), or in staggered rows, where the diaphragms in each bay are
located at the same distance from the abutments (Figure 5.9).
Figure 5.7. Skew Bridge with Parallel Diaphragms (applicable to bridges with skew
angles less than 20o)
Figure 5.8. Straight-Skew Bridge with Normal Contiguous Diaphragms
105
Figure 5.9. Skew Bridge with Normal Non-contiguous Diaphragms
5.2.3
Deck
Deck elements are three- and four- node shell elements. Deck elements are assigned a bending
and membrane thickness. Both of these values are equal to the deck thickness. Deck nodes are
located at the “center” of the shell element’s thickness.
5.2.4
Sidewalk
Sidewalk elements are three- and four- node shell elements. Sidewalk elements are assigned a
bending and membrane thickness.
Both of these values are equal to the sidewalk height.
Sidewalk nodes are located at the “center” of the shell element’s thickness.
5.2.5
Barriers
Barriers are rectangular two-node beam elements. Barrier centroids are offset left or right by half
the barrier width towards the interior of the structure.
106
5.2.6
Boundary Conditions
Boundary stiffness is adjusted using a combination of translational and rotational springs in
pound per inch or pound-inches, respectively. Two boundary condition sets may be applied to
the model, “Type I” and “Type II.” Any number of abutments or piers may be given either the
“Type I” or “Type II” classification. This classification is applied to the entire support row, i.e.
the boundary nodes at each girder at an individual pier or abutment. The six global degrees of
freedom (DOF), corresponding to translation or rotation about the longitudinal (along the
girders), transverse (perpendicular to the girders), and vertical directions, may be assigned a free,
fixed, or finite stiffness case. A finite stiffness case results in a translational or rotational springs
applied at all nodes for that boundary classification (see Figure 5.10). Any DOF for the two
classifications may be “linked”; for example, the longitudinal translational stiffness spring for
both “Type I” and “Type II” boundaries may be updated together using the same alpha
coefficient.
Two special fixity cases may be applied that enable the user to have different boundary
conditions for nodes on the same support row (Figure 5.10). The first, “Alignment Bearing,” fixes
the support node at the center girder of the first boundary row, or abutment, in the transverse
direction for translation. The center girder node takes on all other fixity and stiffness parameter
for the designated row classification except for the transverse case. In the case of an even number
of girders, one of the two center girder nodes is used. The figure illustrates the translation
transverse fixity applied to the 2nd girder of a four girder structure. The vertical fixity remains in
place and transverse translational fixity is added to the Type II bearing classification. The second
condition, “Longitudinal Fixity,” fixes the longitudinal translation of the exterior girder support
nodes only and, like the “Alignment” case, is applied only to the first abutment of the structure.
The exterior nodes take on whichever fixity or stiffness was designated for its row classification
excepting the longitudinal transverse fixity. Note in the figure how the “Longitudinal” fixity case
adds translational fixity to the exterior girders for only the first Type II classified bearing row
107
(Figure 5.10). Note that the illustrated case is not the same case used for the study of bias
presented later in this thesis.
Figure 5.10. Illustration of “Alignment” and “Longitudinal” Special Boundary Condition
Cases.
5.2.7
Non-structural Mass
Non-structural mass is applied to the deck nodes between the inside edges of the sidewalks.
Deck overlays are assigned a certain thickness. This thickness is translated to a certain poundage
per node based on the number of deck nodes in the model.
5.2.8
Composite Action
Composite action is enforced by connecting two-node beam elements between the deck nodes
and the girder top flange nodes. These beam elements have adjustable stiffness that may be used
to modulate the degree of composite action. See Section 5.4.3 for an investigation into the effects
108
of using the moment of inertia and shear area of a beam element to simulate variable composite
action and a comparison with using deck concrete modulus.
5.3
5.3.1
Model Creation
Strand7 API
FE models are created by controlling Strand7 via the API with Matlab. The Strand7 API consists
of a dynamic link library (DLL) file as well as a number of header files and include files. (Strand7
2014) Strand7 is packaged with a Matlab-specific API library that contains function calls for
almost every functionality in Strand7. The DLL files include functions that are used to read FE
model data, modify or create FE model element data, launch FE model solvers, and read FE
model result data. The header files allow external programs to communicate with the Strand7
DLL file and contain definitions for constants and function calls used by each supported
language. All functions used in the API are accessed using the Windows function stdcall. The
header files used with Matlab is St7APICall.h.
The constants files used with Matlab is
St7APIConst.m. Use of the Matlab-Strand7 API requires that API calls be made using the built-in
Matlab function calllib. Function names and their arguments are both passed into calllib on the
right hand side while error handling and function-specific outputs are passed on the left hand
side with the following format for the example Strand7 API function St7GetNodeXYZ:
XYZ = zeros(3, 1);
[iErr, XYZ] = calllib(‘St7API’, ‘St7GetNodeXYZ’, uID, NodeNum, XYZ);
109
5.3.2
Node Placement Algorithm
5.3.2.1
Node Element Metadata
Nodes are placed using a Strand7 API call that requires the node location coordinates and node
index. Node information for a given FE model is stored in a Matlab structure: Node. The Node
structure contains a list of information about each node’s index, type, and which elements are
connected to it. The Node structure also contains an MxNx6 3D array that represents the relative
placement of each node in space. The M dimension corresponds to the longitudinal space – or ‘X’
direction in the model’s Cartesian coordinate system, the N dimension corresponds to the
transverse space ( ‘Y’ direction), and third dimension of the array corresponds to vertical space
(‘Z’ direction). The six vertical levels of the model correspond, from top to bottom, to barrier
nodes, sidewalk nodes, deck nodes, girder top flange nodes, girder centroid and diaphragm
nodes, and girder bottom flange and boundary nodes.
5.3.2.2
Deck and Sidewalk Nodes
Deck nodes are placed at zero on the Z axis. The placement of deck nodes guides the placement
of the nodes for all other levels in the node array. Nodes are placed according to the specified
minimum and average mesh size option specified by the user. The “near” and “far” deck edge
nodes are placed first and then nodes are placed longitudinally between by dividing the length
by the average mesh size. If any node is to be placed closer to the deck edge nodes than the
minimum mesh size, it is not placed (Figure 5.12-A) Deck nodes are placed transversely based on
the girder spacing and specified average and minimum mesh sizes.
In the case of multiple spans, each span is created separately. The Node structure is indexed to
each span by Node(i) where i is the ith span. Overlapping nodes at the near and far ends of
adjacent spans in the structure are removed from the Node structure and replaced with the correct
node numbers and other metadata.
110
Deck nodes between the centerlines of the exterior girders are placed according to the skew and
diaphragm configuration of the model. Straight bridges, bridges with skews less than or equal to
20°, and bridges with skews over 20° that have the “staggered” diaphragm configuration (See
Section 0) are all built with deck nodes that are placed parallel with the skew direction (Figure
5.11).
Deck nodes for bridges with skews over 20° with the “contiguous” diaphragm
configuration and unequal skews at the ends are placed normal to the girder lines (Figure 5.12-B).
Overhang nodes are always placed parallel to the skew (Figure 5.13).
Figure 5.11. Deck Node Placement
111
B
A
B
.
Figure 5.12. Deck Node Placement
Figure 5.13. Overhang Deck Node Placement
Sidewalk nodes are placed directly above the deck nodes. Deck nodal meshes that do not allow
for exact placement of sidewalk nodes round the placement of the sidewalk edge nodes to the X
and Y coordinates of the nearest deck node.
112
5.3.2.3
Other Element Nodes
Girder nodes are placed transversely at the centerline of the girder under deck nodes. Nodes are
placed vertically at the centroid of the girders, at the top surface of the top flange, and at the
bottom surface of the bottom flange.
Extra diaphragm nodes are placed halfway between the girder lines at the same vertical level as
the girder centroid nodes for cross-bracing. Extra diaphragm nodes are placed halfway between
the girder lines at the same vertical level as the girder bottom flange nodes for chevron-bracing.
Channel section diaphragm elements connect directly to girder centroid nodes.
Barrier nodes are placed at mid-height for the barriers above the right and left edge sidewalk
nodes.
5.3.3
Beam Element Placement
Girder elements are placed between the girder centroid nodes. Girder centroid node rows in the
node array are identified and girder elements are iteratively placed between them. Diaphragm
elements are placed between either girder centroid nodes in the case of channel section or from
nodes placed halfway between girder centroid to the top and bottom flange nodes. Barrier
elements are placed between the barrier nodes.
5.3.4
Continuity Element Placement
Rigid links are placed at every girder centroid node between the centroid node and the top and
bottom flange nodes. Girder-to-deck connection elements are placed between the top girder
flange node and the deck node directly above it.
113
5.3.5
Shell Placement Algorithm
Deck shell elements are placed by iteratively searching through the deck node dimension of the
node array starting with node [1, 1, 3]; the algorithm searches the local space where the current
node is in array index [i, j, 3] with {i = 1, j = 1} and looks at indices [i+1, j, 3], [i+1, j+1, 3], and [i, j+1,
3]. Deck shell elements are placed according to which bins in the array contain node index
numbers and may be either three- or four- sided shell elements. The algorithm then moves to the
[i+1, j, 3] node and repeats the same process where {i = i+1}. After completing an entire row in the
array the algorithm moves to {i = 1, j = j+1} and repeats the process. Sidewalk shell elements are
placed in the same manner beginning with node [1, 1, 2].
5.4
5.4.1
Property Assignment
Shell Elements
5.4.1.1
Deck and Sidewalks
Deck and sidewalk shell elements are assigned the material properties given in Table 5.1.
114
Table 5.1. Deck and Sidewalk Concrete Material Property Assignment
5.4.2
Material Property
Value
Modulus
Variable
Poisson’s Ratio
0.2
Density
0.0868 lb/in3
Viscous Damping
0
Damping Ratio
0
Thermal Expansion
5.556x10-6 /F
Beam Elements
All beam elements using rolled sections are based on standard AISC steel sections. This includes
girders using wide-flange sections and diaphragms using either angle sections or channel
sections. Dimensions are taken from the AISC Steel Construction Manual Shapes Database V14.1
and are used in accordance with the AISC Naming Conventions for Structural Steel Products for Use
in Electronic Data Interchange (EDI). (AISC 2015) (AISC 2001)
5.4.2.1
Girders
Rolled girders are AISC W-shapes and assigned material properties given in Table 5.2 and section
properties in Table 5.3. Built-up, or welded girders, are doubly symmetric I-shape sections that
are assigned total depth, web thickness, flange thickness, and flange width. For both types of
sections section properties are calculated with the exception of Area for rolled sections. For
sections with cover-plates, the flange thickness (tf), includes the thickness of the flange and coverplate. Steel girders with shear deflection turned “off” have shear area set to an arbitrarily high
value.
115
Table 5.2. Girder Steel Material Property Assignment
Material Property
Value
Modulus
2.9x107 psi
Poisson’s Ratio
0.25
Density
0.2836 lb/in3
Viscous Damping
0
Damping Ratio
0
Thermal Expansion
6.5x10-6 /F
Table 5.3. Girder Steel Section Property Assignment
Section Property
Equation
Section Area
From ASIC Manual or Calculated
I11
I22
J
3
2𝑡𝑡𝑓𝑓 𝑏𝑏𝑓𝑓3 − �𝑑𝑑 − 2𝑡𝑡𝑓𝑓 � (𝑏𝑏𝑓𝑓 − 𝑡𝑡𝑤𝑤 )
− 𝐴𝐴𝑥𝑥̅ 2
3
2
𝑏𝑏𝑓𝑓 𝑑𝑑3 + �𝑑𝑑 − 2𝑡𝑡𝑓𝑓 �𝑡𝑡𝑤𝑤
12
3
𝑡𝑡𝑤𝑤
�𝑑𝑑 − 2𝑡𝑡𝑓𝑓 � + 2𝑏𝑏𝑓𝑓 𝑡𝑡𝑓𝑓3
3
−𝐴𝐴
4
Shear L1
𝑑𝑑
2
Shear L2
Shear A1
Shear A2
𝑑𝑑𝑡𝑡𝑤𝑤 or 1x106
5
𝑏𝑏 𝑡𝑡
3 𝑓𝑓 𝑓𝑓
or 1x106
116
5.4.2.2
Diaphragms
Diaphragms are either AISC C-shapes or L-shapes and assigned a steel material property with
Table 5.4 and section properties with Table 5.5 and Table 5.6.
Table 5.4. Diaphragm Steel Material Property Assignment
Material Property
Value
Modulus
2.9x107 psi
Poisson’s Ratio
0.25
Density
0.2836 lb/in3
Viscous Damping
0
Damping Ratio
0
Thermal Expansion
6.5x10-6 /F
117
Table 5.5. Diaphragm Channel Section Property Assignment
Section Property
Equation
Section Area
From ASIC Manual
I11
I22
J
Shear L1
Shear L2
Shear A1
Shear A2
3
2𝑡𝑡𝑓𝑓 𝑏𝑏𝑓𝑓3 − �𝑑𝑑 − 2𝑡𝑡𝑓𝑓 � (𝑏𝑏𝑓𝑓 − 𝑡𝑡𝑤𝑤 )
− 𝐴𝐴𝑥𝑥̅ 2
3
2
𝑏𝑏𝑓𝑓 𝑑𝑑3 + �𝑑𝑑 − 2𝑡𝑡𝑓𝑓 �𝑡𝑡𝑤𝑤
12
3
𝑡𝑡𝑤𝑤
�𝑑𝑑 − 2𝑡𝑡𝑓𝑓 � + 2𝑏𝑏𝑓𝑓 𝑡𝑡𝑓𝑓3
3
−𝐴𝐴
4
𝑑𝑑
2
𝑑𝑑𝑡𝑡𝑤𝑤
5
𝑏𝑏 𝑡𝑡
3 𝑓𝑓 𝑓𝑓
118
Table 5.6. Diaphragm Angle Section Property Assignment
Section Property
Equation
Section Area
From ASIC Manual
I11
I22
J
Shear L1
Shear L2
Shear A1
Shear A2
5.4.2.3
1
(𝑡𝑡(𝐵𝐵 − 𝑥𝑥̅ )3 + 𝑑𝑑𝑥𝑥̅ 3 ) − (𝑑𝑑 − 𝑡𝑡)(𝑥𝑥̅ − 𝑡𝑡)3 )
3
1
(𝑡𝑡(𝑑𝑑 − 𝑦𝑦�)3 + 𝐵𝐵𝑦𝑦� 3 ) − (𝐵𝐵 − 𝑡𝑡)(𝑦𝑦� − 𝑡𝑡)3 )
3
3
𝐵𝐵𝐵𝐵𝑤𝑤
+ (𝑑𝑑 − 𝑡𝑡)𝑡𝑡𝑓𝑓3
3
𝑡𝑡
2
𝑡𝑡
2
2
𝑑𝑑𝑑𝑑
3
2
𝐵𝐵𝐵𝐵
3
Barriers
Barriers are given the material properties found in Table 5.7 and section properties in Table 5.8.
119
Table 5.7. Barrier Concrete Material Property Assignment
Material Property
Value
Modulus
Variable
Poisson’s Ratio
0.2
Density
0.0868 lb/in3
Viscous Damping
0
Damping Ratio
0
Thermal Expansion
5.556x10-6 /F
Table 5.8. Barrier Rectangular Section Property Assignment
Section Property
Equation
Section Area
𝑏𝑏ℎ
I11
I22
J
Shear L1
Shear L2
Shear A1
Shear A2
𝑏𝑏ℎ^3
12
ℎ𝑏𝑏^3
12
3
𝑡𝑡𝑤𝑤
3
−𝐴𝐴
4
𝑑𝑑
2
𝑑𝑑𝑡𝑡𝑤𝑤
5
𝑏𝑏 𝑡𝑡
3 𝑓𝑓 𝑓𝑓
120
5.4.3
Composite Action Elements
Composite action elements are two-node beam elements with steel material properties (Table 5.9)
and section properties set to an arbitrarily high value except for moment of inertia about the
global transverse axis (Table 5.10).
I22 is set to either an arbitrarily high value to enforce
complete composite action or set to a lower value, determined with sensitivity analysis, to
modulate the degree of composite action.
Table 5.9. Composite Action Element Steel Material Property Assignment
Material Property
Value
Modulus
2.9x107 psi
Poisson’s Ratio
0.25
Density
0.2836 lb/in3
Viscous Damping
0
Damping Ratio
0
Thermal Expansion
6.5x10-6 /F
121
Table 5.10. Composite Section Property Assignment
Section Property
Section Area
I11
I22
J
Shear L1
Shear L2
Shear A1
Shear A2
Equation
1𝑥𝑥107
1𝑥𝑥107
𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉
1𝑥𝑥107
1𝑥𝑥107
1𝑥𝑥107
1𝑥𝑥107
1𝑥𝑥107
A two beam model was constructed to investigate the effect of using a beam element for
composite action. The beam element was given the properties shown in Table 5.10. The moment
of inertia about the transverse axis, I22, was adjusted on a logarithmic scale to determine the
sensitivity and stability of the model to a AASHTO LRFD live load truck and lane combination
(Figure 5.15 and Table 5.11) as well as the sensitivity of live load distribution. Adjusting the
moment of inertia proved effective in changing the live load in both beam under the load path
(Beam 1) as well as the adjacent beam. The same study was carried out for a single point load of
80 kips over mid-span of the right beam (Beam 1) and showed a similar effect and stability.
122
Figure 5.14. Two-Beam Model with Live Load Combination Applied
Live Load Moment [kip-fit]
-14000.00
-12000.00
-10000.00
-8000.00
Beam 1
-6000.00
Beam 2
-4000.00
-2000.00
0.00
0.00001
0.01
10
10000
10000000
Moment of Inertia [in4]
Figure 5.15. Effect of Composite Action Beam Moment of Inertia on Live Load Moment
and Live Load Distribution
123
Table 5.11. Sensitivity Study for Use of Moment of Inertia in Beam Elements as
Composite Action Links Using Two-Beam Model
Live Load Moment
Absolute [kip-ft]
I22 [in4]
% Total
Beam 1
Beam 2
Sum
Beam 1
Beam 2
0.0001
-11430
-6397
-17827
64.12
35.88
0.001
-11076
-6108
-17184
64.46
35.54
0.01
-8930
-4452
-13381
66.73
33.27
0.1
-6026
-2450
-8476
71.09
28.91
1
-5186
-1987
-7174
72.30
27.70
10000000
-5062
-1920
-6983
72.50
27.50
Shear area was also studied for the same effects. Moment of inertia (I22) was set to 1x107 in4 shear
area was adjusted on the same logarithmic scale (Table 5.12).
strong agreement with the effects of using moment of inertia.
Adjusting shear area showed
124
Table 5.12. Sensitivity Study for Use of Shear Area Adjustment in Beam Elements as
Composite Action Links Using Two-Beam Model
Live Load Moment
Absolute [kip-ft]
SA [in2]
% Total
Beam 1
Beam 2
Sum
Beam 1
Beam 2
0.0001
-11430
-6377
-17807
64.12
35.77
0.001
-11129
-6131
-17260
64.76
35.68
0.01
-9923
-4633
-14555
74.15
34.62
0.1
-6180
-2523
-8703
72.91
29.76
1
-5230
-1993
-7223
72.91
27.78
10000000
-5047
-1919
-6966
72.28
27.48
Also investigated was the effect of adjusting both moment of inertia and shear area together.
This resulted in no change in live load moment at the lower and upper bounds of the sensitive
range and negligible changes in live load moment in the middle of the sensitive range.
Deck modulus was also studied for its efficacy as a surrogate for composite action. Adjusting the
deck modulus resulted in similar live load moment variation in the beams (Table 5.13). The effect
on distribution of load between the beams was significantly more sensitive than when adjusting
moment of inertia or shear area when deck modulus was at the lower bounds.
125
Table 5.13. Sensitivity Study for Use of Deck Modulus in Beam Elements as Composite
Action Links Using Two-Beam Model
Live Load Moment
Absolute [kip-ft]
% Total
E [100 psi x 10x]
Beam 1
Beam 2
Sum
Beam 1
Beam 2
1
-17277
-2155
-19432
88.91
11.09
2
-17136
-2250
-19386
88.39
11.61
3
-16578
-2417
-18995
87.27
12.73
4
-13616
-2472
-16088
84.63
15.37
5
-7252
-1906
-9158
79.19
20.81
6
-3931
-2144
-6075
64.71
35.29
5.5
Verification of Automated Finite Element Modeling for the Study of Bias in the
Single Line-Girder Model
Multi-girder bridges have been commonly modeled using 2D grillage, 3D element-level models,
shell element models, and solid element models. An archetypical multi-girder bridge was used
to assess the trade-offs of various modeling approaches and select an appropriate model type.
Specifically, this study examined various model forms, element types, boundary and continuity
conditions, mesh size, results extraction approaches, etc.
The objectives were to determine
appropriate (in terms of both accuracy and efficiency) FE modeling approaches for multi-girder
bridges. In order to investigate the efficacy of element-level and shell-element multi-girder
bridge FE models, two benchmark structures were examined. First, a single 2-span continuous
composite beam was modeled using both of the aforementioned methods. Second a multi-girder
126
system composed of two continuous beams with a contiguous deck as well as cross-bracing
elements was studied to examine the modeling of transverse elements
5.5.1
Common Modeling Approaches For Multi-Girder Bridges
The behavior of common multi-girder bridges is frequently simulated using a wide range
modeling techniques.
The following sections provide a brief overview of commonly used
modeling approaches to provide context to the investigation at hand.
5.5.1.1
Single Line Girder Method of Analysis
This method is the most basic and commonly used approach for the design and performance
evaluation of common bridge types within the U.S. This approach approximates structural
phenomena through various equations to estimate the equivalent demands a single girder within
the structural system will experience. As mentioned previously, this approach has been shown
to under-estimate stiffness, but is generally conservative for the computation of dead and live
load actions.
5.5.1.2
2D Grid Method of Analysis
The 2D grid method borrows assumptions from the classical “plane grid” analysis method, and is
sometimes referred to as a grillage model. The girders and diaphragms are modeled as beam
elements having three degrees of freedom (DOF) per node – specifically, two rotational and one
translational DOF, with no depth information being explicitly represented. The two rotational
degrees of freedom capture each girders’ major axis bending and torsional response. The single
translational degree of freedom captures the vertical displacements of the girder. With this
method, all of the girders, diaphragms, and bearings are located at the same theoretical elevation
127
in the model. Such models only permit the computation of vertical displacements and rotations
within the plane of the bridge model.
5.5.1.3
2D Frame Method of Analysis
Similar to the 2D grid model, the 2D frame method of analysis ignores depth information.
However, in this approach, the beam elements are equipped with six degrees of freedom at each
node, three translational and three rotational. According to White et al. (2012), if there is no
coupling between the degrees of freedom for the conventional 2D-grid and the three additional
degrees of freedom, 2D-frame models actually do not provide any additional information beyond
the ordinary 2D-grid solutions. That is, all of the displacements at the three additional nodal
degrees of freedom will be zero, assuming gravity acts normal to the plane of the structure (i.e.
the bridge does not have a significantly longitudinal slope).
5.5.1.4
Element-Level Method of Analysis
This type of model employs both one-dimensional (frame/beam elements) and two-dimensional
elements (plate or shell elements) to model girders/diaphragms and the deck, respectively. Beam
elements have either 2 or 3 nodes with 6 DOFs each. Plate/shell elements may have 3 (in the case
of triangular elements), 4(in the case of rectangular elements), or up to 9 (in the case of 9-node
rectangular shells) nodes with up to 6 DOFs each. In an effort to remain consistent with the three
dimensional geometry of the structure, various link elements (to connect girders to the deck and
diaphragm elements to the girders) and constraints (to simulate boundaries) are also employed.
This model resolution is commonly termed “element-level” and is the most common class of 3D
FE models employed for constructed systems (ASCE, 2013). The figure below shows a schematic
illustrating how 3D geometry of the bridge is simulated using various elements and links. In an
element level model a girder is discretized into 1D beam elements and the cross-sections are
128
applied through the definition of geometric constants (e.g. area, moment of inertia, etc.) to the
finite elements.
Beam
Shell
Rigid Link
Beam
Figure 5.16. 3D Geometric Element-Level Model
5.5.1.5
Shell Element Method of Analysis
The most significant distinction between element-level and shell element models of multi-girder
bridges is that the beams in shell element models are discretized vertically, laterally, and
longitudinally using shell elements.
5.5.1.6
Conclusions
An element-level FE model can reasonably simulate most bridge responses, it is not without its
shortcomings, specifically: (1) an inability to effectively simulate warping deformation of girders
(associated with torsion), and (2) an inability to simulate localized stresses (i.e. stress
concentrations) associated with geometric discontinuities. While these shortcomings may be
critical in the case of modeling specific construction sequences for complex bridges (White et al.
2012) and advanced fatigue/fracture assessment, they are not relevant for the global limits states
to be investigated in this study. Modeling girders with shell elements, on the other hand, allows
129
for the accurate simulation of warping of the girders due to torsion. Computation time, model
construction, and result extraction activities however, are more time consuming and more
difficult than with element level models.
While the element-level model resolution can reasonably simulate many actual bridge responses,
it is not without its shortcomings. Specifically, the shortcomings include (1) an inability to
effectively simulate the warping deformation (associated with torsion), and (2) an inability to
simulate localized stresses (i.e. stress concentrations) associated with geometric discontinuities.
While these shortcomings may be critical in the case of modeling specific construction sequences
for complex bridges (White et al. 2012) and advanced fatigue/fracture assessment, they are not
relevant for
While an element-level FE model can reasonably simulate most bridge responses, it is not
without its shortcomings, specifically: (1) an inability to effectively simulate warping deformation
of girders (associated with torsion), and (2) an inability to simulate localized stresses (i.e. stress
concentrations) associated with geometric discontinuities. While these shortcomings may be
critical in the case of modeling specific construction sequences for complex bridges (White et al.
2012) and advanced fatigue/fracture assessment, they are not relevant for the global limits states
to be investigated in this study.
5.5.2
Effects of Modeling Choice on Performance of Shell and Element Level Model
Types
A composite multi-girder modeling study was undertaken to investigate effect of modeling
decisions on responses of interest. For the two modeling approaches examined, the following
aspects were studied for response convergence and/or consistency with the behavior mechanisms
being simulated:
1.
Boundary conditions
130
2.
Continuity Conditions
3.
Model discretization
4.
Results extraction methods
5.
Beam-element shear deformation and response
6.
Computational efficiency
To examine these modeling aspects, the girder and deck responses for the two benchmark
structures under dead load and support settlement were examined as described in Table 5.14.
Table 5.15 provides the details of the benchmark structures.
Table 5.14. Summary of Demands and Reponses Used in Benchmark Study
Demand: Dead Load
Demand: Support Settlement
Deflection
Total composite section (total fiber) stress
Vertical reaction at the support
Deck stress due to tension
Member actions (shear, axial, moment)
Table 5.15. Benchmark Model Details
Type
Element-level
Element-level
Shell Element
Shell Element
Number of
Girders
1
2
1
2
Deck
Thickness
8 in.
8 in.
8 in.
8 in.
Total Deck
Width
96 in.
96 in.
96 in.
96 in
Span Length
960 in.
960 in.
960 in.
960 in.
Girder
Depth
21 in.
21 in.
21 in.
21 in.
The models used in the investigation were made up of 2-node beam elements with 6 DOFs at
each node, 3-node triangular shell elements with 6 DOFs at each node, and 4-node rectangular
131
shell elements with 6 DOFs at each node. The following sections provide results related to each
of the modeling aspects examined within this study.
5.5.2.1
Effects of Boundary and Continuity Conditions
Boundary conditions for the element-level model were enforced by restricting all degrees of
freedom except for rotation about the Z axis on each “pin” boundary, and all degrees of freedom
except rotation about Z axis and translation in the X (longitudinal) axis on each “roller”
boundary. Instead of placing the boundary restriction at the beam centroid, boundary nodes
were placed at the bottom fiber of the beam section and rigid link elements were used to connect
the beam element node to the boundary node. This boundary offset more closely mimics that of
a real structure.
Figure 5.17. Element-level Model Continuity and Boundary Conditions
Girder/cross-bracing continuity was enforced for the two-girder models by the same rigid link
construction. At each boundary as well as intermediate cross-bracing points, rigid links were
132
connected to nodes located at the top and bottom of the girder cross-section.
Deck/girder
continuity was connecting rigid links between the nodes located half at the top surface of the
beam flange to the nodes of the deck shell elements located directly above. Boundary and
continuity construction for a sample two-girder element-level model is shown in Figure 5.17 .
Boundary conditions on the shell element model were enforced by restricting all degrees of
freedom except for rotation about the Z axis on each “pin” boundary, and all degrees of freedom
except rotation about Z axis and translation in the X axis on each “roller” boundary. Because
shell element models are susceptible to local distortion due to point loads, rigid links were placed
along the edge of each exterior shell element at each girder end, essentially rendering the crosssection at each boundary vertically rigid.
Girder/cross-bracing continuity was enforced for the two-girder models by the same rigid link
construction: at each boundary as well as intermediate cross-bracing points, rigid links were
connected between each node in the girder cross-section. Deck/girder continuity was enforced by
connecting rigid links between each node of the top flange shells and each node of the deck shells
located directly above. Boundary and continuity construction for a sample two-girder shell
element model may be seen in Figure 5.18.
Figure 5.18. Shell Element Model Continuity and Boundary Conditions
133
5.5.2.2
Effects of Model Discretization
Model discretization, or element size, was studied for both element-level and shell-element
models to determine response convergence. Five levels of discretization were studied for the
element-level models and three levels of discretization were studied for the shell-element models.
Beam element sizes were given to the model building software as a target length. In the case of
irregular geometry, skews, etc., the software will shorten or lengthen an element according to
maximum and minimum element size criteria. No shell elements had an aspect ratio greater than
2:1, with most shell elements having an aspect ratio around 1:1. Table 5.16 provides the element
sizes examined for both element-level and shell element models; the average element size is
given as a length as well as the ratio of the beam depth to the element size. Figure 5.19, Figure
5.20, Figure 5.21, and Figure 5.22 show schematics of each model included within this study for
both modeling approaches and both benchmark structures.
Table 5.16. Element Sizes for Discretization Study
Element-Level
[in]
2.5”
5”
10”
20”
40”
Element-Level
[ratio]
8
4
2
1
0.5
Shell Element
[in]
2.5”
5”
10”
-
Shell Element
[ratio]
8
4
2
Early in the discretization study it was evident that shear deformation of the beam elements
within the element-level models was producing divergent results in many load cases. Due to
this, the discretization study was modified to include the effects of shear deformation on mesh
size convergence. The following sections provide representative results from these studies.
134
Figure 5.19. Discretization Levels of Single-Girder Element-level Model
135
Figure 5.20. Discretization Levels of Two-Girder Element-level Model
136
Figure 5.21. Discretization Levels of Single Girder Shell Element Model
137
Figure 5.22. Discretization Levels for Two-Girder Shell Element Model
5.5.2.3
Effect of Shear Deformation Calculation
Element-level models are susceptible to mesh dependency due to the shear deformation of the
beam elements. Discretization levels strongly influence the degree of composite action, and
responses such as deformation under dead load as well as moment in the beam, axial force in the
beam, and shear in the beam due to support settlement vary with mesh size. In order to assess
the degree to which shear deformation causes divergence issues, a study was performed to
compare the aforementioned responses when shear deformation in the beam elements was
“turned” ON or OFF. In order to do this, the shear area of the beam element cross section was
replaced with an artificially higher number (in the case of this study, 1x109 in2 was used).
Beam elements were discretized to the sizes listed in the previous section. For plotting purposes,
the element size to beam depth ratio is used instead of the absolute value. This provides a more
138
widely applicable metric for all future studies. It is the intention of this study to determine the
relative element size that will provide a converged solution while also allowing for
computational efficiency.
Figure 5.23, Figure 5.24, Figure 5.25 illustrates the shear force, moment, and axial force in the
girder versus relative element size in response to vertical settlement at one abutment, and
indicates that that if shear deformation of the beam elements is considered, the results do not
converge. In contrast, when shear deformation of the beam elements is ignored, convergence
was observed at a beam depth to element length ratio of approximately four. In addition, when
shear deformation of the beam elements is ignored, the shear force computed for the girders
represents the majority of the shear force on the composite cross-section, which is consistent with
the mechanics of materials solutions. The results presented here are derived the two-girder
element-level models. The single girder model results were almost identical to these and are not
shown.
3.000
Shear Force [k]
2.500
Shear
Deformation OFF
2.000
1.500
Shear
Deformation ON
1.000
0.500
0.000
0
5
10
15
Beam Depth/Element Length
Figure 5.23. Effect of Shear Deformation Calculation on Shear Force Convergence as a
Function of Beam Depth to Element Length Ratio Subject to Vertical Abutment
Settlement in the Two-Girder Model
139
-430.000
Moment [k-in]
-435.000
-440.000
Shear
Deformation ON
-445.000
-450.000
Shear
Deformation OFF
-455.000
-460.000
-465.000
-470.000
0
5
10
Beam Depth/Element Size
15
Figure 5.24. Effect of Shear Deformation Calculation on Moment Convergence as a
Function of Beam Depth to Element Length Ratio Subject to Vertical Abutment
Settlement in the Two-Girder Model
-60.000
Axial Force [k]
-60.500
-61.000
Shear Deformation
ON
-61.500
-62.000
Shear Deformation
OFF
-62.500
-63.000
-63.500
-64.000
0
5
10
15
Beam Depth/Element Length
Figure 5.25. Effect of Shear Deformation Calculation on Axial Force Convergence as a
Function of Beam Depth to Element Length Ratio Subject to Vertical Abutment
Settlement in the Two-Girder Model
140
Figure 5.26, Figure 5.27, Figure 5.28 illustrates the shear force, moment, and axial force in the
girder versus relative element size in response to a point load at mid-span in the center of the
deck elements between the two girders.
10.000
9.000
Shear Force [k]
8.000
Shear Deformation
OFF
7.000
6.000
5.000
Shear Deformation
ON
4.000
3.000
2.000
1.000
0.000
0
5
10
15
Beam Depth/Element Length
Figure 5.26. Effect of Shear Deformation Calculation on Shear Force Convergence as a
Function of Beam Depth to Element Length Ratio Subject to Point Load at Mid-Span in
the Two-Girder Model
141
510.000
500.000
Moment [k-in]
490.000
Shear Deformation
ON
480.000
470.000
Shear Deformation
OFF
460.000
450.000
440.000
430.000
420.000
0
5
10
15
Beam Depth/Element Length
Figure 5.27. Effect of Shear Deformation Calculation on Moment Convergence as a
Function of Beam Depth to Element Length Ratio Subject to Point Load at Mid-Span in
the Two-Girder Model
62.00
61.50
Shear Deformation
OFF
Axial Force [k]
61.00
60.50
Shear Deformation
ON
60.00
59.50
59.00
0
5
10
15
Beam Depth/Element Length
Figure 5.28. Effect of Shear Deformation Calculation on Axial Force Convergence as a
Function of Beam Depth to Element Length Ratio Subject to Point Load at Mid-Span
142
5.5.3
Comparison of Results Convergence Agreement Between Shell Element and
Element-Level Model Types
This portion of the study examined the convergence of various responses for both element-level
and shell element models and agreement between model types. Figure 5.29 shows convergence
plots of deck stress for the element-level and shell element models. This figure illustrate three
key points: (1) In the element-level model, with shear deformation ignored, deck stress converges
at an element ratio of approximately four; (2) Ignoring shear deformation results in convergence
at a coarser mesh size; and (3) There is good agreement (under 5%) between the element-level
and shell element model. This agreement is slightly better when shear deformation of the beam
elements is ignored.
5.5.3.1
Effect of Mean Element Length on Deck Stress Under Vertical
Settlement in Two-Girder Beam Element Model
The other responses listed in Table 5.14 were also examined for both the element-level and shell
element models, and the results were consistent with the trends shown in this section.
Specifically, results convergence occurred around an element ratio of four, and ignoring shear
deformation within the element-level model produced more consistent results with the shell
element model. The results illustrating this can be found in Figure 5.29.
143
80.00
Maximum with
Shear Def. Off
79.00
Maximum with
Shear Def. ON
Stress [psi]
78.00
Shell Element
Maximum
77.00
Mean with Shear
Def. Off
76.00
Mean with Shear
Def. ON
75.00
74.00
Shell Element
Mean
0
2
4
6
8
Beam Depth/Element Length
Figure 5.29. Effect of Element Size on Deck Stress Under a Vertical Settlement in a TwoGirder Shell Element Model
5.5.4
Investigation of Automated Analysis and Results Extraction Methods
In order to ensure reliable analysis for large groups of FE models, the requirements of extracting
static load and displacement results in a repeatable and automated fashion through the API were
investigated for element level and shell element models. Analysis times using different model
types and discretization levels were also compared.
5.5.4.1
Element-Level Model
Results from the element-level models were extracted at the nodes above the center support.
Stress results for the beam elements were computed using the calculated moment and the
geometric constants that describe the cross-section. Shear in the beams were extracted in two
ways: (1) It was taken as the shear within the beam element; (2) It was assumed to be equal to the
144
reaction at the support. The first method of extracting shear proved to be dependent on the mesh
size, and thus was deemed unreliable. The second method was conservative (in that it assumed
the total shear force on the cross-section was resisted by the girder alone) and this approach
converged at the element aspect ratio of four.
Deck stress over the center support was also computed based on the beam element response as
opposed to directly extracted from the model. This was done to avoid the anomalous stress
concentrations observed in the shell elements where the rigid links connect them to the beam
elements. By assuming strain compatibility for the composite section, the nominal stresses in the
deck were computed by calculating the strain diagram for the beam and extrapolating this to the
top of the deck (and the transforming this to stress using the elastic modulus of concrete).
5.5.4.2
Shell Element Model
The shell element model investigation required that the normal and shear forces from each
element were extracted and then summed to obtain various member actions (moment, axial,
shear). Because each girder is made up of individual shell elements, there is no direct way to
extract member actions. This is in contrast to the element level model, where member actions are
solved for directly.
Local stress concentrations are also an issue with shell elements. For example, Figure 5.30 shows
examples of how localized loading (either through a support reaction, point load, or rigid link
connection) causes local distortion and stress concentrations. Such responses are strongly meshdependent and are not realistic, since in the actual bridge structure such point loadings do not
exist. As a result, when extracting results from shell elements, the analyst has to be careful to
capture nominal responses and not those associated with these localized effects. This may be
done by following Saint-Venant’s Principle, i.e., extracting results at a sufficient distance from the
concentrated force.
145
(a)
(b)
Figure 5.30. Stress Contour in the Principle XX Direction in a Shell Element Model Due
to Point Load at Mid-Span. (a) Deck (b) Beam Web
5.5.4.3
Computational Efficiency
A study was conducted in order to compare the computing time for element-level and shell
element models. As shown in Figure 5.31, the shell elements were found to be much more
computationally expensive (e.g. requiring 800% more time for modeling with an element ratio of
four). While the absolute times for these models may appear small, this factor is critical when
large sensitivity studies are carried out.
146
100.000
Double Girder Element-Level
Time [sec]
10.000
Double Girder Shell-Element
Single Girder Element-Level
1.000
Single Girder Shell-Element
0.100
0
2
4
6
8
Beam Depth/Element Length
Figure 5.31. Computation Time as a Function of Element Discretization Level
5.5.5
Summary and Conclusions of the Composite Beam Modeling Study
Based on the results of this modeling study, it is concluded that the element-level modeling
approach (where the shear deformation of the beam elements is ignored) is the most appropriate
for simulation of multi-girder bridges. In comparing the element-level modeling approach to the
shell element modelling approach, the following observations were made:
1.
The different approaches provided consistent results (approximately 5%
difference) for the demands and responses examined
2.
Both modeling approaches converged at element ratios of approximately four
(when the shear deformation of the beam elements was ignored)
3.
The element-level model allowed for easier and more consistent results
extraction approaches
147
4.
The element-level modeling approach required less 15% of the computational
time required by the equivalent shell element modelling approach
5.5.6
Final Investigation into Model Form Using Benchmark Full-Bridge Models
To extend the element-level model selected through the study in the previous section to complete
bridge systems, an additional mesh sensitivity study is required. This study was carried out on a
two-span continuous benchmark multi-girder bridge with 100 ft. spans. Since a 1:1 aspect ratio
for shell elements was desired, a girder spacing of 10 ft. (which is a factor of the span length) was
selected. The skew was set to zero to eliminate variably sized deck elements and to avoid nonrectangular elements. Boundary conditions were defined so as to provide the least restraint
possible. This was achieved by restraining all supports in the vertical direction, restraining the
exterior girders in the longitudinal direction at one abutment, and restraining the central girder
in the transverse direction at that same abutment. In this manner local self-equilibrating forces
were avoided.
Figure 5.32. Restrained Boundary Degrees of Freedom
148
Shear deformation was turned off by drastically increasing the shear area of the girders. As
shown in the previously presented benchmark study, shear cannot be accurately modeled with
beam elements and this change will have little effect on moment and axial forces. Shear forces
will be determined directly from the support reactions.
The bridge was designed with five 45 in. deep beams of 36 ksi steel. As used with the single and
dual-girder composite models, an 8 in. deep concrete deck was connected to the girders through
the use of rigid links to enforce composite action. Diaphragms were placed every 20 ft. Girders
and barriers were modeled as beam elements with the dimensions shown in Figure 5.31. The
deck and sidewalk were modeled as shell elements.
Figure 5.33. Steel W-Shape (I-Beam)
Figure 5.34. Typical Multi-Girder Bridge Cross Section
149
Six levels of discretization were investigated, including 60 in., 30 in., 15 in., 12 in., 10 in., and 6 in.
A separate model was created for each, with all other geometry and properties remaining
constant. These models were analyzed under three loading cases: dead load, a point load, and a
settlement. The dead load was based upon the material densities and geometry. A single point
load of 32 kips was applied over the center girder 40 feet from the abutment. The vertical
settlement was applied to each girder at the abutment and was set to one inch.
Linear static analysis was performed on each of these models under the three loading scenarios
using Strand7 FE software. Moment and axial forces, as well as reactions were recorded for
comparison. These results were located in the exterior and central girders over the center pier for
dead load and settlement member actions. Under the point load, member actions of the center
girder were recorded at the location of the applied point force. The reactions for the load
scenarios were taken at the abutments.
Results of this study revealed that a girder spacing to mesh size ratio of ten (corresponding to a
12 in. discretization) provides convergence, with any further refinement not significantly
effecting the results. Figure 5.35, Figure 5.36, and Figure 5.37 illustrate this convergence by
providing a sample of the results obtained. As can be seen from these figures, the percentage
differences quickly fall below one percent, thus indicating that little is gained from the increased
discretization. As a result of this study it is concluded that a girder spacing to mesh size ratio of
ten is more than sufficient to provide convergence of straight bridge.
150
1.8%
1.6%
Moment Ext
Percent Change
1.4%
Moment Int
1.2%
Axial Ext
1.0%
Axial Int
0.8%
Reaction Ext
0.6%
Reaction Int
0.4%
0.2%
0.0%
0
5
10
15
Girder Spacing/Mesh Size
20
25
Figure 5.35. Percent Change in Response to Support Settlement with Decreasing Mesh
Size
8%
Percent Change
7%
6%
Moment Ext
5%
Moment Int
Axial Ext
4%
Axial Int
Reaction Ext
3%
Reaction Int
2%
1%
0%
0
5
10
15
Girder Spacing/Mesh Size
20
25
Figure 5.36. Percent Change in Response to Dead Load with Decreasing Mesh Size
151
5.0%
4.5%
4.0%
Percent Change
3.5%
3.0%
Moment
2.5%
Axial
2.0%
Reaction
1.5%
1.0%
0.5%
0.0%
0
5
10
15
20
25
Girder Spacing/Mesh Size
Figure 5.37. Percent Change in Response to Point Load with Decreasing Mesh Size
5.5.6.1
Conclusions
Through a comparison of both single and two-girder element level and shell element model
systems, it was determined that the element-level model was the best choice for the large
parametric study. This conclusion was based on (1) the good agreement (approximately 5%
difference) between the element-level model and the more refined shell element model, (2) the
more straightforward manner in which results may be extracted from the element-level model,
and (3) the drastically reduced computational time associated with the element-level model.
In addition, to the selection of the general model approach, the multi-girder modeling studies
also provided insight into various implementation strategies for the element-level model. In
particular, the following modeling strategies are recommended for the large parametric study:
152
•
Shear deformation of the beam elements within the element-level models should be ignored
to ensure proper convergence of results.
•
Boundary conditions that provide minimum restraint (such as those detailed in earlier in this
chapter) should be used to minimize extraneous inputs associated with local, selfequilibrating forces.
•
Support reactions should be used to conservatively estimate the shear force in the girders, as
the computed shear force in the beam elements is mesh dependent.
•
Deck stresses should be approximated by extrapolating the strain in the girders to the top of
the deck to avoid local stress concentrations exist in the vicinity of rigid links.
•
Analysis methods and results extraction that eliminates concerns over shear deformation
sensitivity are to be preferred.
Using web girder area and vertical support reaction to
estimate shear forces in steel girders is preferred. Deck stresses due to tension should be
approximated from total composite section stress using interpolation and plane sections
assumptions.
153
6. Automated Finite Element A nalysis and Simulation
This chapter presents methods for the automated analysis and load
rating of finite element models developed using software described in the
preceding chapters.
Load application, results extraction, and the
computation of AAHSTO Load and Resistance Factor Rating Factors and
resultant tolerable support movements are detailed for simply supported
and two-span continuous steel multi-girder bridges.
6.1
Introduction
AASHTO LRFD load rating is achieved using a semi-automated load application, response
extraction, and rating computation tool developed using Matlab and the Strand7 Application
Programming Interface (API). A 3D geometric element level FE model is used as analysis model
to calculate demands on structural members. The capacity of each member is derived from the
AASHTO LRFR code and is the same capacity used in this research in LRFD girder design
(Chapter 4). This chapter presents the methods by which FE models are utilized for dead load,
live load, and support movement simulation. It also presents the methods for extracting member
responses, processing results, and computing AASHTO live load rating.
The FE rating software analyzes the FE model for applied dead and live loads by selectively
turning “on” or “off” the mass and stiffness of various elements and applying point loads to
targeted areas on the model. The loads are analyzed using the linear static solver in Strand7. The
responses at critical locations are extracted from the FE model and combined as per the AASHTO
MBE to develop rating factors for each applicable limit state.
154
6.2
Load Application
The load rating software applies forces and adjusts parameter stiffness values to simulate dead
and live load conditions. Deck concrete, girder steel, and diaphragm dead load is simulated by
turning off the deck shell element stiffness and barrier and sidewalk stiffness and mass.
Superimposed dead load is simulated by turning off deck concrete, girder steel and diaphragm
steel mass and by turning on barrier and sidewalk concrete mass while keeping barrier and
sidewalk stiffness off. Dead load due to additional components such as a deck overlay is then
simulated by turning the mass off and stiffness on all structural elements and barriers and
sidewalks. Live load is simulated by applying vertical point loads to deck shell element. Truck
point loads are placed on deck shell elements. Lane loads are simulated by placing point loads
over a lane patch. Member and node responses are extracted from the model. The following
outline lists the steps required for load application.
1. Optional new parameters assigned
2. Dead Load
2.1. Load Application
2.1.1. Due to self-weight at time of deck pour
2.1.1.1. Deck weight
2.1.1.2. Girder weight
2.1.1.3. Diaphragm weight
2.1.1.4. Deck stiffness turned off
2.1.2. Due to self-weight after deck has cured
2.1.2.1. Sidewalk weight
2.1.2.2. Barrier weight
2.1.2.3. Sidewalk stiffness turned off
2.1.2.4. Barrier stiffness turned off
2.1.3. Due to self-weight of additional components
155
2.1.3.1. Wearing Surface as point masses
3. Live Load
3.1. Load Application
3.1.1. Truck Loads (each individually)
3.1.1.1. 12’ lanes
3.1.1.2. Wheel point loads two feet from lane boundary
3.1.1.3. Three transverse lane positions
3.1.1.4. Positioned longitudinally for maximum effect
3.1.2. Lane Loads
3.1.2.1. Point loads at each node within 10’ wide load
path
3.1.2.2. Sum to AASHTO specified load
6.2.1
Dead Load
The given model is first analyzed under the dead load case. Three separate dead load cases are
analyzed separately as shown in Table 6.1.
The first is due to the self-weight of all the
components existing at the time the deck is poured including the weight of the deck. For this
analysis the stiffness of the deck is not considered because the deck concrete would not yet have
cured. The second case includes the self-weight of only those components that come after the
deck, specifically the sidewalks and barriers. Again, the stiffness of these components is not
considered, as the bridge experiences this load before the concrete of these components has
cured. The final dead load case is only analyzed when there are additional components that are
later added to the bridge, such as a wearing surface, that could be supported in part by the
sidewalks and barriers. If a wearing surface is to be included in the rating, the additional mass is
modeled as point masses on the deck nodes. The magnitude of these point masses corresponds
to the mass of the pavement occupying the tributary area surrounding the node.
156
Table 6.1. Dead Load Stage Parameter Modifications
Stage
Parameter Modifications
Deck
Girder
1
Dead Load
Diaphragm
Barriers and Sidewalks
Deck
Girder
2
Superimposed
Dead Load
Diaphragm
Barriers and Sidewalks
Deck
3
Additional
Components Dead
Load
Girder
Diaphragm
Barriers and Sidewalks
Stiffness Off
Weight On
Stiffness On
Weight On
Stiffness On
Weight On
Stiffness Off
Weight Off
Stiffness On
Weight Off
Stiffness On
Weight Off
Stiffness On
Weight Off
Stiffness Off
Weight On
Stiffness On
Weight Off
Stiffness On
Weight Off
Stiffness On
Weight Off
Stiffness On
Weight Off
157
6.2.2
Live Load
6.2.2.1
Definition of Truck and Lane Loads
The model is analyzed for live load by positioning truck and lane loads in individual lanes and .
Both of these load types are positioned in lanes.
Only loading corresponding to a single truck or
single lane load is analyzed at a time. In this way the results can be investigated under all
possible load combinations (through superposition) to identify the “worst case” loading scenario.
The truck loads consist of six point loads corresponding to each tire group and with magnitudes
defined by AASHTO as shown in
Table 6.2. The lane loads are defined by AASHTO and are modeled as point loads distributed to
the deck nodes such that the sum of the point loads equals the total load prescribed by
AASHTO’s specified uniform load.
158
Table 6.2. Live Load Application Types
Live Load Type
Details
1.
HL-93 Design Truck
2.
Wheel loads as point loads on
shell element surface
3.
Placed
transversely
within
lanes to maximize loading for
Truck Load
interior and exterior girders
4.
Placed
to
longitudinally
maximize to moments and
shear
1.
640 lb/ft distributed over a 10’
width
Lane Load
6.2.2.2
2.
Point loads on deck nodes
3.
Placed in center of lane
Transverse Load Positioning
The number of lanes is determined by the road width such that the maximum number of
complete 12’ wide lanes is assigned. If the number of 12’ lanes that fit onto the deck is less than
two, the software tries to fit two lanes with widths between 10’ and 12’. If the deck cannot fit two
10’ lanes, one 12’ lane is used (See Table 6.3). Lanes are apportioned to the model and are placed
159
in three positions: centered, shifted to the extreme right, and shifted to the extreme left (Figure
6.1).
Table 6.3. Load Rating Lanes
Deck Width
Number of Lanes
Lane Width
> 24’
Width/12’
(rounded down to
nearest whole number)
12’
20’ <= w <= 24’
2
w/2
< 20’
1
12’
The truck loads are positioned in the center of the lanes, as well as shifted to either side to within
two feet of the edge of the lane as per the AASHTO MBE (Figure 6.1). Lane loads are positioned
in a 10’ load path at the center of the lane.
Figure 6.1. Transverse Lane Positions
160
6.2.2.3
Longitudinal Truck and Load Placement
The magnitude and spatial distribution of truck and lane loads are defined by the AASHTO
LRFD Specifications. Truck loads consist of six point loads corresponding to each tire group. Per
the AASHTO LRFD Specifications, trucks are placed to maximize moments and shear as shown
in Figure 6.2 and Figure 6.3. Trucks are placed only facing one direction. This produces the
greatest responses in all transversely symmetric (symmetric about the centerline) bridges. Dual
trucks are run facing in opposite direction to result in the largest negative moment over the pier
in two-span continuous structures. Table 6.4 details the longitudinal placement of each truck
along the span.
161
Table 6.4. Demand and Truck Load Locations
Demand
Truck Location
Continuity
Type Affected
Skew Types
Affected
Abutment Shear
Rear Axle Over
Abutment
Simply
Supported
and Two-Span
Continuous
Non-skewed
Abutment Shear
Closest Rear Wheel
Over Abutment
Simply
Supported
and Two-Span
Continuous
Skewed
Pier Shear
Rear Axle Over Pier
Two-Span
Continuous
Non-Skewed
Pier Shear
Centerline of Truck
and Rear Axle Over
Pier Line
Two-Span
Continuous
Skewed
Positive Bending
Centroid of Truck at
0.5 L
Simply
Supported
Skewed and
Non-skewed
Positive Bending
Centroid of Truck at
0.4 L or 0.6 L
Two-Span
Continuous
Skewed and
Non-skewed
Negative Bending
Centroid of Truck
Over Pier
Two-Span
Continuous
Skewed and
Non-skewed
Negative Bending
Centroid of Dual
Trucks Over Pier
Two-Span
Continuous
Skewed and
Non-skewed
The lane loads are modeled as point loads distributed to the deck nodes such that the sum of the
point loads equals the total load prescribed by AASHTO’s specified uniform load. Each load case
(corresponding to a single truck or single lane load) is analyzed individually. The results are
then combined to investigate all possible live load combinations in order to capture the “worst
case” loading scenario. Figure 6.1 through Figure 6.3 depict truck and lane load positioning for
simply supported as well as two-span continuous bridges.
Figure 6.4 through Figure 6.6
illustrate the point load positioning on the FE model deck shell elements.
162
Figure 6.2. Truck Positions for Simply Supported Bridges
163
Figure 6.3. Truck positions for Two-span Continuous Bridges
164
Figure 6.4. Truck Point Loads on FE Model Shell Element Faces
Figure 6.5. Lane Point Loads on FE Model Shell Element Vertex Nodes
165
Figure 6.6. Simulated Load Combination. Actual Load Combinations are Calculated
Using Superimposed Results
6.2.3
Support Movement
Two types of support movement have been coded in the software for static load analysis. The
two support movement types are summarized in Table 6.5.
166
Table 6.5. Support Movement Types
Bridge Configuration
Support Movement
Considered
Simple Spans – For
support movements that
induce forces.
Transverse Rotation –
about an axis
longitudinal to the
bridge.
Representation
Vertical Translation
Continuous Spans – For
all support movements
Transverse Rotation –
about an axis
longitudinal to the
bridge
Super-structure tolerance to longitudinal translation and longitudinal rotation of supports is
closely related to the Service I limit state.
The two types of support movement considered are vertical translation and transverse rotation
about the longitudinal axis. Two-span continuous bridges are subject to responses from vertical
translation and transverse rotation occurring at the abutment and at the pier.
Both abutments
are analyzed for settlement. This is due to the nature of the live load rating software. Truck
loads are run in one direction only. For non-skewed bridges the support settlement may be run
on either end of the bridge to obtain the greatest superposition of forces from live load and
settlement.
Skewed bridges, however, are non-symmetric about the longitudinal axis and
therefore must be analyzed for tolerable support settlement by moving both abutments. For this
same reason, simply supported bridges are analyzed with transverse rotation at both abutments.
167
Vertical translation of a support was simulated in the model by applying a unit translation in the
vertical downward direction to each bearing node at the support location of interest.
For
transverse rotation, the first bearing node is held stationary. A vertical translation is then applied
to subsequent nodes in increments until a maximum unit vertical translation is applied at the last
node. Two cases were analyzed for transverse rotation. The first analyzed rotation in the
clockwise direction by holding the left most bearing node stationary and settling the right side of
the support (Figure 6.7). The second analyzed rotation in the counterclockwise direction by
holding the right most bearing node stationary and settling the left side of the support (Figure
6.8).
0.25
0.5
0.75
1.0
Figure 6.7. “Clockwise” Transverse Rotation Support Movement
0.25
0.5
0.75
1.0
Figure 6.8. “Counter-Clockwise” Transverse Rotation Support Movement
168
6.3
6.3.1
Results Extraction
Response Locations of Interest
Response locations of interest refer to the likely points of critical response along a member. The
responses for beam element forces are recorded along the entire member however certain
“zones” or locations of interest for specific types of responses are analyzed as a filtering step.
Load rating and settlement response locations, while intrinsically linked due to tolerable support
movements including rating factors, will have different locations for interest.
The critical
response in a member due to some support movements may be located in a location other than
that of the limiting rating factor. These locations of interest are a range of values for length along
the entire structure.
For example, the negative moment region of interest for two-span
continuous bridges is approximately 20% of the span length on either side of the pier. The
maximum negative moment due to dead and live load effects is located directly over the pier,
and most support settlements will also result in maximum forces over the pier. It is unknown,
however, if the superposition of different live load and settlement forces for a highly skewed
bridge or a bridge with certain diaphragm configurations will result in the location for all critical
limit states to be directly over the pier. This filtering step prevents local spikes in responses due
to the model form from polluting rating or tolerable support results. It also condenses large
amounts of data into smaller memory requirements for large population studies.
169
Table 6.6. Response Types and Locations of Interest
Response Type
Response
Location
Continuity
Type Affected
Load Rating
Shear Rating
Abutment
Simply
Supported
and Two-Span
Continuous
Load Rating
Positive Moment
Rating
Mid-span
Simply
Supported
Load Rating
Shear Rating
Pier
Two-Span
Continuous
Load Rating
Negative Moment
Rating
Centerline of Truck
and Rear Axle Over
Pier Line
Two-Span
Continuous
Tolerable Settlement
Shear
Abutment
Simply
Supported
and Two-Span
Continuous
Tolerable Settlement
Shear
Pier
Two-Span
Continuous
Tolerable Settlement
Positive Moment
Mid-span
Two-Span
Continuous
Tolerable Settlement
Negative Moment
Pier
Two-Span
Continuous
For simple span bridges, only a single boundary condition (pinned-free) is considered.
A
transverse rotation of a single abutment about the longitudinal axis of the bridge induces shear in
the beams and negative moment in beams. The negative moment in the beams counteracts any
dead and live load forces. For this reason only shear over the abutments is considered for
tolerable settlement analysis. Live loads ratings for positive moment are analyzed at the midspan region for positive moment while shear rating are analyzed at the supports.
170
For two-span continuous bridges, the settlement response locations of interest are dependent on
the location of the support movement (i.e. movement of the abutment, movement of the pier) and
location of truck loads. Live load rating response locations are located over the pier for negative
moment and shear while positive moment is located in a mid-span region. The limiting rating
factor for all two-span bridges in located in the negative moment region however the rating for
positive moment is investigated due to its requirement for the settlement study. The schematic
representations of a two-span continuous bridge in the Figure 6.9 and Figure 6.10 depict the
locations of interest for the two support movement types examined this quarter.
Table 6.7. Support Movement Locations and Resultant Response Locations of Interest
Bridge
Configuration
Support
Movement
Considered
Support
Movement
Location
Response
Location
Response
Type
Simple Spans
– For support
movements
that induce
forces.
Transverse
Rotation –
about an
axis
longitudinal
to the
bridge.
Abutment
Abutment
Shear
Abutment
Pier
Shear and
Moment
Mid-span
Moment
Abutment
Shear
Pier
Shear and
Moment
Mid-span
Moment
Abutment
Shear
Vertical
Translation
Continuous
Spans – For all
support
movements
Transverse
Rotation –
about an
axis
longitudinal
to the
bridge
Pier
Abutment
Pier
171
Movement of the abutment induces a negative moment in the structure for each support
movement type. This serves to decrease the positive moment in regions between the supports
caused by dead load and live load, while increasing the negative moment in the region over the
interior pier. Further, the abutments experience an uplift reaction force, reducing shear demand
in those locations, while the reaction at the pier increases shear in that location. Given these
observations, the response location of interest for all settlement types occurring at the abutment is
the region over the pier, as shown in Figure 6.9.
Figure 6.9. Response Locations of Interest for Support Movements Occurring at the
Abutment.
Downward movement of the pier induces a positive moment in the structure for each support
movement type. This serves to decrease the negative moment over the support caused by dead
load and live load, while increasing the positive moment in all other regions. Bending responses
are considered in the regions where the structure experiences maximum dead load and live load
demand as shown in Figure 6.10.
The pier experiences an uplift reaction force from the
172
downward support movement, reducing the shear demand over the pier; however, the reactions
at the abutments increase due to support movement at the pier. Shear responses are considered
at both abutments for all support movements occurring at the pier.
Figure 6.10. Response Locations of Interest for Support Movements Occurring at the
Pier.
6.3.2
Live Load and Dead Load Results Extraction Steps
The live loads are also analyzed by a linear static solver. The results are recorded at the same
points of interest described previously for dead load. These results, along with the dead load
results are post processed to appropriately combine different loads and to apply various factors
including load combination factors, multi-presence factors, and impact factors. Results are then
filtered to return the maximum response for each girder and at each point of interest. The
following outline details the analysis steps and what is extracted from the FE model.
173
1.1. Dead Load Analysis
1.1.1. Linear static solver
1.1.2. Results pulled over entire length of all beam elements
1.1.2.1. Bending moment (in & out of plane)
1.1.2.2. Axial Force
1.1.3. Results from support nodes
1.1.3.1. Vertical Reaction
1.2. Live Load Analysis
1.2.1. Linear Static Solver
1.2.2. Results recorded along entire beam
1.2.2.1. Bending moment (in & out of plane)
1.2.2.2. Axial Force
1.2.3. Results recorded from support nodes
1.2.3.1. Vertical Reaction
6.4
Computation of Rating Factors
Load rating is performed by taking all analysis results and manipulating them to sum all possible
truck and lane load combinations for all lanes, lane positions, and spans. Load combination and
multi-presence factors are applied along with impact factors. The composite bending moment
and composite total fiber stress is calculated using composite section properties and FE model
demands. Rating factors for each beam along the entire span length are then computed using the
calculated capacity and demands. The minimum load rating for each limit state is recorded and
the location of the minimum rating factor is recorded. Table 6.8 shows the steps for load rating
calculation.
174
Table 6.8. Load Factor Calculation Steps
1. Rating Factor Computation
1.1. Analyses Results Manipulated
1.1.1. Lane and truck load results are summed in all possible
combinations
1.1.2. Factors applied
1.1.2.1. Load Combination
1.1.2.2. Multi-presence
1.1.2.3. Impact
1.1.3. Composite moment calculated from bending moment and
axial force
1.1.4. Stresses computed from internal forces using appropriate
section moduli
1.1.5. Maximum responses recorded for each girder at every POI
1.2. Girder capacity determined
1.3. Rating factors computed
1.3.1. Steel (Inventory & Operating)
1.3.1.1. Strength I – moment for compact sections; stress for noncompact
1.3.1.2. Service II – stress
1.4. Minimum Rating Factor Located and Recorded
6.4.1
Member Response
The composite moment is calculated by summing the bending moment in the beam element with
the product of the axial force in the beam element and the sum of half the depth of the beam
section property and half the depth of the deck (Equation 6.1). This assumes a linear and fully
175
composite planar composite section for both beam element and deck shell element.
The
superposition of the axial forces and bending moment is shown in Figure 6.11.
𝑀𝑀𝐶𝐶 = 𝑀𝑀 + 𝐹𝐹𝐴𝐴
𝑑𝑑𝐺𝐺 + 𝑡𝑡𝐷𝐷
2
6.1
Figure 6.11. Composite Section Stress Superposition
Section total fiber stress is computed using the major axis and minor axis bending moments
divided by their corresponding section moduli and axial force divided by the total area of steel
obtained from the FE beam element (Equation 6.2).
𝜎𝜎𝑇𝑇𝑇𝑇 =
𝑀𝑀1 𝑀𝑀2 𝐹𝐹𝐴𝐴
+
+
𝑆𝑆1
𝑆𝑆2
𝐴𝐴
6.2
176
Shear is calculated by using the conservative assumption that the entirety of shear stress in the
composite section is carried by the girder web (Equation 6.3). The vertical reaction from each
support node is extracted and used as a substitute for shear force. This assumption is further
made conservative by the fact that in a FE model the reaction over the pier of a two-span
continuous structure is equal to difference of the shear forces in the two adjacent beam elements.
Because shear is a difficult quantity to measure at a nodal point in adjoining beam elements, the
entirety of the vertical reaction is used. If the shear force in one of the adjacent beams were to be
zero, the shear force in the second adjacent beam would be equal to the entirety of the reaction
force.
𝜎𝜎𝑆𝑆 =
6.4.2
𝑅𝑅
𝐴𝐴𝑤𝑤
6.3
Load Rating Factors
Rating factors are calculated using the AASHTO live load rating factor equation (Equation 6.4).
The capacity used in calculating the load rating is dependent on the demand quantity used.
When stresses are used as demands, the capacity is the yield strength of the steel.
When
composite moments are used as demands, the capacity is the nominal moment capacity of the
composite section.
177
𝑅𝑅𝑅𝑅 =
𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙 − 𝛾𝛾𝐷𝐷𝐷𝐷 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷
𝛾𝛾𝐿𝐿𝐿𝐿 𝐿𝐿𝐿𝐿𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷
6.4
The limit states investigated by the load rating include Strength I and Service II. The quantities
used in the load rating calculation for Strength I limit state are bending moments if the section is
deemed compact by AASHTO standard. If the section is deemed non-compact, stresses are
instead used. For calculation of the Service II load rating, stresses are always employed.
The load ratings are computed for both limit states shown in Table 6.9 and for every point of
interest on each girder using the listed load and resistance factors for Inventory and Operating
rating. The minimum ratings for both limit states are located and reported as the overall bridge
load ratings.
Table 6.9. Load and Resistance Factor Rating Limit States
γDL
γLL (Inv)
γLL (Op)
Compact: Comp. Moment
Non-compact: Stress
𝝓𝝓
1.00
1.25
1.75
1.35
Stress
0.95
1.00
1.30
1.0
Limit State
Responses
Strength I
Service II
6.4.3
Tolerable Support Settlement
Tolerable support movements (φs) are calculated for Strength I, Service II Limit States for bending
and Strength I for shear using dead load (DL), live load (LL) and support movement demands.
178
The tolerable support movements for these three limit states are calculated using the Equations
6.5 through 6.7 below. For Strength I and Service II, the composite moment demands were
obtained from the moment and axial forces within the beam elements of the FE model. Reactions
for DL, LL and support movement are conservatively used in the calculation of tolerable support
movement for shear.
Tolerable Settlement Factor for the Strength I limit state (Equation 6.5):
𝜑𝜑𝑠𝑠_𝑆𝑆𝑆𝑆1 =
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 − 1.25 ∗ 𝐷𝐷𝐷𝐷 − 1.75 ∗ 𝐿𝐿𝐿𝐿
𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑜𝑜𝑜𝑜 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀
6.5
Tolerable Settlement Factor for the Service II limit state (Equation 6.6):
𝜑𝜑𝑠𝑠_𝑆𝑆𝑆𝑆2 =
0.95 ∗ 𝐹𝐹𝑦𝑦 − 1.0 ∗ 𝐷𝐷𝐷𝐷 − 1.3 ∗ 𝐿𝐿𝐿𝐿
𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀
6.6
Tolerable Settlement Factor for the Service A limit state (Equation 6.7):
𝜑𝜑𝑠𝑠_𝑆𝑆𝑆𝑆1𝑉𝑉 =
𝑉𝑉𝑛𝑛 − 1.25 ∗ 𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 − 1.75 ∗ 𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿
𝑅𝑅𝑅𝑅𝑅𝑅𝑅𝑅 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑡𝑡 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀
6.7
178
7. Investigation of Inherent Bias in the AASHTO Single LineGirder Model for Steel Multi-Girder Bridges
Presented in this chapter are the preliminary findings of research into the
effect of bias and variability in the AASHTO single line-girder structural
analysis model.
First discussed are the qualifications for sample
population convergence and acceptance. Following is a presentation of
the variability and bias of the single line-girder model for simply
supported structures; this section contains a discussion of the population
single line-girder ratings, finite element model ratings, controlling
girders, effects of diaphragm stiffness on load rating, and the ratio of
finite element ratings and dead and live load moment demands to those
predicted with the single line-girder model. An overview of results from
the investigation into rating and tolerable support movement of two-span
continuous structures is found at the end of the chapter.
7.1
Introduction
Bridge design constraints - length, width, skew, girder spacing, and span length to girder depth
ratio - were sampled to create a suite of likely geometries for common steel simply-supported
multi-girder bridges. These parameters were then used to generate likely girder designs based
on the LRFD single line-girder design methodology using automated girder design software
developed as part of this research. Finite element models were constructed using automated
software. These models were analyzed for dead load and live load, and AASHTO live load
rating factors based on finite element analysis were developed in order to study the inherent bias
between finite-element based live load ratings and single line-girder ratings.
This chapter
presents the findings of this research.
7.2
Sample Population Evaluation
Figure 7.1 through Figure 7.3 show the sample space distribution for the continuous parameters
for bridge Suites 1, 2, and 3. Suites 1 and 2 did not pass initial results convergence tests, therefore
179
a third suite was created. Suites 1 and 2 were combined and passed convergence tests when
compared with the third suite. The mean girder design time per bridges was 11.5 seconds with a
standard deviation of 12.57 (Figure 7.4). The mean number of iterations needed to design a
girder was 2.45 with a standard deviation of 1.82 (Figure 7.5).
Figure 7.1. Distribution of Sample Space for Continuous Parameters for Bridge Suite 1
180
Figure 7.2. Distribution of Sample Space for Continuous Parameters for Bridge Suite 2
181
Figure 7.3. Distribution of Sample Space for Continuous Parameters for Bridge Suite 3
182
Figure 7.4. Girder Design Time
Figure 7.5. Number of Required Design Iterations to Achieve Solution
183
7.2.1
Results Convergence
Results convergence was studied by comparing the cumulative density of the ratios of finite
element rating factors to single line-girder rating factors for both Strength I and Service II LRFR
rating limit states. The empirical cumulative density function (ECDF) for the ratios for each limit
state were developed using the Matlab function ecdf.
The ECDFs were first compared using a two-sample Kolmogorov-Smirnov (KS) test with the
Matlab function kstest2. The KS test returns the test decision for the null hypothesis that the data
in two given vectors are from the same continuous distribution and returns the test decision 1 if
the test rejects the null hypothesis at the 5% significance level (Chakravarti 1967). The twosample KS test is a nonparametric hypothesis test that evaluates the difference between the
ECDFs of two distributions. The test statistic is (Equation 7.1):
𝐷𝐷∗ = 𝑚𝑚𝑚𝑚𝑚𝑚 ��𝐹𝐹�1 (𝑥𝑥𝑖𝑖 ) − 𝐹𝐹�2 (𝑥𝑥𝑖𝑖 )��
1≤𝑖𝑖≤𝑁𝑁
7.1
Where 𝐹𝐹�1 (𝑥𝑥𝑖𝑖 ) and 𝐹𝐹�2 (𝑥𝑥𝑖𝑖 ) are the empirical distribution functions at each ordered data point xi of
the first and second sample, respectively, and N is the total number of data points to be
compared.
Suites 1 and 2 were compared using the two-sample KS test and failed when comparing the FE to
SLG rating factor ratios for the Service II limit state both with and without the inclusion of barrier
stiffness for a total of four two-sample KS tests. 100 additional sets of bridge design parameters
were sampled, and girders were sized for each sample. These 100 bridges, composing Suite 3,
were then analyzed for live and dead load. An ECDF for the FE to SLG rating factors for Strength
I and Service II limit states was then developed for Suite 3. This ECDF was compared to the
184
combined ECDF for Suites 1 and 2 and passed the two-sample KS test for both limit states.
Figure 7.6through Figure 7.9 show the ECDFs for the combined Suites 1 and 2 and Suite 3.
Figure 7.6. Suite 1, 2, and 3 Empirical Cumulative Distribution Function of Ratio of FE to
SLG LRFR Strength I Rating with Barrier Stiffness Off
185
Figure 7.7. Suite 1, 2, and 3 Empirical Cumulative Distribution Function of Ratio of FE to
SLG LRFR Strength I Rating with Barrier Stiffness On
186
Figure 7.8. Suite 1, 2, and 3 Empirical Cumulative Distribution Function of Ratio of FE to
SLG LRFR Service II Rating with Barrier Stiffness Off
187
Figure 7.9. Suite 1, 2, and 3 Empirical Cumulative Distribution Function of Ratio of FE to
SLG LRFR Service II Rating with Barrier Stiffness On
The ECDFs for all three suites were compared using a quantile-quantile (q-q) plot. The q-q plot
shows the distribution of two independent samples against each other; if the samples have close
to the same distribution the plot will be linear. The adjusted R-squared value for residuals the qq plotted points when compared to the function f(x) = x were calculated. The error residuals
were plotted underneath each q-q plot. The adjusted R-squared value is unity minus the ratio of
the sum of squares of the error (SSE) (Equation 7.2) to the total sum of squares (SST) (Equation
7.3).
188
𝑛𝑛
2
𝑆𝑆𝑆𝑆𝑆𝑆 = � 𝑤𝑤𝑖𝑖 (𝑦𝑦𝑖𝑖 − 𝑦𝑦�)
𝚤𝚤
7.2
𝑖𝑖=1
The SSE measures the total deviation of the response variable of the fit to the response variable of
the data.
𝑛𝑛
𝑆𝑆𝑆𝑆𝑆𝑆 = � 𝑤𝑤𝑖𝑖 (𝑦𝑦𝑖𝑖 − 𝑦𝑦�)2
7.3
𝑖𝑖=1
The SSR measures the total deviation of the response variables of the data to the mean of the data.
The adjusted R-squared value (Equation 7.4) is the square of the correlation between the response
values and the predicted response values.
𝐴𝐴𝐴𝐴𝐴𝐴. 𝑅𝑅𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 1 −
𝑆𝑆𝑆𝑆𝑆𝑆(𝑛𝑛 − 1)
𝑆𝑆𝑆𝑆𝑆𝑆(𝑣𝑣)
7.4
Where v = n – m, n is the number of response value data points and m is the number of
coefficients fitted. When n >> m the adjusted r-squared value is approximately equal to the rsquared value. The adjusted R-squared values is on a scale from 0 – 1, with 1 indicating a better
fit. If the r-squared value is less than 0 it indicates that the fit is worse than using a horizontal
line. The RMSE value is the square root of the mean square error, which itself is the ratio of the
SSE to v.
RMSE is on scale from 0 – 1 with 1 indicating the worst fit. All four q-q plots had an
adjusted R-squared valued greater than 0.85 with a 95% confidence interval with the largest
disagreement between populations found at the upper tail of the distributions (Figure 7.10
through Figure 7.13).
189
Figure 7.10. Quantile-Quantile Plot of Empirical Cumulative Distribution Function of
Ratio of FE to SLG LRFR Strength I Rating with Barrier Stiffness On with Residual Error
Bars
190
Figure 7.11. Quantile-Quantile Plot of Empirical Cumulative Distribution Function of
Ratio of FE to SLG LRFR Strength I Rating with Barrier Stiffness off with Residual Error
Bars
191
Figure 7.12. Quantile-Quantile Plot of Empirical Cumulative Distribution Function of
Ratio of FE to SLG LRFR Service II Rating with Barrier Stiffness On with Residual Error
Bars
192
Figure 7.13. Quantile-Quantile Plot of Empirical Cumulative Distribution Function of
Ratio of FE to SLG LRFR Service II Rating with Barrier Stiffness off with Residual Error
Bars
7.2.2
Linear Regression Analysis
Rating factor bias and dead and live load moment demands for the sample bridge populations
were observed for linear and polynomial trends using bivariate analysis. Rating factor bias and
demands were investigated as a function of the independently sampled parameters: length,
width, skew, girder spacing, and span length to girder depth ratio. Rating factor and demands
were also studied for trends as a function of two dependent parameters: skew ratio and SLG
distribution factors.
Regression analysis, a commonly used method for curve fitting, allows for the estimation of the
relationship between variables using an iterative approach that seeks to minimize some function,
usually the sum of the squares of the residuals. One limit to this method is that it assumes the
193
predictors are linearly independent, however many of the configuration parameters that will be
used for this portion of the study are interrelated. Linear or polynomic trends were fitted to each
bivariate plot. The Matlab function fit was used to fit a polynomial up to the 3rd order (Matlab
2015). The fit function uses a regression analysis to minimize the least-squares error of each fitted
polynomial. The option ‘Robust’ was used with the function to ignore outliers in the fitting
process. Polynomials with coefficients that had 95% confidence intervals that spanned from
negative to positive – implying non-zero probability that the coefficient could be zero or
negligible – were not accepted. The accepted polynomials were compared using the RMSE as
well as the adjusted R-squared value. A higher-order polynomial could only be chosen over a
lower order polynomial if both the adjusted R-squared value as well as the RMSE were better
than that calculated for the lower-order polynomial.
7.3
Population-Based Comparison of Single Line-Girder and Finite Element Model
Demands for Simply Supported Structures
7.3.1
Single Line Girder Ratings
Single line-girder (SLG) ratings were developed as part of the simulation of the AASHTO LRFD
SLG design process. Figure 7.14 and Figure 7.16 show frequency plots for the SLG rating for
interior and exterior girders.
Horizontal whisker plots detailing the mean and standard
deviation for each distribution are shown.
Vertical lines mark the median value for each
distribution. Plots marked with an asterisk indicate samples outside the bounds of the plot that
were placed in the next lowest bin that lied within the plot bounds. Strength I limit state exhibits
a slightly right-skewed distribution with a mean rating factor of 2.64 for interior girders and 2.99
for exterior girders with standard deviations of 0.61 and 0.68. Service II limit state exhibits a
slightly right-skewed distribution with a mean rating factor of 2.69 for interior girders and 3.37
for exterior girders with standard deviations of 0.76 and 0.88. The inclusion of infinite fatigue life
194
in the design criteria raises the minimum controlling rating for any design above 1.5. Neglecting
fatigue life criteria in design results in Strength I limit state rating factors of 1.32/1.26 with
standard deviation 0.08/0.11; Service II SLG ratings have mean 1.00 with standard deviation 0.
Neglecting fatigue life in design allows the optimization algorithm to find plate girder
dimensions that exactly satisfy Service II criteria.
7.3.1.1
Strength I Limit State
Figure 7.14. Frequency of Single Line-Girder LRFR Strength I Rating
195
7.3.1.2
Strength I Limit State without Infinite Fatigue Life Design Criteria
Figure 7.15. Frequency of Single Line-Girder LRFR Strength I Rating without
Consideration of Infinite Fatigue Life Design Criteria
196
7.3.1.3
Service II Limit State
Figure 7.16. Frequency of Single Line-Girder LRFR Service II Rating
197
7.3.1.4
Service II Limit State without Infinite Fatigue Life Design Criteria
Figure 7.17. Frequency of Single Line-Girder LRFR Service II Rating
7.3.2
Finite Element Rating Controlling Girder – Nominal Diaphragm Stiffness
Figure 7.18 through Figure 7.21 show frequency plots for the controlling girder from finite
element (FE) ratings. Exterior girders control the majority of time for the Strength I limit states.
Including barrier stiffness contributions in the Strength I rating factor results in almost all bridges
having an exterior girder for controlling rating. Interior girders control in the majority of bridges
198
when barrier stiffness is not included for Service II rating factors. Inclusion of barrier stiffness
contributions leads to exterior girders controlling in the majority of bridges for Service II ratings.
7.3.2.1
Strength I Limit State
Figure 7.18. Frequency of Finite Element LRFR Strength I Rating Controlling Girder
199
Figure 7.19. Frequency of Finite Element LRFR Strength I Rating Controlling Girder
Order from Center Girder
200
7.3.2.2
Service II Limit State
Figure 7.20. Frequency of Finite Element LRFR Service II Rating Controlling Girder Order
from Center Girder without Inclusion of Out of Plane Moment
201
Figure 7.21. Frequency of Finite Element LRFR Service II Rating Controlling Girder Order
from Center Girder without Inclusion of Out of Plane Moment
7.3.3
Finite Element Ratings – Nominal Diaphragm Stiffness
Figure 7.22 and Figure 7.30 show frequency plots for the minimum FE rating for each bridge for
LRFR Strength I and Service II limit states. Figure 7.23 and Figure 7.31 show frequency plots for
the ratio of the minimum FE rating to the minimum SLG rating for each bridge for LRFR Strength
I and Service II limit states. Figure 7.24 and Figure 7.32 show frequency plots for the ratio of the
minimum FE rating to the minimum SLG rating for the interior girder of each bridge for LRFR
Strength I and Service II limit states. Figure 7.25 and Figure 7.33 and show frequency plots for
the ratio of the minimum FE rating to the minimum SLG rating for the exterior girder of each
202
bridge for LRFR Strength I and Service II limit states. Figure 7.34 shows the ratio for the Service
II limit state to when out of plane bending is also included in stress demand analysis. Horizontal
whisker plots detailing the mean and standard deviation for each distribution are shown.
Vertical lines mark the median value for each distribution. Appendix B contains the FE to SLG
rating ratio plots for Service II limit state when out of plane bending is included.
7.3.3.1
Strength I Limit State
The distribution of FE Strength I ratings factors is a normal distribution with a mean of 2.89 and
3.20 for barrier stiffness “off” and “on,” respectively. Inclusion of barrier stiffness contributions
increases standard deviation from 0.53 to 0.71. The ratios of Strength I FE rating factor to SLG
rating factor are normally distributed with means of 1.12 and 1.24 with standard deviation of 0.09
and 0.15. Interior girders exhibit a normal distribution of mean 1.24/1.30 and standard deviation
0.10/0.15 while exterior exhibit a normal distribution with mean 0.99/1.16 and standard deviation
0.11/0.22.
203
Figure 7.22. Frequency of Finite Element LRFR Strength I Rating
204
Figure 7.23. Frequency of Ratio of LRFR Strength I Finite Element Rating to Single LineGirder Rating
205
Figure 7.24. Frequency of Ratio of LRFR Strength I Finite Element Rating to Single LineGirder Rating for Interior Girders
206
Figure 7.25. Frequency of Ratio of LRFR Strength I Finite Element Rating to Single LineGirder Rating for Exterior Girders
207
7.3.3.2
Strength I Limit State without Infinite Fatigue Life Design Criteria
Figure 7.26. Frequency of Finite Element LRFR Strength I Rating without Consideration
of Infinite Fatigue Life Design Criteria
208
Figure 7.27. Frequency of Ratio of LRFR Strength I Finite Element Rating to Single LineGirder Rating without Consideration of Infinite Fatigue Life Design Criteria
209
Figure 7.28. Frequency of Ratio of LRFR Interior Girder Strength I Finite Element Rating
to Single Line-Girder Rating without Consideration of Infinite Fatigue Life Design
Criteria
210
Figure 7.29. Frequency of Ratio of LRFR Exterior Girder Strength I Finite Element Rating
to Single Line-Girder Rating without Consideration of Infinite Fatigue Life Design
Criteria
7.3.3.3
Service II Limit State
The distribution of FE Service II ratings factors is slightly skewed left with a mean of 3.21 and
3.27 for barrier stiffness “off” and “on,” respectively. Inclusion of barrier stiffness contributions
increases standard deviation from 0.82 to 0.99. The ratios of Service II FE rating factor to SLG
rating factor are slightly skewed left with means of 1.21 and 1.30 with standard deviation of 0.08
and 0.13. Interior girders exhibit a normal distribution of mean 1.23/1.30 and standard deviation
0.10/0.13 while exterior girders exhibit both normal and slightly skewed left distributions with
mean 1.04/1.26 and standard deviation 0.11/0.25. The frequency of the ratio of the FE ratings
211
when out of plane bending is not included (for normal Service II ratings) to when it is included
shows no ratio less than unity (Figure 7.34).
When barrier stiffness is not considered the
distribution shows a large number of structures with no change (a high frequency at unity) and
then a strongly right-skewed distribution. The total distribution has a mean of 1.05 and standard
deviation of 0.06. Inclusion of barrier stiffness results in a strongly right-skewed distribution
with no frequency spike at unity and a mean of 1.13 and standard deviation of 0.07.
Figure 7.30. Frequency of Finite Element LRFR Service II Rating
212
Figure 7.31. Frequency of Ratio of LRFR Service II Finite Element Rating to Single LineGirder Rating
213
Figure 7.32. Frequency of Ratio of LRFR Service II Finite Element Rating to Single LineGirder Rating for Interior Girders
214
Figure 7.33. Frequency of Ratio of LRFR Service II Finite Element Rating to Single LineGirder Rating for Exterior Girders
215
Figure 7.34. Frequency of Ratio of LRFR Service II Finite Element Rating to Finite
Element Rating Including Out of Plane Bending
7.3.3.4
Service II Limit State without Infinite Fatigue Life Design Criteria
The distribution of FE Service II ratings factors is slightly skewed left with a mean of 1.30 and
1.47 for barrier stiffness “off” and “on,” respectively. Inclusion of barrier stiffness contributions
increases standard deviation from 0.35 to 0.43. The ratios of Service II FE rating factor to SLG
rating factor are slightly skewed left with means of 1.27 and 1.43 with standard deviation of 0.28
and 0.36. Interior girders exhibit a normal distribution of mean 1.25/1.38 and standard deviation
0.16/0.25 while exterior girders exhibit both normal and slightly skewed left distributions with
mean 1.29/1.64 and standard deviation 0.30/0.46.
216
Figure 7.35. Frequency of Finite Element LRFR Service II Rating without Consideration of
Infinite Fatigue Life Design Criteria
217
Figure 7.36. Frequency of Ratio of LRFR Service II Finite Element Rating to Single LineGirder Rating without Consideration of Infinite Fatigue Life Design Criteria
218
Figure 7.37. Frequency of Ratio of LRFR Interior Girder Service II Finite Element Rating
to Single Line-Girder Rating without Consideration of Infinite Fatigue Life Design
Criteria
219
Figure 7.38. Frequency of Ratio of LRFR Exterior Girder Service II Finite Element Rating
to Single Line-Girder Rating without Consideration of Infinite Fatigue Life Design
Criteria
7.3.4
Effect of Diaphragm Stiffness on FE Rating Factors
The effect of diaphragm stiffness on FE rating factors was studied for two populations of bridges
created with the population modeling software. Diaphragms in the software are selected based
solely on the slenderness ratio criteria outlined in the AASHTO design codes and the demands of
lateral wind loads on the structure as discussed in Chapter 4. The effect of using only the
minimum diaphragm size on FE rating factors was investigated in this research by analyzing the
change in live load rating factors in a population of bridges after increasing the Young’s modulus
220
of the diaphragms. Placement of diaphragms along the length of the girder was also included in
this study.
This study was performed for bridges with a chevron or cross-bracing diaphragm configuration.
Bridges with channel section diaphragms are designed using separate criteria and therefore were
not good candidates for this study. Channel section diaphragms are chosen using the ratio of
girder web depth to channel depth and bridges with this diaphragm type have shown greater
mean FE to SLG rating factor ratios than those with other diaphragm configurations. 14% of the
bridges in Suite I were designed using a channel section diaphragm.
For bridges designed with fatigue life criteria, the mean FE to SLG rating factor ratio for bridges
with channel section diaphragms is 1.20 and 1.29 for Strength I and Service II limit states,
respectively, with standard deviations of 0.06 and 0.08.
For bridges with angle section
diaphragms designed to the minimum criteria given in the AASHTO LRFD code, the ratio has
mean 1.11 and 1.19 for Strength I and Service II limit states, with standard deviations of 0.09 and
0.06.
For bridges designed without fatigue life criteria, the mean FE to SLG rating factor ratio for
bridges with channel section diaphragms is 1.29 and 1.32 for Strength I and Service II limit states,
respectively, with standard deviations of 0.06 and 0.08.
For bridges with angle section
diaphragms designed to the minimum criteria given in the AASHTO LRFD code, the ratio has
mean 1.13 and 1.26 for Strength I and Service II limit states, with standard deviations of 0.23 and
0.30.
7.3.4.1
Normalizing Diaphragm Stiffness Contribution
In order to study the effect of diaphragms on the ability of a structure to distribute dead and live
loads, the increase in lateral stiffness of the diaphragms was normalized for each bridge by
comparing the flexibility of the composite girder section to the flexibly of the diaphragm though
221
a simplified model. This conceptual model resulted in the creation of an effective flexibility ratio
that quantifies the ratio of the longitudinal flexibility of a girder to the combined transverse and
longitudinal flexibility of connected diaphragms and adjacent girders. The simplified model
consists of three simply supported girders connected at mid-span with a single diaphragm
section with a unit point load at the center girder at mid-span.
Figure 7.39. Simplified Model for Diaphragm Flexibility Contribution
The flexibility contribution of the center girder is assumed to be (Equation 6.2):
𝑓𝑓𝐺𝐺 =
𝐿𝐿3
48𝐸𝐸𝐸𝐸
7.5
With L as the span length and EI the flexural stiffness of the short term composite section. The
flexibility of the connecting diaphragms and adjacent girders is assumed to be (Equation 7.6)
222
3
1
√𝑠𝑠 2 + 𝑑𝑑2
𝑓𝑓𝐷𝐷 = �𝑓𝑓𝐺𝐺 +
�
2
𝐴𝐴𝐴𝐴𝑑𝑑 2
7.6
Where s is the girder spacing and d is the girder depth and AE the axial stiffness of the
diaphragm angle section for cross-bracing configurations. S is taken as half the girder spacing for
chevron bracing configurations. For structures with skew angle less than 20° the girder spacing is
equal to the distance on center between girders multiplied by sin(θskew). It is also assumed that
only a single diagonal of the diaphragm truss is takes the entire transverse load. The effective
flexibility ratio of the center girder to the adjacent diaphragm/girder construction is (Equation
7.7):
𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒 =
𝑓𝑓𝐺𝐺
𝑓𝑓𝐷𝐷
7.7
This term is modified for each structure by according to the distance of the nearest diaphragm
row to mid-span (Equation 7.8):
𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒 =
𝑓𝑓𝐺𝐺
𝑙𝑙𝐷𝐷 𝑓𝑓𝐷𝐷
7.8
223
Where lD is (Equation 7.9):
𝑙𝑙𝐷𝐷 = 1 −
𝑆𝑆𝐷𝐷
𝐿𝐿/2
And SD is the distance from mid-span to the nearest diaphragm row.
7.9
For bridges with
diaphragms at mid-span lD equals zero.
7.3.4.2
Case Study for Effective Flexibility Ratio
Before investigating the effect of the effective flexibility ratio, feff, on a population of bridges, two
structures were selected from Suite I for in-depth investigation. Two structures with skews less
than 5° were chosen. The two structures were chosen for having a live load distribution factor
that was either similar to or different from the theoretical minimum distribution factor. The
theoretical minimum distribution factor is the number of lanes (the number of whole 12’ lanes
that can fit in the clear distance between barriers) multiplied by the LRFR live load multipresence
factor divided by the number of girders (Equation 7.10):
𝐷𝐷𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 =
𝑓𝑓𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙 𝑁𝑁𝑙𝑙𝑙𝑙𝑙𝑙𝑙𝑙
𝑁𝑁𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔𝑔
7.10
These two structures were chosen to uncouple to the effect of the flexibility ratio from the
distribution factor equations. No discernable trend was uncovered when comparing the ratio of
224
the theoretical minimum distribution factor to the ratio of FE to SLG live load rating for
individual structures (Figure 7.40); however increasing the nominal diaphragm stiffness by 30X
did appear to increase interior girder FE ratings and decrease exterior girder ratings on average
(Figure 7.41). Table 7.1 details the effective flexibility ratio statistics for all populations.
Table 7.1. Effective Flexibility Ratio for Bridges Suites 1 and 4
Bridge
Suite
feff
Fatigue Diaphragm
Design
Stiffness
Interior Girder Exterior Girder
1
Y
4
N
1
10
30
1
10
30
μ
1.422
1.654
1.675
1.536
1.687
1.701
σ
0.343
0.397
0.404
0.307
0.333
0.337
μ
0.711
0.827
0.838
0.768
0.844
0.850
σ
0.171
0.199
0.202
0.154
0.167
0.169
225
Figure 7.40. Ratio of FE to SLG Strength I Rating Factor as a Function of the Ratio of the
Minimum Theoretical Distribution Factor to the Maximum Live Load Moment
Distribution Factor with Nominal Diaphragm Stiffness
226
Figure 7.41. Ratio of FE to SLG Strength I Rating Factor as a Function of the Ratio of the
Minimum Theoretical Distribution Factor to the Maximum Live Load Moment
Distribution Factor with 30X Nominal Diaphragm Stiffness
The first structure, Bridge #83, was chosen because of it had a DFmin maximum interior girder DF
to ratio of 0.50, indicating that the minimum possible live load distribution of an interior girder is
one half of a standard AASHTO HL-93 or design tandem truck. Bridge #61 had a DFmin to DF
ratio of 0.75, indicating that the minimum possible live load distribution of an interior girder is
0.75 design trucks.
First, the effect of diaphragm stiffness on the effective flexibility ratio was quantified by
adjusting the diaphragm E for both models from 2.9x107 psi to 2.9x109 (Figure 7.42). Effective
flexibility asymptotically increases with increasing diaphragm stiffness for both interior and
exterior girders. Flexibility ratio approaches a limit at about 9.0 x 108 psi, or 30x diaphragm
stiffness. The change in diaphragm stiffness was also compared to the change in limiting rating
227
factor for interior and exterior girders (Figure 7.43 and Figure 7.44).
Both bridges saw an
asymptotic increase in FE rating factor with increasing diaphragm stiffness. Exterior girders
asymptotically decreased with increasing diaphragm stiffness for Bridge #83 and increased for
Bridge #61. The change in rating factor was then compared to the change in effective flexibility
ratio (Figure 7.45 and Figure 7.46).
FE ratings factors increased or decreased in the same
direction as seen with increasing diaphragm stiffness. Each structure exhibits low sensitivity to
the flexibility ratio at the lower ranges of flexibility, then rapidly increasing FE rating factors until
the effective flexibility reaches an upper limit. Note that diaphragm angle sections listed in the
AISC database range in area from 0.48 in2 to 16.7 in2. This corresponds to an approximate 30x
increase from the lower to upper limit of angle area. Because of this limit, and the fact that the
effective flexibility ratio reaches an approximate upper limit at 30x 2.9 x 107, the population study
will be limited to diaphragm stiffness of 30x nominal.
% Change in Effective Flexibility
80
70
60
Bridge #61 Interior
Girder
Bridge #61 Exterior
Girder
Bridge #83 Interior
Girder
Bridge #83 Exterior
Girder
50
40
30
20
10
0
29,000,000
2,029,000,000
4,029,000,000
Diaphragm E [psi]
Figure 7.42. Percent Change in Effective Flexibility Ratio as a Function of Change in
Diaphragm Stiffness
228
% Change in Ratio of FE to SLG Rating
Factor
35
30
25
20
15
Strength I Interior
10
Strength I Exterior
5
Service II Interior
0
Service II Exterior
-5
-10
-15
2.90E+07
1.03E+09
2.03E+09
3.03E+09
Diaphragm E [psi]
Figure 7.43. Percent Change in Ratio of FE to SLG Rating Factor as a Function of
Diaphragm Stiffness for Bridge #83
229
18
% Change in Ratio of FE to SLG Rating
Factor
16
14
Strength I Interior
12
10
Strength I Exterior
8
Service II Interior
6
Service II Exterior
4
2
0
2.90E+07
1.03E+09
2.03E+09
3.03E+09
Diaphragm E [psi]
Figure 7.44. Percent Change in Ratio of FE to SLG Rating Factor as a Function of
Diaphragm Stiffness for Bridge #61
% Change in Ratio of FE to SLG
Rating Factor
35
30
25
20
Strength I Interior
15
10
Strength I Exterior
5
Service II Interior
0
Service II Exterior
-5
-10
-15
0
20
40
60
80
% Change In Effective Flexiblitity
Figure 7.45. Percent Change in FE to SLG Rating Factor as a Function of Effective
Flexibility Ratio for Bridge #83
% Change in Ratio of FE to SLG Rating
Factor
230
18
16
14
12
Strength I Interior
Strength I Exterior
Service II Interior
Service II Exterior
10
8
6
4
2
0
0
20
40
60
80
% Change in Effective Flexibility
Figure 7.46. Percent Change in FE to SLG Rating Factor Ratio as a Function of Effective
Flexibility Ratio for Bridge #61
7.3.4.3
Effect of Effective Flexibility Ratio on FE Rating
The effect of effective flexibility ratio on a two populations of bridges was studied by increasing
diaphragm stiffness for each bridge in the population by 10 and 30 times and analyzing the
resulting structures for FE rating factors. Suite 1 from this research was used along with a second
suite of bridges using the same input parameters that were then designed without the fatigue
limit state. During the design of this second suite, 97 passing designs solutions were found by
the design algorithm. The total number of data points from the two bridge suites with 10x and
30x diaphragm stiffness brings this study to 394 data points. Figure 7.47 shows the % increase in
interior girder Strength I FE rating factor for each bridge as a function of % increase in effective
flexibility ratio. All girders exhibited an increase in rating factor that showed similar behavior to
the two structures analyzed in the previous section. The same pattern is shown for the Service II
limit state in Figure 7.49.
Likewise, rating factor for exterior girders both increased and
231
decreased with increasing effective flexibility ratio for both limit states (Figure 7.48 and Figure
7.50).
Table 7.2 and Table 7.3 display the population statistics for both bridge suites with nominal
(design stiffness), 10x, and 30x diaphragm stiffness. The tables indicate whether fatigue design
was used in the population. The tables note the mean and standard deviation SLG and FE
ratings, the mean and standard deviation FE to SLG rating ratios, the percentage of bridges in
each population with a FE to SLG ratio less than unity, and the mean plus two times standard
deviation of the FE to SLG rating ratios. Bridges design without fatigue criteria exhibited higher
mean flexibility ratios at each level of diaphragm stiffness when compared to the fatigue design
population. Interior girders exhibited a larger mean FE to SLG ratio for both fatigue and nonfatigue designed structures. Increasing the flexibility ratio of structures resulted in a mean
population increase in FE to SLG rating ratio. Exterior girders saw large numbers of structures
with FE to SLG rating ratios less than one for the Strength I limit state, especially with the fatigue
designed structures. Exterior girders did not see the population FE to SLG ratio mean increase
with increasing flexibility ratio.
232
Figure 7.47. Percent Change in Interior Girder Strength I Finite Element Rating Factor as
a Function of Percent Increase in Effective Flexibility Ratio
233
Figure 7.48. Percent Change in Exterior Girder Strength I Finite Element Rating Factor as
a Function of Percent Increase in Effective Flexibility Ratio
234
Figure 7.49. Percent Change in Interior Girder Service II Finite Element Rating Factor as
a Function of Percent Increase in Effective Flexibility Ratio
235
Figure 7.50. Percent Change in Exterior Girder Service II Finite Element Rating Factor as
a Function of Percent Increase in Effective Flexibility Ratio
Fatigue
Design
Y
N
Fatigue
Design
Y
N
Bridge
Suite
1
4
Bridge
Suite
1
4
1.32
2.64
FE Absolute
Rating Factors
Strength I
μ
σ
2.89 0.56
0.61 2.93 0.60
2.95 0.65
1.57 0.42
0.08 1.62 0.55
1.65 0.61
SLG Rating
Factors
Strength I
μ
σ
μ
1.13
1.14
1.15
1.15
1.19
1.21
σ
0.09
0.10
0.10
0.23
0.32
0.37
All
% < 1 μ+2σ
5 1.31
7 1.35
5 1.35
16 1.61
13 1.84
12 1.94
FE to SLG Rating Factor Ratio
Strength I
Int
μ
σ
% < 1 μ+2σ
1.23 0.10
3 1.42
1.33 0.10
1 1.52
1.36 0.11
1 1.58
1.23 0.18
3 1.59
1.32 0.23
3 1.78
1.35 0.27
1 1.89
μ
0.99
0.98
0.98
1.11
1.13
1.14
1
10
30
1
10
30
Diaphragm
Stiffness
1.00
2.69
FE Absolute
Rating Factors
Service II
μ
σ
3.21 0.86
0.76 3.34 0.82
3.37 0.87
1.36 0.46
0.00 1.41 0.58
1.44 0.63
SLG Rating
Factors
Service II
μ
σ
μ
1.21
1.26
1.27
1.27
1.32
1.35
σ
0.07
0.10
0.10
0.28
0.44
0.51
All
% < 1 μ+2σ
0 1.35
0 1.46
0 1.47
0 1.83
1 2.21
0 2.36
FE to SLG Rating Factor Ratio
Service II
Int
μ
σ
% < 1 μ+2σ
1.22 0.09
3 1.40
1.32 0.10
1 1.53
1.36 0.12
0 1.59
1.26 0.16
3 1.58
1.36 0.28
2 1.92
1.40 0.34
1 2.08
μ
1.03
1.01
1.02
1.28
1.25
1.25
Table 7.3. Population Statistics for Effect of Effective Flexibility Ratio on Service II Rating Factor
1
10
30
1
10
30
Diaphragm
Stiffness
Table 7.2. Population Statistics for Effect of Effective Flexibility Ratio on Strength I Rating Factor
σ
0.11
0.11
0.11
0.30
0.40
0.45
Ext
% < 1 μ+2σ
42 1.25
52 1.24
48 1.24
8 1.89
8 2.05
7 2.15
Ext
σ
% < 1 μ+2σ
0.11
61 1.20
0.11
64 1.20
0.11
66 1.21
0.25
24 1.61
0.33
21 1.78
0.37
20 1.88
236
237
7.3.5
Effect of Deck Thickness on FE Rating Factors
A set of bridges were designed with a 7.5” deck using the same sample set used for the creation
of bridge suite 1. The FE to SLG rating factor ratios were compared to the population created
with a 9” deck from the input parameters of bridge suite 1. These bridges were designed without
the inclusion of infinite fatigue life criteria as well as the nominal diaphragm stiffness required by
the LRFD design code in order to minimize the contribution of any other factors to design
conservatism.
The mean FE Strength I limit state rating for interior girders was 1.91 with a standard deviation
of 0.73 while the mean exterior girder rating was 1.55 with a standard deviation of 0.57 (Figure
7.51). The mean FE Service II limit state rating for interior girders was 1.63 with a standard
deviation of 0.81 while the mean exterior girder rating was 1.44 with a standard deviation of 0.66
(Figure 7.52). The ratio of FE to SLG rating factor for the Strength I limit state has means of 1.33
and 1.15 for interior and exterior girders, respectively, with standard deviations of 0.26 and 0.37
(Figure 7.53). The mean ratios found for the Service II limit state were found to be 1.37 and 1.27
for interior and exterior girders while the standard deviations were 0.29 and 0.45 (Figure 7.54).
These values may be compared to those used in the suite of bridges with a 9” deck: The FE
Strength I rating factor had a mean of 2.05 and 1.97 for interior and exterior girders with a
standard deviation of 0.64 and 0.62. The Service II rating factors had a mean of 1.89 and 1.60 with
standard deviations of 0.76 and 0.71. The ratio of FE to SLG rating factor had a mean of 1.23 and
1.12 for interior and exterior girders with standard deviations of 0.18 and 0.25 for the Strength I
rating factor. The mean rating ratio for the Service II rating factor was 1.29 and 1.25 with
standard deviations of 0.16 and 0.30.
238
This comparison shows that when designing a bridge with a 7.5” deck instead of a 9” deck – or
decreasing deck thickness in the design phase - interior girders exhibit a mean increase in FE
rating with decreasing standard deviation while exterior girders show an increase in mean rating
with an increase in standard deviation. The mean conservatism of the single line-girder model
(illustrated through the ratio of FE to SLG rating factor) decreases with decreasing deck thickness
although the variance of this conservatism also decreases. Note that, when compared to
Figure 7.51
239
Figure 7.52
240
Figure 7.53
241
Figure 7.54
7.3.6
Bivariate Analysis of Ratio of FE and SLG Rating Factors – Nominal
Diaphragm Stiffness and Inclusion of Infinite Fatigue Life Design Criteria
Bivariate scatter plots for the ratio of the minimum FE rating to the minimum SLG rating for each
bridge for LRFR Strength I and Service II limit states as a function of the input design parameters
are shown in Figure 7.55 through Figure 7.61. Included in these plots are also those that detail
the relationship of the FE to SLG ratio with the skew ratio. The influence of distribution factors is
also shown.
242
Bivariate plots present the ratio of FE to SLG rating as a function of each independent parameter.
Each plot presents the findings for when Barrier stiffness is “on” or “off” during live load. Error
bars showing the residual from the fit line to the dependent variables of the data points are
shown where a fit was appropriate.
Error bars show the residual in rating factor ratio
normalized by the SLG rating, i.e. a fit line predicting a FE to SLG rating ratio of two while the
data shows a rating ratio of unity would result in a residual of negative unity divided by the SLG
rating for that data point. If a strong trend was not detected between the bias of the SLG model
and the design parameter, the plot is shown in Appendix B.
7.3.6.1
Strength I Limit State
The relation between the Strength I FE to SLG rating factor shows a cubic relationship to length.
The conservatism of the SLG model diminishes with span length (Figure 7.55). Skew ratio
exhibits no effect for skew ratios under 0.5, while skew ratios over 0.5 show what may be a linear
or quadratic trend with increasing skew ratios resulting in increasing conservatism of the SLG
model (Figure 7.56). Note that the lowest skew ratios over 0.5 result in FE to SLG rating ratios
less than unity. The FE to SLG rating ratios increase linearly with increasing exterior girder
distribution factors (Figure 7.57).
243
Figure 7.55. Ratio of LRFR Strength I FE Rating to SLG Rating as a Function of Length
244
Figure 7.56. Ratio of LRFR Strength I FE Rating to SLG Rating as a Function of Skew
Ratio
245
Figure 7.57. Ratio of LRFR Strength I FE Exterior Girder Rating to SLG Rating as a
Function of Exterior Girder Distribution Factor
7.3.6.2
Service II Limit State
Service II limit state plots are shown for Service II without the inclusion of out of plane moment.
Plots with the inclusion of out of plane moment are found in the Appendix. The ratio of FE to
SLG rating factors decreased along a quadratic with increasing length (Figure 7.58). Likewise
skew exhibited a linear decreasing effect on the ratio of FE to SLG ratings, with an increase in
variance with increasing skew (Figure 7.59). Skew ratio exhibited a similar effect as seen with
Strength I rating factors, with the lowest FE to SLG ratios occurring between skew ratios of 0.5
and 0.75 (Figure 7.60). The bias of the SLG model for exterior girders increased linearly with
exterior girder distribution factor (Figure 7.61).
246
Figure 7.58. Ratio of LRFR Service II FE Rating to SLG Rating as a Function of Length
247
Figure 7.59. Ratio of LRFR Service II FE Rating to SLG Rating as a Function of Skew
248
Figure 7.60. Ratio of LRFR Service II FE Rating to SLG Rating as a Function of Skew Ratio
249
Figure 7.61. Ratio of LRFR Service II FE Exterior Girder Rating to SLG Rating as a
Function of Exterior Girder Distribution Factor
7.3.7
Moment Demands – Nominal Diaphragm Stiffness and Inclusion of
Infinite Fatigue Life Design Criteria
The moment demands predicted using the single line-girder model were compared to the
maximum moment demands calculated with the finite element models for interior and exterior
girders. Dead load, superimposed dead load, and live load moment demands were investigated.
7.3.7.1
Dead Load
The distribution of the ratio of FE to SLG dead load moment demand is normally distributed
with a mean of 1.06 and 0.86 for interior and exterior girders, respectively, with standard
250
deviations of 0.06 and 0.08 (Figure 7.62). It is likely that interior girder demands are higher and
exterior girder demands are lower than that predicted by SLG due to the actual tributary widths
for girder being larger than those calculated using the SLG model. The SLG model apportions
dead load to each girder equally, regardless of the disparity between girder spacing and
overhang width.
Figure 7.62. Frequency of Ratio of Predicted SLG Dead Load Moment Demand to
Maximum FE Dead Load Moment Demand
251
7.3.7.2
Superimposed Dead Load
The distribution of the ratio of maximum FE to SLG superimposed (barrier mass) dead load
moment demand is skewed right with a mean of 1.21 and standard deviation of 0.38 for interior
girders and normally distributed for exterior girders with a mean of 2.20 and standard deviation
of 0.54 (Figure 7.63). The high ratio for exterior girders is a result of the localization of the barrier
dead load over the exterior girders
Figure 7.63. Frequency of Ratio of Predicted SLG Superimposed Dead Load Moment
Demand to Maximum FE Superimposed Dead Load Moment Demand
252
7.3.7.3
Live Load
The distribution of the ratio of FE to SLG live load moment demands for interior girders is leftskewed a mean of 0.82 and 0.78 for barrier stiffness “off” and “on,” respectively (Figure 7.64).
Inclusion of barrier stiffness contributions results in a change of standard deviation from 0.06 to
0.07. The distribution of the ratio of FE to SLG live load moment demands for exterior girders is
normally distributed with a mean of 1.03 and 0.90 for barrier stiffness “off” and “on,”
respectively (Figure 7.65). Inclusion of barrier stiffness contributions results in an increase in
standard deviation from 0.10 to 0.14. Interior girder live load demands less than one indicate that
the live load distribution factors are conservative for interior girders. A total of three outlier
structures exhibit a ratio greater than 1. A mean of close to unity for exterior girder indicates that
the live load distribution factors for exterior girders may be non-conservative for some bridge
configurations, with some exterior girders seeing up to 30% more live load moment demand than
predicted with the SLG model.
253
Figure 7.64. Frequency of Ratio of Predicted SLG Live Load Moment Demand to
Maximum FE Live Load Moment Demand for Interior Girders
254
Figure 7.65. Frequency of Ratio of Predicted SLG Live Load Moment Demand to
Maximum FE Live Load Moment Demand for Exterior
7.3.8
Bivariate Analysis of Ratio of FE and SLG Moment Demands – Nominal
Diaphragm Stiffness and Inclusion of Infinite Fatigue Life Design Criteria
Bivariate scatter plots for the ratio of dead load, superimposed dead load, and live load moment
demands as a function of the input design parameters are shown in Figure 7.66 through Figure
7.76.
Bivariate plots present the ratio of the maximum moment demand from FE analysis to the
maximum predicted SLG moment demand as a function of the five independent parameters
(Length, Width, Skew, Girder Spacing, and Span Length/Girder Depth) and three dependent
255
parameters (Skew Ratio, Interior Girder Distribution Factor, and Exterior Girder Distribution
Factor). Each plot presents the findings for exterior and interior girders as well as when barrier
stiffness is “on” or “off” during live load. Error bars showing the residual from the fit line to the
dependent variables of the data points are shown where a fit was appropriate. Error bars show
the residual of the ratio between FE and SLG demands. If a strong trend was not detected
between the bias of the SLG model and the design parameter, the plot is shown in Appendix B.
7.3.8.1
Dead Load Moment Demand
Exterior girder dead load demand conservatism decreases linearly with span length while
interior girder dead load conservatism did not show a trend (Figure 7.66).
Interior girder dead
load conservatism exhibits a quadratic increase with increasing girder spacing (Figure 7.67).
Figure 7.66. Ratio of Maximum FE Dead Load Demand to SLG Dead Load Demand as a
Function of Span Length
256
Figure 7.67. Ratio of Maximum FE Dead Load Demand to SLG Dead Load Demand as a
Function of Girder Spacing
7.3.8.2
Superimposed Dead Load Moment Demand
The ratio of maximum FE to maximum SLG superimposed dead load demand exhibits an
increasing linear trend with increasing width for exterior girders (Figure 7.68).
The
superimposed dead load ratio decreases on a quadratic girder spacing for interior girders (Figure
7.69). The ratio of FE to SLG demands increase linearly with skew ratio for exterior girders
(Figure 7.70).
257
Figure 7.68. Ratio of Maximum FE Superimposed Dead Load Demand to SLG
Superimposed Dead Load Demand as a Function of Width
258
Figure 7.69. Ratio of Maximum FE Superimposed Dead Load Demand to SLG
Superimposed Dead Load Demand as a Function of Girder Spacing
259
Figure 7.70. Ratio of Maximum FE Superimposed Dead Load Demand to SLG
Superimposed Dead Load Demand as a Function of Skew Ratio
7.3.8.3
Live Load Moment Demand for Interior Girders
The ratio maximum live load FE demand to maximum SLG demand increases along a quadratic
trend with span length for both barrier stiffness types (Figure 7.71). The demand ratio linearly
decreases with increasing skew ratio (Figure 7.72), as well as interior girder distribution factor
(Figure 7.73).
260
Figure 7.71. Ratio of Maximum Interior Girder FE Live Load Demand to SLG Live Load
Demand as a Function of Length
261
Figure 7.72. Ratio of Maximum Interior Girder FE Live Load Demand to SLG Live Load
Demand as a Function of Skew Ratio
262
Figure 7.73. Ratio of Maximum Interior Girder FE Live Load Demand to SLG Live Load
Demand as a Function of Interior Live Load Distribution Factor
7.3.8.4
Live Load Moment Demand for Exterior Girders
The ratio maximum live load FE demand to maximum SLG demand increases along a quadratic
trend with span length for both barrier stiffness types (Figure 7.74). The demand ratio peaks at
skew ratios about 0.5 and then decreases with greater skew ratios (Figure 7.75). The demand
ratio linearly increases with increasing distribution factor (Figure 7.76).
263
Figure 7.74. Ratio of Maximum Exterior Girder FE Live Load Demand to SLG Live Load
Demand as a Function of Length
264
Figure 7.75. Ratio of Maximum Exterior Girder FE Live Load Demand to SLG Live Load
Demand as a Function of Skew Ratio
265
Figure 7.76. Ratio of Maximum Exterior Girder FE Live Load Demand to SLG Live Load
Demand as a Function of Exterior Girder Live Load Distribution Factor
266
7.4
Population-Based Investigation of Single Line-Girder Bias for Two-Span
Continuous Structures
Horizontal whisker plots detailing the mean and standard deviation for each distribution are
shown. Vertical lines mark the median value for each distribution. Plots marked with an asterisk
indicate samples outside the bounds of the plot that were placed in the next lowest bin that lied
within the plot bounds.
All analysis for two-span continuous was performed with barrier
stiffness contribution turned “off.” Population statistics are located in Table 7.4.
Table 7.4. Two-Span Continuous Rating Factor Population Statistics
SLG
Neg
Strength I
Pos
Neg
Service II
Pos
7.4.1
Rating Factor
FE
μ
σ
FE/SLG
μ
σ
μ
σ
Ext
Int
Ext
Int
1.27
1.31
3.30
3.64
0.39
0.42
0.68
0.66
1.95
1.90
3.80
4.06
0.52
0.62
0.69
0.80
1.55
1.45
1.16
1.11
0.16
0.11
0.06
0.06
Ext
Int
Ext
Int
1.77
1.83
3.68
3.91
0.50
0.54
0.78
0.81
2.74
2.68
4.28
4.28
0.65
0.80
0.88
1.01
1.57
1.48
1.17
1.09
0.23
0.24
0.09
0.07
Single Line Girder Ratings
Figure 7.77 through Figure 7.80 show frequency plots for the SLG rating for interior and exterior
girders.
267
7.4.1.1
Strength I Limit State
Figure 7.77. Frequency of Single Line-Girder Rating of Positive Moment Region for
Strength I Limit State
268
Figure 7.78. Frequency of Single Line-Girder Rating of Negative Moment Region for
Strength I Limit State
269
7.4.1.2
Service II Limit State
Figure 7.79. Frequency of Single Line-Girder Rating of Positive Moment Region for
Service II Limit State
270
Figure 7.80. Frequency of Single Line-Girder Rating of Negative Moment Region for
Service II Limit State
7.4.2
Finite Element Ratings
Figure 7.81 through Figure 7.88 present frequency plots for positive and negative moment region
FE ratings and FE to SLG rating factor ratios for LRFR Strength I and Service II limit states.
271
7.4.2.1
Strength I Limit State
Figure 7.81. Frequency of Finite Element Rating of Positive Moment Region for Strength
I Limit State
272
Figure 7.82. Frequency of Finite Element Rating of Negative Moment Region for Strength
I Limit State
273
Figure 7.83. Frequency of Ratio of Finite Element Rating to Single Line-Girder Rating of
Positive Moment Region for Strength I Limit State
274
Figure 7.84. Frequency of Ratio of Finite Element Rating to Single Line-Girder Rating of
Negative Moment Region for Strength I Limit State
275
7.4.2.2
Service II Limit State
Figure 7.85. Frequency of Finite Element Rating of Positive Moment Region for Service II
Limit State
276
Figure 7.86. Frequency of Finite Element Rating of Negative Moment Region for Service
II Limit State
277
Figure 7.87. Frequency of Ratio of Finite Element Rating to Single Line-Girder Rating of
Positive Moment Region for Service II Limit State
278
Figure 7.88. Frequency of Ratio of Finite Element Rating to Single Line-Girder Rating of
Negative Moment Region for Service II Limit State
7.4.3
Tolerable Support Movement
Frequency of tolerable support movement for shear and bending moment demands for the
Strength I and Service II limit state are presented in Figure 7.89 through Figure 7.100.
Table 7.5. Population Statistics for Tolerable Support Settlement
279
Settlement
Location Type
Abutment
Pier
Abutment
Strength I
Pier
Abutment
Pier
Abutment
Pier
Abutment
Service II
Pier
Abutment
Pier
R
T
R
T
R
T
Response
Location
Type
Tolerable Settlement [in]
μ
σ
min
Pier
M
7.64
4.81
1.39
Mid-span
M
28.29
10.37
11.87
Pier
M
8.32
5.87
1.07
Mid-span
M
28.42
10.67
11.68
Pier
V
19.36
11.31
4.94
Abutment
V
28.22
21.42
3.86
Pier
V
32.75
14.78
9.44
Abutment
V
44.01
27.13
7.04
Pier
M
11.52
4.75
3.74
Mid-span
M
24.77
9.05
8.78
Pier
M
12.72
6.02
3.16
Mid-Span
M
25.22
9.67
7.84
280
7.4.3.1
Strength I Limit State
Figure 7.89. Frequency of Strength I Tolerable Support Movement Under Transverse
Rotation of Abutment (in Inches) – Bending Response Over Pier
281
Figure 7.90. Frequency of Strength I Tolerable Support Movement Under Transverse
Rotation of Pier (in Inches) – Bending Response at Mid-Span
282
Figure 7.91. Frequency of Strength I Tolerable Support Movement Under Vertical
Translation of Abutment (in Inches) – Bending Response Over Pier
283
Figure 7.92. Frequency of Strength I Tolerable Support Movement Under Vertical
Translation of Pier (in Inches) – Bending Response at Mid-Span
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Figure 7.93. Frequency of Strength I Tolerable Support Movement Under Transverse
Rotation of Abutment (in Inches) – Shear Response at Pier
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Figure 7.94. Frequency of Strength I Tolerable Support Movement Under Transverse
Rotation of Pier (in Inches) – Shear Response at Abutment
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Figure 7.95. Frequency of Strength I Tolerable Support Movement Under Vertical
Translation of Abutment (in Inches) – Shear Response at Pier
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Figure 7.96. Frequency of Strength I Tolerable Support Movement Under Vertical
Translation of Pier (in Inches) – Shear Response at Abutment
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7.4.3.2
Service II Limit State
Figure 7.97. Frequency of Service II Tolerable Support Movement Under Transverse
Rotation of Abutment (in Inches) – Bending Response at Pier
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Figure 7.98. Frequency of Service II Tolerable Support Movement Under Transverse
Rotation of Pier (in Inches) – Bending Response at Mid-Span
290
Figure 7.99. Frequency of Service II Tolerable Support Movement Under Vertical
Translation of Abutment (in Inches) – Bending Response at Pier
291
Figure 7.100. Frequency of Service II Tolerable Support Movement Under Translation of
Pier (in Inches) – Bending Response at Mid-Span
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8. Finite Element Model Calibration
A discussion of the software tools developed to assist in rapid model
parameter estimation for structures with dynamic experimental data is
contained within this chapter. First presented are details for the specific
optimization algorithm used for parameter estimation as well as the
method for interfacing finite element models developed as part of this
research with the parameter estimation tool. Discussed at the end of the
chapter is the graphical user interface developed for rapid structural
identification, including software tools for parameter editing, parameter
sensitivity studies, model/experimental data comparison, and parameter
estimation.
8.1
Overview
Model fitting implemented through parameter estimation allows for the gap between bridge and
FEM model to be narrowed. A model’s predictive ability can be enhanced by model-experiment
correlation: obtaining structural response measurements from the structure of interest and then
updating a set of parameters – boundary conditions, continuity conditions, and material
properties – in order to bring the model into closer agreement with the behavior of the physical
structure. A model with parameters that are assumed before any empirical evidence is known is
termed an a priori model. The a priori model is then updated using empirical knowledge – or
experimental data – so that it may more closely match the behavior of the experimental subject.
This process, known as structural identification, has historically been used to develop a single
parameter set and model for a given structure; however this approach has been steadily replaced
by multiple model methods and Bayesian parameter estimation (Dubbs 2012). Both of these
approaches provide a distribution of likely models, parameters, and analytical outputs for a
model updated to more closely resemble a given experimental input.
In general, two types of data are used for model updating of typical highway bridge structures:
strains and displacements obtained through truck load tests or modal parameters (frequencies
and mode shapes) obtained through ambient monitoring or forced vibration testing. Once the
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data are obtained, an error function (or objective function) is defined to compare the measured
responses to those computed from the simulation model. The goal of the process is to minimize
this error function, which is done through perturbing a set of select parameters.
This research effort focused on the use of modal parameters for updating purposes for two
reasons. First, these properties carry information related to global load-carrying mechanisms
(e.g. boundary/ continuity conditions) and global stiffnesses (transverse, longitudinal, etc.), which
are well-suited to update models for capacity predictions. Second, due to the nature of these
properties, they may be obtained in a rapid manner, which results in lower costs and minimal
traffic disruptions, and thus is compatible with widespread implementation.
The software developed for this project utilizes a single model, deterministic updating approach
using the API link between Strand7 and Matlab. All FE models built with the software may be
updated for any number of parameters. These parameters are correlated to experimental data
using a gradient-based minimization algorithm that utilizes a trial and error approach to find the
slope and curvature of an n-dimensional parameter space to find local minima.
This chapter details this single model updating process, the algorithm used by the software, the
method by which the software uses the updating algorithm to adjust parameters, and the
graphical user interface (GUI) developed to allow the updating process to be applied to any
model created with the software in an efficient manner. Two methods of parameter estimation,
or model updating are discussed: global identification of unknown parameters and local – or
regionally varying – redistribution of a known parameter, namely, mass.
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8.2
Single Model Optimization Methods for Updating Parameters
In this research the model correlation is achieved by deterministic updating. In deterministic
updating, each parameter is assumed to have a single value, and the purpose of updating is to
solve for this value that minimizes the objective (or error) function using an iterative process.
Given for the potential for gradient-based optimization approaches to get stuck in local minima,
the starting point of the parameter values can influence the updating results. To guard against
this possibility, it is common to perform the updating multiple times from different starting
points.
In the case of RAMPS, the nonlinear gradient-based minimization with constraints
algorithm, lsqnonlin, is used to adjust parameters. For each calibration run, the frequencies and
mode shapes (the deformed shapes of the structure while vibrating at certain frequencies) of the
FE model are compared to those from the experiment and the differences are minimized in an
iterative process as shown in Figure 8.1. The details of this process are described in the following
sections.
Figure 8.1. Schematic of Iterative Parameter Identification Process
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8.2.1
Gradient-based Least Squares Minimization
8.2.1.1
Lsqnonlin
Lsqnonlin stands for “Least-squares non-linear” and is a built-in gradient-based objective function
minimization algorithm in Matlab (Matlab 2015). The Matlab function finds a vector x that is a
local minimizer to the function that is a sum of the squares of the differences between the
simulated and experimentally observed responses. It solves nonlinear least-squares curve fitting
problems of the following form (Equation 8.1):
𝑚𝑚𝑚𝑚𝑚𝑚
𝑚𝑚𝑚𝑚𝑚𝑚
‖𝑓𝑓(𝑥𝑥)‖22 =
(𝑓𝑓1 (𝑥𝑥)2 + 𝑓𝑓2 (𝑥𝑥)2 + ⋯ 𝑓𝑓𝑛𝑛 (𝑥𝑥)2 )
𝑥𝑥
𝑥𝑥
8.1
Where:
‖𝑓𝑓(𝑥𝑥)‖22 is the squareroot sum of the squares of the function 𝑓𝑓𝑛𝑛 (𝑥𝑥), and
𝑓𝑓𝑛𝑛 (𝑥𝑥) is the nth error normal for the objective value function at for the vector of parameter x
The function takes an x0 vector as the starting point for each parameter (Matlab 2015). The
function takes the upper and lower bounds for each parameter as a vector of bounds, lb and ub,
respectively, so that the solution is always in the range lb <= x <= ub (Matlab 2015). The function
utilizes either the Trust Region Reflective or the Levenberg-Marquardt algorithm to iteratively
find the best solution for x that resides within the given bounds (Matlab 2015).
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The Trust Region Reflective algorithm is what is known as a “Large-Scale” algorithm. This
means it uses linear algebra that does not operate on full matrices and instead uses sparse
matrices for computations whenever possible (Matlab 2015).
The Trust Region Reflective algorithm iteratively takes a point x in n-space (the vector of the error
normal), and seeks to “improve” that location within that space by finding a new x that results in
a lower objective function value (Matlab 2015). The “trust region” feature in this minimization
algorithm approximates the behavior of the response space in the local region around the current
x, known as N, with a simplified function q (Matlab 2015). The local region N is defined by both
slope and curvature (Matlab 2015). This is determined by varying each parameter in the vector x
while holding all other values constant. The algorithm uses this local region to estimate the
simplified function q, which is then used to propose a trial step s that results in a lower value of q
and, hopefully, f (Matlab 2015). This new point (x + s) is accepted if the f(x+s) < f(x) (Matlab 2015).
If f(x+s) >= f(x), the current point x remains unchanged, the space N is shrunk, and a new q is
estimated. This process repeats until the algorithm tolerances are satisfied or no better solution
may be found (Matlab 2015).
The algorithm may be given three tolerances, TolX, TolFun, and MinDiffX, where:
•
XTol is the relative tolerance of the parameter values to be adjusted
•
TolFun is the relative tolerance of the function f(x)
•
MinDiffX is the minimum change in x permitted to the algorithm,
These values may be set to any number, however the default value for each in Matlab is 1x10-6
(Matlab 2015).
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8.2.1.2
Objective Function
The objective function is a vector describing a comparison between simulation and
experimentally measured responses. In this research it is composed of the natural frequencies
and model shapes derived from experimental dynamic testing and those obtained from FE model
analysis. Minimization of the objective function results brings the simulated modal parameters
into better agreement with those experimentally measured from the structure. The objective
function consists of a vector of the error normal of two model-experiment comparisons: the
Modal Assurance Criterion value – based on the difference between the experimental and
analytical mode shapes, and error fraction of the experimental and analytical natural frequencies
as seen in Equations 8.2 and 8.3, respectively (Pastor, 2012).
𝑁𝑁
�[(1 − 𝑀𝑀𝑀𝑀𝑀𝑀)]
8.2
𝑖𝑖=𝑖𝑖
𝑁𝑁
𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒𝑖𝑖 − 𝑓𝑓𝑎𝑎𝑎𝑎𝑎𝑎𝑖𝑖
� ��
��
𝑓𝑓𝑒𝑒𝑒𝑒𝑒𝑒𝑖𝑖
8.3
𝑖𝑖=𝑖𝑖
The MAC value error normal is unity minus the MAC value for a given model-experiment pair.
The frequency error fraction is the difference between paired frequencies divided by the
experimental frequency of each pair.
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Natural frequencies are matched using a basic algorithm (discussed in the following section) in
order for the parameter estimation software to iteratively calculate the objective function in
response to changes in the parameter vector x.
These natural frequency pairs are used to
calculate the error for both frequency fractional difference and the MAC value. The MAC value,
shown in Equation 8.4, is used as the criterion by which frequency pairs are assigned, and is
calculated as the normalized scalar product of two sets of modal vectors, {𝛷𝛷𝐴𝐴 } and {𝛷𝛷𝐸𝐸 }.
Computing the MAC value of a modal vector with itself will always result in a MAC value of
unity. The MAC value of a set of modal vectors that are orthogonal to each other will result in a
MAC value of zero (Pastor, 2012). The resulting scalars are arranged into the MAC matrix:
MAC(r, q) =
2
�{𝛷𝛷𝐴𝐴 }𝑇𝑇
𝑟𝑟 {𝛷𝛷𝐸𝐸 }𝑞𝑞 �
8.4
𝑇𝑇
�{𝛷𝛷𝐴𝐴 }𝑇𝑇
𝑟𝑟 {𝛷𝛷𝐴𝐴 }𝑟𝑟 ��{𝛷𝛷𝐸𝐸 }𝑞𝑞 {𝛷𝛷𝐸𝐸 }𝑞𝑞 �
Where:
n
=
Number of matching mode pairs. In the case of this research, this is the
number of experimentally determined mode shapes.
𝛷𝛷𝐸𝐸
=
The modal displacement vector from the experimental modal analysis.
=
The modal displacement vector from the analytical modal analysis.
r
=
The index of the analytical mode
q
=
The index of the experimental mode
𝛷𝛷𝐴𝐴
As stated previously, a MAC value of zero indicates no correlation between mode shapes
whereas a value close to unity indicates a high correlation. As a result, high MAC values are
used to pair frequencies. Figure 8.2 shows a color MAC plot comparing two sets of modal
matrices. These modal matrices may consist of any number of modal vectors and each numbered
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row or column is referenced to a mode shape/natural frequency. The first set (e.g. experimentally
determined), indicated on the Y axis, consists of 8 mode shapes, while the second set (e.g.
simulated) consists of 12 mode shapes. Each square in the grid relates to the MAC value for a
given modal pair. Note how the number of mode shapes on each axis of the MAC plot are not
equal, since in general the user must include far more analytical modes due to the presence of
local and/or numerical modes. The colorbar to the right of the plot shows the scale for the MAC.
Figure 8.2. MAC Matrix Plot
8.2.1.3
Mode Pairing Algorithm
The algorithm used to pair frequencies and mode shapes for the objective function has the
following structure:
A. 1.
To ensure a common spatial grid, analytical (FE model) nodes (points of interest) are
paired with experimental nodes using a nearest-neighbor approach. Priority in the case of
equally distant nodes is given to the first analytical node paired. This is never the case in
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this software as the node density (approximately 1 ft.) is much higher than any sensor
density practical for a global structural experiment.
B. 1.
Obtain natural frequency and mode shape results from the FE model. The number of
natural frequencies and associate mode shapes may be set to any number less than or
equal to the analytical degrees of freedom. For practical purposes this is general set to 3 to
4 times the number of experimentally determined frequencies and mode shapes to
account for the presence of local and/or numerical modes.
B. 2.
Compute the MAC value between each analytical mode shape and each experimental
mode shape (using all paired nodes).
B. 3.
Iteratively pair each experimental mode shape with an analytical mode shape: Pair the
ith experimental mode with the analytical mode shape with the highest experimentalanalytical MAC. Remove that analytical mode shape from the available list of pairing
modes. Pair the ith+1 experimental mode shape using the MAC matrix with the best
remaining analytical mode shape. Iterate until all experimental modes have been paired
with an analytical mode.
B. 4.
B. 5.
Compute the frequency fraction errors for each mode pair.
Form the objective function vector using the paired frequency fraction errors and 1-MAC
values for paired experimental frequencies.
C. 1.
Iterate B.1. through B.5 until the minimization algorithm converges.
Figure 8.3 shows the evolution of the MAC values of a model-experiment pair through multiple
iterations of the parameter estimation algorithm. The MAC values should improve to unity as
the algorithm seeks to minimize the differences between the frequencies and mode shapes.
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8.2.1.4
Using Frequencies without Associate Mode Shapes for Updating
Not every experimental test will produce reliable mode shapes due to test constraints and the
inability to properly integrate individual forced vibration tests. In these cases, the mode shapes
may be sufficient for pairing analytical and experimental natural frequencies, but may not be
reliable enough to include directly within the objective function.
To address this case, the
software allows for just the frequency errors to be included in the objective function. In no cases
may a mode shape be used in the objective function without its associated frequency error
formal.
See DeVitis (2015) for a detailed explanation of closely spaced modes and their
usefulness for mode parameter estimation.
Figure 8.3. Development of MAC Matrix Plot with Model Updating
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8.3
Parameter Configuration
The following section provides the salient details associated with the two normalized updating
approaches adopted for parameters as well as the standard parameters included within the
updating software developed.
8.3.1
Unknown Global Parameters
8.3.1.1
Alpha Coefficients
To help normalize the updated parameters, alpha coefficients, which are multiplied by the
parameter values, are adjusted in the updating software as opposed to directly updating the
parameter values. These coefficients are used to normalize the parameter values to the same
order of magnitude and scale. Any parameter sharing the same lower and upper alpha bounds
may be updated using the same alpha or may be updated individually. Alpha coefficients may
be updated using a linear or log scale. The updating algorithm, lsqnonlin, does not differentiate
between the updating scales therefore this process was coded as part of the parameter application
and assignment code in Matlab (see Chapter 5).
Log scale parameters are beneficial for parameters with a sensitive range over several orders of
magnitude. For example, composite action is modeled using beam elements connected between
a node located at the top flange of the beam element’s extruded section and the centerline of the
deck shell elements. Link elements connect the top flange node to the actual beam element as
explained in Chapter 5. These elements serve to enforce composite action and typically have a
sensitive range of Moment of Inertia between 0.001 and 10000 in, which spans many orders of
magnitude. Equation 8.5 illustrates the application of the log scale alpha value, where 𝑥𝑥∝𝑖𝑖 is the
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resultant parameter value, ∝𝑖𝑖 is the current alpha coefficient, and x0 is the starting parameter
value.
𝑥𝑥∝𝑖𝑖 = 𝑥𝑥0 × 10∝𝑖𝑖
8.5
A linearly scaled alpha parameter would likely neglect the lower area of the sensitive range or
move too slowly within the upper area of the sensitive range. Linearly scaled parameters are
appropriate for parameters with a sensitive range over the same order of magnitude, such as
deck stiffness, which may range from 1000 to 8000 f’c. Equation 8.6 illustrates the application of
the alpha coefficient.
𝑥𝑥∝𝑖𝑖 = ∝𝑖𝑖 𝑥𝑥
8.3.1.2
8.6
Material and Section Properties
Each stiffness parameter updated by the software must take the form of a material property,
element section property assigned in the case of an element, or a translational or rotational
stiffness in the case of a boundary condition.
The material property may be modified by
adjusting the Young’s Modulus, E, of the material. The section property may be modified by
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either adjusting the Moments of Inertia or cross sectional area within the software. The following
list details the stiffness parameters that are by default adjustable using the software and their
associated parameters:
7.
Deck: The deck stiffness is adjusted using the modulus of concrete in pounds
per square inch (psi).
8.
Girders: The girder stiffness is adjusted by the Moment of Inertia, Ix, about the
major bending axis (the transverse global axis of the structure using the right
hand rule).
Interior and exterior girders may be adjusted individually or
together. Positive and negative moment regions of the girder may be adjusted
individually or together.
Updating exterior and interior girders as well as
positive and negative moment regions of the girders results in a total of four
girder stiffness parameters that can be updated simultaneously.
9.
Diaphragms: Diaphragms are adjusted using the Young’s Modulus of steel in
pounds per square inch. This allows for the axial, shear, and bending stiffness of
the diaphragm elements – cross bracing, chevron, or channel section girder, to be
modified simultaneously.
10. Sidewalks: Sidewalks are adjusted using the Young’s Modulus of concrete in
pounds per square inch (psi).
Sidewalks on both sides of the structure are
updated together with the same parameter value.
11. Barriers: Barriers are adjusted using the Young’s Modulus of concrete in pounds
per square inch (psi). Barriers on both sides of the structure are updated together
with the same parameter value.
12. Boundary Conditions: This is discussed in Chapter 5, Section 1.2.6 – “Boundary
Conditions.”
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8.3.2
Mass Redistribution
Mass is redistributed over the deck nodes of the FE models in order to minimize the natural
frequency and mode shape error normal between model and experiment as previously described
in this chapter. Updating with mass redistribution utilizes the inverse relationship between mass
and stiffness in the basic equation of natural frequency (Equation 8.8), where fn is first natural
frequency in hertz, in order to substitute mass, m, for stiffness, k, as the adjusted parameter in the
minimization problem.
𝑓𝑓𝑛𝑛 =
1 𝑘𝑘
�
2𝜋𝜋 𝑚𝑚
8.7
As the mass, m, of the deck is considered a known quantity in the FE models used by the
software presented in this research, it may be used as a substitute for stiffness, k. The ability to
divide the deck into an unlimited number of “zones” allows the redistribution of mass, and
consequently, vertical stiffness, to any areas of the structure, as seen in Figure 8.4 and Figure 8.5.
The total mass of the deck may be held to a constant while the division of mass among “zones” is
adjusted with the Matlab algorithm, fmincon (Matlab 2015). The software developed in this
research allows for the total mass of all zones to range between a set of bounds if the user selects
that option. In order to redistribute mass between zones, the mass of the deck shell elements is
set to zero and an equivalent non-structural mass is added to the deck shell element nodes. This
non-structural mass may be included in the natural frequency analysis solver in Strand7 (Strand7
2015). The nonstructural masses on each node are adjusted using linear alpha coefficients as
mentioned earlier in this chapter.
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In the case of a FE model exhibiting higher frequencies in the transversely dominated modes or
high transverse stiffness, such as first torsion, the mass would redistribute away from the
transverse center of the model (Figure 8.4). Greater mass at the lateral edges of the model results
in lower frequencies for those modes with greater mass participation (or amplitude) from the
outside of the structure. Likewise, in a FE model exhibiting lower longitudinally dominated
natural frequencies than those shown in experimental data (i.e. lower stiffness in the model than
implied via dynamic testing), the mass would be increased about the longitudinal center of the
model (Figure 8.5). Greater mass concentrated at mid-span counteracts the global stiffness in
model and results in lower natural frequencies. In both of the above cases, mass redistributes
symmetrically according to global stiffness errors in the model and is influenced primarily by the
frequency errors in either longitudinal or transversely domination modes. Mass redistribution
may also take place asymmetrically where the error may derive equally from both mode shape
and frequency. In Figure 8.6 mass is redistributed to the right side of the FE model, indicating
that there is a local stiffness deficit on that side of the model when compared to experimental
data. As in the previous examples, frequencies pertaining to primarily transverse mode shapes
will be affected, however the relative amplitudes of the mode shapes along the right side of the
structure will also be greater than those on the left side for both longitudinally and transversely
dominated modes.
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Figure 8.4. Symmetric Lateral Redistribution of Deck Mass
Figure 8.5. Symmetric Longitudinal Redistribution of Deck Mass
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Figure 8.6. Asymmetric Lateral Redistribution of Deck Mass
8.4
Graphical User Interface for Parameter Estimation
This software module provides users with the ability to perform simplified model-experiment
correlation of various uncertain parameters. RAMPS utilizes the bridge dynamic properties
determined through experimental testing for the model calibration process.
FE model
parameters are adjusted so that the frequencies and mode shapes of the model are more closely
aligned to those from the experiment.
The model calibration software graphical user interface
consists of a number of interconnected modules which are used simultaneously to assist the user
in the model experiment correlation. The model updating process follows a basic workflow:
1.
Import experimental results
2.
Choose experimental results to use for updating
3.
Choose internal parameters for updating
4.
Perform internal parameter sensitivity study
5.
Choose external parameter for updating (e.g. BCs)
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6.
Perform external parameter sensitivity study
7.
Update internal and external parameters
8.4.1.1
Parameter Editing
The parameter editing GUI window allows the user to edit parameter values as well as the
starting, minimum, and maximum alpha and parameter values. Parameters may be grouped
together to use the same alpha value. The alpha scale may be selected as a linear or logarithmic
scale. Figure 8.7 through Figure 8.11 shows the parameter GUI window and highlights different
parts of the edit table.
Figure 8.7. Parameter Edit GUI Window – Parameter Group Number
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Figure 8.8. Parameter Edit GUI Window – Parameter a Priori Value
Figure 8.9. Parameter Edit GUI Window – Update Logic Box
Figure 8.10. Parameter Edit GUI Window – Starting, Minimum, Maximum Alpha Values
and Alpha Scale
311
Figure 8.11. Parameter Edit GUI Window – Starting, Minimum, Maximum Parameter
Value
8.4.1.2
Sensitivity Studies
Sensitivity studies may be performed to determine the sensitive range of the each internal or
external parameter value (Figure 8.12 through Figure 8.19). This tool allows the sensitivity of
frequency and mode shapes to changes in a single parameter value as well as live load rating
factor to be determined. The parameter of interest is selected in a drop down box and the upper
and lower bounds may be edited as well as the parameter step (Figure 8.12). Upper and lower
bounds may also be selected by clicking within the frequency sensitivity window (Figure 8.14).
The frequencies to be plotted for sensitivity can be selected using the list box shown in Figure
8.13. Sensitivity analysis may be performed using both linear and logarithmic alpha scales, as
shown in Figure 8.12 and Figure 8.18, respectively.
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Figure 8.12. Parameter Sensitivity GUI Window - Deck Stiffness Sensitivity with a Linear
Alpha Scale
313
Figure 8.13. Parameter Sensitivity GUI Window - Deck Stiffness Sensitivity with a Linear
Alpha Scale
Figure 8.14 indicates the frequency sensitivity window. Each frequency (Y-axis) is plotted over
the parameter range.
Each mode shape is tracked over the range of parameter value and
frequency values by matching the first mode shape from the parameter range to each subsequent
parameter values mode shapes. For example, the first parameter value is applied to the model
and a number of natural frequencies are solved for. The next parameter value is then applied to
the model and those natural frequencies and mode shapes are obtained. The frequencies from
this second parameter value are matched to those from the first parameter value by the MAC
value of their associated mode shapes. A MAC matrix is plotted for a single selected frequency
for all parameter values (Figure 8.15). This matrix indicates the change in mode shape for a
single mode over the range of parameter values for a single parameter. As indicated in Figure
8.15 there is little difference in mode shape from parameter value to parameter value for the deck
as the majority of the matrix plot is near a value of unity.
Figure 8.18 shows a slightly more
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sensitive mode shape for composite action. The “dog-bone” shape and MAC matrix values
below 0.95 indicate that there is a slight change in mode shape between the bounds for composite
action.
Figure 8.14. Parameter Sensitivity GUI Window - Deck Stiffness Sensitivity with a Linear
Alpha Scale
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Figure 8.15. Parameter Sensitivity GUI Window - Deck Stiffness Sensitivity with a Linear
Alpha Scale
Load rating sensitivity may be investigated from the same GUI window. The upper and lower
bounds are set in the same manner as with natural frequency sensitivity. The sensitivity for
AASHTO live load truck ratings for both ASR and LRFR design codes may be used (Figure 8.17).
LRFR sensitivity analysis results in plots for both Strength I and Service II limit states for flexure.
ASR sensitivity analysis gives only the ASR rating factor for flexure. In each case the rating factor
for both interior and exterior girders is displayed (Figure 8.17).
Rating factor sensitivity may include the effects of transverse and longitudinal stiffeners on
girder web plates. LRFR ratings may also include the moment gradient factor, Cb. The number of
crawl steps and lane divisions may also be adjusted.
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Figure 8.16. Parameter Sensitivity GUI Window - Deck Stiffness Sensitivity with a Linear
Alpha Scale
317
Figure 8.17. Parameter Sensitivity GUI Window - Deck Stiffness Sensitivity with a Linear
Alpha Scale
318
Figure 8.18. Parameter Sensitivity GUI Window – Composite Action Sensitivity with a
Logarithmic Alpha Scale
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Figure 8.19. Parameter Sensitivity GUI Window – Boundary Rotational Spring Sensitivity
with a Logarithmic Alpha Scale
8.4.1.3
Experimental Data Comparison
The experimental data comparison GUI window allows the user to compare the natural
frequency and mode shape content between experimental data and an FE model (Figure 8.20).
This tool displays the mode shapes, natural frequencies, MAC matrices, and COMAC plots for
both model and experiment.
A)
Experimental Mode Shape. The experimental mode shape is plotted with
circles to denote the experimental node location (usually an accelerometer) as
well as an interpolated mode shape surface. The surface is calculated using a
fourth-order approximation function within the griddata function in Matlab
(Mathworks 2015) using the edge nodes of an associated FE model as
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boundaries (noted as black dots in the figure) and scaled using a user-supplied
coefficient.
B)
Analytical Mode Shape. This mode shape surface is first developed using a
Delaunay triangulation of the FE model deck node mesh then plotted using the
trimesh function (Mathworks 2015).
It is scaled using a user-supplied
coefficient. The experimental mode shape using only the experimental nodes
is overlayed.
C)
Experimental MAC Matrix.
The module displays an Experimental-
Experimental MAC matrix plot.
This plot indicates how unique each
experimental mode shape is from all other experimental mode shapes. Note
the high correlation of mode shapes for experimental modes 8, 9, and 10 (see
Figure X), which may indicate spatial aliasing or other issues during the
experimental test.
D)
Experimental-Analytical
MAC
Matrix.
This
module
displays
an
Experimental-Analytical MAC matrix plot. This plot indicates the degree of
similarity between individual experimental and analytical mode shapes. The
gray bars at the bottom of the matrix plot show that these modes have been not
selected for use in the parameter estimation objective function (see E below).
E)
Objective Function Experimental Frequency Selection Table. This module
lists all experimental and analytical frequencies. The first list of logic boxes
allows a user to use an experimental frequency in the objective function. The
second list of logic boxes sets whether the associated mode shape for a given
frequency is to be used to only pair experimental and analytical frequencies or
to be used for pairing and in the objective function. Unselecting the frequency
321
will automatically unselect the associate mode shape. This frequency line will
be greyed out from experimental-analytical MAC matrix plot.
F)
Frequency Pairing and MAC Value Table. This module lists all experimental
frequencies that are paired with analytical frequencies and lists the MAC
values for each paired mode as well as the percent difference in the
frequencies.
G)
Experimental COMAC Plot. This module plots the COMAC (COordinate
Modal Assurance Criterion) for the experimental data.
The COMAC is a
Degree of Freedom-wise calculation, shown in Equation 8.8, which illustrates
the relative agreement over a given set of mode pairs (Allemang 2003). This
plot shows spatially where a set of mode shapes provides greater or less
information. This plot only uses experimental modes that were selected to be
used in the objective function.
𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝑝𝑝 =
G)
�
𝑛𝑛
𝑖𝑖=1
�
𝑛𝑛
2
�𝜓𝜓𝑝𝑝𝑝𝑝 𝜙𝜙𝑝𝑝𝑝𝑝 �
𝑖𝑖=1
∗
𝜓𝜓𝑝𝑝𝑝𝑝 𝜓𝜓𝑝𝑝𝑝𝑝 �
𝐿𝐿
𝑖𝑖=1
𝜙𝜙𝑝𝑝𝑝𝑝 𝜙𝜙𝑝𝑝𝑝𝑝
∗
8.8
Analytical COMAC Plot. This module plots the COMAC (COordinate Modal
Assurance Criterion) for the analytical data (Allemang 2003).
This plot is
updated by using only the analytical mode shapes that have been paired with
experimental modes.
B
A
H
G
C
D
E
F
322
Figure 8.20. Model-Experimental Comparison GUI Window
323
8.4.1.4
Parameter Estimation
The parameter estimation GUI window displays the parameter alpha values (A), frequency (B)
and MAC (C) residuals, and total objective function value (D). The horizontal slider bars below
each alpha value window allows the user to select the alpha group value to display.
The
parameter alpha value table E displays the current alpha value, starting alpha value, percent
change in alpha value, average alpha value for all iterations and the current percent change in
alpha value from the starting alpha value to current value for each parameter group. The user
can specify the number of analytical modes to solve for during the iterative FE model updating
process, frequency weighting scheme, and updating algorithm tolerances. The “Re-Pair Modes”
options box allows the user to repair the experimental mode shapes with new analytical modes
after every function iteration. Unchecking this box will only pair modes at the first iteration.
A
A
B
C
A
E
D
Figure 8.21. Parameter Estimation GUI Window
324
9. Case Studies for Rapid Model Calibration Usi ng U nknown and
Known Parameters
The findings from two case studies on parameter estimation on a simply
supported steel multi-girder bridge are discussed. A single structure is
modified to simulated global or local damage and used to develop
simulated experimental data. Globally distributed parameter estimation
for unknown parameters is used to update the model as well as locally
varying mass distribution.
The response of global parameters to
simulated structural anomalies as well as mass redistribution to
determine model fitness and anomalous experimental behavior are
presented.
9.1
Overview
In this chapter findings are presented from a series of case studies on the effects of local and
global structural abnormalities on model updating behavior, specifically local mass
redistribution. Two cases of change in structural characteristics are examined in this chapter: 1)
global loss of composite action of all girders in a simply supported steel multi-girder bridge and
2) local loss of composite action along two adjacent girders in a simply support steel multi-girder
bridge. The case study experimental data was developed by simulating the change in composite
action in a FE model and extracting frequencies and mode shapes from a grid of nodes. The same
model, with the composite action set to the “normal” value, was then used as the base for mass
redistribution.
Also considered is the use of mass updating to assess the model accuracy for an actual structure.
A FE model was created using the automated modeling tool presented in Chapter 5 in this thesis.
Unknown parameters were updated using a gradient-based optimization algorithm also
presented in Chapter 8 in this thesis. Mass was redistributed in the updated model to assess
model fitness.
The work in this chapter is based upon the initial findings of Pan (2007) where a known
parameter, the Young’s modulus of steel, was updated deterministically to assess the fitness of a
325
FE model when compared to natural frequency analysis data. In that study, unknown model
parameters were first updated in order to bring the natural frequency analysis data of a FE model
in closer alignment with experimental results. The parameter values that resulted in the lowest –
or best – objective function value were chosen as resulting in the most representative model.
Next, the E of steel of the updated model was then adjusted globally over a set range of values
and the objective functions related to each value of E was compared. The study found that the E
of steel that resulted in the lowest objective function value was ~1.15 times the known E of steel,
indicating that the model was lacking stiffness globally. This led investigators to refine the FE
model and re-update the unknown parameters. The best set of parameter values were again
used with the FE model while the E of steel was adjusted; the E of steel that resulted in the lowest
objective function value was ~1.00, indicating that the current model form and parameter set was
appropriate and no further changes were needed.
This research utilizes the mass of the FE model deck nodes in place of the E of steel as the known
parameter and adds a layer of complication to the process by dividing the deck into a set of mass
“zones” that the total mass of the deck may be divided among. Mass may be used in place of
stiffness for updating a model with natural frequency analysis data as mass and stiffness have an
inverse effect on frequencies and mode shapes.
9.2
Mossy Creek Bridge
Mossy Interchange Bridge is located in Mossy, West Virginia, just off the West Virginia Turnpike
(I-77). It is a 3-span, simply supported steel stringer bridge with a cast-in-place composite
concrete deck. The center span is approximately 52 feet long with 2 rows of internal diaphragms
oriented perpendicular to the girders. The out-to-out width is 35 feet. The superstructure is
composed of five girders spaced at 8’-1” and has zero skew. The center span girders are rolled
326
sections. The interior girders are W36X170 and the exterior girders are W36X150. The southern
end of the center span rests on steel rocker bearings, while the northern end rests on pinned
bearings. These bearings are supported by reinforced concrete hammerhead piers. Figure 9.1
and Figure 9.2 show the structure.
Figure 9.1. Topside of Mossy Creek Bridge
327
Figure 9.2. Underside of Mossy Creek Bridge
9.3
FE Model
An element-level FE model of the bridge was created using the semi-automated model creation
software described in this thesis (Figure 9.3 and Figure 9.4).
328
Figure 9.3. 3D FE Model of Mossy Creek Bridge
Figure 9.4. 3D FE Model of Mossy Creek Bridge Shown without Deck Shell Elements
329
9.4
Simulation of Experimental Data
Experimental results were extracted for the nine deck nodes shown in Figure 9.5. Nodes were
located along every girder line at 0.25, 0.50 and 0.75 times the length of the structure. In order to
develop the experimental modes, the FE model was analyzed using the natural frequency
analysis solver in Strand7 with the a priori parameter values shown in Table 9.1. Boundary
conditions were set according to with corresponding girder numbers shown in Figure 9.5 and in
Table 9.2 and Table 9.3.
Figure 9.5. Location of “Experimental” Nodes and Girder Numbers
330
Table 9.1. A Priori Parameter Values
Parameters
Name
Type
Value
Deck Stiffness
f'c [psi]
4000
Girder Stiffness
E [psi]
2.9x107
Barrier Stiffness
f'c [psi]
2500
Diaphragm Stiffness
E [psi]
2.9x107
Composite Moment of Inertia (Fully Composite)
I [in4]
1.0x107
Composite Moment of Inertia (Loss of Composite Action)
I [in4]
1.0x10-3
Table 9.2. Fixed Bearing Degrees of Freedom
Girder
Fixed Bearings Degree of Freedom
ux
uy
uz
rx
ry
rz
1
fixed
free
fixed
fixed
free
fixed
2
fixed
free
fixed
fixed
free
fixed
3
fixed
free
fixed
fixed
free
fixed
4
fixed
free
fixed
fixed
free
fixed
5
fixed
free
fixed
fixed
free
fixed
6
fixed
free
fixed
fixed
free
fixed
Table 9.3. Expansion Bearings Degrees of Freedom
Girder
Expansion Bearings Degree of Freedom
ux
uy
uz
rx
ry
rz
1
free
free
fixed
fixed
free
fixed
2
free
free
fixed
fixed
free
fixed
3
free
free
fixed
fixed
free
fixed
4
free
free
fixed
fixed
free
fixed
5
free
free
fixed
fixed
free
fixed
6
free
free
fixed
fixed
free
fixed
331
9.5
Case 1: Global Loss of Composite Action
The first case studied was the complete loss of composite action along every girder in the
structure. Loss of composite action was simulated by setting the moment of inertia about the
transverse axis of every composite action link to 0.003 in4, which is the lower bound of the
sensitive natural frequency range for the a priori FE model. The first ten natural frequencies and
mode shapes were used for model updating in all following discussions. MAC matrix plots show
the comparison of the eleventh and twelfth modes for completeness.
9.5.1
Initial Model-Experiment Comparison
Comparing the initial a priori model to the experimental data indicates greater stiffness in the
model for all modes (Table 9.4). The MAC matrix plot is shown in Figure 9.6 and shows no clear
diagonalization past mode 3. The modal order of the “experimental” model results in the mispairing of a priori mode 5 with “experimental” mode 4 instead of that mode being paired with
“experimental” mode 5. Visually all other mode shapes are in good agreement, however the
MAC values for mode shapes 4 through 7 are less than 0.53 (Figure 9.7 through Figure 9.20). The
modal order of a priori modes 8 through 10 is also incorrect. The MAC values do not indicate
either a primarily longitudinally or laterally dominated source of error. Frequency errors are all
positive, indicating a general global lack of stiffness. The total objective function for the modelexperiment comparison is 2.36.
332
Table 9.4. A Priori Model-Experiment Comparison for Global Loss of Composite Action
Exp Freq
A Priori Model
Ana Freq
% Diff
Mode #
MAC
1
5.08
10.93 115.14
1 0.968
2
7.76
12.74
64.13
2 0.998
3
11.21
16.63
48.33
3 0.905
4
15.48
26.21
69.34
5 0.175
5
19.96
30.39
52.23
10 0.426
6
22.34
36.24
62.17
6 0.527
7
23.55
39.39
67.29
4 0.338
8
28.09
39.58
40.94
8 0.842
9
33.16
40.92
23.42
11 0.970
10
35.59
43.96
23.52
7 0.920
OBJ
2.36
Figure 9.6. A Priori MAC Matrix for Global Loss of Composite Action
333
Figure 9.7. A Priori Mode 1
Figure 9.8. “Experimental” Mode 1
Figure 9.9. A Priori Mode 2
Figure 9.10. “Experimental” Mode 2
334
Figure 9.11. A Priori Mode 3
Figure 9.13. A Priori Mode 5
Figure 9.12. “Experimental” Mode 3
Figure 9.14. “Experimental” Mode 4
335
Figure 9.16. “Experimental” Mode 5
Figure 9.15. A Priori Mode 10
Figure 9.17. A Priori Mode 6
Figure 9.18. “Experimental” Mode 6
336
Figure 9.19. A Priori Mode 4
Figure 9.21. A Priori Mode 8
Figure 9.20. “Experimental” Mode 7
Figure 9.22. “Experimental” Mode 8
337
Figure 9.23. A Priori Mode 11
Figure 9.25. A Priori Mode 7
Figure 9.24. “Experimental” Mode 9
Figure 9.26. “Experimental” Mode 10
338
9.5.2
Mass Redistribution as Model Fitness Check for a Priori Model
Mass was redistributed among deck zones in three different configurations on the a priori before
updating unknown parameters in the model: five lateral zones (Figure 9.27), five longitudinal
zones (Figure 9.28), and a 3x3 grid (Figure 9.29).
Figure 9.27. 5 Lateral Mass Zones
Figure 9.28. 5 Longitudinal Mass Zones
339
Figure 9.29. 3x3 Grid Mass Zones
Table 9.5 shows the final mass multipliers, or alpha coefficients, for each zone at the end of mass
redistribution. Using five lateral mass zones results in a change of mass between -7% and +6%.
Contrast this with the greater change in mass seen when using five longitudinal mass zones,
where the zone nearest the fixed support has a greater than 99% decrease in mass and the zone
nearest the expansion bearing has a 93% decrease in mass. Further, the middle three show a %15
to 161% increase in mass. Clearly the errors in the longitudinal stiffness components of the
model have a greater effect on model error than the transverse components.
Further, the
decrease in mass in the longitudinal center of the model indicates that the longitudinal stiffness
of the model is higher than that of the experimental subject. In the case of the 3x3 grid, mass is
primarily redistributed to zones four, five, and six, indicating to greater longitudinal stiffness in
the model, further pointing to less longitudinal stiffness in the experimental than a priori FE
model. The objective function of the a priori model drops from 2.36 to 2.35, 1.89, and 2.00 for the
five lateral, five longitudinal, and 3x3 grid mass zones, respectively. The greatest reduction in
objective function resulting from mass redistribution among five longitudinal zones points to the
main source of model error deriving from a difference in longitudinal stiffness.
340
Table 9.5. Mass Zone Multipliers at End of Redistribution for Global Loss of Composite
Action
Mass
Zone
Prior to Parameter Est.
5 lat.
5 long.
3x3 grid
1
0.973
0.006
0.016
2
1.054
1.160
0.016
3
0.929
1.154
0.016
4
1.061
2.610
2.408
5
0.984
0.070
1.484
6
2.471
7
0.980
8
0.643
9
0.965
341
Table 9.6. Model-Experiment Comparison after Initial Mass Redistribution for Global
Loss of Composite Action
Exp
Freq
Ana
Freq
5 Lateral Mass Zones
Mode
% Diff #
MAC
5 Longitudinal Mass Zones
Ana
%
Mode
Freq
Diff
#
MAC
Ana
Freq
3x3 Grid Zones
Mode
% Diff #
MAC
1
5.08
10.93
115.08
1
0.968
9.36
84.12
1
0.959
8.41
65.54
1
0.977
2
7.76
12.75
64.25
2
0.998
10.99
41.54
2
0.996
12.48
60.73
2
0.997
3
11.21
16.74
49.28
3
0.905
14.51
29.41
3
0.882
17.47
55.82
3
0.939
4
15.48
25.94
67.55
5
0.177
22.68
46.54
5
0.203
25.07
61.93
5
0.258
5
19.96
30.39
52.24
10
0.432
28.64
43.45
10
0.200
28.39
42.19
10
0.571
6
22.34
36.07
61.43
4
0.528
33.47
49.79
6
0.572
33.15
48.36
4
0.530
7
23.55
39.53
67.90
6
0.331
33.72
43.21
4
0.693
37.65
59.89
8
0.394
8
28.09
39.82
41.77
8
0.926
36.39
29.57
9
0.859
41.75
48.65
9
0.896
9
33.16
40.92
23.43
11
0.960
37.01
11.63
11
0.841
42.46
28.07
11
0.882
10
35.59
43.95
23.49
7
0.958
41.61
16.93
7
0.891
43.10
21.12
6
0.898
OBJ
9.5.3
2.35
1.89
2.00
Parameter Estimation
Four parameters were updated in the model using the software described in Chapter 8.
Updating was performed with a single set of parameter starting values and was stopped once the
algorithm determined no sensitivity in the objective function above 0.001 or parameter values
above 0.001α. Composite action was the only parameter to exhibit greater than an order of
magnitude change in value (Table 9.7). Figure 9.30 and Figure 9.32 show the MAC matrix plot
for the model-experiment comparison at the fifth iteration and final (tenth) iteration of parameter
estimation, respectively. Table 9.7 shows frequency errors less than +/- 1.04% and MAC values
greater than 0.991 for the final updated model with an objective of 0.02. The MAC matrix plot at
the fifth iteration shows the beginning of convergence of mode pairing and diagonalization and
the final MAC matrix plot shows complete diagonalization. Comparing the final model to the
“experimental” model indicates that the model updating process was able to find an adequate
342
solution to the minimization problem by updating global parameters, specifically global
composite action.
Table 9.7. A Priori and Converged Parameter Values
Lower
Bound
Upper
Bound
Start
End
%Δ
Barrier Stiffness [f'c]
1000
8000
2500
3297
31.88
Composite Action [in4]
0.001
22000
22000
0.0021
-100.00
Deck Stiffness [f'c]
1000
8000
4000
3491
-12.73
Diaphragm Stiffness [psi]
1000
2.90E+07
2.90E+07
2.59E+07
-10.66
Table 9.8. Model Experiment Comparison
Exp.
Mode
Exp
Freq
A Priori Model
Ana Freq
% Diff
Updated Model
Mode #
MAC
Ana Freq
% Diff
Mode #
MAC
1
5.08
10.93
115.14
1
0.968
5.10
0.34
1
1.000
2
7.76
12.74
64.13
2
0.998
7.77
0.13
2
1.000
3
11.21
16.63
48.33
3
0.905
11.28
0.57
3
0.997
4
15.48
26.21
69.34
5
0.175
15.55
0.44
4
0.992
5
19.96
30.39
52.23
10
0.426
19.76
-1.03
5
0.999
6
22.34
36.24
62.17
4
0.527
22.20
-0.63
6
1.000
7
23.55
39.39
67.29
6
0.338
23.39
-0.67
7
1.000
8
28.09
39.58
40.94
8
0.842
28.07
-0.08
8
1.000
9
33.16
40.92
23.42
11
0.970
33.22
0.20
9
0.999
10
35.59
43.96
23.52
7
0.920
35.73
0.40
10
1.000
OBJ
2.36
0.02
343
Figure 9.30. MAC Matrix Plot at 5 Iterations
Figure 9.31. MAC Matrix Plot at Parameter Convergence (10 Iterations)
344
9.5.4
Mass Redistribution as Model Fitness Check for Updated Model
After updating unknown global parameters, the fitness of the final updated model was checked
by again using the mass redistribution algorithm. Like the initial model error check, mass was
redistributed using three schemes: five lateral zones, five longitudinal zones, and a 3x3 grid.
Table 9.9 shows the results of mass redistribution for five lateral and 3x3 grid mass zones while
Figure 9.32 and Figure 9.33 show the final MAC matrix plots. The response surface space was
insufficiently sensitive to the use of five longitudinal zones and the algorithm was unable to
proceed past the initial iteration. Comparison of natural frequency data indicates that the model
exhibited behavior less like that shown in the "Experimental" data with mass redistribution; in
other words, globally distributed parameters were sufficient for model calibration and no further
refinement would be needed to use the model to investigate global or system-level structural
characteristics.
345
Table 9.9. Model Experiment Comparison of Mass Redistribution Solution after
Parameter Estimation for Global Loss of Composite Action
Exp
Mode
Exp
Freq
5 Lateral Mass Zones
Ana Freq
% Diff
Mode #
3x3 Grid Zones
MAC
Ana Freq
% Diff
Mode #
MAC
1
5.08
5.13
0.88
1
1.000
5.14
1.18
1
1.000
2
7.76
7.72
-0.48
2
1.000
7.75
-0.11
2
0.999
3
11.21
11.11
-0.94
3
0.999
11.26
0.40
3
0.999
4
15.48
15.41
-0.45
4
0.993
15.67
1.22
4
0.987
5
19.96
19.96
-0.04
5
0.999
20.45
2.44
5
0.996
6
22.34
22.39
0.22
6
0.995
23.06
3.22
6
0.990
7
23.55
23.51
-0.14
7
0.993
23.99
1.88
7
0.990
8
28.09
27.70
-1.39
8
0.998
28.19
0.35
8
0.993
9
33.16
33.02
-0.40
9
0.996
34.31
3.47
9
0.986
10
35.59
35.90
0.87
10
0.998
37.02
4.04
10
0.992
OBJ
0.03
0.08
Figure 9.32. Final MAC Matrix Plot for Mass Redistribution Convergence with 5 Lateral
Zones for Global Loss of Composite Action
346
Figure 9.33. Final MAC Matrix Plot Mass Redistribution Convergence with 3x3 Grid
Zones for Global Loss of Composite Action
9.6
Case 2: Local Loss of Composite Action along Two Girders
Next, the use of mass redistribution to diagnose model errors related to unknown local structural
characteristics was investigated. Like the previous case, complete loss of composite action was
used as the structural abnormality, however in this instance only girders 1 and 2 had a loss of
composite action while girder 3, 4, and 5 were held to completely composite (Figure 9.34).
347
Figure 9.34. Isometric View of 3D FE Model of Mossy Creek Bridge Indicating Two
Girders with Total Loss of Composite Action
9.6.1
Initial Model-Experiment Comparison
Table 9.10 shows the model-experiment comparison with the a priori model and Figure 9.35
shows the corresponding MAC matrix plot. Frequency errors are up to 75% and MAC values
range from 0.921 to 0.919. Modes are paired correctly up to “experimental” mode 8; however
comparing the mode shapes visually shows that the a priori model form is unable to properly
describe the local variation in degree of composite action.
348
Table 9.10. A Priori Model Experiment Comparison for Local Loss of Composite Action
Exp Freq
A Priori Model
Ana Freq
% Diff
Mode #
MAC
1
6.26
10.93
74.67
1
0.637
2
8.62
12.74
47.74
2
0.683
3
15.15
16.63
9.80
3
0.919
4
21.16
26.21
23.86
5
0.384
5
23.62
30.39
28.67
6
0.360
6
24.36
36.24
48.73
4
0.921
7
30.94
39.39
27.30
11
0.291
8
35.75
39.58
10.74
10
0.295
9
37.23
40.92
9.92
7
0.845
10
38.47
43.96
14.26
9
0.406
OBJ
1.92
Figure 9.35. A Priori Mac Matrix Plot for Local Loss of Composite Action
349
Figure 9.36. A Priori Mode 1
Figure 9.38. A Priori Mode 2
Figure 9.37. “Experimental” Mode 1
Figure 9.39. “Experimental” Mode 2
350
Figure 9.40. A Priori Mode 3
Figure 9.41. “Experimental” Mode 3
Figure 9.42. A Priori Mode 5
Figure 9.43. “Experimental” Mode 4
351
Figure 9.44. A Priori Mode 6
Figure 9.46. A Priori Mode 4
Figure 9.45. “Experimental” Mode 5
Figure 9.47. “Experimental” Mode 6
352
Figure 9.48. A Priori Mode 11
Figure 9.50. A Priori Mode 10
Figure 9.49. “Experimental” Mode 7
Figure 9.51. “Experimental” Mode 8
353
Figure 9.52. A Priori Mode 7
Figure 9.53. “Experimental” Mode 9
Figure 9.55. “Experimental” Mode 10
Figure 9.54. A Priori Mode 9
Figure 9.56. Example of Software GUI for Model-Experimental Comparison with Local Loss of Composite Action
354
355
9.6.2
Mass Redistribution as Model Fitness Check for a Priori Model
Mass was redistributed in the model in the same manner as before as a fitness check for the a
priori model. Using both five lateral zones and a 3x3 grid result in concentration of mass over the
two girders with a complete loss of composite action (Table 9.11). The algorithm was unable to
find a better solution when using five longitudinal zones and did not progress past the first
iteration. The objective function of the a priori model drops from 1.92 to 1.42 and 1.40 for the five
lateral and 3x3 grid mass zones, respectively. These results indicate that the primary source of
model error is due to a discrepancy in the transverse stiffness, specifically lower stiffness in the
“experimental” model than in the a priori model on the side of model near lateral zone one.
Table 9.11. Final Mass Redistribution Coefficients for Local Loss of Composite Action
Prior to Parameter
Est.
5 lat
3x3 grid
1
1.606
1.667
2
1.270
0.863
3
1.074
0.376
4
0.710
1.788
5
0.340
0.890
6
0.561
7
1.543
8
0.975
9
0.336
356
Table 9.12. Model-Experiment Comparison of Local Loss of Composite Action to Initial
Mass Updating
Exp
Freq
5 Lateral Mass Zones
Ana
Freq
%
Diff
Mode
#
MAC
Ana
Freq
%
Diff
Mode
#
MAC
1
6.26
10.36
65.52
1
0.910
10.31
64.68
1
0.939
2
8.62
13.45
55.99
2
0.961
13.60
57.70
2
0.971
3
15.15
17.26
13.93
3
0.865
17.03
12.42
3
0.930
4
21.16
26.46
25.02
5
0.649
26.60
25.71
5
0.751
5
23.62
29.12
23.31
6
0.464
29.52
24.99
6
0.449
6
24.36
34.76
42.68
4
0.932
35.40
45.31
4
0.923
7
30.94
39.85
28.78
8
0.557
40.57
31.12
7
0.674
8
35.75
40.44
13.13
9
0.665
41.54
16.22
9
0.641
9
37.23
41.57
11.66
7
0.938
42.43
13.98
8
0.883
10
38.47
44.16
14.79
10
0.776
45.07
17.16
10
0.756
OBJ
9.6.3
Grid of 9 Zones
1.42
1.40
Parameter Estimation
Four parameters were updated in the model in the same manner as seen in the previous case
study. Again, composite action was the only parameter to exhibit an order of magnitude change
in value (Table 9.7), however both deck and barrier stiffness was reduced by 60-75 percent. The
model-experiment comparison (Table 9.14 and Figure 9.57) shows that the primary reduction in
objective function comes from reduction in frequency errors. The a priori model has higher
frequencies for all modes while the updated model has frequency errors of approximately -15%
to +16%. MAC values increased slightly with greater increases seen in the “experimental” modes
that had MAC values less than 0.500. A lower objective function and better frequency agreement
by a reduction in stiffness of the global components indicates that the “experimental” model has
lower stiffness – either global or local - than the a priori model. The objective function decrease,
357
however, comes primarily from the reduction in frequency error, indicating that a local source of
error may still be present.
Table 9.13. A Priori and Converged Parameter Values for Local Loss of Composite
Action
Lower Bound
Upper Bound
Start
End
%Δ
Barrier Stiffness [f'c]
1000
8000
2500
1000
-60.00
Composite Action [in4]
0.001
22000
22000
3.058
-99.86
Deck Stiffness [f'c]
1000
8000
4000
1025
-74.38
Diaphragm Stiffness [psi]
1000
2.90E+07
2.90E+07
2.84E+07
-2.07
Table 9.14. Model Experiment Comparison for Local Loss of Composite Action
Exp Freq
A Priori Model
Updated Model
Ana Freq
% Diff
Mode #
MAC
Ana Freq
% Diff
Mode #
MAC
1
6.26
10.93
74.67
1
0.637
7.34
17.20
1
0.685
2
8.62
12.74
47.74
2
0.683
9.79
13.53
2
0.690
3
15.15
16.63
9.80
3
0.919
14.16
-6.49
3
0.935
4
21.16
26.21
23.86
5
0.384
21.95
3.74
5
0.541
5
23.62
30.39
28.67
6
0.360
23.40
-0.91
6
0.461
6
24.36
36.24
48.73
4
0.921
28.61
17.42
4
0.926
7
30.94
39.39
27.30
11
0.291
30.16
-2.52
8
0.573
8
35.75
39.58
10.74
10
0.295
30.86
-13.67
7
0.459
9
37.23
40.92
9.92
7
0.845
31.91
-14.28
9
0.988
10
38.47
43.96
14.26
9
0.406
35.26
-8.34
11
0.614
OBJ
1.92
1.21
358
Figure 9.57. MAC Matrix Plot at Parameter Convergence for Local Loss of Composite
Action
9.6.4
Mass Redistribution as Model Fitness Check for Updated Model
Mass was redistributed in the updated model using the three zone types as noted previously,
however the algorithm was able to progress beyond the first iteration with five lateral zones only
(Table 9.15). Redistributing mass in the updated model reduced the objective function from 1.21
to 0.62, or 46%. Frequency errors did not change, however MAC values were improved (Table
9.15 and Figure 9.58), with the lowest MAC value equal to 0.650 and five MAC vales over 0.900.
These results indicate that global parameter estimation is inadequate for characterizing the
experimental data. More refined methods, such as local unknown parameter estimation would
be appropriate for FE model simulation. In the case of an actual structure, in situ investigation of
the region of the structure near girders 1 and 2 would provide greater understanding of the local
variance in structural characteristics.
359
Table 9.15. Model-Experiment Comparison for Local Loss of Composite Action for Mass
Zone Updating
Exp
Freq
5 Lateral Mass Zones
Ana Freq
% Diff
Mode #
MAC
1
6.26
7.15
14.21
1
0.906
2
8.62
9.78
13.47
2
0.967
3
15.15
14.86
-1.89
3
0.980
4
21.16
22.08
4.32
5
0.837
5
23.62
22.78
-3.55
6
0.650
6
24.36
28.02
15.02
4
0.931
7
30.94
29.88
-3.45
8
0.829
8
35.75
32.47
-9.16
10
0.837
9
37.23
33.57
-9.84
9
0.961
10
38.47
35.04
-8.93
11
0.736
OBJ
0.62
Figure 9.58. MAC Matrix Plot at Mass Redistribution Convergence with 5 Lateral Zones
for Local Loss of Composite Action
360
10. Conclusions and F uture Work
10.1
Summary of Research Objectives and Scope
The overarching aim of the research reported herein is to establish a framework whereby realistic
simulations and structural identification may be brought to bear on furthering the understanding
of performance of large populations of bridges.
More specifically, the following research
objectives were defined and adopted to guide this research effort:
1.
Develop and validate an automated design/modeling tool capable of developing
realistic
estimates
of
the
structural
characteristics/responses
for
broad
populations of bridges. This tool should be capable of (a) sizing members as per
the current AASHTO LRFD Bridge Design Specifications for different bridge
configurations, (b) constructing 3D FE models of common bridge types as per
best practices approaches, (c) simulating a wide range of demands (including
dead load, superimposed dead load, live load, etc.) as per current design
practice, and (d) automating the response extraction process for the various
considered demands.
2.
Using the tool developed in (1), establish inherent bias and variability in the
LRFD analysis model and the extent to which common design assumptions can
result in deterministic trends of structural characteristics within populations of
bridges. The specific design assumptions selected for this study include (a) the
use of distribution factors to estimate the transverse distribution of live load and
(b) the equal distribution of superimposed dead load across all girders.
361
3.
Using the tool developed in (1), examine how the current practice of bridge
design (inclusive of the conservatism introduced through common assumptions)
may produce bridges that are capable of meeting demands that were not
explicitly considered during member sizing. The demand selected for this study
was differential vertical and rotational support movement within continuous
bridges.
4.
Develop and validate a streamlined parameter identification tool capable of
reliably improving the representative nature of simulation models through the
use of field measurements. To permit the reliable implementation to populations
of bridges, this tool must provide the user with the ability to quickly and
effectively identify and diagnose error sources that may compromise the model
updating process and distort the representative nature of the model.
10.2
Conclusions
The following conclusions are drawn from the work presented herein, and are organized based
on the five primary objectives outlined at the beginning of this research.
10.2.1
Objective 1: Development of Automated Design, Modeling, and Simulation
Tool
10.2.1.1
Development of Automated Design
A tool was developed as part of this research that replicates the SLG girder design process as per
AASHTO ASD and LRFD limit states. The options that may be selected by the user include: the
ability to define arbitrary rounding rules for cross-sectional dimension or to instead select
362
member properties with zero additional margin; design continuous-span structures with doublysymmetric cover plate sections in the negative moment region; and to design interior and exterior
girders separately or so that a single section can meet both interior and exterior design criteria.
•
Design software was validated by an independent team at the University of
Delaware. This validation included a line by line check of all calculations used in
computing SLG demands, section capacity, and section proportion limits. This
validation also included a comparison with redundant designs.
•
Existing constrained non-linear optimization solvers are suitable for the task of
replicating the linear and iterative girder design process, especially for sizing the
individual components of welded/fabricated plate girders as per AASHTO LRFD
limit states.
•
Girder designs for simply-supported steel multi-girder bridges required a mean
time of 11.5 seconds. The distribution of design times is strongly skewed left.
The maximum design time was approximately 50 seconds.
The number of
iterations, or trials, needed to find a solution for girder designs had a mean of
2.45. This distribution was also strongly left-skewed and had a maximum of nine
iterations.
10.2.1.2
Development of Automated Modeling and Simulation
Automated creation of 3D element-level FE models, simulation of demands, and extraction of
results is possible through the use of common scripting or programming languages and
application programming interfaces provided by the developers of many FE solvers. A software
tool was developed for the Matlab programming environment that utilizes the Strand7 (a
commercially available FE solver) API to create FE models for simulations. Options in the
software for model construction include: Live load demand simulation models can be maximized
363
by shifting lanes upon the clear deck area between curbs as well as shifting truck positions within
lanes to “crowd” truck loads over girders of interest; Simulation of types of dead load and
superimposed dead load can be achieved by the selective inclusion of various features and
members or by modifying section and material properties; diaphragm section type and
orientation – including contiguous or staggered diaphragm rows for highly skewed-structures may be configured; combinations of asymmetric boundary conditions may be used to ensure
accurate simulation of load phenomena.
A series of strategies for automating the results
extraction process of the automated modeling tool were also identified in order to avoid
anomalies with results repeatability across populations and general reliability in the absence of
direct human user oversight.
•
Through a comparison of both single and two-girder element level and shell
element model systems, it was determined that the element-level model was the
best choice for the automated modeling tool for the simulation of multi-girder
bridges. This conclusion was based on (1) the good agreement (approximately
5% difference) between the element-level model and the more refined shell
element model, (2) the more straightforward manner in which results may be
extracted from the element-level model, and (3) the drastically reduced
computational time associated with the element-level model.
364
•
To automate the results extraction the following strategies should be employed:
(a) Shear deformation of the beam elements within the element-level models
should be ignored to ensure proper convergence of results; (b) Boundary
conditions that provide minimum restraint should be used to minimize
extraneous inputs associated with local, self-equilibrating forces; (c) Support
reactions should be used to conservatively estimate the shear force in the girders,
as the computed shear force in the beam elements is mesh dependent; (d) Deck
stresses should be approximated by extrapolating the strain in the girders to the
top of the deck to avoid local stress concentrations exist in the vicinity of rigid
links.
10.2.2
Objective 2: Establish the Bias, Trends, and Variability in Performance Due to
the LRFD Design Model and Common Design Assumptions
10.2.2.1
•
Sampling and Study Design
Latin Hypercube Sampling allows for efficient sampling of parameter space. For
the five parameters examined within certain bounds the results converged by 200
samples.
Individual sample sets showed similar variance and shape of
distribution. The comparing the ECDF of the FE to SLG rating ratios of a third
suite of 100 bridges to the ECDF of FE to SLG ratio of the combination of two
sample sets of 100 each showed convergence by passing the KolmogorovSmirnov test.
365
10.2.2.2
•
Rating Factor of Simply Supported Steel Multi-girder Bridges
without Diaphragms and with the Consideration of Infinite Fatigue
Life Design Criteria
For these bridges the SLG ratings for interior girders/exterior girders had a mean
of 2.64/2.99 for the Strength I limit state and 2.67/3.37 for the Service II limit state
with standard deviations of 0.61/0.68 and 0.76/0.88, respectively.
•
The cause of the above phenomena is that fatigue criteria for simply supported
structures are the limiting factor when fatigue is considered in design in all cases
and the associated limiting fatigue moments are considerably larger than actual
moment demands.
•
The exterior girder controls for Strength I FE ratings in approximately 55% of
bridges, while the 1st interior girder is controlling for 5%, and other interior
girders control for 40%.
•
The exterior girder controls for Service II FE ratings in approximately 35% of
bridges, while the 1st interior girder controls for 5%, and other interior girders
control for 65%.
•
The ratio of controlling girder FE to SLG ratings is 1.12/1.21 for Strength I/Service
II limit states with standard deviations of 0.08/0.08.
Rating factor ratios for
interior girders have mean 1.24/1.23 and standard deviation 0.10/0.10, for
Strength I/Service II.
deviation 0.10/0.11.
Exterior girders have mean 0.99/1.04 with standard
This reflects the appropriateness of the SLG model for
design of steel multi-girder bridges.
366
•
High skew ratios result in high conservatism in load rating for SLG designs.
Skew ratios under 0.5 show no discernable effect. SLG designs for bridges with
skew ratios between 0.5 and 0.75 may be less conservative than other designs.
•
The remainder of partial lanes (the width of the roadway surface less the total
width of all lanes) has no discernable effect on the ratio of FE to SLF rating ratio.
10.2.2.3
•
Rating Factor of Simply Supported Steel Multi-girder Bridges
without Diaphragms and without the Consideration of Fatigue
Neglecting fatigue life criteria in design results in Strength I limit state SLG
rating factors of 1.32/1.30 with standard deviation 0.08/0.11; Service II SLG
ratings have mean 1.00 with standard deviation 0. Neglecting fatigue life in
design allows the optimization algorithm to find plate girder dimensions that
exactly satisfy Service II criteria. The ratio of controlling girder FE to SLG ratings
is 1.26/1.27 for Strength I/Service II limit states with standard deviations of
0.23/0.28.
Rating factor ratios for interior girders have mean 1.23/1.25 and
standard deviation 0.18/0.16, for Strength I/Service II.
Exterior girders have
mean 1.12/1.29 with standard deviation 0.25/0.30. The Service II limit state was
the controlling limit state for most designs and may be the cause of the
additional conservatism illustrated by higher FE to SLG rating factor ratios.
•
When designing without fatigue life Service II limit state criteria is the
controlling limit state in the large majority of cases; it is reasonable to assume
that in those cases when Service II was not the controlling limit that the
optimization algorithm was stuck in a local minimum for designs. The use of
doubly symmetric sections adds unneeded area to sections when designing for
fatigue.
367
10.2.2.4
•
Effect of Diaphragms on Transverse Load Sharing
The nominal angle section sizes used in cross- and chevron-braced diaphragms
that are required to satisfy exterior girder transverse wind load demands and
minimum slenderness ratio requirements are significantly smaller than the actual
sections chosen by contractors to meet construction stability requirements. The
use of stiffer diaphragm sections has a strong effect on the FE to SLG rating
factor ratio of simply supported structures.
•
Diaphragm stiffness contribution can be normalized among a population of
bridges with varying girder sections by using the ratio of effective flexibilities of
the longitudinal and transverse load paths.
•
The effect of increasing diaphragm stiffness on bridges that were designed
without infinite fatigue life criteria is greater in magnitude than the effect on
bridges designed with fatigue life criteria.
•
Interior girder rating factors increase with increasing diaphragm stiffness. The
majority of structures see a decrease in exterior girder rating factor, while some
structures see an increase in exterior girder rating factor.
•
The effective flexibility ratio of interior girders for bridges designed with
consideration of IFL increases from a mean of 1.42 for nominal diaphragm
stiffness to 1.68 for 30x diaphragm stiffness. The mean ratio for exterior girders
increases from 0.71 to 0.84. When IFL criteria is not used in design the flexibility
ratio increases from 1.54 to 1.70 for interior girders and from 0.77 to 0.85 for
exterior girders.
368
10.2.2.5
•
Distribution Factors
Exterior girder distribution factors become more conservative as they increase
for both Strength I and Service II limit states. Service II rating factors become
more conservative with decreasing span length.
SLG design conservatism
variance increases with increasing skew for the Service II limit state.
•
The difference between the calculated SLG live load distribution factors and the
theoretical minimum distribution factor has no effect on the ratio of FE to SLG
rating factors.
10.2.2.6
•
Demand Ratios
The ratio of FE to SLG dead load moment demand has a mean of 1.06 and 0.86
for interior and exterior girders, respectively, with standard deviations of 0.06
and 0.08. Superimposed dead load demand ratios have mean 1.21/2.20 with
standard deviation 0.38/0.54. Live load demand ratio means are 0.82/1.03 with
standard deviation 0.06/0.10.
•
Dead load moment demands are under predicted by the SLG model for interior
girders while they are over-predicted for exterior girders. Dead load prediction
becomes less conservative for shorter bridges. Dead load demand is slightly less
conservative for interior girders with higher girder spacing.
369
•
Superimposed dead load demands are non-conservative for all exterior girders.
The ratio of predicted to actual superimposed dead load demands increases with
increasing bridge width for both exterior and interior girders, however about one
third of bridges showed maximum interior girder superimposed dead load
demands that were less than predicted by the SLG model.
Neglecting
superimposed dead load during design when using the SLG model results in
significantly higher superimposed dead load demands on the exterior and 1st
interior girders and significantly lower superimposed dead load demands on
other interior girders. Superimposed dead load demands are more likely to be
under predicted for interior girders with smaller girder spacing.
•
Live load demand predictions with the SLG model are less conservative for
longer bridges.
Demand predictions may be non-conservative for exterior
girders on some longer structures. Live load demand predictions with the SLG
model are less conservative for bridges with a higher skew ratio. The ratio of
SLG to FE live load demands may be greater or less than unity depending on
distribution factor. The ratio of predicted live load moment demand to FE live
load demands decreases with increasing distribution factor for interior girders.
•
Inclusion of barrier stiffness always results in higher FE rating factors, due the
mean decrease in FE to SLG live load moment demand. Mean FE to SLG live
load demand is 0.82 without barrier stiffness and 0.78 with barrier stiffness
contributions for interior girders. For exterior girders the mean demand ratio
drops from 1.03 to 0.90. Standard deviations increase from 0.06 to 0.07 and from
0.10 to 0.14 for interior and exterior girders, respectively.
370
10.2.2.7
•
Rating Factor of Two-span Continuous Steel Multi-girder Bridges
without Diaphragms and with the Consideration of Infinite Fatigue
Life Design Criteria
The mean of negative moment region Strength I limit state SLG rating factors for
interior/exterior girders was 1.31/1.27 with standard deviation 0.42/0.39; Service
II SLG ratings have mean 1.83/1.77 with standard deviation 0.54/0.60.
Both
Strength I rating factors exhibited the same pattern as found with Service II
rating factors in simply supported bridges designed without fatigue criteria: the
large majority of ratings were exactly 1.00 with small tail of outliers. Rating
factor ratios for interior girders have mean 1.45/1.48 and standard deviation
0.11/0.24, for Strength I/Service II. Exterior girders have mean 1.55/1.57 with
standard deviation 0.16/0.23.
•
Positive moment region rating factors were similar to those found with simply
supported structures designed with infinite fatigue life criteria.
•
Due to higher moment demands in the negative moment region, the flange
thickness required to satisfy strength and stiffness criteria in the negative
moment region due to dead and live load demands was sufficient in most cases
to satisfy fatigue criteria, therefore no extra capacity was added.
•
Demands were first estimated with the assumption of continuous cross-sections
over the entire length of the girders and then iteratively alternating between
updating SLG demands based on the optimized section and re-optimizing the
section until de.
371
•
The AASHTO LRFD design code and SLG model result in live load distribution
factors that are less conservative for both simply supported structures and twospan continuous structures for positive moment demands than they are for
negative moment demands.
10.2.3
Objective 3: Examine Resiliency for Extraneous Demands due to Inherent
Conservatism in Bridge Design Practice
A population of two-span continuous multi-girder bridges were examined for tolerable support
movement due to vertical translation and transverse rotation of abutments and piers.
•
The following minimum tolerable settlements were calculated for the Strength I
limit state for moment demands: (a) The minimum tolerable settlement for
positive moment demand at mid-span with vertical translation of the center pier
was 11.68 in. (b) Minimum tolerable settlement of the abutment was 1.07 in.
with the controlling negative demand location at the center pier. (c) Rotation of
the pier has a minimum tolerable settlement of 11.87 in. with controlling
response at mid-span. (d) Rotation of the abutment has a minimum tolerable
settlement of 1.39 in. with the controlling location over the center pier.
•
The following minimum tolerable settlements were calculated for the Strength I
limit state for shear demands: (a) The minimum tolerable settlement for shear at
mid-span due to vertical translation of the center pier was 7.04 in. (b) Minimum
tolerable settlement of the abutment was 9.44 in. with the controlling demand
location at the center pier. (c) Rotation of the pier has a minimum tolerable
settlement of 3.86 in. with controlling response at mid-span. (d) Rotation of the
abutment has a minimum tolerable settlement of 3.74 in. with the controlling
location over the center pier.
372
•
The following minimum tolerable settlements were calculated for the Service II
limit state for moment demands: (a) The minimum tolerable settlement for
positive moment at mid-span due to vertical translation of the center pier was
7.84 in. (b) Minimum tolerable settlement of the abutment was 3.16 in. with the
controlling negative moment demand location at the center pier. (c) Rotation of
the pier has a minimum tolerable settlement of 8.78 in. with controlling response
at mid-span. (d) Rotation of the abutment has a minimum tolerable settlement of
3.74 in. with the controlling location over the center pier.
10.2.4
Objective 4: Development of a Streamlined Parameter Estimation Tool
FE models were used to develop controlled and simulated “experimental” data to investigate the
response of deterministic updating of global unknown parameters and local known parameters
to changes in structural characteristics. Nonstructural mass elements were redistributed among
deck zones in various configurations to determine the source of error.
Global structural
characteristics, such as total loss of composite action were investigated as well as the regional loss
of composite action along two girders. The following conclusions may be drawn from these two
case studies:
•
Use of deterministic updating of known parameters by local mass redistribution
is a suitable method for the investigation of error screening models that were
updated for global unknown parameters
•
Use of this method may also be used to pinpoint local anomalous behaviors that
may indicate global parameter estimation or a chosen model form is
inappropriate for sufficient investigation into bridge performance evaluation
373
10.3
Future Work
The following are suggestions for future work based on the research presented herein:
1.
Determine the extent to which the evolution of the AASHTO deign codes effect
load ratings of existing structures developed under previous design codes.
2.
Use existing infrastructure databases to influence the developed probability
densities of bias and variability of the SLG design code. Determine whether
structures at the tails of the population distribution exist in practice.
3.
Use the software tools developed as part of this research to investigate the
dynamic properties of the existing infrastructure population. Investigate the
effects of future developments to the design code on the dynamic characteristics
of structures.
4.
Utilize the tools developed as part of this research to develop maintenance
prioritization schemes to assist infrastructure stakeholders in targeting structures
that are most susceptible to unforeseen environmental inputs.
Investigate
methods that can use the information learned from this research in targeting the
neediest structures in light of limited infrastructure budgets.
5.
Use the software developed as part of this research in concert with modern
geographic information system technology to better understand the resilience of
roadway networks to catastrophic events.
6.
Further investigate the use of mass redistribution techniques for model error
checking.
374
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Appendix A. Support Settlement Sensitivity
A.1
Total Composite Section Stress: Shear Deformation Off – Barrier and Sidewalk
Stiffness On
Figure A.1. Effect of Deck Strength on Total Composite Section Stress
380
Figure A.2. Effect of Deck Thickness on Total Composite Section Stress
Figure A.3. Effect of Girder Spacing on Total Composite Section Stress
381
Figure A.4. Effect of Span Length on Total Composite Section Stress
Figure A.5. Effect of Span Length Normalized by Length on Total Composite Section
Stress
382
Figure A.6. Effect of Skew Angle on Total Composite Section Stress
Figure A.7. Effect of Span Length to Girder Depth Ratio on Total Composite Section
Stress
383
A.2
Total Composite Section Stress: Shear Deformation Off – Barrier and Sidewalk
Stiffness Off
Figure A.8. Effect of Deck Strength on Total Composite Section Stress
384
Figure A.9. Effect of Deck Thickness on Total Composite Section Stress
Figure A.10. Effect of Girder Spacing on Total Composite Section Stress
385
Figure A.11. Effect of Span Length Normalized by Length on Total Composite Section
Stress
Figure A.12. Effect of Span Length on Total Composite Section Stress
386
Figure A.13. Effect of Skew Angle on Total Composite Section Stress
Figure A.14. Effect of Span Length to Girder Depth Ratio on Total Composite Section
Stress
387
A.3
Deck Stress: Shear Deformation Off – Barrier and Sidewalk Stiffness On
Figure A.15. Effect of Deck Strength on Deck Stress
388
Figure A.16. Effect of Deck Thickness on Deck Stress
Figure A.17. Effect of Girder Spacing on Deck Stress
389
Figure A.18. Effect of Span Length on Deck Stress
390
Figure A.19. Effect of Span Length on Deck Stress
Figure A.20. Effect of Skew Angle on Deck Stress
391
Figure A.21. Effect of Span Length to Girder Depth Ratio on Deck Stress
392
A.4
Deck Stress: Shear Deformation Off – Barrier and Sidewalk Stiffness Off
Figure A.22. Effect of Deck Strength on Deck Stress
393
Figure A.23. Effect of Deck Thickness on Deck Stress
Figure A.24. Effect of Girder Spacing on Deck Stress
394
Figure A.25. Effect of Span Length on Deck Stress
395
Figure A.26. Effect of Span Length Normalized by Length on Deck Stress
396
A.5
Vertical Reaction at the Support: Shear Deformation Off – Barrier and Sidewalk
Stiffness On
Figure A.27. Effect of Deck Strength on Vertical Support Reaction
397
Figure A.28. Effect of Deck Thickness on Vertical Support Reaction
Figure A.29. Effect of Girder Spacing on Vertical Support Reaction
398
Figure A.30. Effect of Span Length on Vertical Support Reaction
399
Figure A.31. Effect of Span Length Normalized by Length on Vertical Support Reaction
Figure A.32. Effect of Skew Angle on Vertical Support Reaction
400
Figure A.33. Effect of Span Length to Girder Depth Ratio on Vertical Support Reaction
401
A.6
Vertical Reaction at the Support: Shear Deformation Off – Barrier and Sidewalk
Stiffness Off
Figure A.34. Effect of Deck Strength on Vertical Support Reaction
402
Figure A.35. Effect of Deck Thickness on Vertical Support Reaction
Figure A.36. Effect of Girder Spacing on Vertical Support Reaction
403
Figure A.37. Effect of Span Length on Vertical Support Reaction
404
Figure A.38. Effect of Span Length Normalized by Length on Vertical Support Reaction
Figure A.39. Effect of Skew Angle on Vertical Support Reaction
405
Figure A.40. Effect of Span Length to Girder Depth Ratio on Vertical Support Reaction
406
Appendix B. Supplemental Material to the Invest igation of Bias
in the AASHTO Si ngle Li ne-Girder Model
B.1
Finite Element Rating Controlling Girder
B.1.1
Service II Limit State Including Out of Plane Bending
Figure B.41. Frequency of Finite Element LRFR Service II Rating Controlling Girder with
Inclusion of Out of Plane Moment
407
Figure B.42. Frequency of Finite Element LRFR Service II Rating Controlling Girder Order
from Center Girder with Inclusion of Out of Plane Moment
408
B.2
Finite Element Ratings – Nominal Diaphragm Stiffness
B.2.1
Service II Limit State Including Out of Plane Bending
Figure B.43. Frequency of FE LRFR Service II Rating Including Out of Plane Bending
409
Figure B.44. Frequency of Ratio of Service II FE Rating Factor to SLG Rating Factor
Including Out of Plane Bending
410
Figure B.45. Frequency of Ratio of Service II FE Rating Factor to SLG Rating Factor
Including Out of Plane Bending for Interior Girders
411
Figure B.46. Frequency of Ratio of Service II FE Rating Factor to SLG Rating Factor for
Exterior Girders Including Out of Plane Bending
412
B.3
Finite Element Ratings – 10x Nominal Diaphragm Stiffness
B.3.1
Strength I Limit State
Figure B.47. Frequency of Finite Element LRFR Strength I Rating without Consideration
of Infinite Fatigue Life Design Criteria
413
Figure B.48. Frequency of Ratio of LRFR Strength I Finite Element Rating to Single LineGirder Rating without Consideration of Infinite Fatigue Life Design Criteria
414
Figure B.49. Frequency of Ratio of LRFR Interior Girder Strength I Finite Element Rating
to Single Line-Girder Rating without Consideration of Infinite Fatigue Life Design
Criteria
415
Figure B.50. Frequency of Ratio of LRFR Exterior Girder Strength I Finite Element Rating
to Single Line-Girder Rating without Consideration of Infinite Fatigue Life Design
Criteria
416
B.3.2 Strength I Limit State without Infinite Fatigue Life Design Criteria
Figure B.51. Frequency of Finite Element LRFR Strength I Rating
417
Figure B.52. Frequency of Ratio of LRFR Strength I Finite Element Rating to Single LineGirder Rating
418
Figure B.53. Frequency of Ratio of LRFR Strength I Finite Element Rating to Single LineGirder Rating for Interior Girders
419
Figure B.54. Frequency of Ratio of LRFR Strength I Finite Element Rating to Single LineGirder Rating for Exterior Girders
420
B.3.3 Service II Limit State
Figure B.55. Frequency of Finite Element LRFR Service II Rating
421
Figure B.56. Frequency of Ratio of LRFR Service II Finite Element Rating to Single LineGirder Rating
422
Figure B.57. Frequency of Ratio of LRFR Service II Finite Element Rating to Single LineGirder Rating for Interior Girders
423
Figure B.58. Frequency of Ratio of LRFR Service II Finite Element Rating to Single LineGirder Rating for Exterior Girders
424
B.3.4 Service II Limit State without Infinite Fatigue Life Design Criteria
Figure B.59. Frequency of Finite Element LRFR Service II Rating
425
Figure B.60. Frequency of Ratio of LRFR Service II Finite Element Rating to Single LineGirder Rating
426
Figure B.61. Frequency of Ratio of LRFR Service II Finite Element Rating to Single LineGirder Rating for Interior Girders
427
Figure B.62. Frequency of Ratio of LRFR Service II Finite Element Rating to Single LineGirder Rating for Exterior Girders
428
B.4
Finite Element Ratings – 30x Nominal Diaphragm Stiffness
B.4.1
Strength I Limit State
Figure B.63. Frequency of Finite Element LRFR Strength I Rating without Consideration
of Infinite Fatigue Life Design Criteria
429
Figure B.64. Frequency of Ratio of LRFR Strength I Finite Element Rating to Single LineGirder Rating without Consideration of Infinite Fatigue Life Design Criteria
430
Figure B.65. Frequency of Ratio of LRFR Interior Girder Strength I Finite Element Rating
to Single Line-Girder Rating without Consideration of Infinite Fatigue Life Design
Criteria
431
Figure B.66. Frequency of Ratio of LRFR Exterior Girder Strength I Finite Element Rating
to Single Line-Girder Rating without Consideration of Infinite Fatigue Life Design
Criteria
432
B.4.2 Strength I Limit State without Infinite Fatigue Life Design Criteria
Figure B.67. Frequency of Finite Element LRFR Strength I Rating
433
Figure B.68. Frequency of Ratio of LRFR Strength I Finite Element Rating to Single LineGirder Rating
434
Figure B.69. Frequency of Ratio of LRFR Strength I Finite Element Rating to Single LineGirder Rating for Interior Girders
435
Figure B.70. Frequency of Ratio of LRFR Strength I Finite Element Rating to Single LineGirder Rating for Exterior Girders
436
B.4.3 Service II Limit State
Figure B.71. Frequency of Finite Element LRFR Service II Rating
437
Figure B.72. Frequency of Ratio of LRFR Service II Finite Element Rating to Single LineGirder Rating
438
Figure B.73. Frequency of Ratio of LRFR Service II Finite Element Rating to Single LineGirder Rating for Interior Girders
439
Figure B.74. Frequency of Ratio of LRFR Service II Finite Element Rating to Single LineGirder Rating for Exterior Girders
440
B.4.4 Service II Limit State without Infinite Fatigue Life Design Criteria
Figure B.75. Frequency of Finite Element LRFR Service II Rating
441
Figure B.76. Frequency of Ratio of LRFR Service II Finite Element Rating to Single LineGirder Rating
442
Figure B.77. Frequency of Ratio of LRFR Service II Finite Element Rating to Single LineGirder Rating for Interior Girders
443
Figure B.78. Frequency of Ratio of LRFR Service II Finite Element Rating to Single LineGirder Rating for Exterior Girders
444
B.5
Bivariate Analysis of Ratio of FE and SLG Rating Factors
B.5.1
Strength I Limit State
Figure B.79. Ratio of LRFR Strength I FE Rating to SLG Rating as a Function of Width
445
Figure B.80. Ratio of LRFR Strength I FE Rating to SLG Rating as a Function of Girder
Spacing
446
Figure B.81. Ratio of LRFR Strength I FE Rating to SLG Rating as a Function of Span
Length to Girder Depth Ratio
447
Figure B.82. Ratio of LRFR Strength I FE Interior Girder Rating to SLG Interior Girder
Rating as a Function of Interior Girder Distribution Factor
448
B.5.2 Service II Limit State
Figure B.83. Ratio of LRFR Service II FE Rating to SLG Rating as a Function of Width
449
Figure B.84. Ratio of LRFR Service II FE Rating to SLG Rating as a Function of Girder
Spacing
Figure B.85. Ratio of LRFR Service II FE Rating to SLG Rating as a Function of Span
Length to Girder Depth Ratio
450
Figure B.86. Ratio of LRFR Service II FE Interior Girder Rating to SLG Rating as a
Function of Interior Girder Distribution Factor
451
B.5.3 Service II Limit State with the Inclusion of Out of Plane Bending Moment
Figure B.87. Ratio of LRFR Service II FE Rating with the Inclusion of Out of Plane
Bending to SLG Rating as a Function of Length
Figure B.88. Ratio of LRFR Service II FE Rating with the Inclusion of Out of Plane
Bending to SLG Rating as a Function of Width
452
Figure B.89. Ratio of LRFR Service II FE Rating with the Inclusion of Out of Plane
Bending to SLG Rating as a Function of Skew
453
Figure B.90. Ratio of LRFR Service II FE Rating with the Inclusion of Out of Plane
Bending to SLG Rating as a Function of Girder Spacing
454
Figure B.91. Ratio of LRFR Service II FE Rating with the Inclusion of Out of Plane
Bending to SLG Rating as a Function of Span Length to Girder Depth Ratio
455
Figure B.92. Ratio of LRFR Service II FE Rating with the Inclusion of Out of Plane
Bending to SLG Rating as a Function of Span Length to Girder Depth Ratio
456
Figure B.93. Ratio of LRFR Service II FE Interior Girder Rating with the Inclusion of Out
of Plane Bending to SLG Rating as a Function of Interior Girder Distribution Factor
457
Figure B.94. Ratio of LRFR Service II FE Exterior Girder Rating with the Inclusion of Out
of Plane Bending to SLG Rating as a Function of Exterior Girder Distribution Factor
458
B.6
Bivariate Analysis of Ratio of FE and SLG Moment Demands
B.6.1
Dead Load
Figure B.95. Ratio of Maximum FE Dead Load Demand to SLG Dead Load Demand as a
Function of Width
459
Figure B.96. Ratio of Maximum FE Dead Load Demand to SLG Dead Load Demand as a
Function of the Ratio of Span Length to Girder Depth
Figure B.97. Ratio of Maximum FE Dead Load Demand to SLG Dead Load Demand as a
Function of Span Length to Girder Depth Ratio
460
Figure B.98. Ratio of Maximum FE Dead Load Demand to SLG Dead Load Demand as a
Function of Skew Ratio
461
B.6.2 Superimposed Dead Load Moment Demand
Figure B.99. Ratio of Maximum FE Superimposed Dead Load Demand to SLG
Superimposed Dead Load Demand as a Function of Skew
Figure B.100. Ratio of Maximum FE Superimposed Dead Load Demand to SLG
Superimposed Dead Load Demand as a Function of Span Length to Girder Depth Ratio
462
Figure B.101. Ratio of Maximum FE Superimposed Dead Load Demand to SLG
Superimposed Dead Load Demand as a Function of Skew Ratio
B.6.3 Live Load Moment Demand for Interior Girders
Figure B.102. Ratio of Maximum FE Live Load Demand to SLG Live Load Demand as a
Function of Width
463
Figure B.103. Ratio of Maximum FE Live Load Demand to SLG Live Load Demand as a
Function of Girder Spacing
Figure B.104. Ratio of Maximum FE Live Load Demand to SLG Live Load Demand as a
Function of Span Length to Girder Depth Ratio
464
B.6.4 Live Load Moment Demand for Exterior Girders
Figure B.105. Ratio of Maximum FE Live Load Demand to SLG Live Load Demand as a
Function of Width
465
Figure B.106. Ratio of Maximum FE Live Load Demand to SLG Live Load Demand as a
Function of Skew
Figure B.107. Ratio of Maximum FE Live Load Demand to SLG Live Load Demand as a
Function of Girder Spacing
466
Figure B.108. Ratio of Maximum FE Live Load Demand to SLG Live Load Demand as a
Function of Span Length to Girder Depth Ratio
467
Vita
David Robert Masceri Jr. was born in Philadelphia, Pennsylvania on May 3rd, 1982. He attended
the Rochester Institute in Technology and graduated Cum Laude with a Bachelor’s of Fine Arts in
Illustration Photography and a Minor in Philosophy in 2004. He attended Drexel University and
graduated Cum Laude with a Bachelor’s of Science in Structural Engineering 2011. While at
Drexel University he received the distinction of placement on the Dean’s List in all four years of
undergraduate study and was admitted to Chi Epsilon, the Civil Engineering Honor Society, in
2011.
David began his dissertation in 2011 and taught the undergraduate structural analysis laboratory
at Drexel (CIVE 301) in both Winter 2012 and Winter 2013. During graduate study he published
the following: Overview and Preliminary Validation of a Self-Contained Rapid Modal Testing System for
Highway Bridges; Preliminary Validation of a Rapid Modal Testing Prototype for Population –Based
Condition Assessment of Highway Bridges; Rapid structural identification methods for highway bridges:
towards a greater understanding of large populations; Integration of Isolated Sources of Bridge Data Using
Population Modeling; and Rapid Bridge Modal Analysis for Global Structural Assessment.
He received his PhD in Structural Engineering from Drexel University in 2015.
469
Figure B.108. Ratio of Maximum FE Live Load Demand to SLG Live Load Demand as a
Function of Span Length to Girder Depth Ratio
470
Vita
David Robert Masceri Jr. was born in Philadelphia, Pennsylvania on May 3rd, 1982. He attended
the Rochester Institute in Technology and graduated Cum Laude with a Bachelor’s of Fine Arts in
Illustration Photography and a Minor in Philosophy in 2004. He attended Drexel University and
graduated Cum Laude with a Bachelor’s of Science in Structural Engineering 2011. While at
Drexel University he received the distinction of placement on the Dean’s List in all four years of
undergraduate study and was admitted to Chi Epsilon, the Civil Engineering Honor Society, in
2011.
David began his dissertation in 2011 and taught the undergraduate structural analysis laboratory
at Drexel (CIVE 301) in both Winter 2012 and Winter 2013. During graduate study he published
the following: Overview and Preliminary Validation of a Self-Contained Rapid Modal Testing System for
Highway Bridges; Preliminary Validation of a Rapid Modal Testing Prototype for Population –Based
Condition Assessment of Highway Bridges; Rapid structural identification methods for highway bridges:
towards a greater understanding of large populations; Integration of Isolated Sources of Bridge Data Using
Population Modeling; and Rapid Bridge Modal Analysis for Global Structural Assessment.
He received his PhD in Structural Engineering from Drexel University in 2015.