COMPARATIVE LOAD RATING STUDY UNDER LRFR AND LFR
METHODOLOGIES FOR ALABAMA HIGHWAY BRIDGES
Except where reference is made to the work of others, the work described in this thesis is
my own or was done in collaboration with my advisory committee. This thesis does not
include proprietary or classified information.
________________________
Michael Murdock
Certificate of Approval:
________________________ ________________________
Robert W. Barnes Hassan H. Abbas, Chair
Associate Professor Assistant Professor
Civil Engineering Civil Engineering
________________________ ________________________
J. Michael Stallings George T. Flowers
Professor Dean
Civil Engineering Graduate School
COMPARATIVE LOAD RATING STUDY UNDER LRFR AND LFR
METHODOLOGIES FOR ALABAMA HIGHWAY BRIDGES
Michael Brawner Murdock
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
August 10, 2009
iii
COMPARATIVE LOAD RATING STUDY UNDER LRFR AND LFR
METHODOLOGIES FOR ALABAMA HIGHWAY BRIDGES
Michael Brawner Murdock
Permission is granted to Auburn University to make copies of this thesis at its discretion,
upon request of individuals or institutions and at their expense. The author reserves all
publication rights.
________________________
Signature of Author
________________________
Date of Graduation
iv
VITA
Michael Brawner Murdock, son of Harry Mike and Judy Kay (Moore) Murdock,
was born November 21, 1984, in Cornwall, England. He graduated from Apopka High
School with honors in May, 2003. In September of 2003, Michael entered Auburn
University where he received the Bachelor of Science in Civil Engineering in May, 2007.
He married Jennifer Short on December, 16 2006, also a graduate of Auburn University.
Michael entered the graduate school of Auburn University in September, 2007 to seek the
Masters of Science in Civil Engineering, focusing on structural engineering.
v
THESIS ABSTRACT
COMPARATIVE LOAD RATING STUDY UNDER LRFR AND LFR
METHODOLOGIES FOR ALABAMA HIGHWAY BRIDGES
Michael Murdock
Master of Science, August 10, 2009
(B.S., Auburn University, 2007)
336 Typed Pages
Directed by Hassan H. Abbas
Currently, the Alabama Department of Transportation (ALDOT) uses the load
factor rating (LFR) methodology of the American Association of State Highway and
Transportation Officials (AASHTO) Manual for Condition Evaluation (MCE) of Bridges
(1994) in load rating of highway bridges across the state. With the introduction of the
new AASHTO MCE and Load and Resistance Factor Rating (LRFR) of Highway Bridges
(2003), the need arose to assess the impact of implementing the new manual on
ALDOT?s current bridge rating practices. To this end, a comparative study was
performed between ALDOT?s current rating practices utilizing the older LFR
methodology, according to the AASHTO MCE (1994), and the new LRFR methodology.
This comparative study was performed on a representative sample of 95 bridges
from Alabama?s state and county owned bridge inventory at all three primary levels of
vi
LRFR rating: Design, Legal and Permit rating levels. The load models that were utilized
in the rating analysis were the AASHTO design load models, AASHTO standard legal
loads, ALDOT state legal loads, and a sample of ALDOT overweight loads. The bridges
were modeled in AASHTO BridgeWare?s Virtis version 5.6 (2007) and analyzed in
BRASS-GIRDER LRFR and LFR analysis engines (2007). Rating results were
generated for interior and exterior girders of each bridge analyzed as well as for moment
and shear load effects.
The rating data at all three primary levels of rating indicated that the LRFR
methodology produces lower rating factors than the LFR. It was therefore concluded that
adopting the AASHTO MCE LRFR (2003) can have a significant impact on the rating
practices of ALDOT.
Comparisons were additionally made between the LRFR and LFR rating data, at
the Design rating level, in the context of estimated probability of failure for a bridge
based on the Monte Carlo simulation technique. This comparison showed that rating
factors produced under the LRFR methodology have strong correlation to a bridge?s
estimated probability of failure, whereas rating factors under the LFR methodology
showed only sporadic correlation to a bridge?s estimated probabilities of failure.
vii
ACKNOWLEDGMENTS
I would like to sincerely thank Dr Hassan H. Abbas for his continuous support,
encouragement and insights throughout my time working with him at Auburn University.
I would like to thank my committee members Dr. J. Michael Stallings and Dr. Robert W.
Barnes for their time and invaluable advice.
I would like to additionally thank the Alabama Department of Transportation for
their support and guidance. In practical I would like to thank James Boyer, Daniel Jones,
Eric Christie, and George Conner for their help on this project.
I would additionally like to convey my sincere thanks and love for my wife
Jennifer S. Murdock for her never-ending love and support throughout my time at
Auburn University.
The research described herein has been sponsored by Auburn University Highway
research Center (Dr. Frazier Paker, Director). The findings, opinions, and conclusions
expressed in this thesis are those of the author and not necessarily reflect the views of the
sponsor or others acknowledged herein.
viii
Style manual or journal used The Chicago Manual of Style 15th Edition
Computer software used Microsoft Word, Microsoft Excel, Adobe Photoshop CE,
BRIDGEWares Virtis, Mathcad, MatLab, Microsoft Visual Basic 2008 Express,
ix
TABLE OF CONTENTS
LIST OF TABLES........................................................................................................... xiv
LIST OF FIGURES .......................................................................................................xxiii
Chapter 1 INTRODUCTION ...................................................................................... 1
1.1 Overview........................................................................................................... 1
1.2 Motivation......................................................................................................... 2
1.3 Research Objectives and Scope ........................................................................ 3
1.4 Approach........................................................................................................... 4
1.5 AASHTO Specifications................................................................................... 4
1.6 Thesis Organization and Presentation............................................................... 5
Chapter 2 BACKGROUND ........................................................................................ 7
2.1 Overview of Bridge Rating............................................................................... 7
2.2 Rating Methodologies....................................................................................... 7
2.3 Rating Equations............................................................................................... 8
2.4 LRFR Condition and System Factors ............................................................. 13
2.5 Live Load Factors ........................................................................................... 15
2.6 Load Combinations......................................................................................... 17
2.7 Rating Levels .................................................................................................. 19
2.7.1 LRFR Design Load Rating .................................................................... 22
2.7.2 LRFR Legal Load Rating....................................................................... 22
x
2.7.3 LRFR Permit Load Rating..................................................................... 23
2.8 Posting............................................................................................................. 23
2.9 Live Load Models........................................................................................... 25
2.9.1 Design .................................................................................................... 25
2.9.2 Legal ...................................................................................................... 27
2.9.3 Permit..................................................................................................... 29
2.10 Previous Research........................................................................................... 31
2.10.1 Lichtenstein Consulting Engineers (2001)............................................. 32
2.10.2 Mertz (2005) .......................................................................................... 34
2.10.3 Rogers and J?uregui (2005) ................................................................... 38
Chapter 3 BRIDGE SAMPLE................................................................................... 40
3.1 Determining Bridge Sample .............................................................................. 40
3.1.1 Standard Bridge Sample ........................................................................ 45
3.1.2 Unique Bridge Sample........................................................................... 46
3.1.3 Permit Bridge Sample............................................................................ 47
3.2 Bridge Sample Information................................................................................ 48
Chapter 4 ANALYSIS TOOLS................................................................................. 50
4.1 Analysis Software .............................................................................................. 50
4.1.1 Virtis ...................................................................................................... 50
4.1.2 In-House Rating Tools........................................................................... 52
4.1.2.1 AASHTO Rating Example Comparisons ........................................... 53
4.1.2.2 Output Sorting Programs .................................................................... 59
Chapter 5 RATING RESULTS ................................................................................. 62
xi
5.1 Overview............................................................................................................... 62
5.2 Design Level Rating Results................................................................................. 66
5.2.1 Standard Bridges.................................................................................... 66
5.2.2 Unique Bridges ...................................................................................... 77
5.2.3 Combined Sample Comparison ............................................................. 85
5.2.4 Summary................................................................................................ 87
5.3 Legal Load Rating Results.................................................................................... 88
5.3.1 AASHTO Load Models and ALDOT Legal Loads Comparison .......... 89
5.3.2 Standard Bridge Sample ........................................................................ 96
5.3.2.1 L? Bounding Study Results................................................................ 97
5.3.2.2 ?c and ?s Bounding Study Results .................................................... 101
5.3.2.3 Varying ADTT Bounding Study Results.......................................... 106
5.3.3 Unique Bridge Sample......................................................................... 108
5.3.3.1 Overall Summary.............................................................................. 109
5.3.3.2 Bridge Age........................................................................................ 125
5.3.3.3 Span Length and Girder Spacing ...................................................... 129
5.3.3.4 LRFR Load Posting Recommendations............................................ 132
5.3.4 Summary.............................................................................................. 137
5.4 Permit Load Rating............................................................................................. 138
5.4.1 Permit Bridge Sample.......................................................................... 138
5.4.2 Permit Rating Results .......................................................................... 140
5.4.3 Summary.............................................................................................. 155
5.5 Analysis of Rating Results.................................................................................. 156
xii
Chapter 6 BRIDGE RELIABILITY........................................................................ 166
6.1 Introduction................................................................................................... 166
6.2 Background Information............................................................................... 166
6.2.1 Analysis Method ..................................................................................... 172
6.2.2 Previous Research................................................................................... 174
6.3 Analysis Tools .............................................................................................. 177
6.4 Results........................................................................................................... 181
6.5 Summary and Conclusion............................................................................. 193
Chapter 7 Summary, Conclusions and Recommendations...................................... 195
7.1 Summary............................................................................................................. 195
7.2 Conclusions......................................................................................................... 195
7.3 Recommendations............................................................................................... 197
REFERENCES ............................................................................................................... 199
APPENDICES ................................................................................................................ 201
APPENDIX A: Sample Distributions............................................................................ 202
APPENDIX B1: Slab Bridges Mathcad File ................................................................. 215
APPENDIX B2: Example Problem A1: Steel I Girder ................................................. 220
APPENDIX B3: Example Problem A2: Reinforced Concrete Tee Beam..................... 229
APPENDIX B4: Example Problem A3: Prestressed Concrete I Girder........................ 234
APPENDIX B5: Example Problem A2: Results Summary........................................... 239
APPENDIX B6: Example Problem A3: Results Summary........................................... 241
APPENDIX C.1: Design Inventory Standard Bridge Rating Data................................ 243
APPENDIX C.2: Design Inventory Unique Bridge Rating Data .................................. 256
xiii
APPENDIX D: ALDOT Legal Load Rating Data......................................................... 269
APPENDIX E: ALDOT Legal Load Posting Data........................................................ 282
APPENDIX F1: MatLAB Beta Analysis Program........................................................ 290
APPENDIX F2: Excel Beta Analysis Program ............................................................. 296
APPENDIX F3: Percent Difference Between 1 Million and 10 Million Simulations .. 300
xiv
LIST OF TABLES
Table 2 - 1: Differences Between the LRFR and LFR .................................................... 11
Table 2 - 2: Recommended Condition Factor Values According to AASHTO MCE
LRFR (2003)............................................................................................................. 13
Table 2 - 3: Recommend System Factor Values According to AASHTO MCE
LRFR (2003)............................................................................................................. 15
Table 2 - 4: Live Load Factors as a Function of ADTT (AASHTO 2003) ...................... 16
Table 2 - 5: Live Load Factors for Permit Loads Based on Bridge?s ADTT
(AASHTO 2003)....................................................................................................... 17
Table 2 - 6: Rating Ratio Results From Lichtenstein Consulting Engineers (2001) ....... 33
Table 2 - 7: Controlling Load Effect Data From Lichtenstein Consulting
Engineers (2001)....................................................................................................... 34
Table 2 - 8: Virtis 5.1 Default LRFR Factors (Mertz 2005)............................................ 36
Table 2 - 9: Rating Ratio Results Rrom Dennis Mertz (2005) ........................................ 37
Table 3 - 1: Material Type Distribution of the Reduced SCOMB Database 44
Table 3 - 2: Material Type Distribution of Standard Bridge Sample .............................. 45
Table 3 - 3: Material Type Distribution of Unique Bridge Sample................................. 46
Table 3 - 4: Structural System Type Breakdown for each Material Type....................... 47
Table 4 - 1: Description of AASHTO MCE Example Bridges (AASHTO 2003)........... 53
Table 4 - 2: Steel I-Girder Example Eead Load Results.................................................. 55
xv
Table 4 - 3: Steel I-Girder Example Live Load Moment Results.................................... 56
Table 4 - 4: Steel I-Girder Example Live Load Shear Results ........................................ 57
Table 4 - 5: Steel I-Girder Example Capacity Comparison............................................. 57
Table 4 - 6: Steel I-Girder Example Rating Comparison ................................................ 58
Table 5 - 1: LRFR Rating Factors Generated for the Standard Bridge Sample at the
Design Inventory Rating Level................................................................................. 68
Table 5 - 2: LFR Rating Factors Generated for the Standard Bridge Sample at the Design
Inventory Rating Level ............................................................................................. 69
Table 5 - 3: Mean and Standard Deviation Data at the Design Inventory Level for the
Standard Bridge Sample ? Interior Girder Moment Rating Data ............................. 72
Table 5 - 4: Mean and Standard Deviation Data at the Design Inventory Level for the
Standard Bridge Sample ? Interior Girder Shear Rating Data.................................. 73
Table 5 - 5: Mean and Standard Deviation Data at the Design Inventory Level for the
Standard Bridge Sample ? Exterior Girder Moment Rating Data ............................ 73
Table 5 - 6: Mean and Standard Deviation Data at the Design Inventory Level for the
Standard Bridge Sample ? Exterior Girder Shear Rating Data................................. 74
Table 5 - 7: Mean and Standard Deviation Data at the Design Inventory level for the
Standard Bridge Sample ? Interior to Exterior LRFR Moment Rating Comparison 75
Table 5 - 8: Controlling Load Effect Comparison, Design Inventory Level for the
Standard Bridge Sample ........................................................................................... 76
Table 5 - 9: LRFR Rating Factors Generated for the Unique Bridge Sample at the Design
Inventory Rating Level ............................................................................................. 77
xvi
Table 5 - 10: LFR Rating Factors Generated for the Unique Bridge Sample at the Design
Inventory Rating Level ............................................................................................. 78
Table 5 - 11: Mean and Standard Deviation Data at the Design Inventory Level for the
Unique Bridge Sample ? Interior Girder Moment Rating Data................................ 82
Table 5 - 12: Mean and Standard Deviation Data at the Design Inventory Level for the
Unique Bridge Sample ? Exterior Girder Moment Rating Data............................... 82
Table 5 - 13: Mean and Standard Deviation Data at the Design Inventory Level for the
Unique Bridge Sample ? Interior Girder Shear Rating Data .................................... 83
Table 5 - 14: Mean and Standard Deviation Data at the Design Inventory Level for the
Unique Bridge Sample ? Exterior Girder Shear Rating Data ................................... 84
Table 5 - 15: Controlling Load Effect Comparison, Design Inventory Level for the
Unique Bridge Sample.............................................................................................. 85
Table 5 - 16: Controlling Load Effect and Rating Methodology at the Design Inventory
Level ......................................................................................................................... 87
Table 5 - 17: Controlling AASHTO Legal Loads ........................................................... 90
Table 5 - 18: Controlling ALDOT Legal Loads .............................................................. 90
Table 5 - 19: Standard Bridge Varying ADTT Values.................................................. 107
Table 5 - 20: Standard Bridge Sample Rating Results, Legal level, Varying ADTT
Structural System Type........................................................................................... 108
Table 5 - 21: Controlling ALDOT Truck Comparison at Legal Level for the Unique
Bridge Sample......................................................................................................... 109
Table 5 - 22: LRFR Rating Factors Generated for the Unique Bridge Sample at the Legal
Load Rating Level................................................................................................... 111
xvii
Table 5 - 23: LFR Rating Factors Generated for the Unique Bridge Sample at the Legal
Load Rating Level................................................................................................... 112
Table 5 - 24: Mean and Standard Deviation Data at the Legal Load Level for the Unique
Bridge Sample ? Interior Girder Moment Rating Data .......................................... 120
Table 5 - 25: Mean and Standard Deviation Data at the Legal Load Level for the Unique
Bridge Sample ? Exterior Girder Moment Rating Data ......................................... 121
Table 5 - 26: Mean and Standard Deviation Data at the Legal Load Level for the Unique
Bridge Sample ? Interior Girder Shear Rating Data............................................... 121
Table 5 - 27: Mean and Standard Deviation Data at the Legal Load Level for the Unique
Bridge Sample ? Exterior Girder Shear Rating Data.............................................. 122
Table 5 - 28: Controlling Load Effect Comparison, Legal Load Level for the Unique
Bridge Sample......................................................................................................... 123
Table 5 - 29: Controlling Load Effect and Rating Methodology at the Legal Load Level
................................................................................................................................. 125
Table 5 - 30: Summary ALDOT Legal Loads Weights................................................. 134
Table 5 - 31: ALDOT Legal Loads LRFR Posting Weights ......................................... 134
Table 5 - 32: ALDOT Legal Loads LFR Posting Weights............................................ 135
Table 5 - 33: Material Type Breakdown of the Permit Bridge Sample......................... 140
Table 5 - 34: Controlling Permit Vehicles..................................................................... 141
Table 5 - 35: LRFR Rating Factors Generated for the Permit Bridge Sample at the Permit
Rating Level, Part 1 ................................................................................................ 142
Table 5 - 36: LRFR Rating Factors Generated for the Permit Bridge Sample at the Permit
Rating Level, Part 2 ................................................................................................ 143
xviii
Table 5 - 37: LFR Rating Factors Generated for the Permit Bridge Sample at the Permit
Rating Level, Part 1 ................................................................................................ 144
Table 5 - 38: LFR Rating Factors Generated for the Permit Bridge Sample at the Permit
Rating Level, Part 2 ................................................................................................ 145
Table 5 - 39: Mean and Standard Deviation Data at the Permit Level for the Permit
Bridge Sample ? Interior Girder Moment Rating Data .......................................... 150
Table 5 - 40: Mean and Standard Deviation Data at the Permit Level for the Permit
Bridge Sample ? Exterior Girder Moment Rating Data ......................................... 150
Table 5 - 41: Mean and Standard Deviation Data at the Permit Level for the Permit
Bridge Sample ? Interior Girder Shear Rating Data............................................... 151
Table 5 - 42: Mean and Standard Deviation Data at the Permit Level for the Permit
Bridge Sample ? Exterior Girder Shear Rating Data.............................................. 152
Table 5 - 43: Mean and Standard Deviation Data at the Permit level for the Permit
Bridge Sample ? Interior to Exterior LRFR Moment Rating Comparison............. 152
Table 5 - 44: Controlling Load Effect Comparison, Permit Level for the Permit Bridge
Sample..................................................................................................................... 153
Table 5 - 45: Controlling Load Effect and Rating Methodology at the Permit Level for
the Permit Bridge Sample ....................................................................................... 155
Table 6 - 1: Bias Factors and Coefficients of Variation Used in Reliability Analysis
(Nowak 1999) ......................................................................................................... 180
Table 6 - 2: Probability of Failure Methods Comparison.............................................. 182
Table 6 - 3: Ten Repetitive One-Million-Run Simulations Comparison....................... 183
xix
Table A - 1: Structural System Type Distribution for Reinforced Concrete Simply
Supported Bridges According to SCOMB Distribution ......................................... 202
Table A - 2: Structural System Type Distribution for Reinforced Concrete Continuously
Supported Bridges According to SCOMB Distribution ......................................... 203
Table A - 3: Structural System Type Distribution for Steel Simply Supported Bridges
According to SCOMB Distribution ........................................................................ 203
Table A - 4: Structural System Type Distribution for Steel Continuously Supported
Bridges According to SCOMB Distribution........................................................... 204
Table A - 5: Structural System Type Distribution for Prestressed Concrete Simply
Supported Bridges According to SCOMB Distribution ......................................... 204
Table A - 6: Structural System Type Distribution for Prestressed Concrete Continuously
Supported Bridges According to SCOMB Distribution ......................................... 205
Table A - 7: Proposed Unique Bridge Sample Part 1 .................................................... 206
Table A - 8: Proposed Unique Bridge Sample Part 2 .................................................... 207
Table A - 9: Standard Bridge Sample Bridge Descriptions........................................... 208
Table A - 10: Unique Bridge Sample Matrix................................................................. 208
Table A - 11: Additional Sample Information Table 1.................................................. 210
Table A - 12: Additional Sample Information Table 2.................................................. 211
Table A - 13: Additional Sample Information Table 3.................................................. 212
Table A - 14: Additional Sample Information Table 4.................................................. 213
Table C1 - 1: Design Inventory Standard Bridge Output, Exterior Girder LRFR Part 1
................................................................................................................................. 244
xx
Table C1 - 2: Design Inventory Standard Bridge Output, Exterior Girder LRFR Part 2
................................................................................................................................. 245
Table C1 - 3: Design Inventory Standard Bridge Output, Exterior Girder LRFR Part 3
................................................................................................................................. 246
Table C1 - 4: Design Inventory Standard Bridge Output, Exterior Girder LFR Part 1. 247
Table C1 - 5: Design Inventory Standard Bridge Output, Exterior Girder LFR Part 2. 248
Table C1 - 6: Design Inventory Standard Bridge Output, Exterior Girder LFR Part 3. 249
Table C1 - 7: Design Inventory Standard Bridge Output, Interior Girder LRFR Part 1 250
Table C1 - 8: Design Inventory Standard Bridge Output, Interior Girder LRFR Part 2 251
Table C1 - 9: Design Inventory Standard Bridge Output, Interior Girder LRFR Part 3 252
Table C1 - 10: Design Inventory Standard Bridge Output, Interior Girder LFR Part 1 253
Table C1 - 11: Design Inventory Standard Bridge Output, Interior Girder LFR Part 2 254
Table C1 - 12: Design Inventory Standard Bridge Output, Interior Girder LFR Part 3 255
Table C2 - 1: Design Inventory Unique Bridge Output, Exterior Girder LRFR Part 1. 257
Table C2 - 2: Design Inventory Unique Bridge Output, Exterior Girder LRFR Part 2. 258
Table C2 - 3: Design Inventory Unique Bridge Output, Exterior Girder LRFR Part 3. 259
Table C2 - 4: Design Inventory Unique Bridge Output, Exterior Girder LFR Part 1 ... 260
Table C2 - 5: Design Inventory Unique Bridge Output, Exterior Girder LFR Part 2 ... 261
Table C2 - 6: Design Inventory Unique Bridge Output, Exterior Girder LFR Part 3 ... 262
Table C2 - 7: Design Inventory Unique Bridge Output, Interior Girder LRFR Part 1.. 263
Table C2 - 8: Design Inventory Unique Bridge Output, Interior Girder LRFR Part 2.. 264
Table C2 - 9: Design Inventory Unique Bridge Output, Interior Girder LRFR Part 3.. 265
Table C2 - 10: Design Inventory Unique Bridge Output, Interior Girder LFR Part 1 .. 266
xxi
Table C2 - 11: Design Inventory Unique Bridge Output, Interior Girder LFR Part 2 .. 267
Table C2 - 12: Design Inventory Unique Bridge Output, Interior Girder LFR Part 3 .. 268
Table D - 1: Legal Load Level Unique Bridge Output, Exterior Girder Flexure, LRFR
Part 1 ....................................................................................................................... 270
Table D - 2: Legal Load Level Unique Bridge Output, Exterior Girder Flexure, LRFR
Part 2 ....................................................................................................................... 270
Table D - 3: Legal Load Level Unique Bridge Output, Exterior Girder Flexure, LRFR
Part 3 ....................................................................................................................... 271
Table D - 4: Legal Load Level Unique Bridge Output, Exterior Girder Flexure, LFR
Part 1 ....................................................................................................................... 272
Table D - 5: Legal Load Level Unique Bridge Output, Exterior Girder Flexure, LFR
Part 2 ....................................................................................................................... 273
Table D - 6: Legal Load Level Unique Bridge Output, Exterior Girder Flexure, LFR
Part 3 ....................................................................................................................... 274
Table D - 7: Legal Load Level Unique Bridge Output, Interior Girder Flexure, LRFR
Part 1 ....................................................................................................................... 276
Table D - 8: Legal Load Level Unique Bridge Output, Interior Girder Flexure, LRFR
Part 2 ....................................................................................................................... 276
Table D - 9: Legal Load Level Unique Bridge Output, Interior Girder Flexure, LRFR
Part 3 ....................................................................................................................... 277
Table D - 10: Legal Load Level Unique Bridge Output, Interior Girder Flexure, LFR
Part 1 ....................................................................................................................... 278
xxii
Table D - 11: Legal Load Level Unique Bridge Output, Interior Girder Flexure, LFR
Part 2 ....................................................................................................................... 279
Table D - 12: Legal Load Level Unique Bridge Output, Interior Girder Flexure, LFR
Part 3 ....................................................................................................................... 280
Table E - 1: Legal Load Posting Data for Exterior Girders, LRFR Moment Data........ 282
Table E - 2: Legal Load Posting Data for Exterior Girders, LRFR Shear Data ............ 282
Table E - 3: Legal Load Posting Data for Interior Girders, LRFR Moment Data......... 283
Table E - 4: Legal Load Posting Data for Interior Girders, LRFR Shear Data ............. 284
Table E - 5: Legal Load Posting Data for Exterior Girders, LFR Moment Data .......... 285
Table E - 6: Legal Load Posting Data for Exterior Girders, LFR Shear Data............... 286
Table E - 7: Legal Load Posting Data for Interior Girders, LFR Moment Data............ 287
Table E - 8: Legal Load Posting Data for Interior Girders, LFR Shear Data ................ 288
Table F3 - 1: Exterior Girder Moment Standard Bridge Sample................................... 301
Table F3 - 2: Exterior Girder Moment Unique Bridge Sample ..................................... 301
Table F3 - 3: Exterior Girder Shear Standard Bridge Sample ....................................... 302
Table F3 - 4: Exterior Girder Shear Unique Bridge Sample ......................................... 303
Table F3 - 5: Interior Girder Moment Standard Bridge Sample.................................... 304
Table F3 - 6: Interior Girder Moment Unique Bridge Sample ...................................... 305
Table F3 - 7: Interior Girder Shear Standard Bridge Sample ........................................ 306
Table F3 - 8: Interior Girder Shear Unique Bridge Sample........................................... 307
xxiii
LIST OF FIGURES
Figure 2 ? 1: Load and Resistance Factor Rating Flow Chart From the AASHTO ......... 21
Figure 2 - 2: AASHTO Legal Load Models (AASHTO 1994) ....................................... 28
Figure 2 - 3: ALDOT Legal Load Models....................................................................... 29
Figure 2 - 4: ALDOT permit load models part 1............................................................. 30
Figure 2 - 5: ALDOT permit load models part 2.............................................................. 31
Figure 3 - 1: Structural System Types ............................................................................. 42
Figure 3 - 2: Virtis C - Channel Cross Section Conversion to T - Section...................... 42
Figure 4 - 1: Virtis Output Sorter User Interface (Murdock? 2008) .............................. 60
Figure 5 - 1: LFR Versus LRFR Region Plot 1 ............................................................... 63
Figure 5 - 2: LFR Versus LRFR Region Plot 2 ............................................................... 64
Figure 5 - 3: LFR Versus LRFR Region Plot 3 ............................................................... 65
Figure 5 - 4: Moment Rating Factor Comparison at the Design Inventory Level for the
Standard Bridge Sample ........................................................................................... 70
Figure 5 - 5: Shear Rating Factor Comparison at the Design Inventory Level for the
Standard Bridge Sample ........................................................................................... 71
Figure 5 - 6: Moment Rating Factor Comparison at the Design Inventory Level for
Unique Bridge Sample.............................................................................................. 80
Figure 5 - 7: Shear Rating Factor Comparison at the Design Inventory Level for Unique
Bridge Sample........................................................................................................... 81
xxiv
Figure 5 - 8: Controlling Rating Factor Comparisons at the Design Inventory Level .... 86
Figure 5 - 9: LRFR Moment Rating Factor under AASHTO and ALDOT Legal Loads 91
Figure 5 - 10: LRFR Shear Rating Factor under AASHTO and ALDOT Legal Loads .. 92
Figure 5 - 11: LRFR Moment Rating Factor under HL-93 Load Model and ALDOT
Legal Loads............................................................................................................... 94
Figure 5 - 12: LRFR Shear Rating Factor under HL-93 Load Model and ALDOT Legal
Loads......................................................................................................................... 95
Figure 5 - 13: Effect of varying L? on Moment Rating at the Legal Level for Interior
Girders....................................................................................................................... 98
Figure 5 - 14: Effect of varying L? on Shear Rating at the Legal Level for Interior
Girders....................................................................................................................... 99
Figure 5 - 15: Effect of varying L? on Moment Rating at the Legal Level for Exterior
Girders..................................................................................................................... 100
Figure 5 - 16: Effect of varying L? on Shear Rating at the Legal Level for Exterior
Girders..................................................................................................................... 101
Figure 5 - 17: Effect of varying ?c and ?s, on Moment Rating at the Legal Level for
Interior Girders........................................................................................................ 103
Figure 5 - 18: Effect of varying ?c and ?s, on Shear Rating at the Legal Level for Interior
Girders..................................................................................................................... 104
Figure 5 - 19: Effect of varying ?c and ?s, on Moment Rating at the Legal Level for
Exterior Girders ...................................................................................................... 105
xxv
Figure 5 - 20: Effect of varying ?c and ?s, on Shear Rating at the Legal Level for Exterior
Girders..................................................................................................................... 106
Figure 5 - 21: Moment Rating Factor Comparison at the Legal Load Level for the
Unique Bridge Sample............................................................................................ 113
Figure 5 - 22: Moment Rating Factor Material Type Comparison at the Legal Load Level
for the Unique Bridge Sample Interior Girders ...................................................... 115
Figure 5 - 23: Moment Rating Factor Material Type Comparison at the Legal Load Level
for the Unique Bridge Sample Exterior Girders ..................................................... 116
Figure 5 - 24: Shear Rating Factor Comparison at the Legal Load Level for the Unique
Bridge Sample......................................................................................................... 117
Figure 5 - 25: Shear Rating Factor Material Type Comparison at the Legal Load Level
for the Unique Bridge Sample Interior Girders ...................................................... 118
Figure 5 - 26: Shear Rating Factor Material Type Comparison at the Legal Load Level
for the Unique Bridge Sample Exterior Girders ..................................................... 119
Figure 5 - 27: Controlling Rating Factor Comparisons at the Legal Load Level .......... 124
Figure 5 - 28: Bridge Age and Moment Rating Factor Comparison at the Legal Load
Level for the Unique Bridge Sample ...................................................................... 126
Figure 5 - 29: Bridge Age and Moment Rating Factor Comparison, Material Level, at the
Legal Load Level for the Unique Bridge Sample................................................... 127
Figure 5 - 30: Bridge Age and Shear Rating Factor Comparison at the Legal Load Level
for the Unique Bridge Sample ................................................................................ 128
Figure 5 - 31: Bridge Age and Shear Rating Factor Comparison, Material Level, at the
Legal Load Level for the Unique Bridge Sample................................................... 129
xxvi
Figure 5 - 32: Span length and Moment Rating Factor Comparison, Material Level, at
the Legal Load Level for the Unique Bridge Sample............................................. 130
Figure 5 - 33: Girder Spacing and Moment Rating Factor Comparison, Material Level, at
the Legal Load Level for the Unique Bridge Sample............................................. 130
Figure 5 - 34: Span length and LRFR to LFR Ratio Comparison, Material Level, at the
Legal Load Level for the Unique Bridge Sample................................................... 131
Figure 5 - 35: Girder Spacing and LRFR to LFR Ratio Comparison, Material Level, at
the Legal Load Level for the Unique Bridge Sample............................................. 132
Figure 5 - 36: Posting Weight Fraction Compared to Rating Factor............................. 133
Figure 5 - 37: Permit Bridge Sample Diagram .............................................................. 139
Figure 5 - 38: Moment Rating Factor Comparison at the Permit Level for the Permit
Bridge Sample......................................................................................................... 146
Figure 5 - 39: Moment Rating Factor Material Type Comparison at the Permit Level for
the Permit Bridge Sample Interior Girders ............................................................. 147
Figure 5 - 40: Shear Rating Factor Comparison at the Permit Level for the Permit Bridge
Sample..................................................................................................................... 148
Figure 5 - 41: Moment Rating Factor Material Type Comparison at the Permit Level for
the Permit Bridge Sample Interior Girders ............................................................. 149
Figure 5 - 42: Controlling Rating Factor Comparisons at the Permit Level for the Permit
Bridge Sample......................................................................................................... 154
Figure 5 - 43: LRFR to LFR Component Ratio Comparisons for Exterior Girder Moment
Rating Factors ......................................................................................................... 157
xxvii
Figure 5 - 44: LRFR to LFR Capacity Ratio Comparisons for Exterior Girder Moment
Rating Factors ......................................................................................................... 158
Figure 5 - 45: LRFR to LFR Live Load Component Ratio Comparisons for Standard
Bridge Sample Exterior Girder Moment Rating Factors ........................................ 161
Figure 5 - 46: LRFR to LFR Live Load Component Ratio Comparisons for Unique
Bridge Sample Exterior Girder Moment Rating Factors ........................................ 161
Figure 5 - 47: LRFR to LFR Component Ratio Comparisons for Exterior Girder Shear
Rating Factors ......................................................................................................... 163
Figure 5 - 48: LRFR to LFR Capacity Ratio Comparisons for Exterior Girder Shear
Rating Factors ......................................................................................................... 164
Figure 6 - 1: Probability of Failure Depiction (Nowak and Collins 2000) 167
Figure 6 - 2: Graphical Representation of ? (Nowak and Collins 2000)....................... 170
Figure 6 - 3: Relationship Between ? and the Pf............................................................ 171
Figure 6 - 4: Methods of Calculating Probability of Failure (Nowak and Collins 2000)
................................................................................................................................. 174
Figure 6 - 5: Reliability analysis results from Mertz Task 122 Report (2005).............. 176
Figure 6 - 6: Reliability Index compared to Rating Factor from Mertz Task 122
Report (2005).......................................................................................................... 177
Figure 6 - 7: Percent Difference in Estimated Probability of Failure and Beta Between
One Million and Ten Million Run Simulations for Interior Girders in Flexure ..... 184
Figure 6 - 8: Probability of Failure and Rating Factor Comparison, Interior Girders
Moment Load Effect............................................................................................... 186
xxviii
Figure 6 - 9: Probability of Failure and Rating Factor Comparison, Exterior Girders
Moment Load Effect............................................................................................... 187
Figure 6 - 10: Probability of Failure and Rating Factor Comparison, Interior Girders
Shear Load Effect ................................................................................................... 188
Figure 6 - 11: Probability of Failure and Rating Factor Comparison, Exterior Girders
Shear Load Effect ................................................................................................... 189
Figure 6 - 12: ? and Rating Factor Comparison, Interior Girders Moment Load Effect190
Figure 6 - 13: ? and Rating Factor Comparison, Exterior Girders Moment Load Effect
................................................................................................................................. 191
Figure 6 - 14: ? and Rating Factor Comparison, Interior Girders Shear Load Effect ... 192
Figure 6 - 15: ? and Rating Factor Comparison, Exterior Girders Shear Load Effect .. 193
Figure B5 - 1: Example A2 Comparisons Part 1 ........................................................... 239
Figure B5 - 2: Example A2 Comparisons Part 2 ........................................................... 240
Figure B6 - 1: Example A3 Comparisons Part 1 ........................................................... 241
Figure B6 - 2: Example A3 Comparisons Part 2 ........................................................... 242
1
Chapter 1 INTRODUCTION
1.1 Overview
In 1994 the American Association of State Highway and Transportation Officials
(AASHTO) Load and Resistance Factor Design (LRFD) Bridge Design Specifications
was introduced (Minervino et al. 2004). The AASHTO LRFD introduced a new limit
state design philosophy based on structural reliability. The bridge design philosophy of
the time was load factor design (LFD) or allowable stress design (ASD) as found in the
AASHTO Standard Specifications for Highway Bridges (Sivakumar 2007). The main
advantage of the LRFD over ASD and LFD is that it aims to achieve a more uniform
level of reliability in bridge design among the various types of materials and systems
employed (Minervino et al. 2004).
The AASHTO design specifications are intended to provide guidelines for the
design of new bridges. To assist in the evaluation of existing bridges, AASHTO
developed guidelines for bridge condition evaluation as well. This evaluation involves a
process that is often referred to as bridge rating. The specifications for bridge rating are
found in the AASHTO Manual for Condition Evaluation. The second edition of the
AASHTO Manual for Condition Evaluation of Bridges, published in 1994, provides
guidelines for evaluating existing bridges according to the allowable stress and load
factor methodologies (Sivakumar 2007). With the introduction of the new AASHTO
LRFD Bridge Design Specifications, a new methodology of evaluation and rating was
2
needed for consistency with the new limit state design philosophy (Minervino et al.
2004). In March 1997 the National Cooperative Highway Research Program (NCHRP)
Project 12-46 was initiated and resulted in a rating manual based on the load and
resistance factor approach (Lichtenstein 2001). The end result of NCHRP?s Project 12-
46 was the AASHTO Manual for Condition and Evaluation and Load and Resistance
Factor Rating (LRFR) of Highway Bridges, hereafter referred to as AASHTO MCE
LRFR (Minervino et al. 2004).
Currently, the Alabama?s Department of Transportation (ALDOT) uses the
AASHTO MCE (1994) for bridge rating. With the introduction of the new AASHTO
MCE LRFR (2003) ALDOT expressed concern over how the new bridge rating system
would affect state rating practices. In order to address this concern, a comparative bridge
rating study between the existing AASHTO MCE (1994) and the new AASHTO MCE
LRFR (2003) was conducted on a sample of Alabama?s state bridges.
1.2 Motivation
The study described in this thesis is in response to concerns expressed by ALDOT
over how adopting the new AASHTO MCE LRFR (2003) would affect their current
bridge rating practices in regard to legal load posting and the issuance of overweight
permits. For the legal load posting, ALDOT is interested in evaluating how the number
of bridges required to be posted and the degree to which they are posted would change
under the new rating methodology. For the overweight permits, ALDOT is interested in
evaluating how the number of bridges that overweight loads are allowed on will be
affected and how the allowances for overweight permits will be affected.
3
1.3 Research Objectives and Scope
The research described in this thesis compares the two rating methodologies
LRFR and LFR on a select sample of Alabama State and County owned and maintained
bridges. The research objectives for this study can be broken into primary and secondary
objectives
The primary objectives are as follows:
1. Generate and compare LRFR and LFR rating factor results at the Design
Inventory level of rating
2. Generate and compare LRFR and LFR rating factor results at the Legal
load level of rating for AASHTO and ALDOT legal loads and provide
LRFR load postings
3. Generate and compare LRFR and LFR rating factor results at the Permit
level of rating for ten ALDOT permit trucks
The secondary objectives are as follows:
1. Compare the effect of ALDOT state legal loads to the effect of
AASHTO typical legal loads and design load model on the rating results
2. Compare the rating factor results of the LRFR and LFR in the context of
bridge reliability, as discussed in Chapter 6
The research presented within this thesis is limited to the selected sample of
Alabama State and County owned and maintained bridges described in Chapter 3. The
rating factors used in all comparisons within the study were generated through the use of
software and with the assumptions listed in Chapters 2 and 4.
4
1.4 Approach
The approach taken to accomplish the research objectives outlined above can be
broken into the following briefly described tasks:
1. Review previous research comparing the LRFR and LFR methodologies.
2. Select representative bridge samples for use in the rating analysis from
Alabama?s State and County owned and maintained bridge inventory.
3. Develop experience modeling and rating bridges in AASHTO BridgeWare?s
Virtis Version 5.6.0.
4. Model the selected bridge samples in Virtis and rate bridges at the Design,
Legal, and Permit levels of rating.
5. Review and analyze LRFR and LFR results at the Design, Legal, and Permit
levels of rating.
6. Develop LRFR load posting based on Legal level rating results as described in
the AASHTO MCE LRFR (2003).
7. Perform a reliability study on the selected bridge sample and compare to
LRFR and LFR rating results at the Design Inventory level of rating.
8. Prepare final report on the research findings.
1.5 AASHTO Specifications
Several AASHTO publications are referred to in this study. The AASHTO Bridge
Design Specifications used in the study are as follows:
5
1. AASHTO Standard Specification for Highway Bridges 17th Edition, 2002.
This document will be referred to as AASHTO Standard Specifications 2002.
2. AASTHO Load and Resistance Factor Design Bridge Design Specification 4th
Edition, 2007. This document will be referred to as AASHTO LRFD 2007.
The AASHTO Manuals for Condition Evaluation used in the study are as follows:
1. AASHTO Manual for Condition Evaluation of Bridges Second Edition, 1994,
with revisions and interims through 2003. This document will be referred to
as AASHTO MCE 1994.
2. AASTHO Manual for Condition Evaluation and Load and Resistance Factor
Rating of Highway Bridges, 2003 with 2005 interim. This document will be
referred to as AASHTO MCE LRFR 2003.
1.6 Thesis Organization and Presentation
The thesis is organized into seven chapters. Chapter 1 provides an introduction to
the research objectives and the research approach used. Chapter 2 provides the
background information on the two rating methodologies compared in the research, a
listing of the different live load models used at each rating level, and a summary of the
previous comparative research done. Chapter 3 details how the bridge samples used in
the research were selected as well as descriptions of the bridges included in the samples.
Chapter 4 presents an overview of the analysis software used in the study, a detailed
rating example, and a description of the in-house tools developed to aid in the research.
6
Chapter 5 presents the rating results for Design, Legal, and Permit rating levels and the
comparisons and trends found between the LRFR and the LFR. Chapter 6 provides an
introduction to bridge reliability as well as a comparison between LRFR and LFR factors
of reliability at the Design Inventory level of rating. Chapter 7 presents a summary of the
research findings as well as conclusions and recommendations based on the comparative
study.
7
Chapter 2 BACKGROUND
2.1 Overview of Bridge Rating
The purpose of bridge rating is to provide a measure of a bridge?s ability to carry
a given live load in terms of a simple factor, referred to as the rating factor. These bridge
rating factors can be used by bridge owners to aid in decisions about the need for load
posting, bridge strengthening, overweight load allowances, and bridge closures
(AASHTO 2003). The way that these rating factors are calculated depends on the rating
methodology used. The AASHTO MCE (1994) provides guidelines as to how to
calculate rating factors based upon load factor rating and allowable stress rating
methodologies (Minervino et al. 2004). The load factor rating and allowable stress rating
methodologies are commonly referred to as LFR and ASR, respectively. With the
introduction of the AASHTO LRFD 1994, which was based on structural reliability
methods, a new rating methodology was also needed. The AASHTO MCE LRFR was
developed based on the same limit state philosophies as the AASHTO LRFD (Minervino
et al. 2004). The Load and Resistance Factor Rating methodology is more commonly
referred to as the LRFR.
2.2 Rating Methodologies
The basic concept of the load factor rating (LFR) methodology is to analyze a
structure at its ultimate load level under multiples of the actual dead and live loads. The
8
load factors used to accomplish this are specified in the AASHTO MCE (1994) and are
based on engineering judgment and not on statistical studies or probability of failure
(Sivakumar 2007). The factors were developed assuming normal traffic and overload
conditions. The AASHTO MCE (1994), however, does not provide any additional
guidance as to how to adjust the load factors to more accurately reflect actual conditions.
In essence, the load factor methodology represents a ?tried and true approach? to the
rating problem (Sivakumar 2007).
The load and resistance factor rating methodology, LRFR, was developed under
the NCHRP project 12-46 to be a rating methodology consistent in philosophy with the
AASHTO LRFD Bridge Design Specifications in its use of reliability-based limit states
(Lichtenstein 2001). The goal of the design philosophy in the AASHTO LRFD was to
achieve a more uniform level of reliability in bridge design. With the introduction of the
AASHTO MCE LRFR (2003), the new methodology of rating provided a systematic and
flexible approach to bridge rating based on reliability. The LRFR rating philosophy
allows for a realistic assessment of a bridge?s actual safe load capacity as opposed to the
?tried and true approach? in the LFR (Sivakumar 2007).
2.3 Rating Equations
The general load rating equations for both the LFR and LRFR are arranged in the
same way to provide a ratio of the live load capacity of a member to its live load demand.
As shown in Equation 2 - 1.
9
Effect Load Live
Effect Load DeadCapacity Factor Rating ?= Equation 2 - 1
The numerator of each equation represents the live load capacity of a member, the
difference between the factored capacity and the applied factored dead load effect. The
denominator of each equation consists of the factored live load model?s effect. For the
LFR methodology found the in the AASHTO MCE (1994) the rating factor is given as:
)1(2
1
ILA
DACRF
+
?= Equation 2 - 2
where,
RF = Rating factor
C = Factored Capacity
1A = Factor for dead loads
D = Dead load effect
2A = Factor for live load
L = Live load effect
I = Impact factor
For the LRFR methodology found the in the AASHTO MCE LRFR (2003) the
rating factor is given as:
))((
))(())(())((
IMLL
PDWDCCRF
L
PDWDC
+
???=
?
??? Equation 2 ? 3
where,
10
RF = Rating factor
C = Capacity, defined as ?c?s?Rn for the strength limit state and fR
for the service limit states
DC? = LRFD load factor for structural components and attachments
DC = Dead-load effect due to structural components and attachments
DW? = LRFD load factor wearing surface and utilities
DW = Dead-load effect due to wearing surface and utilities
P? = LRFD load factor for permanent loads
P = Permanent loads other than dead loads
L? = Evaluation live-load factor
LL = Live-load effect
IM = Dynamic load allowance
While the general form of both the LRFR and LFR rating equations is the same,
there are several distinct differences between the two, as summarized in Table 2 - 1. The
first difference is the inclusion of two new resistance factors in the LRFR equations: the
condition factor, ?c, which deals with the amount of deterioration a member has
experienced, and the system factor, ?s, which deals with the global structural redundancy
of the bridge (Lichtenstein 2001). The ? sign associated with the permanent loads in
Equation 2 - 3 accounts for the favorable or unfavorable effect that permanent loads can
have on the live load capacity. The resistance for both the LRFR and the LFR is
calculated differently as well. The resistance for the LRFR capacity is calculated
11
according to the LRFD Bridge Design specifications according to the AASHTO MCE
LRFR (2003). The capacity for the LFR is calculated according to the Standard
Specification for Highway Bridges based on the LFD principles according to the
AASHTO MCE (1994).
Table 2 - 1: Differences Between the LRFR and LFR
Another difference between the two equations is that the LRFR equation separates
the dead loads into two parts: structural components / attachments and the wearing
surface. This allows for unique load factors to be applied to the each of the categories
based on their variable statistics (Lichtenstein 2001). Under the LFR load factor, 1A , was
specified as 1.3 for all dead loads (AASHTO 1994). In the LRFR the load factor DC? is
specified as 1.25 and DW? as 1.5 unless the in-place thickness of the wearing surface can
12
be verified by field measurements. Then, the factor DW? can be reduced to 1.25
(AASHTO 2003).
The live load factors for the two methodologies are also different. The 2A factor
in the LFR is fixed at 2.17 for Inventory rating and 1.3 for Operating rating for all traffic
conditions and vehicle loadings. The differences between these rating levels are
discussed in a later section. The LRFR, however, uses calibrated live load factors which
vary based on the vehicular loadings, bridge ADTT and rating level (Lichtenstein 2001).
In addition to differing live load factors, both the LRFR and LFR use different live load
distribution factors. The live load distribution factor accounts for how live load effects
are passed through the deck to the supporting structural element of a bridge (Lichtenstein
2001). The LFR uses the live load distribution factors from the AASHTO Standard
Specification which accounts for the distribution of the live load across the deck using a
simplistic ?S over? approach, S referring to girder spacing. LRFR uses the reevaluated
live load distribution equations found in the AASHTO LRFD, which accounts for
additional effects in transverse load distribution such as the deck stiffness. The changes
made to the live load distribution equations in AASHTO LRFD result in a more complex
but supposedly more accurate live load distribution factor (Lichtenstein 2001).
The impact factor is also calculated differently for each of the rating equations.
The LFR impact factor is based on a formula where the impact factor increases with a
bridge?s span length. The dynamic load allowance, or impact factor, of the LRFR is
fixed at 33% for all legal loads; however, the code allows for the factor to be lowered
based upon riding surface conditions (Lichtenstein 2001).
13
2.4 LRFR Condition and System Factors
The resistance factor, ?, as defined in the AASHTO LRFD 2007, is usually a
reduction factor applied to the nominal resistance of a new member to account for the
uncertainties associated with its resistance. As an existing member experiences
deterioration, the uncertainties associated with its resistance increase and can no longer
be accounted for solely through the use of the design resistance factor. The condition
factor, ?c, was introduced to provide an additional estimated reduction to a member?s
resistance to account for the added uncertainties caused by the deterioration a member
has experienced and that it is likely to experience between inspections (Minervino et al.
2004).
The recommended values for the condition factor found in Table 2 - 2, are from
the AASHTO MCE LRFR (2003), and are related to the Superstructure Condition Rating
number found in the bridge?s inspection report. While the condition factor is related to
the structural condition of a member, it only accounts for deterioration from natural
causes, such as corrosion, and not from incident-oriented damage.
Table 2 - 2: Recommended Condition Factor Values According to AASHTO MCE
LRFR (2003)
14
The superstructure of a bridge is composed of multiple structural members
interacting with one another to form a single structural system. A bridge?s redundancy is
the capacity of the structural system to carry loads after one or more of its structural
members has been damaged or has failed. The purpose of the system factor, ?s, is to be a
multiplier applied to the nominal resistance of a member to account for the redundancy of
the full superstructure system. As a result, bridges that are less redundant have a lower
system factor, which lowers each individual member?s factored capacities and ratings
(Minervino et al. 2004).
The recommended values for the system factor according to the AASHTO MCE
LRFR (2003) are shown Table 2 - 3. The recommended system factor values are based
on a bridge?s superstructure type as described in the AASHTO MCE LRFR (2003). The
system factor ranges from 1.00 for redundant systems, such as bridges with more than
four girders, to 0.85 for non-redundant systems, such as truss bridges, arch bridges, or
bridges with two girders or less. If the presence of adequate redundancy can be
demonstrated, the system factor can be different from those presented and can exceed 1.0,
but is limited to a maximum value of 1.2 according to NCHRP Report 406 (1998). The
simplified system factors can only be used when checking the flexural and axial effects
under the strength limit states. When checking shear under the strength limit states, a
system factor of 1.0 is recommended for all superstructure types according to the
AASHTO MCE LRFR (2003).
15
Table 2 - 3: Recommend System Factor Values According to AASHTO MCE
LRFR (2003)
The minimum value of the combined effect of the condition and system factors on
an individual member?s capacity shall not be made less than 0.85 according to the general
load rating procedure provided in the AASHTO MCE LRFR (2003).
2.5 Live Load Factors
The live load factors are unique for each of the two rating methodologies. LFR
has fixed factors of 2.17 for Inventory rating and 1.3 for Operating rating (AASHTO
1994), whereas LRFR uses varying calibrated live load factors. The LRFR factors vary
not only with rating level, but also with vehicle type and bridge ADTT (Lichtenstein
2001). For design level rating in the LRFR the live load factor, L? , is specified as 1.75
for Inventory rating and 1.35 for Operating rating (AASHTO 2003). Live load factors for
Legal loads vary based upon a bridge?s ADTT, ranging from 1.4 to 1.8 (AASHTO 2003).
16
Table 2 - 3 from the AASHTO MCE LRFR (2003) shows the specified values for L?
based on a bridge?s ADTT.
Table 2 - 4: Live Load Factors as a Function of ADTT (AASHTO 2003)
Permit loadings have a L? that is based upon several variables, the permit type,
number of trips, whether the permit truck is allowed to be mixed with traffic, ADTT of
the bridge, and total weight of the permit truck. Table 2 - 4 from the AASHTO MCE
LRFR (2003) summarizes these variables and shows the corresponding L? for a given
permit truck?s situation; this factor can range from 1.15 to 1.85 (AASHTO 2003).
17
Table 2 - 5: Live Load Factors for Permit Loads Based on Bridge?s ADTT
(AASHTO 2003)
2.6 Load Combinations
The AASHTO Standard Specifications 2002 and the AASHTO LRFD 2007 both
specify a series of load combinations that new bridge designs must satisfy. The different
load combinations for each specification allow for a structure to be designed for a degree
of different loading conditions.
The AASHTO Standard Specifications 2002 specifies load combinations in two
main groups: service load combinations and load factor design combinations. Each load
combination in the two groups has different loads and load factors that are evaluated
against the design capacity of a member. For evaluating existing bridges according to the
AASHTO MCE (1994) under the LFR methodology, the terms service load combinations
and load factor design combinations are not used. Instead, two load combinations are
specified. The first corresponds to the LFR Inventory level of rating, and the second
18
corresponds to the Operating level of rating. Both Inventory and Operating levels of
rating are discussed in greater detail in the following Section 2.7. The load factors
associated with these load combinations are based on a ?tried and true approach? and are
not calibrated (Sivakumar 2007).
The AASHTO LRFD (2007) specifies load combinations in four different
categories: strength, service, fatigue, and extreme event. Each of the load combinations
under the LRFD methodology are calibrated specifically for the loading condition and
limit state under evaluation (Minervino et al. 2004). Strength load combinations relate to
limit states associated with the strength of a member. Service load combinations relate to
operational effects a structure will experience. The fatigue load combinations relate to
the effect of repetitive live loads. The extreme event load combinations relate to
structural response under extreme loading conditions such as an earthquake loading
(AASHTO 2007). The load combinations that are utilized in the LRFR rating
methodology are taken from the AASHTO LRFD (2007). However, the LRFR load
combinations are limited to the strength, service and fatigue categories, according to the
AASHTO MCE LRFR (2003). Within this comparative study all of the LRFR rating
analysis is performed using the Strength I load combination at the Design and Legal
levels of rating and using the Strength II load combination at the Permit level of rating.
These levels of rating are discussed in greater detail in Section 2.7. The Strength I load
combination is defined as the ?[basic] load combination relating to the normal vehicular
use of a bridge without wind? according to the AASHTO LRFD (2007). The Strength II
load combination is defined as the ?[load] combination relating to the use of the bridge
19
by Owner-Specified special design vehicles, evaluation of permit vehicles, or both
without wind? according to the AASHTO LRFD (2007).
2.7 Rating Levels
The rating systems for both the LFR and LRFR are broken down into a series of
levels that bridges can be evaluated under, each level corresponding to a different level of
safety. The LFR has a simple two-level system, where LRFR has a more complex
three-level system.
The two levels of the LFR?s rating system are the Inventory and Operating levels.
The Inventory level of rating is the highest level of safety corresponding to ?a live load
which can safely utilize an existing structure for an indefinite period of time?, according
to the AASHTO MCE (1994). Rating results under the HS-20 design truck at this level
are used in reporting to the Federal Highway Administration (FHWA) for the National
Bridge Inventory, NBI (Lichtenstein 2001). The operating rating level is a secondary
lower level of safety corresponding to ?the maximum permissible live load to which the
structure may be subjected?, according to the AASHTO MCE (1994). The results from
the Operating level of rating can be used for determinations of load postings, bridge
strengthening, and possible closure (AASHTO 1994). Permitting is recommended to
only be allowed on bridges that are found to be satisfactory at the operating level of
rating under the HS-20 load model (AASHTO 1994).
The three levels that make up the LRFR rating system are the design, legal and
permit load rating levels. Each of these three levels of rating are discussed in detail in
20
immediately following sections. The procedure that the LRFR uses in its rating system is
shown in the flow chart in Figure 2 ? 1 as given in the AASHTO MCE LRFR (2003).
The process starts with a bridge first being rated at the design Inventory level under HL-
93 load model. If the bridge is found to be satisfactory at this level of rating, it?s
considered not to require posting for ?AASHTO legal loads and state legal loads within
the LRFD exclusion limits?, according to the AASHTO MCE LRFR (2003), and hence
the bridge can be evaluated directly for permit load vehicles.
However if the rating factor at the Design Inventory level is found to be less than
1.0, the bridge must be evaluated under either the Design Operating level or the Legal
load level. At these levels of rating if the bridge is found to be satisfactory it is
considered not to require posting for ?AASHTO legal loads and state legal loads having
only minor variations form the AASHTO legal loads?, according to the AASHTO MCE
LRFR (2003), and the bridge can be evaluated for permit load vehicles. If, however, the
bridge is found to be not satisfactory, load posting will be required for legal loads and no
permit analysis is allowed. There is however the option for higher forms of evaluation,
such as load testing of the bridge or the use of finite element modeling, for when a bridge
is found to be unsatisfactory at the Legal load level and the engineer feels the bridge may
not require posting.
21
Figure 2 ? 1: Load and Resistance Factor Rating Flow Chart From the AASHTO
MCE LRFR (2003)
22
2.7.1 LRFR Design Load Rating
The design load rating is the first level of the LRFR rating system that a bridge
undergoes. The design level of rating is intended to measure the performance of an
existing bridge relative to the LRFD Bridge Design Specifications. The load model used
for this rating level is HL-93 live load model, discussed in Section 2.9.1. Design load
analysis can be done at one of two sublevels either the Inventory level, checking design
level reliability, or the Operating level, checking a second lower level of reliability
(Minervino et al. 2004). The LRFR shares the limit state philosophy of its design
counterpart, the LRFD. The design level of rating analysis is primarily checked at the
strength limit state (Lichtenstein 2001). The main difference between the two sublevels
of the design level rating is a difference in the L? factor. The Inventory level uses a L?
factor of 1.75 calibrated that a passing bridge at this level would correspond to reliability
index of 3.5 or greater (Minervino et al. 2004). A reliability index of 3.5 represents a
probability of failure of two hundred and thirty three in one million. The Operating level
uses a L? factor of 1.35, which was calibrated to a reliability index of 2.5 (Minervino et
al. 2004). A reliability index of 2.5 corresponds to a probability of failure of six thousand
two hundred and ten in one million. The results of an Inventory level of rating under the
HL-93 load model are used in the reporting to the NBI (Lichtenstein 2001).
2.7.2 LRFR Legal Load Rating
The second level of rating in the LRFR is legal load rating. At this rating level,
live load factors are selected based upon bridge ADTT values and are used in conjunction
23
with AASHTO and state Legal Loads. The legal load?s L? factors are ?calibrated by
reliability methods to provide a uniform [level of safety] over varying traffic exposure
conditions,? according to Lichtenstein Engineers (2001). Rating results at this level can
be used for the purpose of load posting and making decisions on potential bridge
strengthening needs or closures. The strength limit state is the primary limit state used
for evaluation at this level (Lichtenstein 2001).
2.7.3 LRFR Permit Load Rating
The third level of bridge rating in the LRFR system is the permit load rating for
overweight vehicles. This level of rating is only available to bridges that have at least the
capacity to carry AASHTO or state legal loads. Strength and service limit states are
typically used in evaluations at this rating level.
2.8 Posting
When a bridge is found to be unsatisfactory under Legal loads load posting may
be required, which restricts the weight of legal loads for the bridge. Posting procedures
differ between the LFR and LRFR methodologies. Under the LFR, bridge owners are
given a wide range of freedom as to how posting is performed. The AASHTO MCE
(1994) recommends that the general procedures outlined for rating in Section 6 of the
code should be followed for determination of need for load posting (AASHTO 1994).
The AASHTO MCE (1994) provides three typical legal loads: type 3, type 3-3 and type
3S2. These models can be used for posting considerations in addition to state legal loads.
24
However, the determination of the exact posting loads and procedure to obtain these
loads is left up to the Bridge owner?s own posting practices.
The LRFR methodology provides a more structured format for load posting than
the LFR, however it also allows Bridge Owners to use their own posting polices. The
LRFR makes an important distinction between bridge inspections and rating, which are
considered ?engineering-related activities? and bridge posting, which is a ?policy
decision made by the Bridge Owner? according to the AASHTO MCE LRFR (2003).
The recommended posting procedure outlined in the LRFR calls for bridges to be rated at
the legal load level under the legal load truck in question. If the rating factor from the
analysis is greater than one, the bridge does not need to be posted for the given truck. If
the rating factor is between 0.3 and 1.0, the AASHTO MCE LRFR (2003) recommends
the following safe posting load based on the rating factor:
Safe Posting Load ]3.0)[(7.0 ?= RFW Equation 2 - 5
Where,
W = Weight of rating vehicle
RF = Legal load rating factor
If the rating factor from the legal load analysis is below 0.3, the AASHTO MCE
LRFR (2003) recommends that the legal truck used in the analysis not be allowed to
cross the bridge. When the rating factors for all three of the AASHTO standard legal
loads is below 0.3, the bridge should be considered for closure (AASHTO 2003).
25
2.9 Live Load Models
The live load models used during rating analysis, for both the LFR or LRFR
methodologies, come from two main sources: the AASHTO, and from individual bridge
rating agencies. The AASHTO MCE (1994) and the AASHTO MCE LRFR (2003)
specify live load models in two categories, design load models and legal load models.
The design load model for the LFR is composed of the HS20 load model and a design
lane load AASHTO MCE (1994). The AASHTO MCE LRFR (2003) for the LRFR
specifies the LRFD?s design load model the HL-93. Each of these design load models is
discussed in greater detail in Section 2.1.5.1. Both AASHTO MCE (1994) and the
AASHTO MCE LRFR (2003), for the LFR and the LRFR respectively, specify the same
three legal load models the Type 3, Type 3-3, Type 3S2, discussed in greater detail in
Section 2.1.5.2. Additionally under both rating methodologies individual bridge rating
agencies can specify their own alternative live load models that can be used in their own
rating practices. The live load models used in this study are those specified in the
AASHTO MCE (1994) and the AASHTO MCE LRFR (2003) and those used in
ALDOT?s own rating practices. A detailed breakdown of each live load model used in
the study is given below. The models are divided into their corresponding rating levels
according to the LRFR methodology.
2.9.1 Design
The LFR analysis at the Design Inventory rating level uses the maximum load
effect from either the HS20 load model or the lane load as defined in the AASHTO
26
Standard Specification (2002) according to the AASHTO MCE (1994). The HS20 load
model consists of three axles weighing 8 kips, 32 kips and 32 kips spaced at 14 feet and
14 to 30 feet respectively. The variable spacing of the last axle is used to maximize the
desired load effect (AASHTO 1994). The lane load according to the AASHTO Standard
Specification (2002) is the combination of a uniform load of 640 lb per linear foot and a
moving concentrated load of 18,000 lbs for investigation of moment load effects and
26,000 lbs for shear load effects.
The LRFR methodology at the Design Inventory rating level uses the HL-93 live
load model as defined in the AASHTO LRFD Specification according to the AASHTO
MCE LRFR (2003). The HL-93 load model is composed of three parts: the design truck,
the design tandem, and the design lane load. The design truck resembles that of the HS20
load model with three axles weighing 8 kips, 32 kips and 32 kips spaced at 14 feet and 14
to 30 feet, respectively. The variable spacing of the last axle is once again used to
maximize the desired load effect. The design tandem is composed of two concentrated
loads of 25 kips spaced at 4 feet. The design lane load is composed of a uniform load of
640 pounds per foot. The live load effect used in rating analysis is the combined
maximum effect of the design lane load with either the design truck or design tandem.
An additional live load model can be considered for negative moment regions in
continuous bridges consisting of the design lane load and two design trucks, with fixed
axle spacings of 14 ft, spaced at no closer than 50 feet to each other (AASHTO 2003).
27
2.9.2 Legal
The LRFR Legal rating level is not explicity defined for the LFR methodology,
although its counter part would be the Operating level of rating. The AASHTO MCE
(1994) does not specify any required load models to be used at the Operating level of
rating. However the AASHTO MCE (1994) does suggest three typical legal load models
to consider: the Type 3, Type 3-3, and the Type 3S2. Figure 2 - 2, below, provides a
depiction of each of these three load models showing their axle weights and
configurations. The LFR methodology, under the AASHTO MCE (1994), additionally
allows agencies to specify their own unique load models for use at the operating level of
rating for posting, strengthening, and closure decisions. The LRFR methodology, under
the AASHTO MCE LRFR (2003), also allows for agencies to specify their own unique
load models at the legal load level of rating as well as specifying the same three
AASHTO standard legal loads found in the AASHTO MCE (1994). The three AASHTO
standard legal load models provide a baseline for legal load rating and posting decisions
and are intended to envelope unique state legal loads that have only minor variations in
axle and weight configurations (Moses 2001).
28
Figure 2 - 2: AASHTO Legal Load Models (AASHTO 1994)
In addition to the three AASHTO standard legal loads ALDOT provided eight
legal load models currently being used in their rating practices to be considered at this
level of the study. Currently ALDOT bases its posting decisions on rating results from
these eight legal load models. This selection of load models consists of the H20, HS20,
two axle tuck, three axle dump truck, concrete truck, 18 wheeler (3S2 Alabama), 6 axle
truck (3S3 Alabama), and school bus. A depiction of each load model showing its unique
axle configuration and weight is shown in Figure 2 ? 3.
29
Figure 2 - 3: ALDOT Legal Load Models
2.9.3 Permit
The AASHTO MCE (1994) and the AASHTO MCE LRFR (2003), for the LFR
and the LRFR respectively, do not specify any overload permit evaluation load models.
30
This is due to permit load models consisting of overweight loads that tend to be unique to
individual rating agencies according to the AASHTO MCE LRFR (2003). ALDOT
provided ten unique overweight load models for uses in the permit rating portion of the
study. These load models were selected by ALDOT to be a representative sample of their
current overload model inventory. A depiction of each load model?s axle weight and
configuration is shown below in Figure 2 - 4 and Figure 2 - 5.
Figure 2 - 4: ALDOT permit load models part 1
31
Figure 2 - 5: ALDOT permit load models part 2
2.10 Previous Research
With the introduction of the LRFR a significant amount of research has been
conducted with regards to its implementation. Keeping within the scope of this project,
only research comparing the LRFR and the LFR is reviewed within this section. The
comparative research published to date is limited to three studies. The first comparative
study was conducted by Lichtenstein Consulting Engineers as reported in their final
NCHRP project C12-46 report, which introduced the new LRFR philosophy and the
AASHTO MCE LRFR (Lichtenstein 2001). This research is discussed in detail in
Section 2.10.1. The second comparative study was performed by Mertz in his final report
on NCHRP project 20-07 Task 122 (Mertz 2005). This research is discussed in detail in
Section 2.10.2. The third comparative study, which is more limited in scope than the
32
previous two, was conducted by Rogers and J?uregui (Rogers and J?uregui 2005). This
research is discussed in detail in Section 2.10.3. In all three comparative studies only the
Strength I limit state for Design and Legal levels and Strength II limit state for rating at
the Permit level were considered for the LRFR methodology.
2.10.1 Lichtenstein Consulting Engineers (2001)
The comparative work done by Lichtenstein Consulting Engineers in their project
C12-46 report was performed on 37 bridges rated at both the Design and Legal ratings
levels. The LRFR analysis was preformed according to the Final Draft Manual of the
AASHTO MCE LRFR in March of 2000. The LFR analysis was performed according to
the AASHTO MCE (1994). The 37 bridge sample used for the study consisted of 17
Steel multi-girder bridges, 9 reinforced concrete T-beam bridges, and 11 prestressed
concrete I-girder bridges. The bridges used in the study were provided by nine different
states, including one bridge from Alabama. Each bridge was analyzed at the Design,
Inventory and Operating levels of rating under the HL-93 and HS-20 load models for the
LRFR and LFR, respectively. A subset of the bridge sample was additionally analyzed
under AASHTO Legal loads at the Inventory and Operating level of rating for the LRFR
and the LFR (Lichtenstein 2001). The entirety of the rating analysis was performed on
interior girders of a bridge with only the flexure data reported and compared. Shear
analysis data, although mentioned in the report, was not reported or directly compared.
Analysis of exterior girders was not performed for either flexure or shear.
During the comparative study, the following assumptions and factors were used.
All LRFR analysis was performed at the Strength I limit state. The system factor, ?s, and
33
condition factor, ?c, for the LRFR were allowed to vary in accordance to the AASHTO
MCE LRFR (2003) per bridge. The live load factors used at the Design Inventory level of
rating were 1.75 for the LRFR and 2.17 for the LFR. The live load factors used at the
Design Operating level of rating were 1.35 for the LRFR and 1.67 for the LFR.
Table 2 - 6 summarizes the rating results of the study. The values displayed here
are the mean LRFR to LFR ratios for each type of bridge evaluated for each rating level
investigated. If the displayed rating ratio is greater then 1.0 than on average LRFR
produced higher rating factors than LFR, and if the ratio is less than 1.0 than LRFR
produced lower rating factors than LFR.
Table 2 - 6: Rating Ratio Results From Lichtenstein Consulting Engineers (2001)
As can be seen from Table 2 ? 6, the LRFR produced lower rating results than the
LFR under the design load model at both the Inventory and Operating levels of rating, for
all bridge types evaluated. Under AASHTO legal loads, however, this trend did not
always hold true. At the Inventory level of rating the LRFR produced larger rating
factors than the LFR. However, legal loads rating results at the Inventory level have little
meaning under the LRFR or the LFR. The operating level though, is the critical level of
rating for legal loads as results here are used by agencies for: posting decisions,
34
availability of the bridge to be evaluated for overweight loads, and decisions on bridge
closure. At the Operating level of rating, under the AASHTO legal loads, it was
observed that the LRFR produced nearly equal or lower rating results.
Table 2 - 7: Controlling Load Effect Data From Lichtenstein Consulting
Engineers (2001)
In addition to the rating factor comparisons reported by Lichtenstein Consulting
Engineers, observations were made about the controlling load effect for the LRFR. The
reported observations were made under the HL93 load model at the Design Inventory
level of rating. Table 2 - 7 provides a summary of this controlling load effect data.
Primarily, it was observed that the majority of the bridges in the sample, 75%, were
controlled by flexure. No comparisons to the controlling load effect for the LFR were
reported however.
2.10.2 Mertz (2005)
In June 2005 Mertz?s report on the findings from his NCHRP Project 20-07 Task
122 were released. The goal of Task 122 was to conduct a comparative study between
35
the LRFR and LFR bridge rating methodologies. The comparative study by Mertz
consisted of 74 different bridges. The bridge sample was composed of reinforced
concrete, prestressed concrete, and steel bridges. Each of the bridges used in the study
was provided by either the New York Department of Transportation or Wyoming
Department of Transportation. Bridges were modeled and analyzed in using AASHTO
Bridgeware?s Virtis Version 5.1 with analysis engines BRASS-GIRDERTM (version 5,
release 08, level 6) and BRASS-GIRDER(LRFD)TM (Version 1, release 5, level 4, beta
version). For more information about AASHTO Bridgeware?s Virtis and analysis
engines BRASS-GIRDERTM see Chapter 4.
The assumptions used in the Task 122 comparative study are summarized in
Table 2 - 8. Virtis Version 5.1 uses these summarized values by default. Using these
default factors can have a profound impact on the study. One example of this can be seen
in the use of 1.35 as the live load factor for legal load rating (Mertz 2005). The
AASHTO MCE LRFR (2003) specifies that the minimal value that the live load factor
for legal loads should be taken as is 1.4 and can range as high as 1.8 for bridges exposed
to large volumes of truck traffic (AASHTO 2003). Therefore, use of a 1.35 factor in the
study resulted in higher LRFR rating factors for legal loads than can even be allowed
under the current code previsions. Additionally using the default values for system
factor, ?s, and condition factor, ?c, results in the highest possible factored resistance
under the AASHTO MCE LRFR (2003); see Section 2.1.2.1 and Section 2.1.2.2 for more
details.
36
Table 2 - 8: Virtis 5.1 Default LRFR Factors (Mertz 2005)
Similar to Lichtenstein Consulting Engineers comparative study, the rating
analysis for Task 122 was performed on interior girders and only for flexure. The entire
74 bridge sample was analyzed at each of the LRFR rating levels and their corresponding
LFR counterparts. For the Design Inventory and Operating rating levels the HL93 live
load model was used for LRFR and the HS20 for LFR. At the Legal load rating level, the
three AASHTO standard legal load models were used for both the LRFR and LFR
analysis. For the permit portion of this study a single overload model was used
consisting of 8 axles with a combined weight of 175 kips.
37
Table 2 - 9: Rating Ratio Results Rrom Dennis Mertz (2005)
A summary of the rating results from Mertz?s Task 122 report is given in Table 2
- 9. The data presented as the mean LRFR to LFR ratio for each material and structural
bridge type for the different rating levels analyzed. As can be seen from Table 2 - 9,
LRFR to LFR ratios in Mertz?s report tend to be larger than those presented by
Lichtenstein Consulting Engineers. At the Design Inventory level the LRFR tended to
produce nearly equal or higher rating factors than the LFR, with the one material and
structural bridge type exception, reinforced concrete slab. Additionally, unlike
Lichtenstein Consulting Engineers finding, at the Legal rating level the LRFR produced
nearly equal or higher rating factors then the LFR. Important to keep in mind while
reviewing the results from Task 122 are the assumptions used during its rating analysis
which can produce higher LRFR rating factors then would be specified by the AASHTO
MCE LRFR. This would tend to bias the LRFR to LFR ratio to be slightly higher than
they truly would be.
In addition to LRFR to LFR rating factor comparative study found in the Task
122 report, Mertz also reported the results from a reliability study. This reliability study
shows the relationship between the LRFR and LFR rating factors and a bridge?s
38
estimated probability of failure. More information about reliability studies in regards to
bridge rating and Mertz?s reliability study will be discussed in Chapter 6.
2.10.3 Rogers and J?uregui (2005)
The third comparative study by Rogers and J?uregui (2005) had a limited scope of
only comparing the LRFR to the LFR for five simply supported prestressed concrete
I-girder bridges. The five bridges included in the study were provided by the New
Mexico Department of Transportation and were selected to provide a range of span
lengths from 38 to 107 feet. Analysis of the bridges was performed only for the interior
girders of the bridges for both flexure and shear. The bridges were evaluated using both
a BRASS rating analysis software and hand calculations. The hand calculations were
done to insure the accuracy of the BRASS rating analysis software. The rating analysis
for the comparative study was performed using the BRASS software, after it was
verified, at the Design Inventory and Operating rating levels. The live load models used
in the analysis were the HL-93 load model for the LRFR and the HS-20 load model for
the LFR.
The results of the rating analysis revealed that the LRFR produced nearly equal to
or lower rating factors when compared to the LFR at the Design Inventory rating level for
both flexure and shear. Additionally, at the Design Operating rating level, the LRFR
produced significantly lower rating factors than the LFR for flexure and nearly equal to
or lower rating factors for shear. However when comparing the load effects it was
discovered that the majority of the bridges in the study were controlled by shear load
effects. The flexural rating results were found to be in agreement with those presented by
39
Lichtenstein Consulting Engineers (2001) and are similar to those presented by Mertz
(2005). Direct comparison for the shear rating results cannot be made due to the
limitations of Lichtenstein Consulting Engineers (2001) and Mertz (2005) studies. To
farther understand the source of the disagreement of the LRFR and LFR rating results,
Rogers and J?uregui studied the individual parameters that make up each the rating
factor, ie the resistance, dead load and live load components. Through this study, Rogers
and J?uregui found that for prestressed concrete I-girders:
? The critical dead load flexural and shear effects of the LRFR and the LFR
showed little disagreement
? The critical flexural resistance of the LRFR and the LFR showed little
disagreement
? The critical shear resistance of the LRFR and the LFR showed varying
degrees of disagreement due to differences in design philosophy
? The critical live load flexural effect was shown to be nearly equally or
higher for the LRFR compared to the LFR
? The critical live load shear effect was shown to be greater for the LRFR
compared to the LFR
40
Chapter 3 BRIDGE SAMPLE
3.1 Determining Bridge Sample
The bridge samples used in the study can be broken into three categories: the
standard, unique, and permit bridge samples. The goal in the development of these
samples was to insure that they would be representative of the Alabama?s state and
county owned and maintained bridge inventory. To achieve this two things are required,
first an understanding of the Alabama?s state and county owned and maintained bridge
inventory and second what limitations would be set on the development of each of the
bridge samples. To assist with the first requirement, in the understanding of Alabama?s
bridge inventory, ALDOT provided two main tools. The first was a set of standard
bridge plans that are commonly used and appear repeatedly in Alabama?s bridge
inventory. The second tool provided by ALDOT was a copy of Alabama?s state and
county owned and maintained bridge database, referred to as the SCOMB database from
here on. The combination of these two tools provided an understanding of the ALDOT
bridge composition. The second requirement, in development of the bridge samples, was
to determine what limitations would be used for the bridge samples. The primary source
of these limitations were dictated by the limitations of the principle modeling and rating
software used for the study, AASHTO BRIDGEWare?s Virtis version 5.6. Virtis version
5.6 was used in this study and is discussed in detail in Chapter 4. A secondary source of
limitations came from ALDOTs own modeling and rating practices using Virtis.
41
Based on the modeling and rating limitations of Virtis version 5.6 and ALDOT?s
own practices using Virtis, a number of useable bridge type categories were identified
and used in selecting the bridge samples. There are two main bridge type categories: the
material type and structural system type of the bridge. These two categories are defined
based on their usage in the SCOMB database. The material type of a bridge denotes, the
material type of the principle structural element of a bridge and the end support
conditions of a bridge. An example of a material type in this context would be a
reinforced concrete simply supported bridge. The structural system type of a bridge
denotes the principle structural element of a bridge. An example of a structural system
type in this context would be a T-Beam. Six material types and five structural system
types were selected to be included in the bridge sample selection criteria.
The six material types included are as follows, and are defined in accordance with
ALDOT?s SCOMB database:
? Reinforced Concrete, Simply Supported
? Reinforced Concrete, Continuously Supported
? Steel, Simply Supported
? Steel, Continuously Supported
? Prestressed Concrete, Simply Supported
? Prestressed Concrete, Continuously Supported
The five structural system types included are as follows, and are defined in
accordance with ALDOT?s SCOMB database; also see Figure 3-1:
? Slab
? Stringer / Multi Beam or Girder
42
? T-Beam
? Box-Beam
? C-Channel
Figure 3 - 1: Structural System Types
Virtis version 5.6 can be used to model all of the material and structural system
types listed above straightforwardly with one exception, the structural system type C-
Channel. Currently this structural system type is not directly supported in Virtis version
5.6. However ALDOT uses a modeling simplification, which allows the inclusion of this
structural system type and has requested this practice to be used within this study.
Figure 3 - 2: Virtis C - Channel Cross Section Conversion to T - Section
43
The modeling simplification developed by ALDOT is depicted in the above
Figure 3 - 2. Here the two outside webs of the C-Channel cross-section are moved
together to form a single web, resulting in a more traditional T-Beam cross-section.
Therefore, using this cross-section transformation, C-Channel bridges can be modeled in
Virtis version 5.6 as T-Beam bridges. The implications of modeling a C-Channel bridge
as a T-Beam bridge were not studied in this research.
With the tools previously described, the standard bridge plans and the SCOMB
database, and the usable material and structural system types listed above, the bridge
samples were selected. The three bridge samples that were created are: the standard,
unique and permit bridge samples. The standard bridge sample is composed of bridges
from standard bridge plans, which are repeatedly used in the Alabama?s bridge inventory.
These standard bridges could have multiple Bridge Identification Numbers, BIN,
associated with each standard bridge. A BIN is a unique number given to identify a
single existing bridge. Standard bridges, therefore, could have multiple BIN numbers
associated with each. The unique bridge sample is composed of bridges that have a
single BIN associated with each of them. The permit bridge sample is composed of a
mixture of the standard and unique bridge samples based on selection criteria discussed
in Section 3.1.3 and Chapter 5.
The standard bridge sample was selected by ALDOT. The sample was composed
of the standard bridge plans that ALDOT desired to include in the comparative study. In
total the standard bridge sample has 50 standard bridges, which are discussed in greater
detail in Section 3.1.2.
44
The unique bridge sample was selected to be representative of the SCOMB
database. The SCOMB database provided a great deal of information about each of the
bridges found in Alabama?s bridge inventory. The following information for each bridge
is included: BIN, material type, structural system type, location, total length, maximum
single span length and whether or not a standard drawing was incorporated the bridge?s
design. The SCOMB database originally contained 15839 bridges however not all the
bridges listed meet the previously described selection criteria for the study. Limiting the
database to the selected material and structural system types reduces the database to 7556
bridges. Before the sample of bridges was selected the SCOMB database was evaluated
with regards to the selected material and structural system types and span length. The
breakdown of the material types within the reduced SCOMB database can be found in
Table 3 - 1 below. A breakdown of the structural system types and span length ranges for
each of the six material types is shown in Tables A - 1 through A - 6, in Appendix A.
These distributions are important as they quantify how the SCOMB database is composed
and provide a guideline as to how a reflective sample should be composed.
Table 3 - 1: Material Type Distribution of the Reduced SCOMB Database
Using the material and structural system type distributions of the SCOMB
database, a matrix was formed detailing what bridge categories should be included in the
45
unique bridge sample with regards to material, structural system type and span length.
The matrix consists of 78 bridge categories covering the full range of usable bridge types
found within the reduced SCOMB database. A copy of this bridge matrix can be seen in
Appendix A, Tables A - 7 and A - 8. Initially, the plan was for the unique bridge sample
to have at least one bridge from each of the bridge categories listed to insure that the full
range of bridge categories would be represented. Then, as time permitted, additional
bridges could be added to different bridge categories. However, due to difficulties in
locating bridge plans, mislabeled bridges within the SCOMB database and time
restrictions, the final unique bridge sample contained only 46 unique bridges spanning 31
bridge categories. A detailed breakdown of the bridges included in the unique bridge
sample is provided in Section 3.1.2.
3.1.1 Standard Bridge Sample
The standard bridge sample was extracted from bridge plans that were provided
by ALDOT and are used repeatedly throughout the SCOMB database. The sample is
composed of 50 standard bridges, from 20 different plans. The material breakdown of
the standard bridge sample is presented in Table 3 - 2 and is compared with the material
distribution of the SCOMB database. As can be seen the material distribution found in
the SCOMB database is well represented in the standard bridge sample. A detailed
description of each bridge including its structural system type as well as span length can
be found in Appendix A, Table A - 9.
Table 3 - 2: Material Type Distribution of Standard Bridge Sample
46
3.1.2 Unique Bridge Sample
The goal of the unique bridge sample was to reflect the SCOMB database in
regards to material type, structural system type, and span length. The sample consists of
46 unique bridges spanning 31 different bridge categories. The material type distribution
of the sample is shown in Table 3 ? 3. Table 3 - 4 displays which structural system types
are found within each material type. While the material type percentages of the unique
bridge sample do not directly reflect that of the SCOMB database the goal of the sample
was to capture as many of the unique bridge categories found within the database as
possible. A matrix showing the material type, structural system type, and span length
breakdown of the sample is shown in Table A ? 10 of Appendix A.
Table 3 - 3: Material Type Distribution of Unique Bridge Sample
47
Table 3 - 4: Structural System Type Breakdown for each Material Type
3.1.3 Permit Bridge Sample
The permit bridge sample is composed of bridges form both the standard and
unique bridge samples. The bridges that were included within the sample were those that
are eligible for overweight load evaluation under either the LFR or LRFR methodologies.
A detailed description of bridges comprising this sample is provided in Section 5.4 of
Chapter 5.
48
3.2 Bridge Sample Information
Information about each of the bridges in both the standard and unique bridge
samples is provided in Appendix A, Tables A - 11 through A - 14. These tables provide
the following additional information for each bridge:
? BIN (bridge identification number)
? Year (fiscal year reported on bridge plans)
? ADTT (average daily truck traffic as reported by ALDOT)
? Live Load Factor, L? (based on bridge ADTT)
? Bridge Span Lengths
? Number of Spans
? Girder Spacing
? Condition Factor, ?c, and System Factor, ?s
? Material Type
? Structural System Type
? Deck Concrete Compressive Strength, cf ?
? Girder Concrete Compressive Strength, cf ? / Structural Steel
Grade
? Reinforcement Grade
? Prestressing Tendon Grade
In the few cases where the provided plans for a bridge did not specifically report a
required material property, the following assumptions were used. For unknown cf ? on
bridges constructed prior to 1954, 2.5 ksi was assumed. For bridges constructed post
49
1954 cf ? was assumed to be 3 ksi. For unknown structural steel grade on bridges
constructed between 1936 and 1963, yield strength of 33 ksi was assumed. Bridges
constructed after 1963, 36 ksi was assumed. For an unknown steel reinforcement grade,
Grade 40 was assumed. These assumptions were provided by ALDOT based on their
current bridge rating practices.
50
Chapter 4 ANALYSIS TOOLS
4.1 Analysis Software
Several computer programs were used for the analysis and rating in the project.
AASHTO BridgeWare?s Virtis Bridge Load Rating software version 5.6 (2007) was the
primary analysis and rating tool for both LRFR and LFR methodologies. Additionally,
in-house rating tools were developed in Mathcad version 14 (2007) for several simply
supported bridge cases to develop a working understanding of the new LRFR
methodology. Two additional programs were developed in Visual Basic to aid in data
collection and organization of the Virtis output files.
4.1.1 Virtis
Virtis is a bridge analysis and rating computer program (BridgeWare 2007). The
program is composed of two major components: the graphical user interface (GUI) used
to model a bridge, and the analysis engines. The modeling of a bridge is done through
the use of several input screens where needed pieces of information about each
component of the bridge is required, including member dimensions, material properties,
member locations, weight, etc. Once a bridge is fully modeled, it can be analyzed under
several different rating methodologies and under a variety of different live load models
(BridgeWare 2007).
While the actual modeling of a bridge is done within Virtis, the analysis is
preformed by a separate analysis engine. During a rating exercise Virtis allows the user
51
to specify what rating methodology to be used as well as what engine to use for the
analysis (BridgeWare 2007). In this study version 5.6.0 of Virtis, released in November
of 2007, was used. This was the first version to include an analysis engine capable of
rating under the LRFR methodology. Version 5.6.0 of Virtis is capable of rating in three
methodologies: ASR, LFR and LRFR. To perform the analysis according to these
different rating methodologies, six analysis engines are available; BRASS ASD, BRASS
LFD, BRASS LRFR, Mandero ASD, Virtis ASD, Virtis, LFD. For this study the
BRASS LFD engine was used for the LFR analysis and the BRASS LRFR engine was
used for the LRFR analysis. The BRASS LFD engine is based on the AASHTO MCE
1994 with interims up 2003 and the AASHTO Standard Specifications of Highway
Bridges 17th edition 2002. The BRASS LRFR Engine is based on the AASHTO MCE
LRFR (2003) with the 2005 interim and the AASHTO LRFD (2007) Bridge Design
Specification (BridgeWare 2007).
Each of the analysis engines that Virtis uses operates in a similar fashion. First,
an influence line analysis is conducted to determine the maximum effect for a given live
load model. The influence line approach by default subdivides each span into 100
increments and moves the specified live load model across the span one increment at a
time to determine the maximum effect. Next, the analysis engine subdivides each span
into 10 equal increments and analyzes the eleven cross section created. For each of the
cross-section the dead load, the maximum live load effect, and resistance are determined
at that specific location. Rating factors for both moment and shear are then produced at
each cross section for the live load model (BridgeWare 2007). The assumption Virtis
makes is that the maximum shear and moment effect will occur at one of its eleven
52
predetermined analysis points. This is however not always the case. Additionally,
different shear provisions allow for the shear at supports to be taken at a specified
distance from the support in reinforced and prestressed concrete members, AASHTO
LRFD (2007) Section 5.8.3.2. These provisions when applied would reduce the shear
effect at the supports. The BRASS LRFR analysis engine however does not use these
provisions by default, and as such all shear rating analysis done at the support uses the
non reduced shear effect. The result of this is slightly lower shear rating factors for the
support analysis points, for those bridges that can make use of these shear provisions.
4.1.2 In-House Rating Tools
In-house rating tools were developed for several simply supported bridge cases
using Mathcad version 14 (2007). Mathcad is a powerful mathematical program that can
be used to develop worksheets that can perform repetitive calculations efficiently. This
allows for analysis problems with constrained variables and predefined calculations to be
repeated with little difficulty through only the change of predefined variables. An
example of this can be seen in the Mathcad worksheet found in Appendix B1 that
performs LRFR analysis for slab bridges. Once the worksheet was developed, analyzing
different slab bridges could be done simply through manipulation of the variables
describing the unique components of a bridge located at the top of the file. In this study,
the use of Mathcad served two purposes. First, to develop an understanding of LRFR
methodology through developing worksheets to perform the rating analysis for simply
supported reinforced concrete, steel, and prestressed bridges. Secondly, to perform the
LRFR analysis for the single slab bridge found in the unique bridge sample. A copy of
53
the final Mathcad worksheets for a simply supported reinforced concrete, steel and
prestressed bridges can be found in Appendix B2, B3, and B4 respectively.
4.1.2.1 AASHTO Rating Example Comparisons
To develop a working understanding of the LRFR methodology, three Mathcad
worksheets were developed to perform Strength I analysis and rating for simply
supported reinforced concrete, steel, and prestressed bridges. These worksheets followed
the rating procedure outlined in the AASHTO MCE LRFR (2003) and the resistance and
load effect calculations as detailed in the AASHTO LRFD (2007) Bridge Design
Specification. To assess the accuracy of the developed worksheets, three example
problems form the AASHTO MCE LRFR (2003) Appendix A were analyzed and the
results were compared. Table 4 ? 1 provides a description of the three bridges used in the
comparison. The example bridges were also modeled and analyzed in Virtis allowing for
an additional point of comparison. The live load model used for the comparison was the
HL-93.
Table 4 - 1: Description of AASHTO MCE Example Bridges (AASHTO 2003)
Comparing the results from the example problem A1 of the AASHTO MCE
LRFR (2003) and the Mathcad file to the results Virtis version 5.6 analysis two
discoveries were made. The first discovery was that a factor of 0.8333 or 5 / 6 was being
54
applied by Virtis to live load before it was used in the general rating equation. This
factor reduced the actual live load effect causing the rating factor to be greater than
anticipated by 1.2 or the reciprocal of the applied factor. Upon further investigation, it
was found that this error originated in Virtis from a provision in the AASHTO LRFD
(2007) which allows for the multiple-presence factor, for a single-lane loaded condition,
1.2 to be removed from the live load distribution factor through the application of a 5 / 6
factor when analyzing under the fatigue limit state, since multiple presence factors should
not be used with the fatigue limit state. However Virtis was using this reduction for all
limit states not just for the fatigue limit state as specified in the code. While this error
was known by AASHTO?s BRIDGEWare and would be corrected in a later release of
Virtis, version 6.0, it was not known to ALDOT or to the researcher until this exercise
was performed. To compensate for this unwanted reduction factor Virtis?s Scale Factor,
used to amplify live loads, was set to 1.20. The product of the scale factor set to 1.2 and
the applied reduction factor of 5/6 is 1.0, so the actual live load effect used during the
rating analysis is correct.
The second discovery that was made dealt with the live load distribution factors,
discussed in Section 2.1.2. While the AASHTO MCE LRFR (2003) example problem,
the Mathcad worksheet, and Virtis all produced the same live load distribution factors,
the factor used in the Virtis analysis was different. Section 4.6.2.2.2 of the AASHTO
LRFD (2007) specifies that the controlling, largest, live load distribution factor should be
used in the analysis. However while the AASHTO MCE LRFR (2003) example and the
worksheet did use the controlling live load distribution factors as specified in the code,
Virtis used the smallest of the factors. Due to using the smaller of the factors Virtis was
55
producing lower than anticipated live load effects and higher than anticipated rating
factors. This error was also know by AASHTO?s Bridge Ware and would be corrected in
a later release of Virtis, version 6.0. The error however was not known to ALDOT or to
the research until this investigation was performed. To compensate for the error during
the research the live load distribution factor was manually set to the controlling factor for
each bridge during its modeling process.
With the inclusion of these two corrections Table 4 ? 2 shows a comparison of the
dead load moment results between the AASHTO MCE LRFR (2003) example problem,
the Mathcad worksheet, and Virtis results. As can been seen there is virtually no
difference between the calculated total dead load moments from the three different
methods.
Table 4 - 2: Steel I-Girder Example Eead Load Results
56
Table 4 ? 3 shows a comparison of the live load moment results for the three
different methods for the HL-93 live load model. The moment and shear results
presented for the AASHTO MCE LRFR (2003) example and the Mathcad worksheet are
un-factored effects. Virtis however only outputs factored live load effects which include
the live load distribution factor, the impact factor and the scale factor, where applicable.
The un-factored live load effects for Virtis were produced by manually removing those
known factors in Microsoft Excel. When comparing the live load moment results from
the AASHTO MCE LRFR (2003) example and the Mathcad worksheet, little difference
can be seen. The live load moment for Virtis however is slightly lower. This is due to
Virtis assuming that the maximum moment effect occurs at one of its eleven
predetermined analysis points. The maximum moment for this case then would occur at
the mid-span analysis point, due to the bridge?s support conditions. When dealing with a
simply supported member and a moving live loads however, the maximum moment
typically occurs just off of mid-span, which occurs here. Thus this causes Virtis to
slightly underestimate the true maximum live load moment as it does not occur at one of
its predefined analysis points.
Table 4 - 3: Steel I-Girder Example Live Load Moment Results
57
The shear results however show very little discrepancy between the three different
methods, Table 4 ? 4. This is due to the maximum shear actually occurring at one of the
Virtis eleven predetermined analysis point, the supports.
Table 4 - 4: Steel I-Girder Example Live Load Shear Results
The factored capacities and a summary of the total factored load effects for the
steel I-girder example problem are shown in Table 4 ? 5. Little disagreement can be
found between the three methods with regards to the flexural capacity and load effects.
The capacities for shear however are different for the AASHTO MCE LRFR (2003)
example compared to the Mathcad worksheet and Virtis. This difference is due to
changes in the AASHTO LRFD Bridge Design Specification 2001 code used in the
AASHTO MCE LRFR (2003) example and the AASHTO LRFD (2007) code used in the
Mathcad worksheet and Virits. In the 2001 code the fillet depth is excluded from the web
depth for shear calculations. The 2007 code however does not exclude the fillet depth for
shear calculations, thus the full web depth is used yielding a large shear capacity.
Table 4 - 5: Steel I-Girder Example Capacity Comparison
58
The Strength I rating results for the three different methods for the steel I girder
example are compared in Table 4 ? 6. The three different methods produce nearly the
same moment rating results and similar shear rating results with the exclusion of the
AASHTO MCE LRFR (2003) example due to code changes. This affirms that the use of
the Mathcad worksheet and Virtis using the two previously noted corrections can produce
reliable rating results with regards to the AASHTO MCE LRFR (2003) and the
AASHTO LRFD (2007) Bridge Design Specification for steel I-girders.
Table 4 - 6: Steel I-Girder Example Rating Comparison
Similar results were found when comparing the AASHTO MCE LRFR (2003)
example reinforced concrete T-beam and the prestressed concrete I-girder problems to
there Mathcad worksheets and Virtis results. Summaries of these comparisons similar to
those presented for the steel I-Girder example can be found in Appendix B5 and B6,
respectively. One important differences to note in the AASHTO LRFD 2001 Bridge
Design code used in the AASHTO MCE LRFR (2003) examples and the AASHTO
LRFD (2007) code used in the Mathcad worksheets and Virtis, deals with the way shear
capacity is calculated for reinforced and prestressed concrete members. The 2007 code
provides a more refined form of analysis in calculating shear capacity allowing for ?, a
factor relating effect of longitudinal strain on shear capacity, and ?, the angle of
59
inclination of diagonal compressive stresses, to be variable with regards to a calculated
longitudinal strain, ?x, at the cross section under analysis. This is referred to as the
general procedure for shear design with tables. A simplified approach still is allowed
which assumes a ? of 2.0 and ? of 45? which tends to yield lower shear capacities than
the general approach. By default Virtis uses the general procedure and the general
procedure was used for all concrete analysis in this research.
4.1.2.2 Output Sorting Programs
One of the goals of the research was to be able to compare both moment and shear
rating factors for LRFR and LFR for each bridge member under every used live load
model. This however created a problem because Virtis only displays the absolute
controlling rating factor after each rating analysis. Moreover, in order to be able to
perform bounding studies for the rating results due to changes in ?L, ?c, and ?s, all the
components of the rating equations need to be known, such as: the resistance, applied
dead and live load, ? factors and ? factors. For more information about the bounding
studies see Chapter 5. All this information is not readily available from the Virtis output
screens; therefore, the data was required to be gathered from Virtis?s output files. This
presented an additional challenge in that the formatting of Virtis output files is not
standard. For example during an LRFR analysis each analysis point generates its own
unique file and the structure of that file changes depending on the material and structural
system type of the bridge. LFR on the other hand puts all its output for each analysis
point into a single file but again the structure of the file changes with each bridge?s
material and structure type.
60
To overcome this problem of gathering the required data two options presented
themselves. The first was to manually gather the data by hand. This however presented
significant problems in locating the controlling data as well as gathering it, this would
introduce two possible sources of human error. The second option, was to write a
computer program that would be able to read through the various types of Virtis output
files and gather the required data. To accomplish this task the Virtis Output Sorter
program was written by the author (2008). The Virtis Output Sorter allows its user to
specify the bridge type, material type, rating methodology and location of the output
file(s) and then using this information the program gathers all the required controlling
data for the bridge and exports it into an organized Microsoft Excel file. Figure 4 - 1
shows the graphical user interface of the program.
Figure 4 - 1: Virtis Output Sorter User Interface (Murdock? 2008)
The Microsoft Excel file that the Virtis Output Sorter exports the data to is
extremely large containing over 450 different pieces of data per bridge. This caused the
61
file to be very cumbersome to work with in regards manipulating the data and graphically
presenting it. Therefore, a need arose to be able to break the data down into smaller
segments allowing it to be worked with easier. The Data Organizer written by the author
(2008) extracts the data from the Microsoft Excel file and splits it into several smaller
Excel files, allowing for the large amount of data gathered over the course of the research
to be broken down into smaller data files. These smaller files allow for the data to be
analyzed more easily and rapidly.
62
Chapter 5 RATING RESULTS
5.1 Overview
In order to facilitate the presentation of the results, the data gathered from the
comparative study has been subdivided into several smaller sections based on the rating
level considered. The data from each LRFR rating level is presented in its own separate
section. The Design Inventory rating data is presented in Section 5.2. The Legal load
rating data is presented in Section 5.3. The Permit load rating data is presented in Section
5.4. Each section will present comparisons between the LRFR and the LFR with regards
to flexure and shear rating factors for interior and exterior girders. The data presented in
each of the sections follows a similar pattern. The data will be presented in two primary
manners, and will be presented in alternate manners when needed. The first manner in
which the data will be presented is through the use of LRFR versus LFR rating factor
plots, which will be described in the following paragraphs. The second manner in which
the data will be presented is through the use of tables providing various statistics of the
data.
The LRFR versus LFR rating factor plots, commonly used in the presentation of
the data in this thesis, can provide a great deal of information in a concise manner.
Examples of this type of plot can be seen in Figures 5 - 1 through 5 - 3. Each of these
example plots have been divided into numbered regions. Data falling into each of the
63
shaded regions holds a specific meaning. To help facilitate the discussion of data
presented shortly, these regions are first defined.
Figure 5 - 1: LFR Versus LRFR Region Plot 1
64
Figure 5 - 2: LFR Versus LRFR Region Plot 2
The horizontal and vertical dashed lines shown in Figure 5 - 1 subdivide the plot
into four regions. These four regions are labeled 1 through 4. The solid diagonal line
further divides the LRFR verses LFR rating factor plot into two more regions as seen in
Figure 5 - 2, creating Regions 5 and 6. These two regions are of high importance. Data
in Region 5 have lower LRFR rating factors than LFR rating factors. Data in Region 6
have higher LRFR rating factors than LFR rating factors. Overlaying Figures 5 - 1 and 5
- 2 a six-region plot is created as found in Figure 5 - 3.
65
Figure 5 - 3: LFR Versus LRFR Region Plot 3
Data found in each of these six regions holds a specific meaning when comparing
the rating factors produced by the LRFR to the LFR. Data found in Region 5 - 1
indicates unsatisfactory rating factors for both the LRFR and the LFR, and lower LRFR
rating factors than the LFR rating factors. Data found in Region 5 - 2 indicates
unsatisfactory rating factors for the LRFR, satisfactory rating factors for the LFR, and
lower LRFR rating than the LFR rating factors. Data found in Region 5 - 3 indicates
satisfactory rating factors for both the LRFR and the LFR, and lower LRFR rating factors
than the LFR rating factors. Data found in Region 6 - 1 indicates unsatisfactory rating
factors for both the LRFR and the LFR, and higher LRFR rating factors than the LFR
rating factors. Data found in Region 6 - 2 indicates satisfactory rating factors for the
LRFR, unsatisfactory rating factors for the LFR, and higher LRFR rating factors than the
66
LFR rating factors. Data found in Region 6 - 3 consists of satisfactory rating factors for
both the LRFR and the LFR, and higher LRFR rating factors than the LFR rating factors.
5.2 Design Level Rating Results
Comparisons at the design level of rating were made between the LRFR?s Design
Inventory level and the LFR?s Inventory level. The live load models used were the HL-
93 live load model for the LRFR and the HS-20 Design truck for the LFR. The live load
factors used at this level of rating were as specified by the AASHTO MCE LRFR (2003)
for the LRFR and the AASHTO MCE (1994) for the LFR, as 1.75 and 2.17, respectively.
Data comparisons for the standard bridge sample are given in Section 5.2.1 and for the
unique bridge sample in Section 5.2.2. Combined comparisons for both the standard and
unique bridge samples are given in Section 5.2.3. A summary of the comparisons for
both samples at the design level of rating is provided in Section 5.2.4.
5.2.1 Standard Bridges
The rating data generated for the standard bridge sample (refer to Section 3.1.1),
at the Design Inventory rating level, are provided in the tables from Appendix C1, Tables
C1 - 1 through C1 - 12. A summary of the rating factors used in the comparisons for this
section are provided in Table 5 - 1 and 5 - 2. Table 5 - 1 provides the moment and shear
rating factors generated for both the interior and exterior girders for each bridge in the
sample, under the LRFR methodology. Additionally, the controlling rating factor for the
interior and exterior girders are identified, as well as the controlling rating factor for the
bridge. Table 5 - 2 provides the same rating factor information as Table 5 -1 but for the
67
LFR methodology. The material and structural system type key for this table is as
follows.
For material types:
? Reinforced Concrete, Simply Supported - 1
? Reinforced Concrete, Continuously Supported - 2
? Steel, Simply Supported - 3
? Steel, Continuously Supported - 4
? Prestressed Concrete, Simply Supported - 5
? Prestressed Concrete, Continuously Supported - 6
For structural system types:
? Stringer / Multi Beam or Girder - 2
? T-Beam - 4
? Box-Beam - 5
? C-Channel - 22
68
Table 5 - 1: LRFR Rating Factors Generated for the Standard Bridge Sample at the
Design Inventory Rating Level
69
Table 5 - 2: LFR Rating Factors Generated for the Standard Bridge Sample at the
Design Inventory Rating Level
The first aspect of the data analyzed was the moment rating factor data for the
interior and exterior girders. This data is plotted on the previously described LRFR
70
verses LFR rating factor plot and is shown in Figure 5 - 4 for the entire sample. As can
be seen from the figure the data points fall in Region 5 of the plot, meaning that the
moment rating factor data shows the LRFR methodology producing lower rating factors
than the LFR. The data points, however, are scattered over Regions 5 - 1, 5 - 2, and 5 ?
3.
0.0
1.0
2.0
3.0
4.0
0.0 1.0 2.0 3.0 4.0
LFR Moment Rating Factor
LR
FR
M
om
en
t R
ati
ng
F
ac
to
r
Ext Girder
Int Girder
Figure 5 - 4: Moment Rating Factor Comparison at the Design Inventory Level for the
Standard Bridge Sample
The shear rating factor data at the Design Inventory level for the standard bridge
sample is presented in Figure 5 - 5 for the exterior and interior girders. The shear rating
factor data differs from the moment rating data in that parts of the data fall in Regions 5
and 6. Bridges with low shear rating factors, below 1.0, seem to be found primarily in
71
Region 6 - 1. Bridges with high shear rating factors, above 2.0, are within Region 5 - 3.
Bridges with rating factors between 1.0 and 2.0 are scattered over Regions 5 - 3 and 6 - 3.
This suggests that LRFR produces higher rating results than LFR for bridges with low
shear rating factors and produces lower rating results than LFR for bridges with high
shear rating factors.
0.0
1.0
2.0
3.0
4.0
0.0 1.0 2.0 3.0 4.0
LFR Shear Rating Factor
LR
FR
S
he
ar
R
ati
ng
Fa
cto
r
Ext Girder
Int Girder
Figure 5 - 5: Shear Rating Factor Comparison at the Design Inventory Level for the
Standard Bridge Sample
Results from a statistical analysis of the rating factor data are presented in Tables
5 - 1 to 5 - 4 for the standard bridge sample at the Design Inventory level. These Tables
provide the mean and standard deviation for the LRFR, LFR, and ratio of LRFR to LFR
rating factor data. The tables provide these statistics for the entire standard as well as for
72
the various bridge categories represented within the sample, as shown. Tables 5 - 3 and 5
- 6 provide the results for the interior girders of the sample for moment and shear
respectively. For moment rating factors for interior girders, Table 5 - 3, the LRFR
always produced lower rating results than LFR. These results are in agreement with those
of Lichtenstein Consulting Engineers who found that for this rating level, LRFR
produced nearly equal or lower rating factors than LFR (Lichtenstein 2001). The same
trend is seen in Table 5 - 5 for the exterior girders. The results form the shear rating
factor analysis showed that the LRFR and LFR produced similar results, as shown in
Tables 5 - 4 and 5 - 6. However, at the material and structural system level, Table 5 - 4
and 5 - 6 , that for reinforced concrete T-beams, prestressed concrete channel and I-girder
bridges, LRFR produced greater or equal rating factors than the LFR, for interior and
exterior girders. An interesting observation is seen for the reinforced concrete channel
bridges, where for exterior girders the LRFR produce considerably larger rating results
than the LFR, Table 5 - 6, but for interior girders the opposite was seen, Table 5 - 4. For
all other bridge types the LRFR produces lower rating factors than LFR, for interior and
exterior girders.
Table 5 - 3: Mean and Standard Deviation Data at the Design Inventory Level for the
Standard Bridge Sample ? Interior Girder Moment Rating Data
73
Table 5 - 4: Mean and Standard Deviation Data at the Design Inventory Level for the
Standard Bridge Sample ? Interior Girder Shear Rating Data
Table 5 - 5: Mean and Standard Deviation Data at the Design Inventory Level for the
Standard Bridge Sample ? Exterior Girder Moment Rating Data
74
Table 5 - 6: Mean and Standard Deviation Data at the Design Inventory Level for the
Standard Bridge Sample ? Exterior Girder Shear Rating Data
Based upon material type alone, prestressed bridges are seen to have the highest
LRFR to LFR ratio for both moment and shear except for exterior girder moment rating.
Reinforced concrete bridges on average tend to have the lowest LRFR to LFR ratio for
75
moment rating factors, with steel bridges having the lowest LRFR to LFR ratio for shear
rating factors.
The statistical data also shows that reinforced concrete C ? Channel bridges,
rating factors for both load effects tend to have a lower than usual LRFR to LFR ratio for
interior girders and a higher than usual LRFR to LFR ratio for exterior girders. The
reason for these usual ratios was not investigated; however it is believed that this may in
part be due to the modeling assumptions made for this bridge type as discussed in
Chapter 3. Further investigation however would be required to determine the exact
reason for the C ? Channel?s unusual LRFR to LFR ratios.
Table 5 - 7 compares the interior with exterior girder?s LRFR moment rating
factor statistical data. The comparison reveals that the exterior girder controls over the
interior girder for all material and structural system types with the exception of
prestressed concrete continuously supported girder bridges. This trend was not observed
for the LFR moment statistical data or the shear statistical data of either methodology, as
shown in Tables 5 - 3 to 5 - 6.
Table 5 - 7: Mean and Standard Deviation Data at the Design Inventory level for the
Standard Bridge Sample ? Interior to Exterior LRFR Moment Rating
Comparison
76
The final point of comparison for this sample of bridges was made to determine
the controlling load effect for each rating methodology. Table 5 - 8 shows the results of
this comparison. The data in this table was constructed by counting the number of times
a rating factor for each load effect controlled for a bridge within the sample. The data
indicates that for the LRFR methodology exterior girder moment load effects primarily
controlled. The controlling load effect for the LFR methodology is seen to be evenly
split between the interior girder moment and exterior girder shear load effects.
Table 5 - 8: Controlling Load Effect Comparison, Design Inventory Level for the
Standard Bridge Sample
77
Note: The standard bridge sample consists of 50 bridges
5.2.2 Unique Bridges
The rating data generated for the unique bridge sample (refer to Section 3.1.2) at
the Design Inventory rating level are provided in the tables from Appendix C2, Tables C2
- 1 through C2 - 12. A summary of the rating factors used in the comparisons for this
section are provided in Table 5 - 9 and 5 - 10. Table 5 - 9 provides the moment and
shear rating factors generated for both the interior and exterior girders for each bridge in
the sample, under the LRFR methodology. Additionally, the controlling rating factors for
the interior and exterior girders are identified, as well as the controlling rating factor for
the bridge. Table 5 - 10 provides the same rating factor information as Table 5 - 9 but
for the LFR methodology.
Table 5 - 9: LRFR Rating Factors Generated for the Unique Bridge Sample at the
Design Inventory Rating Level
78
Table 5 - 10: LFR Rating Factors Generated for the Unique Bridge Sample at the Design
Inventory Rating Level
79
The unique bridge sample yielded similar trends to those of the standard bridge
sample at the Design Inventory rating level. Figures 5 - 6 and 5 - 7 present the LRFR
verses LFR rating factor data for moment and shear effects, respectively.
80
0.0
1.0
2.0
3.0
4.0
0.0 1.0 2.0 3.0 4.0
LFR Moment Rating Factor
LR
FR
M
om
en
t R
ati
ng
F
ac
to
r
Ext Girder
Int Girder
Figure 5 - 6: Moment Rating Factor Comparison at the Design Inventory Level for
Unique Bridge Sample
The moment data for both exterior and interior girders falls primarily within
Region 5 of the plot, indicating that the LRFR rating factors are lower than their LFR
counterparts. Data again is heavily scattered over Regions 5 - 1, 5 - 2 and 5 - 3 as was
seen previously for the standard sample. However, a greater number of data points fall
within Region 2 of the plots, which signify satisfactory ratings under LFR but
unsatisfactory ratings under the LRFR. The potential effect of this would be a greater
number of bridges being reported as unsatisfactory to the NBI under the LRFR as
opposed to the LFR.
81
0.0
1.0
2.0
3.0
4.0
0.0 1.0 2.0 3.0 4.0
LFR Shear Rating Factor
LR
FR
S
he
ar
R
ati
ng
Fa
cto
r
Ext Girder
Int Girder
Figure 5 - 7: Shear Rating Factor Comparison at the Design Inventory Level for Unique
Bridge Sample
The shear data for the unique bridge sample has a greater degree of scatter than
was observed in the standard bridge sample, as shown in Figure 5 - 7. The trend of the
LRFR producing higher shear rating results than LFR for bridges with low shear rating
factors, seen previously for the standard bridge sample, is not as pronounced for the
unique bridge sample. The majority of the data for the unique bridge sample falls within
Region 5 with only portions of the data, with rating factors near 1.0 for LFR, falling
within Region 6.
Results from the statistical analysis of the rating data for the unique bridge sample
produced similar trends to those of the standard bridge sample. The mean and standard
82
deviations for the moment rating results of the interior and exterior girders are presented
in Table 5 - 11 and 5 - 12 respectively. Across all material and structural system types
the LRFR method produced nearly equal or lower rating results compared to the LFR for
flexure for interior girders, Table 5 - 11. This trend was also seen for the standard bridge
sample and concurs with the findings of Lichtenstein Consulting Engineers (Lichtenstein
2001). Similar results can be observed for the exterior girder, Table 5 - 12.
Table 5 - 11: Mean and Standard Deviation Data at the Design Inventory Level for the
Unique Bridge Sample ? Interior Girder Moment Rating Data
Table 5 - 12: Mean and Standard Deviation Data at the Design Inventory Level for the
Unique Bridge Sample ? Exterior Girder Moment Rating Data
83
Shear statistics are reported for the interior and exterior girders in Table 5 - 13
and 5 - 14 respectively. For nearly all material and structural system types, for interior
girders, the LRFR method produced nearly equal or lower rating results when compared
to the LFR for shear. With the exception of prestressed concrete continuously supported
girder bridges where the LRFR tended to produce higher rating factors than the LFR.
Similar results were found for the exterior girders of the unique bridge sample.
Table 5 - 13: Mean and Standard Deviation Data at the Design Inventory Level for the
Unique Bridge Sample ? Interior Girder Shear Rating Data
84
Table 5 - 14: Mean and Standard Deviation Data at the Design Inventory Level for the
Unique Bridge Sample ? Exterior Girder Shear Rating Data
The final point of comparison for this sample of bridges was made to determine
the controlling load effect for each rating methodology. Table 5 - 15 shows the results of
this comparison. The data in this table was constructed by counting the number of times
a rating factor for each load effect controlled for a bridge within the sample. The data
indicates that for the LRFR methodology exterior girder moment load effect mainly
85
controlled, similar to what was seen with the standard bridge sample. However this trend
is less dominant as moment and shear load effects for the interior girder controlled a
larger number of bridges for the unique bridge sample. For the LFR methodology
however, the trend seen for the standard bridge sample is not seen at all as bridges in this
sample were nearly evenly controlled across all load effects.
Table 5 - 15: Controlling Load Effect Comparison, Design Inventory Level for the
Unique Bridge Sample
Note: The unique bridge sample consists of 45 bridges
5.2.3 Combined Sample Comparison
The final comparison made at the Design Inventory level of rating was in
studying the absolute controlling rating factor between the two rating methodologies. For
this comparison, rating factor data was used from both the standard and unique bridge
samples. The absolute controlling rating data used for these comparisons can be found in
the previously shown Tables 5 - 1, 5 - 2, 5 - 9, and 5 - 10. Provided in Figure 5 - 8 is a
LRFR versus LFR plot of the controlling rating data. From this plot it is seen that the
majority of the data falls into Region 5 with only sporadic data found in Region 6. This
indicates that the LRFR produced lower rating results than the LFR in general.
86
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
LFR Controlling Rating Factor
LR
FR
C
on
tro
llin
g R
ati
ng
Fa
cto
r
Figure 5 - 8: Controlling Rating Factor Comparisons at the Design Inventory Level
Additionally, the absolute controlling load effect and rating methodology was
investigated; Table 5 - 16 shows the results of this investigation. The data provided in
Table 5 - 16 is the total number of times each load effect and methodology controlled for
the combined bridge samples. This data indicates that the LRFR exterior girder moment
load effect primarily controlled. This finding is in agreement with the previously
reported results showing the LRFR producing nearly equal or lower rating results than
the LFR, in general. An additional point of observation, however, is that for the few
occasions where the LFR methodology did control it was only for the shear load effect.
87
Table 5 - 16: Controlling Load Effect and Rating Methodology at the Design Inventory
Level
Note: The combined bridge sample consists of 95 bridges
5.2.4 Summary
Analysis of the standard and unique bridge samples at the design rating level
provided the following general findings:
? LRFR methodology produces predominantly lower moment rating factors
than the LFR methodology for exterior and interior girders.
? LRFR methodology produces predominantly lower shear rating factors
than the LFR methodology for exterior and interior girders.
? Flexural rating factors predominantly controlled over shear rating factors
for the LRFR methodology
? Flexural and shear rating factors nearly evenly controlled for the LFR
methodology
? Moment rating factors for the Exterior girders tend to control over
moment rating factors for the interior girders under the LRFR
? Prestressed bridges tend to have the highest LRFR to LFR ratio of the
different material types
88
? C ? Channel bridges tend to have unusual LRFR to LFR ratio when
compared to other structural system types.
5.3 Legal Load Rating Results
The primary objective of this portion of the study is to compare rating factors
produced by LRFR and LFR methodologies under ALDOT?s own legal loads. Before the
results of this primary investigation are presented, the findings of a sub-investigation are
given in Section 5.3.1. This sub-investigation examines how the rating results produced
under ALDOT?s legal loads should be handled in the LRFR procedure; see Section 2.7
for the LRFR procedure description and flowchart. This sub-investigation was performed
through a comparison of ALDOT?s legal loads and AASHTO load models under the
LRFR rating procedure. The rating results of comparisons between the LRFR and LFR
methodologies for ALDOT legal loads are then presented in the following sections. Due
to the unknown ADTT (Average Daily Truck Traffic) values for the standard bridge
sample, a series of bounding studies were performed comparing the LRFR to the LFR.
Results of these studies are presented in Section 5.3.2. ADTT information, however, was
available for the unique bridge sample allowing for more explicit comparisons to be
made for the two rating methodologies under ALDOT?s legal loads. These rating results
are presented in Section 5.3.3. A summary of the findings for all of the investigations
made at the Legal load level of rating are provided in Section 5.3.4.
89
5.3.1 AASHTO Load Models and ALDOT Legal Loads Comparison
To determine whether or not the provisions for state legal loads can be applied to
ALDOT legal loads, as outlined in Section 2.7, a comparison study was performed
between ALDOT legal loads and the AASHTO load models. This comparative study is
broken into two parts. The first part is a comparison between the controlling rating
factors for the AASHTO standard legal loads and ALDOT legal loads at the Legal load
level of rating. The second part is a comparison between the controlling rating factors for
the HL-93 load model at the Design Inventory level of rating and ALDOT legal loads at
the Legal load level of rating. Both parts of this study used the standard and unique
bridge samples.
The first part of the study was conducted at the Legal load level of rating and used
the following assumptions. The condition factor, ?c, is set to 1.0 and system factor, ?s,
was allowed to vary as defined in the AASTHO MCE LRFR (2003). However, for this
study, all the bridges included have a system factor, ?s equal to 1.0 according to the
specification. For the standard bridge sample, the live load factor, L? , was taken as 1.4.
For the unique bridge sample, L? was determined based on each bridge?s unique ADTT
as provided by ALDOT. For bridges from the unique bridge sample with unknown
ADTT values, L? was assumed to be 1.8 according to the AASTHO MCE LRFR (2003).
The first part of the study considered eight ALDOT legal loads and three
AASHTO legal loads, at the Legal load level of the LRFR. However for ease of
comparison only the controlling load from the ALDOT legal loads, see Section 2.9.2, and
AASHTO legal loads, see Section 2.9.2, are compared. The controlling load for a given
bridge is defined as the load which produced the lowest rating factor. Tables 5 - 17 and 5
90
- 18 show the number of times an AASHTO and ALDOT legal load, respectively,
controlled for each load effect. Each of the AASHTO legal loads controlled segments of
the bridge within the study with the Type 3 load controlling the most. Within the
ALDOT legal loads, the Tri-Axle load predominantly controlled across all load effects,
with the 6-Axle load occasionally controlling.
Table 5 - 17: Controlling AASHTO Legal Loads
Table 5 - 18: Controlling ALDOT Legal Loads
The LRFR rating results from the controlling AASHTO and ALDOT legal loads
are compared in Figures 5 - 9 and 5 - 10 for moment and shear, respectively. The plots
are set up in an ALDOT controlling rating factor versss AASHTO controlling rating
factor fashion. Therefore, data falling above the solid diagonal line would indicate
ALDOT legal loads controlled over AASHTO Legal Loads and vice versa for data below
the diagonal line.
91
0.0
1.0
2.0
3.0
4.0
0.0 1.0 2.0 3.0 4.0
ALDOT Controlling Rating Factor
AA
SH
TO
C
on
tro
llin
g R
ati
ng
Fa
cto
r
Ext Girder
Int Girder
Figure 5 - 9: LRFR Moment Rating Factor under AASHTO and ALDOT Legal Loads
92
0.0
1.0
2.0
3.0
4.0
0.0 1.0 2.0 3.0 4.0
ALDOT Controlling Rating Factor
AA
SH
TO
C
on
tro
llin
g R
ati
ng
Fa
cto
r
Ext Girder
Int Girder
Figure 5 - 10: LRFR Shear Rating Factor under AASHTO and ALDOT Legal Loads
For both load effects, and for exterior and interior girders, all the rating factor data
can be found in Region 6 of the plots showing that ALDOT legal loads always produce
lower rating factors than AASHTO legal loads. This indicates that ALDOT legal loads
are not enveloped by AASHTO legal loads. The current LRFR rating procedure,
discussed in Section 2.7, indicates that a bridge may be evaluated for permit loads if it
has a satisfactory rating at the Legal load level for either AASHTO or State legal loads
(AASHTO 2003). Because ALDOT legal loads are not enveloped by AASHTO Legal
93
loads, it is therefore that ALDOT legal loads be used instead of AASHTO Legal loads for
load posting decisions and for determinations on whether a bridge can be evaluated for
overweight loads.
The second part of the comparative study was done between the ALDOT legal
loads and the AASHTO HL-93 live load model, see Section 2.9.1. In this part of the
comparison the same factors as described in first part of the study were used. Rating
results for ALDOT legal loads at the Legal load level are compared to the rating results
from the HL-93 load model at the Design Inventory level, for which L? is equal to 1.75.
As stated before, the comparisons presented here are for the controlling rating factor for
both the ALDOT legal loads to the HL-93 load model. Figures 5 - 11 and 5 - 12 present
the moment and shear rating factor data, respectively.
94
0.0
1.0
2.0
3.0
4.0
0.0 1.0 2.0 3.0 4.0
ALDOT Controlling Rating Factor
HL
-93
C
on
tro
llin
g R
ati
ng
Fa
cto
r
Ext Girder
Int Girder
Figure 5 - 11: LRFR Moment Rating Factor under HL-93 Load Model and ALDOT
Legal Loads
95
0.0
1.0
2.0
3.0
4.0
0.0 1.0 2.0 3.0 4.0
ALDOT Controlling Rating Factor
HL
-93
C
on
tro
llin
g R
ati
ng
Fa
cto
r
Ext Girder
Int Girder
Figure 5 - 12: LRFR Shear Rating Factor under HL-93 Load Model and ALDOT
Legal Loads
Data from these plots can be found in both Region 5 and 6 for each load effect
and for interior and exterior girders. This shows that the HL-93 load model does not
always envelope ALDOT?s legal loads. This observation is in agreement with
Hayworth?s findings in a 2008 study comparing several different states? legal loads to the
different AASHTO load models. Hayworth (2008) discovered that state legal loads are
not always enveloped by the AASHTO legal load models and the HL-93 load model.
96
The implication of this for ALDOT is that even for bridges that are found to be
satisfactory at the Design Inventory level of rating under the HL-93 load model, posting
restrictions for the heavier ALDOT legal loads may still be required. Therefore to insure
a bridge does not require posting, rating analysis at the Legal load level under ALDOT
legal loads will always be required even if the rating factor under the AASHTO design or
legal loads is satisfactory.
5.3.2 Standard Bridge Sample
Comparisons made at the legal load level for the standard bridge sample are
broken into three bounding studies. This was necessary because the bridges in the
standard bridge sample did not have unique ADTT values; due to this the LRFR rating
factor data generated are based on assumed live load factors. However, this allowed for
the effects of several different factors in the LRFR methodology to be studied. The rating
factor data generated for these studies was gathered from rating analysis performed at the
Legal load level of rating for the LRFR and the Operating level of the LFR under
ALDOT legal loads. The three bounding studies were performed by varying the live load
factor, L? , and the product of the condition factor, ?c, and system factor, ?s. The first
bounding study shows the effect of varying L? from 1.4 and 1.8 while keeping the
product of ?c and ?s at 1.0. The second bounding study shows the effect of the product of
?c and ?s varying from 1.0 to 0.85 while keeping L? at 1.4. The third bounding study
shows the possible effect that actual ADTT values can have on L? and the LRFR rating
97
factor. Within this third study, three unique ADTT values were provided for each
standard bridge used, bounding the L? factors for each bridge; while ?c and ?s were held
at 1.0.
5.3.2.1 L? Bounding Study Results
The L? factor for the LRFR at the Legal rating level can range from 1.4 to 1.8
depending on a bridge?s ADTT (AASHTO 2003). This variation of L? can change a
bridge?s rating factor by nearly 30% under the LRFR. This bounding study shows how
this variation in L? can influence LRFR and LFR comparisons. The LRFR and LFR data
that are presented in this section are limited to the controlling ALDOT truck for each load
effect, for both interior and exterior girders.
Results for the interior girder are shown in Figures 5 - 13 and 5 - 14 for moment
and shear load effects, respectively. For moment load effects, the LRFR produced lower
rating results than LFR independent of L? , with all the data falling within Region 5 of the
plot. This trend is similar to what was seen at the Inventory level of rating. The variation
of L? only served to amplify the degree to which the LRFR rating factors are below the
LFR factors. This would be especially important in cases where posting is required (i.e.
for bridges with rating factors below 1.0). The shear results for the majority were also
found within Region 5. However the possible influence L? can have is seen on the few
shear rating results found in region 6 for L? equal to 1.4. Two of these bridges when L?
is increased to 1.8 fall into Region 5 changing the rating method that controlled them
98
from LFR to LRFR Similar results were found for the exterior girders for the standard
bridge sample as seen in Figures 5 - 15 and 5 - 16 for moment and shear, respectively.
Figure 5 - 13: Effect of varying L? on Moment Rating at the Legal Level for
Interior Girders
99
Figure 5 - 14: Effect of varying L? on Shear Rating at the Legal Level for
Interior Girders
100
Figure 5 - 15: Effect of varying L? on Moment Rating at the Legal Level for
Exterior Girders
101
Figure 5 - 16: Effect of varying L? on Shear Rating at the Legal Level for
Exterior Girders
5.3.2.2 ?c and ?s Bounding Study Results
According to the AASHTO MCE LRFR (2003), the product of the condition
factor, ?c, and system factor, ?s cannot be taken less than 0.85. To study the effect that
the product of ?c and ?s can have on the rating results for ALDOT?s legal loads, a
bounding study was performed on the combined effect of the factors. For this study the
LFR Operating level rating results, which remain the same, are compared to the bounded
102
results of the LRFR at the Legal load level with the product of ?c and ?s effect ranging
from 0.85 to 1.0. The live load factor, L? , for this study is fixed at 1.4. The AASHTO
MCE 2005 allows ?s to be greater than 1.0, when higher order analysis is performed to
determine a member specific structural redundancy; however, this effect was not studied.
Comparisons of the rating results for interior girders of the standard bridge sample
are shown in Figures 5 - 17 and 5 - 18 for moment and shear, respectively. The effect of
the product of ?c and ?s on the rating comparisons between LRFR and LFR is very
similar to what was seen in the bounded study of L? . All the moment rating data is once
again found in Region 5 of Figure 5 - 17 for the combined ?c?s effect equal to 0.85 and
1.0. The only change lowering the combined ?c?s effect had, was to increase the degree
to which LRFR produced lower factors than LFR. Unlike changes in L? , which had a
fixed effect on a bridge?s rating factor, changes to the product of ?c and ?s have varying
impacts on different bridge?s rating factors due their effect on the factored resistance of a
member. Shear data predominately was found within Region 5 of Figure 5 - 18 with few
exceptions in Region 6. Similar to the trend seen for L? when ?c?s is reduced, the parts
of the data found in Region 6 shift to Region 5. Similar trends were seen for exterior
girders as seen in Figures 5 - 19 and 5 - 20 for moment and shear, respectively.
103
Figure 5 - 17: Effect of varying ?c and ?s, on Moment Rating at the Legal Level for
Interior Girders
104
Figure 5 - 18: Effect of varying ?c and ?s, on Shear Rating at the Legal Level for
Interior Girders
105
Figure 5 - 19: Effect of varying ?c and ?s, on Moment Rating at the Legal Level for
Exterior Girders
106
Figure 5 - 20: Effect of varying ?c and ?s, on Shear Rating at the Legal Level for
Exterior Girders
5.3.2.3 Varying ADTT Bounding Study Results
The standard bridge sample is composed of bridges that are used repeatedly
throughout Alabama?s bridge inventory. As a result, each bridge in the sample does not
have unique ADTT, average daily truck traffic, data. To provide a reference point as to
how actual ADTT values on standard bridges will affect rating results under the LRFR a
small sample of bridges were analyzed using multiple ADTT values. ALDOT provided
107
three different ADTT values for four standard bridges for the study, shown in Table 5 ?
19. Additionally, in Table 5 ? 19 are the corresponding L? values for the ADTT data
provided by ALDOT. Bridges ?STD C2411 34? and ?STD PC34 RC 24R? have a the
same L? due to the relationship between ADTT and L? ; this relationship is described in
Section 2.5.
Table 5 - 19: Standard Bridge Varying ADTT Values
The lack of change in L? factors provided in Table 5 - 19 is due to the low ADTT
values for the studied bridges. The selected bridges, using their unique L? factors, were
analyzed under ALDOT?s legal loads for LRFR at the Legal load level of rating. The
rating results from the controlling legal load are provided in Table 5 - 20 along with
corresponding LFR factors. Similar to what has been seen before, the LRFR produced
lower rating factors than the LFR. Additionally, as L? increased due to higher ADTT
values, the difference between the LRFR and LFR factors increase.
108
Table 5 - 20: Standard Bridge Sample Rating Results, Legal level, Varying ADTT
Structural System Type
5.3.3 Unique Bridge Sample
Comparisons made for the unique bridge sample are presented between the LRFR
at the Legal Load level and the LFR at the Operating level under ALDOT legal loads.
The following assumptions were used for the LRFR analysis:
? ?c set to 1.0
? ?s as specified in the AASHTO MCE LRFR 2003
( 1.0 for every bridge in sample )
? L? based on bridge-specific ADTT
The rating comparisons made in this section are for the controlling ALDOT legal
load. Similar to the results found for the standard bridge studies, the Tri-Axle and 6-Axle
loads produced the lowest rating factor results, or controlling rating factors. Table 5 ? 21
109
provides a summary of the number of times each truck controlled for LRFR and LFR, for
each load effect, and for both interior and exterior girders.
Table 5 - 21: Controlling ALDOT Truck Comparison at Legal Level for the Unique
Bridge Sample
The presentation of the data for the unique bridge sample is broken down into the
following sections to highlight the different trends that were found. Section 5.3.3.1
provides an overall summary of the LRFR and LFR rating factor data compared at this
Legal load level under ALDOT Legal loads. Section 5.3.3.2 looks at how the age of the
bridge may influence the rating data. Section 5.3.3.3 briefly discusses the relationships
between span length and girder spacing on the rating results. Section 5.3.3.4 presents the
suggested load posting for the unique bridge sample based on the AASHTO MCE LRFR
(2003) recommendations.
5.3.3.1 Overall Summary
A summary of the rating factors used for the unique bridge sample (refer to
Section 3.1.2) comparisons at the Legal load level of rating are provided in Table 5 - 22
and 5 - 23. Table 5 - 22 provides the moment and shear rating factors generated for both
110
the interior and exterior girders for each bridge in the sample, under the LRFR
methodology. Additionally, the controlling rating factor for the interior and exterior
girders are identified, as well as the controlling rating factor for the bridge. Table 5 - 23
provides the same rating factor information as Table 5 - 22 but for the LFR methodology.
111
Table 5 - 22: LRFR Rating Factors Generated for the Unique Bridge Sample at the
Legal Load Rating Level
112
Table 5 - 23: LFR Rating Factors Generated for the Unique Bridge Sample at the Legal
Load Rating Level
Presented in Figure 5 -21 is the LRFR versus LFR moment rating factor data for
the controlling ALDOT legal load data for exterior and interior girders. Similar to what
has been seen before, the majority of the data is found to be within Region 5 of the plot,
113
indicating that for the unique bridge sample at the legal load level the LRFR produces
lower rating results than LFR. Of key interest to ALDOT however would be the portions
of the data that fall into Regions 5 - 1 and 5 - 2. Data found in Regions 5 - 1 signify
bridges that would require posting in both the LFR and the LRFR; under the LRFR the
posting loads are likely to be lower. Data found in Region 5 - 2 corresponds to bridges
that are not required to be posted under the LFR but would require posting under the
LRFR. Data found in Regions 5 - 3 has no impact on posting under either rating system.
0.0
1.0
2.0
3.0
4.0
0.0 1.0 2.0 3.0 4.0
LFR Moment Rating Factor
LR
FR
M
om
en
t R
ati
ng
F
ac
to
r
Ext Girder
Int Girder
Figure 5 - 21: Moment Rating Factor Comparison at the Legal Load Level for the
Unique Bridge Sample
114
Breaking the information found in Figure 5 - 21 down into material types can
reveal which types of bridges may be more prone to be found in Regions 5 - 1 and 5 - 2.
A material type plot of the data for the interior girder is shown in Figure 5 - 22. From
this we find that the majority of bridges in Regions 5 - 1 and 5 - 2 are simply and
continuously supported reinforced concrete bridges. The continuously supported steel
bridges appear to be broken into two groups. One group found in Region 5 - 3 and the
second primarily in Region 5 - 2. Upon inspection of the two groups, it was discovered
that bridges with high rating factor had span lengths over 140 whereas the bridges with
lower rating factors had span lengths under 100 feet. The majority of simply and
continuously supported prestressed concrete bridges were found to be in Region 5 - 3.
Similar trends were found for the exterior girder, see Figure 5 - 23.
115
0.0
1.0
2.0
3.0
4.0
0.0 1.0 2.0 3.0 4.0
LFR Moment Rating Factor
LR
FR
M
om
en
t R
ati
ng
Fa
cto
r
RC SS
RC Con
Steel SS
Steel Con
PS SS
PS Con
Figure 5 - 22: Moment Rating Factor Material Type Comparison at the Legal Load
Level for the Unique Bridge Sample Interior Girders
116
0.0
1.0
2.0
3.0
4.0
0.0 1.0 2.0 3.0 4.0
LFR Moment Rating Factor
LR
FR
M
om
en
t R
ati
ng
Fa
cto
r
RC SS
RC Con
Steel SS
Steel Con
PS SS
PS Con
Figure 5 - 23: Moment Rating Factor Material Type Comparison at the Legal Load
Level for the Unique Bridge Sample Exterior Girders
Analysis of the shear ratings factor data produced similar trends to moment rating
data. An overview of the shear ratings for both interior and exterior girders is presented
in Figure 5 - 24. A material breakdown of this shear data for the interior and exterior
girders is shown in Figure 5 - 25 and 5 - 26, respectively. Similar trends to what were
seen in the moment data are found in the shear data with one exception. Prestressed
concrete, simply and continuously supported, bridges tend to transition from Region 5 - 3
into Region 6 - 3 as the rating factors for each method increase. Reinforced concrete,
simply and continuously supported, bridges tend to transition from Region 5 - 2 into
117
Region 5 - 3 as the rating factors for each method increase. Steel simply and
continuously supported bridges primarily were found in Region 5 - 3 These trends would
indicates that as shear rating factors increase so do the LRFR to LFR ratios.
0.0
1.0
2.0
3.0
4.0
0.0 1.0 2.0 3.0 4.0
LFR Shear Rating Factor
LR
FR
S
he
ar
R
at
ing
Fa
ct
or
Ext Girder
Int Girder
Figure 5 - 24: Shear Rating Factor Comparison at the Legal Load Level for the Unique
Bridge Sample
118
0.00
1.00
2.00
3.00
4.00
0 1 2 3 4
LFR Shear Rating Factor
LR
FR
S
he
ar
R
ati
ng
Fa
cto
r
RC SS
RC Con
Steel SS
Steel Con
PS SS
PS Con
Figure 5 - 25: Shear Rating Factor Material Type Comparison at the Legal Load Level
for the Unique Bridge Sample Interior Girders
119
0.00
1.00
2.00
3.00
4.00
0 1 2 3 4
LFR Shear Rating Factor
LR
FR
S
he
ar
R
ati
ng
Fa
cto
r
RC SS
RC Con
Steel SS
Steel Con
PS SS
PS Con
Figure 5 - 26: Shear Rating Factor Material Type Comparison at the Legal Load Level
for the Unique Bridge Sample Exterior Girders
In addition to the material level of behavior between the LRFR and the LFR, the
structural system type differences were studied. The results are presented in the same
form previously discussed. Table 5 - 24 and 5 - 25 summarize the data for the flexural
load effects for interior and exterior girders respectively. Similar to what has been seen
before, LRFR for all material and structural system types produced lower rating results
than the LFR. Prestressed concrete bridges have the highest LRFR to LFR ratio ranging
from 0.62 to 0.82 between different structural systems, for interior girders. C ? Channel
bridges are observed to have the significantly lowest LRFR to LFR ratio for interior
120
girders and the highest LRFR to LFR ratio for exterior girders; similar to the trends
previously observed. Table 5 - 26 and 5 - 27 summarizes the data for the shear load
effects for interior and exterior girders respectively. Similar to the trends previously
stated, the LRFR produced nearly equal or lower rating results than the LFR for the shear
load effect.
Table 5 - 24: Mean and Standard Deviation Data at the Legal Load Level for the Unique
Bridge Sample ? Interior Girder Moment Rating Data
121
Table 5 - 25: Mean and Standard Deviation Data at the Legal Load Level for the Unique
Bridge Sample ? Exterior Girder Moment Rating Data
Table 5 - 26: Mean and Standard Deviation Data at the Legal Load Level for the Unique
Bridge Sample ? Interior Girder Shear Rating Data
122
Table 5 - 27: Mean and Standard Deviation Data at the Legal Load Level for the Unique
Bridge Sample ? Exterior Girder Shear Rating Data
An additional point of comparison was made for the controlling load effect for
each rating methodology. Table 5 - 28 shows the results of this comparison. The data in
this table was constructed by counting the number of times a rating factor for each load
effect controlled for a bridge within the sample. The data indicates that for the LRFR
methodology exterior girder moment load effect mainly controlled. For the LFR
methodology it can be observed that the sample was nearly evenly controlled across all
load effects with the exception of the exterior girder shear load effect.
123
Table 5 - 28: Controlling Load Effect Comparison, Legal Load Level for the Unique
Bridge Sample
Note: The unique bridge sample consists of 45 bridges
The final point of comparison was on the absolute controlling rating factor
between the two rating methodologies. The absolute controlling rating data used for this
comparison can be found in the previously shown Tables 5 - 22 and 5 - 23. Provided in
Figure 5 - 27 is a LRFR verses LFR plot of the absolute controlling rating data. From this
plot it is seen that the majority of the data falls into Region 5 with only a single data point
found in Region 6. This indicates that the LRFR produced lower rating results than the
LFR in general.
124
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
LFR Controlling Rating Factor
LR
FR
C
on
tro
llin
g R
at
ing
Fa
cto
r
Figure 5 - 27: Controlling Rating Factor Comparisons at the Legal Load Level
Additionally, the absolute controlling load effect and rating methodology was
investigated; Table 5 - 29 shows the results of this investigation. The data provided in
Table 5 - 29 is the total number of times each load effect and methodology controlled for
the bridge sample. This data indicates that the LRFR exterior girder moment load effect
primarily controlled. This finding is in agreement with the previously reported results
showing the LRFR producing nearly equal or lower rating results than the LFR, in
general.
125
Table 5 - 29: Controlling Load Effect and Rating Methodology at the Legal Load Level
Note: The unique bridge sample consists of 45 bridges
5.3.3.2 Bridge Age
The potential effect of bridge age on the rating results was also investigated. The
age of the bridge used in this portion of the study was assumed to be the fiscal year of the
bridge as indicated on each bridge?s set of plans. The fiscal year corresponds to the year
in which the plans for the bridge were produced. Figure 5 - 28 shows a plot of interior
girder moment rating factor, for both the LRFR and LFR, against the bridge?s fiscal year.
As can be seen, a trend emerges that progressively newer bridges have the tendency to
produce a higher rating factor for each methodology. Additionally, only two bridges
built after the mid-1980s yielded unsatisfactory rating results for either rating system, for
the flexural load effect.
126
0.0
1.0
2.0
3.0
4.0
1920 1940 1960 1980 2000 2020
Bridge Fiscal Year
Mo
m
en
t R
at
ing
Fa
cto
r
LRFR
LFR
Figure 5 - 28: Bridge Age and Moment Rating Factor Comparison at the Legal Load
Level for the Unique Bridge Sample
A material breakdown of just the LRFR data is seen in Figure 5 - 29, which shows
additional trends. In general, the trend of the fiscal year of the bridge increasing along
with the moment rating factor of a bridge can be seen. On the material level, this trend
can be well observed in continuously supported steel bridges. Reinforced concrete
simply and continuously supported bridges, however, tend to have similar rating factors
under the LRFR independent of their fiscal age. Similar trends were seen for exterior
127
girders. These trends suggest that a correlation between a bridge?s moment rating factor
and its fiscal age does exist.
0.0
1.0
2.0
3.0
4.0
1920 1940 1960 1980 2000 2020
Bridge Fiscal Year
Mo
me
nt
R
ati
ng
Fa
cto
r
LRFR RC SS
LRFR RC Con
LRFR Steel SS
LRFR Steel Con
LRFR PS SS
LRFR PS Con
Figure 5 - 29: Bridge Age and Moment Rating Factor Comparison, Material Level, at
the Legal Load Level for the Unique Bridge Sample
Analyzing a bridge?s fiscal year compared to shear rating data yielded less
apparent trends than when compared to the moment rating data, as can be seen in Figure
5 - 30. Comparing the LRFR and LFR factors to a bridge?s age produced a large degree
of scatter with no apparent trends for the shear rating factor data. Breaking the data
down into its material level for the LRFR yielded no additional trends, as shown in
Figure 5 ? 31 for interior girders. Similar results were found for exterior girders. This
128
suggests that for both LRFR and LFR little correlation exists between a bridge?s shear
rating factor and its fiscal age.
0.0
1.0
2.0
3.0
4.0
1920 1940 1960 1980 2000 2020
Bridge Fiscal Year
Sh
ea
r R
ati
ng
Fa
cto
r
LRFR
LFR
Figure 5 - 30: Bridge Age and Shear Rating Factor Comparison at the Legal Load Level
for the Unique Bridge Sample
129
0.0
1.0
2.0
3.0
4.0
1920 1940 1960 1980 2000 2020
Bridge Fiscal year
Sh
ea
r R
ati
ng
Fa
cto
r
LRFR RC SS
LRFR RC Con
LRFR Steel SS
LRFR Steel Con
LRFR PS SS
LRFR PS Con
Figure 5 - 31: Bridge Age and Shear Rating Factor Comparison, Material Level, at the
Legal Load Level for the Unique Bridge Sample
5.3.3.3 Span Length and Girder Spacing
The potential effect of span length and girder spacing on the rating results was
investigated. Figures 5 - 32 and 5 - 33 show the interior girder moment rating factors for
LRFR versus span length and girder spacing, respectively. Little correlation between
span length and the LRFR moment rating factor can be observed. The one exception
however, is for continuously supported steel bridges for which the rating factor is
observed to increase with span length.
130
0.00
1.00
2.00
3.00
4.00
0 40 80 120 160 200
Span Length (ft)
Mo
me
nt
R
ati
ng
Fa
cto
r
LRFR RC SS
LRFR RC CON
LRFR Steel SS
LRFR Steel Con
LRFR PS SS
LRFR PS CON
Figure 5 - 32: Span length and Moment Rating Factor Comparison, Material Level, at
the Legal Load Level for the Unique Bridge Sample
0.00
1.00
2.00
3.00
4.00
5 6 7 8 9 10 11
Girder Spacing (ft)
Mo
me
nt
R
ati
ng
Fa
cto
r
LRFR RC SS
LRFR RC CON
LRFR Steel SS
LRFR Steel Con
LRFR PS SS
LRFR PS CON
Figure 5 - 33: Girder Spacing and Moment Rating Factor Comparison, Material Level,
at the Legal Load Level for the Unique Bridge Sample
131
Little correlation between girder spacing and LRFR moment rating factors was
found with one exception for continuously supported steel bridges, for which the rating
factor is observed to increase with girder spacing. For the completeness the same
variables, span length and girder spacing, are plotted against the LRFR / LFR in Figures
5 - 34 and 5 - 35; however, little additional information was learned. Similar results were
found for both moment and shear for interior and exterior girders.
0.00
0.50
1.00
1.50
2.00
0 40 80 120 160 200
Span Length (ft)
LR
FR
/ L
FR
LRFR RC SS
LRFR RC CON
LRFR Steel SS
LRFR Steel Con
LRFR PS SS
LRFR PS CON
Figure 5 - 34: Span length and LRFR to LFR Ratio Comparison, Material Level, at the
Legal Load Level for the Unique Bridge Sample
132
0.00
0.50
1.00
1.50
2.00
5 6 7 8 9 10 11
Girder Spacing (ft)
LR
FR
/ L
FR
LRFR RC SS
LRFR RC CON
LRFR Steel SS
LRFR Steel Con
LRFR PS SS
LRFR PS CON
Figure 5 - 35: Girder Spacing and LRFR to LFR Ratio Comparison, Material Level, at
the Legal Load Level for the Unique Bridge Sample
5.3.3.4 LRFR Load Posting Recommendations
The load posting recommendations found in the AASHTO MCE LRFR (2003),
were applied to the ALDOT legal loads for the unique bridge sample of this study. The
recommended posting procedure under the LRFR uses the controlling rating factor for a
bridge and a legal load?s weight to determine the posting load, as described in Section
2.8. For comparison purposes ALDOT?s posting load procedure was used to determine
LFR, load posting data. ALDOT?s current posting load procedure uses the controlling
LFR legal load rating factor and a legal load?s weight. The posting load is determined by
multiplying a load?s controlling rating factor by the weight of the load, in units of tons.
Figure 5 - 36 graphically presents the differences in the LRFR posting load equation and
133
ALDOT?s posting load procedure. As Figure 5 - 36 shows, for a given rating factor
LRFR load postings will be lower than an LFR load posting, calculated by ALDOT?s
procedure. Load posting is only required for when loads produce rating factors below 1.0
for both methods.
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.2 0.4 0.6 0.8 1.0
Rating Factor
Po
sti
ng
W
eig
ht
Fr
ac
tio
n
LRFR
LFR
Figure 5 - 36: Posting Weight Fraction Compared to Rating Factor
The ALDOT legal loads weights are summarized in Table 5 - 30. Using this
LRFR load posting procedure, the load postings found in Table 5 - 31 were developed.
Load posting information for each truck per load effect for interior and exterior girders
according to the LRFR procedure can be found in Appendix E Tables E - 1 through E - 4.
Using this ALDOT?s LFR load posting procedure, the load postings found in Table 5 - 32
were developed. Load posting information for each truck per load effect for interior and
134
exterior girders according to the LFR procedure can be found in Appendix E Tables E - 5
through E - 8. As expected the load posting data generated under the LRFR procedure,
Table 5 - 31, is lower than the LFR load posting data, Table 5 - 32. Additionally the
differences in the number of bridges requiring load posting under the two methods is
seen. From the unique bridge sample 23 bridges, just over half the sample, required load
posting under the LRFR methodology. From the unique bridge sample only 8 bridges
required load posting under the LFR methodology. Therefore, the number of bridges that
require load posting under the LRFR is triple the number that require load posting under
the LFR.
Table 5 - 30: Summary ALDOT Legal Loads Weights
Table 5 - 31: ALDOT Legal Loads LRFR Posting Weights
135
Table 5 - 32: ALDOT Legal Loads LFR Posting Weights
136
137
5.3.4 Summary
Analysis of the standard and unique bridge samples at the Legal Load level of
rating led to the following general findings:
? ALDOT legal loads are not enveloped by AASHTO typical legal loads
? ALDOT legal loads are not enveloped by the HL-93 live load model
? Moment rating factors for exterior girders tend to control over interior
girders under the LRFR, as opposed to no dominant load effect per girder
was observed for the LFR
? For moment load effects the LRFR methodology produces generally
lower rating results than the LFR methodology
? For shear load effects the LRFR methodology in general produces equal
or lower rating results than the LFR methodology
? Variations in L? and ?c?s only amplify the degree to which LRFR
produces lower rating results than LFR
? Newer bridges tend to have higher LRFR and LFR factors
? Load posting values produced under the LRFR were found to be
significantly lower than load postings values under the LFR
? The number of bridges requiring load posting for the unique bridge
sample was found to be much larger for the LRFR than the LFR
138
5.4 Permit Load Rating
Work conducted at the permit load level consisted of two main tasks. The first
task was the selection of the permit bridge sample. The second task was a comparison of
LRFR and LFR rating factor data. Section 5.4.1 describes the permit bridge sample and
its selection. Section 5.4.2 compares the rating factor data of the permit sample.
5.4.1 Permit Bridge Sample
The permit bridge sample is a collection of bridges, from both the unique and
standard bridge samples, that are eligible for overweight load evaluation under at least
one of the rating methodologies. Initially, all 95 bridges from the unique and standard
bridge samples were considered for inclusion in the permit bridge sample. However, as
the rating criteria was checked, the sample size decreased. To aid in the discussion of the
permit bridge sample, the Venn diagram shown in Figure 5 ? 28 is used. Each region of
this figure refers to a different set of bridges. Region ?A? represents the set of all 95
bridges from the unique and standard bridge samples. Region ?B? represents the set of
bridges that are allowed to be permitted under the LFR. In the LFR rating methodology,
bridges are allowed to be permitted if they are found to be satisfactory at the Operating
level under the HS-20 design truck (AASHTO 2003). Of the 95 bridges, 76 were found
to be allowed to be permitted under the LFR. Region ?C? represents the set of bridges
that are allowed to be permitted under the LRFR. Permitting allowance under the LRFR
is determined based on whether a bridge is found to be satisfactory at the Legal load level
under at least the AASHTO standard legal loads, as shown in Section 2.7. Of the 95
bridges considered, 60 meet the LRFR criteria for permit allowance. Region ?D?
139
represents the set of bridges that are allowed to be permitted under both rating
methodologies, which consists of 59 bridges. Therefore, the permit bridge sample
consists of a total of 77 bridges, which as indicated above, is the number of bridges that
are allowed to be permitted under at least one of the rating methodologies.
Figure 5 - 37: Permit Bridge Sample Diagram
A material type breakdown of the permit bridge sample is provided in Table 5 -
24 along with material type breakdowns of the B, C, and D regions of the sample. This
shows that while the LRFR allows fewer bridges to be permitted than the LFR, there is
no material type that is more susceptible to not being allowed under either of the two
systems.
140
Table 5 - 33: Material Type Breakdown of the Permit Bridge Sample
5.4.2 Permit Rating Results
The permit bridge sample was analyzed under the ten ALDOT permit trucks
previously described in Section 2.9.3. The trucks were analyzed at the Operating level of
the LFR with a live load factor of 1.3. Analysis under the LRFR was done at the Permit
level with a live load factor of 1.15. The LRFR live load factor of 1.15 corresponds to
the lower bound of the possible live load factors for permit trucks and assumes a single
trip frequency with the permit truck being escorted and no other vehicles on the bridge
during crossing.
The comparisons made in this section in regards to the ALDOT permit loads are
for the controlling permit vehicle. Table 5 ? 34 presents the breakdown of which permit
vehicle controlled for each rating method, interior and exterior girder, and load effect.
Data is only presented for Vehicles 1, 3, 4, 7, and 8 due to these five vehicles controlling
all of the permitting analysis. Vehicle 4 was found to predominately control in moment
141
rating factors with Vehicle 3 and 4 largely controlling in shear rating factors for both
methodologies.
Table 5 - 34: Controlling Permit Vehicles
The LRFR to LFR comparisons are presented in a similar format as before. A
summary of the controlling rating factors used in the comparisons at the Permit level of
rating are provided in Table 5 - 35 through 5 - 38. Table 5 - 35 and 5 - 36 provides the
moment and shear rating factors generated for both the interior and exterior girders for
each bridge in the sample, under the LRFR methodology. Additionally, the controlling
rating factor for the interior and exterior girders are identified, as well as the controlling
rating factor for the bridge. Table 5 - 37 and 5 - 38 provides the same rating factor
information but for the LFR methodology.
142
Table 5 - 35: LRFR Rating Factors Generated for the Permit Bridge Sample at the
Permit Rating Level, Part 1
143
Table 5 - 36: LRFR Rating Factors Generated for the Permit Bridge Sample at the
Permit Rating Level, Part 2
144
Table 5 - 37: LFR Rating Factors Generated for the Permit Bridge Sample at the Permit
Rating Level, Part 1
145
Table 5 - 38: LFR Rating Factors Generated for the Permit Bridge Sample at the Permit
Rating Level, Part 2
Comparisons of the moment rating factors are presented in Figure 5 - 38. Similar
to previous results, the majority of data is found within Region 5, with only a few data
points found in Region 6, indicating that for the LRFR produces nearly equal or lower
rating results when compared to the LFR at the permit level. Bridges found within
146
Region 5 - 1, while having lower LRFR factors, are found to be unsatisfactory for both
LRFR and LFR for the controlling permit truck. This results in the controlling load not
being permitted for bridges found within this region. Bridges within Region 5 - 3, while
having lower LRFR factors, are found to be satisfactory for both LRFR and LFR for the
controlling permit load. This results in permits being granted under LRFR and LFR for
all bridges within this region. Data found in Region 5 - 2 are satisfactory under LFR but
not under LRFR. These represent bridges where permits would be under LFR, but not
LRFR.
0.0
1.0
2.0
3.0
4.0
0.0 1.0 2.0 3.0 4.0
LFR Moment Rating Factor
LR
FR
M
om
en
t R
ati
ng
F
ac
to
r
Ext Girder
Int Girder
Figure 5 - 38: Moment Rating Factor Comparison at the Permit Level for the Permit
Bridge Sample
147
Breaking down the LRFR interior girder moment rating data into its material
types produced little additional information. As Figure 5 - 39 shows, nearly all the
material types can be found in Regions 5 - 1, 5 - 2, and 5 - 3. However a large amount of
the simply supported steel and presstresed concrete bridges can be found in Region 5 ? 3.
Additionally only simply supported prestressed concrete bridges and steel continuously
supported bridges were found in Region 6 ? 3. Similar trends were found for the exterior
girders.
0.00
1.00
2.00
3.00
4.00
0 1 2 3 4
LFR Moment Rating Factor
LR
FR
M
om
en
t R
ati
ng
Fa
ct
or
RC SS
RC Con
Steel SS
Steel Con
PS SS
PS Con
Figure 5 - 39: Moment Rating Factor Material Type Comparison at the Permit Level for
the Permit Bridge Sample Interior Girders
148
Figure 5 - 40 shows the LRFR to LFR shear rating data for the interior and
exterior girders. As can be seen there are large portions of the data in both Region 5 and
6 of the plot. Additionally a trend can be seen that as rating factors become greater than
2.0 for either rating methodology the data primarily falls in Region 5. However, for shear
rating factors less than 2.0 for either rating methodology, the data falls in both Regions 5
and 6.
0.0
1.0
2.0
3.0
4.0
0.0 1.0 2.0 3.0 4.0
LFR Shear Rating Factor
LR
FR
S
he
ar
R
ati
ng
F
ac
to
r
Ext Girder
Int Girder
Figure 5 - 40: Shear Rating Factor Comparison at the Permit Level for the Permit Bridge
Sample
Breaking down the LRFR interior girder shear rating data into its material types
produced some additional information as seen in Figure 5 - 41. From this plot it is shown
149
that all the simply and continuously supported steel bridges can be found in Region 5.
Additionally all the reinforced and prestressed concrete bridges tend to be near the border
of Region 5 with Region 6.
0.00
1.00
2.00
3.00
4.00
0 1 2 3 4
LFR Shear Rating Factor
LR
FR
S
he
ar
R
ati
ng
Fa
cto
r
RC SS
RC Con
Steel SS
Steel Con
PS SS
PS Con
Figure 5 - 41: Moment Rating Factor Material Type Comparison at the Permit Level for
the Permit Bridge Sample Interior Girders
Statistical analysis was performed on the Permit level rating factor data and the
results are provided in Tables 5 - 39 through 5 - 42. Tables 5 - 39 and 5 - 40 provide the
moment rating factor data analysis for the interior and exterior girders. Similar to
previous findings the LRFR is shown to produce nearly equal or lower rating results
when compared to the LFR. The C-Channel structural system type produced unusual low
150
and high LRFR to LFR ratios for the interior and exterior girders, respectively, as seen
previously.
Table 5 - 39: Mean and Standard Deviation Data at the Permit Level for the Permit
Bridge Sample ? Interior Girder Moment Rating Data
Table 5 - 40: Mean and Standard Deviation Data at the Permit Level for the Permit
Bridge Sample ? Exterior Girder Moment Rating Data
151
Tables 5 - 41 and 5 - 42 provide the shear data analysis for the interior and
exterior girders. Similar to pervious data the LRFR is shown to generally produce nearly
equal or lower rating factors when compared to the LFR. There are a few exceptions to
this with regards to a few reinforced and prestressed concrete bridge types showing the
LRFR produced slightly higher rating factors than the LFR.
Table 5 - 41: Mean and Standard Deviation Data at the Permit Level for the Permit
Bridge Sample ? Interior Girder Shear Rating Data
152
Table 5 - 42: Mean and Standard Deviation Data at the Permit Level for the Permit
Bridge Sample ? Exterior Girder Shear Rating Data
Additionally, as was observed with the Design Inventory level rating data, the
Permit level rating data suggests that the exterior girder produces lower flexural rating
factors than the interior girder for the LRFR methodology, as shown in Table 5 - 43.
Table 5 - 43: Mean and Standard Deviation Data at the Permit level for the Permit
Bridge Sample ? Interior to Exterior LRFR Moment Rating Comparison
153
An additional point of comparison was made for the controlling load effect for
each rating methodology. Table 5 - 44 shows the results of this comparison. The data in
this table was constructed by counting the number of times a rating factor for each load
effect controlled for a bridge within the sample. The data indicate that, for the LRFR
methodology, exterior girder moment load effects mainly controlled. For the LFR
methodology, the sample was more heavily controlled by moment load effects for both
exterior and interior girders.
Table 5 - 44: Controlling Load Effect Comparison, Permit Level for the Permit Bridge
Sample
Note: The permit bridge sample consists of 77 bridges
Additionally, the absolute controlling rating factor between the two rating
methodologies was compared. The absolute controlling rating factor data used for this
comparison can be found in the previously shown Tables 5 - 35 through 5 - 38. Provided
in Figure 5 ? 42 is a LRFR verses LFR plot of the absolute controlling rating data. From
this plot it is seen that the majority of the data falls into Region 5 with only two data
points found in Region 6. This indicates that the LRFR produced lower rating results
than the LFR in general.
154
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0
LFR Controlling Rating Factor
LR
FR
C
on
tro
llin
g R
at
ing
Fa
cto
r
Figure 5 - 42: Controlling Rating Factor Comparisons at the Permit Level for the Permit
Bridge Sample
The final point of comparison was on the absolute controlling load effect and
rating methodology, Table 5 - 45 shows the results of this comparison. The data
provided in Table 5 - 45 is the total number of times each load effect and methodology
controlled for the bridge sample. This data indicates that the LRFR exterior girder
moment load effect primarily controlled.
155
Note: The permit bridge sample consists of 77 bridges
Table 5 - 45: Controlling Load Effect and Rating Methodology at the Permit Level for
the Permit Bridge Sample
5.4.3 Summary
Analysis of the Permit bridge samples at the Permit level of rating provided the
following general findings:
? The LFR allows a slightly greater number of bridges to be considered for
permitting compared to the LRFR
? Permit Vehicle 3 and 4 largely controlled the rating analysis for both
LRFR and LFR for both load effects
? LRFR tends to produce nearly equal or lower moment rating factors
compared to the LFR
? LRFR tends to produce nearly equal or lower shear rating factors
compared to the LFR, with a few exceptions
? Exterior girders tend to control over interior girders for moment load
effects under the LRFR
156
5.5 Analysis of Rating Results
As shown in the previous sections, reviewing the results at each of the LRFR
rating levels reveals that on average the LRFR produces nearly equal or lower rating
results when compared to the LFR. To gain additional insight into the observed trends,
the results at the legal load level were analyzed in greater detail. In particular, the
variation of each of the of the components of the fundamental rating equation, Equation 2
- 1, (i.e. the factored capacity, C, factored dead load effect, D, and factored live load
effect, L) with the rating factor was investigated. Figure 5 - 43, shows a plot of LRFR to
LFR component ratios verses the LRFR to LFR moment rating factor ratio, for the
standard and unique bridge samples at the Legal load level of rating. The live load data
used in this study was from the ALDOT Tri-Axle load model. Three sets of data are
shown on the y-axis. The first set is for the LRFR to LFR capacity ratio, denoted as ratio
C ratio in the figure. The second set is for the LRFR to LFR dead load effect ratio,
denoted as D ratio in the figure. The third set is for LRFR to LFR live load effect ratio,
denoted as L ratio in the figure.
157
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
0.20 0.40 0.60 0.80
LRFR to LFR Moment Rating Ratio
LR
FR
to
LF
R
Co
mp
on
en
t R
ati
o
C ratio
L ratio
D ratio
Figure 5 - 43: LRFR to LFR Component Ratio Comparisons for Exterior Girder
Moment Rating Factors
Examining Figure 5 - 43 two important observations are made. First, it can be
observed that the D ratio for the two methodologies is nearly constant for all LRFR to
LFR moment rating factor ratios. This is expected because there is be no difference in
the way the dead load is calculated between the two methodologies. The observation that
the D ratios is slightly less than 1.0 is due to the difference in dead load factors for the
two methodologies (i.e. dead load factor for LRFR is equal to1.25 and 1.3 for the LFR).
The second observation is that the C ratio for the two methodologies is relativity
constant, being equal to 1.0 or slightly greater; with the exception of two data points, as
discussed next.
158
0.50
0.75
1.00
1.25
1.50
0.20 0.40 0.60 0.80
LRFR to LFR Moment Rating Ratio
LR
FR
to
LF
R
Ca
pa
cit
y R
ati
o
RC SS
RC Con
Steel SS
Steel Con
PS SS
PS Con
Figure 5 - 44: LRFR to LFR Capacity Ratio Comparisons for Exterior Girder Moment
Rating Factors
Breaking the resistance ratios studied previously into their material types, as
shown in Figure 5 - 44, reveals additional information about differences between the two
rating methodologies. In general, it can been seen that the resistance ratios for
reinforced concrete simply and continuously supported bridges and steel simply
supported bridges are constant, indicating that little difference in the capacities between
the LRFR and the LFR is observed for these material types. Prestressed concrete simply
and continuously supported bridges tended to exhibit a C ratio of about 1.1 suggesting
that LRFR capacities are roughly ten percent higher than the LFR capacities. Steel
continuously supported bridges show a C ratio closer to 1.3 suggesting that the LRFR
capacities are on the order of thirty percent greater than the LFR capacities. The
159
differences in the capacities shown for these material types can be attributed to
differences in the capacity calculation guidelines found in the AASTHO LRFD Bridge
Design Specification (2007) and the AASTHO Standard Specification for Bridge Design
(2002), used for the LRFR and LFR respectively.
Examining the L ratio data from Figure 5 - 43 a decaying trend can be observed.
To investigate this trend the LRFR to LFR rating factor, RF, ratio is examined in greater
detail. The LRFR to LFR RF ratio can be written in the following form:
LFR
LFR
LRFR
LRFR
LFR
LRFR
L
DC
L
DC
RF
RF
)(
)(
?
?
= Equation 5 - 1
Equation 5 - 1 can be written in the following form:
))()( )((
LRFR
LFR
LFR
LRFR
LFR
LRFR
L
L
DC
DC
RF
RF
?
?= Equation 5 - 2
Since the capacity, C, and the dead load effect, D, have been shown to be
consistent between the rating methodologies, the ratio of the subtraction of the two can be
approximated as a constant, so that:
)1)(Constant(
LFR
LRFRLFR
LRFR
L
LRF
RF ? Equation 5 ? 3
Equation 5 - 3 can be written in the following form:
)1)(Constant(
LFR
LRFRLFR
LRFR
RF
RFL
L ? Equation 5 - 4
Examining Equation 5 - 4 reveals that when C and D are constant the ratio of L is
related to the ratio of RF through a decaying function. Therefore the decaying trend for
160
the L ratio data seen in Figure 5 - 43 can be expected when the C ratio and D ratio are
constant. To farther understand the decaying trend observed for the L ratio data, the
components of the L were investigated to determine a possible source for the trend.
For the investigation of the components of the live load effect, L, the standard and
unique bridge samples are studied separately. The samples are separated to study the
effect that the live load factor may have on the observed decaying trend. For this
investigation bridges in the standard bridge sample have a fix live load factor of 1.4 and
bridges in the unique bridge sample have a varying live load factor based on bridge
ADTT. Figure 5 - 45 and 5 - 46, shows the plots of LRFR to LFR live load component
ratios verses the LRFR to LFR rating factor ratio for the standard and unique bridge
samples, respectively. There are again three sets of data shown on the y-axis for these
plots. The first set is for the LRFR to LFR factored live load effect ratio, denoted as L
ratio in the figure. The second set is for the LRFR to LFR live load factor ratio, denoted
as A ratio in the figure. For the standard bridge sample, the A ratio is constant and is
equal to 1.08 as seen in Figure 5 - 45. The third set is for LRFR to LFR unfactored live
load effect, without live load factor, ratio, denoted as B ratio in the figure.
161
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
0.20 0.40 0.60 0.80
LRFR to LFR Moment Rating Ratio
LR
FR
to
LF
R
Liv
e L
oa
d
C
om
po
ne
nt
R
ati
o
L ratio
A ratio
B ratio
Figure 5 - 45: LRFR to LFR Live Load Component Ratio Comparisons for Standard
Bridge Sample Exterior Girder Moment Rating Factors
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
0.20 0.40 0.60 0.80
LRFR to LFR Moment Rating Ratio
LR
FR
to
LF
R
Liv
e L
oa
d
C
om
po
ne
nt
R
ati
o
L ratio
A ratio
B ratio
Figure 5 - 46: LRFR to LFR Live Load Component Ratio Comparisons for Unique
Bridge Sample Exterior Girder Moment Rating Factors
162
Examining Figure 4 - 45 reveals that when the live load factor ratio, A ratio, is
constant the decaying trend is observed to be in the B ratio, or the live load effect without
live load factor. Examining Figure 5 - 46 reveals that when the live load factor ratio, A
ratio, is variable the decaying trend is observed is not seen for the B ratio, or the live load
effect without live load factor. This implies the observed trend in the live load effect
ratio, L ratio, is due to the combined effects of the components live load effect (i.e. the
live load factor, live load distribution factor, and impact factor). This indicates that
variations in moment rating factors produced by the LRFR and LFR methodologies can
be contributed to the components of the live load effect.
A similar investigation was conducted for the shear load effects at the Legal load
level for the ALDOT Tri-Axle load on the unique and standard bridge samples. Figure
5 - 47 shows the plot of LRFR to LFR component ratios versus the LRFR to LFR shear
rating factor ratio. Three sets of data are shown on the y-axis. The first set is for the
LRFR to LFR capacity ratio, denoted as ratio C ratio in the figure. The second set is for
the LRFR to LFR dead load effect ratio, denoted as D ratio in the figure. The third set is
for LRFR to LFR live load effect ratio, denoted as L ratio in the figure.
163
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
0.20 0.40 0.60 0.80 1.00 1.20 1.40
LRFR to LFR Shear Rating Ratio
LR
FR
to
LF
R
Co
mp
on
en
t R
ati
o C ratio
L ratio
D raio
Figure 5 - 47: LRFR to LFR Component Ratio Comparisons for Exterior Girder Shear
Rating Factors
Examining Figure 5 - 47 three important observations are made. First, it
can be observed that the D ratio for the two methodologies is constant for all LRFR to
LFR shear rating factor ratios. This is expected because there is be no difference in the
way the dead load is calculated between the two systems, and is in agreement with the
moment rating factor analysis previously reviewed. The second observation is that the C
ratio for the two methodologies is no longer constant. The C ratio is observed to
increase with increasing rating factor ratios. A material type breakdown of the C ratio
data is provided in Figure 5 - 48. The third observation is that the decaying trend
164
previously observed for the L ratio data is no longer seen. This is due to the C ratio and
D ratio no longer being constant; therefore, the decaying trend would not be expected.
0.00
0.50
1.00
1.50
2.00
2.50
0.20 0.40 0.60 0.80 1.00 1.20 1.40
LRFR to LFR Moment Rating Ratio
LR
FR
to
LF
R
Ca
pa
cit
y R
ati
o
RC SS
RC Con
Steel SS
Steel Con
PS SS
PS Con
Figure 5 - 48: LRFR to LFR Capacity Ratio Comparisons for Exterior Girder Shear
Rating Factors
From Figure 5 - 48 it can be seen that the steel material type bridges had a
constant C ratio across different rating factor ratios. However the reinforced concrete
and prestressed concrete material type bridges showed a varying C ratio. This difference
in shear capacity for the two methodologies can be attributed to the new shear provisions
found in the AASHTO LRFD (2007) relating shear capacity for reinforced concrete and
prestressed concrete members. This indicates that variations in shear rating factors
produced by the LRFR and LFR methodologies can be contributed to variations in shear
capacities and live load effect.
165
Investigating the effects the components of the fundamental rating equation on the
rating factors generated for the Tri-Axle load model at the Legal load level for the
standard and unique bridge samples produced the following findings:
? Moment capacities and dead load effects calculated from the LRFR
and LFR methodologies are similar
? Variations in moment rating factors produced by the LRFR and LFR
methodologies can be contributed to the components of the live load
effect (i.e. the live load factor, live load distribution factor, and impact
factor)
? Dead load effects calculated from the LRFR and LFR methodologies
are similar
? Shear capacities for steel bridges calculated from the LRFR and LFR
methodologies are similar
? Shear capacities for reinforced concrete and prestressed concrete
bridges calculated from the LRFR and LFR methodologies show
significant variation
? Variations in shear rating factors produced by the LRFR and LFR
methodologies can be contributed to variations in shear capacities and
the live load effect
166
Chapter 6 BRIDGE RELIABILITY
6.1 Introduction
The goal of the development of the AASHTO MCE LRFR (2003) was to have a
bridge rating specification consistent with the philosophy of the AASHTO LRFD (2007)
in its use of reliability-based limit states. This allows the LRFR to produce a more
rigorous assessment of a bridge?s actual safe load capacity when compared to the LFR
(Sivakumar 2007). To show how the rating results of the LRFR compare to those of the
LFR in the context of a bridge?s reliability, reliability analyses were performed on both
standard and unique bridge samples. In this analysis a bridge?s reliability was assessed
through the use of the Monte Carlo simulation technique (Nowak and Collins 2000).
6.2 Background Information
In structural design, the capacity and applied loads for a member are not
deterministic in nature. There are varying degrees of uncertainty associated with each.
Structures are therefore designed in a manner to fulfill their requirements with an
acceptable degree of probability of failure based on these uncertainties. One way to
define failure is when the applied load effect exceeds the capacity of the structure. The
load effect and capacity can be defined as two continuous random variables Q and R,
respectively. Q and R then would have unique probability density functions (PDF)
similar to the ones found in Figure 6 - 1. Failure then could be expressed as when R ? Q
167
< 0 (Nowak and Collins 2000). Using this terminology, a performance function, g, can
be defined for a given structural member as (Nowak and Collins 2000):
QRQRg ?=),( Equation 6 ? 1
Where,
g = Performance Function
R = Capacity (Resistance)
Q = Demand (Load Effect)
Figure 6 - 1: Probability of Failure Depiction (Nowak and Collins 2000)
When the performance function g ? 0, then the capacity is greater than or equal to
the demand and the member is considered safe, having adequate capacity for the demand.
When g < 0, then the capacity is less than the demand and the member is considered
unsafe, not having adequate capacity for the demand. Therefore, the probability of
failure of the member would be equal to the probability of g < 0 (Nowak and Collins
2000). This can be expressed mathematically as (Nowak and Collins 2000):
168
)0()0( <= number of desired simulations
The results obtained from this procedure will vary with the bias factor and
coefficients of variation. In this thesis, the bias factors and coefficients of variation used
in both programs for the resistance and load were adopted from Nowak?s NCHRP report
368 on Calibration of the LRFD Bridge Design Code and are listed in Table 6 - 1 (Nowak
1999). Note that the dead and live load effects are assumed to be normally distributed
whereas the resistance is assumed to follow a lognormal distribution.
Table 6 - 1: Bias Factors and Coefficients of Variation Used in Reliability Analysis
(Nowak 1999)
A key component of any Monte Carlo simulation is the generation of uniformly
distributed numbers between 0 and 1. These numbers are generated by computer
subroutines, which vary between software packages. Nowak warns that the use of such
181
built-in number generators should be done with caution as some tend to work better than
others (Nowak 2000).
Comparing the probability of failure estimates between the two computer
programs that were developed for the reliability portion of this research, a 5 % difference
was found on a series of test bridges. Investigating the source of this difference revealed
that the random number generator algorithms for Excel and MatLAB differed enough to
produce the 5 % difference. Therefore, it was decided to perform the entire reliability
study using a single algorithm. Due to the desire to produce probabilities of failure based
on all three methods previously described, the MatLAB program was chosen to perform
all the analysis. For the ten million simulations exercise, which was to be performed
using the Excel Macro, a modified version of the MatLAB program was used estimating
the probability of failure only by Method 1.
6.4 Results
The probability of failure for each bridge was calculated by three different
methods for one million simulations and by one method for ten million simulations.
Comparing the three different methods used for calculating the probability of failure at
one million simulations revealed that Methods 1 and 3 produced very comparable results.
The similarity of the results from Method 2 with Methods 1 and 3 was found to be
depend on the probability of failure. As the estimated probability of failure increased,
the results from Method 2 increasingly matched Methods 1 and 3. This trend can be seen
in the data provided in Table 6 - 2. Consequently, the reliability index calculated from
each method?s probability of failure is presented as well which demonstrates a similar
182
trend. The reason for all three methods not always producing similar probability of
failures is due to the PDF of g not truly being Gaussian (i.e. normally distributed).
Therefore the data when plotted on probability paper does not form a perfectly linear,
straight, line and as such the best-fit linear approach, Method 2, does not always agree
with the other approaches. Nowak and Collins (2000) suggest, when a large enough
number of simulations are present, to use the Method 1 approach for estimating
probability of failure. Consequently Method 1?s estimated of probability of failures and ?
values are therefore used in the comparative portion of the study.
Table 6 - 2: Probability of Failure Methods Comparison
To demonstrate the reproducibility of the results using Method 1 for estimating
the probability of failure, three bridges were selected and 10 unique one-million
simulations were performed. The results of these 10 simulations were compared with
regards to their averages and standard deviations, shown in Table 6 - 3. It was found that
for bridges having a significant number of failures, the estimated probability of failure
and ? were highly reproducible. The cut off point for when the one million run
simulation results were no longer reproducible was taken to be 30 failures in 1,000,000.
This roughly corresponds to a ? of 4.0 which would be considered a relatively safe
bridge, targeted ? for design when calibrating the AASHTO LRFD (2007) was 3.5
183
(Nowak 1999). Therefore bridges that produced failure rates lower than 30 in 1,000,000
were considered safe and are not considered in this portion of the study.
Table 6 - 3: Ten Repetitive One-Million-Run Simulations Comparison.
To verify the adequacy of the 30 in 1,000,000 breakpoint for reproducing ? results
of 4.0 or less, a comparison between ? values generated from one million simulations and
ten million simulations were compared. This was done on both the unique and standard
bridge samples. The thought behind this comparison was that if the ? values produced by
the two simulations were comparable, showing little difference, then the chosen
breakpoint would be adequate. To illustrate this comparison the percent difference for ?
and probability of failure between the one million simulations and ten million simulations
184
is plotted versus the number of failures for the one million simulation analysis, shown in
Figure 6 - 7 for interior girders moment load effect. As Figure 6 - 7 shows that even
when the probability of failure showed significant percent differences, larger than 50 %,
? showed little difference with the increase in number of simulations, less than 3%. This
would indicate that the 30 in 1,000,000 breakpoint would adequately capture, allow the
reproduction of, ? values of 4.0 or less. Tables presenting the percent difference for
probability of failure and ? between the one million and ten million run simulations are
presented in Appendix F3 for both interior and exterior girders in flexure and shear.
Figure 6 - 7: Percent Difference in Estimated Probability of Failure and Beta Between
One Million and Ten Million Run Simulations for Interior Girders in Flexure
185
The first set of comparisons made between the LRFR and the LFR with regards to
probability of failure is shown in Figure 6 - 8. In Figure 6 - 8 the probability of failure is
plotted versus moment rating factors produced by both the LRFR and the LFR
methodologies for interior girders. Similar to the results presented by Mertz (2004), the
rating factors produced by the LRFR have a direct correlation with a bridge?s estimated
probability of failure. However, rating factors produced under the LFR are shown to not
be well correlated to estimated probabilities of failure. Additionally the range of
probabilities of failure observed for a given rating factor is greater for the LFR than the
LRFR. This suggests that the rating factors produced under the LFR may not be an
appropriate representation of a bridges adequacy under a given loading. Bridges with
rating factors greater than one are even shown to have probabilities of failure as high as
50% under the LFR. Similar results were found for exterior girders in flexure, as shown
in Figure 6 - 9.
186
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
Moment Rating Factor
Es
tim
ate
d P
ro
ba
bil
ity
of
F
ail
ur
e
LRFR
LFR
Figure 6 - 8: Probability of Failure and Rating Factor Comparison, Interior Girders
Moment Load Effect
187
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
Moment Rating Factor
Es
tim
ate
d P
ro
ba
bil
ity
of
Fa
ilu
re
LRFR
LFR
Figure 6 - 9: Probability of Failure and Rating Factor Comparison, Exterior Girders
Moment Load Effect
In Figure 6 ? 10 the probability of failure is plotted verse shear rating factors
produced by both the LRFR and the LFR methodologies for interior girders. Different
from the data presented for interior girders for moment load effects, the scatter seen for
both the LFR and the LRFR is greatly reduced for interior girder in shear. The correlation
between the LRFR and failure rate is strongly shown for interior girders in shear. The
LFR however shows only sporadic correlation to a probability of failure. Rating factors
less than one are shown to have very low probabilities of failure in some cases, while
188
rating factors greater than one have very high probabilities of failure. Additionally, it is
important to note the sharp increase in probability of failure for rating factors less than
0.8 for both the LRFR and LFR methodologies. Similar results were found for exterior
girders as seen in Figure 6 - 11.
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
Shear Rating Factor
Es
tim
ate
d P
ro
ba
bil
ity
of
Fa
ilu
re
LRFR
LFR
Figure 6 - 10: Probability of Failure and Rating Factor Comparison, Interior Girders
Shear Load Effect
189
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
Shear Rating Factor
Es
tim
ate
d P
ro
ba
bil
ity
of
Fa
ilu
re
LRFR
LFR
Figure 6 - 11: Probability of Failure and Rating Factor Comparison, Exterior Girders
Shear Load Effect
The LRFR and LFR rating factors were also compared to the ? values calculated
from the estimated probabilities of failure. Figure 6 - 12 presents the data for the interior
girders for moment load effect. This comparison demonstrates again the correlation ? has
with the rating factors produced under the LRFR. Additionally, the LFR is shown to
have little to no correlation to ? with a large scatter across the plot. It is important to note
that rating factors equal to 1.0 appear to correlate with a ? of 2.5 instead of the intended
targeted ? of 3.5 for this level of rating in the LRFR. Similar results were reported by
190
Mertz in his Task 122 report (2004) as indicated before. Exterior girders produced
similar results as shown in Figure 6 - 13.
-5
-4
-3
-2
-1
0
1
2
3
4
5
0 0.5 1 1.5 2 2.5 3
Moment Rating Factor
????
LRFR
LFR
Figure 6 - 12: ? and Rating Factor Comparison, Interior Girders Moment Load Effect
191
-5
-4
-3
-2
-1
0
1
2
3
4
5
0 0.5 1 1.5 2 2.5
Moment Rating Factor
????
LRFR
LFR
Figure 6 - 13: ? and Rating Factor Comparison, Exterior Girders Moment Load Effect
The shear data for the interior girders is presented in Figure 6 - 14. Similar to the
flexural results, the LRFR rating factors are seen to have a strong correlation with ? while
the LFR does not. Additionally, the trend of rating factors of 1.0 correlating to a ? of 2.5,
instead of the intended targeted ? of 3.5 for the design rating level, is seen for the interior
girders in shear as well. Similar results are found for the exterior girders in shear as seen
in Figure 6 - 15.
192
-3
-2
-1
0
1
2
3
4
5
0 0.5 1 1.5 2 2.5 3
Shear Rating Factor
????
LRFR
LFR
Figure 6 - 14: ? and Rating Factor Comparison, Interior Girders Shear Load Effect
193
-3
-2
-1
0
1
2
3
4
5
0 0.5 1 1.5 2 2.5 3
Shear Rating Factor
????
LRFR
LFR
Figure 6 - 15: ? and Rating Factor Comparison, Exterior Girders Shear Load Effect
6.5 Summary and Conclusion
The reliability analysis of the standard and unique bridge samples at the Design
Inventory level of rating provided the following findings:
? Rating factors produced by the LRFR are well correlated to estimated
probability of failure for interior and exterior girders in moment and shear
? Rating factors produced by the LFR are not well correlated to estimated
probability of failure for interior and exterior girders in moment and shear
194
? Rating factors equal to 1.0 under the LRFR were shown to correspond to a
reliability index of approximately 2.5 which is significantly lower than the
targeted reliability index of 3.5 for the Design Inventory rating level
Based on the reliability analysis of the unique and standard bridge samples the
LRFR rating methodology is shown to produce a more rigorous assessment of a bridge?s
level of safety compared to the LFR rating methodology. The LFR rating methodology
however showed a poor correlation between estimated probabilities of failure and LFR
rating factors. This suggests that the LFR rating methodology produces rating factors
that do not consistently reflect a bridge?s level of safety.
195
Chapter 7 Summary, Conclusions and Recommendations
7.1 Summary
Adopting the AASHTO MCE LRFR (2003) can have a profound effect on the
rating practices of ALDOT with regards to Alabama?s State and County owned bridges.
In order to assess how the new rating methodology would affect Alabama?s bridge
inventory, a comparative study was done by the Auburn University Highway Research
Center between the LRFR and LFR. This study was conducted on a representative
sample of 95 bridges from Alabama?s State and County owned bridge inventory. Rating
factors were compared between the two rating methodologies at the Design load level,
Legal load level, and Permit load level of rating. AASHTO design and standard legal
load models were used in addition to eight Alabama State Legal Loads and ten State
permit trucks. In addition to the comparative study of rating methodologies, a reliability
study was done to evaluate how rating factors at the Design Inventory level of rating
compared to the estimated probability of failure of a bridge.
7.2 Conclusions
The comparative study provided in this thesis showed how the new LRFR rating
methodology compares to the LFR rating methodology on a sample of 95 bridges from
Alabama State and County owned and maintained bridge inventory. The conclusions
from this study are as follows:
196
? Rating factors produced under the LRFR at all levels of rating were shown
to be nearly equal or lower to the LFR rating factors for exterior and
interior girders as well as for moment and shear.
? Moment rating factors under the LRFR methodology tend to control over
shear rating factors at all levels of rating; for the LFR methodology,
moment and shear rating factors were seen to control more or less evenly
? Load rating under the LRFR methodology was predominantly controlled
by exterior girder moment rating; for the LFR methodology load rating
was not dominated by any particular load effect or girder
? ALDOT legal loads are not enveloped by either the AASHTO legal loads
or the HL-93 design load model
? Load rating under ALDOT legal loads for the unique bridge sample,
which consisted of 45 bridges, showed that 23 bridges require posting
under the LRFR and 8 bridges require posting under the LFR
? Posting loads under the LRFR tend to be significantly lower than posting
loads under the LFR
? The LRFR allows a slightly fewer number of bridges to be considered for
permitting compared to the LFR
? Differences in moment rating factors produced by the LRFR and the LFR
can be attributed to differences in live load distribution factor, live load
factor, and dynamic load allowance factor.
197
? Differences in shear rating factors produced by the LRFR and the LFR can
be attributed to differences in live load distribution factor, live load factor,
dynamic load allowance factor and capacity.
? The structural system type C ? Channel bridges were shown to produce
unusual LRFR to LFR rating factor ratios when compared to other
structural system types at all levels of rating.
? Moment and shear rating factors produced by the LRFR at the Design
Inventory level of rating are well correlated to the estimated probability of
failure for interior and exterior girders
? Moment and shear rating factors produced by the LFR at the Inventory
level of rating are not well correlated to the estimated probability of failure
for interior and exterior girders
7.3 Recommendations
The findings of the comparative study showed that in most cases the LRFR
produces lower rating factors than the LFR for Alabama?s State and County owned and
maintained bridges. However, while the LRFR may produce lower rating factors, the
rating factors produced were found to be well correlated to a bridge?s estimated
probability of failure which adds credence to the LRFR methodology.
Based on this observation the following recommendations are suggested to
ALDOT. From an implementation point of view:
? It is recommended that ALDOT uses the LRFR for rating new bridges
designed to the AASHTO LRFD (2007) at all rating levels
198
? It is recommended that ALDOT uses both the LRFR and LFR
methodologies for rating existing bridges at all rating levels. When RF >
1.0 for LRFR and for LFR, a bridge can be considered satisfactory. When
RF < 1.0 for LRFR and for LFR, a bridge can be considered
unsatisfactory. When RF < 1.0 for LRFR and RF > 1.0 for LFR, further
investigation of the safety of the bridge is recommended according to
ALDOT current policies.
In addition, the following recommendations for further investigations are also
made:
? It is recommended that further research be conducted to understand and
identify factors affecting the observed differences between the LRFR and
the LFR. Factors to investigate may include, but are not limited to: the
live load distribution factor and live load factor.
? Based on unusual LRFR to LFR rating factor ratios produced by the C-
Channel bridges during the study it is recommend that further research be
conducted in regards to the modeling simplifications incorporated within
this study, with special attention given to the live load distribution factors
used during the rating analysis.
199
REFERENCES
AASHTO Load and Resistance Factor Design (LRFD) Bridge Design Specifications 4th
Edition. 2007. American Association of State Highway and Transportation Officials.
Washington, D.C.
AASHTO Manual for Condition Evaluation and Load and Resistance Factor Rating
(LRFR) of Highway Bridges 2003 with 2005 Interim Revisions. 2003. American
Association of State Highway and Transportation Officials. Washington, D.C.
AASHTO Manual for Condition Evaluation Bridges, 1994 Second Edition with 2001 and
2003 Interim Revisions. 1994. American Association of State Highway and
Transportation Officials. Washington, D.C.
AASHTO Standard Specification for Highway Bridges 17th Edition. 2002. American
Association of State Highway and Transportation Officials. Washington, D.C.
BridgeWare. Virtis, Bridge Load Rating Software version 5.6. AASHTO. November
2007.
Hayworth, Rebecca, X., Sharon Huo, and Lei Zheng. ?Effects of State Legal Loads on
Bridge Rating Results Using the LRFR Procedure.? ASCE Journal of Bridge
Engineering. November/December 2008: 565-572.
Lichtenstein Consulting Engineers, Inc. ?NCHRP Web Document 28: Contractor?s Final
Report: Manual for Condition Evaluation and Load Rating of Highway Bridges Using
Load and Resistance Factor Philosophy.? Transportation Research Board Project No.
C12-46. May 2001. .
200
Mertz, Dennis R. ?Load Rating by Load and Resistance Factor Evaluation Method.?
Transportation Research Board NHCRP Project No. 20-07, Task 122. June 2005.
.
Minervino, Charles, Bala Sivakumar, Fred Moses, Dennis Mertz, and William Edberg.
?New AASHTO Guide Manual for Load and Resistance Factor Rating of Highway
Bridges.? ASCE Journal of Bridge Engineering. January/February 2004: 43-54.
Moses, Fred. ?Calibration of Load Factors for LRFR Bridge Evaluation.? NCHRP
Report 454. Transportation Research Board. 2001. Washington, D.C.
Nowak, Andrzej S. ?Calibration of LRFD Bridge Design Code.? NCHRP Report 368.
Transportation Research Board. 1999. Washington, D.C.
Nowak, Andrzej S. and Kevin R. Collins. 2000. Reliability of Structures. Boston,
McGraw-Hill Companies, Inc.
Rogers, Brandy J. and David V. J?uregui. ?Load Rating of Prestressed Concrete Girder
Bridges, Comparative Analysis of Load Factor Rating and Load Resistance Factor
Rating?. Transportation Research Record: Journal of the Transportation Research
Board. No 1928. (2005): 53-63.
Sivakumar, B. ?The New AASHTO Manual for Bridge Evaluation: A Single Standard for
Bridge Evaluation.? 4th New York Bridge Conference. August 2007.
Virtis Version 5.6 Manual. 2008 American Association of State Highway and
Transportation Officials. Washington, D.C.
201
APPENDICES
202
APPENDIX A: Sample Distributions
Presented in Tables A - 1 through A - 6 is the structural system type breakdown of
the SCOMB inventory for each material type. Each table provides a summary of number
of bridges in each structural system, and span length ranges included in each structural
system type.
Table A - 1: Structural System Type Distribution for Reinforced Concrete Simply
Supported Bridges According to SCOMB Distribution
203
Table A - 2: Structural System Type Distribution for Reinforced Concrete
Continuously Supported Bridges According to SCOMB Distribution
Table A - 3: Structural System Type Distribution for Steel Simply Supported
Bridges According to SCOMB Distribution
204
Table A - 4: Structural System Type Distribution for Steel Continuously Supported
Bridges According to SCOMB Distribution
Table A - 5: Structural System Type Distribution for Prestressed Concrete Simply
Supported Bridges According to SCOMB Distribution
205
Table A - 6: Structural System Type Distribution for Prestressed Concrete
Continuously Supported Bridges According to SCOMB Distribution
Tables A - 7 and A - 8 present the proposed unique bridge sample?s material and
structural system types. Table A - 9 presents the final standard bridge sample?s material
and structural system types and additional sample information. Table A - 10 presents the
final unique bridge sample?s material and structural system types.
206
Table A - 7: Proposed Unique Bridge Sample Part 1
207
Table A - 8: Proposed Unique Bridge Sample Part 2
208
Table A - 9: Standard Bridge Sample Bridge Descriptions
Table A - 10: Unique Bridge Sample Matrix
209
210
Provided in Tables A ? 11 thorugh A ? 14 is additional information about each
bridge in both the unique and standard bridge samples. Information included about each
bridge is the following: BIN, Year, AADT, Live Load Factor, Span Length(s), Girder
Spacing, Condition and System Factors, Material Type, Structural System Type, Deck
and Girder Concrete Strength, Reinforcement Grade, and Structural Steel Grade.
Table A - 11: Additional Sample Information Table 1
211
Table A - 12: Additional Sample Information Table 2
212
Table A - 13: Additional Sample Information Table 3
213
Table A - 14: Additional Sample Information Table 4
214
215
APPENDIX B1: Slab Bridges Mathcad File
Presented in this Appendix is the Mathcad file used for slab bridges for rating
under the LRFR methodology.
Input
L 19:= ft slab length, center to center of bearings
W 51.5:= ft entire bridge width
w 36:= ft roadway width
fc 2.5:= ksi compressive strength of concrete, 28 day
fy 40:= ksi yield strength of reinforcement
h 15.5:= in depth of slab
As 1.44:= in2 area of positive flexural reinforcement per-foot
df 1.4375:= in distance from extreme tension fiber to centriod of flexural reinforce.
wb .61:= kip/ft barrier self weight and walkway
wbear 69:= in distance from edge to face of barrier
?c .15:= kip/ft^3 weight of concrete
Live-Load Strip Width
One Lane Loaded
L1 min L 60, ( ) 19=:= ft W1 min W 30, ( ) 30=:= ft
E1 10 5 L1 W1?( )?+ 129.373=:= in
Two Lanes Loaded
L1 min L 60, ( ) 19=:= ft W1 min W 60, ( ) 51.5=:= ft
NL trunc w12??? ??? 3=:=
E2 min 84 1.44 L1 W1?+ 12.0 WN
L
?, ??
?
??
?
129.045=:= in
E min E1 E2, ( ) 129.045=:= in
in
Slab Bridge LRFR File BIN 001541 Mike Murdock 12/08
AASHTO 4.6.2.3
AASHTO 4.6.2.3-1
AASHTO 4.6.2.3-2
AASHTO 3.6.1.1.1
216
Dead Load Calculations
Assumptions:
barrier load spread over width of live-load edge strip
MDCext ?c h12??? ???? 1? wbEdgeStrip
12
??
?
??
?
+??
??
??
??
L2
8
??
?
??
??:= MDCint ?c h12??? ???? 1???? ??? L
2
8
??
?
??
??:=
VDCint 0.5 ?c h12??? ??????? ???? L?:= VDCext 0.5 ?c h12??? ???? wbEdgeStrip
12
??
?
??
?
+??
??
??
??
? L?:=
MDCint 8.743= kip-ft / ft MDCext 13.862= kip-ft / ft
VDCint 1.841= kip / ft VDCext 2.918= kip / ft
Live Load Calculations
Assumptions:
in-house MatLab line load analysis use for moment and shear calculations
impact included
HS 20-44
H 20-44
Tandem
Triaxle M
max
240.14
240.14
239.4
333.83
299.25
209.48
240.98
122.86
??
?
?
?
?
?
?
?
??
??
?
?
?
?
?
?
?
??
:= kip-ft Vmax
64.4
54.4
57.02
77.15
72.52
52.73
55.68
16.17
??
?
?
?
?
?
?
?
??
??
?
?
?
?
?
?
?
??
:= kip
Concrete
18 Wheeler
6-Axel
School Bus
MLL
Mmax
E
12
??
?
??
?
:= kip-ft / ft VLL
Vmax
E
12
??
?
??
?
:= kip / ft
Strength I
Moment Resistance
217
?1 0.85 fc 4?if
0.85 .05 fc 4?( )??[ ] otherwise
:= ?1 0.85=
c As fy?( )0.85 fc? ?
1? 12?
2.657=:= in
a ?1 c? 2.259=:= in ds h df? 14.063=:= in
Mn As fy? ds a2???? ???? 112??? ???? 62.079=:= kip-ft / ft
?T 0.003 ds c?( )c??? ???? 0.013=:=
? 0.9 ?T 0.005>if
0.75 ?T .002if
break
otherwise
:=
Mn FindMn Dt Dp, Mp, ( ) 2874.011=:=
Nominal Shear Resistance Vn 6.10.9.2
Vp .58Fy Dweb? tw? 380.145=:= kips
TranStif 0:= Stiffeners spacing
findK Dw TranStif, ( ) 5 TranStif 0if
5 5
TranStif
Dw
??
?
??
?
2
+ otherwise
:=
k findK Dw TranStif, ( ) 5=:=
findc Dweb tw, E, k, Fy, ( ) 1.0 Dwebtw 1.12 E kFy???? ?????if
1.12
Dweb
tw
E kFy???? ??????
??
??
??
Dweb
tw 1.4 E
k
Fy?
??
?
??
???if
1.57
Dweb
tw
E kFy???? ???? otherwise
otherwise
:=
C findc Dweb tw, E, k, Fy, ( ) 1=:=
Vn Vp C? 380.145=:= kips
228
Design Load Rating:
q 1.0:= qc 1.0:= qs 1.0:=
A) Strength I Limit State
a) Inventory Level
Load Load Factor
DC 1.25
LL 1.75
Flexure: RFif q qc? qs? Mn? 1.25 Mdc1 Mdc2+( )??[ ]
1.75 Mll?:=
RFif 1.295=
Shear: RFis q qc? qs? Vn? 1.25 Vdc1 Vdc2+( )??[ ]
1.75 Vll?:=
RFis 2.439=
b) Operating Level
Load Load Factor
DC 1.25
LL 1.35
Flexure: RFof RFif 1.75
1.35
??
?
??
??:=
RFof 1.679=
Shear: RFos RFis 1.75
1.35
??
?
??
??:=
RFos 3.162=
229
APPENDIX B3: Example Problem A2: Reinforced Concrete Tee Beam
Reinforced Concrete T-Beam
L 26:= ft ? .9:= ?s 1.0:= ?c 1.0:=
fc 3:= ksi
RoadWay 22:= ft
Girders 4:= S 6.52:= ft Girder Spacing
ta 5:= in Asphalt thickness As 6.89:= in2 Area of Reinforcement
tb 15:= in Beam thickness dr 26.61:= in
db 24:= in Beam depth fy 33:= ksi Reinforcement Yield
ts 6:= in Slab thickness ? .85:=
h db ts+:=
Mdl 54.1:= kft Vdl 7:= kip
Mtruck 208:= kft Vtruck 41.4:= kip
Mtan 275:= kft Vtan 41.9:= kip
IM 1.33:=
Dead Load Analysis
DC
Structural Concrete:
SC .150 612??? ??? 6.52? 1.25 2?+ 2 .5 .5? .5?( )?+??? ???? 0.902=:= k/ft
Railing And Curb
RC .2 12? 0.1=:= k/ft
DW
?DW 1.25:=
Asphalt Overlay:
AO ta12??? ??? RoadWay( )? .144? 1Girders? 0.33=:=
Live Load Analysis
Distribution Factors
230
4.6.2.2.2
n 1.0:=
I 112??? ??? tb( )? db3? 1.728 104?=:=
in4 A tb db? 360=:= in2
eg .5 db ts+( ) 15=:=
Kg n I A eg( )2?+?? ??? 9.828 104?=:= in
Moment Distribution Factors
One Lane Loaded:
gm1 0.06 S14??? ???
.4 S
L
??
?
??
?
.3 Kg
12 L? ts3?
??
?
??
?
.1
+ 0.565=:=
Two or More Lanes Loaded:
gm2 0.075 S9.5??? ???
.6 S
L
??
?
??
?
.2 Kg
12 L? ts3?
??
?
??
?
.1
+ 0.703=:=
Use: gm max gm1 gm2, ( ) 0.703=:=
Shear Distribution Factors
One-Lane Loaded:
gv1 0.36 S25+ 0.621=:=
Two or More Lanes Loaded:
gv2 0.2 S12??? ???+ S35??? ???
2.0
? 0.709=:=
Use: gv max gv1 gv2, ( ) 0.709=:=
Maximum Live Load Effects
MaxMoment Mdl max Mtruck Mtan, ( ) IM?+:=
MaxMoment 419.85=
Mll MaxMoment gm? 295.273=:=
MaxShear Vdl max Vtruck Vtan, ( ) IM?+:=
MaxShear 62.727=
Vll MaxShear gv? 44.45=:=
231
Effective Flange Width 4.6.2.6.1
Efw1 14 L? 12?:=
Efw2 ts 12? max tb 12 ts?, ??? ???+:=
Efw3 S 12?:=
Efwidth min Efw1 Efw2, Efw3, ( ):=
Efwidth 78= in
Compute Distance to Neutral Axis c: 5.7.3.2
c As fy?( ).85 ?? fc? Efwidth?( ) 1.345=:= in
Need to add option for when c is in web
a c ?? 1.143=:= in
Mn As fy? dr a2???? ???? 112? 493.363=:= kft
Mr ? Mn? 444.027=:= kft
Compute Nominal Shear Resistance 5.8.2.9
Stirrups: #5 bars @9in
232
Av 2 3.14164??? ???? 58??? ???
2
0.614=:= in2 s 9:= in
dv1 Mn 12?As fy?( ) 26.038=:=
dv2 0.72 h? 21.6=:=
dv3 0.9 dr? 23.949=:=
dv max dv1 dv2, dv3, ( ) 26.038=:= in
bv tb 15=:= in 5.8.2.9
Simple Procedure: 5.8.3.3
?v 2:= ? 45:=
Vc 0.0316?v? fc.5( )? bv? dv? 42.755=:=
Vs Av fy? dv?
cot ? 3.1416? 180??? ?????? ???
s? 58.582=:=
Vn Vc Vs+ 101.337=:=
Vr ? Vn? 91.203=:=
MCE Procedure: 5.8.3.3
?v 2:= ? 45:= dv 23.949:= Conservative Assumption
Vc 0.0316?v? fc.5( )? bv? dv? 39.324=:=
Vs Av fy? dv?
cot ? 3.1416? 180??? ?????? ???
s? 53.881=:=
Vn Vc Vs+ 93.205=:=
Vr2 ? Vn? 83.885=:= Virtis uses more Complex method by default
233
Ratings
Mdc SC RC+( ) L
2( )
8 84.627=:=
Vdc SC RC+( ) L2??? ??? 2612???? ???? 10.85=:=
Mdw AO( ) L
2( )
8 27.885=:=
Vdw AO( ) L2??? ??? 2612???? ???? 3.575=:=
?DW 1.25:= ?DC 1.25:= ?LL 1.75:=
Moment
RF ?s ?c? Mr? ?DC Mdc?? ?DW Mdw??( )?LL Mll? 0.587=:=
Shear
RF ?s ?c? Vr2? ?DC Vdc?? ?DW Vdw??( )?LL Vll? 0.847=:=
?DW 1.5:= ?DC 1.25:= ?LL 1.75:=
Moment
RF ?s ?c? Mr? ?DC Mdc?? ?DW Mdw??( )?LL Mll? 0.574=:=
234
APPENDIX B4: Example Problem A3: Prestressed Concrete I Girder
Reinforced Concrete T-Beam
L 80:= ft ? 1.0:= ?s 1.0:= ?c 1.0:=
fc 4:= ksi fy 60:= ksi
fpc 5:= ksi
fpci 4:= ksi fpu 270:= ksi
RoadWay 27:= ft ? .85:=
Girders 4:= As 0:=
S 8.5:= ft Girder Spacing As1 .153:= in2 Area one strand
ts 8.5:= ybar 3.75:= NumSR1 12:= dr1 61.5:=
tb 8:= NumSR2 12:= dr2 59.5:=
bf 20:=
db 54:= NumSR3 8:= dr3 57.5:=
Hun 1:=
Mdl 512:= kft Vdl 22.3:= kip K .28:= Low Relax
Mtruck 1160:= kft Vtruck 58.8:= kip dp db Hun+ ts+ ybar? 59.75=:=
Mtan 950:= kft Vtan 45.4:= kip
IM 1.33:=
Dead Load Analysis
DC1
GirderSW .822:= k/ft
DiaphSW .15:= k/ft
Slab .925:= k/ft
DC1 GirderSW DiaphSW+ Slab+:=
DC1 1.897= k/ft
DC2
DC2 2 .5Girders? 0.25=:= k/ft
DW
DW 2.512??? ??? 27? .144? .25? 0.203=:= k/ft
Aps As1 NumSR1 NumSR2+ NumSR3+( )?:=
235
Live Load Analysis
Distribution Factors 4.6.2.2.2
n 1.12:=
I 260741:= in4
A 789:= in2
eg 34.52:=
Kg n I A eg( )2?+?? ??? 1.345 106?=:= in
Moment Distribution Factors
One Lane Loaded:
gm1 0.06 S14??? ???
.4 S
L
??
?
??
?
.3 Kg
12 L? ts3?
??
?
??
?
.1
+ 0.514=:=
Two or More Lanes Loaded:
gm2 0.075 S9.5??? ???
.6 S
L
??
?
??
?
.2 Kg
12 L? ts3?
??
?
??
?
.1
+ 0.724=:=
Use: gm max gm1 gm2, ( ) 0.724=:=
Shear Distribution Factors
One-Lane Loaded:
gv1 0.36 S25+ 0.7=:=
Two or More Lanes Loaded:
gv2 0.2 S12??? ???+ S35??? ???
2.0
? 0.849=:=
Use: gv max gv1 gv2, ( ) 0.849=:=
Maximum Live Load Effects
MaxMoment Mdl max Mtruck Mtan, ( ) IM?+:=
MaxMoment 2.055 103?=
Mll MaxMoment gm? 1.487 103?=:=
MaxShear Vdl max Vtruck Vtan, ( ) IM?+:=
MaxShear 100.504=
Vll MaxShear gv? 85.363=:=
236
Effective Flange Width 4.6.2.6.1
Efw1 14 L? 12?:=
Efw2 ts 12? max tb 12 ts?, ??? ???+:=
Efw3 S 12?:=
Efwidth min Efw1 Efw2, Efw3, ( ):=
Efwidth 102= in
Compute Distance to Neutral Axis c: 5.7.3.2
c Aps fpu?( )
.85 ?? fc? Efwidth? K Aps? fpudp?+??? ???
4.392=:= in
a c ?? 3.733=:= in
fps fpu 1 K cdp????? ???? 264.443=:=
Mn Aps fps? dp a2???? ???? 112? 6.245 103?=:= kft
Mr ? Mn? 6.245 103?=:= kft ? 1= 5.5.4.2.1
237
Compute Nominal Shear Resistance 5.8.2.9
Stirrups: #4 bars @9in
Av 2 .2( )? 0.4=:= in2 s 9:= in
dv 58.4:= in
bv tb 8=:= in 5.8.2.9
Simple Procedure: 5.8.3.3
?v 2:= ? 45:=
Vc 0.0316?v? fpc.5( )? bv? dv? 66.024=:=
Vs Av fy? dv?
cot ? 3.1416? 180??? ?????? ???
s? 155.733=:=
Vn Vc Vs+ 221.757=:=
Vr2 ? Vn? 221.757=:=
238
Ratings
Mdc DC1 DC2+( ) L
2( )
8 1.718 10
3?=:=
Vdc DC1 DC2+( ) L2??? ??? 64.412???? ???? 74.358=:=
Mdw DW( ) L
2( )
8 162=:=
Vdw DW( ) L2??? ??? 64.412???? ???? 7.013=:=
?DW 1.5:= ?DC 1.25:= ?LL 1.75:=
Moment
RF ?s ?c? Mr? ?DC Mdc?? ?DW Mdw??( )?LL Mll? 1.481=:=
Shear
RF ?s ?c? Vr2? ?DC Vdc?? ?DW Vdw??( )?LL Vll? 0.792=:=
239
APPENDIX B5: Example Problem A2: Results Summary
Provided in this appendix is a summary of example problem A2 results from the
MCE, Mathcad and Virtis.
Figure B5 - 1: Example A2 Comparisons Part 1
240
Figure B5 - 2: Example A2 Comparisons Part 2
241
APPENDIX B6: Example Problem A3: Results Summary
Provided in this appendix is a summary of example problem A3 results from the
MCE, Mathcad and Virtis.
Figure B6 - 1: Example A3 Comparisons Part 1
242
Figure B6 - 2: Example A3 Comparisons Part 2
243
APPENDIX C.1: Design Inventory Standard Bridge Rating Data
Presented in this appendix is the extracted data from Virtis for the standard bridge
sample at the Design Inventory level for the LRFR under the HL-93 load model and the
Inventory level of the LFR under the HS-20. Tables C.1 - 1 through C.1 - 12 provide the
following information: BIN, Material Type, Structural Type, Number of Spans, Span
Length, Dead Load Factors, Live Load Factors, Resistance Factors, Condition Factor,
System Factor, Controlling Vehicle, Unfactored Capacity, Unfactored Dead Load,
Unfactored Live Load, Virtis Rating Factor, and Excel Calculated Rating Factor. The
Excel rating factor is calculated using the provided information and the rating
methodologies rating equation as provided in Chapter 2.
244
Table C1 - 1: Design Inventory Standard Bridge Output, Exterior Girder LRFR Part 1
245
Table C1 - 2: Design Inventory Standard Bridge Output, Exterior Girder LRFR Part 2
246
Table C1 - 3: Design Inventory Standard Bridge Output, Exterior Girder LRFR Part 3
247
Table C1 - 4: Design Inventory Standard Bridge Output, Exterior Girder LFR Part 1
248
Table C1 - 5: Design Inventory Standard Bridge Output, Exterior Girder LFR Part 2
249
Table C1 - 6: Design Inventory Standard Bridge Output, Exterior Girder LFR Part 3
250
Table C1 - 7: Design Inventory Standard Bridge Output, Interior Girder LRFR Part 1
251
Table C1 - 8: Design Inventory Standard Bridge Output, Interior Girder LRFR Part 2
252
Table C1 - 9: Design Inventory Standard Bridge Output, Interior Girder LRFR Part 3
253
Table C1 - 10: Design Inventory Standard Bridge Output, Interior Girder LFR Part 1
254
Table C1 - 11: Design Inventory Standard Bridge Output, Interior Girder LFR Part 2
255
Table C1 - 12: Design Inventory Standard Bridge Output, Interior Girder LFR Part 3
256
APPENDIX C.2: Design Inventory Unique Bridge Rating Data
Presented in this appendix is the extracted data from Virtis for the unique bridge
sample at the Design Inventory level for the LRFR under the HL-93 load model and the
Inventory level of the LFR under the HS-20. Tables C.1 - 1 through C.1 - 12 provide the
following information: BIN, Material Type, Structural Type, Number of Spans, Span
Length, Dead Load Factors, Live Load Factors, Resistance Factors, Condition Factor,
System Factor, Controlling Vehicle, Unfactored Capacity, Unfactored Dead Load,
Unfactored Live Load, Virtis Rating Factor, and Excel Calculated Rating Factor. The
Excel rating factor is calculated using the provided information and the rating
methodologies rating equation as provided in Chapter 2.
257
Table C2 - 1: Design Inventory Unique Bridge Output, Exterior Girder LRFR Part 1
258
Table C2 - 2: Design Inventory Unique Bridge Output, Exterior Girder LRFR Part 2
259
Table C2 - 3: Design Inventory Unique Bridge Output, Exterior Girder LRFR Part 3
260
Table C2 - 4: Design Inventory Unique Bridge Output, Exterior Girder LFR Part 1
261
Table C2 - 5: Design Inventory Unique Bridge Output, Exterior Girder LFR Part 2
262
Table C2 - 6: Design Inventory Unique Bridge Output, Exterior Girder LFR Part 3
263
Table C2 - 7: Design Inventory Unique Bridge Output, Interior Girder LRFR Part 1
264
Table C2 - 8: Design Inventory Unique Bridge Output, Interior Girder LRFR Part 2
265
Table C2 - 9: Design Inventory Unique Bridge Output, Interior Girder LRFR Part 3
266
Table C2 - 10: Design Inventory Unique Bridge Output, Interior Girder LFR Part 1
267
Table C2 - 11: Design Inventory Unique Bridge Output, Interior Girder LFR Part 2
268
Table C2 - 12: Design Inventory Unique Bridge Output, Interior Girder LFR Part 3
269
APPENDIX D: ALDOT Legal Load Rating Data
Presented in this appendix is the extracted data from Virtis for the unique bridge
sample at the Legal load level for the LRFR under the controlling ALDOT legal load and
the Operating level of the LFR under the controlling ALDOT legal load. Tables C.1 - 1
through C.1 - 12 provide the following information: BIN, Material Type, Structural Type,
Number of Spans, Span Length, Dead Load Factors, Live Load Factors, Resistance
Factors, Condition Factor, System Factor, Controlling Vehicle, Unfactored Capacity,
Unfactored Dead Load, Unfactored Live Load, Virtis Rating Factor, and Excel
Calculated Rating Factor. The Excel rating factor is calculated using the provided
information and the rating methodologies rating equation as provided in Chapter 2.
270
Table D - 1: Legal Load Level Unique Bridge Output, Exterior Girder Flexure, LRFR
Part 1
Table D - 2: Legal Load Level Unique Bridge Output, Exterior Girder Flexure, LRFR
Part 2
271
Table D - 3: Legal Load Level Unique Bridge Output, Exterior Girder Flexure, LRFR
Part 3
272
Table D - 4: Legal Load Level Unique Bridge Output, Exterior Girder Flexure, LFR
Part 1
273
Table D - 5: Legal Load Level Unique Bridge Output, Exterior Girder Flexure, LFR
Part 2
274
Table D - 6: Legal Load Level Unique Bridge Output, Exterior Girder Flexure, LFR
Part 3
275
276
Table D - 7: Legal Load Level Unique Bridge Output, Interior Girder Flexure, LRFR
Part 1
Table D - 8: Legal Load Level Unique Bridge Output, Interior Girder Flexure, LRFR
Part 2
277
Table D - 9: Legal Load Level Unique Bridge Output, Interior Girder Flexure, LRFR
Part 3
278
Table D - 10: Legal Load Level Unique Bridge Output, Interior Girder Flexure, LFR
Part 1
279
Table D - 11: Legal Load Level Unique Bridge Output, Interior Girder Flexure, LFR
Part 2
280
Table D - 12: Legal Load Level Unique Bridge Output, Interior Girder Flexure, LFR
Part 3
281
282
APPENDIX E: ALDOT Legal Load Posting Data
Presented in this appendix is the legal load posting data for the moment and shear rating
factors, for interior and exterior girders, for each rating methodology. For more information on
posting see Chapter 2 and Chapter 5.
Table E - 1: Legal Load Posting Data for Exterior Girders, LRFR Moment Data
Table E - 2: Legal Load Posting Data for Exterior Girders, LRFR Shear Data
283
Table E - 3: Legal Load Posting Data for Interior Girders, LRFR Moment Data
284
Table E - 4: Legal Load Posting Data for Interior Girders, LRFR Shear Data
285
Table E - 5: Legal Load Posting Data for Exterior Girders, LFR Moment Data
286
Table E - 6: Legal Load Posting Data for Exterior Girders, LFR Shear Data
287
Table E - 7: Legal Load Posting Data for Interior Girders, LFR Moment Data
288
Table E - 8: Legal Load Posting Data for Interior Girders, LFR Shear Data
289
290
APPENDIX F1: MatLAB Beta Analysis Program
Presented in this appendix is the MatLab beta analysis program. For more
information on its use and construction see Chapter 6.
% Beta Analysis Program
% Mike Murdock
% 1-29-08
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Cal. Beta
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function beta() %function name
gD = 1.05;
gL = 1.3;
gRS = 1.12;
gRC = 1.12;
gRP = 1.05;
vD = 0.1;
vL = 0.18;
vRS = 0.1;
vRC = 0.13;
vRP = 0.075;
Fail = 0;
NumberOfRuns = 1000;
i = 1
ii=1;
iii = 0;
clc; %clears the screen of any random text
291
fprintf('\n');
fprintf('\n');
[Numbers] = xlsread('testing.xls');
NumberOfBridges = size(Numbers);
while ii <= NumberOfBridges(1)
inputBin = num2str(Numbers(ii, 1));
Mat = Numbers(ii, 2);
Mr = Numbers(ii, 3);
DC = Numbers(ii, 4);
DW = Numbers(ii,5);
Ml = Numbers(ii,6);
if Mat < 2
gR = gRC;
vR = vRC;
else
if Mat < 3
gR = gRS;
vR = vRS;
else
gR = gRP;
vR = vRP;
end
end
ud = gD * (DC + DW);
ul = gL * Ml;
ur = gR * Mr;
qr = (log(vR ^ 2 + 1)) ^ 0.5;
while i < NumberOfRuns
Di = ud + vD * ud * NormSInv(rand(1));
Li = ul + vL * ul * NormSInv(rand(1));
Ri = exp((log(ur) - 0.5 * (qr ^ 2)) + qr * NormSInv(rand(1)));
Ya(i) = Ri - (Di + Li);
292
if Ya(i) < 0
Fail = Fail + 1;
end
i = i + 1;
end
YaSorted = sort(Ya);
i = 1;
V1 = 0;
V2 = 0;
V3 = 0;
V4 = 0;
PaB = 0;
YaB = 0;
while i < NumberOfRuns
Pa(i) = NormSInv(i / NumberOfRuns);
V1 = V1 + YaSorted(i) * Pa(i);
V2 = V2 + YaSorted(i);
V3 = V3 + Pa(i);
V4 = V4 + (YaSorted(i)) ^ 2;
i = i + 1;
end
PaB = V3 / (NumberOfRuns - 1);
YaB = V2 / (NumberOfRuns - 1);
slope = (NumberOfRuns * V1 - V2 * V3) / (NumberOfRuns * V4 - (V2 ^ 2));
betaLine = -(PaB - slope * YaB);
i =1;
while YaSorted(i) < 0
i=i+1;
end
if i < 3
NearZero(1) = 0;
NearZero(2) = 0;
NearZero(3) = 0;
293
NearZero(4) = 0;
else
NearZero(1) = Pa(i-2);
NearZero(2) = Pa(i-1);
NearZero(3) = Pa(i);
NearZero(4) = Pa(i+1);
end
FailRate = Fail / NumberOfRuns;
if Fail > 0
Beta1 = -1 * NormSInv(FailRate);
else
Beta1 = 0;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Results
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Displaying Max Momment and Max Shear values found
fprintf('\n');
fprintf('\n');
fprintf('\n');
fprintf('\n');
fprintf('\n');
fprintf('\n');
fprintf('Matlab Beta Analysis for Bin: %s \n',inputBin);
fprintf('\n');
fprintf('Slope: %1.5f \n',slope);
fprintf('Beta from line: %1.3f \n',betaLine);
fprintf('\n');
if NearZero(1) == 0
fprintf('Betas around Y = 0: --- \n');
fprintf('Betas around Y = 0: --- \n');
fprintf('Betas around Y = 0: --- \n');
fprintf('Betas around Y = 0: --- \n');
else
fprintf('Betas around Y = 0: %1.3f \n',NearZero(1));
fprintf('Betas around Y = 0: %1.3f \n',NearZero(2));
fprintf('Betas around Y = 0: %1.3f \n',NearZero(3));
fprintf('Betas around Y = 0: %1.3f \n',NearZero(4));
end
294
fprintf('\n');
fprintf('Failures: %d \n',Fail);
fprintf('Failure Rate: %1.5f \n',FailRate);
if Beta1 > 0
fprintf('Beta: %1.3f \n',Beta1);
else
fprintf('Beta: --- \n');
end
filename = strcat(inputBin,'.txt');
file_1 = fopen(filename,'w');
file_2 = fopen('resultsfull.txt','a');
fprintf(file_1,'Matlab Beta Analysis for Bin: %s \n\n',inputBin);
fprintf(file_1,'Slope: %1.5f \n',slope);
fprintf(file_1,'Beta from line: %1.3f \n',betaLine);
fprintf(file_1,'\n');
if NearZero(1) == 0
fprintf(file_1,'Betas around Y = 0: --- \n');
fprintf(file_1,'Betas around Y = 0: --- \n');
fprintf(file_1,'Betas around Y = 0: --- \n');
fprintf(file_1,'Betas around Y = 0: --- \n');
else
fprintf(file_1,'Betas around Y = 0: %1.3f \n',NearZero(1));
fprintf(file_1,'Betas around Y = 0: %1.3f \n',NearZero(2));
fprintf(file_1,'Betas around Y = 0: %1.3f \n',NearZero(3));
fprintf(file_1,'Betas around Y = 0: %1.3f \n',NearZero(4));
end
fprintf(file_1,'\n');
fprintf(file_1,'Failures: %d \n',Fail);
fprintf(file_1,'Failure Rate: %1.5f \n',FailRate);
if Beta1 > 0
fprintf(file_1,'Beta: %1.3f \n',Beta1);
else
fprintf(file_1,'Beta: --- \n');
end
fprintf(file_1,'\n');
fprintf(file_1,'\n');
fprintf(file_1,' Y = R - Q Standard Normal Variate \n\n');
i=1;
while i < NumberOfRuns
temp = [YaSorted(i), Pa(i)];
295
fprintf(file_1,' %5.2f %1.5f\n',temp);
i=i+1;
end
fprintf(file_2,'%s',inputBin);
fprintf(file_2,' %5.5f',slope);
fprintf(file_2,' %5.5f',betaLine);
fprintf(file_2,' %i',Fail);
fprintf(file_2,' %5.5f',FailRate);
fprintf(file_2,' %5.5f\n',Beta1);
fclose(file_1);
ii=ii+1;
i=1;
iii=0;
Fail = 0;
FailRate=0;
end
fclose(file_2);
296
APPENDIX F2: Excel Beta Analysis Program
Presented in this appendix is the Excel beta analysis program. For more
information on its use and construction see Chapter 6.
' Beta Analysis Program
' By Mike Murdock 8 / 25 / 08
Dim gD As Double
Dim gL As Double
Dim gRS As Double
Dim gRC As Double
Dim gRP As Double
Dim vD As Double
Dim vL As Double
Dim vRS As Double
Dim vRC As Double
Dim vRP As Double
Dim Di As Double
Dim Li As Double
Dim Ri As Double
Dim Yi As Double
Dim Fail As Double
Fail = 0
Dim Mr As Double
Dim DC As Double
Dim DW As Double
Dim Ml As Double
Dim gR As Double
Dim vR As Double
297
Dim NumberOfRuns As Double
NumberOfRuns = 10000000
Dim Mat As Integer
Dim Ya As Double
Dim Pa As Double
Dim V1 As Double
Dim V2 As Double
Dim V3 As Double
Dim V4 As Double
Dim PaB As Double
Dim VaB As Double
Dim Slop As Double
Dim i As Double
Dim ii As Integer
Dim iii As Integer
iii = 0
Dim Rand As String
Rand = "=RAND()"
Dim Blank As String
Blank = ""
rowNum = 3
i = 1
ii = Cells(rowNum, 1).Value
Do While ii > 0
If (Cells(rowNum, 7).Value > 1) Then
gD = 1.05
gL = 1.3
gRS = 1.14
gRC = 1.2
gRP = 1.15
vD = 0.1
vL = 0.18
vRS = 0.105
vRC = 0.155
vRP = 0.14
Else
gD = 1.05
298
gL = 1.3
gRS = 1.12
gRC = 1.14
gRP = 1.05
vD = 0.1
vL = 0.18
vRS = 0.1
vRC = 0.13
vRP = 0.075
End If
Mr = Cells(rowNum, 7).Value
DC = Cells(rowNum, 8).Value
DW = Cells(rowNum, 9).Value
Ml = Cells(rowNum, 10).Value
Mat = Cells(rowNum, 2).Value
If Mat < 3 Then
gR = gRC
vR = vRC
Else
If Mat < 5 Then
gR = gRS
vR = vRS
Else
gR = gRP
vR = vRP
End If
End If
ud = gD * (DC + DW)
ul = gL * Ml
ur = gR * Mr
qr = (Log(vR ^ 2 + 1)) ^ 0.5
Do While i < NumberOfRuns
Cells(rowNum, 17).Value = Rand
Di = ud + vD * ud * NormSInv(Cells(rowNum, 17).Value)
Cells(rowNum, 17).Value = Rand
Li = ul + vL * ul * NormSInv(Cells(rowNum, 17).Value)
299
Cells(rowNum, 17).Value = Rand
Ri = Exp((Log(ur) - 0.5 * (qr ^ 2)) + qr * NormSInv(Cells(rowNum, 17).Value))
Ya = Ri - (Di + Li)
If Ya < 0 Then
Fail = Fail + 1
End If
i = i + 1
Loop
Cells(rowNum, 17).Value = Blank
Cells(rowNum, 11).Value = Fail
Cells(rowNum, 12).Value = Fail / NumberOfRuns
If Fail > 0 Then
Cells(rowNum, 15).Value = -1 * NormSInv(Cells(rowNum, 12).Value)
Else
Cells(rowNum, 15).Value = " --- "
End If
iii = iii + 2
Fail = 0
i = 1
rowNum = rowNum + 1
ii = Cells(rowNum, 1).Value
Loop
End Sub
300
APPENDIX F3: Percent Difference Between 1 Million and 10 Million Simulations
Presented in this section are table summaries of the precent difference from 1
million and 10 million run probability of failure simulations. The data is divided
in-between load effect, girder and sample.
301
Table F3 - 1: Exterior Girder Moment Standard Bridge Sample
Table F3 - 2: Exterior Girder Moment Unique Bridge Sample
302
Table F3 - 3: Exterior Girder Shear Standard Bridge Sample
303
Table F3 - 4: Exterior Girder Shear Unique Bridge Sample
304
Table F3 - 5: Interior Girder Moment Standard Bridge Sample
305
Table F3 - 6: Interior Girder Moment Unique Bridge Sample
306
Table F3 - 7: Interior Girder Shear Standard Bridge Sample
307
Table F3 - 8: Interior Girder Shear Unique Bridge Sample
308