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( <=<?= gPQRPPf    Equation 6 ? 2 
Where,  
fP  =    Probability of Failure 
P  =    Probability 
g =    Limit State Function 
R  =    Capacity (Resistance)   
Q  =    Demand (Load Effect) 
 
Since R and Q are defined as continuous random variables each having a PDF, g 
would also be a random variable with is own unique PDF.  Moreover, if the PDF for R, 
fR, and the PDF for Q, fQ, are Gaussian (i.e. having a normal distribution) then the PDF 
for g, fg, is Gaussian as well.  The mean for fg then could be defined as: 
gm = Rm - Qm     Equation 6 ? 3 
Where, 
 gm =    Mean of fg 
 Rm =    Mean of fR 
 Qm =    Mean of fQ 
The standard deviation for fg could be defined based on the standard deviations of 
fR and fQ as:   
?g2 = ?R2 + ?Q2    Equation 6 ? 4 
Where, 
 ?g =    Standard Deviation of fg 
 169 
 ?R =    Standard Deviation of fR 
 ?Q =   Standard Deviation of fQ 
Since fg is Gaussian, it is convenient to use standard normal distribution tables to 
evaluate Pf.  However, these tables are prepared for a PDF having a mean of 0 and a 
standard deviation of 1.0.  However, since fg does not necessarily have a mean of 0 and a 
standard deviation of 1.0, the following transformation is used: 
g
mggu
?
?=     Equation 6 ? 5 
Where, u is a standard normal variate of the random normal variable U, which has 
a PDF with a mean of 0 and a standard deviation of 1.0.  Setting g to 0 in Equation 6 - 5, 
this corresponds to the condition of failure. Then, 
?? ?=?=
g
mgu     Equation 6 ? 6 
 This provides a relationship between the standard normal variate, u, and ?. Where, 
? defined as: 
g
mg
?? =     Equation 6 ? 7 
The significance of ? is graphically represented in Figure 6 - 2, as can be seen, as 
? increases the Pf decreases and vise versa.  ? is commonly referred to as the reliability 
index or safety index in the literature when in reference to the probability of failure. 
 170 
 
Figure 6 - 2:  Graphical Representation of ? (Nowak and Collins 2000) 
 
Hence, by using the transformation of Equation 6 - 5, Pf. can be related to ? by: 
)( ?? ?=?<=
g
m
f
gUPP     Equation 6 ? 8 
Pf , therefore, can be evaluated using the standard normal distribution tables under 
this formulation, or alternatively ? can be evaluated from Pf using (Nowak and Collins 
2000): 
uPf ?=??= ? )(1?     Equation 6 ? 9 
where, (Nowak 2000) 
?  =    Reliability Index 
1??  =    Inverse of the Standard Normal Distribution Function 
fP  =    Probability of Failure 
u  =    Standard Normal Variate 
 171 
 The formulation for ? as it relates to the Pf through the inverse of the standard 
normal distribution function (Equation 6 - 9), is used for the estimated ? values presented 
in the results section of this chapter.  A graphical representation of the relationship 
between ? and the Pf can be seen in Figure 6 - 3. 
-5
-4
-3
-2
-1
0
1
2
3
4
5
0 0.2 0.4 0.6 0.8 1
Probability of Failure
????
 
Figure 6 - 3:  Relationship Between ? and the Pf 
 Several important pieces of information can be gathered from Figure 6 - 3:   
? As the Pf increases ? decreases 
? ? equal to 0 corresponds to a Pf  of 0.5 (or 50 percent) 
? Positive ? corresponds to Pf  less than 0.5 (or 50 percent) 
? Negative ? corresponds to Pf  greater than 0.5 (or 50 percent) 
? ? is more sensitive to changes in very low or high Pf 
 172 
This formulation for ? is dependent on the assumption that both fR and fQ are both 
Gaussian.  For cases where fR and fQ are of a different type of distribution, different 
formulations ? are required.  For additional information on the formulations of ? for 
different distribution types see Nowak and Collins (2000). 
 
6.2.1   Analysis Method 
The goal of this analysis is to estimate the probability of failure of an existing 
bridge and relate it to its rating factor.  Since R and Q are not deterministic, there are an 
infinite number of combinations of R and Q than can arise for any given structural 
member.  Therefore, the performance function, g, can assume an infinite number of 
values as well.  However, with a sufficient number of tests, it is possible to estimate to a 
certain degree of confidence the probability of failure of a member.  However, physical 
testing would not be feasible due to its destructive nature and the cost involved.  
Therefore, an artificial simulation technique is needed. One such commonly utilized 
simulation technique is the Monte Carlo method. The Monte Carlo method can generate 
results numerically without the need of any physical testing. An example of a basic 
Monte Carlo procedure is as follows (Nowak and Collins 2000): 
1. Randomly generate a value for R (using a the nominal resistance, assumed 
normal bias factor and coefficient of variation)  
2. Randomly generate a value for Q (using a load effect, assumed normal 
bias factor and coefficient of variation)  
3. Calculate g = R ? Q 
4. Store g values (Each simulation will produce a single value for g) 
 173 
5.  Repeat steps 1 ? 4 until sufficient number of g values has been generated 
6. Estimate the probability of failure as the number of times g < 0 divided by 
the total number simulations.  
 
The g values generated from a Monte Carlo simulation can be used in ways to 
calculate the probability of failure.  One way, as indicated previously, is to take the 
number of times g < 0 and divide it by the total number of simulations.  This will be 
refered to as Method 1 subsequently.  This method works well only when a significant 
number of g < 0 values are produced.  The number of g < 0 values needed depends upon 
how precise an estimate of probability of failure is desired.  Method 2 requires that a plot 
of cumulative probability distribution versus g be constructed on probability paper, as 
shown in Figure 6 - 4.  Then, a straight line is fitted through the data by linear regression 
analysis.  The probability of failure then would be defined as the probability-axis 
intercept.  Method 3 picks the probability of failure by simply observing where the data 
crosses the probability-axis, as shown in Figure 6 - 4.  Ideally, if the assumptions that the 
PDF for the R, Q and g are Gaussian (i.e. normally distributed) are true, the probabilities 
of failure calculated by all three methods for a single data set would be similar (Nowak 
and Collins 2000). 
 174 
 
Figure 6 - 4:  Methods of Calculating Probability of Failure (Nowak and Collins 2000) 
 
 
6.2.2   Previous Research 
In his Task 122 report, Mertz (2005) presented a limited comparative study 
between the probability of failure estimated for a bridge and its corresponding LRFR and 
 175 
LFR factors at the Design Rating level.  The approach used to calculate the probability of 
failure for each bridge was a modified Monte Carlo simulation, performed using Method 
1 of estimating the probability of failure, the ratio of the number of g <0 values to the 
total number of g values.  The main modification used in the Monte Carlo procedure 
outlined previously in the study was to assume a lognormal distribution for the resistance, 
R, instead of assuming R as being Gaussian (Mertz 2004).  Assuming R to be lognormal 
is believed to accurately describe the PDF of R (Nowak and Collins 2000).  However, 
changing the distribution type of R results in the PDF of g no longer being Gaussian as 
assumed before.  The resulting distribution for g though, is similar enough to Gaussian 
for the previously described Method 1 of calculating probability of failure to still be valid 
(Nowak and Collins 2000).    
The findings of Mertz?s study showed a strong correlation between the rating 
factors produced by the LRFR method and a bridge?s corresponding probability of 
failure, which seemed to follow a noticeable trend, as shown in Figure 6 - 5.  However, 
this correlation was not seen with the LFR method, where the data seemed to scatter from 
the trend observed for the LRFR data.  Under the LFR method, multiple bridges produced 
significantly high failure rates, even when their rating factors were found to be above 1.0.  
Mertz concluded that the rating factors produced under the LFR method are not 
appropriate and that continued use of the LFR method was irrational (Mertz 2004). 
 176 
 
Figure 6 - 5:  Reliability analysis results from Mertz Task 122 Report (2005) 
 
In addition, Mertz studied comparisons between the LRFR and the LFR factors 
and a bridge?s corresponding reliability index, as shown in Figure 6 ? 6, where ? were 
calculated from Equation 6 - 9.  As expected, a strong correlation is seen between the 
reliability index and corresponding LRFR rating factor.  Additionally, the rating factors 
produced by the LFR were observed to show little correlation when compared to the 
reliability index of a bridge.  Another important observation made by Mertz was that the 
targeted reliability index of 3.5 at Design level rating for the LRFR was not being 
reached.  The data showed that for rating factors equal to 1.0, a bridge?s observed 
reliability index would be closer to 2.5 (Mertz 2005).   
 177 
 
Figure 6 - 6:  Reliability Index compared to Rating Factor from Mertz Task 122 
Report (2005) 
 
6.3   Analysis Tools 
The reliability analysis for this research was performed using two computer 
programs developed to estimate the probability of failure through the use of the Monte 
Carlo simulation technique.  The first program that was developed to estimate the 
probability of failure of a bridge by all three methods described above. To do this the g = 
Q ? R data for each simulation were stored and sorted.  The need to store and sort the data 
limited the programming packages available due to the desire for the program to be able 
 178 
to perform at least 1 million simulations per bridge.  The ability for the program to 
perform at least 1 million simulations per bridge was desirable so that the results 
generated could be directly compared to those presented by Mertz (2005). With this 
limitation MatLAB (2008) was chosen as the most suited framework for programming.  
A copy of the MatLAB Reliability Analysis program can be found in Appendix F1.  The 
program works by extracting a bridge?s resistance, dead load effect, and live load effect 
from a Microsoft Excel file and then performs a Monte Carlo Simulation for the data one 
million times.  Using the data gathered during these simulations, the probability of failure 
is then calculated based on the three methods previously discussed, and exported to a 
unique file.    The second program was designed only to estimate the probability of 
failure by the Method 1, described above, but for ten million simulations.  Since Method 
1 does not require the data for g to be sorted and stored, a Visual Basic Macro was 
developed in Microsoft Excel.  This macro uses the same Monte Carlo simulation 
procedure as the MatLAB program but for ten million simulations per bridge.  A copy of 
the Reliability Analysis Macro can be found in Appendix F2.   
The reliability of a bridge was determined based on the HL-93 design load model 
effects and resistances calculated based on the AASHTO LRFD (2007).  The Monte 
Carlo simulation procedure used for the determination of the probability of failure for 
each program is similar to the Monte Carlo procedure Mertz used in his Task 122 Study 
(Mertz 2005).  The Monte Carlo simulation involves the following ten steps: 
1. Gather the nominal dead load, Dn, nominal live load plus impact, Ln, and 
nominal resistance, Rn, for a bridge according to the AASHTO LRFD 
Bridge Design Specifications 
 179 
2. Assume i = 1 
3. Generate a uniformly distributed random number Di? , between 0 and 1 
4. Calculate )(1 DiDDiD ??? ??+=  
Where, 1??   =  is the inverse standard normal distribution function 
 D?     =  ND D?  
 D?     =  DDV ?  
Where, D? is the dead load bias factor and DV  is the dead load 
coefficient of variation 
5. Generate a uniformly distributed random number Li? , between 0 and 1 
6. Calculate )(1 LiLLiL ??? ??+=  
Where, 1??   =  is the inverse standard normal distribution function 
 L?     =  NL L?  
 L?     =  LLV ?  
Where, L? is the live load bias factor and LV  is the live load 
coefficient of variation 
7. Generate a uniformly distributed random number Ri? , between 0 and 1 
8. Calculate ))(exp( 1lnln RiRRiR ??? ??+=  
Where, 1??   =  is the inverse standard normal distribution function 
 Rln?     =  2ln2/1)ln( RR ?? ?  
 Rln?     =  2/12 ))1(ln( +RV  
 180 
Where, R? is the resistance bias factor and RV  is the resistance 
coefficient of variation 
9. Calculate gi = Ri ? (Di ? Li) 
10.  assume i = i + 1, loop to step 3 until i > 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.  <http:// trb.org/>. 
 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.  
<http:// trb.org/>. 
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 .002<if
0.65 0.15 dsc 1???? ????+ otherwise
otherwise
:=  
? 0.9=  
? Mn? 55.871=  kip-ft / ft 
Shear Resistance 
? 2:=  ? 45:=  dv ds:=  in 
Vc 0.0316?? fc? dv? 12? 16.863=:=  kip 
Vn Vc 16.863=:=  kip 
? v 0.9:=  
? v Vn? 15.177=  kip 
* assumes shear is only resisted by the concrete section  
AASHTO 5.7.2.2 
AASHTO 5.7.3.2.2 
AASHTO 5.7.3.2.2 
AASHTO 5.5.4.2 
AASHTO 5.8.3.3 
AASHTO 5.8.4.1 
AASHTO 5.8.3.3 
AASHTO 5.5.4.2 
 218 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
LRFR  Analysis  
? c 1.0:=  condition factor 
? s 1.0:=  system factor 
? DC 1.25:=  load factor for structural components and attachments 
? L 1.4:=  evaluation live-load factor 
RFEXTmoment
? c ? s? ?? Mn? ? DC MDCext??( )
? L MLL?:=  
RFEXTshear
? c ? s? ? v? Vn? ? DC VDCext??( )
? L VLL?:=  
HS 20-44  HS 20-44  
H 20-44  H 20-44  
Tandem Tandem 
Triaxle Triaxle RF
EXTmoment
1.233
1.233
1.237
0.887
0.989
1.413
1.229
2.41
??
?
?
?
?
?
?
?
??
??
?
?
?
?
?
?
?
??
=  RFEXTshear
1.375
1.628
1.553
1.148
1.221
1.679
1.59
5.476
??
?
?
?
?
?
?
?
??
??
?
?
?
?
?
?
?
??
=  
Concrete  Concrete  
18 Wheeler 
6-Axel         6-Axel         
School Bus 
RFINTmoment
? c ? s? ?? Mn? ? DC MDCint??( )
? L MLL?:=  
RFINTshear
? c ? s? ? v? Vn? ? DC VDCint??( )
? L VLL?:=  
  
MCE LRFR 6.4.2.3 
MCE LRFR 6.4.2.4 
MCE LRFR 6.4.2.2 
MCE LRFR 6.4.4.2.3 
18 Wheeler 
School Bus 
Controlling Rating Results 
 219 
 
 
 
 
 
 
 
 
 
 
 
 
 
RFEXTmomentMin min RFEXTmoment( ) 0.887=:=  
RFEXTshearMin min RFEXTshear( ) 1.148=:=  
RFINTmomentMin min RFINTmoment( ) 1.034=:=  
RFINTTshearMin min RFINTshear( ) 1.282=:=  
 220 
 
 
 
 
 
APPENDIX B2:  Example Problem A1: Steel I Girder 
Presented in this Appendix is the Mathcad file used for slab bridges for rating 
under the LRFR methodology. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Simple Span Composite Steel Stringer Bridge 
L 65:=  ft W33x130 Section Pro. 
Fy 36:=  ksi Dw 33.1:=  in Aw 38.26:=  in2 Sweight 130:=   
fc 3000:=  psi bf 11.51:=  in tf .855:=  in 
ADDT 1000:=  Dweb Dw 2 tf??:=  tw .58:=  in Iw 6699:=   
Dweb 31.39=  wc .145:=  
PL 5/8 in x 10 1/2 in E 29000:=  
tpl 58:=  in bpl 10.5:=  in Apl tpl bpl?:=  
Loads: 
DesignLaneLoadMoment 338:=  kipft DesignLaneLoadShear 20.8:=  kip 
DesignTruckMoment 890.9:=  kipft DesignTruckShear 61.6:=  kip 
TandemAxlesMoment 726.9:=  kipft TandemAxlesShear 48.4:=  kip 
 
Section Properties: 
y1
Dw
2 tpl+
??
?
??
? Aw?
tpl
2
??
?
??
? Apl( )?+
??
?
??
?
Aw Apl( )+[ ]:=
 Distance to C.G. 
in y1 14.706=  from bottom of section 
Ix Iw Aw Dw2 y1? tpl+??? ???
2
?+ Apl y1 tpl2???? ???
2
?+:=  
Ix 8291.803=  
Sb Ixy1:=  St IxDw tpl+ y1?( ):=  Sb 563.832=  
St 435.978=  
 221 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Effective Flange Width min. of: LRFD  4.6.2.6.1 
tdeck 7.25:=  in wdeck 22:=  ft spans 3:=  
Efw1 14 L? 12?:=  
Efw2 tdeck 12? max tw 12 bf?, ??? ???+:=  
Efw3 wdeckspans 12?:=  
Efwidth min Efw1 Efw2, Efw3, ( ):=  
Efwidth 88=  
Modular Ratio (n): 
n round 29000000
57000 fc( ).5?
0, ??
?
??
?
:=  n 9=  
Short-Term Composite (n) 
in 
Efwidth
n 9.778=
 Width of concrete transformed section 
Long-Term Composite (3n) 
in 
Efwidth
3 n? 3.259=
 Width of concrete transformed section 
Transformed Slab Short-Term Pro. 
yslabs
Dw
2 tpl+
??
?
??
? Aw?
tpl
2
??
?
??
? Apl( )?+
Efwidth
n
??
?
??
? tdeck? Dw tpl+
tdeck
2+
??
?
??
??+
??
?
??
?
Aw Apl+ Efwidthn??? ??? tdeck?+
:=  
yslabs 28.579=  from bottom of section 
 Ixslabs 22682.186=  in4 
Stslabs 4407.365=  
in3 
Stslabs IxslabsDw tpl+ yslabs?( ):=  
Sbslabs 793.678=  
 222 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Sbslabs Ixslabsyslabs( ):=  in3 
Transformed Slab Long-Term Pro. 
yslabl
Dw
2 tpl+
??
?
??
? Aw?
tpl
2
??
?
??
? Apl( )?+
Efwidth
3 n?
??
?
??
? tdeck? Dw tpl+
tdeck
2+
??
?
??
??+
??
?
??
?
Aw Apl+ Efwidth3 n???? ??? tdeck?+
:=  
yslabl 22.523=  from bottom of section 
 
Ixslabl 16328.84=  in4 
Stslabl IxslablDw tpl+ yslabl?( ):=  Stslabl 1457.645=  in3 
Sbslabl Ixslablyslabl( ):=  Sbslabl 724.992=  in3 
Summary of Section Properties at MidSpan 
a. Steel Section Only: 
St 435.978=  in3 at top of Steel section 
Sb 563.832=  in3 at bottom of Steel section 
b. Composite Section - Short Term: 
Stslabs 4407.365=  in3 at top of Steel section 
Sbslabs 793.678=  in3 at bottom of Steel section 
c. Composite Section - Long Term: 
Stslabl 1457.645=  in3 at top of Steel section 
Sbslabl 724.992=  in3 at bottom of Steel section 
 223 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Dead Load Analysis - Interior Stringer 
1. Components and Attachments   DC 
a. Non-Composite Dead Loads:  DC1 
DeckLoad wdeckspans??? ??? tdeck12??? ???? .15? 0.665=:=  kipft  
StringLoad Sweight 1.061000? 0.138=:=  kipft  
CPload tpl bpl? .490144??? ???? 1.06? 3865? 0.014=:=  kipft  
DiaLoad 3 .0427? 7.33? 1.0665? 0.015=:=  kipft  
TotalDC1 DeckLoad StringLoad+ CPload+ DiaLoad+:=  
TotalDC1 0.832=  kipft  
Mdc1 TotalDC1 L
2( )?
8 439.154=:=
 kip ft?  
Vdc1 TotalDC1 L2? 27.025=:=  kip 
b. Composite Dead Loads:  DC2 
All Permanent loads on the deck are uniformly distributed among the beams. 
Curbe 1 1012??? ???? .15? 24??? ???? 0.063=:=  kipft  
Parapet 18 12?( )144 6 19?144+??? ??? .15? 24? 0.172=:=  kipft  
Railing : Assume .02 k/f / 2 
TotalDC2 Curbe Parapet+ .01+ 0.244=:=  kipft  
Mdc2 TotalDC2 L
2( )?
8 129.061=:=
 kip ft?  
Vdc2 TotalDC2 L2? 7.942=:=  kip 
2. Wearing Surface DW = 0 
DeckLoad 65
2
8? 350.983=
 
 224 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Live-Load Analysis - Interior String 
1. Compute Live-Load Distribution Factors (Type (a) cross section). 
Longitudinal Stiffness Parameter Kg 
Ed 33000 wc1.5( )? fc1000??? ???
.5?
?? ??? 3155.924=:=  ksi 
Eb 29000:=  ksi 
eg .5 tdeck? Dweb+ tpl+ y1? 20.934=:=  
Kg EbEd??? ??? Ix Apl Aw+( ) eg2?+?? ??:=  
Kg 2.567 105?=  in4 Kg 290000:=  
a) Distribution Factor for Moment  gm 
S wdeckspans 7.333=:=  
One-Lane Loaded: 
gm1 0.06 S14??? ???
.4 S
L
??
?
??
?
.3 Kg
12 L? tdeck3?
??
?
??
?
.1
+ 0.46=:=  
Two or More Lanes Loaded: 
gm2 0.075 S9.5??? ???
.6 S
L
??
?
??
?
.2 Kg
12 L? tdeck3?
??
?
??
?
.1
+ 0.627=:=  
Use: gm max gm1 gm2, ( ) 0.627=:=  
b) Distribution Factor for Shear 
One-Lane Loaded: 
gv1 0.36 S25+ 0.653=:=  
Two or More Lanes Loaded: 
gv2 0.2 S12??? ???+ S35??? ???
2.0
? 0.767=:=  
Use: gv max gv1 gv2, ( ) 0.767=:=  
 225 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
2. Compute Maximum Live Load Effects. 
a) Maximum Design Live Load (HL-93) Moment at midspan 
IM 1.33:=  
MaxMoment 1522.897=  kipft 
b) maximum Design Live Load Shear at Beam ends 
MaxShear DesignLaneLoadShear max DesignTruckShear TandemAxlesShear, ( ) IM?+:=  
MaxShear 102.728=  kip 
Distributed Live-Load Moments and Shears 
Design Live-Load (HL-93): 
Mll MaxMoment gm? 954.884=:=  kipft 
Vll MaxShear gv? 78.814=:=  kip Mll 954.884=  
Compute Nominal Resistance of Section At Midspan 
Plastic Forces: 
Ps .85 fc? Efwidth? tdeck1000? 1626.9=:=  kips Force from Deck 
Pc Fy bf? tf? 354.278=:=  kips Force of top of W steel 
Pw Fy Dweb? tw? 655.423=:=  kips Force from web of W steel 
Pt Fy bf tf? Apl+( )? 590.528=:=  kips Force from bottom steel 
Ptr 0:=  Force from slab reinforcement 
Pbr 0:=  (will add later)  
Crb 0:=  
Crt 0:=  
MaxMoment DesignLaneLoadMoment max DesignTruckMoment TandemAxlesMoment, ( ) IM?+:=  
 226 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Ybar Calcs 
This program uses the formulas given in Table 6D.1-1 in the AASHTO LRFD to 
determine Ybar based different section forces. 
 
Case One Ybar taken from top of web of W section 
Case Two Ybar taken from top of Top Flange of W section 
Remaining Cases Ybar taken from top of concrete deck 
Ybar FindYbar Ps Pc, Pw, Pt, Ptr, Pbr, Dweb, tf, tdeck, Crb, Crt, ( ):=  
Ybar 7.131=  in 
Plastic Moment Mp 
This program uses the formulas given in Table 6D.1-1 in the AASHTO LRFD to 
determine Mp based different section forces and Ybar. 
 
Case One Ybar taken from top of web of W section 
Case Two Ybar taken from top of Top Flange of W section 
Remaining Cases Ybar taken from top of concrete deck 
Mp FindMp Ps Pc, Pw, Pt, Ptr, Pbr, Dweb, tf tpl+, tf, tdeck, Crb, Crt, Ybar, ( )12 3031.1=:=  kft 
 227 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Nominal Flexural Resistance Mn 6.10.7.1.2 
Dt tpl Dw+ tdeck+ 40.975=:=  
Dp Ybar 7.131=:=  
FindMn Dt Dp, Mp, ( )
Mp
break
Dp 0.1 Dt??if
Mp 1.07 .7 DpDt??? ???????? ???? Dp 0.1 Dt?>if
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