RE-EXAMINATION OF THE SIMPLIFIED METHOD (HENRY’S
METHOD) OF DISTRIBUTION FACTORS FOR LIVE LOAD
MOMENT AND SHEAR
FINAL REPORT
Project No. TNSPR-RES 1218 Contract No. CUT 265
Submitted to
Tennessee Department of Transportation Suite 900, James K. Polk Building Nashville, Tennessee 37243-0334
By
X. Sharon Huo, Ph.D., P.E. Assistant Professor of Civil Engineering
Stewart O. Conner, EIT Graduate Research Assistant
Rizwan Iqbal Graduate Research Assistant
June 2003
Tennessee Technological University P.O. Box 5032
Cookeville, Tennessee 38505
This privileged Document is prepared solely for the appropriate personnel of the Tennessee Department of Transportation and the Federal Highway Administration in review and comment. The opinions, findings, and conclusions expressed here are those of authors and not necessarily those of the Tennessee Department of Transportation and/or the Federal Highway Administration. The document is not to be released without permission of the Tennessee Department of Transportation.
Technical Report Documentation Page 1. Report No. TNSPR-RES 1218
2. Government Accession No. 3. Recipient’s Catalog No.
4. Title and Subtitle Re-Examination of the Simplified Method (Henry’s Method) of Distribution Factors for Live Load Moment and Shear
5. Report Date 6/25/03 6. Performing Organization Code
7. Author(s) X. Sharon Huo, Stewart Conner and Rizwan Iqbal 8. Performing Organization Report No. 9. Performing Organization Name and Address Center for Electric Power Box 5032, Tennessee Technological University Cookeville, TN 38505-0001
10. Work Unit No. (TRAIS)
11. Contract or Grant No. CUT 265
12. Sponsoring Agency Name and Address Structures Division Tennessee Department of Transportation James K. Polk Building, Suite 1100 505 Deaderick Street, Nashville, TN 37243-0339
13. Type of Report and Period Covered November 1, 2001 to June 30, 2003 14. Sponsoring Agency Code
15. Supplementary Notes
16. Abstract The Henry’s Equal Distribution Factor (EDF) method is a simplified method for calculating the distribution factor of live load moment and shear. The method has been in use in Tennessee since 1963. This method assumes that all beams, including interior and exterior beams, have equal distribution of live load effects. Parameters in this method are limited to only roadway width, number of girders, and a load intensity factor. Because Henry’s method is less restrictive, it can be applied to different types of superstructures without any difficulty. The main objective of this study was to carefully reexamine the simplified method (Henry’s method) for live load moment and shear distribution factors in highway bridge design. To pursue this objective, a comparison study was conducted to investigate the differences among the distribution factors in actual bridges calculated using Henry’s method, the AASHTO Standard, the AASHTO LRFD, and finite element analysis (FEA). Twenty-four Tennessee bridges of six different types of superstructures were selected for detailed analysis and comparison. Finite element analysis was pursued to determine the moment and shear distribution factors for each of these bridges. Based on the comparison and evaluation, it was found that the Henry’s distribution factors were in a good agreement with the moment distribution factors obtained from FEA and the LRFD method and were consistently unconservative for shear distribution factors compared to the FEA results. Therefore, modifications to Henry’s method, especially for shear distribution factors, were necessary. Two sets of modification factors were proposed. In the first set, the original structure type multiplier in Henry’s method was expended to more types of superstructures and one single shear factor was introduced. The second set of modification factor included the structure type factors as well as skew angle and span length correction factors to account for the effects of skew angle for skewed bridges and span length for longer span bridges. It was found that, with proper modifications, Henry’s EDF method could produce very reasonable and reliable distribution factors of live load moment and shear. The modified Henry’s method would offer advantages in simplicity, flexibility, reliability and cost savings.
17. Key Words Bridges, live load distribution, distribution factor, Henry’s method, finite element analysis, moment distribution, and shear distribution
18. Distribution Statement
19. Security Classif. (of this report) Unclassified
20. Security Classif. (of this page) Unclassified
21. No. of Pages 254
22. Price
Form DOT F 1700.7 (8-72) Reproduction of Completed page authorized
TABLE OF CONTENTS
LIST OF TABLES...................................................................................................................ix LIST OF FIGURES...............................................................................................................xiii
Chapter Page
1. INTRODUCTION................................................................................................................1
2. LITERATURE REVIEW.....................................................................................................4 2.1 AASHTO Standard Method...............................................................................................4 2.2 AASHTO LRFD Method...................................................................................................6 2.3 Henry’s Equal Distribution Factor (EDF) Method............................................................8 2.4 Other Simplified Method Studies on Distribution Factors..............................................10 2.5 Finite Element Analysis ...................................................................................................17 2.6 Field Load Verifications / Model Verifications...............................................................31
3. SELECTED TWENTY-FOUR BRIDGES AND SPECIFIED DISTRIBUTIONFACTOR METHODS...................................................................................40
3.1 Description of Selected Bridges.......................................................................................40 3.1.1 Precast Concrete Spread Box Beam Bridges.....................................................40 3.1.2 Precast Concrete Bulb-Tee Beam Bridges.........................................................42
3.1.3 Precast Concrete I-Beam Bridges ......................................................................45 3.1.4 Cast-In-Place Concrete T-Beam Bridges...........................................................47 3.1.5 Cast-In-Place Concrete Multicell Box Beam Bridges.......................................49 3.1.6 Steel I-Beam Bridges..........................................................................................51 3.1.7 Steel Open Box Beam Bridges...........................................................................53
3.2 Distribution Factors for Selected Bridges........................................................................55 3.2.1 Precast Concrete Spread Box Beams.................................................................55
3.2.2 Precast Prestressed Concrete Bulb-Tee or I-Beams, Steel I-Beams, and Cast-In-Place Concrete T-Beams...........................................................................59
3.2.3 Cast-In-Place Concrete Multicell Box Beams...................................................62 3.2.4 Steel Open Box Beams.......................................................................................65 3.2.5 AASHTO LRFD Skew Reduction Factors For Live Load Moment ................67 3.2.6 AASHTO LRFD Skew Modification Factors For Live Load Shear ................67
3.2.7 Special Analysis for Exterior Beams .................................................................69 3.2.8 Summary of Live Load Moment and Shear Distribution Factors for Selected
Bridges ...........................................................................................................71
4. FINITE ELEMENT ANALYSIS OF SELECTED TWENTY-FOUR BRIDGES .........75 4.1 ANSYS 5.7/6.1 Finite Element Program.........................................................................75 4.2 ANSYS 5.7/6.1 Elements.................................................................................................75 4.2.1 BEAM44 Element Description ..........................................................................76
4.2.2 SHELL63 Element Description .........................................................................77
v
Chapter Page
4.3 Live Load for Distribution Factors ..................................................................................78 4.4 Two-Dimensional Modeling Procedure...........................................................................83 4.5 Three-Dimensional Modeling Procedure.........................................................................85 4.6 Diaphragm Modeling........................................................................................................88 4.7 Individual Modeling Procedures ......................................................................................89
4.7.1 Precast Concrete Spread Box Beam Bridges.....................................................89 4.7.2 Precast Concrete Bulb-Tee and I-Beam Bridges...............................................90 4.7.3 Cast-In-Place Concrete T-Beam Bridges...........................................................90 4.7.4 Cast-In-Place Concrete Multicell Box Beam Bridges.......................................91 4.7.5 Steel I-Beam Bridges..........................................................................................92 4.7.6 Steel Open Box Beam Bridges...........................................................................93
4.8 Finite Element Analysis Output .......................................................................................94 4.8.1 Live Load Moment .............................................................................................94
4.8.2 Live Load Shear..................................................................................................96 4.9 Finite Element Analysis Results ....................................................................................101
5. COMPARISON AND EVALUATION OF MOMENT AND SHEAR DISTRIBUTION FACTORS OBTAINED .............................................................................103
5.1 Finite Element Analysis vs. Henry’s Method for Live Load Moment .........................103 5.1.1 Precast Concrete Spread Box Beam Bridges...................................................103 5.1.2 Precast Concrete Bulb-Tee Beam Bridges.......................................................104
5.1.3 Precast Concrete I-Beam Bridges ....................................................................105 5.1.4 Cast-In-Place Concrete T-Beam Bridges.........................................................106 5.1.5 Cast-In-Place Concrete Multicell Box Beam Bridges.....................................107 5.1.6 Steel I-Beam Bridges........................................................................................107 5.1.7 Steel Open Box Beam Bridges.........................................................................108
5.2 Finite Element Analysis vs. Henry’s Method for Live Load Shear..............................109 5.2.1 Precast Concrete Spread Box Beam Bridges...................................................109 5.2.2 Precast Concrete Bulb-Tee Beam Bridges.......................................................110
5.2.3 Precast Concrete I-Beam Bridges ....................................................................111 5.2.4 Cast-In-Place Concrete T-Beam Bridges.........................................................112 5.2.5 Cast-In-Place Concrete Multicell Box Beam Bridges.....................................113 5.2.6 Steel I-Beam Bridges........................................................................................113 5.2.7 Steel Open Box Beam Bridges.........................................................................114
5.3 Summary of Finite Element Analysis vs. Henry’s method...........................................115 5.3.1 Live Load Moment ...........................................................................................115
5.3.2 Live Load Shear................................................................................................117 5.4 Summary of AASHTO LRFD vs. FEA and Henry’s Method ......................................118
5.4.1 Live Load Moment ...........................................................................................118 5.4.2 Live Load Shear................................................................................................121 5.5 Key Parameters...............................................................................................................129 5.5.1 Span Length ......................................................................................................129
5.5.2 Skew Angle.......................................................................................................131 5.5.3 Beam Spacing ...................................................................................................134
vi
Chapter Page
5.5.4 Slab Thickness .............................................................................................................135 5.5.5 Beam Stiffness .............................................................................................................137
6. MODIFICATION OF HENRY’S EQUAL DISTRIBUTION FACTOR METHOD ...139 6.1 Discussion of Database #2 .............................................................................................139 6.2 Preliminary Modification Factor for Live Load Moment (Set 1) .................................140
6.2.1 Precast Concrete Spread Box Beam Bridges for Live Load Moment ............140 6.2.2 Precast Concrete Bulb-Tee Beam Bridges for Live Load Moment ................142 6.2.3 Precast Concrete I-Beams for Live Load Moment..........................................144 6.2.4 Cast-In-Place Concrete T-Beam Bridges for Live Load Moment ..................146 6.2.5 Cast-In-Place Concrete Multicell Box Beam Bridges for Live Load
Moment ........................................................................................................148 6.2.6 Steel I-Beam Bridges for Live Load Moment .................................................150 6.2.7 Steel Open Box Beam Bridges for Live Load Moment ..................................152
6.3 Summary of Set 1 Modification Factors for Live Load Moment .................................154 6.4 Preliminary Modification Factors (Set 2) for Live Load Moment................................156 6.5 Preliminary Modification Factors (Set 1) for Live Load Shear ....................................164
6.5.1 Precast Concrete Spread Box Beam Bridges for Live Load Shear.................165 6.5.2 Precast Concrete Bulb-Tee Beam Bridges for Live Load Shear.....................167 6.5.3 Precast Concrete I-Beam Bridges for Live Load Shear ..................................169 6.5.4 Cast-In-Place Concrete T-Beam Bridges for Live Load Shear.......................170 6.5.5 Cast-In-Place Concrete Multicell Box Beam Bridges for Live Load Shear...172 6.5.6 Steel I-Beam Bridges for Live Load Shear......................................................174 6.5.7 Steel Open Box Beam Bridges for Live Load Shear.......................................176
6.6 Summary of Preliminary Modification Factors (Set 1) for Live Load Shear...............178 6.7 Final Modification Factors for Live Load Shear (Set 1) ...............................................180 6.8 Modification Factors for Live Load Shear (Set 2).........................................................183
6.8.1 Skew Correction Factor....................................................................................184 6.8.2 Structure Factors for Live Load Shear (Set 2) .................................................186
7. CONCLUSIONS AND DESIGN RECOMMENDATIONS .........................................192 7.1 Conclusions.....................................................................................................................192 7.2 Design Recommendations..............................................................................................197 7.2.1 Modification Factors - Set 1.............................................................................198
7.2.1.1 Modification Factors for Live Load Moment.................................198 7.2.1.2 Modification Factors for Live Load Shear .....................................199 7.2.1.3 Procedures of the Modified Henry’s Method for Live Load Moment
and Shear Distribution Factors (Set 1)........................................200 7.2.2 Modification Factors - Set 2.............................................................................201
7.2.2.1 Modification Factors for Live Load Moment.................................201 7.2.2.2 Modification Factors for Live Load Shear .....................................202 7.2.2.3 Procedures of Modified Henry’s Method for Moment and Shear
(Set 2)...........................................................................................203 7.3 Final Remarks .................................................................................................................204
vii
Chapter Page
BIBLIOGRAPHY ................................................................................................................206
APPENDIX A ......................................................................................................................210
APPENDIX B ......................................................................................................................242
viii
LIST OF TABLES
Table Page
3.1 Precast Concrete Spread Box Beam Bridge Information................................................42 3.2 Precast Concrete Bulb-Tee Beam Bridge Information....................................................45 3.3 Precast Concrete I-Beam Bridge Information .................................................................47 3.4 Cast-In-Place Concrete T-Beam Bridge Information......................................................49 3.5 Cast-In-Place Concrete Multicell Box Beam Bridge Information..................................51 3.6 Cross-Sectional Properties of Steel I-Beam.....................................................................52 3.7 Steel I-Beam Bridge Information.....................................................................................53 3.8 Steel Open Box Beam Bridge Information......................................................................55 3.9 Distribution Factors for Live Load Moment by Special Analysis ..................................70 3.10 Distribution Factors for Live Load Shear by Special Analysis.....................................71 3.11 Multiple Presence Factors “m” ......................................................................................72 3.12 Summary of Distribution Factors for Live Load Moment ............................................73 3.13 Summary of Distribution Factors for Live Load Shear.................................................74 4.1 FEA Results, Live Load Moment ..................................................................................101 4.2 FEA Results, Live Load Shear.......................................................................................102 5.1 Comparison of Precast Concrete Spread Box Beam Moment Distribution Factors ....104 5.2 Comparison of Precast Bulb-Tee Beam Moment Distribution Factors........................105 5.3 Comparison of Precast Concrete I-Beam Moment Distribution Factors ......................106 5.4 Comparison of CIP Concrete T-Beam Moment Distribution Factors ..........................106 5.5 Comparison of CIP Multicell Box Beam Moment Distribution Factors ......................107 5.6 Comparison of Steel I-Beam Moment Distribution Factors .........................................108 5.7 Comparison of Steel Open Box Beam Moment Distribution Factors ..........................109 5.8 Comparison of Precast Concrete Spread Box Beam Shear Distribution Factors .........110 5.9 Comparison of Precast Bulb-Tee Beam Shear Distribution Factors.............................111 5.10 Comparison of Precast Concrete I-Beam Shear Distribution Factors ........................112 5.11 Comparison of CIP Concrete T-Beam Shear Distribution Factors.............................112 5.12 Comparison of CIP Concrete Multicell Box Beam Shear Distribution Factors.........113 5.13 Comparison of Steel I-Beam Shear Distribution Factors............................................114 5.14 Comparison of Steel Open Box Beam Shear Distribution Factors.............................115 5.15 Summary of FEA/Henry’s Method Results for Live Load Moment ..........................116 5.16 Summary of FEA/Henry’s Method Results for Live Load Shear...............................118 5.17 Summary of FEA/LRFD Results for Live Load Moment ..........................................119 5.18 Summary of LRFD/Henry’s Method Results for Live Load Moment .......................121 5.19 Summary of FEA/LRFD Results for Live Load Shear ...............................................122 5.20 Summary of LRFD/Henry’s Method Results for Live Load Shear............................124 5.21 Moment Distribution Factors – FEA vs. Henry’s Method, Database #1....................125 5.22 Shear Distribution Factors – FEA vs. Henry’s Method, Database #1 ........................126 5.23 Moment Distribution Factors – LRFD vs. Henry’s Method, Database #1.................127 5.24 Shear Distribution Factors – LRFD vs. Henry’s Method, Database #1 .....................128 5.25 Effect of Skew Angle on Shear Distribution Factors ..................................................134
ix
Table Page
6.1 Precast Concrete Spread Box Beam, FEA vs. Modified Henry’s Method forMoment, Database #1 ..................................................................................141
6.2 Precast Concrete Spread Box Beam, LRFD vs. Modified Henry’s Method forMoment, Database #1 ..................................................................................141
6.3 Precast Concrete Spread Box Beam, LRFD vs. Modified Henry’s Method forMoment, Database #2 ..................................................................................142
6.4 Precast Concrete Bulb-Tee Beam, FEA vs. Modified Henry’s Method for Moment, Database #1..................................................................................................143
6.5 Precast Concrete Bulb-Tee Beam, LRFD vs. Modified Henry’s Method for Moment,Database #1..................................................................................................143
6.6 Precast Concrete Bulb-Tee Beam, LRFD vs. Modified Henry’s Method for Moment,Database #2..................................................................................................144
6.7 Precast Concrete I-Beam, FEA vs. Modified Henry’s Method for Moment, Database #1..................................................................................................145
6.8 Precast Concrete I-Beam, LRFD vs. Modified Henry’s Method for Moment, Database #1..................................................................................................145
6.9 Precast Concrete I-Beam, LRFD vs. Modified Henry’s Method for Moment, Database #2..................................................................................................146
6.10 Cast-In-Place Concrete T-Beam, FEA vs. Modified Henry’s Method for Moment, Database #1..................................................................................................147
6.11 Cast-In-Place Concrete T-Beam, LRFD vs. Modified Henry’s Method for Moment, Database #1..................................................................................................147
6.12 Cast-In-Place Concrete T-Beam, LRFD vs. Modified Henry’s Method for Moment, Database #2..................................................................................................148
6.13 Cast-In-Place Concrete Multicell Box Beam, FEA vs. Modified Henry’s Method forMoment, Database #1 ..................................................................................149
6.14 Cast-In-Place Concrete Multicell Box Beam, LRFD vs. Modified Henry’s Method for Moment, Database #1 ..................................................................................149
6.15 Cast-In-Place Concrete Multicell Box Beam, LRFD vs. Modified Henry’s Method for Moment, Database #2 ..................................................................................150
6.16 Steel I-Beam, FEA vs. Modified Henry’s Method for Moment, Database #1 ...........151 6.17 Steel I-Beam, LRFD vs. Modified Henry’s Method for Moment, Database #1 ........151 6.18 Steel I-Beam, LRFD vs. Modified Henry’s Method for Moment, Database #2 ........152 6.19 Steel Open Box Beam, FEA vs. Modified Henry’s Method for Moment,
Database #1..................................................................................................153 6.20 Steel Open Box Beam, LRFD vs. Modified Henry’s Method for Moment,
Database #1..................................................................................................153 6.21 Steel Open Box Beam, Beams, LRFD vs. Modified Henry’s Method for Moment,
Database #2..................................................................................................154 6.22 Final Structure Type Modification Factors for Live Load Moment (Set 1) ...............155 6.23 FEA vs. Modified Henry’s Method for Live Load Moment (Set 1), Database #1.....155 6.24 Preliminary Modification Factors for Live Load Moment (Set 2)..............................158 6.25 FEA vs. Modified Henry’s Method for Live Load Moment (Set 2), Database #1.....158 6.26 LRFD vs. Modified Henry’s Method for Live Load Moment (Set 2), Database #1..158 6.27 LRFD vs. Modified Henry’s Method for Live Load Moment (Set 2), Database #2..159
x
Table Page
6.28 Final Modification Factors for Live Load Moment (Set 2) ........................................160 6.29 Summary of FEA vs. Modified Henry’s Method for Moment (Final Sets 1 and 2) ..161 6.30 Summary of LRFD vs. Modified Henry’s Method for Moment (Final Sets 1& 2) ...162 6.31 Precast Concrete Spread Box Beam, FEA vs. Henry’s Method for Shear, Database
#1..................................................................................................................165 6.32 Precast Concrete Spread Box Beam, LRFD vs. Henry’s Method for Shear, Database
#1..................................................................................................................166 6.33 Precast Concrete Spread Box Beam, LRFD vs. Henry’s Method for Shear, Database
#2..................................................................................................................166 6.34 Precast Concrete Bulb-Tee Beam, FEA vs. Henry’s Method for Shear, Database
#1..................................................................................................................167 6.35 Precast Concrete Bulb-Tee Beam, LRFD vs. Henry’s Method for Shear, Database
#1..................................................................................................................168 6.36 Precast Concrete Bulb-Tee Beam, LRFD vs. Henry’s Method for Shear, Database
#2..................................................................................................................168 6.37 Precast I-Beam, FEA vs. Henry’s Method for Shear, Database #1 ............................169 6.38 Precast I-Beam, LRFD vs. Henry’s Method for Shear, Database #1 .........................170 6.39 Precast I-Beam, LRFD vs. Henry’s Method for Shear, Database #2 .........................170 6.40 CIP T-Beam, FEA vs. Henry’s Method for Shear, Database #1 ................................171 6.41 CIP T-Beam, LRFD vs. Henry’s Method for Shear, Database #1..............................171 6.42 CIP T-Beam, LRFD vs. Henry’s Method for Shear, Database #2..............................172 6.43 CIP Concrete Multicell Box Beam, FEA vs. Henry’s Method for Shear, Database
#1..................................................................................................................173 6.44 CIP Concrete Multicell Box Beam, LRFD vs. Henry’s Method for Shear, Database
#1..................................................................................................................173 6.45 CIP Concrete Multicell Box Beam, LRFD vs. Henry’s Method for Shear, Database
#2..................................................................................................................174 6.46 Steel I-Beam, FEA vs. Henry’s Method for Shear, Database #1................................175 6.47 Steel I-Beam, LRFD vs. Henry’s Method for Shear, Database #1 .............................175 6.48 Steel I-Beam, LRFD vs. Henry’s Method for Shear, Database #2 .............................176 6.49 Steel Open Box Beam, FEA vs. Henry’s Method for Shear, Database #1.................177 6.50 Steel Open Box Beam LRFD vs. Henry’s Method for Shear, Database #1 ...............177 6.51 Steel Open Box Beam, LRFD vs. Henry’s Method for Shear, Database #2 ..............178 6.52 Preliminary Structure Modification Factors (Set 1) for Live Load Shear ..................178 6.53 FEA vs. Modified Henry’s Method (Set 1) for Live Load Shear, Database #1 .........179 6.54 LRFD vs. Modified Henry’s Method (Set 1) for Live Load Shear, Database #1 ......179 6.55 Final Shear Factors (Set 1) ...........................................................................................181 6.56 FEA vs. Modified Henry’s Method, (Final Set 1) for Live Load Shear, Database
#1..................................................................................................................181 6.57 LRFD vs. Modified Henry’s Method (Final Set 1) for Live Load Shear, Database
#1..................................................................................................................182 6.58 Modification Factors for Live Load Shear (Set 2) ......................................................187 6.59 FEA vs. Modified Henry’s Method (Set 2) for Shear, Database #1 ...........................187 6.60 LRFD vs. Modified Henry’s Method for Shear (Set 2), Database #1 ........................188 6.61 LRFD vs. Modified Henry’s Method for Shear (Set 2), Database #2 ........................189
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Table Page
6.62 Distribution Factors for Live Load Moment, Database #1 .........................................190 6.63 Distribution Factors for Live Load Shear, Database #1..............................................191 7.1 Common Deck Superstructures Covered in this Research............................................198 7.2 Structure Type Modification Factors for Live Load Moment (Set 1)...........................199 7.3 Modification Factors for Shear Distribution (Set 1)......................................................200 7.4 Modification Factors for Live Load Moment (Set 2) ....................................................202 7.5 Modification Factors for Live Load Shear (Set 2).........................................................203 A1 Precast Spread Box Beam Distribution Factors for Live Load Moment ......................215 A2 Precast Spread Box Beam Distribution Factors for Live Load Shear...........................216 A3 Precast Concrete Bulb-Tee Distribution Factors for Live Load Moment.....................220 A4 Precast Concrete Bulb-Tee Distribution Factors for Live Load Shear .........................220 A5 Precast Concrete I-Beam Distribution Factors for Live Load Moment........................225 A6 Precast Concrete I-Beam Distribution Factors for Live Load Shear ............................225 A7 Cast-In-Place T-Beam Distribution Factors for Live Load Moment ............................230 A8 Cast-In-Place T-Beam Distribution Factors for Live Load Shear.................................231 A9 CIP Multicell Box Beam Distribution Factors for Live Load Moment........................234 A10 CIP Multicell Box Beam Distribution Factors for Live Load Shear ..........................234 A11 Steel I-Beam Distribution Factors for Live Load Moment .........................................238 A12 Steel I-Beam Distribution Factors for Live Load Shear..............................................239 A13 Steel Open Box Beam Distribution Factors for Live Load Moment ..........................240 A14 Steel Open Box Beam Distribution Factors for Live Load Shear...............................241 B1 Precast Concrete Spread Box Beam Bridges, Database #2 ...........................................243 B2 Precast Concrete Bulb-Tee Beam Bridges, Database #2...............................................244 B3 Precast Concrete I-Beam Bridges, Database #2.............................................................245 B4 Cast-In-Place Concrete T-Beam Bridges, Database #2.................................................246 B5 Cast-In-Place Concrete Multicell Box Beam Bridges, Database #2 .............................248 B6 Steel I-Beam Bridges, Database #2................................................................................250 B7 Steel Open Box Beam Bridges, Database #2.................................................................254
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LIST OF FIGURES
Figure Page
2.1 Two-Plate Mesh Discretization Example ........................................................................19 2.2 Typical Concrete Deck and Beam Elements (Case a).....................................................21 2.3 Typical Cross Section Through Part of Finite-Element Model (Case c) ........................22 2.4 Discretization: (a) Exterior Beam and (b) Interior Beam................................................24 2.5 Position of Truck Wheel Loads: (a) Load Case 1; (b) Load Case 2; and (c) Load
Case 3 .............................................................................................................26 2.6 Cross Section of Test Bridge with Loading Cases ..........................................................33 3.1 Bridge #3 Elevation ..........................................................................................................41 3.2 Bridge #3 Plan View.........................................................................................................41 3.3 Cross Section of Precast Box Beam.................................................................................41 3.4 Bridge #3 Typical Cross Section......................................................................................42 3.5 Bridge #5 Elevation ..........................................................................................................43 3.6 Bridge #5 Plan View.........................................................................................................43 3.7 Cross Section of Bulb-T Beam ........................................................................................44 3.8 Bridge #5 Typical Cross Section......................................................................................44 3.9 Bridge #7 Elevation ..........................................................................................................46 3.10 Bridge #7 Plan View ......................................................................................................46 3.11 Cross Section of AASHTO Type III Beam ...................................................................46 3.12 Bridge #7 Typical Cross Section....................................................................................47 3.13 Bridge #10 Elevation......................................................................................................48 3.14 Bridge #10 Plan View ....................................................................................................483.15 Bridge #10 Cross Section ...............................................................................................48 3.16 Bridge #10 Haunch Profile Near Support......................................................................49 3.17 Bridge #14 Elevation......................................................................................................50 3.18 Bridge #14 Plan View ....................................................................................................503.19 Bridge #14 Typical Cross Section .................................................................................50 3.20 Bridge #17 Elevation......................................................................................................52 3.21 Bridge #17 Plan View ....................................................................................................523.22 Cross Section of Steel I-Beam .......................................................................................52 3.23 Bridge #17 Typical Cross Section .................................................................................53 3.24 Bridge #20 Elevation......................................................................................................54 3.25 Bridge #20 Plan View ....................................................................................................543.26 Bridge #20 Cross Section ...............................................................................................54 4.1 BEAM44 3-D Elastic Tapered Un-symmetric Beam......................................................76 4.2 SHELL63 Elastic Shell.....................................................................................................78 4.3 AASHTO Standard HS20-44 Truck ................................................................................80 4.4 Sample Loading Patterns for Live Load Moment: (a) Non-Skewed Bridge and (b)
Skewed Bridge...............................................................................................80 4.5 Sample Loading Patterns for Live Load Shear: (a) Non-Skewed Bridge and (b) Skewed
Bridge ............................................................................................................81
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Figure Page
4.6 Sample Loading Conditions: (a) Steel I-Beam; (b) AASHTO Type III, I-Beam; and (c) Concrete Multicell Box..........................................................................................82 4.7 2-D Cast-In-Place Multicell Box Beam Model ...............................................................83 4.8 2-D Steel Open Box Beam Model ...................................................................................84 4.9 Two Dimensional Model Loaded with One Truck for Live Load Moment ...................84 4.10 Two Dimensional Model Loaded with One Truck for Live Load Shear......................84 4.11 Finite Element Model .....................................................................................................85 4.12 SHELL63 Elements for Multicell Box Beam Bridge: (a) Cross Section of Interior
Beam and (b) Entire Structure.......................................................................86 4.13 Modeling Procedure: (a) Keypoints Plotted; (b) Keypoints and Lines Plotted; (c)
Keypoints, Lines, and Areas Plotted; and (d) Mesh View of Bridge...........87 4.14 Model Without Support Diaphragm ..............................................................................88 4.15 Model With Support Diaphragm....................................................................................89 4.16 Precast Concrete Spread Box Beam Model...................................................................90 4.17 Precast Concrete Bulb-T Model.....................................................................................90 4.18 Segmented Beam Elements at Pier ................................................................................91 4.19 Cast-In-Place T-Beam Model ........................................................................................91 4.20 Segmented Web Elements..............................................................................................92 4.21 Cast-In-Place Multicell Box Beam Model.....................................................................92 4.22 Cross Section of Steel I-Beam .......................................................................................93 4.23 Steel I-Beam Model........................................................................................................93 4.24 Steel Open Box Beam Model.........................................................................................93 4.25 Example Composite Beam Section: (a) Typical Cross Section; (b) Partial Elevation;
and (c) Stress Diagram Due to Live Load.....................................................95 4.26 BEAM44 Stress Output..................................................................................................96 4.27 Shear in Beam and Slab Bridges....................................................................................97 4.28 Idealized Beam for Shear Distribution...........................................................................98 4.29 Section of Beams with Nodes: (a) Exterior Beam and (b) Interior Beam ....................99 4.30 Sample Nodal Forces for Shear: (a) Exterior Beam and (b) Interior Beam ...............100 5.1 Histogram of FEA vs. Henry’s Method for Live Load Moment ..................................116 5.2 Histogram of FEA vs. Henry’s Method for Live Load Shear.......................................118 5.3 Histogram of FEA vs. Henry’s Method and AASHTO LRFD Method (Live Load
Moment).......................................................................................................120 5.4 Histogram of LRFD vs. Henry’s Method for Live Load Moment ...............................121 5.5 Histogram of FEA vs. LRFD for Live Load Shear .......................................................123 5.6 Histogram of LRFD vs. Henry’s Method for Live Load Shear ....................................124 5.7 Moment Distribution Factor vs. Span Length ...............................................................130 5.8 Shear Distribution Factors vs. Span Length ..................................................................131 5.9 Moment Distribution Factor vs. Skew Angle ................................................................132 5.10 Shear Distribution Factor vs. Skew Angle...................................................................133 5.11 Moment Distribution Factor vs. Beam Spacing ..........................................................134 5.12 Shear Distribution Factor vs. Beam Spacing...............................................................135 5.13 Moment Distribution Factor vs. Slab Thickness .........................................................136 5.14 Shear Distribution Factor vs. Slab Thickness..............................................................137
xiv
Figure Page
5.15 Moment Distribution Factor vs. Beam Stiffness .........................................................138 6.1 Histogram of Moment Distribution Factor (Set 1), Database #1 ..................................156 6.2 Histogram of Moment Distribution Factor (Set 1 and 2), Database #1 ........................159 6.3 Histogram of Moment Distribution Factor (Final Sets 1 and 2) ...................................161 6.4 Moment Distribution Factor vs. Skew Angle (Set 2), Database #1..............................163 6.5 Moment Distribution Factor vs. Span Length (Set 2), Database #1 .............................163 6.6 Database #1, Histogram (Set 1) for Live Load Shear....................................................179 6.7 Histogram of Shear Distribution Factor (Final Set 1), Database #1 .............................182 6.8 Shear Distribution Factor vs. Skew Angle (Set 1), Database #1...................................183 6.9 Skew Correction Factor vs. Skew Angle (Set 1) ...........................................................186 6.10 Histogram of Shear Distribution Factor (Final Set 2), Database #1 ...........................188 6.11 Shear Distribution Factor vs. Skew Angle (Set 2).......................................................189 A1 Bridge #1 Distribution Factor (1)...................................................................................212 A2 Bridge #1 Lever Rule Method........................................................................................213 A3 Bridge # 1 Distribution Factor (2)..................................................................................215 A4 Bridge #5 Distribution Factor ........................................................................................219 A5 Bridge #6 Distribution Factor.........................................................................................224 A6 Bridge #10 Lever Rule Method......................................................................................227 A7 Bridge # 10 Distribution Factor (2)................................................................................229 A8 Bridge # 10 Distribution Factor (3)................................................................................230 A9 Bridge #12 Lever Rule Method......................................................................................233 A10 Bridge #16 Lever Rule Method....................................................................................236
xv
CHAPTER 1
INTRODUCTION
Distribution of live load moment and shear is crucial to the design of structural
members in bridges. Bridge design engineers have utilized the concept of distribution
factors to evaluate the transverse effect of live loads since the 1930’s. Using wheel load
distribution factors, engineers can predict bridge response by uncoupling the longitudinal
and transverse effects from each other. Currently, the lateral distribution factors for live
load moment and shear in highway bridge design are commonly determined by using the
AASHTO Standard method, the AASHTO LRFD method, or a state specified method.
The simple “S-over” approach of the AASHTO Standard method produces
conservative values for the standard bridges of today and highly conservative values for
skewed bridges. This method contains equation constants developed in the early 1930’s
based on bridges with relatively short spans, close girder spacing, and simple geometry.
In the mid 1980’s the National Cooperative Highway Research Program (NCHRP)
conducted a study to develop more accurate equations to determine live load distribution
factors. These equations were accepted as part of the AASHTO LRFD specifications as
an alternate to the AASHTO Standard method. The equations, while more accurate and
vastly more complex than the Standard method, contain limits of application that
sometimes require even more detailed analysis by the engineer.
Even though the AASHTO LRFD method is typically considered to be a more
accurate method, accuracy declines when applied to continuous bridges of varying
lengths and varying member properties. Multiple variations in the structural parameters
1
of a bridge force the design engineer to make assumptions and or generalizations,
abandon certain limits, or resort to a case-by-case finite element analysis to determine the
distribution factors. Therefore, the design community would welcome simple, less
complex live load distribution factor equations. Furthermore, less limited ranges of
applicability would result in more economy in the design process and less potential error
from application of refined analysis.
One such simple method has been in use in Tennessee since 1963 known as
Henry’s method. This method offers advantages in simplicity of calculation, flexibility in
application, and savings of expenses. Parameters in this method are limited to only
roadway width, number of girders, a load intensity factor, and a multiplier for steel and
prestressed I-girders.
The current research on lateral distribution of live load moment and shear was
sponsored by the Tennessee Department of Transportation. The objective of this research
was to carefully reexamine Henry’s equal distribution factor (EDF) method for live load
moment and shear distribution. To pursue this objective, a comparison study was
conducted for the distribution factors of live load moment and shear in actual bridges
using the AASHTO Standard, the AASHTO LRFD, the EDF method, and finite element
analysis (FEA). Twenty-four Tennessee bridges of six different types of superstructures
were selected for the comparison study. Modification factors for the EDF method were
developed, by comparing its results to those obtained by the finite element analysis and
the accepted accurate LRFD method. The modified EDF method was then applied to a
second larger database of actual bridges analyzed in the NCHRP Project 12-26 to further
examine its accuracy and applicability. It is the intention of this research to show that
2
with proper modification, the modified EDF method can be used to determine reasonable
and reliable distribution factors for both live load moment and shear.
This report is divided into seven chapters. The literature review in Chapter 2
presents the literature studies on the development of live load distribution factor
equations, finite element modeling and analysis of highway bridges, and field
experiments and verifications. Chapter 3 covers the descriptions of the selected 24
Tennessee bridges and various methods used in calculating live load distribution factors
for moment and shear. Chapter 4 gives detailed explanations of finite element analysis of
the selected 24 bridges. It summarizes the element types used, modeling procedure, two
cases of modeling with or without diaphragms, and the results of finite element analysis.
Chapter 5 details the comparison study of the different methods used in calculating
moment and shear distribution factors. The study focuses on the comparison of Henry’s
method verses finite element analysis. Modifications to the Henry’s EDF method based
on the comparison study and statistical analysis of bridges in databases #1 and #2 are
presented in Chapter 6. Chapter 7 summarizes the conclusions made through this study
and the design recommendations for modifications of the Henry’s method. Sample
calculations for each bridge type using the AASHTO Standard method, the AASHTO
LRFD method, and unmodified Henry’s method are shown in Appendix A. Appendix B
lists the essential information of all bridges studied in Database #2.
3
CHAPTER 2
LITERATURE REVIEW
There are a number of structural parameters that influence wheel load distribution
factors. Each one, from the type of superstructure, span length, skew angle, beam
spacing, etc., that make up the bridge geometry to the type of vehicle that crosses it, has
varying effects on the distribution factor. Throughout the years, researchers have
developed, examined, and compared countless methods to determine this critical value.
In doing so, this relatively small but crucial step in the design of a bridge has the capacity
to become as complicated as the engineer will allow.
2.1 AASHTO Standard Method
The simple “S-over” approach of the AASHTO Standard [2] method of
determining live load distribution factors produces conservative values for live load
moment for the standard bridges of today and highly conservative values for skewed
bridges. The formula format in this method is usually presented as
DSDF = (2-1)
Where S = beam spacing, and D = constant based on the type of superstructure. These
simple factors were developed when span lengths were comparatively short (about 60 ft)
and beam spacing was near six feet. However, with the bridges of today having longer
spans and greater beam spacing, the accuracy of these formulas diminishes rapidly
outside their intended parameters. The influence of skew angle on live load moment
distribution factors is also neglected in this method. Without considering the resulting
reduction in beam moments, highly conservative distribution factors are obtained. In
4
addition to the overly conservative nature of the AASHTO Standard method, it contains
parameters that limit its use for certain types of structures such as bridges with concrete
slab on prestressed concrete spread boxes. For this type of bridge, the limit of beam
spacing from 6.57 to 11.00 ft and roadway width from 32 to 66 ft forces designers to
either redefine the limits or find another method.
Article 3.23.1.2 of the AASHTO Standard Specifications specifies the use of the
method prescribed for moment to calculate the lateral distribution of wheel load for shear.
The formulas presented in the AASHTO Standard Specifications although simpler, do not
present very accurate results as demanded by today’s bridge engineers. These formulas
can result in highly unconservative distribution factors in some cases and in other cases
they may result in conservative values. A major shortcoming of this simple method is that
because it was developed several decades ago the numerous changes that have taken
place over this period of time are inconsistent with the conditions back then. These
inconsistencies include inconsistent reduction in load intensity for multiple lane loading,
inconsistent changes in distribution factors for changes in design lane width, and
inconsistencies in determination of wheel load distribution factors for different bridge
types. Upon review of the S/D formulas it was also found that these formulas were
generating valid results for bridges of typical geometry (i.e. short spans, beam spacing
near 6 ft and span length of about 60 ft) but loose accuracy once the bridge parameters
are varied. The AASHTO Standard simplified formulas were developed for non-skewed,
simply supported bridges. Although these specifications state that they can be applied to
the design of normal highway bridges, there are no additional guidelines when these
procedures are applicable. Since it is required that most of the modern bridges are
5
constructed on skew supports, the limitations of these procedures should not be
neglected.
2.2 AASHTO LRFD Method
The National Cooperative Highway Research Program (NCHRP) Project 12-26
[46] was performed to develop specification provisions for distribution of wheel loads in
highway bridges in the mid-1980’s. The more accurate and more meticulous live load
distribution factor equations developed by Project 12-26 were adopted by AASHTO in
the AASHTO LRFD design specification. Different tables of distribution factors are
used for live load moment and shear for interior and exterior beams. For both moment
and shear, different equations are used for different types of superstructures.
Additionally, separate equations are used for bridge with one lane loaded and two or
more lanes loaded. A sample equation to calculate the distribution factor for the interior
beam of a steel or concrete I-beam for live load moment is given by:
1.0
3
2.06.0
0.125.9075.0 ⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛+=
s
g
LtK
LSSMDF (2-2)
This equation is used if the bridge is loaded for two or more design lanes.
Many key parameters are considered by these new formulas such as beam
spacing S, span length L, beam stiffness Kg, and slab thickness ts. Longitudinal beam
stiffness
)( 2gg AeInK += (2-3)
is introduced which integrates beam area A, beam inertia I, beam eccentricity eg, and
modular ratio between beam and slab materials n. Although the AASHTO LRFD
specifications also include reduction factors to account for effects of skewed supports for
6
most types of superstructures, diaphragm effects were not included in the models created
in Project 12-26. Additionally, the models used were of continuous spans of equal length
and uniform beam inertia.
As stated earlier the NCHRP 12-26 project developed the AASHTO LRFD
formulas for both live load shear and moment. According to a study by Zokaie, et al [46]
it was found that the AASHTO LRFD formulas generally produced results that were
within 5% of the results of a finite element analysis. The new formulas in the AASHTO
LRFD for calculating distribution of live load shear are much more complex and also
more accurate in that they include the effects of several parameters. A typical equation in
the AASHTO LRFD specifications to calculate the shear distribution factor for the
interior beam of a steel or concrete I-beam bridge is as follows:
0.2
35122.0 ⎟
⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛+=
SSSDF (2-4)
The same parameters as in moment distribution factor formulas are used in shear
formulas. This method also has a skew increase factor for live load shear, which is to be
applied to calculate distribution factors for bridges when the line of support is skewed.
Although the AASHTO LRFD method is considered to be a more accurate
method, it would be most accurate when applied to bridges with similar restraints such as
uniform beam inertia and equal spans. With the variation in parameters of a bridge, the
accuracy of the formulas is reduced. Also previous studies have revealed that the use of
these factors can lead to conservative or unconservative distribution factors for moment
and shear in the design of continuous bridges. The distribution factor formulas in the
AASHTO LRFD include limited ranges of applicability. It is mandated by the LRFD
specifications that finite element analysis or grillage analysis is required when these
7
ranges are exceeded. This range of applicability and complexity of the equations in this
method have always been a concern for the design community and they would prefer
simpler, less complex equations than the ones presented in the AASHTO LRFD code.
2.3 Henry’s Equal Distribution Factor (EDF) Method
Henry’s simplified and easily applicable EDF method [42] has been in use by the
Tennessee Department of Transportation (TDOT) since 1963. A former engineer of the
Structures Division, TDOT, Henry Derthick, developed this simplified method for
calculating live load moment and shear distribution factors. Henry’s method assumes
that all beams, including interior and exterior beams, have equal distribution of live load
effects. Because Henry’s method requires only the width of the roadway, number of
traffic lanes, number of beam lines, and the intensity factor of the bridge, it can be
applied without difficulty to different types of superstructures and beam arrangements.
For most bridges, the distribution factors obtained from Henry’s method are smaller than
the ones from the AASHTO Standard Specifications. TDOT specifications state that the
designer should use the smaller value of lateral distribution factor of live load determined
from the AASHTO Standard Specifications Article 3.23 or Henry’s method in the design
of primary beams. Thus, the majority of Tennessee bridges have been designed using
Henry’s EDF method. Distribution factors from the EDF method have been verified by
bridge applications showing that the results are reasonable and reliable compared to
distribution factors from the AASHTO Standard method. Its ease of calculation,
flexibility in application, reliability, and savings in bridge cost due to the lower
distribution factors have contributed to the use of Henry’s EDF method at the Tennessee
8
Department of Transportation for nearly four decades. Following are the details of the
equal distribution factor method.
Step 1. Reinforced Concrete I-beams, Reinforced Concrete Box Beams, Precast Box
Beams:
(a) Divide roadway width by 10 ft to determine the fractional number of traffic lanes.
(b) Reduce the value from (a) by a factor obtained from a linear interpolation for intensity to determine the total number of traffic lanes considered for carrying liveload on bridge. From the AASHTO Standard 3.12.1 the intensity factor of live load equals 100% for a two-lane bridge, 90% for a three-lane bridge, or 75% for a four- or more lane bridge.
(c) Divide the total number of lanes by the number of beams to determine the number
of lanes of live load per beam or the distribution factor of lane load per beam.
(d) Multiply the value from (c) by 2 to determine the number of rows of wheels per
beam or the distribution factor of wheel load per beam.
Step 2. Steel and Prestressed I-beams:
(e) Proceed with steps (a) through (d) above.
(f) Multiply the value from (d) by a ratio of 6/5.5 or 1.09 to determine the
distribution factor of wheel load per beam.
The multiplier, 1.09, in 2(f) is used to amplify the distribution factor to steel and
prestressed I-beams only because the live load distribution factor to those types of beams
is expected to be higher than the value obtained in Step 1.
2.4 Other Simplified Method Studies on Distribution Factors
A literary review of other distribution factor studies revealed that as complex and
seemingly thorough as the NCHRP Project 12-26 results are, other methods could
9
produce adequately accurate results. One such study is Tarhini and Frederick’s [41]
comparative examination of wheel load distribution factors in I-beam bridges. Using the
basic principles of the current AASHTO S/5.5 distribution factors, a new equation was
developed based on data produced by ICES STRUDL II finite element models. A typical
bridge design was selected, and one parameter was allowed to vary at a time. The
parameters considered were the size and spacing of steel beams, presence of cross
bracing, concrete slab thickness, span length, single and continuous spans, and composite
and non-composite behavior. However, the quadratic equation developed is based only
on span length and beam spacing and yields accurate results only for single or two-lane
loading of non-skewed, single span bridges, and continuous span bridges with equal
length spans.
Khaleel and Itani [26] performed a study evaluating the behavior of continuous
normal and skewed slab-and-beam bridges (0° < θ < 60°) subject to the AASHTO HS20-
44 loadings. The effect of the relative ratio of the longitudinal bending stiffness of the
composite beam to the transverse bending stiffness of a width of slab (H) was also
examined. That is
*aDIE
H compg= (2-5)
)1(12* 2
3
υ−=
tED s (2-6)
sg
sgsgcomp AA
eAAtAII
+++=
22
12(2-7)
tbA effs 8.0= (2-8)
10
Where a = span length, t = slab thickness, Es = modulus of elasticity of the slab, Ig =
beam moment of inertia about the strong axis, beff = the effective width of the flange, e =
the eccentricity of the center of gravity of the beam with respect to the midsurface of the
slab, Eg = modulus of elasticity of the beam, As = area of the slab, and Ag = area of the
beam.
The composite bridge consisted of a reinforced concrete slab of constant thickness
supported on five equally spaced precast prestressed concrete I-beams. Beams were
simply supported at abutments and piers, while the slab was simply supported at
abutments and continuous over piers. Diaphragms were provided at abutments and piers
only. Effects of curbs were disregarded, and beam spacing was varied from six to nine
feet. The finite element method was used to model the bridges utilizing a skew-stiffened
plate consisting of two thin shell elements and one eccentric beam element connected by
rigid links. The skew reduction factor for positive and negative moments produced in the
study were based on the angle of skew, beam spacing, span length, and H. The standard
AASHTO S/5.5 distribution factor was then multiplied by the resultant. Although the
results from this procedure vary within 8% of the results obtained from finite element
analysis, its application is limited to I-beam bridges of equal span lengths and requires
multiple calculations based on beam and slab geometry unknown at the time of design to
determine expressions for the final equation.
Composite steel-concrete multicell box beam bridges were the focus of a study
done by Sennah and Kennedy [36] to determine a more practical and simple method of
developing distribution factors. Using the finite element method with ABAQUS
software, 120 bridges of various geometries were analyzed. The research studied the
11
effects of number of cells, cell geometries, span length, number of lanes, and cross
bracings. The AASHTO HS20-44 truck load and lane loads were considered. Separate
distribution factors were developed for shear and moment and compared to a simply
supported composite concrete deck-steel three-cell bridge model. Moment distribution
formulas for interior and exterior beams were developed separately. For exterior beams
in one-lane bridges, only the number of cells is required. A separate formula for exterior
beams of two-, three-, and four-lane bridges requires the number of cells, number of
lanes, and span length. For interior beams, two more formulas were developed. For one-
lane bridges, the number of cells and span length are required, whereas for two-, three-,
and four-lane bridges the number of cells, number of lanes, and span length is required.
Shear distribution factors developed in the study also consisted of four separate equations
each requiring the number of cells, number of lanes, and span length. This list of
equations, while highly accurate and beneficial to the design of steel-concrete multicell
box beam bridges, is strictly limited to this very specific type of construction.
Heins and Jin’s [22] study focused on single and continuous curved composite
steel I-beam bridge distribution factors by use of a 3-D space frame formulation. The
primary concern of the study was the effect of bottom lateral bracing. Effects of cross
type diaphragms between beams in no bays, every other bay, and all bays were
considered. Equations for distribution factors were developed based on the variation of
the ratio of curved beam stresses to straight beam stresses and the ratio of span length to
radius. Three formulas were generated to relate the curved beam response to the straight
beam response. One equation was developed for bracing in all bays, one for every other
bay, and one for no bracing. The live loads applied to isolated straight beams are
12
multiplied by these distribution factors to account for system integration. Each equation
is dependent upon diaphragm spacing, span length, radius, and beam spacing.
Additionally, the curved beam response with bracing was compared to the un-braced
curved beam system. Two further distribution factors were developed depending upon
whether or not lateral bracing was present in every bay. These equations are dependent
upon beam spacing, span length, and radius. The equations would prove very useful in
the design process for steel I-beam bridges. However, additional research and equations
would be required to accommodate the many other structure types that would add to the
already complex list of equations at hand.
Heins and Siminou [23] present a further study of the effects of radial curved I-
beam bridge systems by use of slope-deflection techniques. A series of factors were
developed, relative to the internal forces and deformations, comparing the single straight,
single curved, and curved systems. The AASHTO HS20-44 truck loading was used
throughout the study. Systems of equations were created to accurately calculate several
design factors including moment and shear. Separate equations were developed for four,
six, and eight-beam systems utilizing radius and length of span centerline. Two
modification/amplification factors were developed, which were dependent upon beam
spacing. These two factors were limited to 7-8 ft and 9-10 ft. Reduction factors due to
continuity were also developed (limited to two and three spans). Final distribution
factors are a multiplication of basic distribution factor equations,
modification/amplification factors, and possible reduction factors due to continuity. This
obviously highly limited study creates more exclusive parameters than the current
AASHTO Standard methods even within its own I-beam type.
13
Samaan, Sennah, and Kennedy [33] focused on continuous composite concrete-
steel spread-box beam bridge distribution factors. In their study, 60 continuous bridges
with various numbers of cells, roadway width, span length, and cell geometries were
subjected to the AASHTO HS 20-44 truck and lane loadings using ABAQUS finite
element software. The two equal-span bridges evaluated in this study were limited to
those with a span-to-depth ratio of 25, which has been shown to be the most economical.
Distribution factors for maximum positive and negative stress in the bottom flange along
the span were developed requiring the number of lanes, span length, and number of
beams. A distribution factor for maximum shear was also developed with the same
parameters as the ones for stresses. However, this study did not include the effect of
different span lengths in continuous bridges or bridges of more than two spans.
Additionally, no provisions were made for skew or curvature, and results are limited to
steel spread-box beam bridges.
Heins’ [21] study on box beam bridge design presents formulas pertaining to
straight and curved composite concrete-steel box beam bridges. The 82 bridges under
study were single-, two-, and three-span continuous structures of varying and unequal
span lengths between 50 and 250 ft. A live load formula for moment in curved beams
was developed using computer program results from bridges having radii varying from
200 to 10,000 ft. The developed quadratic multiplication factor, dependent upon the
centerline radius of the bridge system, is applied to an AASHTO basic distribution factor
dependent on the number of box beams and roadway width between curves. The
multiplication factor created for curvature in the study was for steel box beam bridge
systems and is therefore limited in its application.
14
Tabsh and Tabatabai [40] created modification factors for bridges subject to
oversized trucks to apply to any specification-based distribution factor. The research
utilized finite element methods to evaluate flexure and shear due to HS20, Ontario
Highway Bridge Design Code (OHBDC), PennDOT, and HTL-57 truck loadings.
Effects of truck configuration and gauge widths were investigated in the study. Three
different single span lengths, each with three different beam spacing, and each with
composite steel superstructures were considered. For each span length, a different slab
thickness and number of beams were examined resulting in a total of 9 bridges. Two
resulting gauge modification factors developed for moment and shear are dependent upon
beam spacing and gauge width. For multiple loaded lanes, the live load moment in an
interior beam is to be determined by multiplying the oversized truck effect by the
developed girder distribution factor (GDF) for one lane of loading and then adding it to
the product of specification-based GDF expressions for multiple and one lane of loading.
The study is limited to simple span bridges consisting of a concrete slab on composite
steel beams. In addition to the limited superstructure type, the three different span
lengths considered varied from only 48 ft to a modest 144 ft. For different types of
construction, much longer span lengths are possible and could cause more prominent
effects on results.
Tarhini and Fredrick [41] presented finite element analysis and modeling
techniques of I-girder highway bridges. A wheel load distribution formula was also
developed using the results of finite element analysis. This formula helped to simplify the
calculations of the distribution factors that take into account the span length of the bridge.
The developed formula was similar in form to the current AASHTO LRFD
15
specifications. The results obtained by the given formula were also compared to recent
researches and it had been found that it was consistent to a satisfactory extent. A common
type of bridge superstructure was the concrete slab on steel girders. Use of the AASHTO
method of load distribution reduced the complex analysis of the bridge subjected to one
or more trucks to a simple calculation. The AASHTO LRFD method has assumed that
the bridge acts like a collection of structural members but in reality the entire bridge
superstructure acts like a single unit. This paper presents the results of a wheel load
distribution study using ICES STRUDL three dimensional finite element analysis models
subjected to static wheel loading. The concrete slab was modeled as an isotropic eight-
node brick element (IPLSCSH) with three degrees of freedom at each node. The girder
flanges and the webs were modeled as three-dimensional quadrilateral four-node plate
elements (SBCR) with five degrees of freedom at each node. The results of the finite
element analysis demonstrated a nonlinear relationship between the beam spacing and the
calculated maximum wheel load distribution factors. For composite and non-composite
bridge decks there also existed a nonlinear relationship between the span length and the
factors. The span effect has been completely ignored in the AASHTO LRFD
specifications but this study suggested that span length was an important parameter that
affected the wheel load factors to a considerable extent. Using the results of the finite
element analysis a formula for the wheel load distribution factors related to the span
length has been introduced.
⎟⎠⎞
⎜⎝⎛ +
−+−=10
725.1021.000013.0 2 SSLLDF (2-9)
16
2.5 Finite Element Analysis
Finite element analysis has been a valuable tool in establishing highly accurate
values to compare with various analysis procedures. The method is generally accepted as
an accurate analysis in determination of live load distribution factors. Many researchers
have used finite element analysis in conducting parametric studies of distribution factors.
The NCHRP 12-26 research project is a primary example of the use of finite element
analysis to develop formulas.
Aswad and Chen [5] studied the differences between the NCHRP Project 12-26
equations, the current AASHTO Standard Specifications, and refined finite element
methods for determining live load distribution factors in prestressed concrete bridges.
Finite element modeling using the ADINA program was utilized to check the validity of
the methods. The bridge deck was modeled using both “shell” and “beam” elements as
shown in Figure 2.1. A quadrilateral (four-node) shell element of constant thickness was
used in modeling the slab. Stiffeners were described using a standard iso-parametric
beam element. Centers of the slab and beam were connected with rigid links to obtain
composite action. However, midspan diaphragms were assumed to be non-integral with
the cast-in-place deck resulting in more conservative results. Additionally, concrete
barriers were also assumed to be non-integral with the deck resulting in larger interior
beam moments. Poisson’s ratio, the St. Venant torsional constant, J, for each beam, and a
separate modulus of elasticity for slab and beams were required to describe the material
properties. The deck was modeled as an orthotropic plate by using an orthotropy factor
based on the ratio of center-to-center spacing to the clear span. Roller supports were used
at each end of the beams to resist vertical movement only. Beams were free to rotate
17
about the transverse axis at ends, but they were assumed to be restrained torsionally.
Displacement in the x-direction was restrained at the right end of the bridges at the slab
edges. The maximum aspect ratio of the quadrilateral elements was maintained at about
2 to 1, or less. Twelve or more subdivisions were made in the longitudinal direction, and
slab elements were S/2 wide in the transverse direction, where S is the beam spacing.
The information generated by the ADINA software was verified by comparing the
results to previous tests and analysis of an existing simple span (68.5 ft) two-lane bridge
consisting of five AASHTO Type III beams, 8 ft on center, and a 7½ in. slab with a 4 ft
overhang. A loading vehicle similar to the HS-20 load was used. Comparing the results
for one lane loaded, the correlation was very good. The average tensile stress in the
bottom fiber was calculated based on number of loaded lanes, mid-span moment per lane,
number of beams, and beam section modulus. This value was compared to the average
value computed from the ADINA program. The relative difference between the two
equations varied between 1 to 1.5 percent. Additionally, eight other AASHTO beams of
various shapes and two spread-box superstructures were investigated. Through this study,
it was determined that the LRFD (NCHRP Project 12-26) equations yielded smaller
distribution factors than the current AASHTO Standard methods. For interior beams, the
LRFD method yielded distribution factors 4 to 11 percent higher than finite element
methods even without considering multi-lane reduction factors.
18
Figure 2.1: Two-Plate Mesh Discretization Example
Including the reduction factors, a three-lane bridge analyzed by LRFD methods
would yield results 18 to 24 percent larger than actual values. For interior spread-box
beams, a 13 to 17 percent difference is noted. However, exterior beams consistently
showed higher distribution factors in the refined method analysis (7 to 15 percent).
LRFD methods were not assessed for exterior beams.
One study of Mabsout, Tarhini, Frederick, and Kesserwan [30] presents the
results of one- and two-span, two-, three-, and four-lane simply supported straight,
composite steel beam bridges. Span length, beam spacing, one-versus two-spans, and
number of lanes were changed and their influences were determined. Varying bridge
widths of 30 ft for two lanes, 38 ft for three lanes, and 54 ft for four lanes were studied.
35, 45.5 56, 77, 98, and 119 ft single span bridges were evaluated. 77, 98, 119, 154, and
196 ft equal two-spans were evaluated. A constant 7½ in. slab thickness was used for
each bridge. Beams were spaced at 6, 8, and 12 ft for each bridge. The AASHTO HS20
design trucks were used to load all lanes of the one- and two-span bridges to produce
maximum bending moments.
One hundred forty-four bridges were analyzed using 3D finite element methods
utilizing the SAP90 computer program. All elements were considered to be linearly
elastic, and the analysis assumed small deformations and deflections. The concrete slab
19
was modeled as a quadrilateral shell element with six degrees of freedom at each node,
and the steel beams were modeled as space frame members with six degrees of freedom
at each node. The centroid of all steel beams coincided with the centroid of concrete slab
elements. External supports were located along the centroidal axes of the beam elements.
Hinges and rollers were assigned at bearing locations. The bending moments for each
beam were calculated using the shell stresses over the effective concrete slab area in
addition to the bending moment in the space-frame member. It was found that the
AASHTO Standard wheel load distribution factors were less conservative than the LRFD
methods for short span bridges up to 60 ft and a beam spacing of 6 ft. However, as the
span length and beam spacing increased, the AASHTO Specifications became
conservative, and the LRFD method correlated with the FEA results. It is noted that
correction factors are not needed to account for continuity in steel beam bridges.
Yet another paper by Mabsout, Tarhini, Frederick, and Tayar [29] compares the
performance of four finite element-modeling techniques to evaluate wheel load
distribution factors for steel beam bridges. A typical one-span, simply supported, two-
lane, composite bridge superstructure was selected for the study. The selected bridge was
30 ft wide and consisted of a 7.5 in reinforced concrete deck supported by four steel
beams spaced at 8 ft. Two AASHTO HS20 design truck-loads were positioned
simultaneously to produce maximum moments in the beams. Two finite element
programs, SAP90 and ICES-STRUDL, were used to create four models. The first finite
element model (case a) modeled the concrete slab as quadrilateral shell elements with
five degrees of freedom at each node and the steel beams as space frame members. The
centroid of each beam coincided with the centroid of concrete slab as shown in Figure
20
2.2. The second finite element model (case b) idealized the concrete slab as quadrilateral
shell elements and beams as eccentrically connected space frame members as in case a,
but rigid links were added to accommodate for the eccentricity of the beams. The third
finite element model (case c) idealized the concrete slab and beam webs as quadrilateral
shell elements and beam flanges as space frame elements, and flange to deck eccentricity
was modeled by imposing a rigid link as shown in Figure 2.3. The fourth finite element
model (case d) idealized the concrete slab using isotropic eight node brick elements with
three degrees of freedom at each node and the steel beam flanges and webs using
quadrilateral shell elements. Hinges and rollers were used in all bridges for supports.
The maximum experimental wheel load distribution factors for seven bridges
were compared with the NCHRP 12-26 and the AASHTO Standard distribution factors.
This study supports previous findings of the AASHTO Standard methods yielding highly
conservative distribution factors for longer span lengths and beam spacing. NCHRP 12-
26 methods, while still conservative for most cases, correlated well with field test data
and with case a and d finite element modeling methods.
Figure 2.2 Typical Concrete Deck and Beam Elements (Case a)
21
Figure 2.3 Typical Cross Section Through Part of Finite-Element Model (Case c)
Another paper by Mabsout, Tarhini, Frederick, and Kesserwan [28] focuses on
another 78 two-equal-span, two-lane, straight, composite steel beam bridges. Span length
and beam spacing were varied as before, and their influence on the bridge continuity was
investigated as before. Two finite element modeling techn
finite element programs used was SAP90. In
the SAP90 program, the concrete slab was modeled as quadrilateral shell elements with
six degrees of freedom at each node, and the steel beams were modeled as space-frame
members with six degrees of freedom at each node. External supports were assumed to
be located at the centroidal axes of the beam elements.
The next finite element program used was ICES STRUDL II. The same bridges
were modeled to obtain stresses at critical sections of the bridge deck. The concrete slab
was modeled as isotropic, eight-node brick elements with three degrees of freedom at
each node. Beam flanges and webs were modeled as quadrilateral shell elements with
five degrees of freedom at each node. The AASHTO HS20 design trucks were placed
side-by-side on the two-lane bridges with 4 ft between loading points. Maximum wheel
load distribution factors were calculated for the various span length and beam spacing
combinations using the FEA method and compared with the current AASHTO (Standard
iques were used to determine
wheel load distribution factors. One of the
22
1996) formula or LRFD methods based on proposed NCHRP 12-26 formulas. As in the
previous research, the AASHTO load distribution factors were less conservative than
LRFD for short-span bridges (up to 60 ft) and a beam spacing of 6 ft. And again, as the
span length and beam spacing increased, it was found that the AA
ative and the LRFD methods were comparative to the FEA results. However,
from the results of the research, the introduction of a 5% correction factor for either
positive or negative moments for continuous bridges was recommended for use with the
LRFD methods. An average reduction factor of 15% was recommended when using the
AASHTO methods for design of continuous steel beam bridges.
Bishara and Soegiarso [8] utilized a three-dimensional finite element algorithm to
analyze three concrete double-T pretopped, tangent, simply supported bridges to
determine the internal forces in their beams and compared those to the AASHTO
Standard Specifications. Two of the three 50 ft bridges were normal bridges and the third
was a 40° skewed bridge. One of the right bridges had end diaphragms on both ends, and
the others had no end diaphragms. Using the ADINA software program, the flanges of
each beam were discretized as beam elements. Each web was divided into two beam
elements. Properties for each beam element were placed at their individual centroids.
Top and bottom beam elements were connected by truss “link elements,” see Figure 2.4.
End diaphragms were discretized as two-dimensional solid elements. The flexural
strength of the shear connectors was neglected, deformations caused by the internal
forces generated at flange connectors were very small, and linear displacements of nodal
points on each side of a connector were assumed to be identical. Fixed bearings were
SHTO methods became
conserv
assumed to prevent all displacements and rotations except rotation about an axis
23
perpendicular to that of the beams. Centroids of the bearings were assumed to be located
at the centroids of the bottom beam elements adjacent to them. Contribution of the
6"
2'
6"
2' 4' 2'
7" 7"
Top beam element
Bottom beam element
Link elements
Plate elements
Plate elements1'-9" 3'-6" 1'-9"
Bottom beam element
Top beam element
7"7"
6"
2' Link elements
(b)
Figure 2.4 Discretization: (a) Exterior Beam and (b) Interior Beam
Three loading cases were added in addition to the beam self-weight: (1)
superimposed dead load, (2) live load (AASHTO HS20-44 Standard truck-loads), and (3)
loads caused by differential camber. The AASHTO Standard trucks were positioned to
produce the maximum internal forces in the interior and exterior beams and flange
connectors as shown in Figure 2.5. Internal forces at critical sections of the beams
obtained from finite element analysis were compared to the AASHTO Standard
Specifications. It was determined that the superimposed dead loads of the sidewalks,
asphalt-sealing compound placed between flanges of adjacent beams was neglected. The
sidewalk’s width of 2 ft included the barrier, which was 9 in. wide. Weights of the
sidewalks and the barriers were lumped together as a uniform load equal to 225 psf acting
over the 2 ft width. No composite action between the sidewalks and beam was assumed.
6"
(a)
parapets, and wearing surface were not equally distributed contrary to the AASHTO
24
specifications. End diaphragms were determined to help in the distribution of gravity
loads to the different bea
ee bridges investigated were of the same magnitude as the AASHTO Standard
Specifications. However, for the exterior beams, the computed values were only 80-85%
of the AASHTO value.
Hays, Sessions, and Berry [20] carried out a study to compare the results of the
OHBDC and the AASHTO Standard methods to those obtained from the finite element
program, SALOD. Field studies were also conducted to verify the finite element analysis
modeling. For the study, the maximum moment was assumed to occur at mid-span.
Although this is not true for a series of concentrated loads, a previous study showed this
to not significantly affect distribution factors. Bridge skew was also neglected. Linearly
elastic behavior was assumed. All beams, including the exterior beams were assumed to
have the same moment of inertia. Plate bending elements were used for the finite
element model of the bridge deck. Standard frame elements were used to model the
beams and diaphragms. A 7.0 in. slab was used for prestressed beam and steel beam
bridges. A 7.5 in. slab was used for T-beam bridges. Ten elements per half-span were
used in the longitudinal direction for all the finite element models except for flat slabs.
Two elements over the beams and four elements between adjacent beams were used in
the lateral direction for prestressed beam and T-beam bridges. Steel beam bridges h
ld constant at five.
ms. The maximum live-load moment in the interior beams of
the thr
ad
six equally spaced elements between beam centerlines. For T-beam models, the ratio of
beam spacing to beam width was he Composite action between beams
and deck slab was assumed for T-beams and prestressed beams; steel beams could be
25
(b) Load Case 2
(c) Load Case 3
Figure 2.5 Position of Truck Wheel Loads
(a) Load Case 1
26
composite or non-composite. Effective slab width was calculated on the basis of the
AASHTO Standard recommendations. Constant torsional moments were assumed for
each type of bridge. Diaphragms were used only at the span ends for prestressed beam
and steel beam bridges. Intermediate diaphragms were omitted from all models. The
boundary conditions at mid-span were set so that mid-span moments would be taken only
by the beams and the slab moments at mid-span would be neglected.
The bridges presented in the study had four, five, or six Type III prestressed
concrete beams spaced at 5.6, 7.0, or 9.33 ft, respectively, with span lengths of 30, 60, 90,
and 120 ft. In the analyses, three design lane
was loaded with one, two, and three standard H20 vehicles. The SALOD solution
for one H20 vehicle was never critical for interior beams, and the modified three-H20
solutions were never critical for exterior beams. The OHBDC distribution factors were
always slightly unconservative compared to the critical SALOD distribution factors. For
exterior beams, a pronounced effect with changing beam spacing was noticed when
comparing the OHBDC and SALOD. However, for interior beams, both methods
showed essentially the same variation in the distribution factor due to beam spacing
variation. For exterior beams, span length variation produce similar trends in both
methods, with the critical SALOD values 10% higher than those of the OHBDC. For
exterior beams, the AASHTO Standard methods neglect span length and become over
conservative for span lengths greater than 60 ft. SALOD and
in distribution factor from 30 to 120 ft span lengths, differing by about 17% with
the OHBDC methods being unconservative.
s were used in the 34 ft wide bridge. The
bridge
the OHBDC showed a 22%
change
27
Barr, Eberhand and Stanton [7] present an evaluation of live load distribution
factors for concrete girders in a three span bridge in a recent study. The response of the
bridge during a static live load test was used to evaluate the reliability of a finite element
model. The effects of lifts, intermediate diaphragm, end diaphragms, continuity and skew
angle were investigated with the help of the finite element model. It was found that the
distribution factors computed with the finite element models were within 6% of the
factors calculated with the AASHTO LRFD when the bridge geometries were similar to
those considered in the development of the LRFD specifications. However for the
geometry of the bridge that was tested the discrepancy was 28%. The bridge in
consideration was a high performance prestressed concrete bridge havin
dge had three spans with a skew angle of 40 degrees.
The exterior beam and the first interior beam of the bridge were heavily
instrumented and each had two gauges that were installed at the same height in the
bottom flange to calculate the strains. The maximum moments and shears under the live
load during the test were determined. The trucks were moved from left to right of the
bridge at various locations to determine the maximum live load response. The truck was
then turned around and returned along the same line. Strain measurements were taken for
all the truck positions. The live load distribution factors were also calculated using the
AASHTO Standard method, the AASHTO LRFD method and the Ontario Highway
Bridge Design Code. It was found that the distribution factors calculated with the
AASHTO LRFD method were up to 28% larger than the factors calculated with the finite
element model. It was concluded that if the distribution factors from the finite element
g 5 girder lines.
The bri
28
model of the bridge had been used to design the girders instead of the factors from the
FD, the bridge could have been designed for a 39% higher live load.
Ebiedo and Kennedy [16] studied the effects of skew on the live load distribution
tors for shear in simply supported bridges.. The influence of other factors such as
er spacing, bridge aspect ratio, number of lanes, nu
and intermediate cross beams was presented. An experimental program was conducted on
simply supported skew composite steel-concrete bridge models. Results from the
te element analysis showed excellent agreement with the experimental results. The
ametric study was pursued on 400 bridges and based
empirical formulas for shear distribution factors were generated for the OHBDC truck
ding and also for dead loads. An extensive theoretical and experimental investigation
conducted to determine the effect of several variables on the distribution of shear
es. Empirical formulas were also deduced to calculate the shear distribution factor for
different girders of the bridges subjected to eccentric and concentric truck loading a
well as dead load. Based on this study the following conclusions were made:
Skew was the most critical parameter that influenced the shear distribution factor in
composite bridges. Increase in the skew angle reduced the shear d
girder close to obtuse corner, and increased the shear distribution factor for the
girder close to acute corner and for the interior girders. This influence becomes more
significant for skew angles greater than 30 degrees.
• The shear distribution factor was very sensitive to a change in the girder spacing
expressed in terms of the dimensionless factor N, defined as the ratio of number of
LR
fac
gird mber of girders, end diaphragms,
six
fini
par on this parametric study
loa
was
forc
the s
•
istribution factor for
the
29
lanes to the number of girders. An increase in the ratio N significantly reduced the
shear distribution factor for all the girders in the bridge.
• The presence of intermediate transverse diaphragms moment-connected to the
longitudinal girders enhanced the distribution of shear forces between girders. An
increase in the rigidity ratio, R (i.e. the ratio of tran
for continuous skew composite bridges
B
sverse rigidity to longitudinal
rigidity), increased the shear distribution factor for the girder close to the obtuse
corner and reduced it for girder close to the acute corner and for the interior girders.
• An increase in the bridge aspect ratio reduced the shear distribution factor for all the
bridge girders when the truck loading was close to the abutment. The effect of the
bridge aspect ratio increased with an increase in the skew angle.
In another study Ebiedo and Kennedy [17] demonstrated the influence of skew
and other design parameters on the shear and reaction distribution factors in continuous
composite steel-concrete bridges. Test results on three continuous composite steel-
concrete bridge models were used to verify the correctness of the finite element model for
such bridges. The bridge models had two unequal spans and were simply supported at
ends and continuous at pier. An extensive parametric study was also conducted on
prototype continuous skew composite bridges analyzed using a finite element model. The
objective of the parametric study was to determine the influence of all major parameters.
Empirical formulas were developed for the reaction and shear distribution factors at the
pier supports for exterior and interior girders .
ased on this study the following conclusions were made:
30
• The reaction and shear forces at the simply supported ends of a two span continuous
skewed composite bridge could be estimated accurately using shear distribution
factors for simply supported s
distribution of the reactions at the pier support of a two equal-span continuous
span composite bridge was almost uniform and was not sensibly affected by the skew.
However, it was considerably affected by skew in two unequal-span continuous
composite bridges. Increasing the skew increased the reaction of the exterior girder
and decreased it for the interior girder.
• The distribution of shear forces at the pier support was critical for two equal-span as
well as for two unequal-span continuous skew bridges. The shear forces increased at
the exterior girders and decreased at the interior girders with increasing skew.
• Both the reaction and shear distribution factors at the pier support were very sensitive
to the changes in the girder spacing. These factors decreased significantly with
girdersofNumber lanesofNumber
2.6 Field Load Verifications / Model Verifications
Field-testing is an important method used to verify analytical results. By
conducting live load testing on actual bridges or bridge models, researchers and engineers
will have better understanding of actual distribution factors on bridge beams under live
load. The results from live load testing can be used to validate and calibrate the code
equations of distribution factors.
Tiedeman, Albrecht, and Cayes’ [43] study further demonstrates the inaccuracy
kew bridges presented by Ebiedo and Kennedy.
• The
increase in the ratio, =N , which is a measure of girder spacing.
and inconsistency of the AASHTO Standard and the AASHTO LRFD methods.
31
Measured reactions and moments were compared with those calculated by finite element
methods, the AASHTO analysis,
d by the autostress design method (ASD) for the AASHTO HS20 truck loading
and alternate military loading. The prototype bridge consisted of two symmetrical 140 ft
spans. Three steel beams spaced 17 ft apart supported the 48 ft wide concre
ngitudinally. A 0.4-scale model was tested in a FHWA
Diaphragms consisting of rolled members of WT 2 x 6.5 in a V-type cross bracing were
located over each pier and every 10 ft from the end piers per AASHTO requirements. A
single axle of an AASHTO HS20 truck was simulated with a pair of concentrated loads
applied by pulling pairs of rods down through the deck with a hydraulic jacking system.
Loads were placed as close as possible to the exterior beam in
t in that beam. Placing the loads symmetrically about the first interior beam
maximized its maximum moments. For each loading case, one, two and three lanes were
loaded as shown in Figure 2.6. Each loading case was applied twice, and the average
values were recorded to reduce experimental error. Nine strain gauges were mounted at
six locations along each beam. Stresses were obtained by multiplying the strain and the
modulus of elasticity. Moment distribution was then calculated using the measured strain
at that location in the experiment.
Using the same dimensions and material properties, the test bridge was then
modeled with the ANSYS finite element program. Transverse node lines were located at
the diaphragms, instrumented beam sections, and loaded sections of the bridge.
and the LRFD methods. A prototype bridge was
designe
te deck,
composed of precast high-strength concrete panels, prestressed transversely and post-
tensioned lo laboratory.
order to maximize the
momen
32
Figure 2.6 Cross Section of Test Bridge with Loading Cases
In calculating the AASHTO Standard distribution factors, the specified reduction
factor of 0.9 was not used so reactions obtained could be compared to those from the
experiment and finite element analysis. Since the experiment and finite element analysis
demonstrated that three lanes were more critical than two lanes of loading, the reduction
factor of 0.9 included in the LRFD distribution factors is valid.
The stresses predicted by the AASHTO analysis were typically higher than the
measured stresses. Exterior beams stresses varied from 98-125% of experimental values.
The interior beams, however, varied from 132-308% of experimental stresses. The
LRFD results were also highly conservative for single lane loading for interior beams
(241% and 257%) and slightly conservative for multiple lane loading (131% and 128%)
including the 0.9 reduction factor. Stresses for all other loading cases were greatly over
predicted (207-464% of measured results).
Eom and Nowak’s [18] field study on live load distribution factors for a steel
beam bridge further illustrates the inaccuracy of both the AASHTO Standard and the
33
LRFD methods. In their study, strains were measured for 17 steel I-beam bridges in
Michigan, with spans from 10 to 45 m and two lanes of traffic. Strain transducers were
attached at the midspan of all beams to the lower and/or upper surfaces of the bottom
flanges. For some bridges, strain transducers were also installed on selected beams near
supports to measure the moment restraint provided by the supports and at intermediate
span locations to measure variation in moment along the span. Strain data was taken
from the middle span. The measurements were taken under passages of one and two
Michigan three-unit, 11-axle truck with known weight and axle configuration. For each
tested bridge, the trucks were driven at very low speeds to simulate static loads and at
regular speed to obtain dynamic effect on the bridge. The test results were used to
calculate the girder distribution factors (GDF’s) and calibrate the 3D ABAQUS finite
element program models. Three different boundary conditions were considered in the
FEM models: (1) roller-hinge supports; (2) hinge-hinge supports; and (3) partially fixed
supports. It was found that for bridges with ideal simple supports (roller-hinge), code-
specified GDF’s for one lane loading were more realistic. Also, the absolute value of
measured strains were lower than that
was the partial fixity of supports. For one lane of loading, both codes (AASHTO
and LRFD) were conservative. Additionally, for short span beam bridges, the AASHTO
Standard GDF values and the AASHTO LRFD are not excessively permissive and LRFD
methods provide distribution factors closer to the measured values.
Further field test studies results produced similar results. Fu, Elhelbawey, Sahin,
and Schelling’s [19] research yielded wheel load distribution factors based on strain data
collected from four in-service steel I-beam bridges in the state of Maryland. A low pass
predicted by analysis. One of the most important
reasons
34
digital filter with a cut-off frequency of 0.5 Hz was used to extract the static component
from the dynamic strain data. Although digital filtering smoothed the dynamic strains
and reduced the peak values, changing the cut-off frequency due to the dynamic strain,
the change of distribution factors was less than 1%. The strain gauges were installed at
the web and top and bottom flanges as well as at the cross frames. Six different trucks in
two traffic lanes were used to produce the measured strains in bottom flanges of beams.
The results were compared against the average distribution factors determined from 769
vehicles tested on the bridge. It was determined that distribution factors were dependent
on transverse loading position and independent of load configuration or truck weight.
Also, it was observed that for skewed bridges, the strain along transverse lines was not
equal. A parametric study was conducted on the sensitivity of the strain distribution
factors of beam bottom gauges to the different types of vehicles and their corresponding
lane of loading. From the field testing of the four bridge structures under real truck
loading, distribution factors among other data were calculated using the statistically based
root mean square (RMS) method and compared with other methods. The values obtained
from the field data were significantly less than the Imbsen & Associates, Inc. Formula,
the AASHTO Standard, the OHBDC, and the LRFD methods. In this study, the results
showed the NCHRP 12-26 to be close to the test results for the straight bridges but
unconservative for the skewed bridge.
Kim and Nowak’s [27] field study of two simply supported steel I-beam bridges
added more proof of the AASHTO Standard and LRFD inaccuracy and unreliability.
Two bridges were under observation. The first bridge, spanning 14.6 m, carried
southbound and northbound traffic on state route M-50 over the Grand River in Jackson
35
County, Michigan. The bridge consisted of 10 steel I-beam bridges. Extensive leaking
through the slab caused corrosion in beams and possibly in rebar inside the deck, but no
signs of concrete spalling were apparent. Corrosion was significant at the lower flanges
of the steel beams in the middle of the span, but corrosion near the support was minor.
The second bridge (US23/HR), a multi-span bridge, carried northbound traffic only on
US-23 over the Huron River in Michigan. It was composed of six steel I-beams and three
sparsely spaced steel diaphragms. The bridge was in good condition. Eight interior
beams of the first bridge and all six beams of the second bridg
ransducers at the lower surface of the bottom flange of the steel beams at about
mid-spans and were connected to the data acquisition system. Measurements were taken
in the entrance span in the direction of traffic. For both bridges, strains were measured
under normal truck traffic to investigate statistical characteristics of GDF’s. For the first
bridge, 81 trucks in lane 1, and 49 trucks in lane 2 were used to measure strains. For the
second bridge, 621 trucks in lane 1, and 142 trucks in lane 2 were used. One calibration
truck was used for each bridge as well: for bridge one, a four-axle trucks with a gross
weight of 302 kN was used, for bridge two, a three-axle truck with a gross weight of 280
kN was used. Two data acquisition modes were used for the study: continuous time
history data mode and burst time history data mode. The time history mode was used to
capture the strain data under a calibration truck, and the burst time history mode was used
to acquire only the significant part of the strain data under normal traffic. A sampling
rate of 200 Hz was used in conjunction with several computer programs that were
developed for the automated data processing to calculate the beam distribution factors.
GDF’s were calculated from the maximum static strain obtained by filtering the dynam
e were instrumented with
strain t
ic
36
strain a
model was identical to a straight bridge tested and reported in a
previou
t each beam at the same section along the length of the bridge. For the study,
GDF’s were taken to be the ratio of the static strain at the beam to the sum of all the static
strains based on the assumption that all beams had the same section modulus. GDF’s
obtained for bridge one were more uniform for each beam than for bridge two even with
more sparsely spaced beams and diaphragms. This illustrated the ineffectiveness of
closely spaced diaphragms for load distribution. The distribution factors for bridge one
were also much smaller than the code specifications. For two-lane loading, the AASHTO
Standard and LRFD methods gave 16% and 28%, respectively, larger values than
measured maximum factors. On the other hand, bridge two distribution factors from
LRFD methods were lower than Standard methods, and both were substantially higher
(Standard = 24% and LRFD = 19%) than experimental results.
Scordelis and Larsen [35] constructed a 1:2.82 scale model of a curved box-beam
bridge to evaluate the validity of finite element analysis. The model construction was
similar to that used on prototype structures in the filed. Except for its curvature (radius of
282 ft), the bridge
s experiment. Modeling started with the casting of the end abutments and the
center bent, followed by the erection of the soffit and exterior web formwork, and
placement of the instrumented bottom flange rebar and web reinforcement. The inner
cell forms were then inserted and the bottom flange and webs were cast. Steel billets
were then placed in the cells to give the proper dead weight for the bridge model to
simulate the prototype, and then the top slab soffit forms were put in place. These soffits
were carried by steel rods suspended from transverse beams and were later dropped into
the cells so that their interaction with the top slab was avoided. The top slab
37
reinforcement including strain gauges and strain meters was put in place, and the top slab
was cast. The model had a loading frame at each midspan section, enabling live loads to
be applied to each of the beams by means of jacks, singly or in various combinations.
The instrumentation of the bridge model was designed to measure loads,
reactions, strains, and deflections. A total of 18 load cells were used at various sections
throughout the bridge. A low-speed scanner carried out the data acquisition and
recording. Before loads were applied to the bridge, a “conditioning load” was applied to
produce a uniform nominal steel stress at sections of maximum positive and negative
moment to insure that the response under the subsequent loads was independent of the
sequence of these loads. The results of the experiment were compared to three different
analytical computer programs; SAP, CURDI, and CELL. For each program, the bridge
model was assumed to be elastic, homogeneous, isotropic, and uncracked. The study
indicated that the simple three-dimensional frame analysis could be used to predict the
total reactions, longitudinal distribution of moments, and center line deflections. The
transverse distribution of moments to each individual beam can be accurately predicted
by the theory, as can the total moment at any section. It was also noted that exterior
beams of the curved bridge should be designed for a 5%-10% higher moment than that of
the straight bridge.
Huo, Zhu, Ung, Goodwin, and Crouch [24] conducted a live load test on Pistol
Creek Bridge in Blount County, Tennessee. The bridge is a twin-bridge on State Route
162 with five spans and five lines of prestressed concrete beams. The total length of the
bridge is 373 ft and the width of the bridge is 51 ft. The prestressed concrete beams are
AASHTO Type III beams and are placed at a spacing of 10.58 ft center to center. The
38
thickness of cast-in-place concrete deck is 8.75 in. The interior beams in the first and
second span t behavior
of the prestressed beam e used in the live load
st. Both bridges were tested with one-truck and two-truck loadings. During the test,
d by dividing the obtained moments from the test by the moment obtained in
imple beam analysis. The test results showed that the distribution factors obtained in the
second span were very close to the ones from Henry’s EDF method (less than 1%) and
finite element analysis (2%), but slightly smaller than the AASHTO LRFD method (8%)
and much smaller than the AASHTO Standard method (23%).
s of both bridges were instrumented to monitor the time-dependen
s. Two TDOT three-axle dump trucks wer
te
trucks moved along the designated loading lines on the bridges in a slow speed, and then
stopped and stayed at four locations, 0.4 span length, end, quarter span length, and
midspan, for at least five minutes to allow the data acquisition system to record adequate
data. The measured data included temperature and strain at the four instrumented
sections and deflections at midspan sections. The moment of the interior beams was
determined using the measured strain. The moment distribution factors were then
calculate
s
39
CHAPTER 3
SELECTED TWENTY-FOUR BRIDGES AND SPECIFIED
DISTRIBUTION
essary for calculation of distributi
FACTOR METHODS
As mentioned in previous chapters, twenty-four Tennessee bridges were selected
and studied in this research. The bridges selected vary not only in superstructure type,
but in many other structural parameters as well. These parameters include the number of
spans and their lengths, beam spacing, beam depth, skew angle, slab thickness, and the
presence of support diaphragms. Following are the descriptions of selected bridge details
and specified equations nec on factors for each type of
bridge.
3.1 Description of Selected Bridges
3.1.1 Precast Concrete Spread Box Beam Bridges
This study contains a total of four precast concrete spread box beam bridges.
Each one has varied structural parameters as previously discussed. This section will give
illustrated details of one bridge and tabulated details of all bridges of the same type.
Bridge #3 is a six-span, precast box beam bridge carrying State Route 34 over Lick Creek
at a 90-degree angle with the roadway as shown in Figures 3.1 and 3.2. The bridge has
four middle span lengths of 80 ft – 9 in. and two end spans of 81 ft - 5½ in. The 44 ft
wide bridge, containing three lanes of traffic, is supported by four precast box beams
spaced at 11 ft – 3 in. (shown in Figure 3.3) and a 7¾ in. cast-in-place deck slab shown in
Figure 3.4. The typical cross-section of the bridge is shown in the Figure 3.4. This bridge
was built in 1974 to carry an estimated 1994 Average Daily Traffic (ADT) of 5,500
40
vehicles. The concrete compressive strengths of the precast box beam and cast-in-place
deck are 5000 psi and 3000 psi, respectively. See Table 3.1 for the complete list of key
parameters for each bridge of this type.
Figure 3.1 B idg #3 Elevat on
Figure 3.3 Cross Section of Precast Box Beam
r e i
Figure 3.2 Bridge #3 Plan View
41
centerline of beam to inner face of barrier
Figure 3.4 Bridge #3 Typical Cross Section
Table 3.1 Precast Concrete Spread Box Beam Bridge Information
Bridge Number Bridge Name
Number of
Spans
Maximum Span
Length(ft)
SkewAngle(deg)
Number of
Beams
BeamSpacing
(ft)
Beamf'c
(psi)
SlabThickness
(in)
Slabf'c
(psi)
RoadwayWidth
(ft)
Overhang* (ft)
1 S. R. 7 over Leipers Creek 3 60.88 15.00 3 10.58 6000 8.00 3000 30.00 2.67
2
South HarpethRoad. over
South HarpethCreek
3 44.38 0.00 2 13.75 6000 8.75 3000 26.33 5.12
3 S. R. 34 over Lick Creek 6 81.49 0.00 4 11.25 5000 7.75 3000 44.00 3.38
4 Del Rio Pike
over West Harpeth River
1 69.54 48.49 5 5.67 5500 8.25 4000 28.33 1.67
* Note: Overhang = Distance from
3.1.2 Precast Concrete Bulb-Tee Beam Bridges
Four precast concrete bulb-tee beam bridges were included in this study. Bridge
#5 is a four-span bridge carrying State Route 1 over Rocky River. The bridge consists of
four pier-to-pier spans with two spans of 124 ft and two spans of 124 ft – 4 in. as shown
in Figure 3.5. Each supporting abutment or pier is at a 15-degree angle from the vertical
42
with the roadway as shown in Figure 3.6. The 44 ft wide bridge containing three traffic
lanes is supported by five BT-72 beams at 8 ft – 9 in. center-to-center as shown in Figure
3.7 and an 8¼ in. cast-in-place deck slab as shown in Figure 3.8. The typical cross-
section of the bridge is shown in the Figure 3.8. The concrete compressive strengths of
the prestressed bulb tee beam and cast-in-place deck are 6000 psi and 3000 psi,
respectively. See Table 3.2 for the complete list of key structural parameters for each
bridge of this type.
Figure 3.5 Bridge #5 Elevation
Figure 3.6 Bridge #5 Plan View
43
Figure 3.7 Cross Section of Bulb-Tee Beam
Figure 3.8 Bridge #5 Typical Cross Section
44
Table 3.2 Precast Concrete Bulb-Tee Beam Bridge Information
Bridge Number Bridge Name
Number of
Spans
Maximum Span Skew Number Beam Beam Slab
ess)
Slabf'c
(psi)
RoadwayWidth
(ft)
Overhang* (ft) Length
(ft)
Angle(deg)
of Beams
Spacing(ft)
f'c(psi)
Thickn(in
ecast I-beam bridges were included in this study. Bridge
s
ft
f
c
e is shown in the Figure 3.12. The concrete
s of the prestressed concrete I-beam and cast-in-
p
parameters for each bridge of this type.
5 State Route 1 over Rocky
River 4 124.33 15.00 5 8.75 6000 8.25 3000 44.00 2.75
8 State Route 1 over C.S.X.
Railroad 6 115.49 0.00 8 10.29 9000 8.27 4000 80.74 2.60
22 Porter Roadover State Route 840
2 159.00 26.70 4 8.33 10000 8.25 3000 32.00 1.75
23
Hickman Road overState Route
839
2 151.33 17.50 4 8.33 10000 8.25 3000 32.00 1.75
* Note: Overhang = Distance from centerline of beam to inner face of barrier
3.1.3 Precast Concrete I-Beam Bridges
Three pr #7 is a three-
span bridge carrying Interstate-840 over McDaniel Road. This bridge consists of three
pans of varying span length with one controlling span of 76 ft and two end spans of 52
, as shown in Figure 3.9. Each supporting abutment or pier is at a 33.5-degree angle
rom the vertical with the roadway as shown in Figure 3.10. The 44 ft wide bridge
ontaining three traffic lanes is supported by five Type III beams spaced at 9 ft center-to-
center as shown in Figure 3.11 and an 8¼ in. cast-in-place deck slab as shown in Figure
3.12. The typical cross-section of the bridg
compressive strength place deck are 5000
si and 3000 psi, respectively. See Table 3.3 for the complete list of key structural
45
Figure 3.9 Bridge #7 Elevation
Figure 3.10 Bridge #7 Plan View
1 Cross Section of AASHTO TypeFigure 3.1 III Beam
46
Figure 3.12 Bridge #7 Typical Cross Section
Table 3.3 Precast Concrete I-Beam Bridge Information
s
74.33 0.00 5 10.58 10000 8.75
3
#
c
c
in Figure 3.15. The bridge is supported by
Bridge Number Bridge Name
Number of
Spans
Maximum Span
Length(ft)
SkewAngle(deg)
Numberof
Beam
BeamSpacing
(ft)
Beamf'c
(psi)
SlabThickness
(in)
Slabf'c
(psi)
RoadwayWidth
(ft)
Overhang* (ft)
6 I-840
Over Cox Road
3 67.42 21.33 5 9.00 5000 8.25 3000 44.00 2.25
7 I-840 overMcDaniel
Road 3 67.42 33.50 5 9.00 5500 8.25 4000 44.00 2.25
24 Pistol Creek 5 3000 51.26 2.70
* Note: Overhang = Distance from centerline of beam to inner face of barrier
.1.4 Cast-In-Place Concrete T-Beam Bridges
Three cast-in-place concrete T-beam bridges were included in this study. Bridge
10 is a five-span bridge carrying Highland Road over State Route 137. This bridge
onsists of five pier-to-pier spans of varying length as shown in Figure 3.13 with a
ontrolling span length of 96 ft. Each supporting pier is at a 9.83-degree angle from the
vertical with the roadway as shown in Figure 3.14. The bridge is 35 ft – 9 in. wide with a
6 ft sidewalk and 28 ft roadway as shown
three cast-in-place concrete T-beams of varying depth spaced at 12 ft – 7 in. center-to-
47
center as shown in Figure 3.16. The thickness of cast-in-place
omplete list of key param ters for each bridge of this type.
Figure 3.13 Bridge #10 Elevation
Figure 3.14 Bridge #10 Plan View
deck was 9 in. The
concrete compressive strength of beam and deck was 3000 psi. See Table 3.4 for the
c e
Figure 3.15 Bridge #10 Cross Section
48
Figure 3.16 Bridge #10 Haunch Profile Near Support
Table 3.4 Cast-In-Place Concrete T-Beam Bridge Information
pacing(ft)
10 Road overState Route 5 96.00 9.83 3 12.58 3000 9.00 3000 35.75
11 503
4
= Dis
66.00
ce f
0.00
cen
4
terline
8.17
f be
3000
m to
7.00
ner
3000
ce of
34.50
rrie
1.75
t-In-Pla
u bridges were included in this
dy. ridge #14 a e-span bridg carr ing B ffat ill R ad over Inte ate-6
19 and 140 ft as shown in
igure 3.17. Each support is at a 26.23-degree angle from the vertical with the
as shown in Figure 3.18. The 50 ft wide
Bridge Number Bridge Name
Number of
Spans
Maximum Span
Length(ft)
SkewAngle(deg)
Number of
Beams
BeamS
Beamf'c
(psi)
SlabThickness
(in)
Slabf'c
(psi)
RoadwayWidth
(ft)
Overhang* (ft)
9 State Route 137 4 88.50 31.56 3 11.17 3000 7.50 3000 32.33 3.08
Highland
137
2.92
Mabry Hood Road over I-
* Note: Overhang tan rom o a in fa ba r
3.1.5 Cas ce Concrete Multicell Box Beam Bridges
Four cast-in-place concrete m lticell box beam
stu B is thre e y u M o rst 40.
The bridge consists of three varying span lengths of 91, 1
F roadway
bridge carrying three lanes of traffic is
49
supported by a four-cell, cast-in-place box beam as shown in Figure 3.19. Typical web
thickness is 12 inches spaced at 10 ft – 4 in. on center. Top and bottom slab thicknesses
are 9¼ in. and 7 in., respectively for a total typical depth of 7 ft. The concrete
compressive strength of the box beam was 3000 psi. See Table 3.5 for the complete list of
key parameters for each bridge of this type.
Figure 3.17 Bridge #14 Elevation
Figure 3.18 Bridge #14 Plan View
Figure 3.19 Bridge #14 Typical Cross Section
50
Table 3.5 Cast-In-Place Concrete Multicell Box Beam Bridge Information
Bridge Number Bridge Name
Number of
Spans
Maximum Span
Length(ft)
SkewAngle(deg)
Number of
Beams
BeamSpacing
(ft)
Beamf'c
(psi)
SlabThickness
(in)
Slabf'c
(psi)
Roadway
Width
(ft)
Overhang* (ft)
12 Tri-City
Airport Roadover I-81
2 133.83 0.00 4 9.25 3000 8.00 3000 44.00 1.75
13 State Route 137 2 98.75 0.00 4 9.00 3000 8.25 3000 44.00 2.25
14 Buffat Mill
Road over I-640
3 140.00 26.23 9.25 10.33 3000 9.25 3000 50.00 2.58
15 Hill Road 2 110.00 16.50 3 9.50 300
over N huc ver. T ridg nsists four varie
gths as shown re 3.20 tw dle sp ngt 158 ft nd two en
span lengths of 123 ft – 9 in. The bridge is
n of th bridg is shown in Figure 3.23. h ce mpressive stren
ect n. See Tabl 3. r the c mple li f str tural rame
Hurincane 0 8.00 3000 36.00 2.00
* Note: Overhang = Distance from centerline of beam to inner face of barrier
3.1.6 Steel I-Beam Bridges
Four steel I-beam bridges were included in this study. Bridge #17 is a four-span
bridge carrying State Route 81 olic ky Ri he b e co of d
span len in Figu with o mid an le hs of a d
a straight bridge with no skew as shown in
Figure 3.21. The 46 ft wide bridge contains three lanes of traffic and is supported by five
steel I-beams of varying cross-sections as shown in Figure 3.22 and Table 3.6. The
spacing of the girders is 9 ft – 6 in. and the thickness of the cast-in-place deck slab is 8 in.
The typical cross-sectio e e T o gth
of concrete in deck slab is 3000 psi. Table 3.6 lists cross-sectional properties for each
beam s io e 7 fo o te st o uc pa ters for each bridge
of this type.
51
Figure 3.20 Bridge #17 Elevation
Figure 3.21 Bridge #17 Plan View
Cross Section of Steel I-Beam
n
1.125 16 1.125 161.875 16 1.875 16
Figure 3.22
Table 3.6 Cross-Sectional Properties of Steel I-Beam
Distance from Left Support (ft)
T1 (in)
Top Fla ge(in)
T2 (in)
BottomFlange
(in)
Area (in2)
Moment ofInertia (in4)
0 to 25.75 0.875 16 1.000 16 50.07 20106.4525.75 to 71.16 0.875 16 1.375 16 70.81 25360.1271.16 to 92.16 0.875 16 1.125 16 52.11 21047.04
92.16 to 110.16 58.50 23701.20110.16 to 120.16 81.76 35479.29
120.16 to Pier 2.875 16 2.875 16 112.76 49973.76
52
umberof
Maximum Span
Length
Skew Angle
Numberof
BeamSpacing
Slab Thickness
16 over NolichuckyRiver
4 158.00 0.00 5 9.50
17 ROUTE I-840
E 2 143.00
18
19.40 3 9.33 8.25 3000 28.00 2.92
State Rou
840
verhang
3
Two steel open box beam bridges were in
Granby Road over State Route 137. This bridge is a three-span steel box beam bridge
with a skew angle of 31.95-degrees. However, since the steel boxes of the two end spans
were filled with concrete and supported at the bottom, it is actually a single span bridge
with a length of 252 feet as shown in Figures 3.24 and 3.25. The 36 ft wide bridge
containing three traffic lanes is supported by two steel open boxes spaced at 9.38 feet and
a deck slab of 8½ in. as shown in Figure 3.26. The box beam has varied cross-sections
Figure 3.23 Bridge #17 Typical Cross Section
Table 3.7 Steel I-Beam Bridge Information
Bridge Number Bridge Name
N
Spans (ft) (deg) Beams (ft) (in)
Slab f'c
(psi)
RoadwayWidth
(ft)
Overhang* (ft)
State Route 1 8.00 3000 46.00 2.25
Frontage Road A over STAT
18 State Route 840 2 2.00 50.16 4 11.50 9.00 4000 44.00 3.00
19 te 6
over State Route 2 150.00 26.66 9 11.00 9.00 3000 94.00 1.25
* Note: O = Distance from centerline of beam to inner face of barrier
.1.7 Steel Open Box Beam Bridges
cluded in this study. Bridge #20 carries
53
along the span and over supports. Table 3.8 presents the complete list of structural
parameters for each bridge of this type.
Figure 3.24 Bridge #20 Elevation
Figure 3.25 Bridge #20 Pl an View
Half Section at Mid-Span Half Section at Supports
Figure 3.26 Bridge #20 Cross Section
54
Table 3.8 Steel Open Box Beam Bridge Information
Bridge Number
Spans
Maximum Span
Length (ft)
Skew
(deg)
Number
Beams
Beam
(ft)
Slab
(in)
Slab
(psi)
Roadway
(ft)
Overhang*
State Route 137
ier
3.2 Distribution Factors for Selected
ent and
s
for each type of supers
O Standard and Henry’s EDF method. The term DF is used for distribution factor
for both moment and shear. Therefore, the only separate equations for moment and shear
distribution factors presented are those in the AASHTO LRFD metho
M
f
requires the number of d N
span length (L), and
Number Bridge Name of Angle of Spacing Thickness f'c Width (ft)
20 Granby Rd 1 252.00 31.95 4 9.38 8.50 3000 36.00 2.25
21 over State Route 1
1 170.67 4.50 6 9.00 8.50 3000 52.00 1.75
* Note: Overhang = Distance from centerline of beam to inner face of barr
Bridges
This section details the standard methodology for obtaining live load mom
hear distribution factors for each type of bridge discussed in this research. The
AASHTO Standard, the AASHTO LRFD, and Henry’s EDF Method will be presented
tructure and a summary of results is given at the end of the
chapter. The calculation of distribution factor for moment and shear are the same in the
AASHT
d. In this method
DF is denoted as moment distribution factor and SDF is denoted as shear distribution
actor.
3.2.1 Precast Concrete Spread Box Beams
(a) AASHTO Standard Method
For precast concrete box beams, the AASHTO Standard method, Art. 3.28.1
esign traffic lanes ( L), number of beams (NB), beam spacing (S),
roadway width (W) to determine the distribution factor for interior
beams with the ranges of applicability as shown here:
55
For the
(b) AASHTO L
(1) Live Load M
For inte
equations and l
Beam spacing,
For one design
(3-1)2 ⎞⎛ SN
Range of Appli
greater of the le
For two or more
Range of Appli
(3-2)
⎧
≤≤
≤≤
ft66ft32ft00.11
104
W
Nb
exterior beam, Article 3.28.2 states that the distribution factor shall be the
r
2NL/NB (3-3)
RFD Method
oment
rior beams, the AASHTO LRFD Table 4.6.2.2.2b-1 has the following
imitations for one and two design lanes loaded for cross-section type b.
span length, number of beams, and beam depth (d) are required.
lane loaded
⎪⎩
25.0
2
35.0
0.120.3⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=
LSdSMDF
125.0
2
6.0
123.6⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=
LSdSMDF
3in.65in.18ft140ft20ft5.11ft0.6
⎪⎪⎩
⎪⎪⎨
⎧
≥≤≤≤≤≤≤
bNdLS
⎟⎜+= L kDF
07.0 −= Wk
cability: ⎪⎨ ≤≤ft57.6 S
ver rule method o
12.020.0)26.010.0( −−−⎠⎝
BLL
B
NNNLN
:
(3-4)
design lanes loaded:
(3-5)
cability:
56
For exterior beams, the AASHTO LRFD Table 4.6.2.2.2d-1 has the following
pplicability for cro s-section type b. Here, de is the distance
from the exterior web of the beam to the interior face of the parapet.
For one
⎩⎨⎧
≤≤≤≤
ft5.11ft0.6ft5.4ft0
Sde
5.2897.
1.06.0
equations and ranges of a s
design lane loaded: use lever rule method
For two or more design lanes loaded
)(MDFeMDF =
0e =
(2) Live Load Shear
For interior beams, the AASHTO LRFD Table 4.6.2.2.3a-1 has the following equations and limitations for one and two design la
design lane loaded:
0.1210⎟⎠
⎜⎝
⎟⎠
⎜⎝
=L
SDF (3-8)
For two or more design lanes loaded: 1.08.0
⎟⎞
⎜⎛
⎟⎞
⎜⎛=
dSSDF (3-9)
Where S is the beam spacing, d is the beam depth and L is the span length of the bridge and N is the number of beams. B
:
interior
ed+
Range of Applicability:
nes loaded for cross-section type “b”.
For one
⎞⎛⎞⎛ dS
0.124.7 ⎠⎝⎠⎝ L
(3-6)
(3-7)
57
Range⎪
≤≤ ft5.11S
For exterior beams, the AASHTO LRFD Table 4.6.2.2.3b-1 lists the equations y for shear distribution factors for cross-section type “b”.
For one design lane loaded
⎪
ft5.4ft0 ≤≤ ed
108.0
⎧ ft0.6
of Applicability:
⎪⎩
⎪⎨
≥≤≤≤≤
3in.65in.18ft140ft20
bNdL
Lever rule method is used if the number of beams (NB) is equal to three.
and ranges of applicabilit
For two or more design lanes loaded:
( )interiorSDFeSDF =
Where e is the correction factor and de is the e fromrior edge of curb or traffic barrier.
W N IF
( ) ⎟⎟⎠
⎜⎜⎝
=B
roadway
NIFDF
10
3.2.2 Precast Prestressed Concrete Bulb-Tee or I-Beams, Steel I-Beams, and
In-Place Concrete T-Beams
: use lever rule method
ede +=
Range of Applicability: and if S > 11.5 ft, lever rule method is req
distanc the exterior web of ebeam to the inte
(c) Henry’s EDF Method
Henry’s EDF method, as for all other bridge types to come, requires on
roadway width ( roadway), number of beams ( B), and an intensity factor ( ) base
linear interpolation to determine the total number of traffic lanes considered as liv
on bridge. From the AASHTO Standard 3.12.1, the intensity factor of live load
100% for a two-lane bridge, 90% for a three-lane bridge, or 75% for a four- or mo
bridge. Henry’s EDF method for interior and exterior beams is as follows.
⎞⎛W 2
58
(3-10)
(3-11)
(3-12)
Cast-
uired.
xterior
ly the
d on a
e load
equals
re lane
(a) AA
ver” equations as
ollows. For interior beams of prestressed concrete girders and steel I-beams from the
AASHTO Table 3.23.1:
ft14≤S
0.6SDF =
ft10<S
5.5SDF =
SHTO Standard Method
The AASHTO Standard method for prestressed concrete beams, steel I-beams
and cast-in-place concrete T-beams consists of the familiar “S-o
f
Range of applicab
Standard procedure assu
b
(3-13)
Range of Applicability for this type of bridge:
For interior beams of cast-in-place T-beam bridges, from the same Table 3.23.1:
(3.14)
ility for this type of bridge:
For all types of bridges with beam spacing exceeding limitations, the AASHTO
mes the load on each stringer to be the reaction of the wheel
loads, assuming the flooring between the stringers to act as a simple beam. For exterior
eams the lever rule method is used.
b) AASHTO LRFD Method
(1) Live Load Moment
For precast, prestressed concrete bulb-tee or I-beams, cast-in-place T-beams, and
steel I-beams, the AASHTO LRFD uses the same sets of equations. These equations
require span length (L), slab thickness (ts), beam spacing (S), number of beams (NB),
modular ratio (n), moment of inertia of beam (I), area of beam (A), the distance between
the centers of gravity of the basic beam and deck (eg), and the distance from the exterior
web of the exterior beam and the interior edge of curb or traffic barrier (de).
59
For interior beams, the AASHTO LRFD Table 4.6.2.2.2b-1 has the following
equations and limitations for one and two design lanes loaded for cross-section types “a”,
“e”, “k” and also “i” and “j” if sufficiently connected to act as a unit.
For one design lane lo aded : ⎞⎟ ⎠⎟
0.10.4 0.3⎛⎜⎜⎝
Kg
12Lts
S S⎛⎜
⎞⎟
⎛⎜
⎞⎟ (3-15)MDF 0.06+= 3L14⎝ ⎝⎠ ⎠
(3-16)2 )K n(I +Ae=g g
(3-17)EBn =ES
For two or more design lanes loaded:
0.10.6 0.2⎛⎜ ⎝⎜
K3
⎞⎟ ⎠⎟
S S⎛⎜⎝
⎞⎟⎠
⎛⎜⎝
⎞⎟⎠
gMDF = 0.075 (3-18)+L9.5 12Lts
S≤3.5 ft 4.5 in.
≤16.0 ft⎧ ⎪⎪⎨⎪ ⎪⎩
≤ ≤12.0 in.tsRange of Applicability: L20 ft
N ≤ ≤240 ft
≥4B
For exterior beams, Table 4.6.2.2.2d-1 lists the equations and ranges of
applicability for shear distribution factors for the same cross-section types as listed above
for interior beams.
For one design lane loaded: use lever rule method
For two or more design lanes loaded:
60
ft5.5ft0.1 ≤≤− ed
0.25
0.2
35122.0 ⎟
⎠⎞
⎜⎝⎛−+=
SSSDF
⎪
1.977.0
)( interior
ede
MDFeMD
For interior beams, Table 4.6.2.2.3a-1 lists the shear distribution factor equations
and the corresponding ranges of applicability f
girders and cast-in-place T
For one design lane loaded:
36.0 SSDF += (3-21)
For two or more design lanes loaded:
⎪⎧ ≤≤ ft0.16ft5.3 S
Range of Applicability: ⎪⎨ in5.4
⎪⎪⎩ ≥ 4
g
N
Where S is the beam spacing, d is the beam depth and L is the span length of the bridge,
NB is the
rule method is used if
girders and cast-in-place T-beams.
For one design lane loaded: use lever
F
+=
=
Range of Applicability:
(2) Live Load Shear
or prestressed concrete I girders, steel I-
-beams.
(3-22)
⎪
≤≤
≤≤≤≤
in000,000,7in000,10
in.0.12.ft240ft20
44
b
s
K
tL
number of beams, ts is the slab thickness and Kg is the stiffness parameter. Lever
the number of beams (NB) is equal to three.
For exterior beams, Table 4.6.2.2.3b-1 lists the shear distribution factor equations
and the corresponding ranges of applicability for prestressed concrete I girders, steel I-
rule method
(3-19)
(3-20)
61
ft5.5ft0.1 ≤≤− ed
0.7S
( )
0.8SDF =
For two or more design l de
10
interior
ed
place T-beams is the same as for all oth
3.2.3 Cast-In-Place Concret
The distribution a
For one lane loaded:
For two or more traffic lanes loaded:
D (3-26)
anes loa d:
6.0e
SDFeSDF
+=
=
Range of Applicability:
Where e is the correction factor and de is the distance from exterior web of exterior beam
and the interior edge of curb or traffic barrier.
(c) Henry’s EDF Method
Henry’s EDF equation for steel and prestressed concrete I-beams and cast-in-
er types of beams. However, for steel and
prestressed I-beams, an additional structure type multiplier of 1.09 of wheel load per
beam is used.
e Multicell Box Beams
(a) AASHTO Standard Method
f ctors for the interior beams of cast-in-place concrete multicell
box-beam bridges are calculated using the S-over equations from the AASHTO Table
3.23.1.
(3-25)
F =
(3-23)
(3-24)
62
16≤S
25.035.03.01
8.513
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛=
LS
NMDF
c
:
8use8If
14eWMDF =
45.035.0 116.3
75.1 ⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
14eW
MDF =
Range of Applicability: ft
r exterior beams, the lever rule method is required to calculate the distribution
factor
For interior beam
⎜⎛=MDF
3⎪⎩ ≥
c
cN
For exterior beams, Table 4.6.2.2.2d-1 lists the moment distribution factor
equations and ranges of applicability for cast-in-place multicell box beam bridges.
For one design lane loade
For two or more design lanes loaded:
Fo
for the exterior beams by treating the floor in between the beams as a single span.
b) AASHTO LRFD Method
(1) Live Load Moment
s, Table 4.6.2.2.2b-1 lists the moment distribution factor
equations and ranges of applicability for cast-in-place multicell box beam bridges.
For one design lane loaded:
⎝+
cNLS (3-27)
For two or more design lanes loaded:
(3-28)
Range of Applicabilityft240ft60ft0.13ft0.7
=>
⎪⎨
⎧≤≤≤≤
cNN
LS
d:
(3-29)
(3-30)
63
Range of Applicability:
Where is equal to half the web spacing plus the total overhang and is measured in
feet.
(2) Live Load Shear
ll box beam bridges can be calculated using
e following equations.
SWe ≤
eW
:
⎩
For interior beams, from the AASHTO LRFD Table 4.6.2.2.3a-1, the shear
distribution factors for cast-in-place multice
th
For one design lane loaded1.06.0
0.125.9 ⎠⎞
⎝⎛
⎠⎞
⎝⎛
LdS
For two or more design lanes loaded: 1.09.0
0.123.7⎟⎠
⎜⎝
⎟⎠
⎜⎝
=L
SDF
⎪⎪ ft20
⎪⎪ in.35
S is the beam spacing, d is the beam depth and L is the span length of the b
and N is the number of beams.
n-pl bridges can be cal d
the follow
:
⎟⎜⎟⎜=SDF (3-31)
⎞⎛⎞⎛ dS
Range of Applicability ⎨
⎧
≥≤≤
≤≤≤≤
3in.110ft240ft0.13ft0.6
bNdLS
Where
B
For exterior beams, from the AASHTO LRFD Table 4.6.2.2.3b-1, the
distribution factors for cast-i ace multicell box beam culate
ing equations.
For one design lane loaded: use lever rule method
For two or more design lanes loaded:
64
(3-32)
ridge
using
shear
WNRDF 85.07.11.0 ++=
12
5.1264.0
ft0.5
( )interiorSDFeF = (3-33)
Range of Applicability: ft0.2−
Where e is the correctio
(c) Henry’s EDF Method:
group of equivalent I-be th
3.2.4 Steel Open Box Beams
(a) AASHTO Standard Method
The AASHTO Standard equations for steel open bo
Where,
C
W
WN
NR
=
= (3-36)
ede
SD
+=
≤≤ ed
n factor and de is the distance from exterior web of exterior beam
to the interior edge of curb or traffic barrier.
Henry’s equation for the cast-in-place multicell concrete box beam bridges is the
same as for all other types of bridges. However, the multicell box is considered as a
am based on e center-to-center distance between the webs. The
number of equivalent I-beams is counted as the number of beams in the calculation.
x beams require the number of
box girders (NB) and the roadway width between curbs (Wc) in feet. From the AASHTO
Article 10.39.2, the lateral distribution factor for moment and shear is given by:
(3-35)
W
BN
(3-37)
(3-34)
65
5.15.0 ≤≤b
L
NN
L
Lb
L
NNNDF 425.085.005.0 ++=
5.15.0 ≤≤
Range of applicability: .5.15.0 ≤≤ R The value of Nw is reduced to the n
n .
(1) Live Load Moment
For steel open bo
b
L
NNDF 85.005.0 ++=
Range of applicability: bN
(c) Henry’s EDF Method
The equations for Henry’s EDF method are the same
However, for the calculation of distribution factor, a single box with two webs was
treated as two separate web members. For example, two steel open box beams were
treated as four beam lines.
earest whole
umber
(b) AASHTO LRFD Method
x beam bridges, the AASHTO LRFD equations require the
number of lanes loaded (NL) and number of beams (Nb). For interior and exterior beams
regardless of the number of loaded lanes, from Table 4.6.2.2.2b-1:
N 425.0 (3-38)
Range of applicability:
(2) Live Load Shear
The exact same equation 3-36 as for live load moment is used for live load shear
in the AASHTO LRFD. The equation is also the same for interior and exterior beams
regardless of the number of loaded lanes and is given by:
(3-39)
LN
as those for all other types.
66
s, the reduction factor is based on the angle of skew (θ), beam spacing
3.2.5 AASHTO LRFD Skew Reduction Factors For Live Load Moment
AASHTO LRFD specifications also contain skew reduction factors for most types
of superstructures. For bridges with steel or concrete I-beams, concrete T-beams, or
double T-section
( )50250
31
511
12250
tan1..
s
g
.
LS
LtK
.c
θcSRF
⎟⎠⎞
⎜⎝⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛=
−=
4
0.1tan25.005.1 ≤−= θSRF
(S), span length (L), beam stiffness (Kg), slab thickness (ts), and number of beams (Nb).
From Table 4.6.2.2.2e-1:
1
Range of Applicability:ft240ft20⎪
⎨ ≤≤ L⎪⎩
Range of applicability: °≤≤ 600 θ
The AASHTO Standard m
If θ < 30o, then C = 0.0. If θ > 60o, then use θ = 60o.
ft016ft536030
⎪⎪⎧
≥
≤≤°≤≤°
bN
.S.θ
For bridges with concrete spread box beams and cast-in-place concrete box bea
other equation is used based only on the skew angle (θ) as follows:
If θ > 60o, then use θ = 60o.
ethods and Henry’s EDF method do not curren
into account the reduction in live load moment due to skew.
3.2.6 AASHTO LRFD Skew Modification Factors For Live Load Shear
As specified in the AASHTO LRFD article 4.6.2.2.3c, the shear in the
beams at the obtuse corner of the bridge shall be adjusted when the line of su
67
(3-40)
(3-41)
ms, the
(3-42)
tly take
exterior
pport is
⎪
θdLSCF tan
700
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
≥≤≤
≤≤≤≤
≤≤
3in.110in.35ft240ft20ft0.13ft0.6
600
bNdLS
θ οο
⎪⎪⎪
⎩
skewed at an angle with the vertical. The correction factor that is applied depends on the
type of bridge superstructure. This corre
o
1, respectively.
θLtSCFg
s tan0.1220.00.13.03
⎟⎠
⎞⎜⎝
⎛+= (3-43)
⎪⎩
⎪⎪
≤≤≤≤
ft240ft20ft0.16ft5.3
b
LS
.12⎛
For Concrete Spread Box Beam the skew correc
Ld
θS
SCF tan6
0.1 += (3-45)
Range of Applicability: ⎪⎪⎨ ≤≤ ft140ft20 L
ction factor is applied to the distribution factors
btained for the interior and the exterior beams from Tables 4.6.2.2.3a-1 and 4.6.2.2.3b-
For Precast Concrete I-Beams, Cast-In-Place Concrete T-Beams, and Steel I-Beams the
skew correction factor is:
K ⎟⎜
Range of Applicability: ⎨
⎧
≥
≤≤
4
600
N
θ οο
For Multicell Concrete Box Beam, the skew correction factor is:
25.00.1 ⎟⎠⎞
⎜⎝
++= (3-44)
Range of Applicability:
tion factor is:
0.12
⎪⎧
≥≤≤
≤≤≤≤
3in.65in.18
ft5.11ft0.6600
bNd
Sθ οο
68
In determining the end shear in multi-beams bridges the skew corrections at the obtuse
corner shall be applied t
3.2.7 Special Analysis for Exterior Beams
The Article 4.6.2.2.2d of the AASHTO LRFD code specifies that for bridge
superstructures with diaphragm and cross frames, the distribution factor for the exterior
beam shall not be less than that obtained by assuming that the entire cross section rotates
as a rigid body about the longitudinal centerline of the bridge. This provision is made in
the code because the distribution factor for girders in cross section, types “a”, “e”, and
“k” was determined without taking into account the effect of diaphragms and cross
frames. Equation 3.43 describes this LRFD method of live load distribution factor
calculation in bridge superstructures with diaphragm and cross frames for both shear and
moment in exterior beams. (Pile analogy method)
∑
∑+=
b
L
N
N
ext
b x
eX
NR
2
reac exterior beam i of lanes;s
= number of loaded lanes under ruction
o all the beams.
LN
Where:
R= tion on n term
NL const ;
e = eccentricity of a design truck or a design lane load from the center of gravity
pattern of girders (ft)
x = horizontal distance from the center of gravity of the pattern of girders to the e
girder
69
(3-46)
of the
xterior
Xext = horizontal distance from the center of gravity of the pattern of gi
= nu
istribu actors e exte m in th
ge alculated using the pi technique for both live load mb
The table also lists the distribution factors from other methods discussed previously. As
can be se
as s that the cross section deflects and rotates as a rigid assemblage of elements,
which implies some degree of bending stiffness as in plate mechanics. At the same time
the transverse and torsional superstructure stiffnesses associated with plate bending
theory are ignored which may lead to the over conservative nature of this method. Also
many bridges all over the country do not have moment resistive diaphragms or cross
connections thereby rendering the assumption of the pile analogy method inapplicable.
For these reasons the live load distribution factors calculated by the pile analogy analysis
will not be employed in this research for neither moment nor shear.
Table 3.9 Distribution Factors for Live Load Moment by Special Analysis
AASHTO Standard
AASHTO LRFD
Special Analysis Bridge
Exterior beam
Henry's
Exterior beam Exterior Beams
18 Steel I Beam 0.836 0.828 0.848 0.884
rders to the
exterior beams (ft) and
Nb mber of beams or girders
Table 3.9 and 3.10 present the d tion f for th rior bea ree
rid s c le analogy oment and shear.
en in the tables this method produces over conservative results. This method
sume
No. Structure Type Method
22 Precast Concrete BT Beam 0.610 0.711 0.617 0.734
11 CIP Concrete T-Beam 0.602 0.644 0.676 0.720
70
Table 3.10 Distribution Factors for Live Load Shear by Special Analysis
AASHTO AASHTO Special Bridge No.
Exterior beam
Henry's Method
Exterior beam Exterior Beams22 Precast Concrete BT Beam 0.610 0.711 0.699 0.786 18 Steel I Beam 0.836 0.828 1.118 1.16111 CIP Concrete T-Beam 0.602 0.644 0.640 0.720
3.2.8 Summary of Live Load Moment and Shear Distribution Factors for Selected
Bridges
Table 3.12 lists the distribution factors for live load
on factors for live load shear. As can be seen in these tables there 3
tion factors for th exterior beams under the AASHTO LRFD
in ing tw
s
f
f
m
multiple
d
Standard LRFD Analysis Structure Type
moment for each of the
selected twenty-four bridges using each of the previously detailed methods and Table
.13 lists the distributi
are two columns of distribu e
method. In both the tables, the first column from LRFD method shows the distribution
factor for exterior beams determ ed by consider o or more lanes loaded. The
econd column for the exterior beams in these tables corresponds to the distribution
actors that are calculated by the lever rule method when a single lane is loaded. These
actors also include a multiple presence factor of 1.2 to account for the probability that
ore than one lane is actually loaded. The AASHTO LRFD code specifies that the
presence factors have been included in the approximate equations for
istribution factors in Articles 4.6.2.2 and 4.6.2.3, both for single and multiple lanes
loaded. But where the use of lever rule method is specified in Articles 4.6.2.2 and 4.6.2.3,
the engineer must determine the number and location of vehicles and lanes and therefore
include the multiple presence. Table 3.11 lists the multiple presence factors from the
AASHTO LRFD code.
71
When compared, the results from the equations in
method, ibution factors for exterio eams, obtained
ery con s can b en in Ta
lane lo ng con ls most of the time for
th ear and moment for ex beams ter inc poratin the mu tiple p esence
actor Since the distribution f y this ethod highly onserv tive an since
more design lanes, only the distribution factors
lcul ted for two or more lan een us in this dy.
Table 3.1 iple P ence F tors “m
Numb aded ultiple senceFactor m"
0.6
tables and the lever rule
it is found that the distr r b by
incorporating the multiple presence factor, are v servative a e se ble
3.12 and 3.13. It is also observed that the single adi tro
bo sh terior af or g l r
fac . tors b m are c a d
most highway bridges have two or
ca a es have b ed stu
1 Mult res ac ”
er of LoLanes
M Pres "
1 1.20 2 1.00 3 0.85
> 3 5
72
0.711 0.650 0.625 0.73224 Precast Concrete I-Beam 0.962 0.843 0.782 0849 0.905 0.939
* CIP: Cast-In-Place
** The distribution factors in the first column for the exterior beams were calculated using the equationsin Table 4.6.2.2.3b-1 and in the second column were calculated using the lever rule method for onedesign lane loaded and incorporating a multiple presence factor of 1.2.
0.610
Table 3.12 Summary of Distribution Factors for Live Load Momen
Bridge No. Structure Type*
AASHTO Standard Henry's Method
AASHTO LRFD
Interior Beam
Exterior beam
Interior Beam
Exterior beam**
1 Precast Spread Box Beam 0.866 0.839 0.826 0.723 0.780 0.9342 Precast Spread Box Beam 1.186 1.186 1.152 1.186 1.186 1.2113 Precast Spread Box Beam 0.874 0.972 0.759 0.752 0.856 1.0264 Precast Spread Box Beam 0.433 0.643 0.489 0.343 0.494 0.7715 Precast Concrete BT Beam 0.795 0.743 0.663 0.705 0.756 0.8916 Precast Concrete I-Beam 0.818 0.694 0.663 0.762 0.775 0.8327 Precast Concrete I-Beam 0.818 0.694 0.663 0.702 0.714 0.8328 Precast Concrete BT Beam 0.936 0.810 0.790 0.809 0.853 0.9209 CIP Concrete T-Beam 0.925 0.929 0.869 0.802 0.889 0.913
10 CIP Concrete T-Beam 0.944 0.974 0.859 0.913 0.996 1.00011 CIP Concrete T-Beam 0.681 0.602 0.644 0.703 0.676 0.72212 CIP Concrete Box Beam 0.661 0.649 0.608 0.668 0.758 -13 CIP Concrete Box Beam 0.643 0.694 0.608 0.701 0.607 -14 CIP Concrete Box Beam 0.738 0.810 0.698 0.738 0.679 -15 CIP Concrete Box Beam 0.680 0.684 0.701 0.785 0.607 -16 Steel I-Beam 0.863 0.745 0.695 0.661 0.711 0.85317 Steel I-Beam 0.848 0.790 0.851 0.650 0.790 0.94818 Steel I-Beam 1.045 0.836 0.828 0.696 0.848 0.99119 Steel I-Beam 1.000 0.815 0.822 0.724 0.659 0.79020 Steel Open Box Beam 0.556 0.556 0.701 0.556 0.556 -21 Steel Open Box Beam 0.645 0.645 0.606 0.645 0.645 -22 Precast Concrete BT Beam 0.757 0.610 0.711 0.641 0.617 0.73223 Precast Concrete BT Beam 0.757
t
73
Table 3.13 Summary of Distribution Facto hearrs for Live Load S
AASHTO Standard AASHTO LRFDBridge Henry's
Precast Spread Box Beam 0.933 0.838 0.826 1.016 0.932 0.934Precast Spread Box Beam 1.186 1.186 1.152 1.186 1.186 1.211
3 Precast Spread Box Beam 0.874 0.972 0.759 1.027 1.168 1.0264 Precast Spread Box Beam 0.433 0.643 0.489 0.812 0.798 0.7715 Precast Concrete BT Beam 0.795 0.743 0.663 0.900 0.788 0.8916 Precast Concrete I-Beam 0.818 0.694 0.663 0.940 0.775 0.8327 Precast Concrete I-Beam 0.818 0.694 0.663 0.983 0.811 0.8328 Precast Concrete BT Beam 0.936 0.810 0.790 0.970 0.833 0.9209 CIP Concrete T-Beam 0.873 0.828 0.869 0.942 0.784 0.913
10 CIP Concrete T-Beam 0.944 0.974 0.859 0.969 0.863 1.00011 CIP Concrete T-Beam 0.681 0.602 0.644 0.826 0.640 0.72212 CIP Concrete Box Beam 0.661 0.649 0.608 0.899 0.701 0.77813 CIP Concrete Box Beam 0.643 0.694 0.608 0.900 0.738 0.83214 CIP Concrete Box Beam 0.738 0.810 0.698 1.280 1.084 0.91915 CIP Concrete Box Beam 0.680 0.684 0.700 1.086 0.868 0.8216 Steel I-Beam 0.863 0.745 0.695 0.917 0.756 0.85
18 Steel I-Beam 1.045
Steel Open Box Beam 0.556 0.556 0.701 0.556 0.556 -
22 Precast Concrete BT Beam 0.757 0.610 0.711 0.902 0.699 0.732
24 Precast Concrete I-Beam 0.962 0.843 0.782 0.990 0.861 0.939
* CIP: Cast-In-Place
** The distribution factors in the first column for the exterior beams were calculated using the equationsin Table 4.6.2.2.3b-1 and in the second column were calculated using the lever rule method for one
No. Structure Type* Interior Beam
Exterior beam
Method Interior Beam
Exterior beam**
1 2
03
17 Steel I-Beam 0.848 0.790 0.851 0.906 0.807 0.9480.836 0.828 1.242 1.118 0.991
19 Steel I-Beam 1.000 0.815 0.822 1.099 0.796 0.79020 21 Steel Open Box Beam 0.645 0.645 0.606 0.645 0.645 -
23 Precast Concrete BT Beam 0.757 0.610 0.711 0.877 0.680 0.732
design lane loaded and incorporating a multiple presence factor of 1.2.
74
CHAPTER 4
FINITE ELEMENT ANALYSIS OF SELECTED TWENTY-FOUR BRIDGES
This chapter discusses the finite element modeling methods used for this research.
It details the types of elements used to mo
selected bridge, finite element analysis
ional model) and entire bridge model (three dimensional model). Distribution
factors for live load moment and shear were determined by dividing the maximum
moments and shears obtained from the bridge model by the maximum moment and
shears from the single beam model. Modeling methods and loading procedures for two-
dimensional and three-dimensional models are given for each type of superstructure.
lt
model various structures. In this study two types of elements were used primarily, to
del beams, diaphragms, and slabs. For each
was performed for both single beam model (two-
dimens
4.1 ANSYS 5.7/6.1 Finite Element Program
ANSYS 6.1, which is finite-element simulation software, has been used
throughout this research to determine the wheel load distribution factors in highway
bridges. The finite element results depend mainly upon the input variables such as
structural geometry, support conditions and the load applications. It is necessary to
incorporate the input variables of actual bridges in a consistent and effective way. The
program has all the non-linear structural capabilities as well as the linear capabilities to
deliver the reliable structural simulation resu s.
4.2 ANSYS 5.7/6.1 Elements
The ANSYS program has a host of different types of elements that can be used to
model the beams and deck slab in bridge structures. In order to model the precast
75
concrete I-beams, steel I-beams, concrete spread box beams and the concrete T-beams, a
BEAM44 element was used. This element was also used to model the diaphragms. For
the purpose of simulating the cast-in-place concrete deck slab, cast-in-place multicell box
and steel open box a SHELL63 element was used. A detailed description for these
elements is given in the following sections.
4.2.1 BEAM44 Element Description
The BEAM44 element was used for modeling steel I-beams, precast concrete
bulb-tee and I-beams, cast-in-place T-beams, and concrete and steel spread box beams.
BEAM44 is a uniaxial element with tension, compression, torsion, and bending
capabilities. The element has six degrees of freedom at each node: translations in the
nodal x, y, and z directions and rotations about the nodal x, y, and z-axes. Material
properties for the beams were assumed to be linear, elastic, and isotropic.
Figure 4.1 BEAM44 3-D Elastic Tapered Un-symmetric Beam
The geometry, node location, and coordinate system for the BEAM44 element are
shown in Figure 4.1. The beam must not have a zero length or area. The beam can have
any cross-sectional shape for which the moments of inertia can be computed. The
76
element height is used for locating the extreme fibers for the stress calculations and for
computing the thermal gradient. The element real constants describe the beam in terms
of the cross-sectional area, the area moments of inertia, the extr
o
b
s
a
o
b
w
4
SHELL63 elements were used to m
ulticell box beams, and steel open box beams. Material properties were set to be
linear, elastic, and isotropic. Bridge deck slab was modeled with three-node or four-node
shell elements. The web, top flange, bottom flanges and diaphragms of box beams were
modeled as 3-D elastic elements with 4 nodes per element. SHELL63 has both bending
and membrane capabilities. Both in-plane and normal loads are permitted. Four nodes,
four thicknesses, and the orthotropic material properties define the element. The
geometry, node locations, and the coordinate system for this element are shown in Figure
4.2. Orthotropic material directions correspond to the element coordinate directions. If
eme fiber distances from
the centroid, the centroid offset, and the shear constants. The moments of inertia (IZ and
IY) are about the lateral principal axes of the beam. For this study, the torsional moment
f inertia (IX) was either specified in the input or determined by computer based on the
eam dimensions in common section. The shear deflection constants are used only if
hear deflection is to be included. A zero value can be used to neglect shear deflection in
particular direction. The significance of shear deflection effects in the lateral deflection
f beams decreases as the ratio of the radius of gyration of the beam cross section to the
eam length becomes small compared to unity. In this study, the ratio of the radius of
gyration to the beam length was extremely small. Therefore, shear deflection constants
ere taken to be zero.
.2.2 SHELL63 Element Description
odel cast-in-place concrete deck slabs, cast-in-
place m
77
the element thickness is not constant, all four thicknesses at element nodes were input.
The thickness is assumed to vary smoothly over the area of the element. The element
stress directions are parallel to the element coordinate system.
4.3 Live Load for Distribution Factors
The live load moment and shear in highway bridges under consideration are due
to an AASHTO Standard HS20-44 truck loading or HL-93 truck loading in the AASHTO
LRFD specifications detailed in Figure 4.3. For three-dimensional models, as many
trucks as possible were placed on a bridge in the transverse direction depending on the
width of the bridge. Moment and shears were determined after the addi
Figure 4.2 SHELL63 Elastic Shell
tion of each truck
until the maximum values were obtained. The AASHTO Standard intensity reduction
factors were used for three and four truckload results (0.9 and 0.75 respectively). Trucks
were moved in both the longitudinal and lateral directions on each bridge and the moment
78
and shear on the beams was calculated. In the case of live load moment for non-skewed
bridges, once the location of the maximum moment was obtained with a single truckload,
additional trucks were placed alongside the first. For skewed bridges, the first truck was
moved until its location of maximum influence for the beam under investigation was
found. Once the first truck was in place, the second, third, and fourth trucks were placed
alongside the first truck and were moved independently to find the locations of the
maximum influence. Figure 4.4 shows the sample loading patterns for live load moment
on non-skewed and skewed bridges. The ma m shear usually occurs very near to the
abutments or piers. Therefore the trucks were placed at locations near to the supports for
maximum shear effect. The procedure to ob in the maximum shear was similar to that
for moment. The bridge was loaded with one, two or three trucks and the position of
aximum shear was found by moving these truck independently as well as together. If
e width of the bridge allowed, additional trucks were applied and similar procedures
ere carried out to find the maximum shear. In the case of non-skewed bridges it was
ound that the second and the third trucks should be placed alongside the first truck to
produce maximum shear. For ske ond truck had to be placed at a
certain longitudinal distance from the first truck and both trucks were moved
dependently to obtain the maximum shear. Figure 4.5 shows the sample loading
atterns for live load shear on non-skewed and skewed bridges. The three-dimensional
loading examples on different type of bridges are shown in Figure 4.6.
ximu
ta
m
th
w
f
wed bridges, the sec
in
p
79
Figure 4.3 AASHTO Standard HS20-44 Truck
(a) Non-Skewed Bridge
(b) Skewed Bridge
Figure 4.4 Sample Loading Patterns for Live Load Moment
80
(a) Non-Skewed Bridge
(b) Skewed Bridge
Figure 4.5 Sample Loading Patterns for Live Load Shear
81
(a) Steel I-Beam
(b) AASHTO Type III, I-Beam
(c) Concrete Multicell Box
s
Figure 4.6 Sample Loading Condition
82
4.4 Two-Dimensional Modeling Procedure
b
b
the concrete deck slab, s
m
tre
direction along the bridge, and Z re
Figure 4.7 2-D Cast-In-Place Multicell Box Beam Model
Two-dimensional models were created according to the actual single beam
properties. Keypoints were created in the program and lines were drawn to join these
keypoints. These lines were then defined as BEAM44 elements and meshed. The
meshing creates a number of elements and nodes. BEAM44 elements were typically
used for precast concrete I-beams, steel I-beams, cast-in-place T-beams, and precast
concrete box beams. For a cast-in-place concrete multicell box bridge, an equivalent I-
beam of the same size and properties of an interior web of the multicell box was used as a
eam element as shown in Figure 4.7. For steel open box bridges, a single steel open
ox was modeled using SHELL63 elements with the corresponding material properties in
teel web and steel bottom flange as shown in Figure 4.8. These
odels were then loaded by a single AASHTO Standard HS20-44 truck. The maximum
moment and shear were obtained by moving this truck along the span length of the beam.
Nodes coinciding with beam supports were restrained. The first abutment was treated as
a hinge with movement restrained in the X and Z directions. All other supports were
ated as rollers and restrained only in the Z direction. X represents the longitudinal
presents the vertical direction.
83
Figure 4.10 Two Dimensional Model Loaded with One Truck for Live Load Shear
Figure 4.9 Two Dimensional Model Loaded with One Truck for Live Load Moment
Figure 4.8 2-D Steel Open Box Beam Model
Figures 4.9 and 4.10 show the two-dimensional models loaded by a single
AASHTO HS20-44 truck for live load moment and shear, respectively. The maximum
moment and shear due to the live load were determined for each bridge through two-
dimensional analysis. These maximum values were used as base values in the calculation
of distribution factors later on.
84
4.5 Three-Dimensional Modeling Procedure
To define the geometry of each bridge, “keypoints” were input first. Thes e
keypoints were used to segment the bridge deck into multiple areas, especially for the
irregular areas in skewed bridges. A skewed bridge deck area had to be divided into finer
segments near the supports because the maximum shear is obtained very near to the
supports. Then, the keypoints were connected by lines. Once the lines were created, the
BEAM44 properties were assigned to the lines. These lines were then used to create
areas. All areas served as the deck slab and were defined as SHELL63 elements. The
beam lines as well as all the areas were then meshed and elements and nodes were
created. Typically, elements were meshed to approximately 2 ft x 2 ft. In some cases
element sizes of approximately 1 ft x 1 ft were used to facilitate loading patterns. Actual
element sizes varied for each model.
Figure 4.11 Finite Element Model
A typical cross-section of a finite element beam and slab model is shown in
Figure 4.11 in which the beams are modeled as a beam element and the slab as
quadrilateral shell elements. To make the model closer to real life the beams were offset
by a distance from the centroid o f the beam to the center of the slab. For the modeling of
a cast-in-place concrete multicell box beam bridges and steel open box beam bridges,
85
Shell element
only quadrilateral SHELL63 elements were used. Figure 4.12 shows how shell elements
were used to model top and bottom slabs and web members of a multicell box bridge. As
shown in Figure 4.12, an interior beam was considered as an “I” beam with portions of
the top and bottom flanges and web of box beam used. For cast-in-place concrete bridges,
the same material properties bs. However, for the steel
ox bridges different material properties were specified for steel webs and bottom flanges
and concrete deck slab. An example of the modeling procedure is shown in Figure 4.13
(b) Entire Structure
were used in slabs, flanges and we
b
Figure 4.12 SHELL63 Elements for Multicell Box Beam Bridge
(a) Cross Section of Interior Beam
86
(c) Keypoints, Lines, and Areas Plotted
87
(a) Keypoints Plotted
(b) Keypoints and Lines Plotted
(d) Mesh View of Bridge
Figure 4.13 Modeling Procedure
87
4.6 Diaphragm Modeling
Diaphragm effects for most of the bridges were studied in this research. The pier
and abutment supports were modeled in two different ways to observe the effects of
diaphragms.
Case 1: No Diaphragm
case diaphragms were not considered and only nodes on beam lines at
ined. Typically the first beam support was treated
a pi
For this
abutments and piers or bents were restra
as n with movement restrained in the X, Y and the Z directions. All other beam
supports were treated as rollers and restrained in the Y and Z directions. Here, X
represents the longitudinal direction along the bridge, Y represents the lateral direction
across the bridge, and Z represents the out of plane, or vertical direction as shown in
Figure 4.14.
Case 2: With Diaphragm
In this case the lines were coinciding with diaphragms over piers and abutments
modeled as BEAM44 elements with the size and properties of the designated diaphragms.
By using line supports over piers instead of individual point support under beams,
diaphragm effects were considered in this case. As a result, nodes in the deck slab along
Figure 4.14 Model Without Support Diaphragm
88
the pier lines were restrained. The first support was treated as a pin by restricting the
movement in X, Y and the Z directions. All other supports were treated as roller supports
and restrained in the Y and Z directions as shown in Figure 4.15.
4.7 Indiv
s
p
y.
4.7.1 Precast Concrete Spread Box Beam Bridg
The first type of bridge discussed is the precast prestressed concrete spread box
beam bridge. As stated before, four bridges of this type were studied. A BEAM44
common section available in the ANSYS program was used for this beam type. Exact
dimensions were input, and the ANSYS program calculated section properties. Another
method that can be used is that in which the section properties if known can be input
directly into the real constant table for the BEAM44 element. Both these methods
idual Modeling Procedures
This section shows cross-sections and modeling procedures for each type of
uperstructure. Different methods were used to input section properties. Section
roperties could be directly input when they were known. For other cases, common
sections available in the ANSYS program were used. Parapet effects were not taken into
account for this stud
es
Figure 4.15 Model With Support Diaphragm
89
provide the option of offsetting the nodes of beam elements to coincide with the slab
nodes. SHELL63 elements were used to model the slab. Figure 4.16 shows the typical
cross-secti
e
on of the model for precast spread box beams. Two models were created for
ach bridge with and without diaphragms.
Figure 4.16 Precast Concrete Spread Box Beam Model
4.7.2 Precast Concrete Bulb-Tee and I-Beam Bridges
Four bulb-tee and three precast concrete I-beam bridges were studied. Known
section properties for each type of beam were input into the program. BEAM44 elements
were used to define the beams, and SHELL63 elements were used to define the deck slab.
As before, two cases were created for each bridge, one with diaphragm and the other not.
Figure 4.17 shows the cross-section of the finite element model for a typical bulb-tee
bridge.
Figure 4.17 Precast Concrete Bulb-Tee Model
4.7.3 Cast-In-Place Concrete T-Beam Bridges
Three cast-in-place concrete T-beam bridges were included in this study. A
BEAM44 common rectangular section was used to input section properties for these
90
bridges. These properties stayed the same for constant sections. However, for the
haunch portion of the beams, spans were segmented into three-foot sections to match the
actual beam depth changes. These segments varied linearly in depth from end to end
over their three-foot span to match actual beam geometry as shown in Figures 4.18.
SHELL63 elements were used to define the slab. Again, two models were created for
each bridge. Figure 4.19 shows the typical cross-section of a segment for these types of
bridges.
Figure 4.18 Segmented Beam Elements at Pier
Figure 4.19 Cast-In-Place T-Beam Model
4.7.4 Cast-In-Place Concrete Multicell Box Beam Bridges
Four cast-in-place concrete multicell box beam bridges were included in this
study. SHELL63 elements were used to model these bridges. The shell elements used to
model the webs varied in height and thickness according to structural drawings as shown
in Figure 4.20. Shell elements used to model the slabs were set to their appropriate
91
th
Figure 4.21. The diaphragm effects we
4
Four steel I-beam bridge
model the beams. SHELL63 elements were used to model the slab. For each I-
beam, because the depth and the cross-section o
common section was used
. Figure 4.22 shows a sample beam with varying flange thickness and section
properties. Two models were created for each bridge one with diaphragm and the other
without diaphragm. See Figure 4.23 for a typical cross-section of the model.
Shell element
92
ickness. Shell elements from the slab and webs were connected together as shown in
re not considered for this type of a bridge.
Figure 4.20 Segmented Web Elements
Figure 4.21 Cast-In-Place Multicell Box Beam Model
.7.5 Steel I-Beam Bridges
s were included in this study. BEAM44 elements were
used to
f the beams varied along the span, a
for section properties input and exact beam dimensions were
entered
92
Figure 4.22 Cross Section of Steel I-Beam
Figure 4.23 Steel I-Beam Model
4.7.6 Steel Open Box Beam Bridges
Two steel open box beam bridges were included in this study. SHELL63
elements were used to model both the slab and web members as in the cast-in-place
concrete box beam bridges. Steel and
ents and n in F gure 4.24. Only one
variedand 21 TT
concrete properties were assigned to the
appropriate shell elem meshed together as show i
model was created for this type of bridge and the effects of diaphragms were not
considered.
Figure 4.24 Steel Open Box Beam Model
93
4.8 Finite Element Analysis Output
This section discusses the methodologies involved in the determination of the
beam moment and shear from the analysis output.
4.8.1 Live Load Moment
Two methods were used to calculate the live load moment from the finite element
analysis output. The first method was based on a simplified equation. The second was
based on the effective co ses.
Each method is discu
follows:
+=
mposite section taken by the program and the resulting stres
ssed below in detail.
The first method utilized equations developed by Chen and Aswad [12]. The
finite element analysis output included the axial force, P, and moment, Mb, for the beam
element. These allow the stress computation at the centerline of the bottom flange as
b
bb SA
f += (4-1)
Where Sb = non-composite section modulus at bottom fiber, and A = beam cross-sectional
area.
The moment, Mc, carried by one composite cross section is given by
∫b
slabbc0
b
s
lt
ra
MP
dlMMM ' (4-2)
Where b = effective width of the slab, Msla = slab moment, and M’b = beam moment
referenced to a plane within the slab. It is usually very tedious to calculate the integral
term in (4-2) unless the reference plane i set at the level of the slab compression
resultant. Very often the location of the resu ant is not known. However, because of the
general trapezoidal shape of the stress diag m in slab, it is reasonable to assume the
94
resultant plane is near 0.6ts from the top of the basic beam where ts is the slab thickness,
see Figure 4.25. Therefor
bbcc fSM =
e:
)6.0(' stbbc tyPMMM ++≈= (4-3)
Where yt is the distance from the basic beam centroid to its top fiber.
(b) Partial Elevation (c) Stress Diagram Due to Live Load
Figure 4.25 Example Composite Beam Section
The second method to calculate beam moments used the stress output from the
ANSYS program to determine the composite section p
(a) Typical Cross Section
roperties. A reasonable and
effective way of computing Mc is to use the moment formula from the beam theory:
(4-4)
95
Figure 4.26 BEAM44 Stress Output
to SDIR plus SBZB, and the maximum stress at the top surface of the beam element is
b
bc
cb
yIf
4.8.2 Live Load Shear
For beam and slab bridges such as precast concrete spread box beam, precast
concrete I-beam, cast-in-place concrete T-beam, and steel I-beam bridges, the centroid of
Where Sbc = composite section modulus at bottom fiber and fb = bottom fiber stress. To
determine fb value three stress values given in the output were used: SDIR, SBZB, and
SBZT as shown in Figure 4.26. The SDIR value is the axial direct stress, which
represents the stress at the cross-section of the beam element. The SBZT and SBZB
values are the bending stresses at the top and bottom surface of the beam element
respectively. The maximum stress at the bottom surface of the beam element, fb is equal
equal to SDIR plus SBZT. Using the stresses at the top and bottom fibers, the composite
section centroid, ybc, could be located. With the composite centroid known, the
composite moment of inertia, Ic, could be determined. Then the maximum moment could
e calculated using the maximum stress and Equation (4-5).
cM = (4-5)
96
the beam was offset a distance equal to top centroid distance of beam plus half of the slab
thickness to simulate the actua
The shear force in
th
o
l bridge beam arrangement. The shear force in a beam
could be directly obtained from the output results of ANSYS program. The beam shear
was then added to the shear from slab to get the total shear force of the composite section.
the slab was calculated by adding the nodal forces of shell elements in
e section of effective width of slab. The nodal forces in the slab were given in the
utput file also. This method is used for all types of beam and slab bridges. Figure 4.27
shows the nodal forces in the beam and slab of the bridge.
For cast-in-place multicell box beam bridges and steel open box beam bridges, the
box section was divided into multi-I-shaped beams as shown in Figure 4.28. Each
idealized beam consisted of a web and portions of top flange and bottom flange. Since
the model consisted of only shell elements, the shear forces were calculated by adding the
nodal forces of all shell elements in the top flange, web, and bottom flange of the section.
Figure 4.27 Shear in Beam and Slab Bridges
97
4
o
f
section was considered
Figure 4.29 shows an example of the cross-section of an exterior beam and
interior beam. In this example, the exterior beam consists of 11 elements and the interior
beam consists of 14 elements, 6 for top flange, 6 for bottom flange, and 2 for web. Figure
.30 shows the sample nodal forces for shear in the exterior and interior beams. The sum
f all the vertical nodal forces in an exterior beam section was considered as the shear
orce in the exterior beam and the sum of all vertical nodal forces in an interior beam
as the shear force for the interior beam. The maximum shear
occurred near support location.
Figure 4.28 Idealized Beam for Shear Distribution
98
(a) Exterior Beam
(b) Interior Beam
Figure 4.29 Section of Beam with Nodes
99
xterioram
(a) EBe
(b) Interior Beam
Figure 4.30 Sample Nodal Forces for Shear
100
4.9 Finite Element Ana
The following tables summarize the ained through
finite t ana shows th ive load ribution
factors with two cases of diaphragm sit w load
shear w a
Table 4 F om
lysis Results
elemen lysis. Table 4.1
distribution factors obt
e results of l moment dist
uations. Table 4.2 sho s the results for live
ith nd without diaphragms.
.1 EA Results, Live Load M ent
Structure Type* Inter eam Exter eamWith With aphragm aphragm aphragm aphragm
1 Precast Spread Box Bea 0.754 0.815 0.762 0.7972 Precast Spread Box Beam N/A N/A 1.251 1.256 3 Precast Spread Box Beam 0.867 0.904 0.826 0.842 4 Precast Spread Box Beam 0.358 0.399 0.396 0.453 5 Precast Concrete BT Beam 0.683 0.625 0.621 6 Precast Concre 0.715 N/A 0.689 N/A 7 Precast Concre 0.721 0.731 0.654 0.6608 Precast Concre 0.827 0.747 0.7629 CIP Concrete 0.877 N/A 0.870 N/A
10 CIP Concrete T-Beam 0.930 0.936 0.943 0.94711 CIP Concrete T-Beam 0.704 0.724 0.658 0.65612 CIP Concrete Box Beam 0.687 0.708 0.409 0.42313 CIP Concrete Box Beam 0.620 0.620 0.415 0.41514 CIP Concrete Box Beam 0.665 0.665 0.431 0.43115 CIP Concret16 Steel I-Beam
Finite Element Analysis
ior B ior BBridge No.
DiWithout
Di DiWithout
Dim
0.660 te I-Beamte I-Beamte BT Beam 0.812
T-Beam
e Box Beam 0.765 0.765 0.594 0.594 0.690 0.692 0.653 0.655
17 Steel I-Beam 0.749 0.749 0.842 0.841 18 Steel I-Beam 0.857 1.094 0.906 0.910 19 Steel I-Beam 0.830 0.830 0.835 0.835 20 Steel Open Box Beam 0.641 0.641 0.641 0.641 21 Steel Open Box Beam 0.630 0.630 0.685 0.685 22 Precast Concrete BT Beam 0.559 0.570 0.552 0.572 23 Precast Concrete BT Beam 0.537 0.547 0.511 0.521 24 Precast Concrete I-Beam 0.757 N/A 0.791 N/A
* CIP: Cast-In-Place
101
Table 4.2 FE Load ShearA Results, Live
Finite Element Analysis
Interior Beam Ecture Type*
1 Precast Spread Box Beam 0.858 1.042 0.815 0.861 2 Precast Spread Box Beam N/A N/A 1.080 1.170 3 Precast Spread Box Beam 1.010 1.190 0.933 1.050 4 Precast Spread Box Beam 0.452 0.624 0.568 0.610 5 Precast Concrete BT Beam 0.931 1.000 0.730 0.850 6 Precast Concrete I-Beam 0.917 0.918 0.677 0.694 7 Precast Concrete I-Beam 0.770 0.835 0.700 0.712 8 Precast Concrete BT Beam 0.960 1.060 0.784 0.853 9
10
12 CIP Concrete Box Beam 0.896 N/A 0.651 13 CIP Concrete Box Beam 0.931 N/A 0.660 14 CIP Concrete Box Beam 1.076 N/A 0.796 N/A
16 Steel I-Beam 0.921 0.956 0.721 0.736
18 Steel I-Beam 0.971 1.150 0.884 0.922
20 Steel Open Box Beam N/A N/A 0.831
22 Precast Concrete BT Beam 0.933
24 Precast Concrete I-Beam 0.940 1.130 0.841 0.931
xterior BeamBridge No. Stru
With Diaphragm
Without Diaphragm
With Diaphragm
Without Diaphragm
CIP Concrete T-Beam 0.911 1.020 0.762 0.822 CIP Concrete T-Beam 1.090 1.120 0.956 0.985
11 CIP Concrete T-Beam 0.770 0.864 0.665 0.735 N/A N/A
15 CIP Concrete Box Beam 0.833 N/A 0.851 N/A
17 Steel I-Beam 0.875 0.883 0.841 0.852
19 Steel I-Beam 1.017 1.050 0.790 0.820 N/A
21 Steel Open Box Beam 0.819 N/A 0.727 N/A 1.016 0.756 0.761
23 Precast Concrete BT Beam 0.932 1.010 0.727 0.736
* CIP: Cast-In-Place
102
CHAPTER 5
COMPARISON AND EVALUATION OF MOMENT AND SHEAR
DISTRIBUTION FACTORS OBTAINED
This chapter presents the comparison and evaluation studies of Henry’s ED
m
thod fo databa #1 (tw ty-four enness e bridge are co pared with the resul
eme nalys vs. Henr
tio of e mom rs from EA an enry’ F me
w
multiplication of 1.09 for steel I-beams and prestr
Results from FEA analysis with diaphragms were used in the comparison.
5.1.1 Precast Concrete Spread Box Beam Bridges
Four precast concrete spread box beam bridges were analyzed in this study.
Bridges #2 and #3 were both non-skewed, whereas bridges #1 and #4 were skewed. Table
5.1 shows the distribution factors obtained from FEA, the AASHTO LRFD, and Henry’s
method. The ratios of FEA to Henry’s methods and FEA to the LRFD method were
calculated. It can be seen that Henry’s method produced an average value close to FEA.
Henry’s method was typically conservative compared to the FEA values. The average
r
r
F
ethod with other standard methods. The distribution factors obtained from Henry’s
me r se en T e s) m ts
from finite element analysis (FEA), the AASHTO Standard and the AASHTO LRFD
methods.
5.1 Finite El nt A is y’s Method for Live Load Moment
A ra th ent distribution facto F d H s ED thod
as created for comparison for each bridge. Step 2 of Henry’s method (the
essed concrete I-beams) was included.
atio of FEA/Henry’s method was 0.996, which showed the agreement between the
esults from Henry’s method and FEA. From Table 5.1, it can also be seen that results
103
f
fairly reasonable results for exterior beams when compared to FEA results. All o
b
e impact to distribu ion facto
w pan enry's SHTO EA /. le ngth Method RFD enry'sg) (ft)
0.88 terior .767 .826 .7230.88 terior .744 .826 .7804.38 xterior .251 .152 .186 .09 1.05
81.4 In 0.10.75
10.752 .14 1.15 0.
4 48.5 69.54 Interior 0.4 48.5 69.54 Exterior 0.45
Average 0.996 1.04
Four precast concrete bulb-tee beam bridges were analyzed in this study. All of
for these bridges varies from zero to 26.7 degrees. Table 5.2 shows the results of FEA,
and Henry’s EDF methods. Results from Henry’s method and the AASHTO LRFD were
obtained from FEA. The longer the bridge, the more conservative the Henry’s method is.
rom AASHTO LRFD consistently yield unconservative results for interior beams and
f the
ridges of this type had span lengths less than 100 ft. These span lengths did not show
th t rs.
Table 5.1 Comparison of Precast Spread Box Beam Moment Distribution Factors
Bridge No
SkeAng(de
SLe Beam FEA H AA
L
Ratio: F
HMethod
Ratio: FEA / LRFD
1 15.0 6 In 0 0 0 0.93 1.06 1 15.0 6 Ex 0 0 0 0.90 0.95 2 0.0 4 E 1 13 0.0 6 terior 867 9 13 0.0 81.46 Exterior 0.826 0.759 856 1.09 0.96
399 0.489 0.343 0.82 1.16 3 0.489 0.494 0.93 0.92
5.1.2 Precast Concrete Bulb-Tee Beam Bridges
the bridges analyzed of this type had span lengths greater than 100 ft. The angle of skew
the AASHTO LRFD, and Henry’s method for each bulb-tee beam bridge. Finite element
analysis typically produced smaller distribution factors than both the AASHTO LRFD
equally conservative. It can be observed that the span length influences the results
104
ent Dist bution F
Ratio:
ctors
Skew Angl
Span Leng h Be FE Henry LRFD FEA / Ratio:
(deg (ft) Metho enry'
15.0 124.33 Inter r 0.6 0.663 0.705 1.00 0.945 15.0 124.33 Exte r 0.6 0.663 0.756 0.94 0.838 0.0 115.49 Inter r 0.8 0.790 0.809 1.03 1.00
7 0.853 0.95 0.880 11
23 26.7 151.33 Interior 0.537
Average 0.87 0.88
thod is relative conse ive w en compared to F
od
A resu
r these b
as shown
s
.6 perce compa to th perc
s m d. B use bo skew an le and an len h param
f ha litt ffect o distribu n facto s, to fu er imp e He meth
m
Table 5.2 Comparison of Precast Bulb-Tee Beam Mom ri a
Bridge No. e)
t am A 's d
AASHTO
H s Method
FEA / LRFD
5 io 6rio 25io 12
8 0.0 115.49 Exterior 0.74 0.79022 26.7 159. 0 Interior 0.559 0.7 0.649 0.79 0.86 22 26.7 159.00 Exterior 0.552 0.711 0.624 0.78 0.88
0.711 0.650 0.76 0.83 23 26.7 151.33 Exterior 0.511 0.711 0.625 0.72 0.82
5.1.3 Precast Concrete I-Beam Bridges
The three precast concrete I-beam Bridges analyzed in this study show the
AASHTO LRFD method to be typically conservative. The distribution factors obtained
from Henry’s method were very close to the FEA results. The average ratio of
FEA/Henry’s method was 1.03 indicating some slightly higher values being obtained
from FEA. Skew angles for these bridges ranged from zero to 33.5 degrees and span
lengths ranged from 67.42 ft to 76 ft. Henry’s method was slightly unconservative and
the LRFD me ly rvat h E lts in
Table 5.3. Differences between FEA and the LRFD meth fo ridge differed
from –2 percent to 14.4 percent, a range of 17 nt red e 10 ent by
Henry’ etho eca th g sp gt eters are well within the
range o ving le e n tio r rth rov nry’s od the
structure type ultiplier in Step 2 of Henry’s method has to be adjusted.
105
Table 5.3 Comparison of Precast Concrete I-Beam Momen
No.
Skew Angle (deg)
Span Length
(ft) Beam FEA Method
AASHTOLRFD
Ratio:
Henry's
Ratio:
LRFD
6 21.3 67.42 Interior 0.715 0.663 0.762 1.08 0.94
7 33.5 76.00 Interior 0.721 0.663 0.702 1.09 1.03
24 0.0 74.33 Interior 0.757 0.782 0.849 0.97 0.89
Average 1.03 0.92
enry’ method equals to .05. this type
yie er ic resu ts comp red to F A for erior be
5.4 mpar on of IP Con rete T Beam M ment D butio tors
Ratio:
No. (deg) (ft)
9 31.6 88.50 Interior 0.877 0.869 0.802 1.01 1.09 9 31.6 88.50 Exterior 0.87 0.869 0.889 1.00 0.98
10 9.8 96.00 Interior 0.93 0.859 0.913 1.08 1.02 10 9.8 96.00 Exterior 0.943 0.859 0.996 1.10 0.95 11 0.0 66.00 Interior 0.704 0.644 0.703 1.09 1.00 11 0.0 66.00 Exterior 0.658 0.644 0.676 1.02 0.97
Average 1.05 1.00
t Distribution Factors
Bridge Henry's FEA /
Method
FEA /
6 21.3 67.42 Exterior 0.689 0.663 0.775 1.04 0.89
7 33.5 76.00 Exterior 0.654 0.663 0.714 0.99 0.92
24 0.0 74.33 Exterior 0.791 0.782 0.905 1.01 0.87
5.1.4 Cast-In-Place Concrete T-Beam Bridges
The three cast-in-place concrete T-beam bridges analyzed in this study yielded
similar results to the precast concrete I-beam sections. As shown in Table 5.4, Henry’s
method was found to be typically unconservative for each of the cast-in-place concrete T-
beam bridges. As the angle of skew increased, Henry’s method became closer to FEA
values. The average ratio of FEA/H s 1 For of
bridge, the AASHTO LRFD results were very close to FEA results. However, the LRFD
method lded rat l a E int ams.
Table Co is C c - o istri n Fac
Bridge Skew Angle
Span Length Beam FEA Henry's
Method
AASHTOLRFD FEA /
Henry's Method
Ratio: FEA / LRFD
106
5.1.5 Cast-In-Place Concrete Multicell Box Beam Bridges
Four cast-in-place concrete multicell box beam bridges were analyzed in this
study. Because of the different cross-sectional properties of interior and exterior webs,
even in the same bridge the FEA results were much smaller than the results from the
Henry’s method. For the purpose of evaluation, only interior beams were analyzed. As
seen in Table 5.5, Henry’s m
extremely conservative for exterior beams and the LRFD metho
a sk angle f e26 d ees and maxi um span
istribution
length o
ctors in in
40 ft. T
rior beams
e long r spane
th a rger kew ang e could e the re son for is cons tive d ion fa
He ’s met od.
le 5 ompa ison of IP Mul cell Bo Beam oment ribut tors
Bri SkewAngle
Span Length Beam EA enry's SHTO EA /F M LRFD enry's A /
FD(deg)
12 0 133.83 Interior 0.6813 0 98.75 Interior 0.14 26.2 140 Interior 0.665 0.698 0.738 0.95 0.90
Average 1.05 0.95
5.1.6 Steel I-Beam Bridges
Four steel I-beam bridges were analyzed in this study. Each one had a different
skew angle ranging from zero to 50 degrees and span length ranging from 143 ft to 182
ft. Results from Henry’s method and the AASHTO LRFD method for these bridges differ
from trends seen in other bridge types. The data from Table 5.6 show that Henry’s
ethod was slightly unconservative for interior beams and
d was slightly
conservative. The only bridge to have conservative d fa te
had ew o gr a m f 1 h
leng nd la s l b a th erva istribut ctor
from nry h
Tab .5 C r C ti x M Dist ion Fac
dgeNo. (ft)
Hethod
AARatio: F
HMethod
Ratio: FELR
7 0.608 0.668 1.13 1.03 6200 0.608 0.701 1.02 0.88
15 16.5 110 Interior 0.765 0.701 0.785 1.09 0.97
method accurately predicted the moment distribution factors for steel I-beams. The
107
average ratio of FEA/Henry’s method was 0.994. In contrast the LRFD method
underestimated the st
this bridge type, Henry’s method compared much better to FEA than the LRFD m
d be to a c
he sligh
mb tioina of ske angle pan lens
e interio
, and th
eam of
act that
ridge #17
idge #1
nly nterio eam.
e 5. par n of St el I-Beam Mom t Distr tion Fa s
Sk S HTO R :'s
(deg) (ft) Method
0.0 158.00 Exterior 0.653 0.695 0.711 0.94 0.92 19.5 143.00 Interior 0.749 0.851 0.650 0.88 1.15
17 19.5 143.00 Exterior 0.842 0.851 0.790 0.99 1.07 18 50.2 182.00 Interior 0.850 0.828 0.696 1.03 1.22 18 50.2 182.00 Exterior 0.901 0.828 0.848 1.09 1.06 19 26.7 150.00 Interior 0.830 0.822 0.724 1.01 1.15 19 26.7 150.00 Exterior 0.835 0.822 0.659 1.02 1.27
Average 0.99 1.11
the distribution factors obtained from FEA, LRFD and the Henry’s method. The
average ratio FEA/Henry’s method was 0.998, indicating the results from Henry’s
method and FEA were in good agreement. It was observed from Table 5.7 that for bridge
# 20, a very long bridge with a skew angle, Henry’s method predicts slightly conservative
results. In contrast to Henry’s method, the AASHTO LRFD results as shown in Table 5.7
were typically unconservative. This was most likely due to the fact that the AASHTO
di ribution factors for moment and gave unconservative results. For
ethod
for nearly every case. T tly conservative result for th r b b
coul due o n w , gth e f br 7
has o one i r b
Tabl 6 Com iso e en ibu ctor
Bridge No.
ewAngle
panLength Beam FEA Henry
Method
AASLRFD
atioFEA /
Henry's
Ratio: FEA / LRFD
16 0.0 158.00 Interior 0.690 0.695 0.661 0.99 1.04 1617
5.1.7 Steel Open Box Beam Bridges
Only two steel open box beam bridges were analyzed in this study. Table 5.7
shows
108
LRFD equation for steel open box beams was directly adopted from the AASHTO
Standard method.
Table 5.7 Comparison of Steel Open Box Beam Moment Distribution Factors
Skew Span Henry's AASHTO Ratio
No. Angle (deg)
Length (ft)
Beam FEA Method LRFD Henry's LRF
202121
32.4.54.5
252.170.6170
Inter 0.60.701 0.6060.606
0.55650.64
0.645
0.911.041.13
0
1.15 0.981.06
Aver 1.0 1.09
in lem Analy is vs. He ry’s M hod fo ive Loa hear
ompa n stud was pur ued for ear dis ution f rs obta from F
He s met . The rocedure for the mparis or live d shear simila
o o mth
o
enry’s method.
5.2.1 Precast Concrete Spread Box Beam Bridges
Distribution factors for FEA, the AASHTO LRFD and Henry’s method are
tabulated in Table 5.8. From the results it can be seen that Henry’s method produced
unconservative shear distribution factors while the AASHTO LRFD method predicted
very conservative distribution factors compared to FEA results. The average ratio of
FEA/Henry’s method is 1.085 and that of FEA/LRFD is 0.801. The LRFD method
overestimated the shear distribution f
AASHTO LRFD m
Bridge :
FEA /
Method
Ratio: FEA /
D
20 32.0 252.00 Interior 0.641 0.701 0.556 0.91 1.15 0 00 Exterior 0.641
7 ior 30.67 Exterior 0.685
age
5.2 F ite E ent s n et r L d S
A c riso y s sh trib acto ined EA
and nry’ hod p co on f loa was r to
at f r live l ad moment. The ultiplier in Step 2 of Henry’s method (the multiplication
f 1.09 to steel I-beams and prestressed concrete I-beams) was considered in the results
from H
actors in every bridge case studied. Because the
ethod incorporated a skew correction factor, the distribution factors
109
for shear were amplified for beams in skewed bridges. A larger discrepancy b
Henry’s m
odifica n to H nry’s method would be neces ry in order t
the tribut factor rom Hen ’s method closer to the FEA results.
C Ple om rison of recast pread x Bea hear D ributio ctors
RatioSkew Span Henry' FEA Ratio:No. Angle Length Beam FEA Method LRFD Henry(deg) (ft) Metho1 5 1.016 1.039 0.844
2 0.0 44.38 Exterior 1.080 0.0 81.46 Interior 1.010 0.759 1.027 1.331 0.983
4 48.5 69.54 Interior 0.452 0.492 0.812 0.919 0.557 4 48.5 69.54 Exterior 0.568 0.492 0.798 1.154 0.712
Average 1.085 0.801
5.2.2 Precast Concrete Bulb-Tee Beam Bridges Table 5.9 lists the distribution factors for shear from Henry’s method, the LRFD
and FEA. For this type of bridge, Henry’s method produced very unconservative shear
distribution factors when compared to the FEA results. It can be seen that, from Table
5.9, the shear distribution factors from the AASHTO LRFD method were close to the
FEA results with an average ratio of FEA/LRFD equal to 1.017. The average ratio of
FEA/Henry’s method is 1.179 and higher ratios are observed for interior beams. It can be
concluded that modification factors to Henry’s method w
etween
ethod and the LRFD results was observed when the skew angle became larger.
It can be concluded that a m tio e sa o
get dis ion f ry
Tab 5.8 pa S Bo m S ist n Fa
Bridge s AASHTO:/’sd
FEA/LRFD
1 .0 60.89 Interior 0.858 0.8261 15.0 60.89 Exterior 0.815 0.826 0.932 0.987 0.874
1.153 1.290 0.937 0.837 33 0.0 81.46 Exterior 0.933 0.759 1.168 1.229 0.799
ould be necessary in order to
improve the accuracy of this method.
110
Table 5.9 Comparison of Precast Bulb-Tee Beam Shear Distribution Factors
kewngle
Span enry's A SHTO atio:EA /enry’s
atio:
eg) (ft) thod5 15.0 24.33 nterior .931 .663 .900 404 .0345 15.0 24.33 xterior .730 .663 .788 101 .9268 0.0 15.49 nterior .960 .790 .970 215 .9908 .0 15.49 xterior .784 .790 .833 992 .941
22 26.7 59.00 nterior .933 .710 .902 314 .0340 6 0
00.699 065 .082
recast Concrete I-Beam Bridges Table 5.10 tabulates the shear distribution factors from FEA, LRFD and the
Henry’s method for precast concrete I-beam bridges. It was found that the FEA results
were again greater than the results from Henry’s method. Henry’s method
underestimated the shear distribution factors for this type of bridge and gave the smallest
distribution factors among all three methods studied. All of the ratios of FEA vs. Henry’s
method were greater than 1.0 and the average ratio was equal to 1.152. It can be seen
from Table 5.10 that the AASHTO LRFD method was conservative for shear distribution
factors compared to the FE
closer to the FEA results. Although skew angle was an important parameter fo
n, the ew an
beams Again,
ificant e
odification to Henry’s
t on sh
od shea istribu n factors would neces .
Bridge No.
SA(d
Length Beam FEA HMethod
ALRFD
RF
HMe
RFEA/LRFD
1 I 0 0 0 1. 11 E 0 0 0 1. 01 I 0 0 0 1. 0
0 1 E 0 0 0 0. 01 I 0 0 1. 1
22 26.7 159. 0 Exterior 0.75 0.71 1. 123 26.7 151.33 Interior 0.932 0.710 0.877 1.313 1.063 23 26.7 151.33 Exterior 0.727 0.710 0.680 1.024 1.069
Average 1.170 1.017
5.2.3 P
A method. The results from the LRFD method were relatively
r shear
distributio sk gles for these bridges did not show a sign ffec ear
distribution factors, especially for interior . m
meth for r d tio be sary
111
Table 5.10 Comparison of Precast Concrete I-Beam Shear
eNo.
Skew
(deg)
Span
(ft)
Henry's Method
AASHTOLRFD
Ratio: FEA /
Henry’s Method
Ratio:
LRFD
6 21.3 67.42 Interior 0.917 0.662 0.940 1.385 0.976 6 21.3 67.42 Exterior 0.677 0.662 0.776 1.023 0.872
7 33.5 76.00 Exterior 0.700 0.662 0.811 1.057 0.863
24 0.0 75.00 Exterior 0.841 0.780 0.861 1.078 0.977
5.2.4 Cast-In-Place Concrete T-Beam Bridges
Table 5.11 lists distribution factors for shear from Henry’s method, the AASHTO
LRFD method, and FEA for three cast-in-place concrete T-beam bridges. It can be seen
from Table 5.1
for this type of bridge. The average ratio of FEA/Henry’s method was 1.080, althou
s
t ser re lts tfor exter eams than the in erior be s. It w also fo
ethod w a little nserva for liv ad she ompare
The verage atio of F /LRF is 1.02 indicat the LR results
to t FEA sults.
e 5. Com rison o CIP Co rete T eam Sh r Distr tion Fa rs
No. (deg) (ft)
31.6 88.48 Interior 0.911 0.895 0.942 1.018 0.96731.6 88.48 Exterior 0.762 0.895 0.784 0.851 0.972
10 9.8 96.00 Interior 1.090 0.859 0.969 1.269 1.125 10 9.8 96.00 Exterior 0.956 0.859 0.863 1.113 1.10811 0.0 66.00 Interior 0.770 0.644 0.826 1.196 0.932 11 0.0 66.00 Exterior 0.665 0.644 0.640 1.033 1.039
Distribution Factors
Bridg Angle Length Beam FEA FEA /
7 33.5 76.00 Interior 0.770 0.662 0.983 1.163 0.783
24 0.0 75.00 Interior 0.940 0.780 0.990 1.205 0.949
Average 1.152 0.903
1 that Henry’s method produced unconservative shear distribution factors
gh
howing a better ratio than the previous bridge type. It seemed that Henry’s method
yielded bet u he ior b t am as und
that the AASHTO LRFD m as co tive e lo ar c d to
FEA results. a r EA D 4, ing FD are
close he re
Tabl 11 pa f nc -B ea ibu cto
Bridge Skew Angle
Span Length Beam FEA Henry's
MethodAASHTO
LRFD
Ratio: FEA /
Henry’s Method
Ratio: FEA / LRFD
99
Average 1.08 1.024
112
5.2.5 Cast-In-Place Concrete Multicell Box Beam Bridges
Table 5.12 tabulates the shear distribution factors from FEA, the LRFD and the
Henry’s method for the four cast-in-place concrete multicell box beam bridges. The
distribution factors from Henry’s method were again found to be unconservative, more
for the int
LRFD method predicted the shear distribution factors over conservativ
shea
EA/LR
increa
meth
d with incre in sk angle bu did not
he distr
ry in any
ion fact
d with
sp length. The ave ge rat of FE enry’s thod w .255.
ry’ etho for thes bridges was to nconse tive an efinitel ded so
dific ion.
ble 5 2 Com arison of CIP Concrete ulticell x Beam ear Di ution
No. Angle (deg)
Length (ft)
Bea
0.0 133.83 Interior 0.842 0.608 0.899 1.385 0.937 0.0 133.83 Exterior 0.651 0.608 0.701 1.071 0.929
13 0.0 98.75 Interior 0.856 0.607 0.900 1.410 0.951 13 0.0 98.75 Exterior 0.660 0.607 0.738 1.087 0.894 14 26.2 140.00 Interior 0.975 0.698 1.280 1.397 0.762 14 26.2 140.00 Exterior 0.866 0.698 1.084 1.241 0.799 15 16.5 110.00 Interior 0.883 0.700 1.086 1.261 0.813 15 16.5 110.00 Exterior 0.851 0.700 0.868 1.216 0.980
Average 1.255 0.883
5.2.6 Steel I-Beam Bridges
Table 5.13 lists the shear distribution factors from FEA, LRFD and the Henry’s
method for the four steel I-beam bridges. The results in Table 5.13 show that for this type
erior beams than for the exterior beams. It is also seen from Table 5.12 that the
ely. The average
ratio of F FD od for these types of bridges was 0.883. T ibut ors
for r se the ase ew t va tren the
increase in an ra io A/H me as 1 The
Hen s m d e o u rva d d y nee me
mo at
Ta .1 p M Bo Sh stribFactors
Bridge Skew Spanm FEA Henry’s
MethodAASHTO
LRFD
Ratio: FEA / Henry’s Method
Ratio: FEA /LRFD
1212
of bridge the finite element analysis results were close to the LRFD results. Henry’s
113
method again unconservatively predicted the shear distribution factors but not quite so as
for the other type of bridges. The average ratios of FEA/Henry
are 1.103 and 0.932, respectively. The LRFD produced over conservative res
thod.
8, whi could
eam S r Dist ution tors
Span enry's ASHTO Ratio:FEA / Ratio:
LRFD Henry’s FEA /LRFD
16 0.0 158.00 Exterior 0.721 0.694 0.756
17 19.5 143.00 Exterior
50.2 182.00 Exterior 0.884 0.828 1.118 1.068 0.791150.00 Interior 1.017 0.822 1.099 1.237 0.925
19 26.7 150.00 Exterior 0.790 0.822 0.796 0.961 0.992
5.2.7 Steel Open Box Beam Bridges
’s method and FEA/LRFD
ults for
bridge #1 ch be attributed to a larger skew correction factor in the LRFD
me
Table 5.13 Comparison of Steel I-B hea rib Fac
Bridge No.
Skew Angle (deg)
Length (ft)
Beam FEA HMethod
A
Method 16 0.0 158.00 Interior 0.921 0.694 0.917 1.327 1.004
1.039 0.954 17 19.5 143.00 Interior 0.875 0.850 0.906 1.029 0.966
0.841 0.850 0.807 0.989 1.042 18 50.2 182.00 Interior 0.971 0.828 1.242 1.173 0.782 1819 26.7
Average 1.103 0.932
Two steel open box beam bridges were studied and the results tabulated in Table
5.14. Bridge #20 had two exterior box beams and, for this reason, there are no results for
the interior beams for this bridge. As in most cases, Henry’s method was unconservative
for the interior as well as the exterior beams. The average ratio of FEA/Henry’s method
was 1.246. The ratio suggested that the distribution factors from Henry’s method were
about 25% lower than those from FEA. Therefore Henry’s method needs to be corrected
to have the results closer to the FEA results. The calculation of distribution factors for
shear and moment in the LRFD method is the same as in the AASHTO Standard method.
114
It was found that the LRFD method produced unconservative results for both bridges of
this type. The average ratio of FEA/LRFD method was 1.298.
Table 5.14 Comparison of Steel Open Box Beam Shear Distribution Factors
No. Angle (deg)
Length (ft)
Beam FEA Method LRFD Henry’s FEA / LRF
8.00 Interior0.701
9 0.6060.606
0.645 1.351
1
of Finite Element A . Hen d
ad Moment
c
b
could have an am
Bridge Skew Span Henry's AASHORatio: FEA /
Method
Ratio:
D
20 32.0 251.00 Exterior 0.831 0.556 1.185 1.495 21 4.5 17 0.81 1.270 21 4.5 178.00 Exterior 0.727 0.645 1.200 1.127
Average .246 1.298
5.3 Summary nalysis vs ry’s Metho
5.3.1 Live Lo
Table 5.15 shows a summary of the average ratio of FEA to Henry’s method for
each beam type. Henry’s method accurately predicted live load moment for precast
concrete box beams, steel I-beams, and steel open box beams consistently. Results from
Henry’s method were slightly unconservative on average for prestressed concrete I-
beams, cast-in-place concrete T-beams, and cast-in-place concrete multicell box beams.
Figure 5.1 shows the frequency of this ratio for the entire 24-bridge database for live load
moment. From this histogram, it can be seen that while the average ratio of FEA to
Henry’s method is typically close to one, a countable number of the ratios are either
above 1.0 or below 1.0. To reduce the number of conservative and unconservative results
from Henry’s method, minor adjustments should be made by introducing multipliers for
ertain beam types. For example, precast concrete I-beams, precast concrete bulb-tee
eams, cast-in-place concrete T-beams, and cast-in-place concrete multicell box beams
plifier or slightly adjusted multiplier, if there is an amplifier already, to
115
increase the results from
results of Henry’s method and FEA suggest that little to no modification of Henry’s
method is required for steel I-beams, steel open box beams, and concrete spread box
beams.
Table 5.15 Summary of FEA/Henry’s Method Results for Live Load Moment
Structure Type
Average
Henry’s Method
Standard Deviation
Precast Spread Box Beam 0.98 0.12 Precast Concrete BT Beam 0.87 0.12 Precast Concrete I-Beam 1.04 0.06
CIP Concrete Box Beam 1.05 0.08
Steel Open Box Beam 1.00 0.11
Database #1 Histogram
02468
101214
18
Freq
uenc
y
Henry’s method and make them comparable to FEA. The
Ratio: FEA /
CIP Concrete T-Beam 1.05 0.04
Steel I-Beam 0.99 0.06
16
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 More
Ratio
Ratio: FEA / Unmodified Henry's Method
Figure 5.1 Histogram of FEA vs. Henry’s Method for Live Load Moment
116
he bridge types for live loa r. Ta
A to Henry’s metho dard deviation to all the bridges.
his method pr to 25 ative
F
sumed that some bridge types would have
factors.
5.3.2 Live Load Shear
ry’s method to
Henry’
Based on the comparison of Hen the finite element analysis, it was
found that the distribution factors from the s method were highly unconservative
for most of t d shea ble 5.16 shows the summary of average
ratio of FE d and stan Henry’s
method in t oduced about 10% % unconserv values compared to
EA distribution factors for shear. It was also found from the finite element analysis that,
in most cases, interior beams carry more shear force. Relatively speaking, Henry’s
method predicted the shear distribution factors in a less percentage of unconservativity
for precast concrete spread box beams, cast-in-place concrete T-beams and steel I-beams.
For all of other types of bridges Henry’s method was very unconservative. Figure 5.2
shows the frequency histogram of the average ratio for all the 24 bridges. As we can see
from this figure most of the ratios fall in the range greater than one. This fact indicated
again that Henry’s method was unconservative and needed some modification to bring
the ratios closer to one. Introducing multipliers to different types of bridge
superstructures could improve the predication of Henry’s method close to FEA results.
From individual bridge ratios it can be as
higher modification factors and the other bridge types would have lower modification
117
D a tab a s e # 1 H is to g ram
0
Table 5.16 Summary of FEA/Henry’s Method Results for Live Load Shear
Average Ratio:
Method
Precast Concrete BT Beam 1.18 0.15
CIP Concrete T-Beam 1.08 0.15
Steel I-Beam 1.10 0.13
2
4
8
1 0
1 2
1 4
quen
c
0 .8 0 .9 1 1 .1 1 .2 1 .3 1 .4 1 .5 1 .6
e d H e n r
d HenrSHTO LRFD vs. FEA an Metho
Moment
shows a osummary ge ratio of FEA to the AASHTO LRFD
Structure Type FEA / Henry’s Standard Deviation
Precast Spread Box Beam 1.09 0.16
Precast Concrete I-Beam 1.15 0.13
CIP Concrete Box Beam 1.26 0.14
Steel Open Box Beam 1.23 0.08
6
R atio
Fre
y
R a tio : F E A /U nm o d ifi y 's M e th od
Figure 5.2 Histogram of FEA vs. Henry’s Method for Live Load Shear
5.4 Summary of AA y’s d
5.4.1 Live Load
Table 5.17 f the avera
method for each type of beam. Tables 5.15 and 5.17 shows the relationship between
FEA, the AASHTO LRFD, and Henry’s method. For precast concrete spread box beams,
118
precast concrete bulb-tee beams, and cast-in-place concrete T-beams, Henry’s method
and the AASHTO LRFD method yield similar results compared to FEA. For cast-in-
place concrete multicell box beam bridges, the AASHTO LRFD method produced
slightly conservative results in contrast to the slightly unconservative results produced by
Henry’s method. For steel I-beams, Henry’s method produced more accurate results than
the unconservative LRFD method. From Table 5.17 it can also be seen that the LRFD
method accurately predicted distribution factors of live load moment for precast concrete
spread box beams and cast-in-place concrete T-beams. Predictions for precast concrete
bulb-T beams, precast concrete I-beams, and cast-in-place concrete box beams were
conservative while the predictio nd steel open box beams were
unconservative. Figure 5.3 shows the frequency of the ratio of FEA to Henry’s method
and the AASHTO LRFD method for the entire 24-bridge database. From this histogram,
slightly different data distributions of FEA/Henry’s method ratios and FEA/AASHTO
LRFD ratios are observed.
Table 5.17 Summary of FEA/LRFD Results for Live Load Moment
ns for steel I-beams a
Structure Type FEA /
LRFD
Standard
Precast Spread Box Beam 1.02 0.08 Precast Concrete BT Beam 0.88 0.06 Precast Concrete I-Beam 0.92 0.06 CIP Concrete T-Beam 1.00 0.05 CIP Concrete Box Beam 0.95 0.07 Steel I-Beam 1.11 0.11 Steel Open Box Beam 1.09 0.08
Average 1.00 0.07
Ratio:
AASHTO Deviation
119
Database #1 Histogram
121086
24
00.6 0.7 0.8 1.1 1.3
d on the fact that the moment distribution factors from Henry’s method were v
close to the FEA resu
distribution factors for these bridge types. Figure 5.3 shows the frequency histogram for
ratios of the LRFD method vs the Henry’s method for live load moment.
141618
0.9 1 1.2 More
Ratio
Freq
uenc
y
Ratio: FEA / AASHTO LRFD Ratio: FEA / Henry's Method
Figure 5.3 Histogram of FEA vs. Henry’s Method and AASHTO LRFD Method (Live Load Moment)
Table 5.18 shows a summary of the average ratio and standard deviation of the
LRFD to the Henry’s method for each type of beam. When comparing the AASHTO
LRFD and Henry’s method for live load moment it is observed that the LRFD method
produces higher distribution factors for precast concrete I-beams, cast-in-place concrete
T-beams, and cast-in-place concrete box beams than Henry’s method and similar results
for precast bulb-tee beams. As noted in Table 5.17, the LRFD results are slightly
unconservative for precast concrete box beams, steel I-beams, and steel open box beams.
Base ery
lts, it is acceptable that Henry’s method produced reliable
120
Table 5.18 Summary of LRFD/Henry's Method Results for Live Load Moment
Structure Type Average Ratio: LRFD / Henry's
Method Deviation
Precast Spread Box Beam 0.95 0.14 Precast Concrete BT Beam 0.98 0.11 Precast Concrete I-Beam 1.13 0.08 CIP Concrete T-Beam 1.05 0.08 CIP Concrete Box Beam 1.11 0.04 Steel I-Beam 0.90 0.10 Steel Open Box Beam 0.93 0.16
8
12
14
quen
c
Database # 1 Histogram
0
2
4
6Fr
0.7 0.8 1.1 1.4
io
R D/Hen
y’s M oad
Standard
10
16
0.9 1 1.2 1.3
Rat
ey
atio:AASHTO LRF ry's Method
Figure 5.4 Histogram of LRFD vs. Henr ethod for Live L Moment
5.4.2 Live Load Shear
Table 5.19 shows the summary of the average ratio of FEA to the AASHTO
LRFD method for each type of beam for live load shear. Tables 5.16 and 5.19 show the
relationships between FEA to the AASHTO LRFD and FEA to Henry’s method for live
load shear, respectively. For precast concrete spread box beams, cast-in-place concrete
121
multicell box beam and steel I-beam, AASHTO LRFD method produced slightly
conservative results in contrast to Henry’s method, which produced unconservative
results for these types of bridges. For precast concrete bulb-tee beams, and cast-in-place
concrete T-beams, the AASHTO LRFD method produced similar results to the FEA
method. Henry’s method on the other hand was unconservative for bulb-tee beam and
cast-in-place concrete T-beam bridges. Both Henry’s method and the AASHTO LRFD
method produced very unconservative results for steel open box beam bridges when
compared to the FEA results. Overall, the LRFD method was a little conservative when
compared to the FEA method, and Henry’s method was found to be consistently
unconservative. F RFD method for
the enti
igure 5.5 shows the frequency of the ratio of FEA to L
re 24-bridge database. From this histogram it can be seen that a majority of ratios
of FEA to LRFD were less than one, while some were equal to one. This fact indicated
that the AASHTO LRFD method is a little conservative for live load shear.
Table 5.19 Summary of FEA/LRFD Results for Live Load Shear
Structure Type Average Ratio: FEA / LRFD Standard Deviation
Precast Concrete BT Beam 1.02 0.05
CIP Concrete T-Beam 1.02 0.08
Steel I-Beam 0.93 0.09
Precast Spread Box Beam 0.80 0.13
Precast Concrete I-Beam 0.90 0.07
CIP Concrete Box Beam 0.88 0.08
Steel Open Box Beam 1.30 0.20
122
14
16
18
Database # 1 Histogram
10eq
uenc
y
4
6
2
00.6 0.7 1 1.4
atio
AASH
es again that a modification to Henry’s method for live load shear is necessary.in
p
u
m
8
12
0.8 0.9 1.1 1.2 1.3R
Fr
Ratio: FEA/ TO LRFD
Figure 5.5 Histogram of FEA vs. LRFD for Live Load Shear
The Henry’s method was also compared to the AASHTO LRFD method for live
load shear. Again it was found that distribution factors from Henry’s method were
smaller than the LRFD results for almost all bridges types. Table 5.20 shows the
summary of average ratio of the AASHTO LRFD to Henry’s method for live load shear
and Figure 5.6 shows the frequency of ratios of the LRFD to Henry’s method. It can be
seen from this figure that most of the ratios are greater than one. As mentioned before,
the AASHTO LRFD method predicted the distribution factors for shear close to the FEA
results, therefore the larger discrepancy between the LRFD method and Henry’s method
dicat If a
roper modification factor were developed, the Henry’s method could become less
nconservative than it currently is; however, it would not be as conservative as the LRFD
ethod for live load shear. Tables 5.21 and 5.22 list the distribution factors for moment
and shear obtained from FEA and Henry’s method as well as the corresponding ratios for
123
Data
all bridges in database #1. Tables 5.23 and 5.24 list the distribution factors for mome
d shear obt from the AAS LRFD
or the bridges in the same tabase.
od Re or Liv ear
Structure Type Average atio: / Sta ationHenr ’s methP st Spread Box Beam 1.37 P st Concrete BT Beam 1.16 P st Concrete I-Beam 1.28 C oncrete T-Beam 1.06 C oncrete Box Beam 1.44 S I-Beam 1.20 S Open Box Beam 0.93
base # 1 istogram
12
10
8ync
6
eque
4Fr
2
00.8 1.1 .2 1 .4 1
Ratio
o:AASHTO RFD/He thod
Fig e 5.6 Histogram of L vs. He
nt
an ained HTO and Henry’s method and the corresponding
ratios f all da
Table 5.20 Summary of LRFD/Henry's Meth sults f e Load Sh
R LRFDy od ndard Devi
reca 0.25 reca 0.14 reca 0.15 IP C 0.14 IP C 0.20 teel 0.21 teel 0.16
H
0.9 1 1 .3 1 .5 1.6
Rati L nry's Me
ur RFD nry’s Method for Live Load Shear
124
Table 5.21 Moment Distribution Factors - FEA vs. Henry’s Method, Database #
Bridge No. Structure Type Beam FEA Henry's Method Ratio: FEA / Henry’s Method
1 Precast Spread Box Beam Interior 0.754 0.826 0.9131 Precast Spread Box Beam Exterior 0.762 0.826 0.9232 Precast Spread Box Beam Exterior 1.251 1.152 1.0863 Precast Spread Box Beam Interior 0.867 0.759 1.1423 Precast Spread Box Beam Exterior 0.826 0.759 1.0884 Precast Spread Box Beam Interior 0.358 0.489 0.7324 Precast Spread Box Beam Exterior 0.396 0.489 0.8105 Precast Concrete BT Beam Interior 0.660 0.663 0.9955 Precast Concrete BT Beam Exterior 0.625 0.663 0.9436 Precast Concrete I-beam Interior 0.715 0.663 1.0786 Precast Concrete I-beam Exterior 0.689 0.663 1.0397 Precast Concrete I-beam Interior 0.721 0.663 1.0877 Precast Concrete I-beam Exterior 0.654 0.663 0.9868 Precast Concrete BT Beam Interior 0.812 0.790 1.0288 Precast Concrete BT Beam Exterior 0.747 0.790 0.9469 CIP Concrete T-Beam Interior 0.877 0.869 1.0099 CIP Concrete T-Beam Exterior 0.870 0.869 1.001
10 CIP Concrete T-Beam Interior 0.930 0.859 1.08310 CIP Concrete T-Beam Exterior 0.943 0.859 1.09811 CIP Concrete T-Beam Interior 0.704 0.644 1.09311 CIP Concrete T-Beam Exterior 0.658 0.644 1.02212 CIP Concrete Box Beam Interior 0.687 0.608 1.130 12 CIP Concrete Box Beam Exterior 0.409 0.608 0.673 13 CIP Concrete Box Beam Interior 0.620 0.608 1.020 13 CIP Concrete Box Beam Exterior 0.415 0.608 0.683 14 CIP Concrete Box Beam Interior 0.665 0.698 0.953 14 CIP Concrete Box Beam Exterior 0.431 0.698 0.617 15 CIP Concrete Box Beam Interior 0.765 0.700 1.093 15 CIP Concrete Box Beam Exterior 0.594 0.700 0.849 16 Steel I-Beam Interior 0.690 0.695 0.99316 Steel I-Beam Exterior 0.653 0.695 0.94017 Steel I-Beam Interior 0.749 0.851 0.88017 Steel I-Beam Exterior 0.842 0.851 0.98918 Steel I-Beam Interior 0.857 0.828 1.03518 Steel I-Beam Exterior 0.906 0.828 1.09419 Steel I-Beam Interior 0.830 0.822 1.01019 Steel I-Beam Exterior 0.835 0.822 1.01620 Steel Open-Box Girder Interior - 0.701 - 20 Steel Open-Box Girder Exterior 0.641 0.701 0.91421 Steel Open-Box Girder Interior 0.630 0.606 1.04021 Steel Open-Box Girder Exterior 0.685 0.606 1.13022 Precast Concrete BT Beam Interior 0.559 0.711 0.78622 Precast Concrete BT Beam Exterior 0.552 0.711 0.77623 Precast Concrete BT Beam Interior 0.537 0.711 0.75523 Precast Concrete BT Beam Exterior 0.511 0.711 0.71924 Precast Concrete I-beam Interior 0.757 0.782 0.96824 Precast Concrete I-beam Exterior 0.791 0.782 1.012
Average 0.960
1
125
Table 5.22 Shear Distribution Factors - FEA vs. Henry’s Method, Data
Bridge No. Structure Type Henry's Method Ratio: FEA / Henry’s Method
1 Precast Spread Box Beam Interior 0.858 0.826 1.0391 Exterior 0.815
Precast Spread Box Beam 0.759
4 Precast Spread Box Beam Interior 0.452 0.489 0.9244 Precast Spread Box Beam Exterior 0.568 0.489 1.1625 Precast Concrete BT Beam Interior 0.931 0.663 1.4045 6
Precast Concrete BT BeamPrecast Concrete I-beam
ExteriorInterior
0.7300.917
0.6630.663
1.1011.383
6 Precast Concrete I-beam Exterior 0.677 0.663 1.0217 Precast Concrete I-beam Interior 0.770 0.663 1.1617 Precast Concrete I-beam Exterior 0.700 0.6638 Precast Concrete BT Beam Interior 0.960 0.790 1.2158 Exterior 0.784 0.790 0.9929 CIP Concrete T-Beam Interior 0.911 0.869 1.0489 CIP Concrete T-Beam Exterior 0.762 0.869 0.877
CIP Concrete T-Beam Interior 1.090 0.859 1.26910 CIP Concrete T-Beam Exterior 0.956 0.859 1.1131111 CIP
CIP Concrete T-BeamConcrete T-Beam
InteriorExterior
0.770665
0.6440.644
1.1961.033
Interior Exterior
0.842 0.651
1.385 1.071 12
13 ncrete Box Beam
CIP Concrete Box Beam Interior 0.856 0.608 0.608 1.408
13 CIP Concrete Box Beam Exterior 0.608 1.086 CIP Concrete Box Beam Interior
1.241
15 Exterior 0.851 0.700 1.216 16 Steel I-Beam Interior 0.921 0.695 1.32516 Steel I-Beam Exterior 0.721 0.695 1.03717 Steel I-Beam
teel I-BeamInterior 0.875 0.851 1.028
17 ExteriorInt
0.841 0.851 0.9881818
Steel I-BeamSteel I-Beam
eriorExterior
0.9710.884
0.8280.828
1.1731.068
19 Steel I-Beam Interior 1.017 0.822 1.237Steel I-Beam Exterior 0.790 0.822 0.961
20 Steel Open-Bo Interior 0.701 - 20 Steel Open-Bo Exterior 0.831 0.701 1.18521 Steel Open-Bo Interior 0.81921 Steel Open-Box Girder
1.31222 Precast Concrete BT Bea Exterior 0.756 0.711 1.06323 Precast Concrete BT Bea Interior
Exterior 0.720.932 0.711 1.311
23 24
Precast Concrete BT Beamoncrete I-beam Interior
Exterior 0.841
base #1
Beam FEA
Precast Spread Box Beam 0.826 0.987 2 Precast Spread Box Beam Exterior 1.080 1.152 0.938 3 Interior 1.010 1.3313 Precast Spread Box Beam Exterior 0.933 0.759 1.229
1.056
Precast Concrete BT Beam
10
0.12 CIP Concrete Box Beam 0.608
CIP Co
0.660 14 0.975 0.698 1.397 14 CIP Concrete Box Beam Exterior 0.866 0.698 15 CIP Concrete Box Beam Interior 0.883 0.700 1.261
CIP Concrete Box Beam
S
19x Girder - x Girder x Girder 0.606 1.351
Exterior 0.727 0.606 1.200 22 Precast Concrete BT Beam Interior 0.933 0.711
mm
7 0.711 1.023 Precast C 0.940 0.782 1.202
24 Precast Concrete I-beam 0.782 1.075 Average 1.154
126
Table 5.23 Moment Distribution Factors – LRFD vs. Henry’s Method, Database #1
Bridge No. Beam LRFD Henry's Method Ratio: LRFD / Henry's Method Structure Type
1 Precast Spread Box Beam Interior 0.723 0.826 0.8751 Precast Spread Box Beam Exterior 0.780 0.826 0.9442 Exterior 1.186 1.152 1.0303 Precast Spread Box Beam Interior 0.7523 Precast Spread Box Beam Exterior 0.856 0.759 1.1284 Precast Spread Box Beam Interior 0.343 0.489 0.7014 Precast Spread Box Beam Exterior 0.494 0.489 1.0105 Precast Concrete BT Beam
mInterior 0.705 0.663 1.063
5 Precast Concrete BT Bea
Exterior 0.775 0.663 1.1697 Precast Concrete I-beam Interior 0.702 0.663 1.0597 Precast Concrete I-beam Exterior 0.714
8 Precast Concrete BT B Exterior 0.853 0.790 1.0809 CIP Concrete T-Beam Interior 0.802 0.869 0.9239 CIP Concrete T-Beam Exterior 0.889 0.869 1.023
10 CIP Concrete T-Beam Interior 0.913 0.859 1.06310 CIP Concrete T-Beam Exterior 0.996 0.859 1.15911 CIP Concrete T-Beam Interior 0.703 0.644 1.09211 CIP Concrete T-Beam Exterior 0.676 0.644 1.05012 CIP Concrete Box Beam Interior 0.668 13 CIP Concrete Box Beam Interior 0.701 0.608 1.153 14 CIP Concrete Box Beam Interior 0.738 0.698 1.057 15 CIP Concrete Box Beam Interior 0.785 1.120 16 CIP Concrete Box Beam Interior 0.661 0.695 16 CIP Concrete Box Beam Exterior 0.711 0.695 1.023 17 CIP Concrete Interior 0.650 0.851 0.764
CIP Concrete Exterior 0.790 0.851 0.928 17 18 Steel I-Beam Interior 0.696 0.828 0.84118 Steel I-Beam Exterior 0.848 0.828 1.02419 Steel I-Beam Interior 0.724 0.822 0.88119 Steel I-Beam 0.659 0.822 0.80220 Steel I-Beam Interior 0.556 0.701 0.79320 Steel I-Beam Exterior 0.556 0.701 0.79321 Interior 0.645 0.606 1.06421 Steel I-Beam Exterior 0.645 0.606 1.06422 Steel Open-Box Girder Interior 0.641 0.711 0.902
Steel Open-Box Girder Exterior 0.617 0.711 0.86823 Steel Open-Box Girder Interior23 Exterior
0.650 0.625
0.711 0.711
0.914 0.879
24 Steel Open-Box Girder Precast Concrete BT Beam Interior 0.849 0.782 1.086
24 Precast Concrete BT Bea Exterior 0.905 0.782 1.263verage 1.000
Precast Spread Box Beam0.759 0.991
Exterior 0.756 0.663 1.140 6 Precast Concrete I-beam Interior 0.762 0.663 1.149 6 Precast Concrete I-beam
0.663 1.077 8 Precast Concrete BT Beam Interior 0.809 0.790 1.024
eam
0.608 1.099
0.701 0.951
Box BeamBox Beam
Exterior
Steel I-Beam
22
mA
127
Table 5.24 Shear Distribution Factors – LRFD vs. Henry’s Method, Database #1
Bridge No. Structure Type Beam LRFD Henry's Method Ratio LRFD / Henry’s Method
1 Precast Spread Box Beam Interior 1.016 0.826 1.230 1 Precast Spread Box Beam Exterior 0.932 0.826 1.128 2 Precast Spread Box Beam Interior 1.290 1.152 1.120 2 Precast Spread Box Beam Exterior 1.290 1.152 1.120 3 Precast Spread Box Beam Interior 1.027 0.759 1.353 3 Precast Spread Box Beam Exterior 1.168 0.759 1.539 4 Precast Spread Box Beam Interior 0.812 0.489 1.661 4 Precast Concrete BT Beam Exterior 0.798 0.489 1.632 5 Precast Concrete BT Beam Interior 0.900 0.663 1.357 5 Precast Concrete I-beam Exterior 0.788 0.663 1.189 6 Precast Concrete I-beam Interior 0.940 0.663 1.418 6 Precast Concrete I-beam Exterior 0.776 0.663 1.170 7 Precast Concrete I-beam Interior 0.983 0.663 1.483 7 Precast Concrete BT Beam Exterior 0.811 0.663 1.223 8 Precast Concrete BT Beam Interior 0.970 0.790 1.228 8 CIP Concrete T-Beam Exterior 0.833 0.790 1.054 9 CIP Concrete T-Beam Interior 0.942 0.869 1.084 9 CIP Concrete T-Beam Exterior 0.784 0.869 0.902
10 CIP Concrete T-Beam Interior 0.969 0.859 1.128 10 CIP Concrete T-Beam Exterior 0.863 0.859 1.005 11 CIP Concrete T-Beam Interior 0.826 0.644 1.283 11 CIP Concrete Box Beam Exterior 0.640 0.644 0.994 12 CIP Concrete Box Beam Interior 0.899 0.608 1.479 12 CIP Concrete Box Beam Exterior 0.701 0.608 1.153 13 CIP Concrete Box Beam Interior 0.900 0.608 1.480 13 CIP Concrete Box Beam Exterior 0.738 0.608 1.214 14 CIP Concrete Box Beam Interior 1.280 0.698 1.834 14 CIP Concrete Box Beam Exterior 1.084 0.698 1.553 15 CIP Concrete Box Beam Interior 1.086 0.700 1.551 15 Steel I-Beam Exterior 0.868 0.700 1.240 16 Steel I-Beam Interior 0.917 0.695 1.319 16 Steel I-Beam Exterior 0.756 0.695 1.088 17 Steel I-Beam Interior 0.906 0.851 1.065 17 Steel I-Beam Exterior 0.807 0.851 0.948 18 Steel I-Beam Interior 1.242 0.828 1.500 18 Steel I-Beam Exterior 1.118 0.828 1.350 19 Steel I-Beam Interior 1.099 0.822 1.337 19 Steel Open-Box Girder Exterior 0.796 0.822 0.968 20 Steel Open-Box Girder Interior 0.556 0.701 0.793 20 Steel Open-Box Girder Exterior 0.556 0.701 0.793 21 Steel Open-Box Girder Interior 0.645 0.606 1.064 21 Precast Concrete BT Beam Exterior 0.645 0.606 1.064 22 Precast Concrete BT Beam Interior 0.902 0.711 1.269 22 Precast Concrete BT Beam Exterior 0.699 0.711 0.983 23 Precast Concrete BT Beam Interior 0.877 0.711 1.233 23 Precast Concrete I-beam Exterior 0.680 0.711 0.956 24 Precast Concrete I-beam Interior 0.990 0.782 1.266 24 Precast Spread Box Beam Exterior 0.861 0.782 1.101
Average 1.263
128
5.5 Key Parameters
Several key structural parameters were studied for database #1 including span
length, skew angle, beam spacing, slab thickness, and beam stiffness. Each of these
parameters’ effect on live load distribution factors was investigated. Based on these
examinations, the determination was made whether or not to propose modification factors
for each parameter.
5.5.1 Span Length
Currently only the AASHTO LRFD specifications consider span length for
calculating live load moment distribution factors. As noted before, these codes are only
applicable for certain limits. It is the intent of this research to propose a method that
considers a notably important factor possibly including span length without imposing
restrict
LRFD specifications.
ions on its use. The span lengths of bridges in this study ranged from 44 ft to 252
ft. Figure 5.7 shows the moment distribution factor vs. span length for the 24 bridges
analyzed in this study. The linear trend line of Henry’s method matches with that of FEA
very well. It can be seen from Figure 5.7 that Henry’s EDF method becomes slightly
conservative for bridges with spans longer than 100 ft. This is evident when compared to
both FEA and the AASHTO
129
Distribution Factor vs Span Length
0.200
0.400
0.600
0.800
1.000
1.200
1.400
0.00 50.00 100.00 150.00 200.00 250.00 300.00
Span Length (ft)
Dis
tribu
tion
Fact
or
AASHTO LRFD AASHTO StandardHenry's Method FEALinear (AASHTO Standard) Linear (Henry's Method)Linear (AASHTO LRFD) Linear (FEA)
Figure 5.7 Moment Distribution Factor vs. Span Length
Figure 5.8 shows the shear distribution factors versus span length for the selected
24 bridges. As can be seen from the figure, the shear distribution factors have little
change with the changes in span length, indicating that span length is not an important
impact factor for shear distribution. The linear trend lines for all the methods are almost
similar except Henry’s method, which gave smaller distribution factors compared to the
results from other methods.
130
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
30 50 70 90 110 130 150 170 190
Span Length (FT)
Dis
tribu
tion
Fact
ors
FEA AASHTO LRFDHenry's Method AASHTO StandardLinear (AASHTO LRFD) Linear (FEA)Linear (Henry's Method) Linear (AASHTO Standard)
Figure 5.8 Shear Distribution Factor vs. Span Length
5.5.2 Skew Angle
Currently, only the AASHTO LRFD method considers the effect of skew angle
on live load moment distribution factors. Again, these reduction factors are applicable
for certain ranges of applicability. The skew angle for bridges in this study ranged from
zero to 50 degrees. Figure 5.9 shows the moment distribution factors obtained for all four
methods versus the skew angle. By comparing each method of calculation with FEA, it
can be seen that all methods, the AASHTO LRFD, the AASHTO Standard, and the
Henry’s method, predicted moment distribution factors fairly close to FEA results. A
very minor difference can be observed between Henry’s EDF method and FEA when
skew angle is greater than 30 degrees, see Figure 5.9
131
Distribution Factor vs Skew Angle
0.200
0.400
0.800
1.000
1.200
Skew Angle (degrees)
Dis
tribu
tac
tor
AASHTO LRFD AASHTO StandardHenry's Method FEA
Linear (FEA) Linear (Henry's Method)
Figure 5.9 Moment Distribution Factor vs. Skew
S
y a precast concrete bulb-tee beam bridge, Bridge #23, was selected
fo
0.600
1.400
0.0 10.0 20.0 30.0 40.0 50.0 60.0
ion
F
Linear (AASHTO Standard) Linear (AASHTO LRFD)
Angle
imilar to Figure 5.9, the shear distribution factors versus skew angle are shown
in Figure 5.10. As can be seen in Figure 5.10, the shear distribution factors from the
AASHTO LRFD method increase with the increase in skew angle. One main reason for
the increase in the AASHTO LRFD results is that this method has included a skew angle
factor for all bridge types. The shear distribution factors from all other methods are
smaller than that from the LRFD method including FEA. The results from Henry’s
method were the smallest among all results. However the linear trend line for Henry’s
method is parallel to that for FEA method. With proper modification, Henry’s method
should be able to determine accurate shear distribution factors.
In this stud
r further analysis of skew angle effects. The bridge has a skew angle of 17.50 degrees.
132
The maxim eam were
determined through finite element analysis. The effect of skew angle on the distribution
factors for live load shear can io of ribution factors
at the acute corner to the factor obtu lcu ented in the
table. From n b that an beam at the btuse corner of a
skewed bridg she at the ac er. It was a served from this
study that the effect of s interior beam prominent. The distribution
factors from the finite element method for bridges in database #1 do not change much
with varied skew angles. This can be attributed to the fact that distribution factors are also
a function of other variables such as beam spacing, number of girders, span length, size
of the beam, slab thickness, and longitudinal stiffness of girders.
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
0 10 20 30 40 50 60Skew Angle
Shea
r Dis
tribu
tion
Fact
ors
FEA AASHTO LRFDAASHTO Standard Henry's MethodLinear (FEA) Linear (AASHTO LRFD )Linear (AASHTO Standard) Linear (Henry's Method)
um shears at both the obtuse and the acute corner of an exterior b
be seen in
at the
Table 5.25. The rat the dist
lated and presse corner was ca
this table it ca e seen exterior o
lso obe carries more ar than ute corn
kew angle on s is less
Figure 5.10 Shear Distribution Factor vs. Skew Angle
133
Distribution Factor vs Girder Spacing
0.200
0.4
Table 5.25 Effect of Skew Angle on Shear Distribution Factors
Bridge # 23
Distribution FactorsMaximum
(lbs) With Diaphragm Acute/Obtuse
Obtuse Corner 50170 0.727
Acute Corner 43012 0.624
Interior Beam 64263 0.932
0.858
00
4.00
AASHTO LRFD AASHTO Standard
Linear (AASHTO Standard) Linear (Henry's Method)Linear (AASHTO LRFD) Linear (FEA)
Beam Type Shear Ratio of
Exterior Beam
Exterior Beam
5.5.3 Beam Spacing
Each of the current methods considers beam spacing to be a crucial parameter in
moment distribution. However, Henry’s EDF method is the only one without limiting
ranges of applicability. Beam spacing for bridges in this study ranged from 5.67 ft to
13.75 ft. All methods followed a similar trend of moment distribution factors increasing
as beam spacing increased as shown in Figure 5.11.
0.600
0.800
1.000
1.200
1.400
6.00 8.00 10.00 12.00 14.00 16.00
Girder Spacing (ft)
Dis
tribu
tion
Fact
or
Henry's Method FEA
Figure 5.11 Moment Distribution Factor vs. Beam Spacing
134
All the current methods use beam spacing in their equations to calculate
distribution factors for shear. As can be seen from Figure 5.12, the trend lines of all the
methods were parallel to each other indicating the distribution factors increase with the
increase in beam spacin
0.00
FEA AASHTO LRFDHenry's Method AASHTO Standard
Linear (Henry's Method) Linear (AASHTO Standard)
5.5.4 Slab Thickness
Slab thickness is considered by the AASHTO LRFD
this parameter comes with its own rang
s
g. The LRFD method predicted the highest shear distribution
factor while Henry’s method gave the lowest distribution factor. According to Figure
5.12 it can be concluded again that Henry’s method needs some modification to have the
results close to FEA results and the AASHTO LRFD results.
0.20
0.40
0.60
0.80
1.00
1.20
1.40
4 6 8 10 12 14 16
Beam Spacing
Dis
tribu
tion
Fact
ors
L inear (FEA) Linear (AASHTO LRFD )
Figure 5.12 Shear Distribution Factor vs. Beam Spacing
specifications only. Again,
e of applicability. Cast-in-place concrete deck
lab thickness ranged from 7 in. to 9.25 in. By comparing moment distribution factors
again, it can be seen that even though other methods do not consider slab thickness, a
135
similar trend of increasing distribution factors with increasing slab thickness is observed
as shown in Figure 5.13 for live load moment. This is because the slab thickness is
directly proportional to the girder spacing. Therefore, no modification is recommended
based on slab thickness.
Distribution Factor vs Slab Thickness
0.200
0.400
0.600
0.800
1.000
1.200
1.400
6.50 7.00 7.50 8.00 8.50 9.00 9.50
Slab Thickness (in)
Dis
tribu
tion
Fact
or
AASHTO LRFD AASHTO StandardHenry's Method FEALinear (AASHTO LRFD) Linear (FEA)Linear (Henry's Method) Linear (AASHTO Standard)
Figure 5.13 Moment Distribution Factor vs. Slab Thickness
Similar to the moment distribution factors, only the AASHTO LRFD method uses
slab thickness in calculation of distribution factors for live load shear. As seen in Figure
5.14 the distribution factors increased for all the methods with an increase in the slab
thickness. Again the LRFD method was over conservative and the Henry’s method was
unconservative. However the trends for all methods were similar. This is because the
slab thickness is directly proportional to the girder spacing.
136
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
6 7 8 9 10
Slab Thickness
Dis
tribu
tion
Fact
ors
FEA AASHTO LRFDHenry's Method AASHTO StandardLinear (FEA) Linear (AASHTO LRFD )Linear (AASHTO Standard) Linear (Henry's Method)
e5
Figure 5.14 Shear Distribution Factor vs. Slab Thickness
.5.5 B am Stiffness
Beam stiffness, Kg, is considered by the LRFD specifications only. This
parameter also comes with a range of applicability and is only used for certain beam
types such as steel I-beams, cast-in-place concrete T-beams, and precast concrete I-
beams. As shown in Figure 5.15, only a slight variance in distribution factor for live load
moment can be attributed to beam stiffness. For this database, the majority of data
suggests that Henry’s method compares well with FEA and the AASHTO LRFD
distribution factors. Therefore, no modification is recommended for beam stiffness.
137
Distribution Factor vs Beam Stiffness (Kg)
0.400
0.500
0.600
0.700
0.800
0.900
1.000
1.100
0.E+00 5.E+05 1.E+06 2.E+06 2.E+06 3.E+06 3.E+06 4.E+06 4.E+06
Beam Stiffness (in4)
Dis
tribu
tion
Fact
or
AASHTO LRFD AASHTO Standard FEA Henry's Method Linear (AASHTO Standard) Linear (AASHTO LRFD) Linear (FEA) Linear (Henry's Method)
Figure 5.15 Moment Distribution Factor vs. Beam Stiffness
138
CHAPTER 6
MODIFICATIONS OF HENRY’S EQUAL DISTRIBUTION
FACTOR METHOD
This chapter introduces two sets of preliminary modification factors to Henry’s
EDF method for both live load moment and shear. The first set consists of structure type
modification factors for live load moment along with a single multiplier to structure type
factors for live load shear. The second set contains a different set of structure type
modification factors for live load moment and shear. The moment modification is in
conjuncture with skew angle and or span length modification factors, while the shear
modification includes the skew correction formula. Each set of modification factors and
their effects for live load moment and shear are detailed below. To further analyze the
effects of these modification factors, a before-and-after comparison to database #2 is
discussed.
6.1 Discussion of Database #2
Database #2 consists of 419 real bridges analyzed in NCHRP Project 12-26 built
between 1920 and 1988. The bridges for Project 12-26 were randomly chosen from 15
different states including Arizona, California, Florida, Indiana, Maine, Minnesota,
Missouri, New York, Ohio, Oklahoma, Oregon, Pennsylvania, Tennessee, Texas, and
Washington. This database contained 35 precast concrete box beam bridges, 66 precast
concrete bulb-tee and I-beam bridges, 69 cast-in-place concrete T-beam bridges, 148
steel I-beam bridges, 82 cast-in-place concrete multicell box beam bridges, and 19 steel
open box beam bridges. The key parameters for database #2 had a much larger range
139
than database #1. Span length varied from 18.75 feet to 281.7 feet. Beam spacing varied
from 2.42 feet to 24 feet. Slab thickness varied from 5 inches to 11 inches. Skew angles
varied from 0 to 61 degrees. Live load moment and shear distribution factors for each
bridge were calculated using AASHTO LRFD and Henry’s EDF method.
6.2 Preliminary Modification Factors for Live Load Moment (Set 1)
The first set of preliminary modification factors includes modification factors for
structure types only. These factors are based on numerical analysis from Chapter 5 and
were obtained by setting the average ratio of FEA to Henry’s method to unity (1.0) and
standard deviation as close to 0.1 as possible for each structure type. To do this, EXCEL
spreadsheets were created. Bridge types were separated, and applicable key structural
parameters were input for each bridge. Distribution factors for each bridge were input for
FEA and Henry’s method. The ratio of FEA to Henrys method was then calculated as
presented previously for each bridge. For each bridge type, the ratios were multiplied by
a common multiplier that would become the structural modification factor. The average
of these modified ratios was then set to 1.0 using the EXCEL tool, “Goal Seek.” The
standard deviation was also calculated for each bridge type before and after the
modification of Henry’s method. Slight adjustments were then made in the modification
factors to minimize the standard deviations for each bridge type to ensure accurate results
compared to FEA.
6.2.1 Precast Concrete Spread Box Beam Bridges for Live Load Moment
Four precast concrete box beam bridges were included in this study. Table 6.1
shows the ratio of live load moment distribution factors from FEA to that from the
unmodified and modified Henry’s method. Preliminary modification factors were derived
140
from database #1 results as stated before. From the statistical analysis, a factor of 0.98
was determined to adequately adjust Henry’s method to fit those of FEA. Results from
the unmodified and the modified Henry’s method differ very little since Henry’s method
was already accurate compared to FEA and required very little modification. Table 6.2
shows the ratio of AASHTO LRFD to the unmodified and modified Henry’s method.
Henry’s method remains slightly conservative compared to FEA results, but not as
conservative as LRFD results.
Table 6.1 Precast Concrete Spread Box Beam, FEA vs. Modified Henry’s Method for Moment, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam FEA Henry's
Method
Ratio: FEA / Unmodified
HM
Modified Henry's
Ratio: FEA / Modified HM
1 15.0 60.88 Interior 0.767 0.826 0.93 0.813 0.94 1 15.0 60.88 Exterior 0.744 0.826 0.90 0.813 0.92 2 0.0 44.38 Exterior 1.251 1.152 1.09 1.133 1.10 3 0.0 81.46 Interior 0.867 0.759 1.14 0.747 1.16 3 0.0 81.46 Exterior 0.826 0.759 1.09 0.747 1.11 4 48.5 69.54 Interior 0.399 0.489 0.82 0.481 0.83 4 48.5 69.54 Exterior 0.453 0.489 0.93 0.481 0.94
Average 0.98 1.00Standard Deviation 0.12 0.12
Table 6.2 Precast Concrete Spread Box Beam, LRFD vs. Modified Henry’s Method for Moment, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam AASHTO
LRFD Henry's Method
Ratio: LRFD / Henry's Method
Modified Henry's
Ratio: LRFD /
Modified HM
1 15.0 60.88 Interior 0.723 0.826 0.88 0.813 0.891 15.0 60.88 Exterior 0.780 0.826 0.94 0.813 0.962 0.0 44.38 Exterior 1.186 1.152 1.03 1.133 1.053 0.0 81.46 Interior 0.752 0.759 0.99 0.747 1.013 0.0 81.46 Exterior 0.856 0.759 1.13 0.747 1.154 48.5 69.54 Interior 0.343 0.489 0.70 0.481 0.714 48.5 69.54 Exterior 0.494 0.489 1.01 0.481 1.03
Average 0.95 0.97Standard Deviation 0.14 0.14
141
Thirty-five precast concrete box beam bridges were included in database #2.
Skew angles ranged from zero to 52.8 degrees. Span lengths ranged from 29.3 ft to 134.2
ft. The number of beams ranged from 2 to 13 with a beam spacing range from 6.4 ft to
11.75 ft. Using the modified Henry’s Method, ratios between AASHTO LRFD and
Henry’s method were calculated for this database. A summary of the results for this
comparison can be seen in Table 6.3. From these results, it can be seen that the modified
Henry’s method becomes slightly more conservative. The standard deviation also
increased a negligible amount.
Table 6.3 Precast Concrete Spread Box Beam, LRFD vs. Modified Henry’s Method for Moment, Database #2
Item LRFD / Unmodified Henry's Method
LRFD / Unmodified Henry's Method
Average Ratio 1.01 1.03 Standard Deviation 0.16 0.17
Based on the analysis of both databases, it was determined that Henry’s method
needs no modification for precast concrete beams. A final structural modification factor
of 1.0 is recommended.
6.2.2 Precast Concrete Bulb-Tee Beam Bridges for Live Load Moment
Four precast concrete bulb-tee sections were analyzed in this study. Table 6.4
shows the ratio of live load moment distribution factors from FEA to those from the
unmodified and modified Henry’s method. Preliminary modification factors were
derived from database #1 results. For precast concrete bulb-tee beams, the 6/5.5
multiplier in step 2 of Henry’s method was omitted. From this statistical analysis, a
factor of 0.95 was determined to adequately adjust the distribution factors from this basic
Henry’s method to fit those of FEA. The modified Henry’s method compares much
better to FEA and has only a slightly higher standard deviation. However, Table 6.5
142
shows the comparison of the LRFD results to the modified Henry’s method. Initial
results showed the unmodified Henry’s method to yield results similar to those from
LRFD specifications. After modification though, the distribution factors from Henry’s
method became smaller. The standard deviation before and after modification remained
approximately the same.
Table 6.4 Precast Concrete Bulb-Tee Beam, FEA vs. Modified Henry’s Method for Moment, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam FEA Henry's
Method
Ratio: FEA /
Henry's Method
Modified Henry's
Ratio: FEA /
Modified HM
5 15.0 124.33 Interior 0.660 0.663 1.00 0.576 1.155 15.0 124.33 Exterior 0.625 0.663 0.94 0.576 1.098 0.0 115.49 Interior 0.812 0.790 1.03 0.686 1.188 0.0 115.49 Exterior 0.747 0.790 0.95 0.686 1.09
22 26.7 159.00 Interior 0.559 0.711 0.79 0.618 0.9122 26.7 159.00 Exterior 0.552 0.711 0.78 0.618 0.8923 17.5 151.33 Interior 0.537 0.711 0.76 0.618 0.8723 17.5 151.33 Exterior 0.511 0.711 0.72 0.618 0.83
Average 0.87 1.00Standard Deviation 0.12 0.14
Table 6.5 Precast Concrete Bulb-Tee Beam, LRFD vs. Modified Henry’s for Moment, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam AASHTO
LRFD Henry's Method
Ratio: LRFD / Henry's Method
Modified Henry's
Ratio: LRFD / Modified Henry's Method
5 15.0 124.33 Interior 0.705 0.663 1.06 0.576 1.22 5 15.0 124.33 Exterior 0.756 0.663 1.14 0.576 1.31 8 0.0 115.49 Interior 0.809 0.790 1.02 0.686 1.18 8 0.0 115.49 Exterior 0.853 0.790 1.08 0.686 1.24
22 26.7 159.00 Interior 0.649 0.711 0.91 0.618 1.05 22 26.7 159.00 Exterior 0.624 0.711 0.88 0.618 1.01 23 26.7 151.33 Interior 0.650 0.711 0.91 0.618 1.05 23 26.7 151.33 Exterior 0.625 0.711 0.88 0.618 1.01
Average 0.98 1.13Standard Deviation 0.11 0.12
143
Thirty-six precast concrete bulb-tee beam bridges were included in database #2.
Skew angles ranged from zero to 47.7 degrees. Span lengths ranged from 45 ft to 136.2
ft. The number of beams ranged from 4 to 15 with a beam spacing range from 4.21 ft to
10.5 ft. Using the modified Henry’s Method, ratios between AASHTO LRFD and
Henry’s method were calculated for this database. A summary of the results for this
comparison can be seen in Table 6.6. A complete list can be seen in Appendix B, Table
B2. Again, results indicate that the LRFD specifications yield higher distribution factors
than the unmodified and modified Henry’s method.
Based on the analysis of both databases, it seems the four bridges analyzed in this
study yielded lower distribution factors from FEA than the AASHTO LRFD methods and
Henry’s method, which compared very well. On the other hand, the results for AASHTO
LRFD methods for bridges from the larger database in NCHRP Project 12-26 yielded
higher distribution factors than Henry’s method. Therefore, a structural factor close to
the original multiplier for bulb-tee beams in Henry’s method, 1.10, was chosen. This
modification factor causes Henry’s method to yield results more conservative than FEA
but not as conservative as LRFD methods.
Table 6.6 Precast Concrete Bulb-Tee Beam, LRFD vs. Modified Henry’s Method for Moment, Database #2
Item Ratio: LRFD / Unmodified Henry's Method
Ratio: LRFD / Modified Henry's Method
Average 0.98 0.99Standard Deviation 0.14 0.14
6.2.3 Precast Concrete I-Beams for Live Load Moment
Three precast concrete I-beams were analyzed in this study. Table 6.7 shows the
ratio of FEA to the unmodified and modified Henry’s methods. Preliminary modification
factors were derived from database #1 results as stated before. For prestressed concrete
144
I-beams, the 6/5.5 multiplier in step 2 of Henry’s method was omitted. From this
statistical analysis, a factor of 1.12 was determined to adequately adjust the distribution
factors from the basic Henry’s method to fit those of FEA. The modified Henry’s
method for live load moment compares better to FEA and has a slightly lower standard
deviation. Additionally, Table 6.8 shows the comparison of the LRFD results to the
modified Henry’s method. Henry’s method also compares better to the LRFD
distribution factors after multiplying Henry’s method with a larger modification factor
than the original 6/5.5 (1.09). The standard deviation also improved by a small amount.
Table 6.7 Precast I-Beam, FEA vs. Modified Henry’s for Moment, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam FEA Henry's
Method
Ratio: FEA / Henry's Method
Modified Henry's
Ratio: FEA / Modified
Henry's Method
6 21.3 67.42 Interior 0.715 0.663 1.08 0.682 1.056 21.3 67.42 Exterior 0.689 0.663 1.04 0.682 1.017 33.5 76.00 Interior 0.721 0.663 1.09 0.682 1.067 33.5 76.00 Exterior 0.654 0.663 0.99 0.682 0.96
24 0.0 74.33 Interior 0.757 0.782 0.97 0.804 0.9424 0.0 74.33 Exterior 0.791 0.717 1.10 0.804 0.98
Average 1.04 1.00Standard Deviation 0.06 0.05
Table 6.8 Precast I-Beam, LRFD vs. Modified Henry’s for Moment, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam AASHTO
LRFD Henry's Method
Ratio: LRFD / Henry's Method
Modified Henry's
Ratio: LRFD /
Modified HM
6 21.3 67.42 Interior 0.762 0.663 1.15 0.682 1.12 6 21.3 67.42 Exterior 0.775 0.663 1.17 0.682 1.14 7 33.5 76.00 Interior 0.702 0.663 1.06 0.682 1.03 7 33.5 76.00 Exterior 0.714 0.663 1.08 0.682 1.05
24 0.0 74.33 Interior 0.849 0.782 1.09 0.804 1.06 24 0.0 74.33 Exterior 0.905 0.717 1.26 0.804 1.13
Average 1.13 1.09Standard Deviation 0.08 0.05
145
Thirty precast concrete I-beam bridges were included in database #2. Skew
angles ranged from zero to 45 degrees. Span lengths ranged from 18.75 ft to 113 ft. The
number of beams ranged from 4 to 13 with a beam spacing range from 3.67 ft to 13.08 ft.
Using the modified Henry’s Method, ratios between the AASHTO LRFD and Henry’s
method were calculated for this database. A summary of the results for this comparison
can be seen in Table 6.9. A complete list can be found in Appendix B, Table B3. Again,
results indicate that the LRFD specifications yield more conservative results than the
unmodified and modified Henry’s method. However, the difference lessens after
modification of Henry’s method. The standard deviation remains unchanged.
Based on the analysis of both databases, it was determined that Henry’s method
needs a slightly larger modification for precast concrete beams than originally. A final
structural modification factor of 1.1 is recommended.
Table 6.9 Precast Concrete I-Beam, LRFD vs. Modified Henry’s Method for Moment, Database #2
Item LRFD / Unmodified Henry's Method
LRFD / Modified Henry's Method
Average Ratio 1.13 1.10 Standard Deviation 0.15 0.15
6.2.4 Cast-In-Place Concrete T-Beam Bridges for Live Load Moment
Three cast-in-place concrete T-beams were analyzed in this study. Table 6.10
shows the live load moment distribution factor ratio of FEA to the unmodified and
modified Henry’s methods. Preliminary modification factors were derived from database
#1 results. From this statistical analysis, a factor of 1.05 was determined to adequately
adjust this basic Henry’s method to fit those of FEA. The modified Henry’s method
compares better to FEA and has an identical standard deviation. Table 6.11 shows the
146
comparison of LRFD results to the modified Henry’s method. The modified Henry’s
method also performs very well compared to the LRFD method. The standard deviation
also slightly improved by using the modification factor.
Table 6.10 Cast-In-Place Concrete T-Beam, FEA vs. Modified Henry’s for Moment, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam FEA Henry's
Method
Ratio: FEA /
Henry's Method
Modified Henry's
Ratio: FEA /
Modified Henry’s Method
9 31.6 88.50 Interior 0.877 0.869 1.01 0.913 0.969 31.6 88.50 Exterior 0.870 0.869 1.00 0.913 0.95
10 9.8 96.00 Interior 0.930 0.859 1.08 0.903 1.0310 9.8 96.00 Exterior 0.943 0.859 1.10 0.903 1.0411 0.0 66.00 Interior 0.704 0.644 1.09 0.677 1.0411 0.0 66.00 Exterior 0.658 0.644 1.02 0.677 0.97
Average 1.05 1.00Standard Deviation 0.04 0.04
Table 6.11 Cast-In-Place Concrete T-Beam, LRFD vs. Modified Henry’s for Moment, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam AASHTO
LRFD Henry's Method
Ratio: LRFD / Henry's Method
Modified Henry's
Ratio: LRFD /
Modified Henry’s Method
9 31.6 88.50 Interior 0.802 0.869 0.92 0.913 0.88 9 31.6 88.50 Exterior 0.889 0.869 1.02 0.913 0.97
10 9.8 96.00 Interior 0.913 0.859 1.06 0.903 1.01 10 9.8 96.00 Exterior 0.996 0.859 1.16 0.903 1.10 11 0.0 66.00 Interior 0.703 0.644 1.09 0.677 1.04 11 0.0 66.00 Exterior 0.676 0.644 1.05 0.677 1.00
Average 1.05 1.00Standard Deviation 0.08 0.07
Sixty-nine cast-in-place concrete T-beam bridges were included in database #2.
Skew angles ranged from zero to 45 degrees. Controlling span lengths ranged from 22.6
ft to 93 ft. The number of beams ranged from 2 to 17 with a beam spacing range from
147
2.42 ft to 16 ft. Using the modified Henry’s Method, ratios between the AASHTO LRFD
and Henry’s method were calculated for this database. A summary of the results for this
comparison can be seen in Table 6.12. A complete list can be seen in Appendix B, Table
B4. Results indicate that the LRFD specifications yield more conservative results than
both the unmodified and modified Henry’s method. However, the difference lessens after
modification of Henry’s method. The standard deviation is also reduced slightly.
Based on the analysis of both databases, it was determined that Henry’s method
needed a modification for cast-in-place concrete T-beam bridges. A final structural
modification factor of 1.05 is recommended.
Table 6.12 Cast-In-Place Concrete T-Beam, LRFD vs. Modified Henry’s Method for Moment, Database #2
Item Ratio: LRFD / Unmodified Henry's Method
Ratio: LRFD / Modified Henry's Method
Average 1.16 1.11Standard Deviation 0.18 0.17
6.2.5 Cast-In-Place Concrete Multicell Box Beam Bridges for Live Load Moment
Four cast-in-place concrete multicell box beam bridges were analyzed in this
study. Table 6.13 shows the ratio of FEA to the unmodified and modified Henry’s
method for live load moment. Preliminary modification factors were derived from
database #1 results. From this statistical analysis, a factor of 1.05 was determined to
adequately adjust the results of Henry’s method to fit those of FEA. The modified
Henry’s method compares better to FEA and has an identical standard deviation. Table
6.14 shows the comparison of LRFD results to the modified Henry’s method. The
modified Henry’s method performs very well compared to the LRFD method. The
standard deviation also slightly improved by using the modification factor.
148
Table 6.13 Cast-In-Place Concrete Multicell Box Beam, FEA vs. Modified Henry’s for Moment, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam FEA Henry's
Method
Ratio: FEA /
Henry's Method
Modified Henry's
Ratio: FEA /
Modified Henry’s Method
12 0.0 133.83 Interior 0.687 0.608 1.13 0.638 1.0813 0.0 98.75 Interior 0.620 0.608 1.02 0.638 0.9714 26.2 140.00 Interior 0.665 0.698 0.95 0.732 0.9115 16.5 110.00 Interior 0.765 0.701 1.09 0.735 1.04
Average 1.05 1.00Standard Deviation 0.08 0.07
Table 6.14 Cast-In-Place Concrete Multicell Box Beam, LRFD vs. Modified Henry’s for Moment, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam AASHTO
LRFD Henry's Method
Ratio: LRFD / Henry's Method
Modified Henry's
Ratio: LRFD /
Modified Henry’s Method
12 0.0 133.83 Interior 0.668 0.608 1.10 0.638 1.05 13 0.0 98.75 Interior 0.701 0.608 1.15 0.638 1.10 14 26.2 140.00 Interior 0.738 0.698 1.06 0.732 1.01 15 16.5 110.00 Interior 0.785 0.701 1.12 0.735 1.07
Average 1.11 1.06Standard Deviation 0.04 0.04
Eighty-two cast-in-place concrete multicell box beam bridges were included in
database #2. Skew angles ranged from zero to 45 degrees. Span lengths ranged from
22.6 ft to 93 ft. The number of beams ranged from 2 to 17 with a beam spacing range
from 2.42 ft to 16 ft. Using the modified Henry’s Method, ratios between AASHTO
LRFD and Henry’s method were calculated for this database. A summary of the results
for this comparison can be seen in Table 6.15. A complete list can be seen in Appendix
B, Table B5. Results indicate that the LRFD specifications yield slightly more
conservative results than both the unmodified and modified Henry’s method. However,
149
the difference lessens after modification of Henry’s method. The standard deviation is
identical for the modified and unmodified ratios.
Based on the analysis of both databases, it was determined that Henry’s method
needs a modification for cast-in-place concrete multicell box beam bridges. A final
structural modification factor of 1.05 is recommended.
Table 6.15 Cast-In-Place Concrete Multicell Box Beam, LRFD vs. Modified Henry’s Method for Moment, Database #2
Item Ratio: LRFD / Unmodified Henry's Method
Ratio: LRFD / Modified Henry's Method
Average 1.09 1.04Standard Deviation 0.11 0.11
6.2.6 Steel I-Beam Bridges for Live Load Moment
Four steel I-beam bridges were analyzed in this study. Table 6.16 shows the ratio
of live load moment distribution factors of FEA to the unmodified and modified Henry’s
method. For steel I-beams, the 6/5.5 multiplier in step 2 of Henry’s method was omitted.
Preliminary modification factors were derived from database #1 results. From this
statistical analysis, a slightly higher factor of 1.10 was determined to adequately adjust
the results of Henry’s method to fit those of FEA. The unmodified and modified Henry’s
method compares very well to FEA and has an identical standard deviation. Table 6.17
shows the comparison of the LRFD results to the modified Henry’s method. Due to the
negligible change in multipliers for this beam type, the modified and unmodified Henry’s
method perform identically conservative compared to the LRFD methods. Additionally,
the standard deviation remains the same.
150
Table 6.16 Steel I-Beam, FEA vs. Modified Henry’s for Moment, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam FEA Henry's
Method
Ratio: FEA /
Henry's Method
Modified Henry's
Ratio: FEA / Modified
Henry's Method
16 0.0 158.00 Interior 0.690 0.695 0.99 0.692 1.0016 0.0 158.00 Exterior 0.653 0.695 0.94 0.692 0.9417 19.5 143.00 Interior 0.749 0.851 0.88 0.847 0.8817 19.5 143.00 Exterior 0.842 0.851 0.99 0.847 0.9918 50.2 182.00 Interior 0.857 0.828 1.04 0.824 1.0418 50.2 182.00 Exterior 0.906 0.828 1.09 0.824 1.1019 26.7 150.00 Interior 0.830 0.822 1.01 0.818 1.0119 26.7 150.00 Exterior 0.835 0.822 1.02 0.818 1.02
Average 0.99 1.00Standard Deviation 0.06 0.06
Table 6.17 Steel I-Beam, LRFD vs. Modified Henry’s for Moment, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam AASHTO
LRFD Henry's Method
Ratio: LRFD / Henry's Method
Modified Henry's
Ratio: LRFD /
Modified Henry's Method
16 0.0 158.00 Interior 0.661 0.695 0.95 0.692 0.9616 0.0 158.00 Exterior 0.711 0.695 1.02 0.692 1.0317 19.5 143.00 Interior 0.650 0.851 0.76 0.847 0.7717 19.5 143.00 Exterior 0.790 0.851 0.93 0.847 0.9318 50.2 182.00 Interior 0.696 0.828 0.84 0.824 0.8418 50.2 182.00 Exterior 0.848 0.828 1.02 0.824 1.0319 26.7 150.00 Interior 0.724 0.822 0.88 0.818 0.8919 26.7 150.00 Exterior 0.659 0.822 0.80 0.818 0.81
Average 0.90 0.91Standard Deviation 0.10 0.10
One hundred forty-eight steel I-beam bridges were included in database #2. Skew
angles ranged from zero to 66.1 degrees. Span lengths ranged from 27 ft to 205 ft. The
number of beams ranged from 3 to 27 with a beam spacing range from 2 ft to 16 ft.
Using the modified Henry’s Method, ratios between the AASHTO LRFD and Henry’s
method were calculated for this database. A summary of the results for this comparison
can be seen in Table 6.18. A complete list can be seen in Appendix B, Table B6. Results
151
indicate that the LRFD specifications yield similar results to both the unmodified and
modified Henry’s method. Since there is very little difference in the new modification
factor and the original 6/5.5 factor from Henry’s method, differences in the average and
standard deviation are negligible.
Based on the analysis of both databases, it was determined that Henry’s method
needs a modification for steel I-beam bridges very similar to its original value. A final
structural modification factor of 1.10 is recommended.
Table 6.18 Steel I-Beam, LRFD vs. Modified Henry’s Method for Moment, Database #2
Item Ratio: LRFD / Unmodified Henry's Method
Ratio: LRFD / Modified Henry's Method
Average 0.98 0.99Standard Deviation 0.14 0.14
6.2.7 Steel Open Box Beam Bridges for Live Load Moment
Two steel open box beam bridges were analyzed in this study. Table 6.19 shows
the ratio of FEA to the unmodified and modified Henry’s method. Preliminary live load
moment modification factors were derived from database #1 results. From this statistical
analysis, it was determined that Henry’s method required no modification to fit results
from FEA. Table 6.20 shows the comparison of the LRFD results to Henry’s method.
Henry’s method yields more conservative results than the LRFD specifications for this
type of beam.
152
Table 6.19 Steel Open Box Beam, FEA vs. Modified Henry’s for Moment, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam FEA Henry's
Method
Ratio: FEA /
Henry's Method
Modified Henry's
Ratio: FEA / Modified
Henry's Method
20 32.0 252.00 Interior 0.641 0.701 0.91 0.701 0.9120 32.0 252.00 Exterior 0.641 0.701 0.91 0.701 0.9121 4.5 170.67 Interior 0.630 0.606 1.04 0.606 1.0421 4.5 170.67 Exterior 0.685 0.606 1.13 0.606 1.13
Average 1.00 1.00Standard Deviation 0.11 0.11
Table 6.20 Steel Open Box Beam, LRFD vs. Modified Henry’s for Moment, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam AASHTO
LRFD Henry's Method
Ratio: LRFD / Henry's Method
Modified Henry's
Ratio: LRFD /
Modified Henry's Method
20 32.0 252.00 Interior 0.556 0.701 0.79 0.701 0.79 20 32.0 252.00 Exterior 0.556 0.701 0.79 0.701 0.79 21 4.5 170.67 Interior 0.645 0.606 1.06 0.606 1.06 21 4.5 170.67 Exterior 0.645 0.606 1.06 0.606 1.06
Average 0.93 0.93Standard Deviation 0.16 0.16
Nineteen steel open box beam bridges were included in database #2. Skew angles
ranged from zero to 60.5 degrees. Span lengths ranged from 61.8 ft to 281.7 ft. The
number of beams ranged from 2 to 7 with a beam spacing range from 7.75 ft to 24 ft.
Using the modified Henry’s Method, ratios between the AASHTO LRFD and Henry’s
method were calculated for this database. A summary of the results for this comparison
can be seen in Table 6.21. A complete list can be seen in Appendix B, Table B7. Results
indicate that the LRFD specifications yield similar results to both the unmodified and
153
modified Henry’s method. Results from the modified Henry’s method are identical to the
unmodified method due to the nearly identical multipliers.
Based on the analysis of database #1, it was determined that Henry’s method
needs no modification for steel open box beam bridges. Henry’s method compares well
to FEA without modification and slightly conservative compared to the AASHTO LRFD
methods. However, the bridges from database #2 were not included in the Project 12-26
analysis to develop the LRFD specifications. As shown in Chapter 3, the LRFD method
is based on the inaccurate AASHTO Standard method. Therefore, a final structural
modification factor of 1.0 is recommended for steel open box beam bridges.
Table 6.21 Steel Open Box Beam, LRFD vs. Modified Henry’s Method for Moment, Database #2
ItemRatio: LRFD /
Unmodified Henry's Method
Ratio: LRFD / Modified Henry's
Method Average 0.92 0.92
Standard Deviation 0.08 0.08
6.3 Summary of Set 1 Modification Factors for Live Load Moment
Table 6.22 shows the final live load moment modification factors for each bridge
type. Each of these factors will enable Henry’s method to produce accurate results for its
respective bridge type. The original factors in Henry’s method, applied in step 2, are also
listed in the table. Figure 6.1 shows the frequency of ratios for database #1 compared to
the unmodified and modified Henry’s method. The modified Henry’s method typically
produces better results than the unmodified Henry’s method compared to FEA. The
exception to this are the lower bound values of the precast concrete bulb-tee beams as
noted before. The modified Henry’s method has an increased number of distribution
factors with a ratio equal within 0.05 to FEA results and a reduction of unconservative
154
and over conservative results. Table 6.23 shows the average ratio and standard deviation
for each beam type. The average ratio for each beam type is typically very close to 1.0
with the exception to the precast concrete bulb-tee beams. The standard deviation is
reasonably small with a typical value near 0.1 or less.
Table 6.22 Final Structure Type Modification Factors for Live Load Moment (Set 1)
Structure Type Modification Factor Original Factor in Henry’s Method
Precast Spread Box Beam 1.00 1.00 Precast Concrete BT Beam 1.10 1.09 Precast Concrete I-Beam 1.10 1.09 CIP Concrete T-Beam 1.05 1.00 CIP Concrete Box Beam 1.05 1.00 Steel I-Beam 1.10 1.09 Steel Open Box Beam 1.00 1.00
Table 6.23 FEA vs. Modified Henry’s Method for Live Load Moment (Set 1), Database #1
Structure Type Average Ratio: FEA / Modified Henry's Method
Standard Deviation
Precast Spread Box Beam 1.00 0.12 Precast Concrete BT Beam 1.00 0.14 Precast Concrete I-Beam 1.00 0.05 CIP Concrete T-Beam 1.00 0.04 CIP Concrete Box Beam 1.00 0.07 Steel I-Beam 1.00 0.06 Steel Open Box Beam 1.00 0.11
155
Database #1 Histogram
0
5
10
15
20
25
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 More
Ratio
Freq
uenc
y
Ratio: FEA / Unmodified Henry's MethodRatio: FEA / Modified Henry's Method (Final Set 1)
Figure 6.1 Histogram of Moment Distribution Factor (Set 1), Database #1
6.4 Preliminary Modification Factors (Set 2) for Live Load Moment
The second set of preliminary modification factors for live load moment includes
modification factors for structure types, skew angle, and span length. These factors are
also based on numerical analysis from Chapter 5. In order to obtain skew angle
modification factors, all bridges with an angle of skew greater than 30 degrees were
placed in a separate database. These bridges were then multiplied by a skew
modification factor. This modification factor was then changed using the “goal seek”
function to obtain an average ratio as close to 1.0 as possible. A skew modification factor
of 0.94 was determined in this manner. Similarly, all bridges with span lengths greater
than 100 ft were placed in a separate database. These bridges were then multiplied by a
length modification factor. This modification factor was also changed using the “goal
seek” function to obtain an average ratio as close to 1.0 as possible. Similar to the skew
156
modification factor, a length modification factor of 0.94 was determined. With these
skew and length modification factors multiplied to appropriate bridges in the entire
database, the structural modification factors were obtained. Again, the goal was to have
average ratios close to 1.0 and standard deviations close to 0.1. Adjustments were made
in the structural modification factors to minimize the standard deviations for each bridge
type to ensure accurate results compared to FEA. The preliminary modification factors
for live load moment for each bridge type are listed in Table 6.24.
Each modification factor was applied to databases #1 and #2. The resulting
averages and standard deviations for each bridge are listed in Tables 6.25 and 6.26.
Again, the average ratio in database #1 for each beam type with these preliminary
modification factors is 1.0. Standard deviations for this modification factor set are
approximately the same as the results for modification factor set 1. Table 6.27 compares
the results for database #2 between the LRFD and the modified Henry’s method. The
data from Tables 6.23 and 6.25 as well as the ones for the unmodified Henry’s method
are presented in Figure 6.2. Using the second set of modification factors, the number of
ratios of FEA results to the modified Henry’s results between 0.95 and 1.05 is increased
slightly compared to results using the first set of modification factors. The number of
unconservative and conservative values is also reduced.
157
Table 6.24 Preliminary Modification Factors for Live Load Moment (Set 2)
Structure Type Modification Factor Precast Spread Box Beam 1.00 Precast Concrete BT Beam 1.01 Precast Concrete I-Beam 1.15 CIP Concrete T-Beam 1.07 CIP Concrete Box Beam 1.12 Steel I-Beam 1.17 Steel Open Box Beam 1.09
θ < 30° θ > 30°Skew Modification Factor 1.00 0.94
L < 100 ft L > 100 ft Length Modification Factor 1.00 0.94
Table 6.25 FEA vs. Modified Henry’s Method for Live Load Moment (Set 2), Database #1
Structure Type Average Ratio: FEA / Modified Henry's Method
Standard Deviation
Precast Spread Box Beam 1.00 0.11 Precast Concrete BT Beam 1.00 0.14 Precast Concrete I-Beam 1.00 0.06 CIP Concrete T-Beam 1.00 0.03 CIP Concrete Box Beam 1.00 0.07 Steel I-Beam 1.00 0.09 Steel Open Box Beam 1.00 0.07
Table 6.26 LRFD vs. Modified Henry’s Method for Live Load Moment (Set 2), Database #1
Structure Type Average Ratio:
LRFD / Modified Henry's Method
Standard Deviation
Precast Spread Box Beam 0.95 0.14 Precast Concrete BT Beam 0.98 0.11 Precast Concrete I-Beam 1.13 0.08 CIP Concrete T-Beam 1.05 0.08 CIP Concrete Box Beam 1.11 0.04 Steel I-Beam 0.90 0.10 Steel Open Box Beam 0.93 0.16
158
Table 6.27 LRFD vs. Modified Henry’s Method for Live Load Moment (Set 2), Database #2
Structure Type Average Ratio: LRFD / Modified Henry's Method
Standard Deviation
Precast Spread Box Beam 0.95 0.14 Precast Concrete BT Beam 0.98 0.11 Precast Concrete I-Beam 1.13 0.08 CIP Concrete T-Beam 1.05 0.08 CIP Concrete Box Beam 1.11 0.04 Steel I-Beam 0.90 0.10 Steel Open Box Beam 0.93 0.16
Database #1 Histogram
0
5
10
15
20
25
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 More
Ratio
Freq
uenc
y
Ratio: FEA / Unmodified Henry's MethodRatio: FEA / Modified Henry's Method (Set 1)Ratio: FEA / Modified Henry's Method (Set 2)
Figure 6.2 Histogram of Moment Distribution Factors (Sets 1 and 2), Database #1
Using the same strategy as in developing the preliminary modification factor set
1, the preliminary modification factor set 2 were adjusted by incorporating the analytical
results of both database #1 and database #2. The preliminary modification factor for
precast concrete bulb-tee beams is increased to 1.15 to match steel I-beams and precast
concrete I-beams based on the analysis of database #2. Cast-in-place concrete T-beam,
159
concrete cast-in-place multicell box beam, and steel open box beam modification factors
were set to be 1.10. Henry’s method for concrete spread box beams still required no
modification. Table 6.28 shows the final modification factors for set 2 and Figure 6.3
shows the histogram with the unmodified Henry’s method, and sets 1 and 2 of the final
modified methods. As before, the modified Henry’s method with modification factor set
2 produces much better results than the method with modification factor set 1 and the
unmodified Henry’s method. More so than any other method, a higher percentage of
results fell between 0.95 and 1.05 using modification set 2. As in Figures 6.1 and 6.2, the
overly conservative values are due to the bulb-tee beam results.
Table 6.29 shows the final ratios of FEA to the modified Henry’s method using
both sets of modification factors. From Table 6.29, it can be seen that using either set of
modification factors with Henry’s method generates accurate and reliable results
compared to finite element analysis. Additionally, Table 6.30 shows the modified
Henry’s methods compared to the AASHTO LRFD methods. Again, the simpler
modified Henry’s method yields similar results to those of the LRFD methods.
Table 6.28 Final Modification Factors for Live Load Moment (Set 2)
Structure Type Modification Factor Precast Spread Box Beam 1.00 Precast Concrete BT Beam 1.15 Precast Concrete I-Beam 1.15 CIP Concrete T-Beam 1.10 CIP Concrete Box Beam 1.10 Steel I-Beam 1.15 Steel Open Box Beam 1.10
θ < 30° θ > 30°Skew Modification Factor 1.00 0.94
L < 100 ft L > 100 ft Length Modification Factor 1.00 0.94
160
Database #1 Histogram
0
5
10
15
20
25
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 More
Ratio
Freq
uenc
y
Ratio: FEA / Unmodified Henry's MethodRatio: FEA / Modified Henry's Method (Set 1)Ratio: FEA / Modified Henry's Method (Set 2)
Figure 6.3 Histogram of Moment Distribution Factors (Final Sets 1 and 2)
Table 6.29 Summary of FEA vs. Modified Henry’s Method for Moment (Final Sets 1 & 2)
Structure Type
Average Ratio: FEA / Modified Henry's
Method (Set 1)
Standard Deviation
(Set 1)
Average Ratio: FEA / Modified Henry's
Method (Set 2)
Standard Deviation
(Set 2)
Precast Spread Box Beam 0.98 0.12 1.00 0.11 Precast Concrete BT Beam 0.86 0.12 0.88 0.12 Precast Concrete I-Beam 1.02 0.05 1.00 0.06 CIP Concrete T-Beam 1.00 0.04 0.97 0.03 CIP Concrete Box Beam 1.00 0.07 1.01 0.08 Steel I-Beam 0.98 0.06 1.02 0.09 Steel Open Box Beam 1.00 0.11 1.00 0.07
161
Table 6.30 Summary of LRFD vs. Modified Henry’s Method for Moment (Final Sets 1 & 2)
Structure Type
Average Ratio: LRFD / Modified Henry's Method
(Set 1)
Standard Deviation
(Set 1)
Average Ratio: LRFD / Modified Henry's Method
(Set 2)
Standard Deviation
(Set 2)
Precast Spread Box Beam 0.95 0.14 0.97 0.13 Precast Concrete BT Beam 0.98 0.10 0.99 0.11 Precast Concrete I-Beam 1.11 0.05 1.08 0.03 CIP Concrete T-Beam 1.00 0.07 0.98 0.05 CIP Concrete Box Beam 1.05 0.04 1.07 0.04 Steel I-Beam 0.89 0.10 0.93 0.11 Steel Open Box Beam 0.93 0.16 0.92 0.12
The difference between the two sets of modification factors for live load moment
can be seen in Figures 6.4 and 6.5. Figure 6.4 is a graph of skew angle versus moment
distribution factor for the AASHTO LRFD, the unmodified Henry’s method, the
modified Henry’s method using set 1 modification factors, the modified Henry’s method
using set 2 modification factors, and FEA. As Figure 6.4 shows, all five methods follow
the same general trend. The modified Henry’s method, using set 2 multipliers, performs
more closely to FEA than any other method as skew angles increase. Figure 6.5 shows
the relationship between span length and live load moment distribution factor for each
method. For all span lengths, the modified Henry’s method, using set 2 modification
factors, followed FEA more closely than any other method. Where the AASHTO LRFD
method became unconservative for the longer span bridges, the modified Henry’s method
remained slightly conservative.
162
Distribution Factor vs Skew Angle
0.200
0.400
0.600
0.800
1.000
1.200
1.400
0.0 10.0 20.0 30.0 40.0 50.0 60.0
Skew Angle (degrees)
Mom
ent D
istri
butio
n Fa
ctor
LRFD Modif ied Henry's (Set 1)Unmodif ied Henry's FEAModif ied Henry's (Set 2) Linear (Modif ied Henry's (Set 1))Linear (LRFD) Linear (FEA)Linear (Unmodif ied Henry's) Linear (Modif ied Henry's (Set 2))
Figure 6.4 Moment Distribution Factor vs. Skew Angle (Set 2), Database #1
Distribution Factor vs Span Length
0.200
0.400
0.600
0.800
1.000
1.200
1.400
0.00 50.00 100.00 150.00 200.00 250.00 300.00
Span Length (ft)
Mom
ent D
istri
butio
n Fa
ctor
LRFD Modified Henry's (Set 1)Unmodified Henry's FEAModified Henry's (Set 2) Linear (Modified Henry's (Set 1))Linear (Unmodified Henry's) Linear (LRFD)Linear (FEA) Linear (Modified Henry's (Set 2))
Figure 6.5 Moment Distribution Factor vs. Span Length (Set 2), Database #1
163
6.5 Preliminary Modification Factors (Set 1) for Live Load Shear
The procedure of developing modification factors for Henry’s method for live
load shear is described in this section. The development of the preliminary modification
factors was initiated by developing the modification factors for structure types. These
factors were developed by performing statistical analysis of the shear distribution factors
obtained from Henry’s method versus that from finite element analysis. Two goals were
set for the statistical analysis: the average ratio of the distribution factors from the FEA to
the Henry’s method was close to unity and the standard deviation was close to 0.1 or as
small as possible. For each bridge type a separate bridge spreadsheet was created based
on their structure type. Then the ratios were multiplied with a common multiplier. This
common multiplier was found by setting the average of modified ratios of FEA/Henry’s
method to 1.0 using the EXCEL tool “Goal Seek”. The common multiplier so obtained
would be the structural modification factor for that particular type of bridge. Slight
adjustments can be made to the modification factors in order to further minimize the
standard deviation. Once the structure modification factors are obtained the modification
for shear distribution was further simplified. The simplification included the introduction
of a single shear factor in conjuncture with the use of moment modification factors of
structure type developed in previous section. The incorporation of moment modification
factors and a single shear factor would present fairly good accuracy as the preliminary
modification factors brought in. As a result, the final shear modification factor for each
bridge type is equal to the structure type factor for live load moment multiplied by the
single shear factor.
164
6.5.1 Precast Concrete Spread Box Beam Bridges for Live Load Shear
Table 6.31 shows the ratios of FEA results versus Henry’s method results for live
load shear. From a preliminary statistical analysis it was found that a factor of 1.09 was
needed to adjust the shear distribution factors from Henry’s method to fit in the FEA
results. Table 6.31 also shows the results of the modified Henry’s method and
corresponding ratios. It can be seen that after modification, the average modified ratio
was 1.0 and the standard deviation was 0.14. Table 6.32 shows a comparison of Henry’s
method as well as the modified Henry’s method to the LRFD method for live load shear
for this type of bridge. From Table 6.32, it can be seen that the modified Henry’s method
produced the distribution factors closer to the LRFD results than those before
modification but still much smaller than the LRFD results. This outcome implied that
some more increase on Henry’s distribution factors might be necessary. However,
because the LRFD method for shear was conservative, the modified Henry’s method did
not need to be too close to the LRFD method.
Table 6.31 Precast Concrete Spread Box Beam, FEA vs. Henry’s Method for Shear, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam FEA Henry's
Method
Ratio: FEA /
Henry’s Method
Modified Henry’s Method
Ratio: FEA / Modified Henry’s Method
1 15.0 60.89 Interior 0.858 0.826 1.039 0.896 0.9571 15.0 60.89 Exterior 0.815 0.826 0.987 0.896 0.9092 0.0 44.38 Exterior 1.080 1.152 0.937 1.251 0.8633 0.0 81.46 Interior 1.010 0.759 1.331 0.824 1.2263 0.0 81.46 Exterior 0.933 0.759 1.229 0.824 1.1334 48.5 69.54 Interior 0.452 0.489 0.919 0.534 0.8474 48.5 69.54 Exterior 0.568 0.489 1.154 0.534 1.064
Average 1.085 1.000Standard Deviation 0.157 0.144
165
1.030 1.257 0.9443 0.0 81.46 Interior 1.027 0.759 1.353 0.827 1.2413 0.0 81.46 Exterior 1.168 0.759 1.539 0.827 1.4124 48.5 69.54 Interior 0.812 0.489 0.536 1.5144 48.5 69.54 Exterior 0.798 0.489 1.622 0.536 1.488
Average 1.322 1.213Standard Deviation 0.248 0.224
To further consider modification, the database #2 was used. Thirty-five precast
concrete box beam bridges were included in database #2, as described in the moment
modification section. Ratios between the AASHTO LRFD and modified as well as
unmodified Henry’s method were calculated for this database. A summary of the results
for this comparison can be seen in Table 6.33. Again a large difference was observed
between the results from Henry’s method and the LRFD method. This fact indicated that
Henry’s method would need more adjustment. Based on the analysis of both databases a
preliminary modification factor of 1.12 was proposed.
Table 6.33 Precast Concrete Spread Box Beam, LRFD vs. Henry’s Method for Shear, Database #2
Item Ratio: LRFD / UnmodifiedHenry's Method
Ratio: LRFD / Modified Henry's Method
Average Ratio 1.45 1.33 Standard Deviation 0.26 0.24
Table 6.32 Precast Concrete Spread Box Beam, LRFD vs. Henry’s Method for Shear, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam AASHTO
LRFD Henry's Method
Ratio: LRFD / Henry’s Method
Ratio: LRFD /
Modified Henry’s Method
1 15.0 60.89 1.016 0.826 1.230 0.900 1.1281 60.89 Exterior 0.932 0.826 1.128 0.900 1.0352 0.0 44.38 Interior 1.186 1.152 1.030 0.9442 0.0 44.38 Exterior 1.186
Modified Henry’s Method
Interior15.0
1.2571.152
1.650
166
6.5.2 Precast Concrete Bulb-Tee Beam Bridges for Live Load Shear
Table 6.34 shows the ratios of shear distribution factors of FEA to Henry’s
method and to the modified Henry’s method for precast concrete bulb-tee beam bridges.
Based on statistical analysis for database #1 it was determined that a modification factor
of 1.18 was needed for this type of bridge. As shown in Table 6.34 the modified Henry’s
method compared much closer to FEA. The average ratio of FEA versus Henry’s method
was improved from 1.179 before modification to 1.00 after modification. The standard
deviation of the ratios was also improved with the modified Henry’s method.
Table 6.34: Precast Concrete Bulb-Tee Beam, FEA vs. Henry’s Method for Shear, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam FEA Henry's
Method
Ratio: FEA / Henry’s
Method
Modified Henry’s Method
Ratio: FEA / Modified
Henry’s Method
5 15.0 124.33 Interior 0.931 0.663 1.404 0.781 1.1915 15.0 124.33 Exterior 0.730 0.663 1.101 0.781 0.9348 0.0 115.49 Interior 0.960 0.790 1.215 0.931 1.0318 0.0 115.49 Exterior 0.784 0.790 0.992 0.931 0.842
22 26.7 159.00 Interior 0.933 0.711 1.314 0.837 1.11522 26.7 159.00 Exterior 0.756 0.711 1.065 0.837 0.90323 26.7 151.33 Interior 0.932 0.711 1.313 0.837 1.11423 26.7 151.33 Exterior 0.727 0.711 1.024 0.837 0.869
Average 1.179 1.000Standard Deviation 0.154 0.131
Table 6.35 shows the comparison of the modified Henry’s method with the LRFD
method for live load shear. The initial ratio of LRFD versus Henry’s method results was
1.160, indicating a larger difference between the shear distribution factors predicted by
the LRFD method and Henry’s method. After modification Henry’s method yielded
slightly smaller distribution factors for shear than the LRFD method for this type of
bridge. The standard deviation was also improved.
167
Table 6.35: Precast Concrete Bulb-Tee Beam, LRFD vs. Henry’s Method for Shear, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam AASHTO
LRFD Henry's Method
Ratio: LRFD / Henry’s Method
Modified Henry’s Method
Ratio: LRFD / Modified
Henry’s Method
5 15.0 124.33 Interior 0.900 0.663 1.357 0.776 1.1605 15.0 124.33 Exterior 0.788 0.663 1.189 0.776 1.0168 0.0 115.49 Interior 0.970 0.790 1.228 0.924 1.0498 0.0 115.49 Exterior 0.833 0.790 1.054 0.924 0.901
22 26.7 159.00 Interior 0.902 0.711 1.270 0.831 1.08622 26.7 159.00 Exterior 0.699 0.711 0.985 0.831 0.84123 26.7 151.33 Interior 0.877 0.711 1.235 0.831 1.05623 26.7 151.33 Exterior 0.680 0.711 0.958 0.831 0.819
Average 1.160 0.991Standard Deviation 0.144 0.123
Thirty-six precast concrete bulb-tee beam bridges were included in database #2.
Ratios between the AASHTO LRFD and both the unmodified and modified Henry’s
method were calculated for this database. A summary of the results for this comparison
can be seen in Table 6.36. From the table it can be seen that the unmodified Henry’s
method produced smaller shear distribution factors compared to the LRFD method for
bridges in database #2 also. After modification, the Henry’s method yielded distribution
factors very close to those from the LRFD method for shear.
Table 6.36: Precast Concrete Bulb-Tee Beam, LRFD vs. Henry’s Method for Shear, Database #2
Item Ratio: LRFD / Unmodified Henry's Method
Ratio: LRFD / Modified Henry's Method
Average 1.23 1.04Standard Deviation 0.24 0.20
Based on these two databases a preliminary modification factor of 1.15 was
introduced for these types of bridges for shear. The 6/5.5 multiplier for precast concrete
bulb-tee beams in step 2 of Henry’s method was included in the analysis.
168
6.5.3 Precast Concrete I-Beam Bridges for Live Load Shear
There were three bridges of this type in Database #1. Analysis of this database,
based on a comparison to the finite element results, revealed that the Henry’s method was
unconservative in predicting shear distribution factors. Table 6.37 shows the ratios of
FEA to the unmodified and modified Henry’s methods. To correct the Henry’s method a
preliminary modification factor of 1.15 was determined to adequately adjust the shear
distribution factors from Henry’s method to fit those of FEA. As can be seen from Table
6.37, the average ratio after modification was adjusted to 1.0 and the standard deviation
was improved. Table 6.38 shows the comparison of Henry’s method to the LRFD
method. It was also found that the modified Henry’s method yielded distribution factors
closer to the LRFD results. For precast concrete I-beams, the 6/5.5 multiplier in step 2 of
Henry’s method was included.
Table 6.37 Precast I-Beam, FEA vs. Henry’s Method for Shear, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam FEA Henry's
Method
Ratio: FEA / Henry’s
Method
Modified Henry’s Method
Ratio: FEA / Modified Henry’s Method
6 21.3 67.42 Interior 0.917 0.663 1.385 0.763 1.2036 21.3 67.42 Exterior 0.677 0.663 1.023 0.763 0.8887 33.5 76.00 Interior 0.770 0.663 1.163 0.763 1.0107 33.5 76.00 Exterior 0.700 0.663 1.057 0.763 0.918
24 0.0 75.00 Interior 0.940 0.782 1.205 0.898 1.04624 0.0 75.00 Exterior 0.841 0.782 1.078 0.898 0.936
Average 1.152 1.000Standard Deviation 0.133 0.115
169
Table 6.38 Precast I-Beam, LRFD vs. Henry’s Method for Shear, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam AASHTO
LRFD Henry's Method
Ratio: LRFD / Henry’s Method
Modified Henry’s Method
Ratio: LRFD / Modified
Henry’s Method
6 21.33 67.42 Interior 0.940 0.663 1.420 0.775 1.2146 21.33 67.42 Exterior 0.776 0.663 1.172 0.775 1.0027 33.50 76.00 Interior 0.983 0.663 1.485 0.775 1.2697 33.50 76.00 Exterior 0.811 0.663 1.225 0.775 1.047
24 0.00 75.00 Interior 0.990 0.782 1.269 0.913 1.08424 0.00 75.00 Exterior 0.861 0.782 1.103 0.913 0.943
Average 1.279 1.093Standard Deviation 0.208 0.178
Thirty precast concrete I-beam bridges were included in database #2. Using the
modified Henry’s Method, ratios between the AASHTO LRFD and Henry’s method were
calculated for this database. A summary of the results for this comparison can be seen in
Table 6.39. Again, results indicate that the LRFD specifications yield more conservative
results than the unmodified and modified Henry’s method. However, the difference
lessens after the modification of Henry’s method. The precast concrete bulb-tee and
precast concrete I-beam bridges were combined together to have the same modification
factors because of their similarities. From the analysis of both databases it was found that
a preliminary modification factor of 1.15 was suitable for these types of bridges.
Table 6.39 Precast I-Beam, LRFD vs. Henry’s Method for Shear, Database #2
Item Ratio: LRFD / Unmodified Henry's Method
Ratio: LRFD / Modified Henry's Method
Average 1.23 1.04Standard Deviation 0.24 0.20
6.5.4 Cast-In-Place Concrete T-Beam Bridges for Live Load Shear
Three cast-in-place concrete T-beam bridges were studied and analyzed. Based on
the comparison in Table 6.40, a modification factor of 1.08 was introduced to adjust the
170
Henry’s method to fit with the FEA results for this type of bridge for live load shear. This
factor was developed based on an analysis on database #1. As seen in the table, the
modified Henry’s method compares better with the FEA results. Table 6.41 compares the
Henry’s method to the LRFD method. As can be seen, after modification, Henry’s
method compared quite well with the LRFD method.
Table 6.40 CIP T-Beam, FEA vs. Henry’s Method for Shear, Database #1
Bridge No.
Skew Angle(deg)
Span Length
(ft) Beam FEA Henry's
Method
Ratio: FEA /
Henry’s Method
Modified Henry’s Method
Ratio: FEA /
Modified Henry’s Method
9 31.6 88.48 Interior 0.911 0.869 1.048 0.967 0.9429 31.6 88.48 Exterior 0.762 0.869 0.877 0.967 0.788
10 9.8 96.00 Interior 1.090 0.859 1.269 0.928 1.17510 9.8 96.00 Exterior 0.956 0.859 1.113 0.928 1.03011 0.0 66.00 Interior 0.770 0.644 1.196 0.696 1.10711 0.0 66.00 Exterior 0.665 0.644 1.033 0.696 0.956
Average 1.089 1.000Standard Deviation 0.137 0.136
Table 6.41 CIP T-Beam, LRFD vs. Henry’s Method for Shear, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam AASHTO
LRFD Henry's Method
Ratio: LRFD / Henry’s Method
Modified Henry’s Method
Ratio: LRFD /
Modified Henry’s Method
9 31.6 88.48 Interior 0.942 0.869 1.084 0.967 0.9759 31.6 88.48 Exterior 0.784 0.869 0.902 0.967 0.811
10 9.8 96.00 Interior 0.969 0.859 1.128 0.928 1.04410 9.8 96.00 Exterior 0.863 0.859 1.005 0.928 0.93011 0.0 66.00 Interior 0.826 0.644 1.283 0.696 1.18811 0.0 66.00 Exterior 0.640 0.644 0.994 0.696 0.920
Average 1.066 0.978 Standard Deviation 0.132 0.117
Sixty-nine cast-in-place concrete T-beam bridges were included in database #2.
Using the modified Henry’s Method, ratios between the AASHTO LRFD and Henry’s
171
method were calculated for this database. A summary of the results for this comparison
can be seen in Table 6.42. Results indicated that the AASHTO LRFD specification
yields more conservative results than both the unmodified and modified Henry’s method.
It seems that a factor larger than the one suggested by the preliminary analysis would be
helpful. For this reason a preliminary modification factor of 1.12 was proposed based on
the two databases for this type of bridge.
Table 6.42 CIP T-Beam LRFD vs. Henry’s Method for Shear, Database #2
Item Ratio: LRFD / Unmodified Henry's Method
Ratio: LRFD / Modified Henry's Method
Average 1.25 1.16Standard Deviation 0.25 0.23
6.5.5 Cast-In-Place Concrete Multicell Box Beam Bridges for Live Load Shear
Four cast-in-place concrete multicell box beam bridges were analyzed in this
study. Table 6.43 shows the comparison of Henry’s method to the finite element method.
Based on this comparison a preliminary modification factor of 1.25 was introduced for
this type of bridge for live load shear. As seen in the table, the standard deviation was
reduced after modification and the average ratio also brought closer to 1. Table 6.44
shows the comparison of Henry’s method to the AASHTO LRFD method. After
modification the average ratio of LRFD results versus Henry’s method results was
improved and the standard deviation reduced. It seems that the modified Henry’s method
still yielded smaller shear distribution factors than the AASHTO LRFD method.
172
Table 6.43 CIP Concrete Multicell Box Beam, FEA vs. Henry’s Method for Shear, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam FEA Henry's
Method
Ratio: FEA / Henry’s
Method
Modified Henry’s Method
Ratio: FEA /
Modified Henry’s Method
12 0.0 133.83 Interior 0.842 0.608 1.385 0.764 1.10112 0.0 133.83 Exterior 0.651 0.608 1.071 0.764 0.85213 0.0 98.75 Interior 0.856 0.608 1.410 0.763 1.12213 0.0 98.75 Exterior 0.645 0.608 1.063 0.763 0.84514 26.2 140.00 Interior 0.975 0.698 1.397 0.878 1.11114 26.2 140.00 Exterior 0.866 0.698 1.241 0.878 0.98715 16.5 110.00 Interior 0.883 0.701 1.261 0.880 1.00315 16.5 110.00 Exterior 0.851 0.701 1.216 0.880 0.967
Average 1.255 0.999Standard Deviation 0.138 0.110
Table 6.44 CIP Concrete Multicell Box Beam, LRFD vs. Henry’s Method for Shear, Database #1
Bridge No.
Skew Angle
Span Length Beam AASHTO
LRFD Henry's Method
Ratio: LRFD / Henry’s Method
Modified Henry’s Method
Ratio: LRFD /
Modified Henry’s Method
12 0.0 133.83 Interior 0.899 0.608 1.479 0.760 1.18312 0.0 133.83 Exterior 0.701 0.608 1.153 0.760 0.92213 0.0 98.75 Interior 0.900 0.608 1.483 0.759 1.18613 0.0 98.75 Exterior 0.738 0.608 1.216 0.759 0.97314 26.2 140.00 Interior 1.280 0.698 1.834 0.873 1.46714 26.2 140.00 Exterior 1.084 0.698 1.553 0.873 1.24215 16.5 110.00 Interior 1.086 0.701 1.551 0.875 1.24115 16.5 110.00 Exterior 0.868 0.701 1.240 0.875 0.992
Average 1.439 1.151Standard Deviation 0.226 0.180
Eighty-two cast-in-place concrete multicell box beam bridges were included in
database #2. The ratios of the AASHTO LRFD distribution factors for shear versus
Henry’s method and modified Henry’s method distribution factors were calculated for
this database. A summary of the results for this comparison can be seen in Table 6.45.
173
Results indicate that the LRFD specifications yield much higher shear distribution factors
than the unmodified Henry’s method. After modification, Henry’s method compared
very well to the LRFD method for database #2. The standard deviation is also reduced
after modification.
Table 6.45 CIP Concrete Multicell Box Beam, LRFD vs. Henry’s Method for Shear, Database #2
Item Ratio: LRFD / Unmodified Henry's Method
Ratio: LRFD / Modified Henry's Method
Average 1.26 1.01Standard Deviation 0.26 0.21
Based on the analysis of two databases and comparisons to the two methods, a
preliminary modification factor of 1.25 was recommended to the Henry’s method for this
type of bridges.
6.5.6 Steel I-Beam Bridges for Live Load Shear
Four steel I-beam bridges were analyzed for this study. Table 6.46 shows the ratio
of FEA to the unmodified and modified Henry’s method. From the statistical analysis it
was found that a preliminary modification factor of 1.10 would be suitable to adjust the
results of Henry’s method to fit those of FEA for this type of bridge. Again for steel I-
beams, the 6/5.5 multiplier in step 2 of Henry’s method was already included in the
calculation of Henry’s method. Table 6.47 shows the comparison of the Henry’s method
with the AASHTO LRFD method. Even after modification, Henry’s method remains not
as conservative as the LRFD method. The average ratio and the standard deviation are
improved after modification.
174
Table 6.46 Steel I-Beam, FEA vs. Henry’s Method for Shear, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam FEA Henry's
Method
Ratio: FEA /
Henry’s Method
Modified Henry’s Method
Ratio: FEA / Modified
Henry’s Method
16 0.0 158.00 Interior 0.921 0.695 1.327 0.765 1.20316 0.0 158.00 Exterior 0.721 0.695 1.039 0.765 0.94217 19.5 143.00 Interior 0.875 0.851 1.029 0.938 0.93317 19.5 143.00 Exterior 0.841 0.851 0.989 0.938 0.89718 50.2 182.00 Interior 0.971 0.828 1.173 0.913 1.06318 50.2 182.00 Exterior 0.884 0.828 1.068 0.913 0.96819 26.7 150.00 Interior 1.017 0.822 1.237 0.907 1.12219 26.7 150.00 Exterior 0.790 0.822 0.961 0.907 0.871
Average 1.103 1.000Standard Deviation 0.129 0.117
Table 6.47 Steel I-Beam, LRFD vs. Henry’s Method for Shear, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam AASHTO
LRFD Henry's Method
Ratio: LRFD / Henry’s Method
Modified Henry’s Method
Ratio: LRFD /
Modified Henry’s Method
16 0.0 158.00 Interior 0.917 0.695 1.321 0.763 1.20116 0.0 158.00 Exterior 0.756 0.695 1.089 0.763 0.99017 19.5 143.00 Interior 0.906 0.851 1.066 0.935 0.96917 19.5 143.00 Exterior 0.807 0.851 0.950 0.935 0.86318 50.2 182.00 Interior 1.242 0.828 1.500 0.911 1.36418 50.2 182.00 Exterior 1.118 0.828 1.350 0.911 1.22719 26.7 150.00 Interior 1.099 0.822 1.336 0.904 1.21519 26.7 150.00 Exterior 0.796 0.822 0.968 0.904 0.880
Average 1.197 1.088 Standard Deviation 0.211 0.192
One hundred forty-eight steel I-beam bridges were included in database #2.
Using the modified Henry’s Method, ratios between the AASHTO LRFD and Henry’s
method were calculated for this database. A summary of the results for this comparison
can be seen in Table 6.48. Results indicate that the LRFD method produced higher shear
distribution factors than the unmodified Henry’s method. However the modified version
175
of Henry’s method compared better to the LRFD method. The average ratio improved
and the standard deviation though not perfect, did improve from before.
From the analysis of database #2 it was found that a modification factor of 1.10
was very reasonable for this type of bridge. Therefore a preliminary modification factor
of 1.10 was recommended for steel I-beam bridges.
Table 6.48 Steel I-Beam, LRFD vs. Henry’s Method for Shear, Database #2
Item Ratio: LRFD / Unmodified Henry's Method
Ratio: LRFD / Modified Henry's Method
Average 1.19 1.08Standard Deviation 0.29 0.27
6.5.7 Steel Open Box Beam Bridges for Live Load Shear
Two steel open box bridges were considered in this study. Table 6.49 shows the
ratios of FEA to the unmodified and modified Henry’s methods. The preliminary
modification factor was derived from database #1 results. From this statistical analysis, it
was determined that a modification factor of 1.25 was needed to adjust Henry’s method
and to fit results from FEA. The results of modified Henry’s method compared very well
to the FEA results as the average ratio and the standard deviation suggested. Table 6.50
shows the comparison of LRFD results to Henry’s method. In controversy to all the other
bridge types, the LRFD method for steel open box bridges produced smaller shear
distribution factors compared to both the unmodified and modified Henry’s method for
live load shear.
176
Table 6.49 Steel Open Box Beam, FEA vs. Henry’s Method for Shear, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam FEA Henry's
Method
Ratio: FEA /
Henry’s Method
Modified Henry’s Method
Ratio: FEA / Modified Henry’s Method
20 32.0 252.00 Exterior 0.831 0.701 1.185 0.873 0.95221 4.5 178.00 Interior 0.819 0.606 1.351 0.755 1.08521 4.5 178.00 Exterior 0.727 0.606 1.200 0.755 0.963
Average 1.246 1.000Standard Deviation 0.092 0.07
Table 6.50: Steel Open Box Beam, LRFD vs. Henry’s Method for Shear, Database #1
Bridge No.
Skew Angle (deg)
Span Length
(ft) Beam AASHTO
LRFD Henry's Method
Ratio: LRFD / Henry’s Method
Modified Henry’s Method
Ratio: LRFD / Modified Henry’s Method
20 32 252 Interior 0.556 0.701 0.793 0.876 0.63420 32 252 Exterior 0.556 0.701 0.793 0.876 0.63421 4.5 178 Interior 0.645 0.606 1.064 0.758 0.8521 4.5 178 Exterior 0.645 0.606 1.064 0.758 0.85
Average 0.928 0.742 Standard Deviation 0.285 0.197
Nineteen steel open box beam bridges were included in database #2. Using the
modified Henry’s Method, ratios between the AASHTO LRFD and Henry’s method were
calculated for this database. A summary of shear distribution factor ratios is shown in
Table 6.51. Similar to the results in database #1, the AASHTO LRFD method produced
smaller shear distribution factors than those from Henry’s method for bridges in database
#2. As indicated in Chapter 3, the LRFD method is adopted directly from the AASHTO
Standard method, which underestimates the distribution factor for both shear and
moment. The modification of Henry’s method was based on the comparison between
FEA and Henry’s method. Therefore a final structure modification factor of 1.25 is
recommended for steel open box beam bridges.
177
Table 6.51 Steel Open Box Beam, LRFD vs. Henry’s Method for Shear, Database #2
Item Ratio: LRFD / Unmodified Henry's Method
Ratio: LRFD / Modified Henry's Method
Average 0.92 0.72Standard Deviation 0.08 0.09
6.6 Summary of Preliminary Modification Factors (Set 1) for Live Load Shear
A complete summary of the modification factors for structure type for live load
shear has been given in this section. Table 6.52 shows the preliminary modification
factors for each bridge type. Figure 6.6 shows the frequency of ratios of FEA results to
the modified Henry’s method. It can be seen that the use of the first set of modification
factors improves the ratios of FEA results to the modified Henry’s method. The range of
ratios has been shifted from 1.00~1.30 to the new range of 0.9 ~1.10 after modification.
Table 6.53 shows the average ratios and the standard deviations for the FEA to modified
Henry’s method based on the structure type and Table 6.54 shows the average ratios and
standard deviations for the LRFD to modified Henry’s method for set 1 modification. The
average ratios between the FEA and Modified Henry’s method are all close to 1.0 and the
standard deviations are acceptable.
Table 6.52 Preliminary Structure Modification Factors (Set 1) for Live Load Shear
Structure Type Modification Factor From the Study
Shear Modification Factor Including Original
Factor in Henry’s Method
Precast Spread Box Beam 1.12 1.12Precast Concrete BT Beam 1.15 1.25Precast Concrete I-Beam 1.15 1.25CIP Concrete T-Beam 1.12 1.12CIP Concrete Box Beam 1.25 1.25Steel I-Beam 1.10 1.20Steel Open Box Beam 1.25 1.25
178
Database # 1 Histogram
0
5
10
15
20
25
0.800 0.900 1.000 1.100 1.200 1.300 1.400
Ratio
Freq
uenc
y
Ratio: FEA Vs Unmodified Henry's MethodRatio: FEA Vs Modified Henry's Method
Figure 6.6 Database #1, Histogram (Set 1) For Live Load Shear
Table 6.53 FEA vs. Modified Henry’s Method (Set 1) for Live Load Shear, Database #1
Structure Type Average Ratio: FEA /
Modified Henry’s Method
Standard Deviation
Precast Spread Box Beam 0.97 0.14Precast Concrete BT Beam 1.02 0.13Precast Concrete I-Beam 1.00 0.11CIP Concrete T-Beam 0.96 0.13CIP Concrete Box Beam 1.00 0.11Steel I-Beam 1.00 0.11Steel Open Box Beam 1.01 0.07
Table 6.54 LRFD vs. Modified Henry’s Method (Set 1) for Live Load Shear, Database #1
Structure Type Average Ratio: LRFD /
Modified Henry’s Method
Standard Deviation
Precast Spread Box Beam 1.23 0.20Precast Concrete BT Beam 1.00 0.12Precast Concrete I-Beam 1.11 0.12CIP Concrete T-Beam 0.96 0.11CIP Concrete Box Beam 1.15 0.18Steel I-Beam 1.08 0.18Steel Open Box Beam 0.80 0.13
179
6.7 Final Modification Factors for Live Load Shear (Set 1)
In order to retain the simplicity of Henry’s method, the shear modification factors
were revised by incorporating the moment modification factors. The modification factors
for live load moment have been listed in Table 6.22.
Table 6.22 Final Moment Structure Type Modification Factors (Set 1)
Structure Type Modification Factor Precast Spread Box Beam 1.00 Precast Concrete BT Beam 1.10 Precast Concrete I-Beam 1.10 CIP Concrete T-Beam 1.05 CIP Concrete Box Beam 1.05 Steel I-Beam 1.10 Steel Open Box Beam 1.00
Based on the moment and preliminary shear modification factors the
simplification process was focused on developing one single factor for shear that could
be multiplied to the moment modification factors to obtain final shear modification
factors. The single shear factor was chosen so that after multiplying with the moment
factors it gave results closer to the preliminary shear modification factors. Table 6.55 lists
the preliminary shear factors, moment modification factors and the single shear factor
(Set 1). A single shear factor of 1.15 was recommended in this set of shear modification
factors. The final shear modification factors that were obtained by multiplying the
moment factors with the single shear factor are also tabulated in Table 6.55.
By comparing the preliminary modification factors with the final shear factor
shown in Table 6.55, it is found that, for some bridge types such as precast concrete
spread box beams, cast-in-place concrete T-beams, precast concrete I-beams and steel I-
beams, the final shear factors were larger than the preliminary factors. These slightly
higher modification factors are considered to be reasonable because in most cases the
180
modified Henry’s method produced the shear distribution factors smaller than those from
the AASHTO LRFD method. Tables 5.56 and 5.57 show the average ratios and the
standard deviations of shear distribution factor of FEA versus the modified Henry’s
method and the LRFD versus the modified Henry’s method based on the final first set of
modification factors. The increase in shear distribution factors would bring the results
from modified Henry’s method close to the LRFD results but not as conservative as the
LRFD method. The final shear modification factors for cast-in-place concrete multicell
box and steel open box beam bridges were slightly smaller than the preliminary factors.
Again, when taking into account the comparison study in database #2, the proposed final
shear modification factors were acceptable.
Table 6.55 Final Shear Factors (Set 1)
Structure Type
PreliminaryModification
Factors (Set 1)
Moment Modification
Factors
Single Shear Factor Set 1
Final Shear Modification
Factors (Set 1)
Precast Spread Box Beam 1.12 1.00
1.15
1.15 Precast Concrete BT Beam 1.25 1.10 1.27 Precast Concrete I-Beam 1.25 1.10 1.27 CIP Concrete T-Beam 1.12 1.05 1.21 CIP Concrete Box Beam 1.25 1.05 1.21 Steel I-Beam 1.20 1.10 1.27 Steel Open Box Beam 1.25 1.00 1.15
Table 6.56 FEA vs. Modified Henry’s Method, (Final Set 1) for Live Load Shear, Database # 1
Structure Type Average Ratio: FEA / Modified Henry’s Method Standard Deviation
Precast Spread Box Beam 0.95 0.13Precast Concrete BT Beam 1.01 0.13Precast Concrete I-Beam 0.99 0.11CIP Concrete T-Beam 0.90 0.11CIP Concrete Box Beam 1.05 0.12Steel I-Beam 0.95 0.11Steel Open Box Beam 1.08 0.07
181
Table 6.57 LRFD vs. Modified Henry’s Method (Final Set 1) for Live Load Shear, Database #1
Structure Type Average Ratio: LRFD / Modified Henry’s Method Standard Deviation
Precast Spread Box Beam 1.20 0.20Precast Concrete BT Beam 0.99 0.12Precast Concrete I-Beam 1.09 0.12CIP Concrete T-Beam 0.89 0.11CIP Concrete Box Beam 1.19 0.18Steel I-Beam 1.02 0.18Steel Open Box Beam 0.85 0.13
Histogram for Live Load Shear, Database #1
0
5
10
15
20
25
0.800 0.900 1.000 1.100 1.200 1.300 1.400Ratio
Freq
uenc
y
Ratio:FEA Vs Unmodified Henry's MethodRatio:FEA Vs Modified Henry's Method Set 1
Figure 6.7 Histogram of Shear Distribution Factor (Set 1), Database #1
Figure 6.7 shows the frequency of the ratios of FEA results to the modified and
unmodified Henry’s methods. As can be seen from this figure, most of the ratios have
been improved to be closer to 1.0 or within a range of +/-0.05. The Henry’s method,
which was very unconservative before, has been brought closer to the FEA results. In
addition, most of the conservative and unconservative ratios were improved after
modification. Figure 6.8 shows the shear distribution factors obtained from all the current
methods as well as the modified Henry’s method with the first set of shear modifications
182
factors versus skew angle for the bridges in database #1. From this figure it can be seen
that the modified Henry’s method compares very well with the finite element analysis
results.
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
0 10 20 30 40 50 60
Skew Angle
Shea
r Dis
tribu
tion
Fact
ors
Modified Henry's method FEAAASHTO LRFD AASHTO StandardUnmodified Henry's method Linear (Modified Henry's method)Linear (FEA ) Linear (AASHTO LRFD)Linear (AASHTO Standard) Linear (Unmodified Henry's method)
Figure 6.8 Shear Distribution Factor vs. Skew Angle (Set 1), Database #1
6.8 Modification Factors for Live Load Shear (Set 2)
The second set of modification factors for live load shear was developed
independently and was not related to the moment modification factors. Therefore,
structure type modification factors for shear are different with the moment factors. These
shear modification factors were introduced based on finite element analysis of the 24
bridges in database #1. AASHTO LRFD method has a skew angle correction factor,
which is an increase factor and is applied to lane fraction of beams when the line of
support is skewed. Different studies on shear distribution in bridges with skew angles,
suggest that there is an increase in the distribution factors with the increase in skew. For
183
this reason the modification factors (set 2) presented in this section consists of a skew
modification factor along with the structure type modification factors. The skew
modification factor in this set should be applied to the bridges with a skewed support line.
6.8.1 Skew Correction Factor
The skew correction factor incorporated in this study was based on the skew
correction factors specified in the AASHTO LRFD code. The LRFD code has specified
different skew correction factors based on the type of bridge superstructure. The skew
correction factor equation is the same for prestressed concrete I-beams, cast-in-place
concrete T-beams and steel I-beam bridges. The equation specified in the code for these
types of bridges is
θK
Lt
g
s tan0.122.00.13.0
3
⎟⎟⎠
⎞⎜⎜⎝
⎛+ (6-1)
In this equation the beam stiffness , span length L, slab thickness tgK s, along with skew
angle θ, are the variables. The term ⎟⎟⎠
⎞⎜⎜⎝
⎛
g
s
KLt. 3012 is a measure of the ratio of transverse to
longitudinal stiffness of the superstructure. Although this ratio is in the equation for the
skew correction factor, it is not a variable in live load distribution calculation for shear.
From this study it was found that this ratio could be effectively removed from the
amplification equation without affecting the accuracy of the equation. The equation after
simplification is shown in Eq. 6-2. This equation will be used as the skew correction
factor for live load distribution of shear in this research.
θtan2.00.1 + (6-2)
184
The validity of this simplified skew correction factor was examined by comparing
the results from the LRFD equation to the results from Eq. 6.2. Figure 6.9 shows skew
correction factors from the simplified equation and LRFD equation for selected bridges in
database #2. These selected bridges included prescast concrete I-beam, cast-in-place
concrete T-beam, and steel I-beam bridges. It can be seen from Figure 6.9 that these two
equations produced correction factors that were very close. The same simplified equation
for skew angle correction was also applied to other types of bridges such as concrete
spread box beam and cast-in-place concrete multicell box beam bridges even though the
LRFD code had different equations for these bridges. Further study showed that the
simplified equation compared well with the LRFD equation for concrete spread box beam
bridges. The specified AASHTO LRFD equation for cast-in-place concrete multicell box
beam bridges produced slightly higher skew correction factors than the simplified
equation presented in this study. However, based on the analysis of the 24 bridges in
database #1 it was found that the simplified equation was adequately close to the LRFD
results. Therefore, a single skew correction factor mentioned above would be used for all
types of superstructures.
185
Comparison of Skew Correction Factors
00.20.40.60.8
11.21.41.6
0 10 20 30 40 50 60 70
Skew Angle
Cor
rect
ion
Fact
ors
LRFD Simplif ied Equation
Linear (Simplif ied Equation) Linear (LRFD)
Figure 6.9: Skew Correction Factor vs. Skew Angle (Set 1)
6.8.2 Structure Factors for Live Load Shear (Set 2)
Once the skew correction factor is obtained, the structure modification factors to
Henry’s method were developed. The procedure to find the structure factors is similar to
the one used for set 1 modification factors. Adjustments were made in the structural
modification factors to minimize the standard deviations for each bridge type to ensure
accurate results compared to FEA. It was found that after including the skew correction
factor, a structure factor of 1.05 was needed for the precast concrete spread box beams
and cast-in-place concrete T-beams, and a factor of 1.10 was needed to correct the
Henry’s method for precast I-beam bridges. The original Henry’s method compared well
for steel I-beam bridges and only a structure factor of 1.06 was applied to this kind of
bridge. The structure factor for cast-in-place concrete multicell box beam bridges is 1.20
and 1.15 for steel open box girder bridges. Table 6.58 lists all the structure modification
factors and also the skew correction factor in this set of modification for live load shear
186
distribution factors. In this table, θ is the angle of skew of a bridge from the vertical to
the bridge support line. Table 6.59 shows the comparison of the average ratio and
standard deviation of FEA to Henry’s method for each type of bridge. Figure 6.10 shows
the frequency histogram of the ratios when the FEA method is compared to the modified
and unmodified Henry’s method. As can be seen from this figure, most of the ratios have
been brought closer to 1.0 or within a range of +/-0.10 of one.
Table 6.58 Modification Factors for Live Load Shear (Set 2)
Structure Type Modification Factors (Set 2)
Shear Modification Factor Including Original
Factor in Henry’s Method
Precast Spread Box Beam 1.05 1.05 Precast Concrete BT Beam 1.10 1.20 Precast Concrete I-Beam 1.10 1.20 CIP Concrete T-Beam 1.05 1.05 CIP Concrete Box Beam 1.20 1.20 Steel I-Beam 1.06 1.15 Steel Open Box Beam 1.15 1.15
Skew Modification Factor θ.. tan2001 +
Table 6.59 FEA vs. Modified Henry’s Method (Set 2) for Shear, Database # 1
Structure Type Average Ratio: FEA /
Modified Henry’s Method
Standard Deviation
Precast Spread Box Beam 0.97 0.19Precast Concrete BT Beam 1.01 0.13Precast Concrete I-Beam 0.98 0.13CIP Concrete T-Beam 0.99 0.16CIP Concrete Box Beam 1.00 0.11Steel I-Beam 0.96 0.15Steel Open Box Beam 1.00 0.11
187
Database # 1 Histogram Set 2
0
5
10
15
20
25
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4
Ratio
Freq
uenc
y
Ratio:FEA vs Unmodif ied Henry's MethodRatio:FEA vs Modif ied Henry's Method Set 2
Figure 6.10 Histogram of Shear Distribution Factor (Set 2) Database #1
Table 6.60 shows the average ratio and the standard deviation of the LRFD results
to the modified Henry’s method results for Database #1 and Table 6.61 shows the
average ratio and standard deviation for Database #2 between the LRFD method and the
modified Henry’s method.
Table 6.60 LRFD vs. Modified Henry’s Method for Shear (Set 2), Database #1
Structure Type Average Ratio: LRFD / Modified Henry’s Method Standard Deviation
Precast Spread Box Beam 1.20 0.17Precast Concrete BT Beam 0.99 0.13Precast Concrete I-Beam 1.09 0.10CIP Concrete T-Beam 0.97 0.15CIP Concrete Box Beam 1.15 0.15Steel I-Beam 1.03 0.15Steel Open Box Beam 0.76 0.17
188
Table 6.61 LRFD vs. Modified Henry’s Method for Shear (Set 2), Database #2
Structure Type Average Ratio: LRFD / Modified Henry’s Method Standard Deviation
Precast Spread Box Beam 1.29 0.18Precast Concrete BT Beam 1.05 0.20Precast Concrete I-Beam 1.05 0.20 CIP Concrete T-Beam 1.13 0.21CIP Concrete Box Beam 1.05 0.22Steel I-Beam 1.04 0.23Steel Open Box Beam 0.79 0.09
Figure 6.11 shows the shear distribution factors plotted against the skew angle.
The data shown in the figure included the results from the AASHTO Standard, the
AASHTO LRFD, FEA, unmodified Henry’s method and the modified Henry’s method
(set 2). As can be seen from Fig. 6.11, the Henry’s method was improved after
modification and its results showed the increase with the increase in skew angle. Tables
6.62 and 6.63 list the distribution factors for live load moment and shear obtained from
FEA, Henry’s method, and the modified Henry’s method for database #1, respectively.
Dis tribution Factor Vs Skew Angle
0.00
0.20
0.40
0.60
0.80
1.00
1.20
1.40
0 10 20 30 40 50 60Skew Angle
She
ar D
istri
butio
n Fa
ctor
s
Modif ied Henry's Method Set 2 FEAAASHTO LRFD AASHTO StandardUnmodif ied Henry's Method Linear (Modif ied Henry's Method Set 2)Linear (FEA ) Linear (AASHTO LRFD)Linear (AASHTO Standard) Linear (Unmodif ied Henry's Method)
Figure 6.11 Shear Distribution Factor vs. Skew Angle (Set 2)
189
Table 6.62 Distribution Factors for Live Load Moment, Database #1
Bridge No. Bridge Type Beam FEA Henry's
Method
Modified Henry's Method (Set 1)
Modified Henry's Method (Set 2)
1 Precast Spread Box Beam Interior 0.767 0.826 0.826 0.8261 Precast Spread Box Beam Exterior 0.744 0.826 0.826 0.8262 Precast Spread Box Beam Exterior 1.251 1.152 1.152 1.1523 Precast Spread Box Beam Interior 0.867 0.759 0.759 0.7593 Precast Spread Box Beam Exterior 0.826 0.759 0.759 0.7594 Precast Spread Box Beam Interior 0.399 0.489 0.489 0.4604 Precast Spread Box Beam Exterior 0.453 0.489 0.489 0.4605 Precast Concrete BT Beam Interior 0.660 0.663 0.668 0.6575 Precast Concrete BT Beam Exterior 0.625 0.663 0.668 0.6576 Precast Concrete I-Beam Interior 0.715 0.663 0.668 0.6996 Precast Concrete I-Beam Exterior 0.689 0.663 0.668 0.6997 Precast Concrete I-Beam Interior 0.721 0.663 0.668 0.6577 Precast Concrete I-Beam Exterior 0.654 0.663 0.668 0.6578 Precast Concrete BT Beam Interior 0.812 0.790 0.797 0.7838 Precast Concrete BT Beam Exterior 0.747 0.790 0.797 0.7839 CIP Concrete T-Beam Interior 0.877 0.869 0.912 0.8999 CIP Concrete T-Beam Exterior 0.870 0.869 0.912 0.899
10 CIP Concrete T-Beam Interior 0.930 0.859 0.902 0.94510 CIP Concrete T-Beam Exterior 0.943 0.859 0.902 0.94511 CIP Concrete T-Beam Interior 0.704 0.644 0.676 0.70811 CIP Concrete T-Beam Exterior 0.658 0.644 0.676 0.70812 CIP Concrete Box Beam Interior 0.687 0.608 0.638 0.62913 CIP Concrete Box Beam Interior 0.620 0.608 0.638 0.62914 CIP Concrete Box Beam Interior 0.665 0.698 0.733 0.72215 CIP Concrete Box Beam Interior 0.765 0.701 0.736 0.72516 Steel I-Beam Interior 0.690 0.695 0.701 0.68916 Steel I-Beam Exterior 0.653 0.695 0.701 0.68917 Steel I-Beam Interior 0.749 0.851 0.858 0.84317 Steel I-Beam Exterior 0.842 0.851 0.858 0.84318 Steel I-Beam Interior 0.850 0.828 0.835 0.77118 Steel I-Beam Exterior 0.901 0.828 0.835 0.77119 Steel I-Beam Interior 0.830 0.822 0.829 0.81419 Steel I-Beam Exterior 0.835 0.822 0.829 0.81420 Steel Open-Box Girder Interior 0.641 0.701 0.701 0.68120 Steel Open-Box Girder Exterior 0.641 0.701 0.701 0.68121 Steel Open-Box Girder Interior 0.630 0.606 0.606 0.62721 Steel Open-Box Girder Exterior 0.685 0.606 0.606 0.62722 Precast Concrete BT Beam Interior 0.559 0.711 0.717 0.70422 Precast Concrete BT Beam Exterior 0.552 0.711 0.717 0.70423 Precast Concrete BT Beam Interior 0.537 0.711 0.717 0.70423 Precast Concrete BT Beam Exterior 0.511 0.711 0.717 0.70424 Precast Concrete I-Beam Interior 0.757 0.782 0.788 0.82424 Precast Concrete I-Beam Exterior 0.791 0.782 0.788 0.824
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0.7685 Precast Concrete BT Beam Interior 0.931 0.663 0.7725 Precast Concrete BT Beam Exterior 0.730 0.663 0.772 0.7686 Precast Concrete I-Beam Interior 0.917 0.663 0.771 0.785
0.663 0.771 0.7856 Precast Concrete I-Beam Exterior 0.6777 Precast Concrete I-Beam Interior 0.770 0.663 0.8250.7717 Precast Concrete I-Beam Exterior 0.700 0.663 0.771 0.8258 Precast Concrete BT Beam Interior 0.960 0.790 0.920 0.8698 Exterior 0.784 0.790 0.920 0.869Precast Concrete BT Beam
Bridge No. Bridge Type Beam FEA Henry's
Method
Modified Henry's Method (Set 1)
Modified Henry's Method (Set 2)
1 Precast Spread Box Beam Interior 0.858 0.826 0.950 0.9141 Precast Spread Box Beam Exterior 0.815 0.826 0.950 0.9142 Precast Spread Box Beam Exterior 1.080 1.152 1.326 1.2113 Precast Spread Box Beam Interior 1.010 0.759 0.873 0.7973 Precast Spread Box Beam Exterior 0.933 0.759 0.873 0.7974 Precast Spread Box Beam Interior 0.452 0.489 0.566 0.6334 Precast Spread Box Beam Exterior 0.568 0.489 0.566 0.633
9 CIP Concrete T-Beam Interior 0.911 0.869 1.051 1.0259 CIP Concrete T-Beam Exterior 0.762 0.869 1.051 1.025
10 CIP Concrete T-Beam Interior 1.090 0.859 1.039 0.93310 CIP Concrete T-Beam Exterior 0.956 0.859 1.03911 CIP Concrete T-Beam Interior 0.770 0.644 0.779 0.67611 CIP Concrete T-Beam Exterior 0.665 0.644 0.779 0.67612 CIP Concrete Box Beam Interior 0.842 0.608 0.736 0.730 12 CIP Concrete Box Beam Exterior 0.651 0.608 0.736 0.730 13 CIP Concrete Box Beam Interior 0.856 0.608 0.734 0.728 13 CIP Concrete Box Beam Exterior 0.660 0.608 0.734 0.728 14 CIP Concrete Box Beam Interior 0.975 0.698 0.845 0.920 14 CIP Concrete Box Beam Exterior 0.866 0.698 0.845 0.920 15 CIP Concrete Box Beam Interior 0.883 0.701 0.847 0.890 15 CIP Concrete Box Beam Exterior 0.851 0.701 0.847 0.890 16 Steel I-Beam Interior 0.921 0.695 0.809 0.73616 Steel I-Beam Exterior 0.721 0.695 0.809 0.73617 Steel I-Beam Interior 0.875 0.851 0.990 0.96517 Steel I-Beam Exterior 0.841 0.851 0.990 0.96518 Steel I-Beam Interior 0.971 0.828 0.965 1.08818 Steel I-Beam Exterior 0.884 0.828 0.965 1.08819 Steel I-Beam Interior 1.017 0.822 0.958 0.95919 Steel I-Beam Exterior 0.790 0.822 0.958 0.95920 Steel Open-Box Girder Exterior 0.831 0.701 0.806 0.90721 Steel Open-Box Girder Interior 0.819 0.606 0.697 0.70821 Steel Open-Box Girder Exterior 0.727 0.606 0.697 0.70822 Steel Open-Box Girder Interior 0.933 0.711 0.827 0.86022 Precast Concrete BT Beam Exterior 0.756 0.711 0.827 0.86023 Precast Concrete BT Beam Interior 0.932 0.711 0.827 0.86023 Precast Concrete BT Beam Exterior 0.727 0.711 0.827 0.86024 Precast Concrete BT Beam Interior 0.940 0.782 0.909 0.85824 Precast Concrete I-Beam Exterior 0.841 0.782 0.909 0.858
0.933
Table 6.63 Distribution Factors for Live Load Shear, Database #1
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CHAPTER 7
CONCLUSIONS AND DESIGN RECOMMENDATIONS
7.1 Conclusions
Lateral distribution of live load moment is an important factor in highway bridge
design. Using wheel load distribution factors, engineers can predict bridge response by
uncoupling the longitudinal and transverse effects from each other. One simple method
of calculating lateral distribution factor is Henry’s equal distribution factor (EDF)
method. In this method, it is assumed that all beams, including interior and exterior
beams, have equal distribution of live load effects. Parameters in this method are limited
to only roadway width, number of beams, a load intensity factor, and a structure type
multiplier for steel and prestressed I-beams. Because of its lack of restrictions, it can be
applied without difficulty to different types of superstructures and beam arrangements.
The method has been used in Tennessee for almost four decades, therefore a re-
evaluation, if possible some modification, is necessary to ensure the accuracy of the
method and comparability with the current AASHTO Specifications.
The primary objective of this research was to carefully reexamine Henry’s
method for live load moment and shear distribution. To pursue this objective, a
comparison study was conducted for the distribution factors of live load moment and
shear in actual bridges using the AASHTO Standard, the AASHTO LRFD, Henry’s
method, and finite element analysis. Twenty-four Tennessee bridges of six different
types of superstructures were selected for the comparison study. These superstructure
types included concrete spread box beams, cast-in-place concrete T-beams, steel I-beams,
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precast concrete I-beams, cast-in-place concrete multicell box beams, and steel open box
beams. The accuracy of Henry’s method was evaluated mainly based on the comparison
between Henry’s method and finite element analysis. A statistical analysis was
performed in developing modification factors to Henry’s method for moment and shear
distribution. It was found that, with proper modifications, modified Henry’s method can
produce very reasonable and reliable distribution factors for live load moment and shear
that are comparable to the distribution factors obtained using finite element analysis and
the AASHTO LRFD method. Furthermore, this method offers advantages in simplicity
of calculation, flexibility in application, and savings of expenses. Following are the
conclusions drawn from this research project.
(1) From this research, it was found that the AASHTO Standard method for
calculating live load moment distribution factors produces conservative results
compared to FEA in nearly every case. When compared to each other, the
AASHTO LRFD and Henry’s method produce very similar results. However, for
the most part, Henry’s method is conservative compared to FEA but not as
conservative as the AASHTO LRFD method. For superstructure types where
Henry’s method is unconservative compared to FEA and the AASHTO LRFD
results, modification factors are recommended. When comparing the distribution
factors for live load shear from the AASHTO LRFD method, AASHTO Standard
method, Henry’s method and the FEA results, it can be said that AASHTO
Standard method produces inconsistent results while the AASHTO LRFD method
produced results closer to FEA results for live load shear for almost all types of
bridges. The Henry’s method was unconservative for most of the bridge types.
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(2) Finite element analysis has long been accepted as an accurate method to
determine live load distribution factors in highway bridges. The bridges in this
study were modeled and analyzed using the finite element analysis program,
ANSYS. Two finite element bridge models, Case 1 and Case 2, were developed
for each bridge as discussed in Chapter 4. One model considered support
diaphragms and the other did not. The model with support diaphragms was
considered to be the most accurate system and was used in the parametric study
because it most closely represented the actual bridge conditions. Comparing
results from this model to those without support diaphragms, noticeable
differences in live load distribution factors can be observed. With the exception
of the steel open box beam bridges, the distribution factors obtained from the
AASHTO LRFD method were consistently higher than the results from finite
element analysis. One primary reason is that the effects of diaphragms were not
considered in the development of the AASHTO LRFD equations. The
distribution factors without diaphragm consideration in structures are normally
larger than those with diaphragms. Therefore, it is essential to include support
diaphragms when conducting FEA of actual bridges.
(3) Results from the FEA reiterate findings from previous studies for skew angle and
span length. Higher skew angles and longer span lengths have been shown to
moderately reduce distribution factors for live load moment. Results from this
study have shown Henry’s method to become conservative compared to FEA and
AASHTO LRFD methods for skew angles greater than 30o and spans lengths
greater than 100 ft for live load moment. For these reasons, skew and length
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reduction factors for live load moment have been developed in the set two
modification factors to Henry’s method. From this study it was also found that
the distribution factors for live load shear would increase due to the effect of skew
angle, which was in agreement with other studies. The increase in span length was
found to have minor or little impact on the distribution factors for shear. The
results from Henry’s method were in a similar trend to that of FEA results but
smaller in values for almost all bridges, especially for bridges with higher skew
angles. A correction factor of skew angle was introduced in set 2 shear
modification factors. The effects of other parameters like bridge width, aspect
ratio, slab thickness and beam spacing were also studied for both live load
moment and shear and the results were plotted.
(4) Results from FEA conducted in this study for steel open box beams show the
AASHTO Standard and the AASHTO LRFD methods to be inaccurate for both
live load moment and shear. As noted before, the AASHTO LRFD equations
were taken directly from AASHTO Standard methods and divided by two.
Henry’s method, on the other hand, compared well to FEA regardless of which set
of modification factors were used.
(5) Two sets of modification factors were developed for Henry’s method for live load
moment based on both the finite element analysis conducted on bridges in
database #1 in this study and results from AASHTO LRFD methods for a second
database of 419 bridges from NCHRP Project 12-26. The modified Henry’s
methods for live load moment had a higher degree of accuracy than the
unmodified method. Furthermore, the modified Henry’s method using the second
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set of modification factors performed the best of all three forms of Henry’s
method. Distribution factor values for this modified Henry’s method were
typically slightly more conservative than FEA as they should be. However, they
were not as conservative as the AASHTO Standard or AASHTO LRFD methods.
This balance of accurate and reliable results between actual bridge response and
the accepted codes is ideal for bridge design. Using a simple, less restrictive, and
accurate method such as the modified Henry’s method is highly desirable to
bridge engineers.
(6) Two sets of modification factors were created in this study for Henry’s method
for live load shear. The first set of modification factors comprised of the structural
modification factors, same as the set 1 moment modification factors and a single
shear modification factor. This shear factor was multiplied to the moment
modification factors to obtain final shear modification factors. The set 2 included
different structure modification factors along with the skew modification
equation. The equation for skew correction factor is expressed as
where is the angle that the bridge makes with the vertical support line. All of
these modification factors were determined based on the comparison of Henry’s
method to the finite element analysis of bridges from database #1 and also to
AASHTO LRFD method for the bridges in database #2. Both sets of modification
factors for Henry’s method compared well to the FEA method than the
unmodified Henry’s method for live load shear.
θtan2.00.1 +
θ
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7.2 Design Recommendations
Modification factors to Henry’s method were introduced based on the comparison
and evaluation of the live-load distribution factors from Henry’s method and finite
element analysis (FEA). The distribution factors from Henry’s method were also
compared with the results from the AASHTO Standard and the AASHTO LRFD
methods. Two bridge databases were used in the study. The first database consisted of
24 selected Tennessee bridges of different superstructure types. Finite element analysis
was pursued to all 24 selected bridges. The second database was the bridge database
used in the NCHRP Project 12-26, consisting of more than 400 bridges across the nation.
The modification factors to Henry’s method were initially developed based on a
comparison between distribution factors from Henry’s method and finite element analysis
for the bridges in database #1. Then, the preliminary modification factors were calibrated
according to the comparison between Henry’s method and the AASHTO LRFD method
for the bridges in database #2. The superstructure types studied in this research are listed
in Table 7.1.
Two sets of modification factors for moment and shear distributions are
recommended through this study. The first set includes moment modification factors of
superstructure type along with a single shear factor to all types of structures. The second
set of modification factors includes separate sets of moment and shear modification
factors. The effects of skew and span length have been considered in the second set of
modification factors. Details of the recommended two sets of modification factors are
presented in the following sections.
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Table 7.1 Common Deck Superstructures Covered in this Research
Supporting Components Type of Deck Typical Cross-Section Steel Beam Cast-in-place concrete slab,
precast concrete slab, steelgrid, glued/spiked panels, stressed wood
Closed Steel or Precast Concrete Boxes
Cast-in-place concrete slab
Open Steel or Precast Concrete Boxes
Cast-in-place concrete slab,precast concrete deck slab
Cast-in-Place Concrete Multicell Box
Monolithic Concrete
Cast-in-Place Concrete Tee-Beam
Monolithic Concrete
Precast Concrete I or Bulb-Tee Sections
Cast-in-place concrete, precast concrete
7.2.1 Modification Factors - Set 1
7.2.1.1 Modification Factors for Live Load Moment
The moment modification factors to Henry’s method, also named as
superstructure type factors, vary with the type of bridge superstructure. The proposed
moment modification factors or superstructure type factors are listed in Table 7.2. Also
listed are the original structure factors in Henry’s method. In the modified Henry’s
method, steel and concrete spread box beams require no modification as the unmodified
Henry’s method. Cast-in-place concrete T-beams and cast-in-place concrete boxes
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require a modification factor of 1.05. The multiplier for steel I-beam and precast
concrete I-beam is increased from 1.09 to 1.10. Each of these factors will enable the
modified Henry’s method to produce accurate moment distribution factors for its
respective bridge type. The modified Henry’s method generally produces better results
than the unmodified Henry’s method compared to FEA results.
Table 7.2 Structure Type Modification Factors for Live Load Moment (Set 1)
Structure Type Modification Factor Original Factor
Precast Spread Box Beam 1.00 1.00 Precast Concrete I-Sections 1.10 1.09 CIP Concrete T-Beam 1.05 1.00 CIP Concrete Box Beam 1.05 1.00 Steel I-Beam 1.10 1.09 Steel Open Box Beam 1.00 1.00
7.2.1.2 Modification Factors for Live Load Shear
It has been observed that distribution factors using Henry’s method are typically
smaller than the shear distribution factors from finite element analysis and the AASHTO
LRFD method. Thus, the introduction of modification factors to shear distribution is
necessary for obtaining accurate results from Henry’s method. To simplify the
modification for shear distribution factor, a single shear factor is introduced in the
modified Henry’s method. In developing the shear factor, the shear distribution factor
from FEA are compared with the modified Henry’s method that have already included
the moment modification factors. Therefore, this single shear modification factor will be
used along with the moment modification factors to Henry’s method in the calculation of
shear distribution factors. Table 7.3 shows modification factors for moment and shear
distribution.
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Table 7.3 Modification Factors for Shear Distribution (Set 1)
Structure Type Moment
Modification Factors
Single Shear Factor
Precast Spread Box Beam 1.00
1.15 Precast Concrete I-Sections 1.10 CIP Concrete T-Beam 1.05 CIP Concrete Box Beam 1.05 Steel I-Beam 1.10 Steel Open Box Beam 1.00
7.2.1.3 Procedures of the Modified Henry’s Method for Live Load Moment and
Shear Distribution Factors (Set 1)
Following are the details of the modified Henry’s method for live load moment and
shear:
Step 1: Basic Equal Distribution Factor
(a) Divide roadway width by ten (10 ft) to determine the fractional number of design
traffic lanes.
(b) Reduce the value from (a) by a factor obtained from a linear interpolation of
intensity factors. This will be the total live load (total number of traffic lanes
carrying live load) on the bridge. The intensity factor (multiple-presence factor)
of live load equals 100% for a two-lane bridge, 90% for a three-lane bridge, or
75% for a four- or more lane bridge.
(c) The basic distribution factor of lane load per beam will be equal to total live load
divided by number of beams.
Step 2: Superstructure Type Modification – Moment Distribution Factors
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(d) Multiply the value from (c) by the appropriate moment modification factor from
Table 7.3 to determine the moment distribution factor.
Step 3: Shear Factor Modification - Shear Distribution Factors
(e) The number obtained from (d) is then multiplied by the shear factor in Table 7.3
to get the shear distribution factor.
7.2.2 Modification Factors - Set 2
The second set of modification factors to Henry’s method have been developed
independently for moment and shear. Separate groups of moment and shear modification
factors are recommended in this set. It was found that for both live load moment and
shear the second set of modification factors produces slightly better results than the first
set.
7.2.2.1 Modification Factors for Live Load Moment
The second set of modification factors for live load moment includes multipliers
for structure type, skew angle, and span length. Results from this method compare very
well to the FEA analysis results. Distribution factors from this method are typically
slightly more conservative than FEA but not as conservative as the AASHTO LRFD
method. The superstructure type modification factors for Set 2 can be seen in Table 7.4.
Results from this study reiterate the findings from other studies for skew angle and span
length. Higher skew angles and longer span lengths have been shown to moderately
reduce distribution factors for live load moment. The Henry’s method becomes
conservative compared to FEA results and AASHTO LRFD method for skew angles
greater than 30o and spans lengths greater than 100 ft. Therefore, skew angle and span
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length modification factors for Henry’s method have been developed and they are also
listed in Tables 7.4. The skew angle is an angle from the vertical with roadway.
Table 7.4 Modification Factors for Live Load Moment (Set 2)
Structure Type Structure Type Modification
Factor
Skew Modification
Factor
Length Modification Factor
θ < 30o θ > 30o L < 100 ft L > 100 ftPrecast Spread Box Beam 1.00
1.00 0.94 1.00 0.94
Precast Concrete I-Sections 1.15 CIP Concrete T-Beam 1.10 CIP Concrete Box Beam 1.10 Steel I-Beam 1.15 Steel Open Box Beam 1.10
7.2.2.2 Modification Factors for Live Load Shear
The second set of modification factors for live load shear is not related to the
moment modification factors. Therefore the structure type modification factors for shear
are different with those for moment. The AASHTO LRFD method has a skew angle
correction factor. The factor is an increase factor and is applied to lane fraction of beams
when the line of support is skewed. Different studies on shear distribution for bridges
with skew angles suggest that the distribution factors increase with the increase in skew.
For this reason the modification factors in Set 2 consist of structure type modification
factors for shear along with a skew modification factor. The skew modification factor is
expressed in an equation that is a function of skew angle. The skew angle is an angle
from the vertical with roadway. Distribution factors from this method are slightly more
conservative than FEA but not as conservative as the AASHTO LRFD method. The Set 2
structure type modification factors for shear as well as the skew angle correction equation
are listed in Table 7.5.
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Table 7.5 Modification Factors for Live Load Shear (Set 2)
Structure Type Structure Type Modification Factor
Skew Modification Factor
Precast Spread Box Beam 1.05
θtan2.00.1 +
Precast Concrete I-Sections 1.20 CIP Concrete T-Beam 1.05 CIP Concrete Box Beam 1.20 Steel I-Beam 1.15 Steel Open Box Beam 1.15
7.2.2.3 Procedures of Modified Henry’s Method for Moment and Shear (Set 2)
Following are the details of the modified Henry’s method with the Set 2 moment and
shear distribution factors.
(1) For Moment Distribution Factor
Step 1: Basic Equal Distribution Factor
(a) Divide roadway width by ten (10 ft) to determine the fractional number of design
traffic lanes.
(b) Reduce the value from (a) by a factor obtained from a linear interpolation of
intensity factors. This will be the total live load (total number of traffic lanes
carrying live load) on the bridge. The intensity factor (multiple-presence factor)
of live load equals 100% for a two-lane bridge, 90% for a three-lane bridge, or
75% for a four- or more lane bridge.
(c) The basic distribution factor of lane load per beam will be equal to total live load
divided by number of beams.
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Step 2: Superstructure Type Modification
(d) Proceed with steps (a) through (c) above. Multiply the value from (c) by an
appropriate structure type modification factor in Table 7.4 to determine the
moment distribution factor.
Step 3: Skew Angle and Span Length Modifications
(e) Multiply the value from (d) by the appropriate skew modification factor and
length factor in Table 7.4 to get the final moment distribution factor.
(2) For Shear Distribution Factor
Step 1: Basic Equal Distribution Factor
The parts (a), (b) and (c) in Step 1 are the same as the ones for moment distribution
factor.
Step 2: Superstructure Type Modification for shear
(d) Multiply the value from (c) by an appropriate structure modification factor in
Table 7.5 to obtain the shear distribution factor.
Step 3: Skew Angle Modification
(e ) Multiply the value from (d) by the appropriate skew modification factor in Table
7.5 for a skewed bridge to get the final shear distribution factor.
7.3 Final Remarks
It can be seen from this study that both versions of the modified Henry’s method
offer obvious advantages over the AASHTO Standard and AASHTO LRFD methods.
For live load moment distribution, Henry’s method will be slightly modified to obtain
more accurate moment distribution factors. For shear distribution, the unconservative
Henry’s method has been brought closer to the accurate finite element analysis through
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the use of modification factors. Both version of modified Henry’s methods are much less
complex and less restrictive than the LRFD method and the results they produce are just
as accurate as other complicated methods like the AASHTO LRFD if not more so.
These simple and reliable methods for calculating live load moment distribution factors
enable the engineer to avoid making approximations and conducting refined modeling
during bridge analysis when the parameters of the bridges are in excess of the range of
applicability set in the AASHTO LRFD. In addition, the modified Henry’s method is
less conservative than the AASHTO methods, which is also an effective way of cost
savings.
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[20] Hays, C., Sessions, L., and Berry, A., “Further Studies on Lateral Load Distribution using a Finite Element Method,” Transportation Research Record, 1072, Washington, D.C., 1986, pp. 6-14.
[21] Heins, C.P., “Box Girder Bridge Design-State-of-the-art,” AISC Engineering Journal, 2, September-December 1978, pp. 126-142.
[22] Heins, C.D., and Jin, J.O., “Live Load Distribution on Braced Curved I-Girders,” Journal of Structural Engineering, Vol.110, No.3, March 1984, pp. 523-530.
[23] Heins, C.D., and Siminou, J., “Preliminary Design of Curved Girder Bridges,” AISC Engineering Journal, Vol.7, No.2, April 1970, pp. 50-61.
207
[24] Huo, Zhu, Ung, Goodwin, and Crouch, “Experimental Study of Optimal Erection Schedule of Prestressed Concrete Bridge Girders,” Final Report, Tennessee DOT Project No. TNSPR-RES 1188, 2002, Tennessee Technological University, Cookeville, Tennessee.
[25] Imbsen, R.A., and Nutt, R.V., “Load Distribution Study on Highway Bridges Using STRUDL Finite Element Analysis Capabilities,” Conference On Computing in Civil Engineering: Proceedings, 1978, pp. 639-655.
[26] Khaleel, M.A., and Itani, R.Y., “Live Load Moments for Continuous Skew Bridges,” Journal of Structural Engineering, Vol.116, No 9, September 1990, pp. 2361-2373.
[27] Kim, S.J., and Nowak, A.S., “Load Distribution and Impact Factors for I-Girder Bridges,” Journal of Bridge Engineering, Vol.2, No.3, August 1997, pp. 97-104.
[28] Mabsout, M.E., Tarhini, K.M, Frederick, G.R., and Kesserwan, A., “Effect ofMultilanes on Wheel Load Distribution in Steel Girder Bridges,” Journal of Bridge Engineering, Vol.4, No.2, May 1999, pp. 99-106.
[29] Mabsout, M.E., Tarhini, K.M., Frederick, G.R., and Tayar, C., “Finite-Element Analysis of Steel Girder Highway Bridges,” Journal of Bridge Engineering, Vol.2, No.3, August 1997, pp. 83-87.
[30] Mabsout, M.E., Tarhini, K.M., Frederick, G.R., and Kesserwan, A., “Effect of Continuity on Wheel Load Distribution in Steel Girder Bridges,” Journal of Bridge Engineering, Vol.3, No.3, August 1998, pp. 103-110.
[31] Nowak, A.S., and Hong, Y-K., “Bridge Live-Load Models,” Journal ofStructural Engineering, Vol.117, No.9, September 1991, pp.2757-2767.
[32] Puckett, J.A., “Comparative Study of AASHTO Load and Resistance Factor Design Distribution Factors for Slab-Girder Bridges,” Transportation Research Record No.1770, Transportation Research Board, Washington, D.C. 2001, pp. 34-37.
[33] Samaan, M., Sennah, K, and Kennedy, J.B., “Distribution of Wheel Loads on Continuous Steel Spread-box Girder Bridges,” Journal of Bridge Engineering, Vol.7, No.3, May 2002, pp.175-183.
[34] Schwarz, M., and Laman, J.A., “Response of Prestressed Concrete I-Girder Bridges to Live Load,” Journal of Bridge Engineering, Vol.6, No.1, January-February 2001, pp.1-8.
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[35] Scordelis, A.C., and Larsen, P.K., “Structural Response of Curved RC box-Girder Bridge,” J. Structural Div., Vol.103, No.8, August 1977, pp.1507-1524.
[36] Sennah, K.M., and Kennedy, J.B., “Load Distribution Factors for CompositeMulticell Box Girder Bridges,” Journal of Bridge Engineering, Vol.4, No.1, February 1999, pp. 71-78.
[37] Sennah, K.M., and Kennedy, J.B., “Shear Distribution in Simply-Supported Curved Composite Cellular Bridges,” Journal of Bridge Engineering, Vol.3, No.2, May 1998, pp.47-55.
[38] Sennah, K.M., and Kennedy, J.B., “State-of-the-art in Design of Curved Box-Girder Bridges,” Journal of Bridge Engineering, Vol.6, No.3, June 2001, pp.159-167.
[39] Shahawy, M., and Huang, D., “Analytical and Field Investigation of Lateral Load Distribution in Concrete Slab-On-Girder Bridges,” ACI StructuralJournal, Vol. 98, No. 4, July/August 2001, pp. 590-599.
[40] Tabsh, S.W., and Tabatabai, M., “Live Load Distribution in Girder BridgesSubject to Oversized Trucks,” Journal of Bridge Engineering, Vol.6, No.1, January-February 2001, pp. 9-16.
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209
APPENDIX A
210
Appendix A
Sample Calculations of Distribution Factors for Live Load Moment And Shear
A1 Precast Concrete Spread Box Beam, Bridge # 1 – State Route 7 over Leipers Creek
Number of beams ( ) = 3 bNNumber of design lanes ( ) = 2 LNSkew angle from the vertical roadway (θ ) = 15 degrees Spacing of beams (S) = 10.58 ft Span of beam (L) = 60.88 ft Edge-to-Edge width of bridge (W) = 30 ft Depth of beam (d) = 30 in
A1.1 AASHTO LRFD for Live Load Moment:
(a) Interior Beam: (Table 4.6.2.2.2b-1)
One Design Lane Loaded:
452.0)88.60)(12()30)(58.10(
0.358.10
0.120.3
25.0
2
35.025.0
2
35.0
=⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=
LSdSMDF
Two or More Design Lanes Loaded:
736.0)88.60(12
)30)(58.10(3.658.10
123.6
125.0
2
6.0125.0
2
6.0
=⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=
LSdSMDF
Range of Applicability:
⎪⎪⎩
⎪⎪⎨
⎧
≥≤≤≤≤≤≤
3in.65in.18ft140ft20ft5.11ft0.6
bNdLS
Skew Reduction Factor
983.0)15tan(25.005.10.1θtan25.005.1
=−=≤−=
SRFSRF
723.0)736.0)(983.0( ==MDF
Range of Applicability: °≤≤ 0600 .θ
211
(b) Exterior Beam: (Table 4.6.2.2.2d-1)
One Design Lane Loaded: (Lever Rule Method)
Figure A1 Bridge #1 Distribution Factor (1)
780.058.10
)25.525.11(21
=⎥⎦⎤
⎢⎣⎡ +
=MDF
Multiple Presence Factor for Single Lane = 1.2
MDF = 0.780(1.2) = 0.936
Two or More Design Lanes Loaded:
731.0)723.0)(011.1(
011.15.28
17.197.0
ft17.175.15.142.45.28
97.0
)( interior
==
=+=
=−−=
+=
=
MDF
e
d
de
MDFeMDF
e
e
Range of Applicability: ⎩⎨⎧
≤≤≤≤
ft5.11ft0.6ft5.40
Sde
212
A1.2 AASHTO LRFD for Live Load Shear:
(a) Interior Beam: (Table 4.6.2.2.3a-1)
One Design Lane Loaded:
751.088.60*0.12
3010
58.100.1210
.1.06.01.06.0
=⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=
LdSFD
Two or More Design Lanes Loaded:
966.088.60*0.12
304.758.10
0.124.7.
1.08.01.08.0
=⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=
LdSFD
Range Of Applicability:
⎪⎪⎩
⎪⎪⎨
⎧
≥≤≤≤≤≤≤
3in.65in.18ft140ft20ft5.11ft0.6
bNdLS
Skew Correction Factor:
016.1)0520.1(966.0
052.115tan583.10*60.12
30*88.60
0.1tan6
0.120.1
==
=+=+=
SDFS
Ld
CF οθ
Range of Applicability: οθ 600 ≤≤
(b) Exterior Beam: (Table 4.6.2.2.3b-1)
One Design Lane Loaded: (Lever Rule Method)
Figure A2 Bridge #1 Lever Rule Method
213
780.058.10
)25.525.11(21
=⎥⎦⎤
⎢⎣⎡ +
=SDF
Multiple Presence Factor for Single Lane = 1.2
SDF = 0.780(1.20) = 0.936
Two or More Design Lanes Loaded:
932.0)0162.1(917.0
917.01017.18.0
108.0
)( interior
==
=+=+=
=
SDF
de
SDFeSDF
e
Range of Applicability: ft5.40 ≤≤ ed
A1.3 AASHTO Standard for Live Load Moment and Shear:
(a) Interior Beam: (Art. 3.28.1)
Note: Since W = 26.5 < 32, the following equations are invalid. Use either lever rule method or the specified equations.
776.088.6058.10255.1
3)2)(2(
21
255.112.0)3(2.0)26.02.0)(2()5.26(07.012.020.0)26.010.0(07.0
2
=⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛+=
=−−−−=−−−−=
⎟⎠⎞
⎜⎝⎛+=
DF
kNNNWk
LSk
NNDF
BLL
B
L
Range of Applicability: ⎪⎩
⎪⎨
⎧
≤≤≤≤
≤≤
ft66ft32ft00.11ft57.6
104
WS
Nb
(b) Exterior Beam: (Art. 3.28.2)
Distribution factor shall be the greater of the lever rule method or 2NL/NB:
214
Figure A3 Bridge #1 Distribution Factor (2)
839.058.10
25.125.525.1121
=⎟⎠⎞
⎜⎝⎛ ++
=DF
-Or-
667.0322
212
21
=⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛=⎥
⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛=
B
L
NNDF
A1.4 Henry’s Method for Live Load Moment and Shear:
826.032)935.0(
10)5.26(
21
935.0)9.01(10
20)5.2630(1)(Factor Intensity
2102
1
65.210/5.2610
LanesofNumber
,ft5.26)75.1(230
=⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛=
=−−−
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
===
=−=
DF
IF
NIF
WDF
W
W
B
roadway
roadway
roadway
Table A1 Precast Spread Box Beam Distribution Factors for Live Load Moment
Bridge Number Structure Type
AASHTO LRFD Method
AASHTO Standard Method Henry's
MethodInterior Beam
Exterior Beam
Interior Beam
Exterior Beam
1 Precast Spread Box Beam 0.723 0.78 0.866 0.839 0.8262 Precast Spread Box Beam 1.186 1.186 1.186 1.186 1.1523 Precast Spread Box Beam 0.752 0.856 0.874 0.972 0.7594 Precast Spread Box Beam 0.343 0.494 0.433 0.643 0.489
215
Table A2 Precast Spread Box Beam Distribution Factors for Live Load Shear
Bridge Number Structure Type
AASHTO LRFD Method
AASHTO Standard Method Henry's
MethodInterior Beam
Exterior Beam
Interior Beam
Exterior Beam
1 Precast Spread Box Beam 1.016 0.932 0.866 0.839 0.8262 Precast Spread Box Beam 1.186 1.186 1.186 1.186 1.1523 Precast Spread Box Beam 1.027 1.168 0.874 0.972 0.7594 Precast Spread Box Beam 0.812 0.798 0.433 0.643 0.489
A2 Precast Concrete Bulb-Tee Beams, Bridge # 5 – State Route 1 over Rocky River
Area of beam (A) = 767 in2
Moment of Inertia (I) = 545894 in4
Compressive strength of concrete (f’c beam) = 6000 psi Compressive strength of concrete (f’c slab) = 3000 psi Modulus of elasticity of beam material (EB) = 4.696x106 psi Modulus of elasticity of slab material (ES) = 3.321x106 psi Number of beams ( ) = 5 bNNumber of design lanes ( ) = 3 LNSkew angle from the vertical roadway (θ ) = 15 degrees Spacing of beams (S) = 8.75 ft Span of beam (L) = 124.333 ft Edge-to-Edge width of bridge (W) = 40.5 ft Depth of concrete slab (ts) = 8.25 in
A2.1 AASHTO LRFD for Live Load Moment:
(a) Interior Beam: (Table 4.6.2.2.2b-1)
One Design Lane Loaded:
626
6
2
1.0
3
3.04.0
10597.2])025.41)(767(545894[10321.310696.4
025.4160.36225.875.972
)(
121406.0
xxxK
e
EEn
AeInK
LtK
LSSMDF
g
g
S
B
gg
s
g
=+⎟⎟⎠
⎞⎜⎜⎝
⎛=
=−−+=
=
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛+=
216
479.0)25.8)(333.124(12
10597.2333.12475.8
1475.806.0
1.0
3
63.04.0
=⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛+=
xMDF
Two or more design lane loads:
Range of Applicability:
⎪⎪⎩
⎪⎪⎨
⎧
≥≤≤
≤≤≤≤
4ft240ft20in.12in.5.4ft0.16ft5.3
B
s
NLt
S
Skew Reduction Factor: (Table 4.6.2.2.2e-1)
θ = 15 degrees < 30 degrees, therefore, SRF = 1.0
(b) Exterior Beam: (Table 4.6.2.2.2d-1)
One Design Lane Loaded: use lever rule method
See AASHTO Standard below.
743.075.8
5.95.321
=⎟⎠⎞
⎜⎝⎛ +
=MDF
Multiple Presence Factor for Single Lane = 1.2
MDF = 0.743(1.2) = 0.892
Two or More Design Lanes Loaded:
756.0)705.0)(072.1(
072.11.9
)75.15.4(77.01.9
77.0
)( interior
==
=⎥⎦⎤
⎢⎣⎡ −
+=+=
=
MDF
dee
MDFeMDF
Range of Applicability: ft5.5ft0.1 ≤≤− ed
702.0)25.8)(333.124)(12(
10597.2333.12475.8
5.975.8075.0
125.9075.0
1.0
3
62.06.0
1.0
3
2.06.0
=⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛+=
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛+=
xMDF
LtK
LSSMDF
s
g
217
A2.2 AASHTO LRFD for Live Load Shear:
(a) Interior Beam: (Table 4.6.2.2.3a-1)
One Design Lane Loaded:
710.02575.836.0
0.2536.0 =+=+=
SSDF
Two or More Design Lanes Loaded:
866.03575.8
1275.82.0
35122.0
0.20.2
=⎟⎠⎞
⎜⎝⎛−+=⎟
⎠⎞
⎜⎝⎛−+=
SSSDF
Range Of Applicability:
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
≥
≤≤
≤≤≤≤≤≤
4
in000,000,7in000,10
in.12in.5.4ft240ft20
ft16ft5.3
44
b
g
s
N
K
tLS
Skew Correction Factor:
900.0)866.0(038.1
038.115tan10*597.2
)25.8)(3.124(0.1220.00.1
10597.2)(
tan0.12
20.00.1
3.0
6
3
62
3.03
==
=⎟⎟⎠
⎞⎜⎜⎝
⎛+=
×=+=
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
SDF
CF
AeInK
KLt
CF
gg
g
s θ
Range of Applicability:
⎪⎪⎩
⎪⎪⎨
⎧
≥≤≤≤≤
≤≤
4ft240ft20
ft16ft5.3600
bNLS
θ ο
(b) Exterior Beam: (Table 4.6.2.2.3b-1):
One Design Lane Loaded: use lever rule method
218
See AASHTO Standard below
743.075.8
5.95.321
=⎟⎠⎞
⎜⎝⎛ +
=SDF
Multiple Presence Factor for Single Lane = 1.2
SDF = 0.743(1.2) = 0.892
Two or More Design Lanes Loaded:
788.0)900.0(875.0
875.01075.26.0
106.0
)( interior
==
=+=+=
=
SDF
de
SDFeSDF
e
Range of Applicability: ft5.5ft0.1 ≤≤− ed
A2.3 AASHTO Standard for Live Load Moment and Shear:
(a) Interior Beam: (Table 3.23.1)
795.05.5
75.821
5.521
=⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛=
SDF
Range of Applicability: ft14≤S
(b) Exterior Beam: use lever rule method
Figure A4 Bridge #5 Distribution Factor
743.075.8
5.95.321
=⎟⎠⎞
⎜⎝⎛ +
=DF
219
A2.4 Henry’s Method for Live Load Moment and Shear:
663.0)5.5/6()52)(75.0(
10)5.40(
21
75.0)(Factor Intensity
2102
1
05.410/5.4010
LanesofNumber
ft5.40)75.1(244
=⎥⎦⎤
⎢⎣⎡=
=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
===
=−=
DF
IFN
IFW
DF
WW
B
roadway
roadway
roadway
A multiplier 6/5.5 is used for precast I-beams and steel I-beams.
Table A3 Precast Concrete Bulb-Tee Distribution Factors for Live Load Moment
Bridge Number Structure Type
AASHTO LRFD Method
AASHTO Standard Method Henry's
Method Interior Beam
Exterior Beam
Interior Beam
Exterior Beam
5 Precast Concrete BT Beam 0.702 0.756 0.795 0.743 0.6638 Precast Concrete BT Beam 0.809 0.853 0.936 0.810 0.790
22 Precast Concrete BT Beam 0.641 0.617 0.757 0.610 0.71123 Precast Concrete BT Beam 0.650 0.625 0.757 0.610 0.711
Table A4 Precast Concrete Bulb-Tee Distribution Factors for Live Load Shear
Bridge Number Structure Type
AASHTO LRFD Method
AASHTO Standard Method Henry's
Method Interior Beam
Exterior Beam
Interior Beam
Exterior Beam
5 Precast Concrete BT Beam 0.900 0.788 0.795 0.743 0.6638 Precast Concrete BT Beam 0.970 0.833 0.936 0.810 0.790
22 Precast Concrete BT Beam 0.902 0.699 0.757 0.610 0.71123 Precast Concrete BT Beam 0.877 0.680 0.757 0.610 0.711
A3. Precast Concrete I-Beams, Bridge # 6 – Interstate 840 over Cox Road
Area of beam (A) = 560 in2
Moment of Inertia (I) = 125,390 in4
Compressive strength of concrete (f’c beam) = 5000 psi Compressive strength of concrete (f’c slab) = 3000 psi Modulus of elasticity of beam material (EB) = 4.287x106 psi
220
Modulus of elasticity of slab material (ES) = 3.321x106 psi Number of beams ( ) = 5 bNNumber of design lanes ( ) = 3 LNSkew angle from the vertical roadway (θ ) = 21.33 degrees Spacing of beams (S) = 9.0 ft Span of beam (L) = 67.42 ft Edge-to-Edge width of bridge (W) = 40.5 ft Depth of concrete slab (ts) = 8.25 in
A3.1 AASHTO LRFD for Live Load Moment:
(a) Interior Beam: (Table 4.6.2.2.2b-1)
One Design Lane Loaded:
Two or More Design Lanes Loaded:
1.0
3
2.06.0
125.9075.0 ⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛+=
s
g
LtK
LSSMDF
762.0)25.8)(42.67)(12(
1028.842.670.9
5.90.9075.0
1.0
3
52.06.0
=⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛+=
xMDF
Range of Applicability:
⎪⎪⎩
⎪⎪⎨
⎧
≥≤≤
≤≤≤≤
4ft240ft20
in.0.12in.5.4ft0.16ft5.3
B
s
NLt
S
546.0)25.8)(42.67)(12(
1028.842.67
914906.0
in1028.8])36.30)(560(390,125[10321.310287.4
in.36.3027.20225.875.945
)(
121406.0
1.0
3
53.04.0
4526
6
2
1.0
3
3.04.0
=⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛+=
=+⎟⎟⎠
⎞⎜⎜⎝
⎛=
=−−+=
=
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛+=
xMDF
xxxK
e
EEn
AeInK
LtK
LSSDF
g
g
S
B
gg
s
g
221
Skew angle, = 21.33 degrees, No skew reduction factor θ
(b) Exterior Beam: (Table 4.6.2.2.2d-1)
One Design Lane Loaded: use lever rule method
See AASHTO Standard method below
694.09
25.925.321
=⎟⎠⎞
⎜⎝⎛ +
=MDF
Multiple Presence Factor for Single Lane = 1.2
MDF = 0.694(1.2) = 0.833
Two or More Design Lanes Loaded:
775.0)762.0)(017.1(
017.11.9
)75.14(77.01.9
77.0
)( interior
==
=⎥⎦⎤
⎢⎣⎡ −
+=+=
=
MDF
de
MDFeMDF
e
Range of Applicability: ft5.5ft0.1 ≤≤− ed
A3.2 AASHTO LRFD for Live Load Shear:
(a) Interior Beam: (Table 4.6.2.2.3a-1)
One Design Lane Loaded:
720.025936.0
0.2536.0 =+=+=
SSDF
Two or More Design Lanes Loaded:
883.0359
1292.0
35122.0
0.20.2
=⎟⎠⎞
⎜⎝⎛−+=⎟
⎠⎞
⎜⎝⎛−+=
SSSDF
222
Range Of Applicability:
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
≥
≤≤
≤≤≤≤≤≤
4
in000,000,7in000,10
in.12in.5.4ft240ft20
ft16ft5.3
44
b
g
s
N
K
tLS
Skew Correction Factor:
Range of Applicability: ο600 ≤≤ θ
940.0)883.0(065.1
065.133.21tan10*28.8
)25.8)(159(0.1220.00.1
in1028.8)(
tan0.1220.00.1
3.0
5
3
452
3.03
==
=⎟⎟⎠
⎞⎜⎜⎝
⎛+=
×=+=
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
SDF
CF
AeInK
θK
LtCF
gg
g
s
(b) Exterior Beam: (Table 4.6.2.2.3b-1):
One Design Lane Loaded: use lever rule method
See AASHTO Standard below
694.09
25.925.321
=⎟⎠⎞
⎜⎝⎛ +
=SDF
Multiple Presence Factor for Single Lane = 1.2
SDF = 0.694(1.2) = 0.833
Two or More Design Lanes Loaded:
775.0)94.0(825.0
825.01025.26.0
106.0
)( nterior
==
=+=+=
=
SDF
de
SDFeSDF
e
i
Range of Applicability: ft5.5ft0.1 ≤≤− ed
223
A3.3 AASHTO Standard for Live Load Moment and Shear:
(a) Interior Beam: (Table 3.23.1)
818.05.50.9
21
5.521
=⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛=
SDF
Range of Applicability: ft14≤S
(b) Exterior Beam: use lever rule method
Figure A5 Bridge #6 Distribution Factor
694.09
25.925.321
=⎟⎠⎞
⎜⎝⎛ +
=DF
A3.4 Henry’s Method for Live Load Moment and Shear:
663.0)5.5/6(52)75.0(
10)5.40(
21
75.0)(Factor Intensity
2102
1
05.410/5.4010
LanesofNumber
ft 5.40)75.1(244
=⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛=
=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
===
=−=
DF
IFN
IFW
DF
W
W
B
roadway
roadway
roadway
A multiplier 6/5.5 is used for precast I-beams and steel I-beams.
224
Table A5 Precast Concrete I-Beam Distribution Factors for Live Load Moment
Bridge Number Structure Type
AASHTO LRFD Method
AASHTO Standard Method Henry's
Method Interior Beam
Exterior Beam
Interior Beam
Exterior Beam
6 Precast Concrete I-Beam 0.762 0.775 0.818 0.694 0.6637 Precast Concrete I-Beam 0.702 0.714 0.818 0.694 0.663
24 Precast Concrete I-Beam 0.849 0.905 0.962 0.843 0.782
Table A6 Precast Concrete I-Beam Distribution Factors for Live Load Shear
Bridge Number Structure Type
AASHTO LRFD Method
AASHTO Standard Method Henry's
Method Interior Beam
Exterior Beam
Interior Beam
Exterior Beam
6 Precast Concrete I-Beam 0.940 0.775 0.818 0.694 0.6637 Precast Concrete I-Beam 0.983 0.811 0.818 0.694 0.663
24 Precast Concrete I-Beam 0.990 0.861 0.962 0.843 0.782
A4 Cast-In-Place T-Beam, Bridge # 9 – Highland Road over State Route 137
Area of beam (A) = 2290 in2
Moment of Inertia (I) = 389837 in4
Compressive strength of concrete (f’c beam) = 3000 psi Compressive strength of concrete (f’c slab) = 3000 psi Modulus of elasticity of beam material (EB) = 3.321x106 psi Modulus of elasticity of slab material (ES) = 3.321x106 psi Number of beams ( ) = 3 bNNumber of design lanes ( ) = 2 LNSkew angle from the vertical roadway (θ ) = 9.83 degrees Spacing of beams (S) = 12.583 ft Span of beam (L) = 96 ft Edge-to-Edge width of bridge (W) = 31.56 ft Depth of concrete slab (ts) = 9.0 in
A4.1 AASHTO LRFD for Live Load Moment:
(a) Interior Beam: (Table 4.6.2.2.2b-1)
225
According to the AASHTO LRFD Table 4.6.2.2.2b-1, when = 3 use lesser of the
values obtained from the equations below for one and two design lanes loaded and the
lever rule.
bN
One Design Lane Loaded:
614.0)9)(96)(12(
10549.196583.12
14583.1206.0
in10549.1])5.22)(2290(389837[321.3321.3
5.2229
236
)(
121406.0
1.0
3
63.04.0
462
2
1.0
3
3.04.0
=⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛+=
=+⎟⎠⎞
⎜⎝⎛=
=+=
=
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛+=
xMDF
xK
INe
EEn
AeInK
LtK
LSSMDF
g
g
S
B
gg
s
g
Two or More Design Lanes Loaded:
1.0
3
2.06.0
125.9075.0 ⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛+=
s
g
LtK
LSSMDF
913.0)9)(96)(12(
10549.196583.12
5.9583.12075.0
1.0
3
62.06.0
=⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛+=
xMDF
Range of Applicability:
⎪⎪⎩
⎪⎪⎨
⎧
≥≤≤
≤≤≤≤
4ft240ft20
in.0.12in.5.4ft0.16ft5.3
B
s
NLt
S
Skew Reduction Factor: (Table 4.6.2.2.2e-1)
θ = 9.83 degrees < 30 degrees, therefore, SRF = 1.0
Lever Rule Method:
226
See AASHTO Standard below
944.0583.12
583.2583.8583.1221
=⎟⎠⎞
⎜⎝⎛ ++
=DF
(b) Exterior Beam: (Table 4.6.2.2.2d-1)
One Design Lane Loaded: use lever rule method
Figure A6 Bridge #10 Lever Rule Method
834.0583.12
5.135.721
=⎟⎠⎞
⎜⎝⎛ +
=MDF
Multiple Presence Factor for Single Lane = 1.2
MDF = 0.834(1.2) = 1.008
Two or More Design Lanes Loaded:
996.0)913.0)(091.1(
091.11.9
)75.1667.4(77.01.9
77.0
)( interior
==
=⎥⎦⎤
⎢⎣⎡ −
+=+=
=
MDF
dee
MDFeMDF
Range of Applicability: ft5.5ft0.1 ≤≤− ed
A4.2 AASHTO LRFD for Live Load Shear:
(a) Interior Beam: (Table 4.6.2.2.3a-1)
One Design Lane Loaded:
Because =3 Use lever rule method. The calculation is the same as the one for the AASHTO Standard method for interior beams.
bN
227
SDF = 0.944
Two Design Lanes Loaded:
In this bridge the number of beams is less than four. As specified in the AASHTO LRFDTable 4.6.2.2.3a-1 the lever rule method is used for the interior as well as the exterior beams. The details of loading and calculations are shown in AASHTO Standard method.
Therefore SDF = 0.944
Range Of Applicability:
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
≥
≤≤
≤≤≤≤≤≤
4
in000,000,7in000,10
in.12in.5.4ft240ft20
ft16ft5.3
44
b
g
s
N
K
tLS
Skew Correction Factor:
( )
969.0)944.0(026.1026.1
83.9tan98.2171801)9)(96(0.1220.00.1
in98.2171801)81.33(4.15248.429235321.3
10*321.3)(
tan0.1220.00.1
3.03
426
2
3.03
===
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
=+=+=
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
SDF
CF
AeInK
θK
LtCF
gg
g
s
Range of Applicability:
⎪⎪⎩
⎪⎪⎨
⎧
≥≤≤≤≤
≤≤
4ft240ft20
ft16ft5.3600
bNLS
θ ο
(b) Exterior Beam: (Table 4.6.2.2.3b-1):
One Design Lane Loaded: use lever rule method
834.0583.12
5.135.721
=⎟⎠⎞
⎜⎝⎛ +
=SDF (Same as the one for moment)
228
Multiple Presence Factor for Single Lane = 1.2
SDF = 0.834(1.2) = 1.008
Two or More Design Lanes Loaded:
863.0)969.0)(891.0(
891.010916.26.0
106.0
)( nterior
==
=+=+=
=
SDF
de
SDFeSDF
e
i
Range of Applicability: ft5.5ft0.1 ≤≤− ed
A4.3 AASHTO Standard for Live Load Moment and Shear:
(a) Interior Beam: (Table 3.23.1)
For S > 10.0 ft, load on each stringer shall be the reaction of the wheel loads, assuming
the flooring between the stringers to act as a simple beam.
Figure A7 Bridge #10 Distribution Factor (2)
944.0583.12
583.2583.8583.1221
=⎟⎠⎞
⎜⎝⎛ ++
=DF
229
(b) Exterior Beam: use lever rule method
Figure A8 Bridge #10 Distribution Factor (3)
974.0583.12
5.135.75.321
=⎟⎠⎞
⎜⎝⎛ ++
=DF
A4.4 Henry’s Method for Live Load Moment and Shear:
859.032)92.0(
10)28(
21
92.0)9.00.1(10
20280.1)(Factor Intensity
2102
1
8.210/2810
LanesofNumber
ft 28
=⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛=
=−⎟⎠⎞
⎜⎝⎛ −
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
===
=
DF
IF
NIF
WDF
W
W
B
roadway
roadway
roadway
Table A7 Cast-In-Place T-Beam Distribution Factors for Live Load Moment
Bridge Number Structure Type
AASHTO LRFD Method
AASHTO Standard Method Henry's
Method Interior Beam
Exterior Beam
Interior Beam
Exterior Beam
9 CIP Concrete T-Beam 0.802 0.889 0.873 0.828 0.86910 CIP Concrete T-Beam 0.913 0.996 0.944 0.974 0.85911 CIP Concrete T-Beam 0.703 0.676 0.681 0.602 0.644
230
Table A8 Cast-In-Place T-Beam Distribution Factors for Live Load Shear
Bridge Number Structure Type
AASHTO LRFD Method
AASHTO Standard Method Henry's
Method Interior Beam
Exterior Beam
Interior Beam
Exterior Beam
9 CIP Concrete T-Beam 0.942 0.784 0.873 0.828 0.86910 CIP Concrete T-Beam 0.969 0.863 0.944 0.974 0.85911 CIP Concrete T-Beam 0.826 0.640 0.681 0.602 0.644
A5 Cast-In-Place Multicell Box Beam, Bridge # 12 – Tri-City Airport Road
Compressive strength of concrete (f’c) = 3000 psi Modulus of elasticity (E) = 3.321x106 psi Number of cells ( ) = 3 cNNumber of design lanes ( ) = 2 LNSkew angle from the vertical roadway (θ ) = 0 degrees Spacing of beams (S) = 9.25 ft Span of beam (L) = 133.83 ft Edge-to-Edge width of bridge (W) = 44 ft
A5.1 AASHTO LRFD for Live Load Moment:
(a) Interior Beam: (Table 4.6.2.2.2b-1)
One Design Lane Loaded:
417.041
83.1331
6.325.975.111
6.375.1
45.035.045.035.0
=⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛ +=⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛ +=
cNLSMDF
Two or More Design Lanes Loaded:
668.083.133
18.525.9
4131
8.513 25.03.025.03.0
=⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=
LS
NcMDF
Range of Applicability: ⎪⎩
⎪⎨
⎧
≥≤≤≤≤
3ft240ft60ft0.13ft0.7
cNLS
(b) Exterior Beam: (Table 4.6.2.2.2d-1)
For Any Number of Design Lanes Loaded:
231
14eW
MDF =
We = half the web spacing, plus the total overhang
580.014125.8
ft512.85.3225.9
==
=+=
MDF
We
Range of Applicability: SWe ≤
A5.2 AASHTO LRFD for Live Load Shear:
(a) Interior Beam: (Table 4.6.2.2.3a-1)
One Design Lane Loaded:
715.083.133*12
665.9
25.90.125.9
1.06.01.06.0
=⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=
LdSSDF
Two or More Design Lanes Loaded:
899.083.133*12
663.7
25.90.123.7
1.06.01.09.0
=⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=
LdSSDF
Range Of Applicability:
⎪⎪⎩
⎪⎪⎨
⎧
≥≤≤
≤≤≤≤
3in.110in.35ft240ft20
ft13ft0.6
cNdLS
(b) Exterior Beam: (Table 4.6.2.2.3b-1):
One Design Lane Loaded: use lever rule method
Follow lever rule method as shown in the AASHTO Standard method for the exterior beam.
SDF = 0.649
Multiple Presence Factor for Single Lane = 1.2
232
SDF = 0.649(1.2) = 0.780
Two or More Design Lanes Loaded:
701.0899.0*78.0
78.05.12
75.164.05.12
64.0
)( interior
==
=+=+=
=
SDF
de
SDFeSDF
e
Range of Applicability: ft0.5ft0.2 ≤≤− ed
A5.3 AASHTO Standard for Live Load Moment and Shear:
(a) Interior Beam: (Table 3.23.1)
Range of Applicability: ft16≤S
661.00.7
25.921
0.721
=⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛=
SDF
(b) Exterior Beam: use lever rule method
Figure A9 Bridge #12 Lever Rule Method
649.025.9
9321
=⎟⎠⎞
⎜⎝⎛ +
=DF
233
A5.4 Henry’s Method for Live Load Moment and Shear:
608.0)52)(75.0(
10)5.40(
21
75.0)(Factor Intensity
2102
1
405.410/5.4010
LanesofNumber
ft5.405.344
=⎥⎦⎤
⎢⎣⎡=
=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
>===
=−=
DF
IFN
IFW
DF
WW
B
roadway
roadway
roadway
Table A9 CIP Multicell Box Beam Distribution Factors for Live Load Moment
Bridge Number Structure Type
AASHTO LRFD Method
AASHTO Standard Method Henry's
Method Interior Beam
Exterior Beam
Interior Beam
Exterior Beam
12 CIP Concrete Box Beam 0.668 0.58 0.661 0.649 0.608 13 CIP Concrete Box Beam 0.701 0.607 0.643 0.694 0.60814 CIP Concrete Box Beam 0.738 0.679 0.738 0.810 0.69815 CIP Concrete Box Beam 0.785 0.607 0.680 0.684 0.701
Table A10 CIP Multicell Box Beam Distribution Factors for Live Load Shear
Bridge Number Structure Type
AASHTO LRFD Method
AASHTO Standard Method Henry's
Method Interior Beam
Exterior Beam
Interior Beam
Exterior Beam
12 CIP Concrete Box Beam 0.899 0.701 0.661 0.649 0.60813 CIP Concrete Box Beam 0.900 0.738 0.643 0.694 0.60814 CIP Concrete Box Beam 1.280 1.084 0.738 0.810 0.69815 CIP Concrete Box Beam 1.086 0.868 0.680 0.684 0.701
A6 Steel I-Beam, Bridge # 16– State Route 81 over Nolichucky River
Area of beam (A) = 63.125 in2
Moment of Inertia (I) = 42425 in4
Compressive strength of concrete (f’c slab) = 3000 psi Modulus of elasticity of beam material (EB) 29 x 106 psi Modulus of elasticity of slab material (ES) = 3.321x106 psi
234
Number of beams ( ) = 5 bNNumber of design lanes ( ) = 3 LNSkew angle from the vertical roadway (θ ) = 0 degrees Spacing of beams (S) = 9.5 ft Span of beam (L) = 158 ft Edge-to-Edge width of bridge (W) = 46 ft Depth of concrete slab (ts) = 8.0 in
A6.1 AASHTO LRFD for Live Load Moment:
(a) Interior Beam: (Table 4.6.2.2.2b-1)
One Design Lane Loaded:
438.0)8)(158)(12(
10289.1158
5.914
5.906.0
in10289.1])0821.40)(125.63(42425[321.329
in.821.4041082.34
)(
121406.0
1.0
3
63.04.0
462
2
1.0
3
3.04.0
=⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛+=
=+⎟⎠⎞
⎜⎝⎛=
=−+=
=
+=
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛+=
xMDF
xK
eEEn
AeInK
LtK
LSSMDF
g
g
S
B
gg
s
g
Two or More Design Lanes Loaded:
1.0
3
2.06.0
125.9075.0 ⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛+=
s
g
LtK
LSSMDF
661.0)8)(158)(12(
10289.1158
5.95.95.9075.0
1.0
3
62.06.0
=⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛+=
xMDF
Range of Applicability:
⎪⎪⎩
⎪⎪⎨
⎧
≥≤≤
≤≤≤≤
4ft240ft20
in.0.12in.5.4ft0.16ft5.3
B
s
NLt
S
235
(b) Exterior Beam: (Table 4.6.2.2.2d-1)
One Design Lane Loaded: use lever rule method
Figure A10 Bridge #16 Lever Rule Method
711.05.9
75.375.921
=⎟⎠⎞
⎜⎝⎛ +
=MDF
Multiple Presence Factor for Single Lane = 1.2
MDF = 0.711(1.2) = 0.853
Two or More Design Lanes Loaded:
672.0)661.0)(017.1(
017.11.9
)75.14(77.01.9
77.0
)( interior
==
=⎥⎦⎤
⎢⎣⎡ −
+=+=
=
MDF
dee
MDFeMDF
Range of Applicability: ft5.5ft0.1 ≤≤− ed
A6.2 AASHTO LRFD for Live Load Shear:
(a) Interior Beam: (Table 4.6.2.2.3a-1)
One Design Lane Loaded:
740.025
5.936.00.25
36.0 =+=+=SSDF
236
Two or More Design Lanes Loaded:
917.035
5.912
5.92.03512
2.00.20.2
=⎟⎠⎞
⎜⎝⎛−+=⎟
⎠⎞
⎜⎝⎛−+=
SSSDF
Range Of Applicability:
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
≥
≤≤
≤≤≤≤≤≤
4
in000,000,7in000,10
in.12in.5.4ft240ft20
ft16ft5.3
44
b
g
s
N
K
tLS
(b) Exterior Beam: (Table 4.6.2.2.3b-1)
One Design Lane Loaded: use lever rule method
711.05.9
75.375.921
=⎟⎠⎞
⎜⎝⎛ +
=SDF
Multiple Presence Factor for Single Lane = 1.2
SDF = 0.711(1.2) = 0.853
Two or More Design Lanes Loaded:
756.0917.0*825.0
825.01025.26.0
106.0
)( interior
==
=+=+=
=
SDF
de
SDFeSDF
e
Range of Applicability: ft5.5ft 0.1 ≤≤− ed
A6.3 AASHTO Standard for Live Load Moment and Shear:
(a) Interior Beam: (Table 3.23.1)
864.05.55.9
21
5.521
=⎟⎠⎞
⎜⎝⎛=⎟
⎠⎞
⎜⎝⎛=
SDF
237
Range of Applicability: S
(b) Exterior Beam: (Art. 3.23.2.3.1.5)
745.0)5.9)(25.0(4
5.921
25.0421
=⎟⎟⎠
⎞⎜⎜⎝
⎛+
=⎟⎠⎞
⎜⎝⎛
+=
SSDF
Range of Applicability: ⎩⎨
A6.4 Henry’s Method for Live Load Moment and Shear:
695.0)5.5/6(52)75.0(
10)5.42(
21
75.0)(Factor Intensity
2102
1
425.410/5.4210
LanesofNumber
ft 5.425.346
=⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛=
=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
>===
=−=
DF
IFN
IFW
DF
WW
B
roadway
roadway
roadway
The multiplier 6/5.5 is used for steel I-beams.
Table A11 Steel I-Beam Distribution Factors for Live Load Moment
Bridge Number Structure Type
AASHTO LRFD Method
AASHTO Standard Method Henry's
Method Interior Beam
Exterior Beam
Interior Beam
Exterior Beam
16 Steel I-Beam 0.661 0.711 0.864 0.745 0.69517 Steel I-Beam 0.65 0.79 0.848 0.790 0.85118 Steel I-Beam 0.696 0.848 1.045 0.836 0.82819 Steel I-Beam 0.724 0.666 1.000 0.815 0.823
ft14≤
⎧≤≤
≥146
4S
NB
238
Table A12 Steel I-Beam Distribution Factors for Live Load Shear
Bridge Number Structure Type
AASHTO-LRFD Method
AASHTO-Standard Method Henry's
Method Interior Beam
Exterior Beam
Interior Beam
Exterior Beam
16 Steel I-Beam 0.917 0.756 0.864 0.745 0.69517 Steel I-Beam 0.906 0.807 0.848 0.790 0.85118 Steel I-Beam 1.242 1.118 1.045 0.836 0.82819 Steel I-Beam 1.099 0.796 1.000 0.815 0.823
A7 Steel Open Box Girders, Bridge # 20 – Granby Road over State Route 137
Number of beams ( ) = 4 bNNumber of Box ( ) = 2 boxNNumber of design lanes ( ) = 2 LNSkew angle from the vertical roadway (θ ) = 31.95 degrees Spacing of webs ( ) = 9.38 ft webSSpan of beam (L) = 252 ft Edge-to-Edge width of bridge (W) = 36 ft
A7.1 AASHTO LRFD for Live Load Moment and Shear:
Interior and Exterior Beams: (Table 4.6.2.2.2b-1)
Regardless of Number of Loaded Lanes:
556.02425.0
2285.005.0
21425.085.005.0
21
=⎥⎦
⎤⎢⎣
⎡+⎟
⎠⎞
⎜⎝⎛+=⎟⎟
⎠
⎞⎜⎜⎝
⎛++=
Lb
L
NNNDF
Range of Applicability: 5.15.0 ≤≤b
L
NN
A7.2 AASHTO Standard for Live Load Moment and Shear:
Interior Beams and Exterior Beams: (Article 10.39.2)
239
556.0285.0)0.1(7.11.0
21
load.for wheel85.07.11.021
0.122
ft32.5feet,in curbsbetween dth roadway wi
212
5.3212
where
=⎥⎦⎤
⎢⎣⎡ ++=
⎟⎟⎠
⎞⎜⎜⎝
⎛++=
==
==
===
=
DF
NRDF
R
W
WN
NNR
W
c
cW
B
W
Range of Applicability: 0
A7.3 Henry’s Method for Live Load Moment and Shear:
701.0)42)(863.0(
10)5.32(
21
2102
1
863.0)75.09.0(10
)305.32(9.0
425.310/5.3210
LanesofNumber
ft 5.325.336
=⎟⎠⎞
⎜⎝⎛=
⎟⎟⎠
⎞⎜⎜⎝
⎛=
=−⎟⎠⎞
⎜⎝⎛ −
−=
>===
=−=
DF
NIF
WDF
IF
WW
B
roadway
roadway
roadway
Table A13 Steel Open Box Beam Distribution Factors for Live Load Moment
Bridge Number Structure Type
AASHTO LRFD Method
AASHTO Standard Method Henry's
Method Interior Beam
Exterior Beam
Interior Beam
Exterior Beam
20 Steel Open Box Beam 0.556 0.556 0.556 0.556 0.70121 Steel Open Box Beam 0.645 0.645 0.645 0.645 0.606
515 .R. ≤≤
240
Table A14 Steel Open Box Beam Distribution Factors for Live Load Shear
Bridge Number Structure Type
AASHTO LRFD Method
AASHTO Standard Method Henry's
Method Interior Beam
Exterior Beam
Interior Beam
Exterior Beam
20 Steel Open Box Beam 0.556 0.556 0.556 0.556 0.70121 Steel Open Box Beam 0.645 0.645 0.645 0.645 0.606
241
APPENDIX B
242
Table B1 Precast Concrete Spread Box Beam Bridges, Database # 2
Bridge No.
Span Length
(ft)
Width (E-E) (ft)
Skew Angle (deg)
Number of
Beams
Girder Spacing
(in)
Beam Depth (in)
Slab Thickness
(in) Year Built
1 114.0 124.1 0.00 13 114 66 7.50 1982 2 95.0 122.5 0.00 13 114 66 7.50 1982 3 136.5 68.8 0.00* 7 114 72 6.50 1978 4 123.0 88.0 0.20 9 120 54 6.00 1968 5 112.0 88.0 30.00 10 96 54 6.00 1974 6 120.0 88.0 30.00 10 96 54 6.00 1974 7 120.8 82.0 0.40 8 120 63 6.00 1974 8 134.2 82.0 0.40 8 120 63 6.00 1974 9 36.0 32.5 30.00 5 77 17 8.00 1985
10 41.0 47.1 30.00 6 93 21 8.00 1985 11 64.0 66.5 18.00 7 120 33 8.00 1985 12 48.0 46.5 22.00 6 96 21 8.00 1983 13 29.3 46.5 0.00 6 96 17 8.00 1985 14 54.0 46.5 15.00 6 99 27 6.50 1981 19 32.0 59.2 0.00 6 108 26 8.50 1984 20 70.0 42.3 0.00 5 113 42 8.50 1979 21 30.0 25.7 0.00 3 108 18 8.00 1985 22 53.8 35.5 30.00 5 87 35 7.50 1987 23 47.0 46.6 25.00 7 80 21 8.00 1988 24 54.3 51.0 28.90 6 104 36 8.00 1986 25 40.0 35.0 0.00 4 108 21 8.00 1986 26 37.4 65.5 14.10 8 109 27 8.60 1984 27 79.5 65.5 14.10 10 80 40 8.00 1984 28 61.5 35.5 0.00 5 86 33 7.50 n/a 29 71.8 45.5 3.00 5 114 42 7.50 n/a 30 64.7 33.5 8.00 4 108 42 7.50 n/a 31 65.9 33.5 45.00 4 106 36 7.50 n/a 32 65.3 33.5 0.00 4 105 39 7.50 n/a 48 40.5 28.5 52.80 4 96 48 8.00 1987 49 88.6 28.5 52.80 4 96 48 8.00 1987 50 32.8 28.5 52.80 4 96 48 8.00 1987 51 53.6 52.3 30.00 5 124 27 8.00 1986 52 56.0 68.0 15.00 6 141 39 8.50 1985 53 76.0 68.0 15.00 6 141 39 8.50 1985 54 58.0 28.3 45.00 3 141 31 8.00 1985 55 100.0 40.0 0.00 4 116 40 7.50 1988
243
Table B2 Precast Concrete Bulb-Tee Beam Bridges, Database # 2
Bridge No.
Span Length
(ft)
Width (E-E) (ft)
Skew Angle (deg)
Number of
Beams
Girder Spacing
(ft)
Beam Depth
(ft)
Slab Thickness
(in)
Year Built
15 84.5 47.1 14.08 7 6.83 4.50 7.00 1970 16 45.0 47.1 14.08 7 6.83 4.50 7.00 1970 17 89.7 95.1 29.12 11 8.58 4.50 8.50 1983 79 79.2 34.0 8.00 5 7.10 5.25 6.00 1961 80 113.0 43.0 0.00 7 6.42 6.33 6.90 1984 81 96.0 58.0 0.00 8 7.50 5.70 6.25 1962 82 70.5 58.0 0.00 8 7.50 5.70 6.25 1962 83 84.0 68.6 0.00 10 7.00 4.60 6.25 1964 87 101.0 45.8 7.00 7 6.90 5.25 7.00 1981 89 72.5 37.0 5.30 5 7.75 4.70 6.25 1967 90 62.3 37.0 5.30 5 7.75 4.70 6.25 1967 91 52.0 53.0 47.70 6 8.83 5.25 6.50 1970 92 76.3 53.0 47.70 6 8.83 5.25 6.50 1970 93 84.2 53.0 47.70 6 8.83 5.25 6.50 1970 94 47.5 43.5 11.04 5 8.75 5.30 7.75 1981 95 74.8 43.5 11.04 5 8.75 5.30 7.75 1981 96 91.5 43.5 11.04 5 8.75 5.30 7.75 1981 97 58.4 41.0 40.00 5 8.25 5.20 6.25 1970 98 94.3 41.0 40.00 5 8.25 5.20 6.25 1970 99 94.3 57.0 47.20 7 8.00 5.20 6.25 1970
100 109.8 66.0 19.15 10 6.64 5.40 6.00 1969 101 79.0 66.0 19.15 8 8.50 5.60 6.40 1969 102 90.0 52.0 14.10 8 6.58 4.40 6.00 1971 103 67.5 46.0 0.00 7 6.83 4.70 6.00 1963 104 97.2 82.0 9.20 10 8.25 6.25 6.25 1971 114 82.0 46.8 0.00 5 9.70 4.50 7.50 1976 128 113.8 70.8 39.26 15 4.50 4.50 7.00 1980 129 113.7 82.8 39.30 13 6.33 4.50 7.00 1980 200 96.0 42.0 19.30 6 6.83 4.50 6.00 1977 203 97.0 50.8 0.00 6 8.75 5.25 8.00 1975 204 96.0 50.8 30.00 7 7.33 4.50 8.00 1975 334 92.0 29.5 0.00 4 8.00 4.50 7.50 1970 352 136.2 53.7 0.00 8 6.75 6.10 7.00 1978 360 48.0 42.2 22.00 6 7.25 6.10 7.00 1971 361 118.0 42.2 22.00 6 7.25 6.10 7.00 1971 362 90.0 42.2 22.00 6 7.25 6.10 7.00 1971
244
Table B3 Precast Concrete I-Beam Bridges, Database # 2
Bridge No.
Span Length
(ft)
Width (E-E) (ft)
Skew Angle
Number of
Beams
Girder Spacing
(ft)
Beam Depth
(ft)
Slab Thickness
(in) Year Built
18 77.0 40.6 20.00 6 7.00 3.75 7.00 1972 19 78.5 35.2 30.00 6 5.58 3.75 6.50 1962 79 79.2 34.0 8.00 5 7.10 5.25 6.00 1961 84 61.6 74.0 0.00 10 7.66 3.70 6.25 1968 85 27.0 74.2 0.00 10 7.66 3.70 6.25 1968 86 80.0 45.0 10.50 6 8.00 5.14 6.25 1964 87 101.0 45.8 7.00 7 6.90 5.25 7.00 1981 88 84.0 190.0 0.00 21 9.10 5.40 7.13 1972
105 67.7 68.7 12.90 12 6.00 3.60 6.00 1963 112 40.0 31.2 0.00 4 9.70 3.00 7.00 1957 113 60.0 31.2 0.00 6 5.83 3.00 7.00 1957 115 32.5 31.0 0.00 4 6.75 3.75 7.00 1960 116 72.0 31.0 0.00 4 6.75 3.75 7.00 1960 117 37.5 43.0 10.11 6 7.83 3.75 7.00 1960 118 64.5 43.0 10.11 6 7.83 3.75 7.00 1960 119 75.5 43.0 2.60 6 7.42 3.75 7.00 1960 120 63.5 31.5 3.50 5 5.50 3.75 7.00 1960 121 38.9 31.0 3.50 4 6.75 3.75 7.00 1960 122 79.0 42.5 0.00 9 4.75 3.75 7.00 1960 123 79.0 31.5 0.00 6 5.42 3.75 7.00 1960 124 47.5 31.5 0.00 4 9.03 3.75 7.00 1960 125 65.2 84.7 0.00 10 8.77 3.75 7.50 1975 126 87.0 84.7 0.00 10 8.77 3.75 7.50 1975 127 46.0 70.7 39.30 9 8.12 3.00 7.00 1980 201 81.0 50.8 0.00 8 6.50 3.75 9.00 1976 202 74.5 47.2 0.00 5 10.25 3.75 8.00 1972 234 63.0 33.3 9.00 8 3.67 3.00 5.25 1957 270 47.0 66.0 0.00 9 7.50 3.00 7.50 1967 301 50.0 49.4 0.00 5 10.50 3.75 8.50 1970 331 18.8 30.8 0.00 7 4.21 1.50 5.25 1975 332 60.0 34.0 35.42 4 9.33 3.75 6.25 1964 333 58.0 34.0 35.42 4 9.33 3.75 6.25 1964 357 113.0 52.0 45.00 8 6.83 4.80 6.50 1966
245
Table B4 Cast-in-Place Concrete T-Beam Bridges, Database # 2
Bridge No.
Span Length
(ft)
Width (E-E) (ft)
Skew Angle (deg)
Number of
Beams
Girder Spacing
(ft)
Beam Depth
(ft)
Slab Thickness
(in) Year Built
1 58.0 32.0 0.00 4 8.00 5.20 6.75 1973 2 71.0 32.0 0.00 4 8.00 5.20 6.75 1973
20 31.2 16.0 33.75 3 7.35 2.30 9.00 1926 21 31.0 30.1 27.50 6 5.75 2.30 6.50 1945 22 31.0 30.1 27.50 6 5.75 2.30 6.50 1945 23 31.0 27.2 0.00 4 7.60 3.00 9.00 1929 24 30.0 53.0 4.00 7 7.92 3.50 6.25 1966 25 29.2 39.0 0.00 5 8.33 3.50 6.50 1966 26 30.0 39.0 0.00 5 8.33 3.50 6.50 1966 27 55.0 53.0 4.00 7 7.92 3.50 6.50 1966 28 60.0 92.0 45.00 12 7.83 4.00 6.25 1967 29 60.0 66.0 45.00 9 7.55 4.00 6.37 1957 30 34.0 45.0 29.30 7 6.50 3.50 6.50 1965 31 71.0 33.2 0.00 4 8.50 5.00 6.62 1953 32 59.0 73.0 29.00 6 13.08 2.75 8.50 1937 33 39.0 41.0 5.00 5 8.25 2.50 6.50 1965 34 38.0 40.5 0.00 5 9.00 3.50 7.00 1966 35 53.0 33.8 24.36 4 8.00 4.75 6.50 1971 36 72.0 33.8 24.36 4 8.00 4.75 6.50 1971 37 43.0 33.8 33.21 6 6.25 3.50 7.25 1932 38 43.0 23.8 33.21 4 6.25 3.50 7.25 1932 39 53.0 33.0 22.00 3 12.75 5.00 9.00 1935 40 30.0 31.5 30.00 6 5.83 2.25 6.50 1956 41 46.0 30.7 0.00 4 8.00 2.33 8.00 19.38 42 34.0 30.7 0.00 4 8.00 2.33 8.00 1938 43 49.0 39.6 12.37 7 6.00 3.33 6.00 1966 44 60.0 39.7 12.37 7 6.00 3.33 6.00 1966 45 48.5 42.2 24.37 7 6.33 4.75 6.37 1954 46 34.0 58.0 0.00 8 7.50 2.50 6.25 1951 47 37.0 68.6 0.00 10 7.00 4.53 5.25 1963 48 46.0 45.8 10.05 6 8.00 5.17 6.50 1962
106 25.0 33.4 0.00 5 7.17 2.00 7.00 1960 178 60.0 34.5 45.00 6 6.50 3.10 5.75 1952 179 58.0 34.7 0.00 6 6.56 3.10 5.75 1947 180 65.0 50.2 0.00 8 6.27 2.50 6.00 1979 181 68.3 42.0 0.00 7 6.00 3.10 6.75 1983 205 39.0 34.3 0.00 4 8.33 3.00 9.00 1934 235 60.0 31.2 0.00 6 4.87 3.90 6.50 1979 236 40.0 32.2 0.00 7 4.96 2.20 6.50 1977 271 50.0 41.7 0.00 6 7.00 2.04 5.00 1984 272 50.0 42.0 0.00 6 7.00 2.90 5.00 1979 273 31.3 33.0 0.00 6 5.87 N/A 7.50 1942
246
Table B4 Cast-In-Place Concrete T-Beam Bridges, Database # 2 (Continued)
Bridge No.
Span Length
(ft)
Width (E-E) (ft)
Skew Angle (deg)
Number of
Beams
Girder Spacing
(ft)
Beam Depth
(ft)
Slab Thickness
(in)
Year Built
274 40.0 33.0 0.00 6 5.87 N/A 7.50 1942 302 70.0 41.4 43.78 5 9.00 4.25 7.50 1962 303 56.0 41.4 43.78 5 9.00 4.25 7.50 1962 304 27.0 41.4 43.78 5 9.00 4.25 7.50 1962 305 48.5 34.8 41.90 4 9.00 4.75 7.00 1961 306 62.0 34.8 41.90 4 9.00 4.75 7.00 1961 307 78.0 34.8 41.90 4 9.00 4.75 7.00 1961 308 56.0 34.8 41.90 4 9.00 4.75 7.00 1961 309 56.0 38.5 0.00 4 9.33 4.00 7.00 1948 310 70.0 38.5 0.00 4 9.33 4.00 7.00 1948 311 50.0 38.5 0.00 4 9.33 4.00 7.00 1948 312 37.0 38.5 0.00 4 9.33 4.00 7.00 1948 313 49.0 28.0 0.00 4 7.00 3.80 6.50 1951 314 65.0 28.0 0.00 4 7.00 3.80 6.50 1951 315 35.0 28.8 0.00 4 7.33 1.92 7.00 1939 316 50.0 28.8 0.00 4 7.33 1.92 7.00 1939 317 47.0 45.5 50.68 4 7.75 2.52 7.75 1979 318 63.0 45.5 50.61 4 7.75 2.52 7.75 1979 319 54.0 45.5 52.98 4 7.75 2.52 7.75 1979 320 35.0 35.2 0.00 8 4.75 2.50 6.00 1954 339 41.4 33.2 0.00 4 10.00 2.50 10.00 1926 340 72.0 28.5 45.00 2 16.00 3.75 8.53 1934 341 50.0 28.5 45.00 2 16.00 2.75 8.53 1934 342 93.0 28.5 45.00 2 16.00 3.75 8.53 1934 343 45.0 26.7 0.00 4 7.17 2.75 6.50 1939 344 22.6 39.4 0.00 5 8.54 2.75 6.50 1946 345 32.0 39.3 45.00 17 2.42 1.53 5.50 1950 346 24.0 39.3 45.00 17 2.42 1.53 5.50 1950 347 25.0 17.4 0.00 3 6.75 2.33 8.00 1930 348 12.0 37.8 0.00 5 7.50 2.54 6.50 1948 349 45.0 37.8 0.00 5 7.50 2.54 6.50 1948 350 40.0 22.8 0.00 2 12.00 4.08 11.00 1926 351 56.0 22.8 0.00 2 12.00 4.08 11.00 1926
247
Table B5 Cast-In-Place Concrete Multicell Box Beam Bridges, Database # 2
Bridge No.
Span Length (ft)
Width E-E (ft)
Number of
Cells
Girder Spacing
(ft)
Beam Depth (in)
Top Slab (in)
Bottom Slab (in)
1 83.5 40.7 5 7.66 57 6.40 5.50 2 89.5 39.3 6 7.00 60 7.50 5.50 3 75.0 60.7 7 8.25 48 6.75 5.75 4 62.0 39.0 4 8.50 48 7.00 5.40 6 105.5 66.0 8 7.75 60 6.50 5.50 8 122.0 28.0 4 7.50 78 6.50 5.50 9 112.0 28.0 3 8.90 72 7.10 6.25
10 106.0 28.0 4 7.10 69 6.25 5.50 11 55.6 80.0 10 8.00 36 6.60 5.50 12 81.3 37.7 5 7.00 54 6.25 5.50 13 108.0 26.0 3 7.50 72 6.60 5.50 14 100.0 66.0 7 9.00 60 7.25 6.50 15 98.0 35.0 4 8.50 66 6.75 5.40 16 71.5 40.0 6 7.50 48 6.60 5.50 19 35.2 40.0 5 8.00 60 6.60 5.50 20 75.6 51.0 7 7.20 51 6.25 5.50 23 91.0 51.0 7 7.10 54 6.25 5.50 24 66.0 41.0 5 8.20 45 6.50 5.50 26 100.0 40.0 5 9.00 74 7.50 6.00 27 104.0 52.3 7 7.00 79 7.00 6.00 29 75.0 26.0 3 7.10 54 7.75 6.00 31 85.0 31.3 3 9.25 54 8.25 6.50 36 46.0 72.0 9 8.20 60 7.75 6.00 37 110.0 72.0 9 8.20 60 7.75 6.00 38 77.8 72.0 9 8.20 60 7.75 6.00 39 100.2 72.0 9 8.20 60 7.75 6.00 40 110.0 22.0 3 7.40 48 7.75 6.50 41 94.0 26.0 3 7.25 60 7.75 6.00 42 94.0 26.0 4 7.25 60 7.75 6.00 44 95.0 26.0 3 7.25 60 7.75 6.00 45 95.0 26.0 3 7.25 60 7.75 6.00 46 95.0 26.0 3 7.20 60 7.75 6.00 47 80.0 26.0 3 7.20 60 7.75 6.00 48 98.0 26.0 3 7.20 60 7.75 6.00 49 61.0 38.0 4 8.50 44 7.00 6.00 50 62.0 34.0 4 8.50 42 7.00 6.00 51 120.0 28.0 3 9.25 69 7.25 6.50 53 88.1 30.3 4 8.60 54 7.00 6.00 54 92.3 37.0 5 8.00 57 6.40 6.00 56 89.0 66.0 9 7.40 54 6.50 5.50 57 78.0 60.0 9 8.00 48 6.60 6.00 59 77.0 51.0 6 8.10 54 6.90 6.00
248
Table B5 Cast-In-Place Concrete Multicell Box Beam Bridges, (Continued)
Bridge No.
Span Length (ft)
Width E-E (ft)
Number of
Cells
Girder Spacing
(ft)
Beam Depth
(in)
Top Slab (in)
Bottom Slab (in)
60 101.2 40.0 5 7.60 66 6.75 6.00 62 90.0 66.0 9 7.50 60 7.00 5.50 63 95.0 51.0 6 7.75 63 6.50 5.50 64 65.6 49.0 7 7.00 42 6.25 5.50 65 80.0 65.0 8 8.50 57 6.75 5.50 68 62.5 64.0 9 7.66 38 6.40 5.50 69 78.0 37.0 5 7.00 48 6.25 5.50 70 123.0 28.0 4 7.30 78 6.50 6.00 71 80.7 32.0 5 8.00 48 6.60 6.00 72 99.0 30.0 4 8.00 60 6.60 6.00 73 70.0 39.7 5 7.30 42 6.25 6.00 74 91.4 37.0 4 8.40 60 6.75 5.75 75 79.0 32.0 4 7.50 48 6.50 6.00 77 122.3 39.0 4 8.70 84 7.00 6.00 79 79.0 51.0 7 6.60 51 6.10 5.50 80 90.8 66.0 7 9.33 60 7.25 6.50 81 75.5 51.0 7 6.62 51 6.10 5.50 82 75.0 78.0 9 8.21 46 6.60 5.75 83 92.0 32.0 4 8.50 69 6.75 5.50 84 137.0 36.0 5 7.00 84 6.25 5.50 85 91.9 40.0 5 8.00 57 6.60 5.50 86 60.5 39.0 5 7.50 48 6.50 5.50 87 99.0 32.0 4 8.50 702 7.00 6.00 88 111.0 24.0 3 9.00 72 7.00 6.00 90 92.0 28.0 3 9.00 60 7.00 6.00 91 87.0 32.0 4 8.50 60 7.00 6.00 92 102.3 52.0 8 6.98 69 6.25 5.50 93 86.0 150.0 18 8.33 57 7.40 5.75 94 52.0 26.0 4 7.42 36 6.25 5.50 95 93.4 40.0 4 9.00 60 8.10 6.25 97 80.5 54.0 7 7.75 72 6.40 5.50 98 86.1 66.7 8 7.91 60 6.50 5.50
100 90.2 138.0 19 7.16 60 6.25 5.50 102 121.0 22.0 3 7.00 72 6.25 5.50 103 75.0 38.0 5 8.00 60 6.50 5.50 105 100.0 22.0 3 7.00 60 6.50 5.50 106 115.1 22.0 3 7.00 84 6.40 5.50 108 92.0 22.0 3 7.16 60 6.25 5.50 109 62.4 100.0 13 7.50 48 6.40 5.50 110 120.0 34.0 3 9.00 79 7.00 6.00 112 84.0 38.0 4 8.63 48 6.50 6.00 113 70.0 76.3 9 8.50 60 8.25 5.75 119 85.0 56.0 6 8.50 56 7.00 6.00
249
Table B6 Steel I-Beam Bridges, Database # 2
Bridge No.
Span Length
(ft)
Width (E-E) (ft)
Skew Angle (deg)
Number of
Beams
Girder Spacing
(ft)
Beam Depth
(ft)
Slab Thickness
(in)
Year Built
3 67.0 34.0 20.00 4 8.83 2.92 7.00 1961 4 67.0 34.0 20.00 4 8.83 2.92 7.00 1961 5 73.0 34.0 20.00 4 8.83 2.92 7.00 1961 6 77.0 34.0 20.00 4 8.83 2.92 7.00 1961 7 86.0 34.0 20.00 4 8.83 2.92 7.00 1961 8 53.0 35.0 30.00 4 9.33 2.75 7.00 1958 9 67.0 35.0 30.00 4 9.33 2.75 7.00 1958
10 46.0 35.2 9.77 4 8.83 3.00 7.50 1959 11 79.0 35.2 9.77 4 8.83 3.00 7.50 1959 12 44.7 25.2 20.00 4 7.00 1.33 9.00 1934 13 30.0 34.0 0.00 5 7.50 2.00 7.75 1937 14 40.0 34.0 0.00 5 7.50 2.00 7.75 1937 49 61.2 48.5 30.00 8 6.50 3.56 6.25 1953 50 47.0 36.0 60.54 7 5.17 2.50 9.00 1936 51 113.2 34.0 0.00 4 8.50 5.21 7.13 1967 52 121.7 33.0 46.96 4 9.33 6.00 7.25 1958 53 58.0 33.3 0.00 4 9.33 4.10 7.00 1955 54 50.0 33.5 30.00 5 7.50 3.75 6.50 1955 55 130.1 33.3 64.20 5 8.33 5.42 6.38 1955 56 92.5 44.0 63.47 6 7.31 4.56 6.75 1956 57 80.7 27.5 0.00 5 5.25 3.51 6.00 1949 58 105.2 33.6 12.57 4 9.20 4.83 7.25 1958 59 187.0 128.0 66.10 15 8.50 6.00 7.25 1962 60 70.5 75.0 40.99 9 8.66 3.79 6.38 1956 61 130.0 41.0 0.00 3 15.50 7.92 9.63 1971 62 155.0 41.0 0.00 3 15.50 7.92 9.63 1971 63 71.0 33.9 20.39 3 11.50 4.70 8.50 1947 64 35.0 33.9 20.39 3 11.50 4.70 8.50 1947 65 68.0 26.3 0.00 4 6.66 3.75 6.75 1954 66 60.0 35.9 2.17 4 9.50 3.58 7.13 1958 67 116.0 33.6 0.00 4 9.00 6.50 7.00 1960 68 140.0 33.3 40.00 3 12.00 8.00 7.75 1959 69 51.3 57.7 0.00 9 6.50 3.25 7.00 1957 70 51.3 33.6 0.00 5 6.50 3.25 6.50 1957 71 100.0 36.2 0.00 4 10.00 5.70 7.25 1951 72 75.3 43.9 0.00 6 7.75 5.00 6.63 1955 73 91.3 43.9 0.00 6 7.75 5.00 6.63 1955 74 65.5 33.3 46.78 4 8.75 3.64 6.88 1958 75 48.8 33.3 46.78 4 8.75 3.64 6.88 1958 76 151.1 33.3 0.00 3 12.00 8.33 7.75 1958 77 75.0 33.3 0.00 3 12.00 8.33 7.75 1958 78 55.0 37.9 15.52 4 10.00 3.66 7.13 1958
250
Table B6 Steel I-Beam Bridges, Database # 2 (Continued)
Bridge No.
Span Length
(ft)
Width (E-E) (ft)
Skew Angle (deg)
Number of
Beams
Girder Spacing
(ft)
Beam Depth
(ft)
Slab Thickness
(in)
Year Built
109 180.0 46.8 22.92 6 8.20 4.67 7.00 1980 110 43.0 30.7 11.55 10 3.00 2.50 10.00 1966 111 49.0 30.7 11.55 10 3.00 2.50 10.00 1966 174 70.0 42.7 0.00 6 7.50 3.00 8.50 1977 175 100.0 33.7 0.00 5 7.00 3.00 8.00 1973 176 201.0 67.2 10.00 8 8.67 9.50 8.50 1977 177 161.0 67.2 10.00 8 8.67 9.50 8.50 1977 182 40.0 28.3 30.00 13 2.29 1.50 10.00 1938 183 56.3 33.3 0.00 7 5.33 2.50 7.25 1920 184 28.0 22.0 0.00 9 2.58 1.25 6.50 1926 185 54.3 30.7 45.00 12 2.62 2.17 7.25 1931 186 76.0 30.3 0.00 7 4.83 2.25 7.00 1935 187 51.0 33.2 0.00 7 5.25 2.50 6.50 1940 188 50.0 34.5 0.00 5 7.00 2.75 6.00 1950 189 68.0 34.0 0.00 5 7.00 3.00 6.00 1955 190 65.0 34.5 0.00 5 7.00 3.00 6.00 1950 191 121.5 64.3 0.00 8 8.08 3.67 9.00 1978 192 47.5 50.3 60.00 6 8.83 2.50 9.00 1977 193 65.0 50.3 60.00 6 8.83 2.50 9.00 1977 194 98.0 45.0 0.00 5 9.83 4.83 8.25 1973 195 125.0 49.0 0.00 5 9.83 4.83 8.25 1973 196 69.8 60.8 56.87 7 8.92 3.50 6.50 1964 197 109.5 60.8 56.87 7 8.92 3.50 6.50 1964 198 69.8 60.8 56.87 7 8.92 3.50 6.50 1964 199 89.0 35.2 0.00 4 9.33 3.00 6.75 1962 206 105.0 47.7 0.00 6 8.67 3.00 12.00 1955 207 130.0 47.7 0.00 6 8.67 3.00 12.00 1955 208 100.7 47.0 0.00 8 6.60 1.50 7.00 1961 209 44.5 80.0 8.29 13 6.58 2.00 7.00 1945 210 96.5 57.8 27.93 8 7.40 1.33 7.50 1968 211 116.5 59.0 51.83 8 7.75 1.67 7.50 1968 212 110.0 81.0 51.83 11 7.60 1.67 7.50 1968 213 87.3 31.3 0.00 5 7.00 3.00 8.50 1970 214 71.7 57.0 10.64 7 8.57 3.00 7.00 1962 215 57.0 56.0 46.83 8 7.33 2.75 7.50 1967 216 86.3 56.0 46.33 8 7.33 2.75 7.50 1967 217 76.3 56.0 46.83 8 7.33 2.75 7.50 1967 218 31.9 33.5 28.00 5 7.51 2.75 7.00 1960 219 58.6 33.5 17.00 5 7.61 2.75 7.00 1960 220 63.0 34.3 52.10 6 5.57 3.00 6.50 N/A
251
Table B6 Steel I-Beam Bridges, Database # 2 (Continued)
Bridge No.
Span Length
(ft)
Width (E-E) (ft)
Skew Angle (deg)
Number of
Beams
Girder Spacing
(ft)
Beam Depth
(ft)
Slab Thickness
(in)
Year Built
223 51.9 70.0 6.43 12 5.55 3.00 7.50 N/A 224 83.6 63.0 13.00 8 8.23 4.00 7.50 1950 225 45.0 32.9 35.17 5 7.00 2.75 7.00 N/A 226 92.7 32.9 35.17 5 7.00 3.00 7.00 N/A 227 50.5 32.9 35.17 5 7.00 2.75 7.00 N/A 228 39.0 53.5 2.06 8 6.33 2.50 7.00 1955 229 84.6 53.5 2.06 8 6.83 3.00 7.00 1955 230 48.7 71.0 16.35 8 9.50 3.00 7.50 1955 231 80.3 71.0 16.35 8 9.50 3.00 7.50 1955 232 41.3 47.8 7.16 6 8.79 2.75 7.25 1957 233 75.7 47.8 7.16 6 8.79 3.00 7.25 1957 237 30.0 35.6 8.42 5 7.25 2.00 7.50 1953 238 38.9 35.7 8.42 5 7.25 2.50 7.50 1953 239 137.0 57.5 61.55 5 12.75 7.38 8.19 1962 240 55.0 44.0 12.00 6 7.87 2.75 8.25 1985 241 39.0 28.5 0.00 5 5.75 2.25 6.75 1941 242 162.0 56.0 58.93 6 10.17 11.71 9.00 1968 243 43.0 28.0 0.00 4 7.50 2.50 8.00 1985 244 12.0 30.0 17.50 11 3.00 0.83 4.42 N/A 245 75.0 75.0 56.84 12 6.75 3.00 7.75 1957 246 63.0 44.5 8.31 6 7.95 3.00 8.75 1964 247 42.5 37.7 0.00 12 3.21 2.00 4.42 1960 248 52.0 72.0 30.06 9 8.56 3.00 8.00 1968 249 72.0 72.0 20.05 9 8.56 3.00 8.00 1968 250 27.0 29.5 0.00 5 5.75 1.75 7.25 1953 251 65.5 29.5 0.00 5 5.75 1.75 7.25 1953 252 63.5 34.3 24.59 4 9.00 3.00 7.75 1964 253 45.0 34.3 24.59 4 9.00 3.00 7.75 1964 254 44.0 40.0 25.00 5 8.87 2.75 7.50 1955 255 55.0 40.0 25.00 5 8.87 2.75 7.50 1955 256 62.6 63.5 2.50 8 8.83 3.00 8.75 1972 257 65.5 59.5 2.50 7 8.83 3.00 8.75 1972 258 64.0 30.0 2.50 4 8.83 3.00 8.75 N/A 259 74.5 58.0 0.00 7 8.33 3.00 8.50 1965 260 66.3 58.0 0.00 7 8.33 3.00 8.50 1965 261 80.0 58.0 0.00 7 8.33 3.00 8.50 1965 262 92.1 58.0 0.00 7 8.33 3.00 8.50 1965 263 56.0 58.0 0.00 7 8.33 3.00 8.50 1965 264 70.0 44.0 30.00 6 7.87 3.00 7.75 1963 265 56.0 44.0 30.00 6 7.87 3.00 7.75 1963 266 80.0 55.2 10.26 7 8.25 3.00 9.00 1959 267 48.0 55.2 10.26 7 8.25 3.00 9.00 1959
252
Table B6 Steel I-Beam Bridges, Database # 2 (Continued)
Bridge No.
Span Length
(ft)
Width (E-E) (ft)
Skew Angle (deg)
Number of
Beams
Girder Spacing
(ft)
Beam Depth
(ft)
Slab Thickness
(in)
Year Built
275 36.0 22.5 0.00 5 5.25 2.17 8.75 1928 276 34.8 22.5 0.00 5 5.25 2.17 8.75 1928 277 50.0 24.3 0.00 5 5.17 2.75 6.50 1938 278 30.0 27.0 0.00 6 4.50 1.75 6.00 1935 279 34.4 24.3 45.00 5 5.17 2.25 6.50 1931 280 36.0 24.3 45.00 5 5.17 2.25 6.50 1931 281 31.7 28.0 30.00 6 4.92 2.00 7.50 1947 282 40.0 28.0 30.00 6 4.92 2.25 7.50 1947 283 61.0 29.0 0.00 6 4.92 2.75 7.50 1936 284 41.3 31.0 0.00 5 6.58 2.50 7.50 1956 285 59.8 31.0 0.00 5 6.58 3.17 7.50 1956 286 32.7 27.0 45.00 6 4.50 1.75 7.50 1935 287 58.9 27.0 45.00 6 4.50 3.00 7.50 1935 288 37.2 22.5 0.00 5 5.25 2.17 8.75 1927 291 38.8 31.0 0.00 5 6.58 2.50 7.50 1956 292 44.0 26.3 30.00 6 4.92 2.25 7.50 1935 293 45.7 26.3 30.00 6 4.92 2.25 7.50 1935 294 31.3 26.5 0.00 5 5.67 2.00 8.00 1932 295 30.0 26.5 0.00 5 5.67 2.00 8.00 1932 296 26.0 43.0 0.00 9 5.25 N/A 5.00 1955 297 72.0 31.0 35.83 5 6.58 2.75 6.00 1954 298 81.50 31.0 35.83 5 6.58 2.75 6.00 1954 299 125.0 41.0 0.00 4 11.00 5.00 10.00 1983 300 150.0 41.0 0.00 4 11.00 5.00 10.00 1983 322 140.0 59.0 0.00 5 13.50 8.81 6.50 1951 323 113.0 76.0 0.00 6 9.00 4.00 7.00 1962 324 142.0 76.0 0.00 6 9.00 4.00 7.00 1962 325 64.0 34.8 30.00 4 11.00 4.50 7.50 1969 326 152.5 34.8 30.00 4 11.00 4.50 7.50 1969 327 53.0 34.8 30.00 4 11.00 4.50 7.50 1969 328 52.5 56.0 0.00 27 2.00 1.50 5.00 1960
253
Table B7 Steel Open Box Beam Bridges, Database # 2
Bridge No.
Span Length
(ft)
Width (E-E) (ft)
Skew Angle (deg)
Number of
Beams
Girder Spacing
(in)
Beam Depth (in)
Slab Thickness
(in) Year Built
15 250.0 60.7 0.00 7 104 N/A 5.00 1973 16 79.7 48.0 0.00 4 144 27 8.50 1980 17 84.5 48.0 0.00 4 144 27 8.50 1980 18 61.8 82.7 6.00 7 144 27 8.50 1988 33 70.0 33.3 3.00 2 198 42 9.00 1987 34 186.2 32.0 60.50 2 198 50 8.20 1982 35 213.0 32.0 0.00 2 198 63 8.50 1982 36 256.5 32.0 0.00 2 198 63 8.50 1982 37 207.5 32.0 0.00 2 198 63 8.50 1982 38 83.5 32.0 0.00 2 198 63 8.50 1982 39 252.0 35.0 31.50 2 225 65 8.50 1982 40 137.5 44.0 0.00 2 288 58 8.50 1982 41 192.0 44.0 0.00 2 288 58 9.50 1980 42 114.8 44.0 0.00 2 288 48 9.50 1982 43 166.1 44.0 0.00 2 288 48 9.50 1982 44 281.7 44.0 0.00 2 288 48 9.50 1982 45 204.1 44.0 0.00 2 288 48 9.50 1982 46 159.5 44.0 0.00 2 288 48 9.50 1982 47 101.8 44.0 0.00 2 288 48 9.50 1982
254