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

xi

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

xii

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.

193

(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

198

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.

199

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.

203

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.

208

[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.

[41] Tarhini, K.M., and Frederick, G.R., “Wheel Load Distribution in I-girder Highway Bridges,” Journal of Structural Engineering, Vol.118, No.5, May 1992, pp. 1285-1295.

[42] Tennessee Structures Memorandum 043, “Lateral Distribution of Structural Loads,” Tennessee Department of Transportation, Nashville, Tennessee 1996.

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[44] Tobias, D.H., Anderson, R.R., Khayyat, S.K., Uzman, Z.B., and Reichers, K.L., “Simplified AADHTO LRFD Girder Live Load Distribution in Illinois,” ASCE Journal of Bridge Engineering, Manuscript Number: BE/2002/022768.

[45] Zokaie, T., “AASHTO-LRFD Live Load Distribution Specifications,” Journal of Bridge Engineering, Vol.5, No.2, May 2000, pp.131-138.

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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