Table Of ContentRE-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. 2. Government Accession No. 3. Recipient’s Catalog No.
TNSPR-RES 1218
4. Title and Subtitle 5. Report Date
Re-Examination of the Simplified Method (Henry’s 6/25/03
Method) of Distribution Factors for Live Load Moment 6. Performing Organization Code
and Shear
7. Author(s) X. Sharon Huo, Stewart Conner and Rizwan Iqbal 8. Performing Organization Report No.
9. Performing Organization Name and Address 10. Work Unit No. (TRAIS)
Center for Electric Power
11. Contract or Grant No.
Box 5032, Tennessee Technological University
CUT 265
Cookeville, TN 38505-0001
12. Sponsoring Agency Name and Address 13. Type of Report and Period Covered
Structures Division November 1, 2001 to June 30, 2003
Tennessee Department of Transportation 14. Sponsoring Agency Code
James K. Polk Building, Suite 1100
505 Deaderick Street, Nashville, TN 37243-0339
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 18. Distribution Statement
Bridges, live load distribution, distribution factor,
Henry’s method, finite element analysis, moment
distribution, and shear distribution
19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of Pages 22. Price
Unclassified Unclassified 254
Form DOT F 1700.7 (8-72) Reproduction of Completed page authorized
TABLE OF CONTENTS
LIST OF TABLES...................................................................................................................ix
LIST OF FIGURES.xi
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 DISTRIBUTION
FACTOR 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
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Table Page
6.1 Precast Concrete Spread Box Beam, FEA vs. Modified Henry’s Method for
Moment, Database #1 ..................................................................................141
6.2 Precast Concrete Spread Box Beam, LRFD vs. Modified Henry’s Method for
Moment, Database #1 ..................................................................................141
6.3 Precast Concrete Spread Box Beam, LRFD vs. Modified Henry’s Method for
Moment, 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 for
Moment, 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
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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
