BEHAVIOR OF LIGHT RAIL BRIDGE: LIVE LOAD DISTRIBUTION AND
SUBSTRUCTURE SETTLEMENT
By
DI WEI
B.S. Northeast Forestry University, China, 2013
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
Of the requirements for the degree of
Master of Science
Civil Engineering
2015
2015
Dl WEI
ALL RIGHTS RESERVED
This thesis for the Master of Science degree by
Di Wei
Has been approved for the
Civil Engineering program
By
Yail Jimmy Kim, Chair
Chengyu Li
Fredrick Rutz
October 10, 2015
Wei Di (M.S., Civil Engineering)
Behavior of light rail bridges: live load distribution and substructure settlement
Thesis directed by Associate Professor Yail Jimmy Kim
ABSTRACT
The live load distribution factors (LDF) provided by AASHTOLRFD Bridge Design
Specification have been in effect for almost 20 years to calculate bending moment and
shear force for highway bridge design. These equations are calibrated based on elastic
finite element analysis. Maximum bending moments and shear forces are determined
by the load distribution factors. Load distribution factors estimated by the LRFD
equations are generally conservative compared with refined analysis results. The use of
the AASHTO LRFD equations may not provide accurate load distribution factors for light
rail bridges because the load configurations of light rail trains are different from those of
highway vehicles. This study examines live load distribution factors for light rail bridges
and proposes new equations.
For this study, standard light rail trains are placed on five different types of light rail
bridges in Denver, Colorado. The focus of this thesis is on developing live load
distribution factor equations for light rail bridges. The behavior of these light rail bridges
subjected to 2, 4 and 6 substructure settlements is also studied.
In this study, 155 light rail bridges are designed and modeled. The load is applied to the
five types of light rail bridges. The moment and shear of each bridge are predicted. This
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research proposes thirtyfive new equations for five different types of light rail bridges
for bending and shear, including interior and exterior girders. The distribution factors
which have been calculated by the developed equations are compared with the
AASHTOLRFD distribution equations. Another comparison is also made against the
responses of the constructed bridges.
The form and content of this abstract are approved. I recommend its publication.
Approved: Yail jimmy Kim
ACKOWLEDGEMENTS
I would like to express my sincerest gratitude to the multitude of individuals who
assisted with this thesis. Dr. Kim, my advisor, who has been instrumental in the
theoretical part of this thesis. Your encouragement, wisdom and support have not gone
unnoticed. You give me the chance to work for you and let me know more about the
design for the bridge. Without your assistance, this thesis would not be where it is
today. Sincere thanks is offered to Dr.Chengyu Li and Dr.Frederick R.Rutz, not only for
being part of my graduate committee but also for the two years teaching and helping.
Also thanks Dr.Li give me the chance to come to this university. Additionally, I want to
say thanks to my teammates Thushara, Yongcheng Ji and Lianjie Liu, They give me so
much help during the last year. I am grateful to acknowledge financial support provided
by the national academy of science. Finally I want to thank my parents, they give me the
financial support for studying in United States.
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CONTENTS
Chapter
1. Introduction.........................................................................1
1.1 Background.....................................................................1
1.2 Objectives.....................................................................2
1.3 Organization ..................................................................3
2. Literature Review....................................................................6
2.1 Bridge design codes ...........................................................6
2.2 Load and load combinations.....................................................8
2.3 Load distribution factors......................................................9
2.4 Live load models..............................................................10
3 .Design and Finiteelement Modeling of Light Rail Bridge.............................15
3.1 Introduction..................................................................15
3.2 Design of the bridge..........................................................15
3.2.1 Types of bridges........................................................16
3.3 FiniteElement modeling.....................................................17
3.3.1 Element types used in modeling..........................................18
3.3.2 Summary of models.......................................................19
4. Load Distribution Analyses for Modeling Bridge......................................31
4.1 Introduction.................................................................31
4.2 Live load distribution factor...............................................31
4.3 Live load model.............................................................33
4.4 Loading placement...........................................................35
4.5 Numerical results...........................................................36
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5. Calibrated Load Distribution Equations for Light Rail Bridge
45
5.1 Development of new equations................................................45
5.1.1 Regression analysis...................................................45
5 .1.2 Summary of new formulas..............................................46
5.2 Comparison of LDF formulas..................................................48
5.2.1 New formulas vs level rule............................................49
5.2.2 New formulas vs site bridges..........................................50
6. Live Load Distribution Factors Influenced by Pier Settlement.......................83
6.1 Introduction for five site light rail bridges in Denver.....................83
6.2 Five bridge models..........................................................84
6.3 Moment and Shear influenced by pier settlement..............................85
6.4 LDF influenced by settlement................................................89
6.5 Results and discuss.........................................................90
7. Conclusions and Recommendations...................................................202
References...........................................................................203
Appendix.............................................................................205
vii
LIST OF FIGURES
Figure
2.1 load distribution on a typical slab girder bridge (a) Steel bridge (b) Bridge with concentrate
load (c) deformation for cross section (d) transversely flexible........................13
2.2 Girder section in slabgirder bridge (a) Exterior section (b) Interior section.... 14
2.3 LRV loading diagram................................................................14
3.1 Details of model....................................................................27
3.2 Boundary elements in bridge model..................................................27
3.3 Rectangular shell element in bridge model..........................................28
3.4 Finite element models for steel plate girder bridge................................28
3.5 Finite element models for steel box girder bridge..................................29
3.6 Finite element models for prestressed concrete I girder bridge.....................29
3.7 Finite element models for prestressed concrete box girder bridge...................30
3.8 Finite element models for reinforced concrete girder bridge........................30
4.1 AASHTO IIS20 Design Truck (AASHTO 2015)...........................................37
4.2. Schematic of procedures for determining the standard live load model of light rail transit...3 8
4.3. European live load model for train (ERRI Committee D192): (a) existing LM71; (b) newly
developed LM 2000 ......................................................................38
4.4. Selected existing live load models and insitu live load for Calibration...........39
4.5. Proposed live load models for light rail transit: (a) standard live load model; (b) alternative
live load models........................................................................39
4.6 Tangent truck ballasted deck dual track bridge....................................40
4.7. Dimensional configurations of the benchmark bridge models (unit in ft; not to scale): (a)
steel plate girder; (b) steel box girder; (c) prestressed concrete I girder.............41
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4.8. Dimensional configurations of the benchmark bridge models (continued; unit in ft; not to
scale): (d) prestressed concrete box girder; (e) reinforced concrete box girder.............42
4.9. Predicted live load distribution of the bridges based on bending moment (PC BOX).......43
4.10. Predicted live load distribution of the bridges based on bending moment (PC I)........43
4.11. Predicted live load distribution of the bridges based on bending moment ( Steel Box)..43
4.12. Predicted live load distribution of the bridges based on bending moment ( Steel Plate )....44
4.13. Predicted live load distribution of the bridges based on bending moment (RC)..........44
5.1 The results of nonlinear regression for the moment of steel girder bridge (interior girder) with
two lanes load..............................................................................58
5.2. Comparison of moment between the finite model prediction and proposed equation: (a)
prestressed concrete box (interior); (b) prestressed concrete box (exterior)................59
5.3. Comparison of moment between the finite model prediction and proposed equation : (a)
prestressed concrete I (interior); (b) prestressed concrete I (exterior)....................60
5.4. Comparison of moment between the finite model prediction and proposed equation : (a) steel
box (interior); (b) steel box (exterior)....................................................61
5.5. Comparison of moment between the finite model prediction and proposed equation: (a) steel
plate (interior); (b) steel plate (exterior)................................................62
5.6. Comparison of moment between the finite model prediction and proposed equation: (a)
reinforced concrete (interior); (b) reinforced concrete (exterior)..........................63
5.7. Comparison of shear between the finite model prediction and proposed equation: (a)
prestressed concrete box (interior); (b) prestressed concrete box (exterior)................64
5.8 Comparison of shear between the finite model prediction and proposed equation: (a)
prestressed concrete I (interior); (b) prestressed concrete I (exterior)....................65
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5.9. Comparison of shear between the finite model prediction and proposed equation: (a) steel
box (interior); (b) steel box (exterior)..................................................66
5.10. Comparison of shear between the finite model prediction and proposed equation: (a) steel
plate (interior); (b) steel plate (exterior)..............................................67
5.11. Comparison of shear between the finite model prediction and proposed equation: (a)
reinforced concrete (interior); (b) reinforced concrete (exterior)........................68
5.12. Comprehensive comparison between the proposed and predicted distribution factors: (a)
moment for exterior girders; (b) moment for interior girders; (c) shear for exterior girders; (d)
shear for interior girders............................................................................70
5.13. LDF comparison between Broadway bridge fieldtesting and FE analysis.......................71
5.14. LDF comparison between Indiana bridge fieldtesting and FE analysis........................71
5.15. LDF comparison between Santa Fe bridge fieldtesting and FE analysis.......................72
5.16. LDF comparison between County Line bridge fieldtesting and FE analysis....................72
5.17. LDF comparison between 6th Ave bridge fieldtesting and FE analysis........................73
5.18. Assessment of existing methods for bending moment: (a) lever rule for exterior girders; (b)
lever rule for interior girders; (c) AASHTO LRFD for exterior girders; (d) AASHTO LRFD for
interior girders.........................................................................75
5.19. Assessment of existing methods for shear: (a) lever rule for exterior girders; (b) lever rule
for interior girders; (c) AASHTO LRFD for exterior girders; (d) AASHTO LRFD for interior
girders..................................................................................77
5.20. Application of the proposed live load distribution factor equations to the five bridges in
Denver: (a) Broadway.....................................................................78
x
5.21. Application of the proposed live load distribution factor equations to the five bridges in
Denver(b) Indiana...........................................................................79
5.22. Application of the proposed live load distribution factor equations to the five bridges in
Denver (Santa Fe)...........................................................................80
5.23. Application of the proposed live load distribution factor equations to the five bridges in
Denver (County Line)........................................................................81
5.24. Application of the proposed live load distribution factor equations to the five bridges in
Denver (6th Ave)............................................................................82
6.1 Cross section of Broadway Bridge......................................................102
6.2 Cross section of 6th Ave Bridge......................................................102
6.3 Cross section of Indiana Bridge.......................................................103
6.4 Cross section of Santa Fe Bridge.....................................................103
6.5 Cross section of County Line Bridge...................................................104
6.6 Pier settlement model for 6th ave Bridge..............................................105
6.7 Pier settlement model for Broadway Bridge.............................................105
6.8 Pier settlement model for County Line Bridge..........................................106
6.9 Pier settlement model for Indiana Bridge..............................................106
6.10 Pier settlement model for Santa Fe Bridge...........................................107
6.11 Moment and Shear data collection points for two span bridges (Broadway Bridge and Santa
Fe Bridge )...............................................................................108
xi
6.12 Moment and Shear data collection points for four span bridges (6th Ave Bridge and County
Line Bridge)................................................................................108
6.13 Moment and Shear data collection points for five span bridge (Indiana Bridge).........109
6.14 Moment of seven points for 6th Ave Bridge with (a) case one (2,0,0) settlement (b) case
two (2,2,0) settlement (c) case three (2,4,2) settlement (d) case four (2,6,2)
settlement...............................................................................Ill
6.15 Moment comparison with different pier settlement cases for 6th Ave Bridge (a) RTD
Loading (b) Settlement (c) Settlement +RTD Loading.......................................112
6.16 Moment of seven points for County Line Bridge with (a) case one (2,0,0) settlement (b)
case two (2,2,0) settlement (c) case three (2,4,2) settlement (d) case four (2,6,2)
settlement..................................................................................114
6.17 Moment comparison with different pier settlement cases for County Line Bridge (a) RTD
Loading (b) Settlement (c) Settlement +RTD Loading..........................................115
6.18 Moment diagram for Broadway Bridge with different pier settlement at midpoint of first
span (a) 2 settlement (b) 4 settlement (c) 6 settlement..................................117
6.19 Moment diagram for Broadway Bridge with different pier settlement at support column (a)
2 settlement (b) 4 settlement (c) 6 settlement...........................................118
6.20 Shear diagram for Broadway Bridge with different pier settlement at abutment (a) 2
settlement (b) 4 settlement (c) 6 settlement..............................................120
6.21 Shear diagram for Broadway Bridge with different pier settlement at support column (a) 2
settlement (b) 4 settlement (c) 6 settlement..............................................121
6.22 Moment diagram for Santa Fe Bridge with different pier settlement at midpoint of first
span (a) 2 settlement (b) 4 settlement (c) 6 settlement..................................123
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6.23 Moment diagram for Santa Fe Bridge with different pier settlement at support column (a) 2
settlement (b) 4 settlement (c) 6 settlement...............................................124
6.24 Shear diagram for Santa Fe Bridge with different pier settlement at abutment (a) 2
settlement (b) 4 settlement (c) 6 settlement...............................................126
6.25 Shear diagram for Santa Fe Bridge with different pier settlement at support column (a) 2
settlement (b) 4 settlement (c) 6 settlement...............................................127
6.26 Moment diagram for County Line Bridge with different pier settlement cases at moment
point 1 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier
settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2....................129
6.27 Moment diagram for County Line Bridge with different pier settlement cases at moment
point 2 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier
settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2....................131
6.28 Moment diagram for County Line Bridge with different pier settlement cases at moment
point 3 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier
settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2....................133
6.29 Moment diagram for County Line Bridge with different pier settlement cases at moment
point 4 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier
settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2....................135
6.30 Moment diagram for County Line Bridge with different pier settlement cases at moment
point 5 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier
settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2...................137
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6.31 Moment diagram for County Line Bridge with different pier settlement cases at moment
point 6 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier
settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2....................139
6.32 Moment diagram for County Line Bridge with different pier settlement cases at moment
point 7 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier
settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2....................141
6.33 Shear diagram for County Line Bridge with different pier settlement cases at shear point 1
(a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement
case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2..............................143
6.34 Shear diagram for County Line Bridge with different pier settlement cases at shear point 2
(a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement
case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2..............................145
6.35 Shear diagram for County Line Bridge with different pier settlement cases at shear point 3
(a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement
case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2..............................147
6.36 Shear diagram for County Line Bridge with different pier settlement cases at shear point 4
(a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement
case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2..............................149
6.37 Shear diagram for County Line Bridge with different pier settlement cases at shear point 5
(a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement
case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2
151
6.38 Moment diagram for 6th Ave Bridge with different pier settlement cases at moment point 1
(a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement
case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2.............................153
6.39 Moment diagram for 6th Ave Bridge with different pier settlement cases at moment point 2
(a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement
case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2.............................155
6.40 Moment diagram for 6th Ave Bridge with different pier settlement cases at moment point 3
(a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement
case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2.............................157
6.41 Moment diagram for 6th Ave Bridge with different pier settlement cases at moment point 4
(a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement
case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2.............................159
6.42 Moment diagram for 6th Ave Bridge with different pier settlement cases at moment point 5
(a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement
case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2.............................161
6.43 Moment diagram for 6th Ave Bridge with different pier settlement cases at moment point 6
(a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement
case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2.............................163
6.44 Moment diagram for 6th Ave Bridge with different pier settlement cases at moment point 7
(a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement
case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2..............................165
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6.45 Shear diagram for 6th Ave Bridge with different pier settlement cases at shear point 1 (a)
Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case
three 2, 4, 2 (d) Pier settlement case four 2, 6, 2....................................167
6.46 Shear diagram for 6th Ave Bridge with different pier settlement cases at shear point 2 (a)
Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case
three 2, 4, 2 (d) Pier settlement case four 2, 6, 2....................................169
6.47 Shear diagram for 6th Ave Bridge with different pier settlement cases at shear point 3 (a)
Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case
three 2, 4, 2 (d) Pier settlement case four 2, 6, 2....................................171
6.48 Shear diagram for 6th Ave Bridge with different pier settlement cases at shear point 4 (a)
Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case
three 2, 4, 2 (d) Pier settlement case four 2, 6, 2....................................173
6.49 Shear diagram for 6th Ave Bridge with different pier settlement cases at shear point 5 (a)
Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case
three 2, 4, 2 (d) Pier settlement case four 2, 6, 2....................................175
6.50 Moment diagram for Indiana Bridge with different pier settlement cases at moment point 1
(a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement
case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2...............................177
6.51 Moment diagram for Indiana Bridge with different pier settlement cases at moment point 2
(a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement
case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2
179
6.52 Moment diagram for Indiana Bridge with different pier settlement cases at moment point 3
(a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement
case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2..............................181
6.53 Moment diagram for Indiana Bridge with different pier settlement cases at moment point 4
(a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement
case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2..............................183
6.54 Moment diagram for Indiana Bridge with different pier settlement cases at moment point 5
(a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement
case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2..............................185
6.55 Moment diagram for Indiana Bridge with different pier settlement cases at moment point 6
(a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement
case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2..............................187
6.56 Moment diagram for Indiana Bridge with different pier settlement cases at moment point 7
(a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement
case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2..............................189
6.57 Shear diagram for Indiana Bridge with different pier settlement cases at shear point 1 (a)
Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case
three 2, 4, 2 (d) Pier settlement case four 2, 6, 2...................................191
6.58 Shear diagram for Indiana Bridge with different pier settlement cases at shear point 2 (a)
Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case
three 2, 4, 2 (d) Pier settlement case four 2, 6, 2...................................193
XVII
6.59 Shear diagram for Indiana Bridge with different pier settlement cases at shear point 3 (a)
Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case
three 2, 4, 2 (d) Pier settlement case four 2, 6, 2....................................195
6.60 Shear diagram for Indiana Bridge with different pier settlement cases at shear point 4 (a)
Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case
three 2, 4, 2 (d) Pier settlement case four 2, 6, 2....................................197
6.61 Shear diagram for Indiana Bridge with different pier settlement cases at shear point 5 (a)
Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case
three 2, 4, 2 (d) Pier settlement case four 2, 6, 2.......................................199
6.62 Load distribution factor under 2 settlement, 4 settlement and 6 settlement for (a)
Broadway Bridge (b) Santa Fe Bridge.............................................................200
6.63 Load distribution factor under case one (2, 0, 0), case two (2, 2, 0), case three (2,
4, 2) case four (2, 6, 2) for (a) County Line Bridge (b) 6th Ave Bridge (c) Indiana
Bridge.........................................................................................201
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LIST OF TABLES
Table
2.1 Load combinations and load factors.......................................................12
3.1 Model matrix for superstructure design...................................................20
3.2. Details of the designed benchmark bridge sections I (steel plate girders)............21
3.3. Details of the designed benchmark bridge sections II (steel plate girders)............21
3.4. Details of the designed benchmark bridge sections I (steel box girders)............22
3.5. Details of the designed benchmark bridge sections II (steel box girders)............22
3.6. Details of the designed benchmark bridge sections I (prestressed concrete I girders).22
3.7. Details of the designed benchmark bridge sections II (prestressed concrete I girders)...23
3.8. Crosssectional area of prestressing steel strands for prestressed concrete I girders....23
3.9. Details of the designed benchmark bridge sections I (prestressed concrete box girders)...23
3.10. Details of the designed benchmark bridge sections II (prestressed concrete box girders)...24
3.11. Crosssectional area of prestressing steel strands for prestressed concrete box girders.24
3.12. Details of the designed benchmark bridge sections I (reinforced concrete girders).......25
3.13. Table 3.13. Details of the designed benchmark bridge sections II (reinforced concrete
girders)......................................................................................25
3.14. Crosssectional area of steel bars for reinforced concrete girders......................26
5.1. Live load distribution calibrated by deterministic standard live load (interior moment).52
5.2. Live load distribution calibrated by deterministic standard live load (exterior moment).54
5.3. Live load distribution calibrated by deterministic standard live load (interior shear)..55
5.4. Live load distribution calibrated by deterministic standard live load (exterior shear)..56
5.5. Load distribution factors calculated by level rule.......................................57
5.6. Sectional Properties of five site Bridges................................................57
xix
5.7. Field testing LDF mean value and standard deviation for each girder.......................57
6.1. Moment for Santa Fe Bridge with different pier settlement................................92
6.2. Shear force for Santa Fe Bridge with different pier settlement..........................92
6.3. Moment for Broadway Bridge with different pier settlement...............................92
6.4. Shear force for Broadway Bridge with different pier settlement..........................93
6.5. Moment for 6th Ave Bridge with case one (2,0,0) pier settlement.......................93
6.6. Moment for 6th Ave Bridge with case two (2,2,0) pier settlement.......................94
6.7. Moment for 6th Ave Bridge with case three (2,4,2) pier settlement.....................94
6.8. Moment for 6th Ave Bridge with case four (2,6,2) pier settlement......................95
6.9. Shear force for 6th Ave Bridge with case one (2,0,0) pier settlement.................95
6.10. Shear force for 6th Ave Bridge with case two (2,2,0) pier settlement.................96
6.11. Shear force for 6th Ave Bridge with case three (2,4,2) pier settlement...............96
6.12. Shear force for 6th Ave Bridge with case four (2,6,2) pier settlement................97
6.13. Moment for County Line Bridge with different cases pier settlement.....................98
6.14. Shear force for County Line Bridge with different cases pier settlement................99
6.15. Moment for Indiana Bridge with different cases pier settlement.........................100
6.16. Shear force for Indiana Bridge with different cases pier settlement....................101
xx
1 Introduction
1.1 Background
AASHTO presents many different ways to analyze bridges like grillage analysis, finite
element analysis, finite strip method and load distribution equation. However, the finite
element analysis to be treated the most accurate method of analyses.
The AASHTO formulas are based on the property of the beam, girder spacing, the
length of the bridge and so on. And the factors in the equation also depends on the
bridge types such as for the concrete deck with steel girder bridges, the AASHTO
specification consider the girder spacing, span length, the thickness of the deck. There
is another factor need to be mentioned is Kg, Kg is longitudinal stiffness parameter to
be taken by modulus of deck and girder and moment of inertia of beam. This is a very
important factor for distribution factor equation. For example in the AASHTO LRFD code
(AASHTO 2014), the load distribution factor equation for concrete slab on steel girder
bridges with two or more lanes loaded is
LDF = 0.075+(S/9.5)A0.6(S/L)A0.2(Kg/12*L*fs)A0.1
where S is girder spacing (ft), L is span length (ft), Kg = n(l+Ae2) is longitudinal stiffness
(in4), ts is slab thickness (in), n is modular ratio between steel and concrete, / is girder
stiffness (in4), A is girder area (in2), and e is eccentricity between centroids of girder
and slab (in). The maximum of girder distribution factors are used for the bridge design.
As we all know, the distribution factors provided by AASHTOLRFD for live load are
over conservative sometimes compared with the finite element model. However the
AASHTO equation only used for highway bridges, and did not include light rail bridges
1
design specification for live load distribution. Therefore, moment and shear load
distribution factor equations for light rail bridge need to be found to fill the gap in
previous studies.
In finite element analysis, Elements may be connected in a twodimensional (2 d) or
threedimensional (3 d) configuration. However, with the load factor we only multiply the
result of onedimensional analyses. Many researches to be published for the live load
distribution factor for various kinds of bridges such as prestressed concrete, spread
boxgirder, steel girder bridges etc. there is a need to use finite element analyses
method for different bridge types to established the equations for light rail bridges. To
ensure safety of the design, the model should be modeled correctly. How to ensure the
finite element is correct is a big problem. Before the researchers do the research, they
need run the models for many times to check the model and analysis the result to make
sure the values are all right. The finite element model used to develop the equation in
AASHTO LRFD did not consider some important factors that may affect the live load
distribution factors. For example the secondary elements like diaphragms, cross bracing
and railings in the bridges, however, these elements will influence the result of the
analysis for LDF equation. Obviously, the AASHTO LRFD equations provided the over
conservative results may influenced by these element.
1.2 Objective
The main purpose of this thesis is to propose new simplified live load distribution factor
equations for the concrete slab on steel plate girder, castinplace concrete multicell
2
box, castinplace concrete tee beam, precast I and closed steel boxes bridges for light
rail. The new equation for the LDF is intended to be as least as conservative compared
with the AASHTO equation. The scope of the study is limited to simple span Bridges.
Additional another purpose of this thesis is to explore the effect of pier settlement on
load distribution factors and shows how the bridges performance under different given
foundation settlement.
1.3 Organization
This thesis contains seven chapters. This chapter presents an introduction of the thesis
outlining the background and the statement of goals.
Chapter 2 is describe for the past and current design code for the highway and railway
bridge design. Include the load, the model and etc. chapter 2 also presents the historical
background of the AASHTO LRFD live load distribution factors and the Sap2000 three
dimensional finite element model background.
Chapter 3 provides the design methods for five different bridges (steel plate girder, cast
inplace concrete multicell box, castinplace concrete tee beam, precast I and closed
steel boxes) with different span length, girder spacing and number of spans. Model all
types of bridges with sap2000, summarize all the bridges and check the model with
HL93 standard load to ensure all the bridges modeled correctly. Second,
3
Chapter 4 is the analysis of the models. First, run the model with the live load which we
developed before. Then got the values for the moment and shear for each girder from
sap2000. Analyzed all the values and got the distribution factors for moment and shear.
Chapter 5 is to simplified equation for wheel load distribution factor, based on the
current AASHTOLRFD formula, all the formula is present in the chapter 5. The
accurate equation we developed is from the finite element model we developed in
chapter 4. The LDF values from proposed equation comparisons of LDF calculated by
AASHTO LRFD, AASHTO level rule and the field bridge testing.
Chapter 6 presents a study of structural performance of light rail bridges under given
foundation settlement at specific column. In this chapter, I choose five site light rail
bridges in Denver, use SAP 2000 model the bridges. Next step is settled the pier with
0, 2, 4 and 6 settlement for two spans bridges and pier settlement combinations were
used for four spans and five spans bridges. The settlement combinations are case one
(2, 0, 0), case two (2, 2, 0), case three (2, 4, 2), case four (2, 6, 2) each value
represent each piers settlement distance. After settled the bridge pier, run the model
and get the moment and shear for each girder under different settlement conditions.
Chapter 7 contains the summery and conclusions of the thesis. Also include some
recommendations for the live load distribution factors for light rail bridge.
4
Appendix A contains the analysis results for multispan light rail bridges under specific
pier settlement from CSIBridge. Appendix B contains the steps for modeling the light rail
bridge with CSIBridge and also showed how to set pier settlement in CSIBridge.
5
2 Literature Review
2.1 Bridge Design Codes
First, we need to know what is the design codes? Why we need design code for the
bridge design? As we all know, when we design a bridge, we need the code to be a
reference. The design codes is to ensure the safety of the bridge like the capability of
the strength, stiffness and the minimum resistance to the demands of force because of
various loads during the bridge design life. Review the country code for unites state, the
first national standard for highway bridge design is Standard Specifications for Highway
Bridge and Incidental Structures which was published by American Association of State
Highway and Transportation Officials (AASHTO). With many researches to be done by
the researchers, the design theory and practice have developed significantly. Before
1970, the design theory for the allowable stress is ASD(allowable stress design). At the
beginning of 1970, a new design theory named LFD (load factor design) was to be
published.
For allowable stress design (ASD) is compared actual and allowable stresses. The
general ASD design equation can be given as:
<
R
n
FS
A factor of safety is also based on some design code or the engineering experienced
judgment. For example, the factor of safety for the recommended strain limits in
A706/A706M bars and bars with splices for seismic zone 3 and 4 is 3 to 4.5 in ultimate
condition in the Standard Specifications (AASHTO 2015).
6
LFD provided the significant philosophic change of realizing that the loads are more
precisely shows the limitation than other factors. The general LFD design equation can
be shows as:
Where Qi = nominal (design) load component. Rn = nominal (design) resistance yi is a
load factor and 0 is the strength reduction factor. For our design the regular limited
factor for the design always based on the stability of the stress loss of the structure. In
many conditions, the resistance always reduced by the strength reduction factors, 0,
which is based on some unrealized situation during the construction or after the
construction, for example the bridge under oversized truck or the material don't have
enough strength for the design. With the reduction factor the design will be more safety,
however, if the reduction factor is not accurate that's may be cause economic problems.
LRFD design equation (Nowak 1995) is given as:
where 4> resistance factor, Rn = nominal (design) resistance, yi = load factor for
given load type, and Qi = nominal (design) load component. The load and resistance
factors of Eq. 2.1 must be calibrated in that way can keep the structure under the safety
situation. There are many limited state for design like the service limit state, strength
limit state and fatigue limit state. The service state takes the limitations on stress, crack
7
Â£y,Â£>, < i?
width under regular service and deformation conditions. The strength limit state of
strength and stability, should be taken to provide specific statistical load resistance
combination of local and global bridge during its design life which is expected to
experienced. Fatigue limit state of stress range will be limited due to a single design
truck happened in expected range of stress cycles.
2.2 Loads and Load Combinations
The categorized of loads was separated into permanent loads and transient loads in the
AASHTO LRFD (7th Edition, 2015). Permanent load is a load that always exist on the
structure, not change during the life time, for example the barriers, the slab and girders.
Transient load is a load that will change during the life time such as the truck, the wind
and pedestrians. In the real life, we always call the transient load as live load. In
AASHTO LRFD (7th Edition, 2015) shows nineteen different kinds of live loads.
When we design the bridge, the load combination is the only way to gather all the loads
together to satisfy the design specification to ensure the bridge is safety. In the
AASHTO LRFD, there are many combinations were used for design such as Strength I,
Strength II, Strength III, Service I, Fatigue I exc. However, fora load combination they
include different loads compared with other combinations. For example, the Strength I
load combination is not consider the wind load, however, the Strength V load
combination take the wind load into consideration. The load factors for the load
combination are proposed based on the structural reliability theory. Larger load factors
represent the higher safety for the design load. Smaller load factors are less uncertainly.
8
In Table 2.1. shows the load combination and load factors in the AASHTO LRFD. The
bridge in this thesis are based on this load combination. For highway bridge design, the
AASHTO use multiple presence factors, however, for the light rail bridge design, we
considered the multiple presence always equals to 1.0.
2.3 Load Distribution Factors
When we analysis the superstructure, the first step is to analyze the longitudinal
direction to calculate the maximum moment and shear, the second step is to analyze
the transverse direction to calculate the load distribution. In the AASHTO LRFD they
discuss the most types of bridges for the load distribution, the slab and slabgirder
bridges. Typical slab girder bridges is shown in Figure 2.1 (a). Figure 2.1 (a) shows a
steel girder bridge with parapet on both side. A concentrated load P was applied on the
slab in Figure 2.1 (b). The load is transfer from the slab to the girders. After applied the
load, girder will take the load and displacement will be appeared shown in Figure 2.1
(c). Beam which directly under the applied load will take the maximum load and other
beams will take different percentage load based on where they located. The beam far
from the load will perform relative small deformation shown in Figure 2.1 (d).
When we calculate the distribution factors, two types of girders need to be considered.
The first one is interior girder the second one is exterior girder. As show in FIG 2.2. In
AASHTO also separate the interior and exterior girder when developed the LDF
equations. The level rules is a method of analysis and always be used for the exterior
girder distribution.
9
2.4 Live Load Models
The most critical concern in light rail bridge design is the absence of a standard live load
model. Current practices are typically based on historical engineering judgment and
experience of individual transit agencies. As such, there is a lack of uniformity in the
level of safety or reliability for bridges designed to resist loadings from light rail transit
trains across the nation. TCRP 155 mentions that the AASHTO Specifications and the
AREMA manual do not provide accurate loading information to designing light rail
bridges (TRB 2012). For example, the wheel spacing of the AREMA loading does not
represent that of light rail trains and the service conditions of freight rail bridges are
different from those of light rail bridges. It is also stated that the conservative bridge
design required by the AREMA manual may not be applicable to light rail bridges. The
types and configurations of light rail trains employed in the US vary depending upon
manufacturers. As a consequence, the loadings actually experienced by structures
entail a certain level of uncertainty or a lack of surety with regard to actual axle load
magnitude and axle spacing, (these loadings and spacings are not deterministic).
Reliabilitybased live load calibration is a tool that allows for the quantification and
management of uncertainty. The developed standard live load model will use a simple
configuration of concentrated and uniform loadings at set spacings that will quantify and
encompass (or envelope) loading effects (shear and moment). A level of reliability or
safety will also be quantified through the development of load factors (in conjunction
with existing factored bridge element capacities) for the standard live load model. The
standard live load model will be flexible enough that it can address potential increase in
10
axle loads and modified train configurations in the future. Fig 2.3 is an existing light rail
truck model which is used in RTD Light rail design.
11
Table 2.1 Load combinations and load factors
Load Combination Limit State DC DD DW EH EV ES EL PS CR SH LL L\{ CE BR PL LS nu ITS WL FR TV TG SE Use One of These at a Time
EQ BL IC CT cv
Strength I (unless noted) yp 1.75 1.00 1.00 0.50/1.20 y to ysi
Strength n y> 1.35 1.00 1.00 0.50/1.20 yro
Strength in yf 1.00 1.4 0 1.00 0.50/1.20 yjKr Yse
Strength IV yp 1.00 100 0.50/1.20
Strength V yf 1.35 1.00 0.4 0 1.0 1.00 0.50/1.20 yro yst
Extreme Event I yr yEQ 1.00 1.00 1.00
Extreme Event H yf 0.50 1.00 1.00 1.00 1.00 1.00 1.00
Service I 1.00 1.00 1.00 0.3 0 1.0 1.00 1.00/1.20 Yro Ysr
Service H 1.00 1.30 1.00 1.00 1.00/1.20
Service in 1.00 0.80 1.00 1.00 1.00/1.20 IT'S ysz
Service IV 1.00 1.00 0.7 0 1.00 1.00/1.20 1.0
Fatigue I LL, I\{ & CE only 1.50
Fatigue H LL.IM&CE only 0.75
12
IT 1__________f]
'ci: _____=e1
rtTfT'i
(c)
Poor Distribution
(d)
Figure 2.1 load distribution on a typical slab girder bridge (a) Steel bridge (B) Bridge with
concentrate load (c) deformation for cross section (d) transversely flexible
13
(a)
(b)
Figure 2.2 Girder section in slabgirder bridge (a) Exterior section (b) Interior section
m
r*
co
m'
CM
*
in
e'
en
CM
*
in
CM
CD
*
in
CM
cd
m
h
co
CM
in
e
co
M"
CM
14
3 Design and FiniteElement Modeling of Light Rail Bridge
3.1 Introduction
This chapter describes the design for five different types of light rail bridges with various
girder spacing, span length and number of spans. In addition, the finite element model
was developed based on the design bridge properties. In this chapter, SAP2000 was to
be used, the basic assumptions and concepts of the finite model in calculate the load
distribution factors were to be considered. The different models for the girders and
boundary conditions of every modeled bridge are summarized in this chapter. 150
various finite element models for light rail bridges are studied. An appropriate five site
bridges models are selected to verify the new LDF equation, ensure its safety.
3.2 Design of the bridge
The purpose of this thesis is to develop the new live load distribution factor equations
for light rail bridges. The first step for this is design the light rail bridge based on light rail
bridge design code. For this thesis, the bridge design are rely on the RTD Light Rail
Design Criteria 2013. The design also reference the design material for Indiana bridge,
Santa Fe bridge, County Line bridge, 6th Ave bridge and Broadway bridge which are five
light rail bridge in Denver. Table 3.1 shows the Model matrix for superstructure design in
this thesis.
All these typical bridges were created to represent the existing site bridges with different
beam spacing, moment of inertia of girder and so on.
15
3.2.1 Types of bridges
Steel plate girder bridges
A total of fortyeight different steel plate girder bridges were designed in this thesis. The
properties of the bridges are presented on Table 3.2. Where I is the moment of inertia
(inA4), L is the length of the span (ft),Gr is the girder spacing (ft), w is the width (in), t is
the thickness (in), h is the height (in). The moment of inertia will be used for the
regression of equation. All the design properties will be used for the modeling part.
Steel Box Girder Bridges
The different Steel Box girder bridge properties showed in table 3.3 A total of twenty
seven different steel box girder bridge have been considered in the analysis. These
bridges are based on the RTD design code.The table 3.3 shows that with the span
length increasing the upper and lower flange width, the web height are raising.
Compared with the simplysupport span with the multiple spans bridge, the height of the
web is increase. Section properties and moment of inertia for each girder spacing and
span length of steel box girder are shown in Table 3.3.
Prestressed concrete I girders
For prestressed concrete I girder light rail bridges the section property details shows in
Table 3.4. For different span length and girder spacing, I choose four types of I girder:
1Type IV, I Type V, 1Type VI and BT84*48. The crosssection area and prestressing
steel strands are shown in Table 3.5.
16
Prestressed concrete box girder
For prestressed concrete box girder, the upper flange width is equal to the deck width.
In Table 3.6 shows the section properties for concrete box girder bridge, what need to
mention is that the upper flange thickness is include the thickness of the deck. With the
increasing of girder spacing and span length, the web height is increased a lot from 56
to 84. Table 3.7 shows the cross section area and tendons for each case.
Reinforced concrete girder
For reinforced concrete girder bridges, we can see from Table 3.8, span length for this
type of bridge is smaller than other bridge. Obviously, the web height is less than other
bridges. The minimum height is 24 and maximum height is 65. Table 3.9 is cross
sectional area of bars for reinforced concrete girders.
3.3 Finite element modeling
The finite element method was used to figure out the analysis details of different bridge
types (steel plate girder bridge, steel box girder bridge, prestressed concrete I girder
bridge, prestressed concrete box girder bridge and reinforced concrete girder bridge)
and multiple continuous spans. The live load used in the CSIBridge modeling were
modeled by a proposed live load model, this will be introduced in chapter 4. Many
important factors need to be considered such as beam spacing, slab thickness, span
length, etc. were showed for every bridge type. In this thesis, the shell element was
used to the bridge deck and beam element were used to the girder modeling or the
upper or lower flanges of girder.
17
3.3.1 Element types used in modeling
As the introduction explained the deck of the bridge was modeled by shell element.
Always the shell element has three dimensional solid element which are takes the plate
bending and shear. For our modeling, I choose the rectangular element for the model
shown in Figure 3.3. Shell element was defined as a unit area that can determine local
stress on the superstructure.
After choose the shell element for the deck, the next step is to connect each element
with link element. Link element may be a common grounding spring or a combination of
twojoint links and is considered composed by six independent springs, each
deformation degree of one degree of freedom including axial, shear and torsion and
pure bending. For our design, the twojoint links to be chose. Three dimensional beam
element with torsion, compression, tension and bending capabilities was used in the
analysis.
Boundary element is very important for the structural analysis. Honestly, when analysis
the bridges to get the maximum moments and shear forces, the boundary element is
the most important part to make sure the values are reliable. Like support shown in
Figure 3.2 (i.e. fixed, pinned, roller) and don't need to consider the foundation system
stiffness. Sometimes the modeling part also use nonlinear spring or damper for some
special cases. Capturing the proper behavior of the structure and determination the
material and the property of the section is also important aspect for the modeling. For
18
this thesis, it is a 3D finite element, the material properties are required to match the
actual structural behavior. The structural based on the uniform material such as steel
and concrete. However, for some bridges also have not uniform material such as
prestressed concrete need to have the limitation. Figure 3.1 shows the detail of model.
3.3.2 Summary of models
Figure 3.4 Figure 3.8 shows eighteen finite element bridge models, steel plate girder,
steel box girder, prestressed concrete I, prestressed concrete box and reinforced
concrete respectively. All these models were modeled by SAP2000 (CSIBridge). In
this study, the finite element model such as deck and barriers, I choose the same
dimensions based on the RTD light rail design criteria. We define girder spacing ,
thickness of the deck, width of the deck and other bridge section data as same as the
values we designed shown in Table 3.1 Table 3.9.
19
Table 3.1. Model matrix for superstructure design
Type Schematic3 Typeb Span length Girder spacing Number of span
80 ft 4ft
^ 100 ft 6ft
Steel plate girder 1l 11 ti L a 140 ft 8 ft 2
160 ft 10ft J
n 80 ft 8 ft 1
Castinplace concrete
multicell box 1UL LJ c 100 ft 10 ft 2
140 ft 12 ft 3
4ft
n n 30 ft 1
Castinplace concrete tee E;,I. T=rfi e 50 ft 6 ft 2
beam u u J u 8 ft
70 ft 3
10 ft
4ft
80 ft
Precast concrete I or bulb p A 6 ft
tee it it i i k 100 ft 8 ft 1
140 ft
10 ft
fl ; 11 80 ft 6ft 1
Closed steel boxes i =TT b 100 ft 8ft 2
140 ft 10 ft 3
20
Table 3.2. Details of the designed benchmark bridge sections I (steel plate girders)
c L (ft) Gr (ft) Upper FL Lower FL Web Span L (ft) Gr (ft) Upper FL Lower FL Web
w (in) t (in) w (in) f (in) h (in) f (in) w (in) t (in) w (in) f (in) h (in) f (in)
Simply supported span 80 4 16 1.75 16 1.75 32 1.0 Multiple spans 80 4 16 1.75 16 1.75 40 1.0
6 16 1.75 16 1.75 32 1.0 6 16 1.75 16 1.75 40 1.0
8 16 1.75 16 1.75 34 1.0 8 16 1.75 16 1.75 43 1.0
10 16 1.75 16 1.75 38 1.0 10 16 1.75 16 1.75 50 1.0
100 4 18 1.75 18 1.75 42 1.0 100 4 18 1.75 18 1.75 55 1.0
6 18 1.75 18 1.75 45 1.0 6 18 1.75 18 1.75 55 1.0
8 18 1.75 18 1.75 48 1.0 8 18 1.75 18 1.75 58 1.0
10 18 1.75 18 1.75 55 1.0 10 18 1.75 18 1.75 66 1.0
140 4 20 2.25 20 2.25 65 1.0 140 4 20 2.25 20 2.25 75 1.0
6 20 2.25 20 2.25 70 1.0 6 20 2.25 20 2.25 75 1.0
8 20 2.25 20 2.25 73 1.0 8 20 2.25 20 2.25 78 1.0
10 20 2.25 20 2.25 76 1.0 10 20 2.25 20 2.25 90 1.0
160 4 24 2.25 24 2.25 76 1.0 160 4 24 2.25 24 2.25 82 1.0
6 24 2.25 24 2.25 78 1.0 6 24 2.25 24 2.25 82 1.0
8 24 2.25 24 2.25 82 1.0 8 24 2.25 24 2.25 85 1.0
10 24 2.25 24 2.25 86 1.0 10 24 2.25 24 2.25 100 1.0
Table 3.3. Details of the designed benchmark bridge sections II (steel plate girders)
Girder
Span length spacing Interior y Interior I Exterior y Exterior I
80 4 28.6622 43800.2782 32.8826 54138.1986
80 6 31.1155 49769.0693 32.0655 52108.4885
80 8 34.3271 59833.9117 34.3271 59833.9117
80 10 38.6959 76693.2345 39.2810 78430.0695
100 4 34.5669 74712.0407 39.5035 91546.7576
100 6 39.3756 95416.1588 40.5629 99759.7662
100 8 43.6782 116474.0971 43.4782 116474.0971
100 10 50.4730 159355.9547 51.2628 163001.8383
140 4 47.2746 206780.2218 53.6401 249519.0840
140 6 53.9327 266172.4792 55.5030 277658.5080
140 8 58.8645 312561.5152 58.8534 312523.1304
140 10 63.3481 359468.3041 64.3872 367909.6503
160 4 52.7748 311950.2491 59.6004 373074.4554
160 6 57.7489 363681.5517 59.3931 378849.7906
160 8 63.4257 433289.4568 63.4229 433243.2493
160 10 68.7305 505196.4610 69.8675 517042.8724
21
Table 3.4. Details of the designed benchmark bridge sections I (steel box girders)
Span L (ft) Gr (ft) Upper FL Lower FL Web Span L (ft) Gr (ft) Upper FL Lower FL Web
w (in) f (in) w (in) f (in) h (in) f (in) w (in) f (in) w (in) f (in) h (in) f (in)
Simplysupported span 80 6 28 1.5 28 1.5 30 0.75 Multiple spans 80 6 28 1.5 28 1.5 32 0.75
8 28 1.5 28 1.5 32 0.75 8 28 1.5 28 1.5 36 0.75
10 28 1.5 28 1.5 34 0.75 10 28 1.5 28 1.5 42 0.75
100 6 38 1.5 38 1.5 38 0.75 100 6 38 1.5 38 1.5 42 0.75
8 38 1.5 38 1.5 40 0.75 8 38 1.5 38 1.5 44 0.75
10 38 1.5 38 1.5 42 0.75 10 38 1.5 38 1.5 52 0.75
140 6 48 1.5 48 1.5 52 0.75 140 6 48 1.5 48 1.5 60 0.75
8 48 1.5 48 1.5 60 0.75 8 48 1.5 48 1.5 68 0.75
10 48 1.5 48 1.5 62 0.75 10 48 1.5 48 1.5 78 0.75
Table 3.5 Details of the designed benchmark bridge sections II (steel box girders)
Girder
Span length spacing Interior y Interior I Exterior y Exterior I
80 6 25.8886 51100.7438 26.6796 53427.7646
80 8 28.7500 62191.5000 28.7500 62191.5000
80 10 31.4601 73893.3549 31.9922 75714.3970
100 6 29.9095 93211.3684 30.8066 97256.1840
100 8 32.9313 110915.5588 32.9511 111097.6897
100 10 35.8090 129376.5753 36.4415 132588.4383
140 6 37.5931 194299.5383 38.6538 202085.6700
140 8 43.4090 258826.0000 43.4100 258826.0000
140 10 46.6475 293946.1308 47.6875 304856.0411
Table 3.6. Details of the designed benchmark bridge sections I
(prestressed concrete I girders)
L (ft) Gr (ft) Girder type L (ft) Gr (ft) Girder type L (ft) Gr (ft) Girder type
80 4 IType IV 100 4 IType V 140 4 BT84 X 48
6 IType IV 6 IType V 6 BT84X 48
8 IType IV 8 IType V 8 BT84 X 48
10 IType V 10 IType VI 10 BT84X 48
22
Table 3.7 Details of the designed benchmark bridge sections II
(prestressed concrete I girders)
Span length Girder spacing Interior y Interior I Exterior y Exterior I
80 4 37.6950 615159.4787 43.5420 777238.6612
80 6 41.0835 708769.9791 42.4033 745452.5221
80 8 43.5420 777238.6612 43.5421 777238.6612
80 10 51.5573 1245249.6781 52.3889 1277052.4167
100 4 43.8347 953163.9692 49.5970 1170626.8280
100 6 47.1058 1076342.9694 48.4304 1126409.9021
100 8 49.5970 1170626.8280 49.5970 1170626.8280
100 10 57.6386 1682394.0500 58.6161 1726106.7416
140 4 58.6137 1719134.5551 65.5790 2064400.4677
140 6 62.5586 1913772.1991 64.1603 1993587.9468
140 8 65.5790 2064400.4677 65.5790 2064400.4677
140 10 67.9801 2184553.9097 69.0051 2235988.7678
Table 3.8 Crosssectional area of prestressing steel strands for prestressed concrete I
girders
L (ft) Gr (ft) Ap (m2) L (ft) Gr (ft) Ap (m2) L (ft) Gr (ft) Ap (m2)
80 4 6.273 (41 tendons) 100 4 8.568 (56 tendons) 140 4 12.699 (83 tendons)
6 6.426 (42 tendons) 6 8.874 (58 tendons) 6 13.005 (85 tendons)
8 6.579 (43 tendons) 8 9.180 (60 tendons) 8 13.464 (88 tendons)
10 7.038 (46 tendons) 10 9.486 (62 tendons) 10 16.830 (110 tendons)
L = span length; Gr = girder spacing; Ap
area of tendon (one standard tendon Aj
0.153 in2)
Table 3.9. Details of the designed benchmark bridge sections I
(prestressed concrete box girders)
Span L (ft) Gr (ft) Upper FL Lower FL Web Span L (ft) Gr (ft) Upper FL Lower FL Web
w (in) t (in) w (in) t (in) h (in) t (in) w (in) t (in) w (in) t (in) h (in) t (in)
Simplysupported span 80 8 384 14 300 10 56 12 Multiple spans 80 8 384 14 300 10 58 12
10 384 14 300 10 58 12 10 384 14 300 10 60 12
12 384 14 300 10 62 12 12 384 14 300 10 64 12
100 8 384 16 300 12 66 12 100 8 384 16 300 12 68 12
10 384 16 300 12 68 12 10 384 16 300 12 70 12
12 384 16 300 12 70 12 12 384 16 300 12 72 12
140 8 384 18 300 14 80 12 140 8 384 18 300 14 82 12
10 384 18 300 14 82 12 10 384 18 300 14 84 12
12 384 18 300 14 84 12 12 384 18 300 14 86 12
23
Table 3.10. Details of the designed benchmark bridge sections II
(prestressed concrete box girders)
Span length Girder spacing Interior y Interior I Exterior y Exterior I
80 8 40.4383 2404131.8890 46.4014 1784409.7548
80 10 41.7560 3165266.6246 49.5783 2375612.7519
80 12 44.1348 4255376.7913 49.1584 2854545.1183
100 8 46.9577 3889801.9155 53.5021 2860500.9872
100 10 48.2632 5074563.7895 55.4443 3557904.8148
100 12 49.5200 6367406.0800 55.3775 4268803.6027
140 8 55.6064 6475256.3192 64.1125 4833345.4750
140 10 56.9117 8365787.9114 68.0157 6322746.9629
140 12 58.1746 10403131.3167 65.1710 6996429.9224
Table 3.11. Crosssectional area of prestressing steel strands for prestressed concrete box
girders
Span L (ft) Gr (ft) AP (m2) Span L (ft) Gr (ft) Ap (m2)
nplysupported span 80 8 16.524 (108 tendons) Multiple spans 80 8 18.36 (120 tendons)
10 17.44 (114 tendons) 10 19.278 (126 tendons)
12 18.36 (120 tendons) 12 19.89 (130 tendons)
100 8 27.54 (180 tendons) 100 8 30.294 (198 tendons)
10 30.294 (198 tendons) 10 33.05 (216 tendons)
12 32.13 (210 tendons) 12 36.72 (240 tendons)
Â£ 36.72 55.08
(240 tendons) (360 tendons)
38.556 57.834
14U 1U (252 tendons) 14U 1U (378 tendons)
38.556 57.834
Iz (252 tendons) Iz (378 tendons)
L = span length; Gr = girder spacing; Ap = area of tendon (one standard tendon ^ = 0.153 in2)
24
Table 3.12. Details of the designed benchmark bridge sections I
(reinforced concrete girders)
Span L (ft) Gr (ft) w (in) frf (in) Span L (ft) Gr(ft) I!7 (in) frf (in)
4 12 24 4 14 25
30 6 13 25 30 6 15 27
03 Oh m 8 15 26 8 16 29
10 16 28 10 18 30
'T3 OJ 4 16 38 03 4 18 40
o 50 6 17 38 Qh m 50 6 19 42
8 19 40 'Oi 8 21 44
i 10 21 42 1 10 23 45
Oh a 4 25 55 4 27 58
& 70 6 27 56 70 6 29 60
8 29 58 8 30 62
10 30 62 10 32 65
Table 3.13. Details of the designed benchmark bridge sections II
(reinforced concrete girders)
Span length Girder spacing Interior y Interior I Exterior y Exterior I
30 4 22.6250 69844.0000 24.8788 84378.5455
30 6 24.5574 91503.6384 24.8295 93509.7064
30 8 25.8000 119826.0000 25.4941 117053.7059
30 10 27.8350 157032.4401 27.9573 158407.5956
50 4 29.5882 231666.1961 33.3236 289672.7462
50 6 31.6501 279862.0683 32.0894 287131.3162
50 8 33.9535 374449.6124 33.4498 364255.4458
50 10 35.9856 483303.5677 36.1932 488398.1156
70 4 35.9097 726423.2087 40.3572 928668.1548
70 6 38.6452 932286.9677 39.1828 959569.9346
70 8 41.3543 1186037.0633 40.7219 1149245.0706
70 10 45.1176 1551137.6471 45.4000 1570377.3333
25
Table 3.14. Crosssectional area of steel bars for reinforced concrete girders
Span L (ft) Gr (ft) A (in2) Span L (ft) Gr(ft) A (in2)
+ve ve +ve ve
03 Oh m 'T3 OJ tn 0 Oh 1 30 4 12 N/A 4ultiple spans 30 4 9 14
6 12 N/A 6 9 14
8 13 N/A 8 9 14
10 14 N/A 10 10 15
50 4 17 N/A 50 4 13 19
6 17 N/A 6 13 19
8 18 N/A 8 13 20
10 20 N/A 10 15 22
Oh a 4 24 N/A 4 17 25
70 6 24 N/A 70 6 17 25
8 25 N/A 8 18 26
10 28 N/A 10 20 30
26
Shell Elements
Figure 3.1 Details of model
Fixed supports
Rollers
Pins
General 3D stiffness matrix (6 X 6)
members
Figure 3.2 Boundary elements in bridge model
27
Figure 3.3 Rectangular shell element in bridge model
4 ft spacing (6 girders)
6 ft spacing (5 girders)
8 ft spacing (4 girders) 10 ft spacing (3 girders)
Figure 3.4 Finite element models for steel plate girder bridge.
28
6 ft spacing (5 girders)
8 ft spacing (4 girders)
10 ft spacing (3 girders)
Figure 3.5 Finite element models for steel box girder bridge
4 ft spacing (6 girders)
6 ft spacing (5 girders)
8 ft spacing (4 girders) 10 ft spacing (3 girders)
Figure 3.6 Finite element models for prestressed concrete I girder bridge
29
8 ft spacing (4 webs)
10 ft spacing (3 webs)
12 ft spacing (3 webs)
Figure 3.7 Finite element models for prestressed concrete box girder bridge
4 ft spacing (6 girders) 6 ft spacing (5 girders)
8 ft spacing (4 girders) 10 ft spacing (3 girders)
Figure 3.8 Finite element models for reinforced concrete girder bridge
4 Load Distribution Analysis for Modeling Bridges
4.1 Introductions
As we all know the slabongirder bridges are the most common bridges in the United
States. Because there is a lack of information about live load distribution factors for light
rail bridges, the lever rule method is frequently employed by practitioners. For the
dimensions of the bridge are based on the RTD light rail bridge design criteria. There is
amount of papers shows that the conservative to use the AASHTO wheel load
distribution factors (Zaher Yousif and Riyadh Hindi,2007 and Erin Hughs and Rola
ldriss,2006). Recently, many researchers did the research about the load distribution
factors, when evaluate the load capacity of an existing bridge or do the research to get
some conclusions finite element analysis can be used to get an accurate and reliable
result of the load distribution factors.
In order to carefully investigate the range of the load distribution factor for the light rail
bridge. Hundreds of light rail bridge models to be analyzed. The properties of bridge
introduced in the chapter 3. Current AASHTO LRFD live load distribution factor
equations consider the bridge span, girder spacing, slab thickness and longitudinal
+stiffness. In order to ensure the safety of the light rail bridges, the parameters on the
live load distribution factors for the light rail bridge was to be investigate in this thesis.
4.2 Live Load Distribution Factor
The lateral load distribution of light rail trains is an important consideration when
designing a bridge superstructure. Test data (Gutkowski et al. 2003) indicated the load
sharing factor of bridge girders was between 16% to 29%, depending upon the position
31
of train load (20% would be the uniform load distribution factor for these girders). It is
worthwhile to note that light rail loading is relatively simple in comparison to standard
highway traffic loading.
The behavior of bridge superstructure will be different when it is loaded in one track or
in two tracks (two tracks would be sufficient for most cases in light rail transit). The
research will quantify such distinct behavior and will compare existing LRFD distribution
equations to assess their applicability. Detailed examinations will contribute to
understanding the application of existing approaches to light rail structures (e.g., the
simple lever rule and empirical formulae of the AASHTO LRFD Specifications). This
endeavor also fulfils the Request of Proposal that has specified the need for load
distribution requirements As a minimum, the specifications shall specify transit load
characteristics (e.g., loads and forces, load distribution, load frequency, dynamic
allowance, and dimensional requirements) In accordance with the development of the
AASHTO LRFD distribution factors (Art. 4.6.2.2 Beamslab bridges), live load
distribution factors for light rail were developed based on simply supported bridges. It is
well understood in the research community that structural continuity does not influence
the distribution of live load (Barker and Puckett 1997). The distribution factors were
obtained by:
LDF = m
where LDF is the live load distribution factor; m is the number of the loaded track; li is
the moment of inertia of the ith girder; n is the number of the supporting girders; and Ri
32
is the response of the superstructure in moment and shear. By including the number of
the loaded track, the results of beamline analysis with a singletrackloaded case can
be expanded to multiple track load cases.
4.3 Live Load Model
A important aspect of the developed standard live load model is its practical
convenience for bridge designers who are familiar with the AASHTO LRFD
Specifications and its standard live load model for highway traffic gravity loadings
(HL20) show in Figure 4.1. With this in mind, the standard light rail load model should
be developed to be as similar as is possible to the HL20 in configuration and application
rather than other load models such as the Cooper E80 from the AREMA manual which
envelopes heavy haul and freight train loadings. The standard live load model for light
rail train loadings to be developed will consider the following aspects:
Integration and consideration of existing light rail train load models used by various
transit agencies across the US Similarity with the live load model of the AASHTO LRFD
Specifications. Familiarity to practicing engineers, and straightforward application and
implementation Convenience for LRFD calibration for the present and future.
The approach to develop live load model is derived in part from the method used by the
International Union of Railways to develop a new train load model (LM 2000), as shown
in Figure. 4.3.
33
The utility of the LM 2000 model is it can cover a wide range of bridge geometries and
accommodate potential increases in train loadings during the projected target bridge
service life of 100 years. Another advantage of the LM 2000 model is its straightforward
application for design (i.e., simple load configurations). It should also be noted that the
configuration of the LM 2000 live load model is essentially analogous to the HL93 in the
AASHTO LRFD Specifications (i.e., a simple combination of concentrated loads with a
uniformly distributed load, which will expedite bridge design). Figure 4.2 illustrates the
layout of the proposed live load model. The model, at this juncture, is believed to be
both general and flexible enough to envelope the structural responses produced by the
various potential loading configurations of articulated light rail trains in conjunction with
the typical operation schedules of transit agencies. For example, some cases with
multiple articulated trains occupying significant portions of a bridge (e.g., 4 to 5 trains
during rush hour) require a uniformly distributed load effect, while some cases with
minimal articulated trains (e.g., 2 trains during offpeak hour) demand a concentrated
load effect. The proposed live load model will be calibrated using the matrix of prototype
bridges. A simple beam line analysis will be conducted to promptly determine flexural
load effects subjected to various existing light rail train loads. The bending and shear
responses of these bridges loaded with the existing live load models and the measured
live loads will be compared until equivalent (and enveloped) responses are achieved
with the proposed live load format (Fig. 4.5). Iterations of the live load model
characteristics (e.g., axle loads and spacings) will be a part of this process. All the
employed live loads will be positioned to generate the maximum load effects according
34
to influence line theory. It is important to note that the effect of small differences in the
existing live load models should be insignificant with regards to the behavior of the
bridges and, as such, the number of finite element models employed should not be
unrealistically large. Four representative live load models are thus selected, and
presented in Fig. 4.4. The calibration work will initially be conducted with all bridges
being simply supported to refine the configuration of the proposed model (i.e., number
of axles, loading magnitudes, and spacings). The initially calibrated load model along
with the representative live loads and measured loads will then be applied to continuous
bridge models to generate maximum bending and shear responses with the aim of
further refinement of the proposed model.
4.4 Loading Placement
The tracks location for the light rail design is shown in Figure 4.6. From the RTD design
criteria, the standard width of the track is 4 feet 81/2 inches. The minimum distance
from the centroid of two tracks is 14. For maintenance or emergency, every track
should have a 26 walkway. The minimum thickness of the deck is 8 inches. The deck
design from RTD design criteria also based on CDOT Bridge Design Manual guidelines.
For the precast concrete deck, need to have a minimum 3.5 inches thickness grout
since the concrete bearing. From two guardrails the minimum distance is 32 for two
tracks bridges. The distance from the inside walk to the centroid of the track is 66.
Based on the dimension requirements list above, figure 4.7 and figure 4.8 shows load
locations for all bridges which I analyzed.
35
4.5 Numerical results
After design and model the bridges, the maximum moment from interior and exterior
were selected, respectively. Use the equation which is shown in 4.2 to get the live load
distribution factors for both moment and shear. The LDF for the moment is shown from
Figure 4.9 to Figure 4.13 for five types of light rail bridges, PC Box, PC I, Steel Box,
Steel Plate and RC respectively. When the bridge stand one track load, LDF graph
shows the girder which under the load obtained the maximum moment and has the
higher LDF compared with other girders which far from the load. For example, the
Figure 4.9 for one track loaded near the left side of the bridge, the LDF shows about
0.41 for left exterior girder and 0.39 for first interior girder and 0.15 for second interior
girder and 0.03 for right exterior girder. For two lanes loaded, LDF for the bridge almost
symmetrical. Such as Figure 4.11 shows the LDF for Steel Box bridge for two lanes
loaded, two interior girders live load distribution factor equals to 0.6 and for two exterior
girder the live load distribution factors are 0.4. Same thing for other types of bridges.
36
W = COMBINED WEIGHT ON THE FIRST TWO AXLES WHICH IS THE SAME
AS FOR THE CORRESPONDING H TRUCK.
V VARIABLE SPACING 14 FEET TO 30 FEET INCLUSIVE. SPACING TO BE
USEO IS THAT WHICH PRODUCES MAXIMUM STRESSES.
CLEARANCE AND
toad LANE width
* iOMT
Figure 4.1 AASHTO HS20 Design Truck (AASHTO 2015)
37
Deterministic
Probabilistic
>=>
Integration
Selection of existing load
models
Numerical parametric
study
Determination of
equivalent live load model
MonteCarlo
simulation for bridge
responses with various
live load effects
Identifying possible
response ranges in
Update the deterministic
live load model based on
the probabilistic analysis
results
Proposal of standard live
load for light rail transit
Comparative assessment
Figure. 4.2. Schematic of procedures for determining the standard live load model of light
rail transit
W=80KN/m 2 Z 2 Z W=80KN/m
H I 1 10 to C\l LO CM to CM I d
Qvk=2x300 kN
cc CC cc c 1.6 b.8  qv,=110 kN/m
6.4 I I ! !'
(a)
(b)
Figure. 4.3. European live load model for train (ERRI Committee D192): (a) existing
LM71; (b) newly developed LM 2000
38
LO LO LO LO
00 00
CN CN o o
CM CN 00 00
m oq m oq o O O o O O
CN CN CN CN CN CN CN CN CN CN
5.9ft 28.8ft 5.9ft 28.8ft 5.9ft 6.0ft 28.0ft 6.0ft 28.0ft 6.0ft
V^kV^kV^k V^kV^kV^k
Minnesota
Utah
m h*. m h*. O LO O LO
m m m m 00 00 CN CN
r*l r*l in in CD CD
CN CN CN CN
y II II h
in in
00 00
6.3ft 16.8ft 6.3ft 16.8ft 6.3ft
'kkV^kV^k
5.9ft 19.6ft 5.9ft 19.6ft 5.9ft
'kkV^kVk
Massachusetts
Colorado
(insitu load: bias factor to be
used after site work)
Figure. 4.4. Selected existing live load models and insitu live load for calibration
30 kips 30 kips 30 kips
JilllllHU
0.90 kip/ft
34 kips 34 kips 34 kips
miiiui
0.96 kip/ft
27 kips 27 kips 27 kips
WIIIIIII
0.82 kip/ft
14ft
14ft
14ft 14ft
L > 100 ft
14ft 14ft
L< 100 ft
(b)
Figure 4.5. Proposed live load models for light rail transit: (a) standard live load model; (b)
alternative live load models
39
Figure 4.6 Tangent truck ballasted deck dual track bridge
40
4.71 9.25 4.71 4.71 9.25 4.71 4.71 9.25 4.71 4.71 9.25 4.71
i i i i i i i i n i i i i i i
'iiiii' 1 i i 1 1 i 1
6 44444 6 .1 w .1 .1 .1 .1 .1 .1 w .1 ., 4 ., 6 ., 6 ., 6 ., 6 ., 4 ., 4 8 ., 8 ., 8 1 4 1 1 6 1 10 10 1 6 1
* 32 ; ^ 32 * 32 * * 32 i
(a)
4.71 9.25 4.71
*k
4.71 9.25 4.71
l^k
4.71 9.25 4.71
V^kl^k
I I 1 I
, 4 6 6 6 6 4,
* 32 *
1 I 1 I
, 4 8 8 8 4 ,
s 32 *
1 I 1 I
, 6 10 10 6 ,
s 32 *
(b)
4.71 9.25 4.71
l^kH^k
' rillITT"
4 6 6 1
4.71 9.25 4.71
4 4 4 4
^TTTTT^
4.71 9.25 4.71
4 4 i i
H H i 1
i 4 i 8 i 8 ., 8 1 4 1
32
4.71 9.25 4.71
' I 1 1
1 6 1 10 10 1 6 1
32 i
(C)
Figure 4.7. Dimensional configurations of the benchmark bridge models (unit in ft; not to
scale): (a) steel plate girder; (b) steel box girder; (c) prestressed concrete I girder
41
4.71 9.25 4.71
4 4 4 4
t 4 l
32
1 4 S
4.71 9.25 4.71
M.
4 4 4 4
10
10
l 6 t
32
4.71 9.25 4.71
M.^*
4 4 4 4
t 4 S
12
12
32
S 4 S
(d)
4.71 9.25 4.71 4.71 9.25 4.71 4.71 9.25 4.71 4.71 9.25 4.71
n 44 44 4 i 4 4 4 i i 4 n
' U U U U U U ' ' U U U U U ' ' u u u u u u u
., 6 ., 4 ., 4 ., 4 ., 4 ., 4 ., 6 ., 4 6 6 6 6 4 1 4 1 8 1 8 ., 8 1 41 1 6 1 10 ., 10 1 6 1
* 32 ; ^ 32 * 32 * 32 *
(e)
Figure 4.8 Dimensional configurations of the benchmark bridge models (continued; unit
in ft; not to scale): (d) prestressed concrete box girder; (e) reinforced concrete box girder
42
_o
T>
,cu
o
co
o
0.8
0.6
0.4
0.2
0
 oTwotrackloaded A Onetrackloaded
""
'''A A
1 2 3 Girder: PC Box 4
Figure 4.9. Predicted live load distribution of the bridges based on bending moment
a
ro
o
~o~ Twotrackloaded
a Onetrackloaded
0.4
0.2
E>
A..
~B~
..A
..A'"
12 3 4
Girder: PC I
0
A
5
Figure 4.10. Predicted live load distribution of the bridges based on bending moment
O
ts
a
c
o
3
JD
CO
A
*D
CO
O
0.8
0.6
0.4 1
Ef
o~ Twotrackloaded
A Onetrackloaded

....A
B.
0.2 
'A.
2
o
1
2
3
Girder: Steel Box
0
4
Figure 4.11. Predicted live load distribution of the bridges based on bending moment
43
0.8
o
B
c 0.6
o
S 4
i
a Twotrackloaded
Onetrackloaded
.....a...........q........
a
A
.........A"
12 3 4
Girder: Steel Plate
Figure 4.12. Predicted live load distribution of the bridges based on bending moment
0.8
o
a
c 0.6
0
D
.Q
1 04
1 0.2
a...
a Twotrackloaded
A Onetrackloaded
...a.........
E3
...A
a
A
....A
2 3
Girder: RC
Figure 4.13. Predicted live load distribution of the bridges based on bending moment
44
5 Calibrated Load Distribution Equation for Light Rail Bridges
5.1 Development new equations
This chapter focused on the calibrate live load distribution factor equations based on the
format of the AASHTO LRFD equations and minimal adjustment was made except the
fact that the lever rule for singlelane loading recommended by the AASHTO LRFD
Specifications was replaced by equations similar to those for multipletrackloaded
cases because the prediction by the lever rule was overly conservative. These
distribution equations for five different types of light rail bridges: steel plate girder
bridge, steel box girder bridge, prestressed concrete I girder bridge, prestressed
concrete box girder bridge and reinforced concrete girder bridge. A total of 35 new
equations for different types of bridges.
5.1.1 Regression analysis
The load distribution factor numbers were get from CSIBridge model. The values
calculated for the five types of light rail bridges were statistically calibrated using the
nonlinear regression analysis use the program named XLSTAT 2015 to develop the
proposed load distribution formulas. XLSTAT statistical analysis provides the data
analysis and statistics abilities. This software can improve the analysis functions of
excel. After regression, excel with XLSTAT can display: coefficient of determination,
sum of square of errors, the new parameters and equation of the model (FIG 5.1).
The R2 corresponds to the % of the variability of the dependent variable (girder
spacing, longitudinal stiffness, span length and slab thickness) that is explained by the
explanatory variable (the load distribution factor obtained from CSIBridge). The closer to
45
1 the R2 is, the better the fit. In this case (FIG 5.2), 99% of the variability of the
dependent variable is explained by the explanatory variable, which is an excellent
result. For the predict value graphs from Figure 5.2 to Figure 5.11, can notice that the
model is well fitted for most cases.
5.1.2 Summary of new formulas
In this study, the load distribution factor equations were proposed for the following five
types of bridges:
steel plate girder bridge
steel box girder bridge
prestressed concrete I girder bridge
prestressed concrete box girder bridge
reinforced concrete girder bridge
The parameters considered in the new equations are following:
de : horizontal distance from the centerline of the exterior web of exterior beam at deck
level to the interior edge of curb or traffic barrier (ft)
g : distribution factor
Kg : longitudinal stiffness parameter (in.4)
L : span of beam (ft)
Nb : number of beams, stringers or girders
Nc : number of cells in a concrete box girder
NL : number of design lanes
S : spacing of beams or webs (ft)
46
ts : depth of concrete slab (in.)
We : half the web spacing, plus the total overhang (ft)
The formulas were developed for five types of bridges separately. There are 35 different
formulas were proposed for the light rail bridge. These formulas are listed in Table 5.1
through Table5.4. For the case of each type of light rail bridge considered as a group,
the equations were proposed separately for the following plots:
bending moment and shear force
single lane and multiple lanes loading
interior and exterior girders
The modified equations (or modification factors) in the AASHTO LRFD specifications
(Art. 4.6.2.2.2 Distribution factor method for moment and shear) were proposed for light
rail trains. The format of these equations are basically be identical to that of the present
AASHTO LRFD Specifications (Art. 4.6.2.2.2) for consistency. Using these load
distribution factor equations, light rail train load will conveniently be allocated to the
exterior and interior girders of a bridge superstructure. Table 5.1 to Table 5.4 shows the
comparison between LDF equations from AASHTO LRFD and proposed equation for
light rail bridge. Table 5.1 and Table 5.2 provided the moment distribution factor
formulas for interior and exterior girder, respectively, with one lane and two or more
lanes load. Table 5.3 and Table 5.4 provided the shear distribution factor formulas for
exterior and interior girder, respectively, with one lane and two or more lanes load.
Figure 5.12 shows the compressive comparison between the proposed and predicted
47
distribution factors for five types light rail bridges in moment and shear for both interior
and exterior girder.
5.2 Comparison of LDF formulas
In this thesis I use five types of site light rail bridges to verify the finite element model is
correct. The five bridges are County Line bridge, Broadway bridge, Indiana bridge,
Santa Fe bridge and 6th Ave bridge. The details of that five site bridge will be introduced
in chapter 6. Compare the fieldtesting LDF data with the finite element analysis LDF
data. The result is shown in Figure 5.13 to Figure 5.17. All the models are based on the
real bridge dimensions. Error bars were used in the comparison. Error bars are come
from the field testing LDF mean value and standard deviation for each girder. Table 5.7
shows the values from the field test. Error bar is the most accurate way to show the
variability of the field testing data, and make an accurate comparison with the finite
element for live load distribution factor analysis. The only different between the LDF
from finite element and fieldtesting is LDF from field is calculated by the midspan
strains from each girder and LDF from finite element is calculated by the midspan
moments from the girders. However, they use the same equation to get the live load
distribution factors. From the comparison, we can see from the graphs the LDF from
finite element is perfectly fit the LDF from site bridge test result except 6th Ave bridge
since when we got the data for 6th Ave bridge theres strong wind load during that
period.
48
After verify the models next step is get the new formulas. The format of the equation are
developed from AASHTO LRFD live load distribution equation and the values are based
on the light rail bridges which modeled in Chapter 4. The maximum live load factor for
interior and exterior girder were selected for every design case. For example,
prestressed I girder bridge has 80ft, 100ft, 140ft span length and for each span length
they have 4ft, 6ft, 8ft and 10ft girder spacing and for each girder spacing they have
different number of girders, more specifically, for 100ft span length, 4ft girder spacing
bridge has six girders and for each girder they can got different LDF values, we choose
the maximum LDF value from interior girder and exterior girder for analysis. Use the
regression method get the similar equation like AASHTO equation for LDF. Table 5.1
shows the calibrated equations with standard live load for interior girder moment live
load distribution factor. Table 5.2 provided the calibrated equations for exterior girder
moment live load distribution factor. The other two are for the shear LDF for interior
girder and exterior girder respectively.
5.2.1 New formulas vs level rule
Because there is a lack of information about live load distribution factors for light rail
bridges, the lever rule method is frequently employed by practitioners. Table 5.5 shows
load distribution factors calculated by level rule. These factors are used for the design
part, however, after we designed and modeled the bridge we can get the new live load
distribution factors based on the finite element analysis. As we all know the level rule is
over conservative in most cases, that will lead to the economical problems. Figure 5.18
and Figure 5.19 shown the assessment of existing methods for bending moment and
49
shear make use of lever rule and AASHTO LRFD equation for both exterior and interior
girders to get the live load distribution factors. In the graph five types of bridges were
under consideration, PC Box, PC I, Steel Box, Steel Plate and RC bridges. Live load
distribution factors from models are traded X axis and level rule and AASHTO LRFD
equation values used for y axis. It is very clearly that the values from level rule are too
conservative than the values from CSIBridge and our proposed equation values.
5.2.2 New formulas vs site bridge
For live load distribution factors, AASHTO LRFD is used for highway bridge design and
Level Rule is used in the RTD Design Criteria. From Figure 5.20 to Figure 5.24 show
the comparison of the live load distribution factors from AASHTO LRFD equations>
Level rule> field testing average measurement values and the proposed equations for
light rail bridges for five site light rail bridges. Since there are lots of factors influence the
field test data, like the test machine error, the wind load and some unexpected
situations. I selected the average values of stress for each girder to calculate the live
load distribution factors. Figure 5.20 to Figure 5.24 show the application of the proposed
live load distribution factor for Broadway bridge, Indiana bridge, Santa Fe bridge,
County Line bridge and 6th Ave bridge respectively. The results show that the Level
Rule method is the most conservative way to calculate live load distribution factors.
Since the proposed LDF equations are based on the AASHTO LRFD equations, the
trend of the live load distribution faction in the figure is almost the same. However, the
moment data for proposed equations is from finite element, so the values of the live
load distribution factors for proposed equations are smaller than the LDF from AASHTO
LRFD equations. The figures from proposed equations are much closer to the
measured average values. Compared the field testing with the theoretical finite element
analysis, some of the measured average are bigger than the proposed LDF, but thats
ok since the values from outside are more flexible because of the weather, the live
loading and some unpredictable errors.
51
Table 5.1. Live load distribution calibrated by deterministic standard live load
(interior moment)
Type of
superstructure
Applicable
cross
section in
AASHTO
LRFD
AASHTO LRFD
Proposed
PC Box
One lane
1.75+AYI
0.35/ \
0.45
3.6 A A
Two or more lanes
KNc;
Onetrackloaded
T1
19.681 ){L
T wotrackloaded
0.223+
Y3A
0.3
v^y
S
1_
5.8 J(L
(7.0 < < 13.0)
(60 < Z < 240)
(Nb>3)
0.25
^ 13 '
v^y
S
9.989
(8.0
(80 < Z < 140)
(Nb>3)
PCI
One lane
Onetrackloaded
S '\0Af s''031'
0.06
14) (L
Two or more lanes
K
,0.1
V12.0 LtSJ
0.19 +
S
0.282 / 0.044
67.569 j
T wotrackloaded
K
12.0 Lf
0.075
S f
K
9.5 J (L
(3.5 < S' < 16.0)
(4.5^ <120)
(20 < Z < 240)
(^^4)
(10,000
V12.0LC,
0.197
S
1.556 / e\0.16 (
O
Z
K
12.734
(4 < S' < 10)
(^=10)
(80 < Z < 140)
(Nb>3)
(500,000
Yl.OLt
v > y
Steel Box
b,c
One lane
AY035
3.0
Sd
One lane
S
12.0 L
Two or more lanes
_A_
6.3
Sd
12.OF2
(6 < S' < 18)
(20
(l8<<7 <65)
(wt>3)
Sd
68.747 ) (12.0L
Two or more lanes
s a926 ( sd
13.945 J [l2.0Z2
(6 < S' < 10)
( 80 < Z < 140 )
(33< d<65)
(3 < Nh < 5 )
Steel Plate
One lane
s' ays\3f
0.06
14) (L
Two or more lanes
K
\ o. 1
v12.0Lt,y
Onetrackloaded
0.119 + ( S
K
\ 0.526 A 0.J58 ,0.096
81.120 KL 42.0 Lt
T wotrackloaded
52
0.075 +
,0.6/ 0 \ 0.2 /
K
,95 J (L
(3.5 < S' < 16.0)
(4.5^ <120)
(20
(^4)
(10,000
v12.0 LtSJ
/ \ 1.639 /_^_\0.02 / \0.027
15.530J T2.0Lf3 '
(4 < S' < 10)
=10)
(80 < Z < 160)
(Nb>3)
(400,000 <7^ <5,000,000)
RC
One lane
Onetrackloaded
0.06 +
S_
,14) (L
Two or more lanes
0.3/
v12.0Lf,y
0.038 +
S
1.954 / 0.085 /
.14.336
T wotrackloaded
0.075 +
S )0'6/
K
,0.1
9.5) \L
(3.5 < S < 16.0)
(4.5^ <120)
(20
(^4)
(l,0000<^g <7,000,000)
v12.0Lf,y
0.074 +
S
K
12.0/7
0.032 (
11.111
(4
(^=10)
(30
(Nb>3)
(50,000
K
\2.0 Lt
53
Table 5.2. Live load distribution calibrated by deterministic standard live load
(exterior moment)
Type of superstructure Applicable cross section in AASHTO LRFD AASHTO LRFD Proposed
PC Box d Regardless of number of lanes W g=^(We
PCI k One lane Lever Rule Two or more lanes Â§ S ^interior e = 0.77 + ^ 9.1 ( ~\.0
Steel Box b,c One lane Lever Rule Two or more lanes Â§ S ^interior d e = 0.97 + e 28.5 (0 < de < 4.5 ) (6.0 18.0) Onetrackloaded Â§ Sginterior e = 0.295+ de 4.765 Twotrackloaded Â§ Sginterior e = 0.797+ dg 25.943 (4 < de < 6 ) (6.0
Steel Plate a One lane Lever Rule Two or more lanes Â§ S ^interior e = 0.77 + ^ 9.1 {\.0
RC e One lane Lever Rule Two or more lanes Â§ S ^interior e = 0.77 + 9.1 (1.0<^<5.5) Onetrackloaded Â§ Sginterior d e = 0.783 + 191362.335 Twotrackloaded Â§ Sginterior e = 0.326+ deL 73.005S (4
54
Table 5.3. Live load distribution calibrated by deterministic standard live load
(interior shear)
Type of superstructure Applicable cross section in AASHTO LRFD AASHTO LRFD Proposed
PC Box d Regardless of number of lanes W g=fiiWeZS) Regardless of number of lanes 8 21.833
PCI k One lane Lever Rule Two or more lanes Â§ Sginterior e = 0.77+ 9.1 (~\.0
Steel Box b,c One lane Lever Rule Two or more lanes Â§ Sginterior e = 0.97 + dg 28.5 (o < de < 4.5 ) (6.0 <Â£< 18.0) Lever Rule (S'>18.0) Onetrackloaded Â§ Sginterior e = 0.295+ dg 4.765 Twotrack1 oaded Â§ Sginterior e = 0.797+ de 25.943 (4 < de < 6 ) (6.0 <Â£< 10.0)
Steel Plate a One lane Lever Rule Two or more lanes Â§ Sginterior e = 0.77+ 9.1 (1.0
RC e One lane Lever Rule Two or more lanes Â§ Sginterior e = 0.77 + ^~ 9.1 (1.0
55
Table 5.4. Live load distribution calibrated by deterministic standard live load
(exterior shear)
Type of superstructure Applicable cross section in AASHTO LRFD AASHTO LRFD Proposed
PC Box d One lane Lever Rule Two or more lanes Â§ Â£ interior e = 0.64+ de 12.5 (2.0
PCI k One lane Lever Rule Two or more lanes Â§ Â£ interior e = 0.6 + 10 (\.0
Steel Box b,c One lane Lever Rule Two or more lanes g 6g interior d e = 0.8 + 10 (o < de < 4.5 ) One lane e = 0.582+ de 2.675 Two or more lanes g ^ginterior d e = 0.726+ e 28.392 (4 < de < 6 )
Steel Plate a One lane Lever Rule Two or more lanes Â§ Â£ interior e = 0.6 + 10 (\.0
RC e One lane Lever Rule Two or more lanes Â§ Â£interior e = 0.6 + 10 (1.0
56
Table 5.5. Load distribution factors calculated by level rule
4ft girder spacing 6ft girder spacing 8ft girder spacing 10ft girder spacing 12ft girder spacing
G1 0.83875 G1 0.5598 G1 0.75 G1 1.4 G1 1.167
G2 0.8225 G2 1.215 G2 1.25 G2 1.2 G2 1.667
G3 0.33875 G3 0.4517 G3 1.25 G3 1.4 G3 1.167
G4 0.33875 G4 1.215 G4 0.75
G5 0.8225 G5 0.5598
G6 0.83875
Table 5.6. Sectional Properties of five site Bridges
Composite sectional properties
Exterior girder Interior girder
Bridge Span Length (ft) Number of girders Girder spacing (ft) Thickness of slab (in) Centroid (in) Moment of inertia (inA4) Centroid (in) Moment of inertia (inA4)
Brodway 119 3 12 10 33.39 86250 35.29 177813
Counry Line 160 4 7 8 41.66 875207 41.66 875207
Indiana 95 2 10 10 50.46 4347653 50.46 4347653
Sanra Fe 155.5 4 7 20 48.32 1932468 52.76 2946503
6th Ave 80 6 6 8 20.23 111689 20.23 111689
Table 5.7. Field testing LDF mean value and standard deviation for each girder
Bridge Grl Gr2 Gr3 Gr4 Gr5 Gr6
Mean value Standard Deviation Mean value Standard Deviation Mean value Standard Deviation Mean value Standard Deviation Mean value Standard Deviation Mean value Standard Deviation
6th Ave 0.256 0.117617 0.258 0.090983 0.339 0.132787 0.311 0.132613 0.293 0.061341 0.301 0.134401
County Line 0.350 0.042901 0.366 0.066622 0.214 0.029321 0.070 0.018291
Santa Fe 0.27 0.038204 0.31 0.066388 0.27 0.025098 0.15 0.078399
Broadway 0.48 0.031337 0.34 0.022129 0.18 0.033279
Indiana 0.56 0.0448 0.45 0.0855
57
Nonlinear regression of variable Yl:
Goodness of fit statistics:
Observatic 16.000
DF 12.000
R2 0.999
SSE 0.000
MSE 0.000
RMSE 0.004
Iterations 13.000
Model parameters:
Parameter Value
prl 0.195
pr2 1.144
pr3 0.345
pr4 0.118
Equation of the model:
Yl = 0.194562920060864+(X1/9.5)A1.14384018464267*(X1/X2)A0.345029176102755*(X3/(12*X2*X4A3))A0.117874487650818
Predictions and residuals:
Observation XI X2 X3 X4 Y1 PredY1) Residuals
Obs1 4.000 80.000 394302.201 10.000 0.314 0.314 0.001
Obs2 6.000 80.000 394302.201 10.000 0.416 0.412 0.004
Obs3 8.000 80.000 445746.069 10.000 0.536 0.534 0.002
Obs4 10.000 80.000 560625.403 10.000 0.682 0.680 0.001
Dbs5 4.000 100.000 749674.128 10.000 0.311 0.310 0.001
Dbs6 6.000 100.000 867619.497 10.000 0.411 0.410 0.001
Obs7 8.000 100.000 996445.879 10.000 0.530 0.531 0.001
Obs8 10.000 100.000 1341655.338 10.000 0.671 0.680 0.009
Obs9 4.000 140.000 2415954.564 10.000 0.307 0.308 0002
ObslO 6.000 140.000 2846532 484 10.000 0.405 0.407 0002
Obs11 8.000 140.000 3126179.037 10.000 0.520 0.524 0004
Obs12 10.000 140.000 3422300.719 10.000 0.662 0.658 0.003
Obs13 4.000 160.000 3860162.299 10.000 0.307 0.308 0 000
Dbs14 6.000 160.000 4089000.907 10.000 0.403 0.403 0.001
Obs15 8.000 160.000 4571313.124 10.000 0.517 0.518 0.001
Obs16 10.000 160.000 5087280.616 10.000 0.858 0.652 0.006
Figure 5.1 The results of nonlinear regression for the moment of steel girder bridge
(interior girder) with two lanes load.
58
1.5 n
T3
O
Q.
O
L_
Q.
1.2
0.9
D 0.6
0.3
cP''
0
0
I1111
0.3 0.6 0.9 1.2 1.5
LDF (model): PC Box (interior)
(a)
1.5 n
T3 1 .2 
O
o 0.9 
w
Q_
Q 0.6 
0.3 
0 +
0
P'
imnfn
ijrrr
11111
0.3 0.6 0.9 1.2 1.5
LDF (model): PC Box (exterior)
(b)
Figure 5.2. Comparison of moment between the finite model prediction and proposed
equation: (a) prestressed concrete box (interior); (b) prestressed concrete box (exterior);
59
1.5
T3 1 .2 
0
0
O
o 0.9 
imm.
Q.
0 0.3 0.6 0.9 1.2 1.5
LDF (model): PC I (interior)
(a)
1.5
T3 1 2 
0
0
O
o 0.9 
0 0.3 0.6 0.9 1.2 1.5
LDF (model): PC I (exterior)
(b)
Figure 5.3. Comparison of moment between the finite model prediction and proposed
equation:(a) prestressed concrete I (interior); (b) prestressed concrete I (exterior);
60
1.5 n
T3 1 .2 
0
0
O
o 0.9 
imm.
Q.
Q 0.6 
0.3 
0 +
0
a'
i1111
0.3 0.6 0.9 1.2 1.5
LDF (model): Steel Box (interior)
(a)
1.5
T3 1 2 
0
0
O
o 0.9 
Q_
Q 0.6 
0.3 
0 +
0
I1111
0.3 0.6 0.9 1.2 1.5
LDF (model): Steel Box (exterior)
(b)
Figure 5.4. Comparison of moment between the finite model prediction and proposed
equation: (a) steel box (interior); (b) steel box (exterior);
61
1.5
T3 1 .2 
a>
w
o
o 0.9 j
a.
Q 0.6
_i
0.3
0
m
0.3 0.6 0.9 1.2
LDF (model): Steel Plate (interior)
(a)
1.5
15 i
T3 1 2 
to
o
o 0.9 
a.
Q 0.6 
0.3 
0 +
0
0.3 0.6 0.9 1.2 1.5
LDF (model): Steel Plate (exterior)
(b)
Figure 5.5. Comparison of moment between the finite model prediction and proposed
equation: (a) steel plate (interior); (b) steel plate (exterior)
62
1.5 n
"u 1.2 l
(D
O
o 0.9 \
imm.
Q.
Q 0.6 I
_i
0.3 
0
O'
B
P'
0'
0.3 0.6 0.9 1.2
LDF (model): RC (interior)
(a)
1.5
~o
d>
w
o
Q.
0 0.3 0.6 0.9 1.2 1.5
LDF (model): RC (exterior)
(b)
Figure 5.6. Comparison of moment between the finite model prediction and proposed
equation: (a)reinforced concrete (interior); (b) reinforced concrete (exterior)
63
1.5
T3 1 .2 
0
0
O
o 0.9 
imm.
Q.
Q 0.6 
_l
0.3 
0 1
0
I1111
0.3 0.6 0.9 1.2 1.5
LDF (model): PC Box (interior)
(a)
1.5
T3 1 2 
0
0
O
o 0.9 
Q_
Q 0.6 
0.3 
0 +
0
I1111
0.3 0.6 0.9 1.2 1.5
LDF (model): PC Box (exterior)
(b)
Figure 5.7. Comparison of shear between the finite model prediction and proposed
equation: (a) prestressed concrete box (interior); (b) prestressed concrete box (exterior);
64
1.5 n
o 1.2 
a>
v>
o
o 0.9 A
Q 0.6 A
0.3 
HO
i in fi
nnf
0.3 0.6 0.9 1.2
LDF (model): PC I (interior)
(a)
1.5
15 I
9 1.2 
a)
w
o
o 0.9 
Q_
q 0.6 
_i
0.3 
0 
.a
o
0.3 0.6 0.9 1.2
LDF (model): PC I (exterior)
1.5
(b)
Figure 5.8 Comparison of shear between the finite model prediction and proposed
equation:(a) prestressed concrete I (interior); (b) prestressed concrete I (exterior)
65
1.5 i
o 1.2
o
o 0.9 
imm.
Q.
D 0.6 
0.3 
0 +
0
I1111
0.3 0.6 0.9 1.2 1.5
LDF (model): Steel Box (interior)
(a)
1.5
T3 1 2 
O
o 0.9 
Q_
q 0.6 
0.3 
0 +
0
11111
0.3 0.6 0.9 1.2 1.5
LDF (model): Steel Box (exterior)
(b)
Figure 5.9. Comparison of shear between the finite model prediction and proposed
equation: (a) steel box (interior); (b) steel box (exterior);
66
1.5 i
* 12 
o
Â§ 0.9 
Q_
S 06 
_i
0.3 
0 +
0
O'
/O
Onetrackloaded
o Twotrackloaded
I I I I I
0.3 0.6 0.9 1.2 1.5
LDF (model): Steel Plate (interior)
(a)
1.5 n
o 1.2 
o
CO
o
Â§0.9 
Q_
0.6 
_i
0.3 
0 +
0
Onetrackloaded
o Twotrackloaded
I1111
0.3 0.6 0.9 1.2 1.5
LDF (model): Steel Plate (exterior)
(b)
Figure 5.10. Comparison of shear between the finite model prediction and proposed
equation:(a) steel plate (interior); (b) steel plate (exterior)
67
1.5
T3 1.2 
0
(/>
O
Â§0.9 
CL
0.3 
0 +
0
0''Q
HE
j?n
i*t&Tr
Onetrackloaded
o Twotrackloaded
0.3 0.6 0.9 1.2 1.5
LDF (model): RC (interior)
Figure 5.11. Comparison of shear between the finite model prediction and proposed
equation: (a) reinforced concrete (interior); (b) reinforced concrete (exterior)
68
1.5
6* 12
w
o
Â§ 0.9
Q_
q 0.6
_i
0.3
0
If
X
PC Box
o PC I
a ST Plate
x ST Box
x RC
0 0.3 0.6 0.9 1.2 1.5
LDF (model): (moment: exterior)
(a)
(b)
69
1.5
e 1.2 
a>
o
Â§ 0.9 
o.
q 0.6 
_i
0.3 
0 i
0
0.3 0.6 0.9
PC Box
o PC I
a ST Plate
x ST Box
x RC
1.2 1.5
LDF (model): (shear: exterior)
(c)
(d)
Figure 5.12. Comprehensive comparison between the proposed and predicted distribution
factors: (a) moment for exterior girders; (b) moment for interior girders; (c) shear for
exterior girders; (d) shear for interior girders
70
1 1
&08
Â§ 0.6 H
A Measured average
B CSI Bridge
t i r
C3
.2 0.4
TD
T3
(0
O
0.2 
0
2
Girder
^4
I
3
Figure 5.13 LDF comparison between Broadway bridge fieldtesting and FE analysis
1.5 n
0 19
(0 1 z 1
4
Â§ 0.9 H
Â§ 0.6 i
T3
T3
(0
O
0.3 
0
AMeasured average
BCSI Bridge
I
2
Box girder web
Figure 5.14 LDF comparison between Indiana bridge fieldtesting and FE analysis
71
Figure 5.15 LDF comparison between Santa Fe bridge fieldtesting and FE analysis
1
O
t<
o
(0
0.8
c
o
t'
0.6
A Measured average
BCSI Bridge
(0
TD
TD
(0
o
0.4
0.2
0 A1r
1 2 3
Girder
a ..
TUI
1
4
Figure 5.16 LDF comparison between County Line bridge fieldtesting and FE analysis
72
Load distribution factor
1
0.8
AMeasured average
B CSI Bridge
n 0
TTTTTT
0.6 
0.4 
L
0.2 
it"
______0
0 Ir
____
___0'
1
ta
=S
1
1
3
1
4
1
5
I
6
Girder
Figure 5.17 LDF comparison between 6th Ave bridge fieldtesting and FE analysis
73
1.5 n
0)
3
1.2 
CD
>
a)
0.9 
u_ 0.6 
Q
0.3 
0
0
CEO
cm ao
o a m>
*
m>m s'
PC Box
A PC I
+ Steel Box
x Steel Plate
o RC
0.3 0.6 0.9 1.2 1.5
LDF (model): exterior girder
(a)
1.5 i
D
1.2 
0.9 
0.6 
0.3 
0.3
$
OS
PC Box
A PCI
+ Steel Box
x Steel Plate
o RC
0.6 0.9 1.2 1.5
LDF (model): interior girder
(b)
74
1.5 n
1.2 
a:
g 0.9 
i
co
$ 0.6 
3 0.3 A
o
o
[%>
0+
nn
PC Box
A PC I
+ Steel Box
x Steel Plate
o RC
0.3 0.6 0.9 1.2
LDF (model): exterior girder
(c)
1.5
1.5 n
Q
fe 12 H
R 0.9 A
x
CO
Â§. 0.6 A
LL
Q
1 0.3 
O
PC Box
A PCI
+ Steel Box
x Steel Plate
o RC
0.3 0.6 0.9 1.2
LDF (model): interior girder
1.5
(d)
Figure 5.18 Assessment of existing methods for bending moment: (a) lever rule for exterior
girders; (b) lever rule for interior girders; (c) AASHTO LRFD for exterior girders; (d)
AASHTO LRFD for interior girders
75
1.5 n *
4DOAK
aT 1.2  on n y'
,, *
0 0.9 
> <*K>m x
ooeA a ^
Ll_ Q 0.6  npn 0*XST PC Box
_l A PC I
0 3  * s + Steel Box
X Steel Plate
O RC
0 
0 0.3 0.6 0.9 1.2 1.5
LDF (model): exterior girder
(a)
1.5 i
3
1.2 
0.9 
>
^ o
AXXX3
LL 0.6 
Q
0.3 
0 l
0
PC Box
A PC I
+ Steel Box
x Steel Plate
o RC
0.3 0.6 0.9 1.2 1.5
LDF (model): interior girder
(b)
76
Q
LL
a:
O
x
co
$
1.5 n + O AX
1.2  O
0.9  , ^ + <# 3S\ >&J rife
0.6  A
0.3  ...feg'B
0 
0 0.3 0.6
0.9
PC Box
A PC I
+ Steel Box
x Steel Plate
o RC
I I
1.2 1.5
LDF (model): exterior girder
(c)
1.5 i
Q
fe 1.2 
O
x
CO
Â§
0.9 
0.6 
Q
* 0.3 
0
0
PC Box
A PCI
+ Steel Box
x Steel Plate
o RC
0.3 0.6 0.9 1.2 1.5
LDF (model): interior girder
(d)
Figure 5.19. Assessment of existing methods for shear: (a) lever rule for exterior girders;
(b) lever rule for interior girders; (c) AASHTO LRFD for exterior girders; (d) AASHTO
LRFD for interior girders
77
Girder
Girder
Girder
Figure 5.20. Application of the proposed live load distribution factor equations to the five
bridges in Denver: (Broadway)
78
o
o
.2
c
o
"5
_o
c/)
"O
CD
O
AAASHTO LRFD B Lever rule
Measured average II II
Proposed
tr g
1 2
Box girder web
Figure 5.21. Application of the proposed live load distribution factor equations to the five
bridges in Denver (Indiana)
79

PAGE 2
2015 DI WEI ALL RIGHTS RESERVED
PAGE 3
ii This thesis for the Master of Science degree by Di Wei Has been approved for the Civil Engineering program By Yail Jimmy Kim, C hair Chengyu Li Fredrick Rutz October 10, 2015
PAGE 4
iii Wei Di (M.S., Civil Engineering) Behavior of light rail bridges: live load distribution and substructure settlement Thesis directed by Associate Professor Yail Jimmy Kim ABSTRACT The live load distribution factors (LDF) provided by AASHTO LRFD Bridge D esign Specification ha ve been in effect for almost 20 years to calculate b ending moment and shear force for highway bridge design These equations are calibrated based on elastic finite element analysis. M aximum bending moments and shear forces are determined by the load distribution factors. Load distribution factor s estimated by the LRFD equation s are generally conservative compared with refined analysis results. The use of the AASHTO LRFD equation s may not provide accurate load distribution factor s for light r ail bridges because the load configurations of light rail trains are different from those of highway vehicles T his study examines live load distribution factors for light rail bridges and proposes new equations. For this study standard light rail trains are placed on five different types of light rail bridges in Denver, Colorado The focus of this thesis is on develop ing live load distribution factor e quations for light rail bridges. The behavior of these light rail bridges subjected to 2, 4 and 6 sub structure settlements is also studied. In this s t udy, 155 light rail bridges are des igned and modeled. The load is applied to the five ty pes of light rail bridges. The moment and shear of each bridge are predicted. This
PAGE 5
iv research proposes thirty five new equation s for five different types of light rail b ridges for bending and shear, including interior and exterior girder s Th e distribution factors which have been calculated by the developed equation s are compared with the AASHTO LRFD distribution equa tions. Another comparison is also made against the responses of the constructed bridges. The form and content of this abstract are approved. I recommend its publication. Approved: Yail jimmy Kim
PAGE 6
v ACKOWLEDGEMENT S I would like to e xpress my sincerest gratitude to the multitude of individuals who assisted with this thesis. Dr. Kim, my advisor who has been instrumental in the theoretical part of this thesis. Your encourage ment wisdom and support have not gone unnoticed. You give me the chance to work for you and let me know more about the design for the bridge. Without your assistance, this t hesis would not be where it is today. Sincere thanks is offered to Dr.Chengyu Li and Dr.Frederick R.Rutz, not only for being part of my graduate committee but also for the two years teaching and helping. Also thanks Dr.Li give me the chance to come to this university. Additionally, I want to say thanks to my teammates Thushara, Yongcheng Ji and Lianjie Liu, They give me so much help during the last year. I am grateful to acknowledge financial support provided by the national academy of science. Finall y I w ant to thank my parents, they give me the financial support for studying in Unite d State s
PAGE 7
vi CONTENTS Chapter 1. Introduction . 1 1.1 Background. 1 1.2 Objectives 2 1.3 Organization .. 3 2. Literature Revie w 6 2.1 Bridge design code s .6 2.2 Load and load combinations .. 8 2.3 Load distribution factors .. .. .9 2.4 Live load model s ..10 3 .Design and Finite el ement Modeling of Light Rail B ri dge ............ ... 1 5 3.1 Introduction .. 1 5 3.2 Design of the bridge ... .. 1 5 3.2.1 Types of bridges .. 1 6 3.3 Finite Element modeling ... ... .. 1 7 3.3 .1 Element types used in modeling .. .. 1 8 3.3 .2 Summa ry of models ...... .. 1 9 4. Load D istribution Analyses for Modeling Bridge ... .. 31 4.1 Introduction ... 31 4. 2 Live load distribution factor .. 31 4.3 Live load model ... 33 4.4 Loading placement 35 4.5 N umerical result s .. 36
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vii 5. Calibrated Load Distribution Equations for Light Rail B ridge 45 5.1 Development of new equations .. ............... 45 5.1.1 Regression analysis .. ... .. 45 5 .1.2 Summary of new formula s .. ... .. 46 5.2 Comparison of LDF formulas ... 48 5.2.1 New formulas vs level rule ... 49 5.2.2 New formulas vs site bridges 50 6 Live Load Distribution Factors Influenced by Pier S ettlement..... ... 83 6.1 Introduction for five site light rail bridges in Denver .. 83 6.2 Five b ridge models .. ... 84 6.3 Moment and Shear influenced by pier settlement ... 85 6.4 LDF influenced by settlement .. .. .89 6.5 Results and discuss .. 90 7 C onclusions and Recommendations.. .. 202 References .. .. 203 Appendix .205
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viii LIST OF FIGURES Figure 2.1 load distribution on a typical slab girder bridge (a) Steel br idge (b ) Bridge with concentrate load (c) deformation for cross section (d) transversely flexible .. ......... 13 2.2 Girder section in slab girder bridge (a) Exterior section (b) Interior section ................. .. ..... 14 2.3 LRV loading diagram................ ........................ ...... 14 3.1 Details of model .27 3.2 Boundary elements in bridge model ............................27 3.3 Rectangula r shell element in bridg e model............................ .................................. ...... ......... 28 3.4 Finite element models for steel plate girder bridge ................. .................................... ...... ....28 3.5 Finite element models for steel box girder bridge ...29 3.6 Finite element models for prestressed concrete I girder bridge ...29 3.7 Finite element models for prestressed concrete box girder bridge..... .................................. . 30 3.8 Finite element models for r einforced concrete girder bridge ....... 30 4.1 AASHTO HS 20 Design Truck (AASHTO 2015).............. ................................... ......... 37 4.2. Schematic of procedures for determining the standard live l oad model of light rail transit .. 38 4.3. European live load model for train (ERRI Committee D192): (a) existing LM71; (b) newly developed LM 2000 ........................... ................................................................ .................. ........ 38 4.4. Selected existin g live load models and in situ live load for Calibration.... ................. .......... 39 4.5. Proposed live load models for light rail transit: (a) standard live load model; (b) alternative live load models.... .................................. ............................ .... .. ............. 39 4.6 Tangent truck ballasted deck dual track bridge................... .................................. .... ......... 40 4.7. Dimensional configurations of the benchmark bridge models (unit in ft ; not to scale): (a) steel plate girder; (b) steel box girder; (c) prestressed concrete I girder ... ........ 41
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ix 4.8. Dimensional configurations of the benchmark bridge models (continued; unit in ft; not to scale): (d) prestressed concrete box girder; (e) reinforced concrete box girder .............. .. ....... 42 4.9. Predicted live load distribution of the bridges based on bending moment (PC BOX).. ......... 43 4.10. Predicted live load distribution of the bridges based on bending moment (PC I)........... ..... 43 4.11. Predicted live load distribution of the bridges based on bending moment ( Steel Box) .... ... 43 4.12. Predicted live load distribution of the bridges based on bend ing moment ( Steel Plate ).... 44 4.13. Predicted live load distribution of the bridges based on bending moment (RC) . 44 5.1 The results of nonlinear regression for the moment of steel girder bridge (interior girder) with two lanes load .. ... ........... 58 5.2. Comparison of moment between the finite mode l prediction and proposed equation: (a) prestressed concrete box (interior); (b) prestressed concrete box (e xterior)........... ....................... 59 5.3. Comparison of moment between the finite model prediction and proposed equation : (a) prestressed c oncrete I (interior); (b) prestressed concrete I (exterior) ...................... .... .... ............ 60 5.4. Comparison of moment between the finite model prediction and proposed equation : (a) steel box (interior); (b) steel box (exterior) ... ..... 61 5.5. Comparison of moment between the finite model prediction and proposed equation: (a) steel plate (interior); (b) steel plate (exterior)...... ..................................................................... .. .......... 62 5.6. Comp arison of moment between the finite model prediction and proposed equation: (a) reinforced concrete (interior); (b) reinforced concrete (exterior) ....... ......................... ...... 63 5.7. Comparison of shear between the finite model prediction and p roposed equation: (a) prestressed concrete box (interior); (b) prestressed concrete box (exterior) .......... 64 5.8 Comparison of shear between the finite model prediction and proposed equation: (a) prestressed concrete I (interior); (b) prestressed c oncrete I (exterior) .. 65
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x 5.9. Comparison of shear between the finite model prediction and proposed equation: (a) steel box (interior); (b) steel box (exterior)......... ............. 66 5.10. Comparison of shear between the finite model prediction and proposed equation: (a) steel plate (interior); (b) steel plate (exterior)... ............................... ........ ......................... ............. 67 5.11. Comparison of shear between the finite model prediction and prop osed equation: (a) reinforced concrete (interior); (b) reinforced concrete (exterior).......................... ................... 68 5.12. Comprehensive comparison between the proposed and predicted distribution factors: (a) moment for exterior girders; (b) moment for interior girders; (c) shear for exterior girders; (d) shear for interior girders ......... 70 5.13. LDF comparison between Broadway bridge field testing and FE analysis... ....... 71 5.14. LDF comparison between Ind iana bridge field testing and FE analysis . 71 5.15. LDF comparison between Santa Fe bridge field testing and FE analysis 72 5.16. LDF comparison between County Line bridge fie ld testing and FE analysis .. 72 5.17. LDF comparison between 6 th Ave bridge field testing and FE analysis .73 5.18 Assessment of existing methods for bending moment: (a) lever rule for exterior girders; ( b ) lever rule for interior girders; ( c ) AASHTO LRFD for exterior girders; (d ) AASHTO LRFD for interior girde rs ........... ........................................................... .............. ........................................... 75 5.19 Assessment of existing methods for shear: (a) lever rule for exterior girders; ( b ) lever rule for interior girders; ( c ) AASHTO LRFD for exterior girders; (d ) AASHTO LRFD for interior girders ....... ............................................................................... 77 5.20 Application of the proposed live load distribution factor equations to the five bridges in Denver: (a) Broadway...................... ............................................... ................... .... ................... 78
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xi 5.21 Application of the proposed live load distribution factor equations to the five bridges in Den ver(b) Indiana............ .................................................... ........... .......................... 79 5.22 Application of the proposed live load distribution factor equations to the five bridges in Denver (Santa Fe ) .. .. 80 5.23 Application of the proposed live load distribution factor equations to the five bridges in Denver (County Line)....... ..................................................... ................................ 81 5.24 Appli cation of the proposed live load distribution factor equations to the five bridges in Denver (6th Ave) .. ......................... 82 6.1 Cross section of Broadway Bridge .. 102 6.2 Cross section of 6 th Ave Bridge ......................... ........... ................ .............. ....... ..................102 6.3 Cross section of Indiana Bridge ......................... ........... ................ ........ .............. .................103 6.4 Cross section of Santa Fe Bridge...103 6.5 Cross section of County Line Bridge.104 6.6 Pier settlement model for 6 th ave B ridge ......................... .......... ........... . .............. ...... ...........1 05 6.7 Pier settlement model for Broadway B ridge ......................... ........... ................ ... .. ......... ...... 105 6.8 Pier settlement model for County Line Bridge ......................... ........... ............ ..................... 106 6.9 Pier settlement model for Indiana B ridge ......................... ......................... ... ....... .................106 6.10 Pier settlement model for Santa Fe Bridge ......................... ...................... ..........................107 6.11 Moment an d Shear data collection points for two span bridges (Broadway Bridge and Santa Fe Bridge )..108
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xii 6.12 Moment and Shear data collection points for four span bridges (6th Ave Bridge and County Line Bridge) .108 6.13 Moment and Shear data collection points for five span bridge ( Indiana Bridge) ... 109 6.14 Moment of seven points for 6 th Ave Bridge with (a) case one (2,0,0) settlement (b) case two (2,2,0) settlement (c) case three (2,4,2) settlement (d) case four (2,6,2) settlement.111 6.15 Moment comparison with different pier settlement cases for 6th Ave Bridge (a) RTD Loading (b) Settlement (c) Settlement +RTD Loading ...112 6.16 Moment o f seven points for County Line Bridge with (a) case one (2,0,0) settlement (b) case two (2,2,0) settlement (c) case three (2,4,2) settlement (d) case four (2,6,2) settlement .114 6.17 Moment comparison with diff erent pier settlement cases for County Line Bridge (a) RTD Loading (b) Settlement (c) Settlement +RTD Loading ...115 6 .18 Moment diagram for Broadway Bridge with different pier settlement at mid point of first span (a) 2 settlement (b) 4 se ttlement (c) 6 settlement ... 117 6.19 Moment diagram for Broadway Bridge with different pier settlement at support column (a) 2 settlement (b) 4 settlement (c) 6 settlement 118 6.20 Shear diagram for Broadway Bridge with different pier settlement at abutment (a) 2 settlement (b) 4 settlement (c) 6 settlement .. 120 6.21 Shear diagram for Broadway Bridge with different pier settlement at support column (a) 2 settlement (b) 4 settlement (c) 6 settlemen t .121 6.22 Moment diagram for Santa Fe Bridge with different pier settlement at mid point of first span (a) 2 settlement (b) 4 settlement (c) 6 settlement ... 123
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xiii 6.23 Moment diagram for Santa Fe Bridge with different pie r settlement at support column (a) 2 settlement (b) 4 settlement (c) 6 settlement .. 124 6.24 Shear diagram for Santa Fe Bridge with different pier settlement at abutment (a) 2 settlement (b) 4 settlement (c) 6 settlement .126 6.25 Shear diagram for Santa Fe Bridge with different pier settlement at support column (a) 2 settlement (b) 4 settlement (c) 6 settlement ................................................ ..127 6.26 Moment diagram for County Line Br idge with different pier settlement cases at moment point 1 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 129 6.27 Moment diagram for County Line Bridge with different pier settlement cases at moment point 2 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 131 6.28 Moment diagram for County Line Bridge with different pier settlement cases at moment point 3 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6 2 ................. 133 6.29 Moment diagram for County Line Bridge with different pier settlement cases at moment point 4 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pi er settlement case four 2, 6, 2 ... ..135 6.30 Moment diagram for County Line Bridge with different pier settlement cases at moment point 5 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case th ree 2, 4, 2 (d) Pier settlement case four 2, 6, 2 137
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xiv 6.31 Moment diagram for County Line Bridge with different pier settlement cases at moment point 6 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 .139 6.32 Moment diagram for County Line Bridge with different pier settlement cases at moment point 7 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 .141 6 .33 Shear diagram for County Line Bridge with different pier settlement cases at shear point 1 (a) Pier settlement case one 2, 0, 0 (b) Pier settle ment case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 .. .143 6.34 Shear diagram for County Line Bridge with different pier settlement cases at shear point 2 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 ..145 6.35 Shear diagram for County Line Bridge with different pier settlement cases at shear point 3 (a) Pier se ttlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 ... 147 6.36 Shear diagram for County Line Bridge with different pier settlement cases at she ar point 4 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 .. .149 6.37 Shear diagram for County Line Bridge with different pier settlement cases at shear point 5 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 ..151
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xv 6.38 Moment diagram for 6 th Ave Bridge with different pier settlement cases at moment point 1 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement ca se four 2, 6, 2 ..153 6.39 Moment diagram for 6 th Ave Bridge with different pier settlement cases at moment point 2 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d ) Pier settlement case four 2, 6, 2 ....... 155 6.40 Moment diagram for 6 th Ave Bridge with different pier settlement cases at moment point 3 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement c ase three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 ..157 6.41 Moment diagram for 6 th Ave Bridge with different pier settlement cases at moment point 4 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 ( c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 ..159 6.42 Moment diagram for 6 th Ave Bridge with different pier settlement cases at moment point 5 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement c ase two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 ..161 6.43 Moment diagram for 6 th Ave Bridge with different pier settlement cases at moment point 6 (a) Pier settlement case one 2, 0, 0 ( b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 .. .163 6.44 Moment diagram for 6 th Ave Bridge with different pier settlement cases at moment point 7 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 ..165
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xvi 6.45 Sh ear diagram for 6 th Ave Bridge with different pier settlement cases at shear point 1 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 .. .167 6.46 Shear diagram for 6 th Ave Bridge with different pier settlement cases at shear point 2 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 ...... .169 6.47 Shear diagram for 6 th Ave Bridge with different pier settlement cases at shear point 3 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pi er settlement case four 2, 6, 2 .. .171 6.48 Shear diagram for 6 th Ave Bridge with different pier settlement cases at shear point 4 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case thr ee 2, 4, 2 (d) Pier settlement case four 2, 6, 2 ... 173 6.49 Shear diagram for 6 th Ave Bridge with different pier settlement cases at shear point 5 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 ... 175 6.50 Moment diagram for Indiana Bridge with different pier settlement cases at moment point 1 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case t wo 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 ..177 6.51 Moment diagram for Indiana Bridge with different pier settlement cases at moment point 2 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 ...... .179
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xvii 6.52 Moment diagram for Indiana Bridge with different pier settlement cases at moment point 3 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 ..181 6.53 Moment diagram for Indiana Bridge with different pier settlement cases at moment point 4 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 ..183 6.54 Moment diagram for Indiana Bridge with different pier settlement cases at moment point 5 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 .. .185 6.55 Moment diagram for Indiana Bridge with different pier settlement cases at moment point 6 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case t hree 2, 4, 2 (d) Pier settlement case four 2, 6, 2 ..187 6.56 Moment diagram for Indiana Bridge with different pier settlement cases at moment point 7 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pi er settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 .. .189 6.57 Shear diagram for Indiana Bridge with different pier settlement cases at shear point 1 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 ..191 6.58 Shear diagram for Indiana Bridge with different pier settlement cases at shear point 2 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 .. .193
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xviii 6.59 Shear diagram for Indiana Bridge with different pier settlement cases at shear point 3 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case four 2, 6, 2 ... 195 6.60 Shear diagram for Indiana Bridge with different pier settlement cases at shear point 4 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pier settlement case f our 2, 6, 2 ... 197 6.61 Shear diagram for Indiana Bridge with different pier settlement cases at shear point 5 (a) Pier settlement case one 2, 0, 0 (b) Pier settlement case two 2, 2, 0 (c) Pier settlement case three 2, 4, 2 (d) Pi er settlement case four 2, 6, 2 .. .199 6.62 Load distribution factor under 2 settlement, 4 settlement and 6 settlement for (a) Broadway Bridge (b) Santa Fe Bridge .200 6.63 Load distribution factor under case one (2 0, 0) case two (2, 2, 0), case three (2, 4, 2) case four (2, 6, 2) for (a) County Line Bridge (b) 6th Ave Bridge (c) Indiana Bridge ...201
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xix LIST OF TABLES Table 2.1 Load combinations and load factors. ........................................... ........................ .... ............... 12 3.1 Model matrix for superstructure design............................................. ............. ... ..................... 20 3.2. Details of the designed benchmark bridge sections I (steel plate girders) .. 21 3.3. Details of the designed benchmark bridge sections II (steel plate girders) ..... ...... ......... ...... 21 3.4. Details of the designed benchmark bridge sections I (steel box girders) .. ............. ....... 22 3.5. Details of the designed benchmark bridge sections II (steel box girders) .... .. ................... 22 3.6. Details of the designed benchmark bridge sections I (prestressed concrete I girders) .. ....... 22 3.7. Details of the designed benchma rk bridge sections II (prestressed concrete I girders) .. 23 3.8. Cross sectional area of prestressing steel strands for prestressed concrete I girders .. .. .......... 23 3.9. Details of the designed benchmark bridge sections I (prestressed concrete box girders) . ... 23 3.10. Details of the designed benchmark bridge sections II (prestressed concrete box girders) ... 24 3.11. Cross sectional area of prestressing steel strands for prestressed concrete box girders ....... 24 3.12. Details of the d esigned benchmark bridge sections I (reinforced concrete girders) 25 3.13. Table 3.13 Details of the designed benchmark bridge sections II (reinforced concrete girders) .. .25 3.14. Cross sectional area of steel bars for reinforced concrete girders .. ..26 5.1. Live load distribution calibrated by deterministic standard live load (interior moment).. .... 52 5.2. Live load distribution calibrated by deterministic standard live load (exterior moment).. ..... 54 5.3. Live load distribution calibrated by deterministic standard l ive load ( interior sh ear )... ......... 55 5.4. Live load distribution calibrated by deterministic standard live load ( exterior shear )..... ...... 56 5.5. Load distribution factors calculated by level rule.................................... ............................... 57 5.6. Sectional Properties of five site Bridges 57
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xx 5.7. Field testing LDF mean value and standard deviation for each girder ... 57 6.1. Moment for Santa Fe Bridge with different pier settlement ... 92 6.2. Shear force for Santa Fe Bridge with different pier settlement .. 92 6.3. Moment for Broadway Bridge with different pier settlement 92 6.4. Shear force for Broadway Bridge with different pier settlement .. ..93 6.5. Moment for 6 th Ave Bridge with case one (2,0,0) pier settlement .. ..93 6.6. Moment for 6 th Ave Bridge with case two (2,2,0) pier settlement ... .94 6.7. Moment for 6 th Ave Bridge with case three (2,4,2) pier settlement ..94 6.8. Moment for 6 th Ave Bridge with case four (2,6,2) pier settlement .. .95 6.9. Shear force for 6 th Ave Bridge with case one (2,0,0) pier settlement ..95 6.10. Shear fo rce for 6 th Ave Bridge with case two (2,2,0) pier settlement .96 6.11. Shear force for 6 th Ave Bridge with case three (2,4,2) pier settlement .. .96 6.12. Shear force for 6 th Ave Bridge with case four (2,6,2) pier settlement .97 6. 13. Moment for County Line Bridge with different cases pier settlement ... ..98 6.14. Shear force for County Line Bridge with different cases pier settlement .99 6.15. Moment for Indiana Bridge with different cases pier settlement .. .100 6.16. Shear force for Indiana Bridge with different cases pier settlement .. .101
