The Pennsylvania State University The Graduate School College of Engineering TRANSVERSE DISTRIBUTION FACTORS FOR HORIZONTALLY CURVED BRIDGES UNDER THE EFFECT OF PERMIT VEHICLES A Thesis in Civil Engineering by Bowen Yang 2018 Bowen Yang Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science May 2018
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The Pennsylvania State University
The Graduate School
College of Engineering
TRANSVERSE DISTRIBUTION FACTORS FOR HORIZONTALLY CURVED
BRIDGES UNDER THE EFFECT OF PERMIT VEHICLES
A Thesis in
Civil Engineering
by
Bowen Yang
2018 Bowen Yang
Submitted in Partial Fulfillment
of the Requirements
for the Degree of
Master of Science
May 2018
ii
The thesis of Bowen Yang was reviewed and approved* by the following:
Jeffrey A. Laman
Professor of Civil Engineering
Thesis Advisor
Ali M. Memari
Professor of Architectural Engineering and Civil Engineering
Hankin Chair of Residential Building Construction
Konstantinos Papakonstantinou
Assistant Professor of Civil Engineering
Patrick J. Fox
Professor of Civil Engineering
Head of the Department of Civil Engineering
*Signatures are on file in the Graduate School
iii
ABSTRACT
Permit vehicles with non-standard gage are increasingly used to carry heavy and
oversized cargos. Currently, approximate methods and evaluation of the transverse, live
load, girder distribution factor (GDF) for horizontally curved, steel, I-girder bridges
subjected to permit vehicles are lacking. Therefore, the effect of permit vehicles on GDFs
for curved bridges needs to be determined to allow rapid and efficient evaluation for issue
of permits. Four permit vehicles obtained from a Pennsylvania Department of
Transportation (PennDOT) database and twenty-seven curved bridges from Kim (2007)
are analyzed with CSiBridge® to conduct the parametric study. The present study
evaluates the influence of key parameters (radius, span length, girder spacing, and gage)
on moment GDFs, determines if GDFs for permit vehicles can be accurately predicted by
modifying AASHTO approximate moment GDF equations, and establishes an
approximate GDFs for the outermost girder. Two approximate moment GDF models
from Kim (2007) are utilized: (1) The single GDF model (SGM); and (2) the combined
GDF model (CGM) to calculate GDF for curved bridges subjected to permit vehicles. A
linear regression analysis is conducted to determine the relationship between AASHTO
approximate GDFs and GDFs for curved bridges subjected to permit vehicles to develop
a proposed, approximate GDF equation for curved girder bridges. Based on the numerical
results from FEM, SGM and CGM, the present study demonstrates that GDFs for curved
bridges cannot be accurately predicted by AASHTO approximate GDFs. The present
study develops a new approximate GDF equation to predict moment distribution in
curved bridges with respect to radius, span length, and vehicle gage. The numerical
iv
analysis results demonstrate that span length and radius have larger effects on GDFs than
girder spacing and vehicle gage. A goodness-of-fit method combined with the linear
regression analysis propose two developed approximate GDF equations (SGM and CGM
equations). Both two developed approximate GDF equations are demonstrated to
accurately predict GDFs for curved bridges compared to FEM results and provide slightly
larger results compared to FEM results. GDFs for HL-93 are also calculated and be
demonstrated to have larger results than GDFs for the evaluated permit vehicles.
v
TABLE OF CONTENTS
Acknowledgements .................................................................................................................. vii
1.1 Problem Statement ..................................................................................................... 3 1.2 Scope of the Research ................................................................................................ 3 1.3 Objectives of the Research ......................................................................................... 5 1.4 Tasks .......................................................................................................................... 5
Chapter 2 LITERATURE REVIEW ........................................................................................ 7
2.1 Introduction ................................................................................................................ 7 2.2 Live Load Distribution Factor Studies for Curved Bridges ....................................... 7 2.3 AASHTO Methods for Curved Bridges ..................................................................... 10 2.4 AASHTO Methods for Straight Bridges .................................................................... 12 2.5 Distribution Factor Studies for Bridges Subjected to Permit Vehicles ...................... 13 2.6 The Finite Element Modeling Method for Curved Bridges ....................................... 16 2.7 Summary .................................................................................................................... 20
Chapter 3 STUDY DESIGN .................................................................................................... 21
3.1 Introduction ................................................................................................................ 21 3.2 Procedure to Obtain GDFs for Curved Bridges ......................................................... 22 3.3 Determination of Parameters ...................................................................................... 23 3.4 Curved Bridge Details ................................................................................................ 24
3.4.1 Curved Bridge Details for the Parametric Study ............................................. 24 3.4.2 Curved Bridges for the Validation of Approximate GDF Equations .............. 27
3.5 Permit Vehicle Information........................................................................................ 28 3.6 Bending and Warping Stresses in Curved I-girder ..................................................... 33 3.7 Load Cases for Curved Bridges Subjected to Permit Vehicles .................................. 34 3.8 AASHTO Approximate GDFs ................................................................................... 34 3.9 Formulation of the GDF Equation ............................................................................. 35
4.1 Introduction ................................................................................................................ 40 4.2 3D Numerical Bridge Model ...................................................................................... 40
4.2.1 Element Types ................................................................................................. 40 4.2.2 Boundary Conditions ....................................................................................... 42 4.2.3 Description of the Bridge Model ..................................................................... 42
4.3 Permit Vehicle Assignment in Numerical Models ..................................................... 44 4.4 2D Straight Bridge Model .......................................................................................... 48 4.5 Summary .................................................................................................................... 48
Chapter 5 DATA PROCESSING ............................................................................................ 50
vi
5.1 Introduction ................................................................................................................ 50 5.2 Single GDF Model ..................................................................................................... 50 5.3 Combined GDF Model ............................................................................................... 52 5.4 Torsional Moment Related to Bending Moment ........................................................ 54 5.5 Summary .................................................................................................................... 58
Chapter 6 ANALYSIS RESULTS AND DISCUSSION ......................................................... 59
6.1 Introduction ................................................................................................................ 59 6.2 Warping Effect on GDFs ........................................................................................... 59 6.3 Modification of AASHTO Approximate GDFs ......................................................... 66 6.4 Strength of Parameters on GDFs................................................................................ 71 6.5 Proposed Approximate GDF (SGM) ......................................................................... 78 6.6 Proposed Approximate GDF (CGM) ......................................................................... 79 6.7 Accuracy of GDF Equations ...................................................................................... 82 6.8 Validation of GDF Equations..................................................................................... 84 6.9 Comparison of Approximate GDFs for Permit Vehicles and HL-93 ......................... 85 6.10 Summary .................................................................................................................. 86
Chapter 7 SUMMARY AND CONCLUSIONS ...................................................................... 87
7.1 Summary .................................................................................................................... 87 7.2 Summary and Conclusions ......................................................................................... 88 7.3 Future Research .......................................................................................................... 90 REFERENCES................................................................................................................. 93 APPENDIX Parameter Effect on GDFs Plots and Residual Plots of SGM and CGM .... 97
vii
ACKNOWLEDGEMENTS
First I thank my thesis advisor, Dr. Jeffrey A. Laman. He is very knowledgeable
and kind. He always gives me the advice immediately when I asked questions about my
research. He also teaches me how to do research and write this thesis. I could not have
imagined having better advisor and mentor for my thesis.
I am grateful to my thesis advising committee that let me know what I should
cover in the thesis and give me some advice about the research.
I also thank my friends Zefeng Dong, Chu Wang, Longji Li, and Meet, for their
support. They really gave me a lot of help.
Finally, I thank my parents, Guang Yang and Yanqiong Peng, for supporting me
to pursue my master degree at Penn State.
1
Chapter 1
INTRODUCTION
Permit vehicles with non-standard configurations are increasingly used to carry
heavy and oversized cargos for economic, military and other special needs. These permit
vehicles must pass over highway bridges to move special loads. Highway bridges are
mainly designed by considering the effect of standard vehicles with a 6 feet gage,
however, the gage of permit vehicles is usually larger than 6 feet and the gross vehicle
weight (GVW) is much heavier than a standard design vehicle. To design or analyze a
curved or straight bridge under live loads, the maximum moment of each girder must be
determined. The live load, girder distribution factor (GDF) is a convenient tool to predict
the maximum moment per girder, which equals the maximum moment per girder divided
by the maximum moment for the entire bridge. Hence, it is very important for bridge
engineers to determine the moment GDF for horizontally curved, steel, I-girder bridges
subjected to permit vehicles.
Evaluating horizontally curved, steel, I-girder bridges subjected to permit vehicles
is more complicated than evaluating straight bridges. Warping normal stresses caused by
bridge girder curvature influence the total girder moments for curved bridges. The most
widely used method to evaluate girder moments for a curved bridge subjected to permit
vehicles is a 3D finite element analysis. However, it is very time-consuming and costly to
use 3D models to get maximum moments for the horizontally curved, steel, I-girder
bridges subjected to permit vehicles.
2
The AASHTO approximate GDFs for straight bridges subjected to standard
vehicles have simplified the process of evaluating girder moments in the straight bridges.
This research is motivated to pursue an approximate method to predict moment GDF for
curved bridges subjected to permit vehicles.
The present study is a parametric study considering key parameters including
radius, span length, girder, and vehicle gage. Twenty-seven curved bridge designs from
Kim (2007), and four different permit vehicles with wide gage and high GVW from a
PennDOT permit vehicle database are used to develop an approximate method to predict
GDF for curved bridges under the effect of permit vehicles.
Regression analysis is used to determine the relationship between GDF for
curved bridges subjected to permit vehicles and AASHTO approximate GDFs for straight
bridges. The regression analysis results show that the AASHTO approximate GDFs
cannot reasonably be modified to accurately predict the GDF for curved bridges
subjected to permit vehicles in this parametric study. Therefore, the present study uses
regression analysis to develop a new approximate GDF equation to predict moments for
curved bridges subjected to permit vehicles.
The developed new approximate GDF equation can be utilized by agencies to
determine whether a specific permit vehicle can pass over a curved bridge without
running 3D finite element analysis, which will considerably increase the evaluation
efficiency.
3
1.1 Problem Statement
There are several approximate methods to predict GDFs for straight bridges
subjected to standard vehicles. However, the evaluation of GDFs for horizontally curved,
steel, I-girder bridges subjected to permit vehicles is lacking. The analysis of curved
bridges is more complex because of the warping effect. A permit vehicle has many more
axles and a wider gage that may influence GDF for horizontally curved, steel, I-girder
bridges. Hence, the effects of permit vehicle gage on GDF for horizontally curved, steel,
I-girder bridges are evaluated in the present study.
1.2 Scope of the Research
This study is limited to the evaluation of girder moment GDF for horizontally
curved, steel, I-girder bridges subjected to four different permit vehicle gages. The permit
vehicle gages considered are 16 ft, 18 ft, and 18.25 ft. Two permit vehicles have the same
gage but different axle spacing.
Curved bridges considered in the present study are simply supported. The
geometry of twenty-seven curved bridges are taken from Kim (2007). The details of
geometry of bridges is provided in Chapter 3.
The parameters considered in the present study are: radius, girder spacing, span
length, and gage. Based on these parameters, the total number of analysis cases in the
present study is 108. The details of analysis cases are provided in Chapter 3. The
variation range of study parameters is provided in Table 1-1.
4
Table 1-1.Study Parameter Values
Parameter Range (ft)
Radius 200, 350, 750
Girder Spacing 10, 11, 12
Span Length 72, 108, 144
Gage 16, 18, 18.25
For a 2D line analysis, three bridges with different span lengths (72 ft, 108 ft, and
144 ft) are modeled as simply supported beams. The parapet and superelevation of the
concrete deck that have been demonstrated to have negligible influence on GDFs are not
considered in 3D models.
Additional limitations in the present study are as follows:
1. All materials remain in the elastic range;
2. No dynamic effect is considered;
3. No centrifugal force is considered;
4. Cross-frame types are “X” type for all curved bridges;
5. Cross-frame spacing is the same for all curved bridges;
6. Concrete deck thickness is the same for all curved bridges; and
7. Girder section for curved brides is composite with concrete deck.
.
5
1.3 Objectives of the Research
The primary objective of the present study is to develop an approximate GDF
equation, based on an extensive parametric study, to predict GDF for the outermost girder
in horizontally curved, steel, I-girder bridges subjected to permit vehicles.
The developed approximate GDF equation can be used by agencies to determine
the best route for a permit vehicle passing over a curved bridge and contribute to the
establishment of a PennDOT permit vehicle database.
1.4 Tasks
Tasks to achieve the objectives of the present study are:
1. Determine key parameters for the parametric study;
2. Gather curved bridges and permit vehicles geometry information;
3. Develop 3D curved bridge models to run four different permit vehicles to
collect maximum total normal stress, bending stress, and warping stress in the
bottom flange of the outmost exterior curved girder for each load case;
4. Develop 2D straight bridge models to run four different permit vehicles to
compute the maximum moment for the entire straight bridge;
5. Compute maximum moment GDFs based on GDF models from Kim (2007);
6. Compute GDFs for straight bridges based on AASHTO approximate GDF
equations;
6
7. Utilize regression analysis to determine the relationship between GDF for
curved bridges subjected to permit vehicles and AASHTO approximate GDFs
for a straight bridge results;
8. Utilize regression analysis to develop a new approximate GDF equation for the
outmost exterior girder in curved bridges subjected to permit vehicles;
9. Evaluate the accuracy of developed GDF equations by comparing to 3D FEM
GDF results;
10. Evaluate the accuracy of developed GDF equations within study range of
parameters; and
11. Compare the developed approximate GDFs for the permit vehicle to GDFs for
HL-93 loading calculated from (Kim, 2007).
7
Chapter 2
LITERATURE REVIEW
2.1 Introduction
This chapter reviews published literature on horizontally curved, steel, I-girder
bridge GDF analysis and the approximate GDF formulas for straight bridges subject to
standard vehicles. Studies of GDFs for straight bridges that are subjected to permit
vehicles are also included. Modeling methods for curved bridges used in published
research is discussed as it relates to the present study.
2.2 Live Load Distribution Factor Studies for Curved Bridges
McElwain and Laman (2000) conducted field tests on three, in-service,
horizontally curved, steel, I-girder bridges subjected to a test truck and to normal truck
traffic. Three numerical grillage models were developed to determine whether the
responses of numerical models were accurate as compared with field test data. The results
presented that grillage models can accurately predict GDFs for curved girder bridges. The
study demonstrated that AASHTO LRFD Bridge Design Specifications (AASHTO 1998)
single lane approximate GDFs are unconservative in some cases, and AASHTO Guide
Specifications for Horizontally Curved Bridges (AASHTO 1993) approximate GDFs are
8
conservative for single truck cases. The study also concluded that the difference between
the S/11 method and V-load analysis method is small.
Depolo and Linzell (2008) examined the influence of live load on the lateral
bending moment distribution in horizontally curved, steel, I-girder bridges. The study
conducted a field test for a curved bridge and a numerical model analysis to determine
the accuracy of the AASHTO Guide Specifications for Horizontally Curved Bridges
(1993) lateral bending distribution factor (LBDF) equation:
2 4[(0.0008 0.13) (0.0022 -0.59 40) 10 ]5.5
Bi
SDF L L L R (2.1)
where BiDF is the LBDF in each curved girder, S is the girder spacing, L is the span
length, and R is the bridge radius. The AASHTO Guide Specifications for Horizontally
Curved Bridges (1993) LBDF results were compared to field responses and FEM results.
The comparison demonstrated that the AASHTO Guide Specifications for Horizontally
Curved Bridges (1993) LBDF is conservative and 20% to 30% deviates from results of
field test and FEM.
Kim and Laman (2007) examined eighty-one curved, steel, I-girder bridges to
study the effect of major parameters on GDFs. Kim and Laman established two different
GDF models; the single GDF model (SGM) and combined GDF model (CGM) to
calculate GDFs. Two methods; averaged coefficient and regression analysis for the
development of GDFs were evaluated. The study demonstrated that regression analysis is
more accurate than the average coefficient to develop an approximate GDF equation.
The study proposed an approximate GDF equation as Eq. (2.2):
1 2 3 4
( )( )( )( )(b b b b
g a R S L X ) (2.2)
9
where a is a scale factor, R is the exterior girder radius, S is the girder spacing, L is
radial span length of the outside girder, and X is the exterior girder cross-frame spacing.
1b , 2b , 3b , and 4b are exponents based on strength of relationships with GDF for the four
major parameters, respectively. The moment per girder in a multilane curved bridge is:
M g Mc s (2.3)
where Ms is the moment per girder in the single lane straight bridge, g is the curved
bridge GDF, and CM is the moment per girder in the multilane curved bridge.
In the SGM, only the maximum, total, normal stress is considered. The SGM
expression proposed by Kim and Laman is presented in Eq. (2.4):
/( )
( )
f I yb w
gb w Ms
(2.4)
where ( )g
b w is the maximum total GDF for the curved girder; Ms is the moment per
girder in the single lane straight bridge; ( )
fb w
is the maximum normal stress in the
curved girder; I is the strong axis bending moment of inertia of the cross section based
on the effective slab width; and y is the distance from the elastic neutral axis of the
section.
In the CGM, the effects of warping and bending are evaluated separately. The
maximum GDF for CGM is the summation of the maximum bending GDF (CGM-B) and
the maximum warping GDF (CGM-W), presented in Eq. (2.5):
( )
g g gb w b w
(2.5)
10
where gb
and gw
are presented in Eq. (2.6) and (2.7):
/( )
f I yb
gb Ms (2.6)
where gb
is the maximum vertical bending GDF; and ( )f
b is the maximum bending
normal stress; and CGM-W expression is presented in Eq. (2.7):
( /( ) ( )
M f f I yc w b w b
gwM Ms s
)
(2.7)
where gw is the warping GDF; and ( )M
c w is the equivalent maximum warping moment.
The study demonstrated that the developed approximate equation by regression analysis
provides the most accurate GDFs compared with field data. GDF results demonstrated
that the bending GDF increases as the span length increases and the warping GDF
increases as the radius decreases. The study found that the span length has the strongest
influence on GDFs and cross-frame spacing has a significant influence on warping GDFs.
2.3 AASHTO Methods for Curved Bridges
AASHTO Guide Specifications for Horizontally Curved Bridges (1993) adopted
the research by Heins and Siminou (1970), and specifies GDFs for vertical bending
moment as follows:
( 3) 0.75.5 4
S Lg N
R
(2.8)
11
where R is the radius ( R >100 ft); N is R /100; and S is the girder spacing
(7 ft ≤ S ≤12 ft). Eq. (2.8) is for the exterior girder and is conservative for other girders.
Eq. (2.8) has been removed from the AASHTO LRFD Bridge Design Specifications
since 2004. Considering the deck thickness, girder spacing, bridge type, number of lanes
loaded, AASHTO Standard Specifications for Highway Bridges (1996) specifies the
general GDF equation as follows:
S
gD
(2.9)
where D is a constant based on bridge type and number of lanes loaded, and S is the
girder spacing.
Equations in the AASHTO Guide Specifications for Horizontally Curved Bridges
(1993) have considered the effect of lateral bracing. The maximum GDF can be
calculated as follows:
Outside exterior girders (all bays with bottom lateral bracing)
3.0 0.06
( ) 0.932
L Lg
RS
(2.10)
Outside exterior girders (bottom lateral bracing in every other bay)
3.0 0.06
( ) 0.9532
L Lg
RS
(2.11)
where L is the exterior girder span length, S is the girder spacing, and R is the radius of
the exterior girder. Equations presented here have excluded any terms relating to cross-
frames, although cross-frames play an important role in resisting lateral bending stresses.
12
2.4 AASHTO Methods for Straight Bridges
The AASHTO LRFD Bridge Design Specifications (2012) presented approximate
GDF equations for steel I-girder bridges subjected to standard vehicles. The moment
approximate GDFs for interior girders are presented as follows:
One design lane loaded:
0.4 0.3 0.1
0.06 ( ) ( ) ( )14 12.0
KS S ggm
L Lts (2.12)
Two and more design lanes loaded:
0.6 0.2 0.1
0.075 ( ) ( ) ( )9.5 12.0
KS S ggm
L Lts (2.13)
where gm is the moment GDF, S is the girder spacing, L is the bridge span length, ts is
the slab thickness, and Kg is the longitudinal stiffness parameter. The expression of Kg is
presented as follows:
2( )gK n I Aeg (2.14)
where n is the modular ratio, I is the moment of inertia of the steel girder, A is the area
of steel girder, and ge is the distance between centers of the gravity steel girder and deck.
Based on these equations, GDFs for interior girders in straight bridges are calculated.
The AASHTO LRFD Bridge Design Specifications (2012) also introduced
several rules and equations to calculate GDFs for exterior girders. For one design lane
loaded, GDFs for exterior girders are obtained from the lever rule, which assumes hinges
are placed in the interior girders locations, and GDFs are reactions for adjacent girders
13
divided by the axle load. Usually, this method provides the upper bound of GDFs. For
two or more design lanes loaded:
int
g e gm (2.15)
0.779.1
dee (2.16)
where gm is the GDF for exterior girders, int
g is the GDF for interior girders, de is the
distance of the exterior girder to the curb.
2.5 Distribution Factor Studies for Bridges Subjected to Permit Vehicles
Goodrich and Puckett (2000) developed a simplified method to predict GDFs for
slab-on-girder bridges subjected to nonstandard wheel gage vehicles. The study
considered 115 bridges from the Distribution of Wheel Loads on Highway Bridges report
on NCHRP Project 12-26 (NCHRP 12-26). Four permit vehicles with two-wheel axle
configurations and twelve permit vehicles with four-wheel axle gage were conducted to
predict GDFs for straight bridges. Numerical modeling was used to calculate GDFs for
these permit vehicles and GDFs were compared to the simplified GDF method. The study
demonstrated that the simplified method provides conservative GDFs and is more
accurate for moment than for shear. The study shows that GDFs for permit and standard
vehicles are different, therefore, this is a need to evaluate how gage influences GDFs for
curved bridges.
Tabsh and Tabatabai (2001) utilized a finite element method to obtain
modification factors for AASHTO Guide Specifications for Distribution of Loads for
14
Highway Bridges (1994) approximate GDFs to predict GDFs for bridges subjected to
permit vehicles. Four different vehicles (HS20-44, PennDOT P-82, OHBD, and HTL-57)
were evaluated in the study. The study considered different gages (6 ft, 8 ft, 10 ft, and 12
ft). Nine bridges with different span lengths (48 ft, 96 ft, and 144 ft) and different girder
spacings (4 ft, 6 ft, and 8 ft) were modeled in the study. The approach proposed in the
study to evaluate the effect of gage one GDFs is presented as Eq. (2.17):
( ( ))G FGDF GD (2.17)
where ( )GGDF is the GDF for gage wider than 6 ft, the is the modification factor that
accounts for gage effect, and ( )GDF is the AASHTO approximate GDF. The finite
element method was used to determine GDFs for permit vehicles and to develop
modification factors. GDFs for an interior girder in the bridge subjected to a single HS20
truck with different gages are presented in Figure 2-1. The finite element results
demonstrate that GDFs decrease with the increase of gage. Figure 2-1 demonstrates that
the NCHRP 12-26 and AASHTO Standard Specifications for Highway Bridges 1996
(1996) predict conservative results. In the study, the HS20-44 truck has the most critical
GDF among the four considered vehicles.
15
Figure 2-1. Effect of gage on GDFs for Bridge with 8 ft Girder Spacing (Tabsh, 2001)
Bae and Oliva (2012) developed new GDF equations for evaluating multi-girder
bridges under the effect of permit vehicles. The study considered the span, girder spacing,
deck depth, girder type, skew, end diaphragm, and number of spans as key parameters
that influence GDFs. 118 multi-girder bridges and 16 load cases were analyzed. Figure 2-
2 demonstrates that developed GDF equations generally predict conservative results as
compared to FEM analysis. The study demonstrated that developed approximate GDF
equations accurately predict GDFs for multi-girder bridges subjected to permit vehicles.
16
Figure 2-2. Comparison of GDFs from Equations and FEM Results (Bae, 2012)
2.6 The Finite Element Modeling Method for Curved Bridges
Al-Hashimy (2005) successfully used SAP2000® to model curved bridges and
examined how study parameters influence GDFs for curved composite bridges. The study
utilized six different element types in SAP2000®: 2D plane element; 3D frame element;
3D shell element; 2D solid element; 3D solid element; and 3D link element. Figure 2-3
presents the model construction with flanges and webs modeled as a four-node shell
element to determine the warping normal stress. The deck slab was also modeled as a
four-node shell element. Truss elements were used for bracing and top and bottom
chords. Interior supports at the right end of the bridge were fixed in all translations. Other
supports at the right end of the bridge were restrained in the vertical and longitudinal
translation direction. For supports at the left end of the bridge, all translation was fixed in
17
the vertical direction and the interior support was also restrained in the transverse
direction.
Figure 2-3. 3D Bridge Model Cross Section (Al-Hashimy, 2005)
Nevling and Linzell (2006) conducted a field test for a three-span, continuous,
steel bridge with five girders to calculate GDFs. Three different levels of numerical
analysis (level 1, level 2, and level 3) were considered. Level 1 numerical analysis
consists of two manual methods: a line girder method from the AASHTO Guide
Specifications for Horizontally Curved Bridges (1993) and the V-load method. Level 2
analysis utilized three programs (SAP2000®, MDX, and DESCUS) to create 2D models.
Nevling and Linzell developed 3D models for level 3 analysis, created in SAP2000® and
BSDI, with flanges and cross-frames modeled as frame elements while deck and webs
were modeled as shell element. Figure 2-4 demonstrates both level 2 and level 3 are
correlated well with field responses. Level 3 analysis is demonstrated to have no
significant increase in accuracy as compared to the level 2 analysis. However, level 3
18
analysis is used in the present study to obtain GDFs for curved bridges subjected to
permit vehicles to do the moving load analysis.
Figure 2-4. Vertical Moment Transverse Distribution, Mid-span Span 2, Level 2 versus
Level 3: (a) Static 3 (Nevling and Linzell, 2006)
Kim (2007) evaluated three different levels of model types to determine a suitable
model type for curved bridge analysis. Figure 2-6 details three evaluated model types.
Figure 2-7 demonstrates that the GDF of the Type I model is conservative and inaccurate
as compared to field test results. Type II and Type III models accurately predict GDFs as
compared to the results of field tests. The difference between Type II and the field test
GDF is 10%, and for Type III is 4%. The study demonstrated that Type III models
increase the accuracy slightly over Type II models, while Type III costs more time and
effort. Therefore, type II models were determined to be used in the study.
19
Figure 2-6. Levels of Analysis (Kim, 2007)
20
Figure 2-7. GDF Comparison of Field versus Numerical Data (Kim, 2007)
2.7 Summary
This chapter reviews GDFs for straight and curved bridges. In the present study,
GDF models from Kim (2007) are used to calculate GDFs. The AASHTO LRFD Bridge
Design Specifications (2012) approximate GDF equations for exterior girders are also
utilized in the study. Based on this review of previous research, the present study
employs 3D FEM to calculate the GDFs for the horizontally curved, steel, I-girder
bridges subjected to permit vehicles. The Type II model from Kim (2007) modeling
girders as shell elements is utilized for the study.
21
Chapter 3
STUDY DESIGN
3.1 Introduction
A parametric study is used to evaluate the effect of permit vehicle gage on
moment GDF for horizontally curved, steel, I-girder bridges. The study parameters for
the present study are the radius, span length, girder spacing, and vehicle gage. Twenty-
seven curved bridge geometries are taken from (Kim, 2007). Four representative permit
vehicles with different gage are obtained from a PennDOT database. The total load case
number is 108. Curved bridges and permit vehicles are modeled in SAP2000® and
CSiBridge® software programs. The maximum bending normal stress and the maximum
total normal stress of the outermost exterior girder are collected for each load case.
Collection of warping normal stress is discussed in this chapter. GDF models from Kim
(2007) are used to calculate GDFs based on collected stresses. The details of GDF
models are presented in Section 2.2. A Linear Regression analysis, based on the least
square method, is used to determine the relationship between GDF for curved bridges and
AASHTO approximate GDFs. The development of approximate GDF equations for the
outmost exterior girder is also based on the regression analysis. Microsoft Excel is
utilized to conduct the linear regression analysis in the present study. R square is used as
a Goodness-of-fit method to evaluate regression results, and to determine the best fit for
the developed approximate GDF equation.
22
3.2 Procedure to Obtain GDFs for Curved Bridges
CSiBridge®, a commercially available and widely recognized software, is used to
calculate bottom flange stresses of the outermost curved girder for each load case. The
bottom flange stresses of each girder are used to calculate girder moments. The method to
obtain warping normal stress and bending stress in curved bridge models is presented in
Section 3.6.
CSiBridge® simulates a vehicle passing over a bridge and collects maximum
bottom flange stresses at any location. The maximum girder stress is determined by an
influence surface analysis. Therefore, CSiBridge® automatically varies transverse and
longitudinal locations of permit vehicles to determine the maximum stresses along the
girder.
The present study utilizes GDF models from (Kim, 2007) to calculate GDFs. The
details of GDF models are discussed in Section 2.2 and Section 5.5. Numerical models
are utilized to establish 2D bridge models to obtain the maximum moment for one lane in
a straight bridge. Details of the studied bridges and permit vehicles are presented in
Section 3.4 and 3.5, respectively. The procedure used to calculate GDFs for curved
bridges subjected to permit vehicles is presented in Figure 3-1.
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Figure 3-1. Flowchart to Obtain GDFs for Curved Bridges
3.3 Determination of Parameters
The span length, girder spacing, radius, and cross-frame spacing have been
demonstrated to be key parameters that influence on GDFs for curved bridges. The
present study maintains cross-frame spacing as a constant, a commonly used spacing
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length, to limit the number of total cases. The gage of permit vehicles is also considered
as a key parameter in the present study, therefore, parameters for the present study are the
span length ( L ), girder spacing ( S ), radius ( R ), and vehicle gage (G ). The vehicle gage
is the widest gage in permit vehicles. Radius is defined as the radius measured at the
outermost exterior girder. Span length is the curve length of the outermost exterior girder
and cross-frame spacing is defined as the curve length between two cross-frame supports
in the outermost exterior girder.
3.4 Curved Bridge Details
3.4.1 Curved Bridge Details for the Parametric Study
Twenty-seven horizontally curved, steel, I-girder bridges are evaluated in the
present study. The geometry of the curved bridges is taken from (Kim, 2007). Design of
these curved bridges by Kim (2007) limits girder spacing for spans greater than 140 ft to
11 ft -14ft, and for span length less than 140 ft to 10 ft - 12 ft. Span length for single span
bridges ranges from 50 ft to 200 ft. Span length was selected to be 72 ft, 108 ft and 144 ft
in Kim (2007). To consider a range of practical radii, the three radii were determined to
be 200 ft, 350 ft and 750 ft, measured at the outermost exterior girder. The cross-frame
spacing is constant and taken as 12 ft in the present study. Table 3-1 presents dimensions
of curved bridges considered in the present study.
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Figure 3-2. Curved Bridges Cross Section in the Parametric Study (Kim, 2007)
Figure 3-2 presents the cross-section detail of the studied bridges. In the present
study, a 1’-6” wide concrete parapet is assumed but was not modeled as the parapet has
been demonstrated to have a negligible influence on GDFs. S is the girder spacing that
ranges from 10 ft to 12 ft for different load cases. The 3’-6” deck overhang, 9” concrete
deck thickness, cross-frame type, and loading area are identical for each load case in the
present study.
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Table 3-1. Curved Bridge Details and Girder Section (Kim, 2007)
Bridge
Type
Span
Length
(ft)
Radius
(ft)
Cross-
Frame
Spacing
(ft)
Girder
Spacing
(ft)
Flange
Thickness
(in)
Flange
Width
(in)
Web
Thickness
(in)
Web
Height
(in)
Curved
144 200 12
10
2.25 24
0.6875 66
11 0.75 69
12 0.75 70
108 200 12
10
1.5 18
0.625 62
11 0.6875 62
12 0.6875 64
72 200 12
10
1.25 15
0.4375 42
11 0.4375 43
12 0.5 43
144 350 12
10
2 24
0.625 62
11 0.6875 65
12 0.6875 66
108 350 12
10
1.5 18
0.5625 56
11 0.625 57
12 0.625 59
72 350 12
10
1.25 15
0.4375 40
11 0.4375 40
12 0.4375 41
144 750 12
10
1.75 21
0.6875 64
11 0.6875 67
12 0.6875 69
108 750 12
10
1.5 18
0.5625 51
11 0.625 53
12 0.625 55
72 750 12
10
1.25 15
0.4375 38
11 0.4375 38
12 0.4375 40
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3.4.2 Curved Bridges for the Validation of Approximate GDF Equations
The range of each parameter is presented in this section. It is necessary to
determine whether the developed approximate GDF equations can accurately predict
GDFs for curved bridges with a geometry in the study ranges. To validate the range of
developed approximate GDF equations, two new preliminary design bridges are
evaluated in the present study. Radii of the two designed bridges are 300 ft and 650 ft,
and span lengths of are 84 ft and 120 ft. Cross-frame spacing is 12 ft and the girder
spacing is 10 ft. Girder spacing is only used 10 ft in the validation process because the
study ranges of girder spacing in the present study is small. The preliminary design of
two curved bridges is based on AASHTO LRFD Bridge Design Specifications (2012)
section design limitations and is presented in Table 3-2. The girder plate yield stress is
taken as 50ksi and applied stress is limited to 80% of yield. Girder section details of the
two designed test bridges are presented in Table 3-3: