transverse distribution factors for horizontally …
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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
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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
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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
Chapter 1 INTRODUCTION ................................................................................................... 1
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
Chapter 4 NUMERICAL MODELING ................................................................................... 40
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
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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)
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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.
23
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
24
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.
25
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.
26
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
27
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:
Table 3-2. Girder Section Design Limitations (AASHTO, 2012)
Section Design Limitations
5025as
yt
L
D F
150w
D
t
12.02
f
f
b
t
6f
Db
1.1f wt t
0.1 10yc
yt
I
I
28
where asL is the curved girder length; D is the depth of steel girder;
ytF is the specified
minimum yield strength of the compression flange (ksi); wt is the web thickness; ft is the
flange thickness; fb is the width of flange; ycI is the moment of inertia of the compression
flange of the steel section about the vertical axis in the plane of the web, and ytI is the
moment of inertia of the tension flange of the steel section about the vertical axis in the
plane of the web.
Table 3-3. Test Bridges Girder Section Details for Validation
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
84 300 12 10 1.25 16 0.4375 43
120 650 12 10 1.5 19 0.5625 47
3.5 Permit Vehicle Information
Four representative and permit vehicles with non-standard gage (16 ft, 18 ft, and
18.25 ft) are obtained from a PennDOT permit vehicle database. The database of the
permit vehicles contains permit truck configurations including weight, length, and width.
Figures 3-3 to 3-6 present the four study permit vehicle dimensions and weights.
29
Figure 3-3. Permit 1 Vehicle Elevation and Plan View (PennDOT Database)
30
Figure 3-4. Permit 2 Vehicle Elevation and Plan View (PennDOT Database)
31
Figure 3-5. Permit 3 Vehicle Elevation and Plan View (PennDOT Database)
32
Figure 3-6. Permit 4 Vehicle Elevation and Plan View (PennDOT Database)
33
3.6 Bending and Warping Stresses in Curved I-girder
The total curved girder normal stress at the bottom flange is the summation of
the warping normal stress and bending normal stress. The maximum total normal stress
exists at the tip of the bottom flange as depicted in Figure 3-7. The bending normal stress
is the average of stresses at bottom flange tips, and the warping normal stress is the
difference between stresses at bottom flange tips. A coupon of the bottom flange is
presented in Figure 3-8. As observed from Figure 3-8, the direction of the shear stresses
and normal stresses are presented and the total normal stress, x , presented in Figure 3-8
(bending normal stress and warping normal stress) is utilized to calculate GDFs.
Figure 3-7. Normal Stress Distribution in Curved I-Girder Flanges: (a) Major Axis
Bending Stress; (b) Warping Stress; (c) Combined Bonding and Warping Stress
(Davidson, 1996)
34
Figure 3-8. Stresses in a Coupon of Bottom Flanges
3.7 Load Cases for Curved Bridges Subjected to Permit Vehicles
Four permit vehicles and twenty-seven curved bridges are evaluated in the present
study and the total number of load cases is 108. Table 3-4 presents the information of
load cases for curved bridges in the present study.
3.8 AASHTO Approximate GDFs
AASHTO approximate GDFs are calculated for load cases in the present study.
Eq. (2.13) and Eq. (2.14) are used to calculate GDFs for interior girders. To obtain GDFs
for exterior girders, Eq. (2.15) and Eq. (2.16) are utilized in the present study.
35
3.9 Formulation of the GDF Equation
A regression analysis is used to develop the GDF equation. As discussed in
Section 2.2, the GDF equation for the curved bridge from (Kim, 2007) is an exponential
function with respect to radius, cross-frame spacing, span length, girder spacing, and a
constant. Therefore, the study anticipates that the resulting GDF equation for the present
study will be exponential. The key parameters considered in developing the GDF
equation are the radius, span length, girder spacing, and gage. The anticipated GDF
equation basic form is Eq. (3.1):
( ) ( ) ( ) ( )b c d e
g a R S L G (3.1)
where a is a constant and b , c , d and e are coefficients for radius ( R ), girder spacing (
S ), span length ( L ), and gage (G ) respectively. Eq. (3.1) is transformed to a logarithmic
form to conduct a linear regression analysis. The logarithmic form of Eq. (3.1) is:
ln ln ln( ) ln( ) ln( ) ln( )g a b R c S d L e G (3.2)
Principle of least squares (PLS), which is used in linear regression analysis, was
used to obtain coefficients. The PLS is presented as follows:
2
1
min ( )n
ii
i
PLS y y
(3.3)
where iy
is the thi predicted value, and iy is the thi dependent value. The P-value is
utilized to evaluate the significance of each coefficient and to determine whether
expected independent variables are related to the dependent variable. The independent
36
variable is strongly related to the dependent variable when the P-value is smaller than
0.05. Eq. (3.4) to Eq. (3.6) are used to calculate the P-value:
An upper-tailed test:
1P (3.4)
A lower-tailed test:
P (3.5)
A two-tailed test:
1 ]2[P (3.6)
where is the cumulative area of a distribution function.
A Goodness-of-fit measures are used to evaluate how well the regression model
describes observations. The coefficient of determination value ( 2R ) is utilized to evaluate
the relationship between the dependent variable and independent variables in regression
models. The 2R value is defined as Eq. (3.7):
2
2 1
2
1
( )
( )
n
i
i
n
i
i
y Y
R
y Y
(3.7)
where Y is the average of dependent variables. The 2R ranges from zero to 1.0. With 2R
equal to 1.0 indicates that a dependent variable is perfectly correlated to the independent
variable in the regression model and the strength of the relationship decreases as the 2R
value decreases.
37
Table 3-4. Load Cases
Case Radius
(ft)
Cross-
Frame
Spacing
(ft)
Span
Length
(ft)
Vehicles
Girder
Spacing
(ft)
1
200 12 72
Permit 1
10
2 11
3 12
4
Permit 2
10
5 11
6 12
7
Permit 3
10
8 11
9 12
10
Permit 4
10
11 11
12 12
13
200 12 108
Permit 1
10
14 11
15 12
16
Permit 2
10
17 11
18 12
19
Permit 3
10
20 11
21 12
22
Permit 4
10
23 11
24 12
25
200 12 144
Permit 1
10
26 11
27 12
28
Permit 2
10
29 11
30 12
31
Permit 3
10
32 11
33 12
34
Permit 4
10
35 11
36 12
38
Table 3-4. Load Cases
Case Radius
(ft)
Cross-
Frame
Spacing
(ft)
Span
Length
(ft)
Vehicles
Girder
Spacing
(ft)
37
350 12 72
Permit 1
10
38 11
39 12
40
Permit 2
10
41 11
42 12
43
Permit 3
10
44 11
45 12
46
Permit 4
10
47 11
48 12
49
350 12 108
Permit 1
10
50 11
51 12
52
Permit 2
10
53 11
54 12
55
Permit 3
10
56 11
57 12
58
Permit 4
10
59 11
60 12
61
350 12 144
Permit 1
10
62 11
63 12
64
Permit 2
10
65 11
66 12
67
Permit 3
10
68 11
69 12
70
Permit 4
10
71 11
72 12
39
Table 3-4. Load Cases
Case Radius
(ft)
Cross-
Frame
Spacing
(ft)
Span
Length
(ft)
Vehicles
Girder
Spacing
(ft)
73
750 12 72
Permit 1
10
74 11
75 12
76
Permit 2
10
77 11
78 12
79
Permit 3
10
80 11
81 12
82
Permit 4
10
83 11
84 12
85
750 12 108
Permit 1
10
86 11
87 12
88
Permit 2
10
89 11
90 12
91
Permit 3
10
92 11
93 12
94
Permit 4
10
95 11
96 12
97
750 12 144
Permit 1
10
98 11
99 12
100
Permit 2
10
101 11
102 12
103
Permit 3
10
104 11
105 12
106
Permit 4
10
107 11
108 12
40
Chapter 4
NUMERICAL MODELING
4.1 Introduction
This chapter presents numerical models established in CSiBridge® and SAP2000®
and introduces the arrangement of permit vehicles in the models. A Type II numerical
bridge model from (Kim, 2007) is utilized to analyze the selected bridges to extract the
warping normal stress and bending normal stress easily.
This chapter introduces the details of element types and boundary conditions used
for numerical models. CSiBridge® requires that the vehicle load be assigned with a
consistent gage for all axles. Discussed here is the approximate method to assign permit
vehicle configuration and its accuracy.
4.2 3D Numerical Bridge Model
4.2.1 Element Types
Several elements are utilized in building the Type II numerical models. Figure 4-1
presents a typical bridge cross-section. Shell elements are used to model the girder
flange, web, and concrete deck. Each shell element used is a four-node element with six
degrees of freedom at each node. Figure 4-1 presents cross-frames are modeled as truss
41
elements, which can only produce tension and compression stresses. Figure 4-1 also
identifies the location of rigid links used to connect the concrete deck and girders top
flange. Figure 4-2 presents a detailed of rigid link. The rigid link used to transfer moment
and shear from deck to girder are fixed all translations and rotations.
Figure 4-1. Bridge Numerical Model Details and Element Type
Figure 4-2. Details of Rigid Link Element
42
4.2.2 Boundary Conditions
Girder boundary conditions are critical and significantly affect bridge behavior
and response to load. Figure 4-3 presents x-axis (tangential) and y-axis (radial) directions.
Figure 4-3 presents G1, G2, G3 and G4 are restrained in the vertical (z-axis) direction.
Tangential and radial translations) are released for G1, G2 and G4. The right end of G3 is
restrained in the radial against translation only and the left end is fixed against all three
translations.
G1
G2
G3
G4
G1
G2
G3
G4
Figure 4-3. Boundary Conditions of the Curved Bridge (Kim, 2007)
4.2.3 Description of the Bridge Model
Figure 4-4 presents the coordinate system of the bridge numerical model and the
details of the model mesh, taken as 1 ft. The aspect ratio in numerical models is
recommended less than 4 from (Logan, 2002). Though the aspect ratio for web and slab
used in numerical models is about 6, the analytical results are accurate compared to the
43
results of lower aspect ratios. Therefore, the model mesh is taken as 1 ft and it is better to
locate where the maximum moment exists in the bridge with small mesh. Plan view of a
144 ft numerical bridge model is presented in Figure 4-5.
X Z
Y
1 ft
Figure 4-4. View of the 144 ft Bridge Numerical Model
44
Figure 4-5. Plan View of the 144 ft Bridge Numerical Model
4.3 Permit Vehicle Assignment in Numerical Models
As previous introduced in Section 3.5, the gage of four permit vehicles
considered is wider than the width of design lane. It is not possible for two permit
vehicles passing over the curved bridge side by side. Figure 4-5 presents only one loading
lane was assigned in the numerical model. Review of Figure 3-2 presents the width of the
loading area. As discussed in Section 3.2, CSiBridge® automatically moves permit
vehicles longitudinally and transversely to calculate the maximum bottom flange stresses.
The exact positions of permit vehicles to produce the maximum bottom flange stresses
are not considered.
This study approximates the actual permit vehicle four wheels per lane in
numerical model to two wheels per lane and approximates the eight wheels per lane to
four wheels per lane in numerical model to simplify the analysis. Figure 4-6 presents the
45
approximation for a permit 2 vehicle axle. This simplification has negligible influence on
results because the distance between two wheels is only 2” as presented in Figure 4-6.
Figure 4-6. Approximation of Permit 2 Vehicle T9 axle
In numerical models, permit vehicles are assigned with a consistent gage.
However, permit vehicles considered are not configured with consistent gage. Permit
vehicles are modeled with consistent gage to approximate the assignment of wheel loads.
Therefore, the difference between the actual vehicle model and the approximate vehicle
model needs to be determined. Figure 4-7 presents the method to equivalently assign a
point load on a shell element. Permit vehicle loads are considered as point loads, and
point loads cannot be directly assigned on a shell element. Therefore, permit wheel loads
are assigned to adjacent four nodes in a shell element proportionally as presented in
Figure 4-7.
46
Figure 4-7. Point Load Distribution to Adjacent Nodes (Kim, 2007)
A static load case is conducted to evaluate the difference between actual and
approximate vehicle assignments. One widest axle lane of permit 1 vehicle is loaded at
the middle of the 144 ft numerical model. Total normal stresses of the actual vehicle
configuration and the approximate vehicle configuration in the model are presented in
Table 4-1. Table 4-1 presents relative errors of total normal stresses between the actual
configuration and approximate configuration. The maximum relative error in Table 4-1 is
0.25% indicates that the approximation has negliable influence on GDF results.
Therefore, permit vehicles were modeled with consistent gage in CSiBridge® to obtain
reasonable GDFs.
47
Table 4-1. Total Normal Stress for the 144ft Numerical Model
Location
from
Left
Supports
(ft)
Actual
Configuration
Total Normal
Stress (ksi)
Approximate
Configuration
Total Normal
Stress (ksi)
Relative
Error
(%)
0 0.34 0.34 0.058
6 2.76 2.76 0.065
12 5.01 5.01 0.048
18 7.89 7.89 0.063
24 9.37 9.37 0.047
30 12.6 12.6 0.056
36 13.5 13.5 0.035
42 17.0 17.0 0.037
48 17.1 17.1 0.001
54 20.6 20.6 0.017
60 20.2 20.2 0.14
66 23.2 23.2 0.16
69 22.5 22.4 0.25
72 22.4 22.4 0.028
78 23.4 23.3 0.25
81 22.1 22.0 0.20
84 20.8 20.8 0.11
90 20.8 20.7 0.037
93 19.2 19.2 0.005
96 17.6 17.6 0.049
102 16.9 17.0 0.045
105 15.4 15.4 0.044
108 13.6 13.6 0.038
114 12.4 12.4 0.058
117 10.9 10.9 0.063
120 9.15 9.15 0.057
126 7.41 7.42 0.073
129 6.08 6.09 0.076
132 4.45 4.45 0.065
138 2.21 2.21 0.082
144 0.23 0.23 0.089
48
4.4 2D Straight Bridge Model
The maximum moment for the entire one lane straight bridge is obtained from a
2D numerical analysis. The frame element is used to model straight bridges. Figure 4-8
presents one of the bridge supports is fixed against horizontal and vertical translations
and another one is only restrained in the vertical direction. As discussed in Section 2.2,
the maximum moment for the entire one lane straight bridge is used to calculate GDFs
for SGM and CGM. The mesh of 2D models is taken as 1 ft to obtain. Figure 4-8 presents
the moment envelope of permit 1 vehicle loaded on the 144 ft 2D model and the
maximum moment of the case is presented directly in SAP2000®.
Figure 4-8. The Moment Envelope of the 144 ft Straight Bridge Model
4.5 Summary
Numerical models for load cases were established to obtain bottom flange stresses.
Shell elements, truss elements, and rigid links were utilized to establish 3D numerical
models. While only the frame element was utilized to model 2D numerical models.
Boundary conditions for numerical models were based on (Kim, 2007). The aspect ratio
utilized in numerical models was larger than 4 but provided accurate and reasonable
49
results. The approximate assignment of vehicle configuration in numerical models was
demonstrated to be accurate. The numerical models established were supposed to provide
accurate results to obtain GDFs.
50
Chapter 5
DATA PROCESSING
5.1 Introduction
The method employed to calculate GDFs is presented here. Based on the
discussions in Section 2.2, SGM and CGM from (Kim, 2007) are utilized to calculate
GDFs. This chapter utilizes a load case to introduce the details of procedures to obtain
SGM and CGM GDFs.
The approximate method, considering the torsional moment as an equivalent
vertical bending moment in the CGM, is introduced in this chapter. A validation of the
approximate method reasonableness is presented here based on published research about
warping.
5.2 Single GDF Model
GDFs based on the SGM are obtained from Eq. (2.4). For SGM, the effects of
warping and bending are considered together rather than separately. Considering only the
total normal stress, the maximum total normal stress exists one tip of the girder bottom
flange. Figure 5-1 presents the total normal stresses along the girder for load case 37. At
1’-0” intervals, the total normal stress collected is determined to be the larger stress at
bottom flange tips.
51
Figure 5-1. The SGM Total Normal Stress Variation (Load Case 37:72 ft)
52
5.3 Combined GDF Model
For CGM, the effects of warping and bending are determined separately. Both the
warping normal stress and bending stress are determined in the CGM. GDFs based on
CGM are calculated from Eq. (2.5) to (2.7). As discussed in Section 3.6, stresses of tips
and center of bottom flanges are used to calculate the bending and warping normal
stresses. Figure 5-2 presents stresses at bottom flange tips and center along the girder for
load case 37 and demonstrates that the location of the maximum total normal stress is
consistent with the location of the maximum bending stress for load case 37. Numerical
analysis results indicate that the location of the maximum total normal stress is consistent
with the location of the maximum bending stress for most load cases. However, the
location of the maximum bending stress is not always same as the location of the
maximum warping normal stress. This is because of the existence of cross-frames in
curved bridges. The difference of the maximum warping normal stress along the girder
and the warping normal stress at the location of the maximum total normal stress is
around 1%. To be conservative, the maximum warping normal stress and the maximum
bending stress anywhere along the girder are superimposed to obtain the maximum total
normal stress in the CGM.
53
Figure 5-2. The CGM Normal Stress Variation (Load Case 37:72 ft)
54
5.4 Torsional Moment Related to Bending Moment
The torsional moment in the CGM is considered approximately as an equivalent
vertical bending moment. However, the torsional moment and bending moment are not
about the same axis. Therefore, it is necessary to demonstrate the strong relationship
between the torsional moment and the vertical bending moment. The validation is
presented as follows:
The bending equilibrium in the curved girder is presented in Figure 5-1
Figure 5-1. Bending Free Body Diagram (Heins and Firmage, 1979)
where GM is the vertical bending moment in the curved girder; h is the distance between
centroids of the two flanges; and F is the internal longitudinal force per flange. The
internal force can be expressed as Eq. (5.2).
GM h F (5.1)
GMF
h (5.2)
55
The warping equilibrium in the curved girder is presented in Figure 5-2
Figure 5-2. Warping Free Body Diagram (Fiechtl, 1987)
where T is torque in the curved girder; h is the distance between centroids of the two
flanges; and the torque causes a lateral flange force, /T h , in the direction of torque. The
lateral force varies along the curved girder. An approximate evaluation of the warping
effect considers torque per unit length for the curved girder. Due to the curvature,
distributed radial forces are developed to establish equilibrium for the curved girder. For
a very small angle, the lateral load can be considered as a uniformly distributed load, q .
Figure 5-3 presents a free body diagram of the girder bottom flange where vF is the
vertical direction component of the internal force; HF is the horizontal direction
component of the internal force; q is the virtual distributed radial force on the flange; and
is the small, curvature angle. Base on Figure 5-3, the equation of equilibrium is
established in Eq. (5.3), where summation of vertical direction forces must equal zero.
56
sinqR F (5.3)
Figure 5-3. Bottom Flange Free Body Diagram (Heins and Firmage, 1979)
For small angles, Eq. (5.4):
sin (5.4)
therefore, the internal longitudinal force and the lateral load, q , can be expressed as Eq.
(5.5) and Eq. (5.6).
qR F (5.5)
Fq
R (5.6)
Substituting Eq. (5.2) into Eq. (5.6), the lateral load, q , in proportion to bending moment,
can be expressed as Eq. (5.7):
GFq
M
R hR (5.7)
57
Lateral loads, q , are equal in magnitude but opposite in direction for the top and bottom
flanges. Therefore, /GM R is represents a torque per unit length. The lateral load in the
flange results in the lateral bending moment,fM , in the curved girder. The lateral bending
moment effect is presented in Figure 5-4. Cross-frames have been demonstrated to reduce
the effect of warping in the curved bridge. Figure 5-4 presents the diaphragm supports are
considered as rigid supports in the curved girder. The lateral load, q , equals /F R as
presented in Figure 5-4. Therefore, the lateral bending moment, fM , can be calculated as
a moment at cross-frames for a continuous beam.
Figure 5-4. Diagram for Flange Distributed Load Analogy (Davidson, 1996)
fM at the diaphragm support can be calculated as shown in Eq. (5.8):
2( )
12f
q XM (5.8)
58
where X is the cross-frame spacing; and q is the lateral load considered as a uniformly
distributed load. The denominator of Eq. (5.8) is 12 for the moment at rigid supports.
Substituting the Eq. (5.7) into (5.8), the lateral bending moment, fM , is expressed as Eq.
(5.9):
2( )
12
Gf
M XM
hR (5.9)
The torsional moment is then a function of the bending moment so that the torsional
moment can be considered as an equivalent bending moment in CGM. The warping
normal stress, w , at the flange tip is calculated by Eq. (5.10) and Eq. (5.11).
f
w
f
M
S (5.10)
21
6f f fS b t (5.11)
where fS is the section modulus of the bottom flange about the vertical axis; fb is the
bottom flange width; and ft is the bottom flange thickness.
5.5 Summary
The procedures of processing stress output to obtain GDFs for SGM and CGM
were discussed here. This chapter presents GDFs based on CGM and are conservative
compared to GDFs based on SGM. The approximate method to evaluate the torsional
moment as an equivalent bending moment in CGM was introduced and demonstrated.
59
Chapter 6
ANALYSIS RESULTS AND DISCUSSION
6.1 Introduction
Parametric study and associated regression analysis results are presented here.
The GDF for each load case and the AASHTO approximate GDF are evaluated. Warping
normal and bending stresses are determined to calculate GDF for each case. The ratio of
warping stress to bending stress used to evaluate the warping effect is also discussed. All
GDF results are calculated based on both the SGM and CGM. The accuracy of each
approximate GDF model is evaluated compared to FEM results. Regression analysis
results are presented to quantify the relationship strength between GDFs for curved
bridges subjected to permit vehicles and AASHTO approximate GDFs. Approximate
GDF models are developed based on regression analysis considering radius, span length,
girder spacing and gage as independent variables. The strength of each parameter is also
evaluated and discussed. Because field test data are not available, results of developed
approximate GDF equations are compared to FEM results to determine the accuracy.
6.2 Warping Effect on GDFs
The maximum bending and warping normal stresses are obtained from 3D
analysis. The 108 analysis cases were conducted to evaluate the warping normal stress
60
influence on total normal stress and the influence strength of each parameter on warping
stress. The maximum bending and total normal stresses for load cases are presented in
Figure 6-1. It can be observed from Figure 6-1 that the x-axis represents analytical cases
based on Table 3-4 and case ranges of three different radii that are R = 200 ft cases (case
1 to 36), R = 350 ft cases (case 37 to 72) and R = 750 ft cases (case 73 to 108). As
observed in Figure 6-1, the three dashed boxes in R = 200 ft cases (case 1 to 36) indicate
the case range of three span lengths which are not shown for clarity for R =350 ft and R =
750 ft. Observations of Figure 6-1, the three dashed boxes in R =350 ft and S =108 ft
cases (case 37 to 61) indicate the case range of three vehicle gages which are not shown
for clarity for R =200 ft and R = 750 ft. Cases 82 to 84 are presented in Figure 6-1 to
indicate the case range of three girder spacings which are not shown for clarity for the
rest load cases based on Table 3-4. The difference between the two lines (total normal
stress and warping normal stress) in Figure 6-1 is the warping normal stress. Observation
of Figure 6-1 indicates the maximum bending stress ranges from 11.2 ksi to 16.7 ksi and
the maximum warping normal stress ranges from 0.87 ksi to 4.38 ksi. Ratios of the
warping normal stress to the bending stress are presented in Table 6-1. Review of Table
6-1 indicates that the warping to bending ratio ranges from 7.3% to 34.2% over the full
range of 108 analysis cases.
Warping normal stress decreases as radius increases with the ranging from 2.11
ksi to 4.38 ksi for R =200 ft while ranges from 0.87 ksi to 1.5 ksi for R =750 ft. As
observed from left to right in Figure 6-1, the warping normal stress decreases as radius
increases from 200 ft to 750 ft. This is because the outermost girder eccentricity being
reduced so that the torsional moment decreases and, therefore, the warping effect
61
decreases. The high and low warping to bending ratios for different radius ranges are also
bolted in Table 6-1. Table 6-1 data also indicates warping to bending ratios for R =750 ft
are less than 10% while ranges from 18.6% to 34.2% for R =200 ft.
Within limited study range of gage, the effect of vehicle gage on warping is
small. Due to permit vehicles with different axles and weights, the warping to bending
ratio instead of the warping normal stress magnitude was utilized to evaluate gage effect
on warping. Table 6-1 indicates that the variation of warping to bending ratio is around
3% for three vehicle gages. To evaluate standard vehicle gage effect on warping to
bending ratio, one HS20 truck was modeled in nine numerical models with different radii
and span lengths. The difference between warping to bending ratios for evaluated permit
vehicles and for HS20 truck is around 8%, indicating the small effect on warping.
The effect of girder spacing on warping is slight that resulted from the range of
girder spacing considered is limited to 10 ft to 12 ft, which is small. Figure 6-1 indicates
that the maximum warping normal stresses are close for each of the three study girder
spacings and the variations of warping normal stress are around 7%.
Span length significantly influences warping normal stress. Besides the effect of
span on warping decreases as radius increases. Figure 6-1 presents that the variations of
the warping normal stress for R =200 ft are larger than variations for R =350 ft and R
=750 ft. As observed from Table 6-1 the variance of warping to bending ratio is 71% for
three study spans for R =200 ft while around 25% and 18% for R =350 ft and R =750 ft.
62
Figure 6-1. Bending and Total Normal Stresses for the Outermost Girder
63
Table 6-1. Warping to Bending Ratio for Analytical Cases
Case Radius
(ft)
Cross-
Frame
Spacing
(ft)
Span
Length
(ft)
Girder
Spacing
(ft)
Vehicles
Warping
to
Bending
Ratio
1
200 12 72
10
Permit 1
0.319
2 11 0.327
3 12 0.342
4 10
Permit 2
0.313
5 11 0.321
6 12 0.336
7 10
Permit 3
0.311
8 11 0.319
9 12 0.334
10 10
Permit 4
0.315
11 11 0.323
12 12 0.338
13
200 12 108
10
Permit 1
0.272
14 11 0.286
15 12 0.295
16 10
Permit 2
0.268
17 11 0.281
18 12 0.290
19 10
Permit 3
0.267
20 11 0.280
21 12 0.288
22 10
Permit 4
0.268
23 11 0.282
24 12 0.290
25
200 12 144
10
Permit 1
0.190
26 11 0.188
27 12 0.183
28 10
Permit 2
0.193
29 11 0.190
30 12 0.186
31 10
Permit 3
0.195
32 11 0.192
33 12 0.188
34 10
Permit 4
0.192
35 11 0.190
36 12 0.186
64
Table 6-1. Warping to Bending Ratio for Analytical Cases
Case Radius
(ft)
Cross-
Frame
Spacing
(ft)
Span
Length
(ft)
Girder
Spacing
(ft)
Vehicles
Warping
to
Bending
Ratio
37
350 12 72
10
Permit 1
0.168
38 11 0.171
39 12 0.176
40 10
Permit 2
0.167
41 11 0.171
42 12 0.175
43 10
Permit 3
0.164
44 11 0.168
45 12 0.172
46 10
Permit 4
0.165
47 11 0.168
48 12 0.172
49
350 12 108
10
Permit 1
0.136
50 11 0.142
51 12 0.148
52 10
Permit 2
0.140
53 11 0.140
54 12 0.145
55 10
Permit 3
0.140
56 11 0.140
57 12 0.144
58 10
Permit 4
0.138
59 11 0.139
60 12 0.144
61
350 12 144
10
Permit 1
0.136
62 11 0.134
63 12 0.129
64 10
Permit 2
0.140
65 11 0.137
66 12 0.132
67 10
Permit 3
0.142
68 11 0.139
69 12 0.134
70 10
Permit 4
0.139
71 11 0.136
72 12 0.132
65
Table 6-1. Warping to Bending Ratio for Analytical Cases
Case Radius
(ft)
Cross-
Frame
Spacing
(ft)
Span
Length
(ft)
Girder
Spacing
(ft)
Vehicles
Warping
to
Bending
Ratio
73
750 12 72
10
Permit 1
0.0747
74 11 0.0736
75 12 0.0734
76 10
Permit 2
0.0765
77 11 0.0750
78 12 0.0744
79 10
Permit 3
0.0770
80 11 0.0756
81 12 0.0751
82 10
Permit 4
0.0753
83 11 0.0743
84 12 0.0741
85
750 12 108
10
Permit 1
0.0810
86 11 0.0796
87 12 0.0782
88 10
Permit 2
0.0855
89 11 0.0832
90 12 0.0812
91 10
Permit 3
0.0851
92 11 0.0829
93 12 0.0810
94 10
Permit 4
0.0826
95 11 0.0810
96 12 0.0796
97
750 12 144
10
Permit 1
0.0896
98 11 0.0868
99 12 0.0840
100 10
Permit 2
0.0939
101 11 0.0901
102 12 0.0867
103 10
Permit 3
0.0947
104 11 0.0910
105 12 0.0875
106 10
Permit 4
0.0913
107 11 0.0883
108 12 0.0855
66
6.3 Modification of AASHTO Approximate GDFs
AASHTO approximate GDFs have been in use for 25 years. It would be very
advantageous if AASHTO approximate GDFs could be modified to predict GDFs for
curved bridges subjected to permit vehicles. A Linear regression analysis was used to
examine the relationship between GDFs of SGM and AASHTO approximate GDFs.
AASHTO approximate GDFs are a function of span length, girder spacing, deck
thickness, and girder stiffness.
Girder stiffness is expressed by a, Kg , discussed in Section 2.4. Kg has a weak
correlation to the AASHTO approximate GDF as evidenced by 0.1 exponent. Therefore,
this variable was calculated based on the geometry of curved bridges.
The present study has determined the SGM GDF to be the dependent variable,
AASHTO approximate GDFs, radius, and vehicle gage as the independent variables in
the linear regression analysis. In order to adapt AASHTO approximate GDF to curved
bridges, the relationship is expected to be an exponential form as follows in Eq. (6.1):
( ) ( ) ( )curved
b c dg a AASHTO GDFs G R (6.1)
where a is a constant; curvedg is the GDF for the outermost curved girder; b , c and d are
regression coefficients; and G is the vehicle gage. Eq. (6.1) is transformed to logarithmic
form to conduct a linear regression analysis. Table 6-2 presents the linear regression
analysis results of modifying AASHTO approximate GDFs. A regression analysis
resulting in a 95% confidence interval, will exhibit a P-value less than 0.05, indicating
the variable is strongly related to the dependent variable. Table 6-2 presents P–values.
67
For AASHTO approximate GDF the P-value is 0.335 that is greater than 0.05, therefore,
the relationship between GDFs for curved bridges subjected to permit vehicles and
AASHTO approximate GDFs is very small based on linear regression analysis.
Therefore, new curved girder approximate GDF equations are needed to predict GDF for
curved bridges subjected to permit vehicles.
Table 6-2 Regression Analysis Results to modify AASHTO Approximate GDFs
Parameters Coefficients Standard Error P-value
Intercept 2.849 0.353 0
Radius -0.228 0.014 0
Gage -0.686 0.121 0
AASHTO GDFs 0.124 0.128 0.335
R Square 0.747 Adjusted R Square 0.739
68
Table 6-3. GDF Results of Parametric Cases and Corresponding AASHTO GDF
Case Vehicles GDF
(SGM)
GDF
(CGM)
GDF
FEM
AASHTO
GDF
1
Permit 1
0.596 0.598 0.486 0.751
2 0.620 0.621 0.501 0.808
3 0.643 0.645 0.514 0.864
4
Permit 2
0.659 0.662 0.527 0.751
5 0.682 0.685 0.54 0.808
6 0.705 0.710 0.552 0.864
7
Permit 3
0.651 0.653 0.524 0.751
8 0.674 0.677 0.537 0.808
9 0.697 0.701 0.549 0.864
10
Permit 4
0.607 0.609 0.487 0.751
11 0.631 0.633 0.502 0.808
12 0.655 0.658 0.515 0.864
13
Permit 1
0.686 0.689 0.564 0.746
14 0.703 0.707 0.571 0.801
15 0.720 0.726 0.578 0.859
16
Permit 2
0.767 0.771 0.614 0.746
17 0.780 0.787 0.617 0.801
18 0.795 0.804 0.621 0.859
19
Permit 3
0.757 0.762 0.606 0.746
20 0.771 0.778 0.61 0.801
21 0.785 0.795 0.614 0.859
22
Permit 4
0.709 0.712 0.571 0.746
23 0.725 0.729 0.577 0.801
24 0.741 0.749 0.584 0.859
25
Permit 1
0.775 0.802 0.665 0.733
26 0.779 0.802 0.665 0.793
27 0.778 0.799 0.663 0.847
28
Permit 2
0.854 0.882 0.71 0.733
29 0.855 0.878 0.707 0.793
30 0.851 0.871 0.702 0.847
31
Permit 3
0.805 0.823 0.698 0.733
32 0.806 0.82 0.696 0.793
33 0.802 0.814 0.692 0.847
34
Permit 4
0.751 0.774 0.665 0.733
35 0.755 0.775 0.665 0.793
36 0.754 0.772 0.663 0.847
69
Table 6-3. GDF Results of Parametric Cases and Corresponding AASHTO GDF
Case Vehicles GDF
(SGM)
GDF
(CGM)
GDF
FEM
AASHTO
GDF
37
Permit 1
0.517 0.518 0.460 0.745
38 0.537 0.538 0.477 0.798
39 0.557 0.558 0.492 0.853
40
Permit 2
0.570 0.573 0.504 0.745
41 0.589 0.591 0.518 0.798
42 0.607 0.610 0.532 0.853
43
Permit 3
0.564 0.565 0.500 0.745
44 0.583 0.585 0.515 0.798
45 0.602 0.604 0.530 0.853
46
Permit 4
0.521 0.522 0.464 0.745
47 0.541 0.541 0.480 0.798
48 0.561 0.561 0.496 0.853
49
Permit 1
0.557 0.560 0.507 0.730
50 0.576 0.578 0.522 0.787
51 0.593 0.595 0.534 0.843
52
Permit 2
0.628 0.630 0.563 0.730
53 0.642 0.642 0.574 0.787
54 0.653 0.657 0.584 0.843
55
Permit 3
0.620 0.624 0.555 0.730
56 0.634 0.636 0.567 0.787
57 0.647 0.650 0.576 0.843
58
Permit 4
0.575 0.581 0.517 0.730
59 0.593 0.595 0.530 0.787
60 0.609 0.611 0.542 0.843
61
Permit 1
0.629 0.643 0.570 0.718
62 0.641 0.654 0.581 0.777
63 0.648 0.660 0.588 0.829
64
Permit 2
0.698 0.714 0.623 0.718
65 0.706 0.720 0.631 0.777
66 0.710 0.722 0.635 0.829
67
Permit 3
0.667 0.672 0.610 0.718
68 0.676 0.679 0.618 0.777
69 0.679 0.681 0.623 0.829
70
Permit 4
0.621 0.629 0.571 0.718
71 0.632 0.639 0.582 0.777
72 0.639 0.645 0.589 0.829
70
Table 6-3. GDF Results of Parametric Cases and Corresponding AASHTO GDF
Case Vehicles GDF
(SGM)
GDF
(CGM)
GDF
FEM
AASHTO
GDF
73
Permit 1
0.455 0.461 0.438 0.738
74 0.474 0.479 0.456 0.791
75 0.494 0.498 0.474 0.850
76
Permit 2
0.503 0.512 0.484 0.738
77 0.521 0.528 0.500 0.791
78 0.539 0.546 0.517 0.850
79
Permit 3
0.497 0.507 0.479 0.738
80 0.515 0.524 0.496 0.791
81 0.535 0.542 0.514 0.850
82
Permit 4
0.458 0.465 0.443 0.738
83 0.476 0.482 0.461 0.791
84 0.496 0.501 0.479 0.850
85
Permit 1
0.487 0.491 0.460 0.725
86 0.506 0.510 0.480 0.774
87 0.523 0.526 0.496 0.830
88
Permit 2
0.555 0.556 0.520 0.725
89 0.571 0.572 0.536 0.774
90 0.584 0.585 0.550 0.830
91
Permit 3
0.548 0.551 0.512 0.725
92 0.564 0.567 0.528 0.774
93 0.577 0.580 0.542 0.830
94
Permit 4
0.505 0.509 0.474 0.725
95 0.522 0.526 0.491 0.774
96 0.538 0.541 0.507 0.830
97
Permit 1
0.529 0.537 0.496 0.710
98 0.546 0.554 0.513 0.767
99 0.559 0.567 0.527 0.821
100
Permit 2
0.595 0.604 0.554 0.710
101 0.608 0.615 0.568 0.767
102 0.617 0.624 0.578 0.821
103
Permit 3
0.571 0.571 0.535 0.710
104 0.584 0.584 0.549 0.767
105 0.594 0.594 0.561 0.821
106
Permit 4
0.529 0.531 0.498 0.710
107 0.545 0.548 0.515 0.767
108 0.558 0.560 0.529 0.821
71
6.4 Strength of Parameters on GDFs
The effect of the four identified parameters on warping normal stress has been
discussed in Section 6.3. The strength of each parameter on GDFs for curved bridges is
also determined based on SGM GDFs in Table 6-3. GDFs for SGM and CGM are
calculated from Eq. (2.4) to Eq. (2.7). Eq. (2.13) to Eq. (2.16) are used to calculate
AASHTO approximate GDFs. It can be observed from Table 6-3 that CGM GDFs are
larger than SGM GDFs.
Permit 2 and permit 3 vehicles have the same gage (16 ft) and have nearly the
same GDFs as observed from Table 6-3. Considering 16 ft gage, only GDF results of the
permit 2 vehicle instead of permit 2 and permit 3 vehicles are presented in Figures 6-2 to
Figures 6-6 to evaluate the effect of each parameter on GDFs.
The GDF significantly increases with increasing span length. It can be observed
from Figure 6-2 that GDF increases as span length increases and the increase is about
18% as spans from 72 ft to 144 ft. This relationship can be explained by the increasing
warping for a given radius as the span increases. The permit vehicles considered are
longer than 144 ft and more axles are present on longer bridges is another reason. As
observed from Figure 6-2, the influence of span length on GDF decreases as radius
increases, because, the warping effect and torsion decreases with increasing radius so that
the effect of span length is more severe for short radius bridges.
Girder spacing, within the limited study range, has no significant effect on GDFs.
As observed from Figure 6-3 that the GDF increases slightly with increasing girder
spacing and the growth of GDF is about 5% over the range of S =10 to 12 ft.
72
Radius has a significant influence on GDFs. As observed from Figure 6-4 that the
GDF decreases considerably as radius increases. From Table 6-3 it can be observed that
GDFs decrease by about 36% on average as radius increases from 200 ft to 750 ft. This
relationship is due to the warping effect decreasing with increasing radius, as previously
discussed.
Within limited study range of gage, GDFs decrease as vehicle gage increases.
Figure 6-5 presents that the GDF decreases about 11% as gage increases from 16 ft to
18.25 ft for L =72 ft and L =108 ft. While for L =144 ft, the GDF for 18.25 ft gage is
slightly larger than the GDF for 18 ft gage. This is resulted from the difference of permit
vehicles length and more axles of 18 ft gage permit vehicle are present on 144 ft span
bridges. To evaluate the effect of standard gage on GDFs, one HS20 truck was also
modeled in nine numerical models with different radii and span lengths. GDFs for HS20
truck are about 37% larger than GDFs for 18.25 ft gage permit vehicle, indicating the
GDF decreases as vehicle gage increases
/L R has been demonstrated to have influence on GDFs based on previous
research and as observed from Figure 6-6 that the GDF increases as /L R increases. It
can be observed from Figure 6-6 that GDF increases about 60% for each of study girder
spacings and vehicle gages as /L R increases from 0.096 to 0.72. In the parametric study,
nine different /L R values were evaluated to determine the effect of /L R on GDFs.
/L R is not considered as one variable but as two independent variables ( L and R ) in the
linear regression analysis.
73
(a) GDF vs Span, S =10 ft
(b) GDF vs Span, S =11 ft
(c) GDF vs Span, S =12 ft
Figure 6-2. Effect of Span Length on GDF ( G =18.25 ft)
74
(a) GDF vs Girder Spacing, L =72 ft
(b) GDF vs Girder Spacing, L =108 ft
(c) GDF vs Girder Spacing, L =144 ft
Figure 6-3. Effect of Girder Spacing on GDF ( G =18.25 ft)
75
(a) GDF vs Radius, S =10 ft
(b) GDF vs Radius, S =11 ft
(c) GDF vs Radius, S =12 ft
Figure 6-4. Effect of Radius on GDF ( G =18.25 ft)
76
(a) GDF vs Gage, S =10 ft
(b) GDF vs Gage, S =11 ft
(c) GDF vs Gage, S =12 ft
Figure 6-5. Effect of Vehicle Gage on GDF ( R =200 ft)
77
(a) GDF vs /L R , G =18.25 ft
(b) GDF vs /L R , G =18 ft
Figure 6-6. Effect of /L R on GDF (Continued)
78
(c) GDF vs /L R , G =16 ft
Figure 6-6. Effect of /L R on GDF
6.5 Proposed Approximate GDF (SGM)
Approximate GDF equations are needed to predict GDFs due to the failure of
modifying AASHTO straight bridge GDFs. A linear regression analysis is used to
develop approximate GDF of SGM. The expected approximate GDF form is Eq. (3.1).
Variables determined to conduct the trial of regression analysis are the radius, span
length, gage, and girder spacing. GDF is the dependent variable in the linear regression
analysis. Results of the regression analysis are presented in Table 6-4. Table 6-4 presents
the P-value for each variable is equal to zero that indicates these four variables are
strongly related to GDF. R square is 96.25% that demonstrates the regression model is
highly accurate and reliable. The accuracy of each variable is higher if its standard error
79
is smaller. Coefficients for variables are presented in Table 6-4. Based on Eq. (3.1), the
proposed approximate GDF of SGM is Eq. (6.2).
0.23 0.27 0.24 0.692.92( ) ( ) ( ) ( )g R S L G (6.2)
Table 6-4. Final Results of Regression Analysis for SGM
Parameters Coefficients Standard Error P-value
Intercept 1.073 0.174 0
Radius -0.231 0.005 0
Span Length 0.241 0.01 0
Girder Spacing 0.267 0.04 0
Gage -0.686 0.047 0
R Square 0.962 Adjusted R Square 0.96
6.6 Proposed Approximate GDF (CGM)
The approximate GDF of SGM has been developed. To present the warping
effect in approximate GDF, the approximate GDF of CGM is needed to be proposed. A
linear regression analysis is utilized to develop two different approximate equations for
GDF in CGM, which are CGM-B and CGM-W. The approximate GDF of CGM is the
summation of CGM-B and CGM-W. The expected form for approximate GDF of CGM
is also Eq. (3.1). Variables to conduct the first trial of regression analysis for CGM-B and
CGM-W are the same as approximate GDF of SGM, which are the radius, span length,
gage, and girder spacing.
80
The procedure to conduct regression analysis trials is excluding variables with P-
value larger than 0.05. After the linear regression analysis, only the intercept was
excluded from approximate GDF (CGM-B).
Final regression analysis results are presented in Table 6-5. Without the intercept,
the R square changes to 96.2%. The higher R square indicates the regression analysis is
more accurate and reliable. P-value of each variable equals to zero that indicates four
variables are strongly related to CGM-B. The coefficient for each variable is presented in
Table 6-5. The proposed approximate CGM-B is presented in Eq. (6.3).
0.12 0.24 0.31 0.68( ) ( ) ( ) ( )g R S L Gb
(6.3)
Table 6-5. Final Results of Regression Analysis for CGM-B
Parameters Coefficients Standard Error P-value
Intercept 0 #N/A #N/A
Radius -0.118 0.007 0
Span length 0.312 0.014 0
Gage -0.684 0.042 0
Girder Spacing 0.24 0.046 0
R Square 0.996 Adjusted R Square 0.986
Final regression analysis results are presented in Table 6-6. All study parameters
and intercept are included in the first trial of regression analysis for CGM-W. Based on
regression analysis trial results, span length and girder spacing were excluded from
approximate CGM-W because the P-values of them are larger than 0.05.
81
Excluding span length and girder spacing, as observed from Table 6-6 the R
square changes to 96.1%. The slight decrease of R square compared to the initial trial
demonstrates span length and girder spacing are not strongly related to the CGM-W. It
can also be observed from Table 6-6 that radius and gage have significant correlation to
CGM-W due to zero P-value. The proposed approximate CGM-W is presented in Eq.
(6.4).
0.98 0.76233( ) ( )g R Gw (6.4)
The proposed approximate GDF of CGM for the outermost exterior girder in
curved bridges then is Eq. (6.5).
0.12 0.24 0.31 0.68 0.98 0.76( ) ( ) ( ) ( ) 233( ) ( )g R S L G R G (6.5)
Table 6-6. Final results of Regression Analysis for CGM-W
Parameters Coefficients Standard Error P-value
Intercept 5.453 0.492 0
Radius -0.984 0.019 0
Gage -0.764 0.169 0
R Square 0.961 Adjusted R Square 0.96
Both approximate GDFs of SGM and CGM have high R square values in the
linear regression analysis. Therefore, two proposed GDFs are supposed to accurately
predict GDFs. The approximate GDF of CGM is the combination of CGM-B and CGM-
W indicates that it can present the warping effect on GDFs. The approximate GDF of
82
SGM is simpler than approximate GDF of CGM but only considers the total bending
GDF, therefore, it cannot indicate the warping effect on GDFs.
6.7 Accuracy of GDF Equations
The accuracy of proposed approximate GDFs (SGM and CGM) is evaluated by
comparing to FEM results. Results of approximate GDF (SGM) are obtained from Eq.
(6.2). Figure 6-7 presents that predicted results are very close to FEM results. The points
on straight line in Figure 6-7 indicates approximate GDF (SGM) perfectly predict FEM
results. Figure 6-7 also indicates approximate GDF (SGM) provides conservative results.
Review of Table 6-3 presents approximate GDF (SGM) predicts 113% of FEM results on
average.
Figure 6-7. Approximate GDF (SGM) and FEM GDF for the Outermost Girder
83
Compared to approximate GDF (SGM), the approximate GDF (CGM) provides
larger values. Results of approximate GDF of CGM are calculated from Eq. (6.5). Figure
6-8 presents most of points are above the straight line indicates approximate GDF (CGM)
provides conservative results. Review of Table 6-3 shows approximate GDF (CGM)
predict 115% of FEM results on average.
Both approximate GDFs (SGM and CGM) proposed by using linear regression
analysis provide close results to FEM results. Therefore, approximate GDFs (SGM and
CGM) can accurately predict GDFs for the outermost exterior girder.
Figure 6-8. Approximate GDF of CGM and FEM GDF for the Outermost Girder
84
6.8 Validation of GDF Equations
Two curved bridges are preliminarily designed in Section 3.4.2 to determine if the
proposed approximate GDFs are applicable for study ranges of parameters. The study
range of girder spacing is small. The permit vehicle with 17 ft gage is not available in the
permit vehicles database. Therefore, the variation of vehicle gage and girder spacing
values is not considered. The preliminary design requirements for curved bridges are
presented in Section 3.4. The geometry of these two bridges is presented in Table 3-3. It
can be observed from Table 6-7 that the relative errors for approximate GDF (SGM) and
SGM GDF results from Eq. (2.4) are smaller than 10%. Table 6-8 presents that the
relative errors for approximate GDF (CGM) and CGM GDF results from Eq. (2.5) are
also smaller than 10%. Therefore, both proposed approximate GDFs (SGM and CGM)
can accurately predict GDFs for the outermost girder in curved bridges.
Table 6-7. Comparison of Proposed Approximate GDF (SGM) and SGM GDF
Radius
(ft)
Gage
(ft)
Girder
Spacing
(ft)
Span
Length
(ft)
SGM
GDF
Approximate
GDF
(SGM)
Relative
Error
(%)
300 18.25 10 84 0.522 0.574 9.96
650 18.25 10 120 0.480 0.523 8.96
300 16 10 84 0.589 0.628 6.62
650 16 10 120 0.541 0.572 5.73
300 18 10 84 0.538 0.579 7.62
650 18 10 120 0.497 0.528 6.24
85
Table 6-8. Comparison of Proposed Approximate GDF (CGM) and CGM GDF
Radius
(ft)
Gage
(ft)
Girder
Spacing
(ft)
Span
Length
(ft)
CGM
GDF
Approximate
GDF
(CGM)
Relative
Error
(%)
300 18.25 10 84 0.529 0.577 9.01
650 18.25 10 120 0.500 0.538 7.42
300 16 10 84 0.597 0.633 5.90
650 16 10 120 0.569 0.589 3.44
300 18 10 84 0.544 0.583 7.10
650 18 10 120 0.521 0.543 4.24
6.9 Comparison of Approximate GDFs for Permit Vehicles and HL-93
It is necessary to evaluate whether the proposed approximate GDFs are greater
than GDFs for standard AASHTO loads. To evaluate the difference between GDFs for
curved bridges subjected to permit vehicles and HL-93, the approximate GDF of CGM
for HL-93 loading from (Kim, 2007) was used and expressed in Eq. (6.6):
0.94 1.3 0.38 0.14 0.350.112( ) ( ) ( ) 0.373( ) ( )g R X L R L (6.6)
where X is the cross-frame spacing. The approximate GDFs of CGM for permit vehicles
are obtained from Eq. (6.5). It can be observed from Figure 6-9 that all of points are
below the perfect correlation line, indicating that GDFs for HL-93 loading are about 30%
larger than GDFs for the evaluated permit vehicles.
86
Figure 6-9. Comparison of GDFs for Permit Vehicles and HL-93
6.10 Summary
Analytical GDF results of SGM and CGM were presented and discussed here.
The effect of study parameters on warping and GDFs for curved bridges is also
evaluated. A linear regression analysis was utilized in this chapter to develop
approximate GDFs (SGM and CGM). The approximate GDFs (SGM and CGM) were
demonstrated to accurately predict GDFs within study range of parameters. The
approximate GDF (SGM) is simpler than approximate GDF (CGM) and only the
approximate GDF (CGM) indicates the warping effect. The comparison of GDFs for HL-
93 from (Kim, 2007) and proposed approximate GDFs for permit vehicles is also
introduced in the chapter.
87
Chapter 7
SUMMARY AND CONCLUSIONS
7.1 Summary
Although GDFs for straight bridges subjected to permit vehicles have been
evaluated recently, research about GDFs for curved bridges under the effect of permit
vehicles is limited. A fundamental objective of this study is to evaluate the effect of
vehicle gage on GDFs and propose approximate moment GDFs for single span, simply
supported, horizontally curved, steel, I-girder bridges.
To evaluate GDFs for curved bridges subjected to permit vehicles, a parametric
study was utilized and the total number of analytical cases is 108. The study parameters,
based on previous research, were determined as radius, girder spacing, span length, and
vehicles gage. The geometry of 27 analyzed bridges is taken from (Kim, 2007) and it is
common geometry in practical. Four representative permit vehicles obtained from a
permit vehicles database of PennDOT were evaluated in the parametric study.
SAP2000® and CSiBridge®, two structural software, were used to calculate
bottom flange stresses for the outermost girder. SGM and CGM, two GDF models from
(Kim, 2007), were utilized to calculate GDFs for each analytical case. A linear regression
analysis was used to determine the relationship between GDFs for curved bridges and
AASHTO approximate GDFs. The approximate GDFs of SGM and CGM were proposed
88
also based on linear regression analysis. A Goodness-of-fit method was utilized to
determine how strong relationship between independent variables and dependent variable
of each regression model. The accuracy of approximate GDFs of SGM and CGM was
evaluated by comparing to FEM results. Two approximate GDFs were preliminarily
validated to accurately predict GDFs within study range of each parameter.
7.2 Summary and Conclusions
The effect of each study parameter on warping and GDFs was evaluated. A linear
regression analysis was conducted and the approximate GDFs of SGM and CGM were
proposed. Based on analysis results presented in Chapter 6, the conclusions are presented
as follows:
1. The maximum bending stress ranges from 11.2 ksi to 16.7 ksi and the maximum
warping normal stress ranges from 0.87 ksi to 4.38 ksi over all analytical cases.
2. The maximum warping to bending stress ratio is 34.2% and the minimum ratio
is 7.3%.
3. Warping normal stress decreases as radius increases and warping normal stress
for R =200 ft is about three times of the warping normal stress for R =750 ft.
4. Within the limited study range, the effect of girder spacing on the magnitude of
warping normal stress is small and the variation of warping normal stress is
about 7% for each of three study girder spacings.
5. Within the study range, the effect of vehicle gage on warping is small and the
variation of warping to bending ratio is around 3% for three vehicle gages.
89
Besides the difference between warping to bending ratios for standard gage (6
ft) and permit 1 vehicle (18.25 ft) is also small and about 8%.
6. Span length significantly influences warping normal stress. Besides the effect
of span length on warping decreases with increasing radius. The variance of
warping to bending ratio is 71% for each of three study spans for R =200 ft but
only 25% and 18% for R =350 ft and R =750 ft.
7. Based on a linear regression analysis, the relationship between GDFs and
AASHTO approximate straight bridge GDFs is determined to be small.
Therefore, new approximate GDFs are needed to be proposed.
8. The GDF increases as span length increases and the effect of span length
decreases with radius increasing. The GDF increases about 18% as span ranges
from 72 ft to 144 ft.
9. The effect of girder spacing on GDFs is small within limited study range. The
GDF increases slightly as girder spacing increases from 10 ft to 12 ft and the
growth is about 5%.
10. Within limited study range of gage, GDFs decrease as vehicle gage increases.
Compared GDFs for a permit vehicle to GDFs for standard gage (6 ft), it
presents that GDFs for standard gage are around 40% larger than GDFs for the
permit vehicle.
11. /L R (central angle) has a significant influence on GDFs. GDF increases about
60% for each of gages and girder sapcings as /L R increases from 0.096 to 0.72.
12. Two approximate GDFs (SGM and CGM) were proposed based on a linear
90
regression analysis. A Goodness-of-fit test was used to evaluate regression
models. The R square values of regression results are higher than 90%,
indicating that developed regression models are accurate and reliable.
13. The accuracy of proposed approximate GDFs (SGM and CGM) was evaluated
by comparing to FEM results. Both approximate GDFs (SGM and CGM) are
demonstrated to accurately predict GDFs. The approximate GDFs (SGM and
CGM) provide about 13% and 15% larger results than FEM results, respectively.
14. Two preliminary designed curved bridges were used to validate the application
range of proposed approximate GDFs (SGM and SGM). The relative error of
approximate GDFs (SGM and SGM) is smaller than 10% compared to expected
SGM and CGM GDFs. Both approximate GDFs are demonstrated to accurately
predict GDFs for curved bridges within study range of each parameter.
15. Approximate GDFs for HL-93 loading calculated from (Kim, 2007) are
demonstrated to have 30% larger results than approximate GDFs for permit 1
vehicle.
7.3 Future Research
Based on the scopes and limitations of the present study, the recommendations for
future research are presented as follows:
1. The present study is mainly based on numerical analysis. The field tests and
experiments can be utilized in the future to evaluate the accuracy of numerical
analysis results and proposed approximate GDFs.
91
2. The GDFs for three or more spans continuous horizontally curved, steel I-girder
bridges subjected to permit vehicles can be evaluated in the future. The present
study only considered single span, simply supported, and four steel I-girder
curved bridges.
3. The girder stiffness, Kg ,in AASHTO approximate GDFs, cross-frame spacing,
central angle and other parameters are needed to be added in the parametric
study to develop more accurate approximate GDF equations. The present study
only considered radius, span length, girder spacing, and vehicle gage as
parameters.
4. More permit vehicles with wider range of gage would be better for the
evaluation of the effect of vehicles gage on GDFs. A wider range of permit
vehicle gage can provide more general approximate GDF equations and more
accurate evaluation of the effect of vehicle gage.
5. A wider range of girder spacing needs to be evaluated in the parametric study.
Girder spacing is supposed to be a major influencing parameter based on
previous research. The effect of girder spacing on GDFs is small within limited
range.
6. The present study only evaluated moment GDFs for curved bridges subjected
to permit vehicles without considering shear GDFs. The shear GDFs affect
girder section design also and needs to be evaluated in the future.
7. The permit vehicles analyzed in the present study have different gage, weight,
and axle length. Therefore, it is not a perfect parametric study to evaluate the
92
effect of gage on GDFs. In the future, a more accurate parametric study about
vehicle gage needs to be conducted.
93
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APPENDIX
Parameter Effect on GDFs Plots and Residual Plots of SGM and CGM
(a) GDF vs Span, S =10 ft
(b) GDF vs Span, S =11 ft
(c) GDF vs Span, S =12 ft
Figure A-1. Effect of Span Length on GDF ( G =18 ft)
98
(a) GDF vs Span, S =10 ft
(b) GDF vs Span, S =11 ft
(c) GDF vs Span, S =12 ft
Figure A-2. Effect of Span Length on GDF ( G =16 ft)
99
(a) GDF vs Girder Spacing, L =72 ft
(b) GDF vs Girder Spacing, L =108 ft
(c) GDF vs Girder Spacing, L =144 ft
Figure A-3. Effect of Girder Spacing on GDF ( G =18 ft)
100
(a) GDF vs Girder Spacing, L =72 ft
(b) GDF vs Girder Spacing, L =108 ft
(c) GDF vs Girder Spacing, L =144 ft
Figure A-4. Effect of Girder Spacing on GDF ( G =16 ft)
101
(a) GDF vs Radius, S =10 ft
(b) GDF vs Radius, S =11 ft
(c) GDF vs Radius, S =12 ft
Figure A-5. Effect of Radius on GDF ( G =18 ft)
102
(a) GDF vs Radius, S =10 ft
(b) GDF vs Radius, S =11 ft
(c) GDF vs Radius, S =12 ft
Figure A-6. Effect of Radius on GDF ( G =16 ft)
103
(a) GDF vs Gage, S =10 ft
(b) GDF vs Gage, S =11 ft
(c) GDF vs Gage, S =12 ft
Figure A-7. Effect of Vehicle Gage on GDF ( R =350 ft)
104
(a) GDF vs Gage, S =10 ft
(b) GDF vs Gage, S =11 ft
(c) GDF vs Gage, S =12 ft
Figure A-8. Effect of Vehicle Gage on GDF ( R =750 ft)
105
(a) Residual Plot of Span Length
(b) Residual Plot of Girder Spacing
(c) Residual Plot of Radius
Figure A-9. Residual Plots for SGM (Continued)
106
(d) Residual Plot of Gage
Figure A-9. Residual Plots for SGM
(a) Residual Plot of Span Length
(b) Residual Plot of Girder Spacing
Figure A-10. Residual Plots for CGM-B (Continued)
107
(c) Residual Plot of Radius
(d) Residual Plot of Gage
Figure A-10. Residual Plots for CGM-B
108
(a) Residual Plot of Radius
(b) Residual Plot of Vehicle Gage
Figure A-11. Residual Plots for CGM-W