TRANSVERSE DISTRIBUTION FACTORS FOR HORIZONTALLY …

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.

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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

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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.

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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


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.

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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.

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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.

.

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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;

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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).

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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

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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)

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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)

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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.

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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

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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

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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.

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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.

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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

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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.

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Figure 2-6. Levels of Analysis (Kim, 2007)

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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.

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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


31


32


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


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


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


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


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


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


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


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


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


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


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


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


Intercept 5.453 0.492 0

Radius -0.984 0.019 0

Gage -0.764 0.169 0


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




Figure A-1. Effect of Span Length on GDF ( G =18 ft)

98




Figure A-2. Effect of Span Length on GDF ( G =16 ft)

99




Figure A-3. Effect of Girder Spacing on GDF ( G =18 ft)

100




Figure A-4. Effect of Girder Spacing on GDF ( G =16 ft)

101




Figure A-5. Effect of Radius on GDF ( G =18 ft)

102




Figure A-6. Effect of Radius on GDF ( G =16 ft)

103




Figure A-7. Effect of Vehicle Gage on GDF ( R =350 ft)

104




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

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