Behavior, Design and Construction of Horizontally Curved Composite Steel Box Girder Bridges

8/22/2019 Behavior, Design and Construction of Horizontally Curved Composite Steel Box Girder Bridges

1/286

Behavior, Design and Construction of Horizontally Curved

Composite Steel Box Girder Bridges

by

Muayad Whyib Aldoori

A thesis submitted in conformity with

the requirements for the degree of

Doctor of Philosophy

Graduate Departm ent o f Civil Engineering

University o f Toronto

Copyright by Muayad A ldoori 2004

roduce d with perm ission o f the copyright owner. Further reproduction prohibited without permission.


2/286

1*1National Library

of Canada

Ac qu isition s and

Bibliographic Services

395 Wellington StreetOttawa ON K1A 0N4Canada

Bibliotheque nationale

du Canada

Acquisis ito ns et

services b ibliographiques

395, rue WellingtonOttawa ON K1A 0N4Canada

Your file Votre reference

ISBN: 0-612-94268-6

Our file Notre reference

ISBN: 0-612-94268-6

The author has granted a non-

exclusive licence allowing the

National Library of Canada to

reproduce, loan, distribute or sellcopies of this thesis in microform,

paper or electronic formats.

The author retains ownership of the

copyright in this thesis. Neither the

thesis nor substantial extracts from it

may be printed or otherwise

reproduced without the author'spermission.

L'auteur a accorde une licence non

exclusive permettant a la

Bibliotheque nationale du Canada de

reproduire, preter, distribuer ouvendre des copies de cette these sous

la forme de microfiche/film, de

reproduction sur papier ou sur format

electronique.

L'auteur conserve la propriete du

droit d'auteur qui protege cette these.

Ni la these ni des extraits substantiels

de celleci ne doivent etre imprimes

ou aturement reproduits sans sonautorisation.

In compliance with the Canadian

Privacy Act some supporting

forms may have been removed

from this dissertation.

While these forms may be includedin the document page count,

their removal does not represent

any loss of content from the

dissertation.

Conformement a la loi canadienne

sur la protection de la vie privee,

quelques formulaires secondaires

ont ete enleves de ce manuscrit.

Bien que ces formulairesaient inclus dans la pagination,

il n'y aura aucun contenu manquant.

Canadaroduce d with perm ission o f the copyright owner. Further reproduction prohibited without permission.


3/286

Behavior. Design and Construction of Horizontally Curved Composite Steel Box Girder Bridges

Doctor o f Philosophy, 2004, M uayad Aldoori

Department of Civil Engineering, Un iversity of Toronto

Abstract

Horizontally curved girder bridges have been used considerably in recent years in highly congested

urban areas. However, although significant research on physical testing and advanced analysis has been

underway fo r the past decade, the practical employment o f many recommendations has not been achieved

by the engineering comm unity nor have standards reflecting this work been brought into practice.

The design process o f curved composite bridges involves tracking the stresses and the potential failure

change in the girders during erection, construction and service loading stages. For structural safety and ser-

viceability, the designer estimates the stresses induced within the bridge and assure tha t they do not exceed

the applicable specified limit state as required in bridge design standards. However, the designer may be

concerned abou t the level o f approximation that is used in his estimate or even the a pplicability of the

underlying theory. To answer this question and provide the designer with more insight into the behavior o f

the curved bridges, the field testing during construction and service loading o f a curved bridge located near

Baltimore, Maryland is reexamined here using linear elastic threedimensional finite element modeling.

Comparisons are made between the finite elem ent results and the measured results.

The level of safety or reliability that would be available during the erection and the construction pro-

cesses of horizontally curved girder bridges represents ano ther major concern for the designer. A three

span continuous curved box girder bridge in Houston, Texas is used in this study as an example reflecting

current detailing and fabricating practice and it is chosen for a detailed evaluation of the structural safety/

reliability during the erection and construction process. This task involves simulating the girder erection

and concrete slab placement sequence o f the bridge using comprehensive nonlinear three dimensional

finite element modeling.

ii

roduced with perm ission of the copyright owner. Further reproduction prohibited without permission .


4/286

Finally, to facilitate the finite element modeling effort for use by a designer, ANSYS Parametric

Design Language (APDL) capabilities are used here to develop an analysis/design tool for "BathTub"

style curved s teel girder bridges. This tool is then used to evaluate the effects o f several important design

variables on th e response and behavior of the girders during the construction phase. This study

demonstrates the ability of finite element modeling to assess the stiffness, serviceability performance,

buckling be hav io r and ultim ate strength o f curved bridges during construction and it is a majo r step

towards a perform ance based approach to design for stability.

iii

roduced with perm ission of the copyright owner. Further reproduction prohibited without permission .


5/286

Acknowledgements

I am indebted to m y supervisor, Professor Peter Birkemoe, for his direction and support throughout my

graduate studies at the University of Toronto. His interest in my work and appreciation of my efforts pro-

vided me with the constant motivation needed to achieve my goal.

I would like to extend my sincere thanks to Professor Todd Helwig and Dr. Zhanfei Fan from Univer-

sity of Houston for providing me with all the data and information about Houston Bridge. I wou ld like also

to thank Professor Chai Yoo for his permission to use the engineering drawing of Baltimore Bridge.

Thanks to Professor Joseph Yura of University o f Texas at Austin for making the software and the thesis of

his students available for me.

I would like to acknowledge the financial support o f the John. Kellerman Fellowsh ip Com mittee and

the University o f Toronto. The support of the Departm ent of Civil Engineering at the University o f Toronto

and the partial support from the operating grants program of the Natural Sciences and Engineering

Research Council of Canada (NSERC) are also gratefully acknowledged. I would like to thank the Cana-

dian Institute of Steel Construction (CISC) for supporting my attendance in the twoday course Steel

Bridges, Design, Fabrication and Construction Based on CHBDCS62000.

Thanks to my wife Sana, my brothers, Waael, Raeed and Yasir, and my sister Eman for their support

and caring and to my beautiful daughter Mina who fills my heart with happiness. Thanks to my colleague

Dr. Deena Dinno for her help, support and pro of reading my thesis.

Finally, I must express my gratitude to my parents. Their encouragement, love and support have

always been a source o f strength and inspiration.

iv



6/286

Table o f Contents

Abstract.............................. ............................ ............................................................... ................ ii

Acknowledgements ...... iv

Table of Co ntents .............................................. ...................... ......... ...... ................ .....................vList of Figures. ..... viii

List of Ta bles ....... xi

Notation ..... xii

Chapter 1

Introduction.................................................................................................................................... 1

1.1 General............................................................................................ .......................................1

1.2 Construction of Box Girder Bridges.................................................................................... 4

1.3 Objectives and Scope.............................................................................................................5

Chapter 2

Literature Review ..........................................................................................................................7

2.1 General Behavior of Curved Box Girders ...........................................................................7

2.2 Behavior of Curved Girder Bridges during Construction ...................................................14

2.3 Static Analysis of Curved Box Girder Bridges ................................................................... 29

2.3.1 The Plane Grid Method................................................................................................ 29

2.3.2 Space Frame Method................................................................................................... 30

2.3.3 Folded Plate Method.................................................................................................... 31

2.3.4 Finite Strip Method...................................................................................................... 32

2.3.5 Finite Element Method................................................................................................ 332.3.6 Thin Walled Curved Beam Theory............................................................................. 36

2.4 Experimental Studies on Elastic Response of Box Girder Bridges ...................................38

2.4.1 Field Studies................................................................................................................. 38

2.4.2 Model Studies............................................................................................................... 40

2.5 Ultimate Response of Box Girder Bridges..........................................................................43

2.6 Design Aids for Curved Box Girder Bridges....................................................................... 51

2.7 Design codes and specifications........................................................................................... 63

2.7.1 AASHTO, Guide Specification for Horizontally Curved Bridges (2003) [40]...... 63

2.7. 1.1. Strength of Curved Top Flange Plates............................................................ 63

2.7. 1.2. Strength of Curved Bottom Flange Plates......................................................67

2.7.2 Canadian Highway Bridge Design Code (CHBDC) (2000) [18]............................ 71

2.7.3 Hanshin Guidelines.......................................................................................................71

Chapter 3

Curved Box Gird er Bridge Field Te st...................................... .............................................. . 76

3.1 Introduction......................................................................................................................... 76

V



7/286

3.2 Description of the Field Tes t........................................... ................................................... 77

3.2.1 Instrumentation.................................................. .........................................................77

3.2.2 Analytical Procedure....................................................................................................86

3.3 Finite Element analysis......................................................................................................... 90

3.3.1 General.......................................................................................................................... 90

3.3.2 Finite Element Method ................................................................................................90

3.4 The Finite Element Program: ANSYS ................................................................................. 93

3.4.1 Shell Elements............................................................................................................. 93

3.5 Model Study By Fam and Turkstra [34] ..............................................................................94

3.6 Finite Element Modeling of the Baltimore Bridge............................................................. 101

3.6.1 NonComposite Bridge Modeling.............................................................................. 101

3.6.2 Composite Bridge modeling ........................................................................................105

3.7 Comparison of Results for Construction Loading .............................................................. 108

3.7.1 Deflection...................................................................................................................... 108

3.7.2 Longitudinal Stress Distribution........................................................... I l l

3.8 Comparisons of Results for the Live Load Tes t..................................................................120

3.8.1 Deflection...................................................................................................................... 120

3.8.2 Longitudinal Stress Distribution..................................................................................123

Chapter 4

Design Safety o f Horizontally Curved Bath-tub Girder Bridges during Erection and

Construction .................................................................................................................................................. 133

4.1 Introduction............................................................................... :.............................. 133

4.2 Objectives............................................................................................................................... 135

4.3 Detailed Description of the Houston Bridge........................................................................ 135

4.4 Nonlinear Finite Element Capabilities o f ANSYS..............................................................141

4.4.1 Shell 181: Characteristics............................................................................................ 1434.4.2 Nonlinear A nalysis.......................................................................................................145

4.4.3 Geometric Nonlinearity: .............................................................................................. 146

4.4.4 Material Nonlinearity:...................................................................................................147

4.4.5 Iterative Solutions......................................................................................................... 149

4.5 Finite Element Modeling of Houston Br idge ...................................................................... 152

4.5.1 Erection of Segment 905.............................................................................................. 157

4.5.2 Erection of Segment 907.............................................................................................. 159

4.5.3 Slab Construction, Stage 1:.......................................................................................... 165



Chapter 5

Behavior of Horizontally Curved Bath-Tub G irder Bridges during Construction ............176

5.1 Introduction............................................................................................................................. 176

5.2 Objectives................................................................................................................................177

5.3 Finite Element Model for Construction Staging.................................................................. 177

5.4 Parametric Study...................................................................... 181

vi



8/286

5.4.1 Span Length ....... ....183

5.4.2 CrossFrame Spacing................................... 184

5.4.3 Girder Depth................................................... 184

5.4.4 Web Inclination:..........................................................................................................187

5.4.5 Top Flange Width:.......................................................................................................187

5.5 Parametric Effects on Ultimate Behavior............................................................................191

5.5.1 Span Length:.................................................................................................................192

5.5.2 Top Flange Width:.......................................................................................................194

5.5.3 CrossFrame Spacing:.................................. 194

5.6 Modes of Failure....................................................................................................................196

5.7 Summary and Conclusions...................................................... 200

Chapter 6

Summary and Conclusions ................................. 201

6.1 Summary of Results, Observations and Conclusions........................................................ 203

6.1.1 Literature Review........................................................................................................204

6.1.2 Linear Elastic Modeling o f Curved BathTub Girder Bridge Structures.................205

6.1.2.1. Fam and Turkstra Experiments.........................................................................205

6.1.2.2. Baltimore Bridge Field Test.............................................................................206

6.1.3 Design Safety of Horizontally Curved BathTub Girder Bridges during

Erection and Construction............................................................................................208

6.1.4 Behavior o f Horizontally Curved BathTub Girder Bridges during Construction 210

6.2 Principal Achievements of the Current Study.................................................................... 211

6.3 Future Research............................................................................... 212

References...................................... 214

APPENDIX I

ANSY S Input File for Two Spans Continuous Curved B ath-tub Girder Bridge during

Construction ........ 223

vii

roduced with permission of the copyright owner. Further reproduction prohibited without permission.


9/286

List of Figures

Fig. 1.1. CrossSection of Box Girder Bridge................................................................................4

Fig. 2.1. General Behavior of an Open Box Section under Gravity Load Showing

Separate Effect................................................................................................................... 8

Fig. 2.2. Normal and Shear Components of Longitudinal Bending Stress ................................ 10Fig. 2.3. Shear Flow in Box Girder from SaintVenant Torsion.................................................10

Fig. 2.4. Bracing System terminology: Common Configurations.............................................. 13

Fig. 2.5. Top Bracing Arrangements.............................................................................................16

Fig. 2.6. Diagonal Brace Forces from Torsion According to (EPM) [50].................................26

Fig. 2.7. Horizontal Component of Applied Loads on the Top Flanges ....................................27

Fig. 2.8. Angular Deformation of a Box Girder Under Distortion............................................. 53

Fig. 2.9. BeamonElasticFoundation Analogy of Box Girder Distortion [50]....................... 54

Fig. 2.10. Box Girder Cross Section Dimensional Parameters...................................................56

Fig. 2.11. Torsional Parameter k Versus Central Angle q for Actual Curved Bridges [60]....59

Fig. 2.12. Maximum Strength of Curved Bottom Flange Plate in Compression.

(AASHTO 1993) [39].................................................................................................... 71Fig. 3.1. Plan View of the Baltimore Bridge................................................................................ 78

Fig. 3.2. Longitudinal View of the Baltimore Bridge..................................................................79

Fig. 3.3. Cross Sectional Dimension at Splice Locations (Baltimore Bridge).......................... 80

Fig. 3.4. Typical Cross Section of the Baltimore Bridge ............................................................ 81

Fig. 3.5. Concrete Placement Sequence........................................................................................ 82

Fig. 3.6. Longitudenal and Transverse strain Gage Locations....................................................84

Fig. 3.7. Lane Locations for Live Load Test................................................................................ 85

Fig. 3.8. Typical Nodal Point Variation for Field Splice............................................................ 88

Fig. 3.9. Analytical Load Point Location for Static Condition of Live Load Test ...................89

Fig. 3.10. The Current FE Model of Fam and Turkstra [34] Plexiglas Curved Box Girder.... 96

Fig. 3.11. Dimensions and Transverse CrossSection Discretization of Fam and Turkstra[34] Model..................... 97

Fig. 3.12. FE (Shell63, Shell93) and Experimental Deflection at Midspan for Fam and

Turkstra [34] Series A Plexiglas Box Girder Bridge ............................................... 98

Fig. 3.13. FE (Shell63, Shell93) and Experimental Deflection at Midspan for Fam and

Turkstra [34] Series B Plexiglas Box Girder Bridge............................................... 98

Fig. 3.14. FE (Shell93, Shell63) and Experimental MidSpan Stresses in Top Flange of

Fam and Turksrta [34] Series A Plexiglas Box Girder Bridge ............................... 99

Fig. 3.15. FE (Shell93, Shell63) and Experimental MidSpan Stresses in Top Flange of

Fam and Turksrta [34] Series B Plexiglas Box Girder Bridge................................99

Fig. 3.16. FE (Shell93, Shell63) and Experimental MidSpan Stresses in Bottom Flange of

Fam and Turksrta [34] Series A Plexiglas Box Girder Bridge ..............................100Fig. 3.17. FE (Shell93, Shell63) and Experimental MidSpan Stresses in Bottom Flange of

Fam and Turksrta [34] Series B Plexiglas Box Girder Bridge .............................. 100

Fig. 3.18. Typical FE Mesh for Baltimore Curved Box Girders ..............................................103

Fig. 3.19. Distributed Loads Used to Simulate the Weight of Concrete Slab ........................ 106

Fig. 3.20. FE Model Used in Simulating Live Load Tests ........................................................ 107

viii

with permission o f the copyright owner . Further reproduction prohibited without permission.


10/286

Fig. 3.21. Comparison o f Vertical Deflections for Exterior Girder Under Concrete

(Pour I) Load..................................................... 109

Fig. 3.22. Comparison of Vertical Deflections for Exterior Girder Under Concrete

(Pour II) load.................................................................................................................110

Fig. 3.23. Comparison o f Vertical Deflections for Exterior Girder Under Concrete

(Pour III) Load..............................................................................................................110

Fig. 3.24. Longitudinal Stress Dead Load Test, Section AAA, Pour I (ksi)..........................114

Fig. 3.25. Longitudinal Stress Dead Load Test, Section BBB, Pour I (ksi)...........................114

Fig. 3.26. Longitudinal Stress Dead Load Test, Section CCC, Pour I (ksi)...........................115

Fig. 3.27. Longitudinal Stress Dead Load Test, Section DDD, Pour I (ksi)..........................115

Fig. 3.28. Longitudinal Stress Dead Load Test, Section AAA, Pour II (ksi).........................116

Fig. 3.29. Longitudinal Stress Dead Load Test, Section BBB, Pour II (ksi)......................... 116

Fig. 3.30. Longitudinal Stress Dead Load Test, Section CCC, Pour II (ksi)......................... 117

Fig. 3.31. Longitudinal Stress Dead Load Test, Section DDD, Pour II (ksi).........................117

Fig. 3.32. Longitudinal Stress Dead Load Test, Section AAA, Pour III (ksi)....................... 118

Fig. 3.33. Longitudinal Stress Dead Load Test, Section BBB, Pour III (ksi)........................118

Fig. 3.34. Longitudinal Stress Dead Load Test, Section CCC, Pour III (ksi)........................119

Fig. 3.35. Longitudinal Stress Dead Load Test, Section DDD, Pour III (ksi)....................... 119

Fig. 3.36. Live Load Deflection, Gage (14), Load Point.1..................................................... 121

Fig. 3.37. Live Load Deflection, Gage (14), Load Point.2.....................................................121



Fig. 3.40. Longitudinal Stress Live Load Test, Section AAA, Load Point 1.........................124




Fig. 3.44. Longitudinal Stress Live Load Test, Section BBB, Load Point 5......................... 126

Fig. 3.45. Longitudinal Stress Live Load Test, Section BBB, Load Point 6......................... 126Fig. 3.46. Longitudinal Stress Live Load Test, Section BBB, Load Point 7......................... 127

Fig. 3.47. Longitudinal Stress Live Load Test, Section BBB, Load Point 8......................... 127

Fig. 3.48. Longitudinal Stress Live Load Test, Section CCC, Load Point 9......................... 128

Fig. 3.49. Longitudinal Stress Live Load Test, Section CCC, Load Point 10....................... 128



Fig. 3.52. Longitudinal Stress Live Load Test, Section DDD, Load Point 13.......................130

Fig. 3.53. Longitudinal Stress Live Load Test, Section DDD, Load Po int 14.......................130

Fig. 3.54. Longitudinal Stress Live Load Test, Section DDD, Load Point 15.......................131

Fig. 3.55. Longitudinal Stress Live Load Test, Section DDD, Load Point 16........... 131

Fig. 4.1. General Layout of Houston Bridge. [50]..................................................................... 136Fig. 4.2. CrossSection o f the Box Girder Bridge. [50]..............................................................136

Fig. 4.3. Erection Sequence of the Interior Girder. [50].............................................................139

Fig. 4.4. Construction Sequence of the Concrete Slab. [50]...................................................... 140

Fig. 4.5. Isotropic and Kinematic Hardening..............................................................................149

Fig. 4.6. Traditional NewtonRaphson Method vs. ArcLength Method Diagrammatic

Description. [4]................................................................................................................152

Fig. 4.7. Typical Finite Element Discretization for the Houston Bridge................................. 154

ix

roduced with perm ission of the copyright owner. Further reproduction prohibited without permission.


11/286

Fig. 4.8. Models Used to Simulate the Erection of Segments 905 and 907. [50] ....... ..156

Fig. 4.9. Distributed Loads Used to Simulate the Weight of Concrete Slab. [50]...... 156

Fig. 4.10. Typical Modeling of StressStrain Curve Including Strain Hardening .................. 157

Fig. 4.11. Erection of Segment 905.............................................................................................160

Fig. 4.12. Applied LoadVertical Deflection Curve for Segment 905 Erection ..................... 161

Fig. 4.13. Deformed Shape of the Houston Bridge (Segment 905 Erection Phase)

at Ultimate..................................................................................................................... 162

Fig. 4.14. Analytical Application of Restraints at the Supports of the Single Interior

Girder Model ................................................................................................... 163

Fig. 4.15. Applied LoadVertical Deflection Curve for Segment 907 Erection ..................... 164

Fig. 4.16. Deformed Shape of Houston Bridge (Segment 907 Erection Phase) at Ultimate. 166

Fig. 4.17. Applied LoadVertical Deflection Curve for Slab Construction, Stage 1..............168

Fig. 4.18. Deformed Shape of the Houston Bridge (Slab Construction, Stage 1)

at Ultimate.................................................................................................................... 169

Fig. 4.19. Applied LoadVertical Deflection Curve for Slab Construction, Stage 2 .............. 171

Fig. 4.20. Deformed Shape of the Houston Bridge (Slab Construction, Stage2)

at Ultimate.................................................................................................................... 173

Fig. 4.21. Applied LoadVertical Deflection for Slab Construction, Stage 3 ......................... 174

Fig. 4.22. Deformed Shape of the Houston Bridge (Slab Construction, Stage 3)

at Ultimate.....................................................................................................................175

Fig. 5.1. CrossSection of the Box Girder Bridge Model.......................................................... 178

Fig. 5.2. Construction Staging in FE Continuous Model...........................................................180

Fig. 5.3. CrossSection of the Bridge Model.............................................................................. 181

Fig. 5.4. Finite Element Discretization, Noncomposite Model............................................... 182

Fig. 5.5. Effect of Span Length on Deflection Ratio................................................................. 185

Fig. 5.6. Effect of Span Length on Bending Stress Ratio ..........................................................185

Fig. 5.7. Effect of CrossFrame Intervals on Deflection Ratio ................................................. 186

Fig. 5.8. Effect of CrossFrame Intervals on WarpingBending Stress Ratio ......................... 186

Fig. 5.9. Effect of Girder Depth on Deflection Ratio.................................................................188

Fig. 5.10. Effect of Girder Depth on Bending Stress Ratio............................................... 188

Fig. 5.11. Effect of Girder Depth on WarpingBending Stress Ratio...................................... 188

Fig. 5.12. Effect of Web Inclination on Deflection Ratio..........................................................189

Fig. 5.13. Effect of Web Inclination on Bending Stress Ratio ....................................... 189

Fig. 5.14. Effect of Web Inclination on WarpingBending Stress Ratio................................. 189

Fig. 5.15. Effect of Top Flange Width on Deflection Ratio......................................................190

Fig. 5.16. Effect of Top Flange Width on Bending Stress Ratio.............................................. 190

Fig. 5.17. Effect of Top Flange Width on WarpingBending Stress Ratio ..............................190

Fig. 5.18. Span Length Effect on LoadDeflection Curves....................................................... 193

Fig. 5.19. Span Length Effect on Ultimate Strength Ratio........................................................193Fig. 5.20. Top Flange Width Effect on Load Vertical Deflection Curve ............................... 195

Fig. 5.21. CrossFrame Spacing Effect on LoadVertical Deflection Curves ......................... 197

Fig. 5.22. LoadVertical Deflection Curve for Model with (L/R=0.2) .................................... 198

Fig. 5.23. Local Lateral Buckling of the External Web of Exterior Girder for Panels Near Mid

Span.................................................................................................................................199

x



12/286

List of Tables

Table 2.1: Curved BoxGirder Methods of Analysis...................................... 37

Table 3.1: Comparison o f Experimental, Analytical and Design Stresses Due to Dead Load... 120

Table 3.2: Comparison of Experimental, Analytical and Design Stresses Due to

Live Load+Impact Factor (28.6%)...................................................................................132

xi



13/286

Notation

Ab = cross sectional area of lateral bracing mem ber (in ) as defined in Eq. 2.9

A j = cross sectional area o f diagonal bracing member

Af = area o f actual top flange plus one quarter o f web as defined in Eq. 2.6

A0 = enclosed area of box section

As = cross sectional area o f the strut

a = spacing between transverse bracing mem bers as defined in Eq. 2.6 and shown in Fig. 2.5

a = longitudenal stiffener span

B = half width o f the bridge deck

b = bottom flange width between webs (in.), (Eq. 2.9)

b = sm aller o f the tw o plate dimensions (Eq. 2.18)

b = width of crosssection at bracing level (Eq. 2.6).

b = dis tance between cente rs of the top flanges (Eq. 2.15b)

bf = top flange width

bs = distance between longitudinal flange stiffeners

Dtot = total forces in the diagonals of the top flange truss

Depm = diagonal forces from the torsional moment determined using EPM (Equivalent Plate

Method) as shown in Fig. 2.6

Dbend = f rces in a diagonal caused by vertical bending o f the box girder

Djat = forces in a diagonal caused by lateral components o f applied loadings

DFC= curved girder load distribution factor

DFg = load distribution factor for straight girders

d = length of diagonal bracing member

xii

roduced with permission o f the copyright owner. Further reproduction prohibited without permission.


14/286

d = depth o f the box (Eq. 2.38)

ds = width o f the given plate in the box

E = m odulu s of elasticity o f steel

Fbc= maximum average stress in the curved flange

Fbs = equ ivale nt straight girder stress

Fws = ratio o f maximum warping stress in a curved bridge to the maximum bending stress in a

straigh t bridge (Eq. 2.7)

Fy = yield stress (psi) (Eq. 2.44)

f = bending norm al stress

fL tot = tota l top flange stress

fx t = longitudinal stress at the middle of the top flange

i) lat = lateral bending stress caused by lateral component o f applied loadings

f| bend = late ral bending stress caused by the axia l forces in the struts that resu lt from the

vertical bending of the box girder.

fv = shear stress associated with moment gradient.

G = shear modulus of the steel

I = moment o f inertia of the section.

ID = distortional warping constant of the box girder

If = m oment o f inertia of actual top flange.

Is = actual mom ent o f inertia o f one longitudinal flange stiffener about an axis parallel to the

flange at base of stiffener (Eq. 2.21)

Is = moment o f inertia of transverse bracing member (strut) which is assum ed to perform like a beam

member (Eq. 2.6)

Iw = warping constant.

xiii



15/286

K = plate buckling coefficient

Kd = disto rtional stiffness of the box girder cross section

Ks = shear buckling coefficient

Kt = torsional constant of the section

L = leng th o f the outside girder (Eq. 2.7)

L = span length

L = length in inches o f the unsupported compression flange between cross frames (Eqs. 2.45 and 2.46)

M = bending moment

Mp = full plastic mom ent

xnD = distortional load on the box girder.

N g = nu mber of box girders

n = num ber o f equally spaced longitudinal flange stiffeners

p = entire lateral component of applied loading as shown in Fig. 2.7

Q = first moment o f area under consideration

q = uniform shear flow (force/length)

R = radius of the outside girder (Eq. 2.7)

R = radius o f curvature

S = section modulus

S = diaphragm spacing, expressed as decimal fraction o f span length (Eqs. 2.23 and 2.24)

S = cross frame spacing (in.) (Eq. 2.37)

S = distance between the center line of the deck and that o f the g irder under consideration (Eqs. 2.40

and 2.41)

Smax = maximum cross frame spacing

Stot = total forces in the struts

xiv



16/286

Sbend = forc es in the struts caused by the vertical bending o f the box girder

Siat = forces in the struts caused by lateral component of applied loading

s = spacing of struts (panel length) (Eq. 2.15b)

T = torque on the cross section of the member

Tp = full plas tic torque

t = thickne ss o f the given plate in the box

tf= top flange thickness

tf = thickness of bottom flange plate ( Eq. 2.21)

tw = web thickness

teq = thickness of equivalent top plate

U = in ternal strain energy

V = shear force

W = external energy

Wc = roadw ay width between curbs

w = bottom flange width or width between longitudinal stiffeners

z = longitudenal axis o f the member

a = acute angle between the top flange and the diagonal (Eq. 2.15b)

a = top flange slenderness

A p = magniude o f the maximum out of plane deformation o f stiffened panels o f the bottom flange

A s = maximum outo f straightness of the longitudinal stiffener

0 = angular distortion of the box girders shown in Fig. 2.9

a) = twist angle of the cross section

A = top flange slenderness as defined in (Eq. 2.43).

v = poissons ratio

XV

r oduce d with perm ission of the copyright owner. Further reproduction prohibited without permission.


17/286

II = total potential energy


18/286

Chapter 1

Introduction

1.1 General

Horizontally curved girder bridges have been used considerably in recent years in highly congested

urban areas. Modem highway bridges are often subjected to severe geometric restrictions, especially in

dense cities where elevated highways and multilevel structures are necessary; therefore, they m ust be built

in curved alignment.

In the past, curved girders were generally composed of a series of straight segments. These segments

were used as chords in forming a curved alignment. Currently curved girders have replaced straight seg-

ments. There are several reasons for using a continuous curved girder over several spans. Even though the

cost of the superstructure for the curved girder is higher, the total cost of the curved girder system is

reduced considerably since the number of intermediate supports, expansion joints and bearing details is

reduced. Using continuous curved girders also permits the use of shallower sections as well as permits a

reduction in the slab overhang at the outside girder [58], The continuous curved girder also provides more

esthetically pleasing structures.

1



19/286

Chapter 1 Introduction

Horizontally curved girders, despite all the advantages mentioned above, are generally more complex

than straight girders. Curved girders are subjected to vertical bending plus torsion caused by the girder cur-

vature. In the sixties, several approximate analysis methods were developed to assist designers to deal with

such complexities.

De Saint Venant published his memoir during the first half of the 19th century [81], whic h has been

considered as the first work on the static analysis o f horizontally curved beams.

Since then, considerable advancement has been done, however, a serious research on the subject of

horizontally curved beams started only at the beginning of the seventies when the Federal Highway

Adm inistration (FHWA) formed The Consortium o f University Research Team (CURT). Resea rchers from

Carnegie Mellon University, The University of Pennsylvania, The University of Rhode Island and Syra-

cuse University participated in (CURT). The study was a funded investigation sponsored by twenty five

state highway departments. The objective of (CURT) was to conduct a comprehensive analytical and

experimental research program on the behavior of horizontally curved highway bridges and to perform a

survey on all published works pertaining to horizontally curved bridges.

In addition, investigations were conducted at the University of Maryland. These research efforts

resulted in the publication o f the first edition of the Guide Specification (a working stress design gu ide) for

Horizontally Curved Bridges by the American Association of State Highway and Transportation Officials

(AASHTO) in 1976 [41]. The load factor design criteria were adopted by (AASHTO) in 1980 was based

on the working stress results. The latest edition o f the guide specification was pub lished in 2003 [40].

Until recently, there has been little research performed on horizontally curved bridges since the early

seventies. The current design methods, which are essentially based on the research conducted m ore than 20

years ago, have a number of deficiencies. As indicated by the National Cooperative Highway Research

Program (NCHRP 1998)[41], many provisions in the Guide Specification are overly conservative, and

many may be difficult to implement. Other provisions lend themselves to misinterpretation, which may

lead to uneconomical designs or designs with a lower factor of safety than intended.

2



20/286


Recently, research projects have been sponsored by a number of highway adm inistrations and organi-

zations to improve design methods for curved highway bridges. For instance, a (NCHRP) project 1238

was done to develop a complete load and resistance factor design (LRFD) specifications for horizontally

curved bridges. The Federal Highw ay Administration (FHWA) sponsored other projects to increase knowl-

edge o f the behavior of curved steel bridges.

There are basically two types of steel cross sections currently being in use for curved alignment: an

open section consisting o f a number o f Ishaped cross sections braced with a heavy transverse bracing sys-

tem. The other type o f section currently in use is a closed section consisting o f few box girders. Steel box

girders that serve to transfer loads directly from the concrete deck to the abutm ents and piers have emerged

as the most common application in North Am erica today [15]. Box girder cross sections may take the form

of single cell (one box), multispline (separate boxes) or multicell with a common bottom flange. Due to

the high torsional stiffness of the closed cross section of the box girders, which often ranges from 100 to

1000 times larger than the torsional stiffness of comparable Ishaped sections, the torsional moment

induced by the curvature of the girder can be resisted by the box girder with much less transverse bracing

than the Ishaped girders. The fabrication o f the box girder is more expensive compared to the Ishaped

girder, but this additional cost is usually balanced by the reduction in substructuring for the box girder.

In addition to the large torsional stiffness, box girders provide higher corrosion resistance because a

high percentage of the steel surface including the top o f the bottom flange is not subjected to the environ-

mental attack. The box girder also has a smooth shape that leads to better bridge aesthetics. The trapezoidal

shape in particular, offers several advantages over rectangular shaped cross section. The trapezoidal box

girder (bathtub girder) provides a norrow bottom flange. Near the abutments where the bending moment

is low, narrow flanges allow for steel savings. The narrow flange as well, provides plate stockiness in the

compression areas of continuous bridge. In add ition, bathtub g irders are more aesthetically pleasing.

3

roduc ed with permission of the copyright owner. Further reproduction prohibited without permission.


21/286


1.2 Construction o f Box Girder Bridges

A multispline composite box girder bridge cross section as shown in Fig. 1.1, usually consists o f two

or more (bathtub) trapezoidal shape steel cross sections. The steel section comm only consists of two top

flanges and one bottom flange, which is connected by the inclined web. The concrete slab that is usually

cast in place is connected to the steel section using shear connectors to ensure full interaction between

them. Therefore during construction, the steel girders are subjected to the wet concrete load in addition to

other construction loads wi thout the composite action that results from the hardened concrete deck.

The open section of the bathtub girder is a major concern because o f its relatively low torsional stiff-

ness. Therefore, the top flanges have to be braced to increase the torsional stiffness of the section. In this

case the section is called quasiclosed section because the it is not fully close d and the top flange is still

susceptible to lateral buckling. Bracing systems commonly consist of a horizontal truss attached to the

girder near its top flange to increase its torsional stiffness. The distortion o f the cross section is reduced by

using internal cross frames and diaphragms. The box girder with the horizontal truss bracing is usually

referred as a quasiclosed box girder. The box girder cross section possesses a high torsional stiffness after

the concrete deck gain its full strength since the cross section is considered as a fully closed section.

Concrete SlabShear ConnectorTop Flange

Web

Bottom Flange

Fig. 1.1. CrossSection of Box G irder Bridge.

4



22/286


13 Objectives and Scope

Construction of horizontally curved steel bridges is generally more complex than the construction of

straight bridges of similar span. Up to date, there is essentially no research or documented construction

plan to help th e fabricators with the construct ion of curved steel girders. The recom mended AA SH TO

specifications [41] indicate that a construction plan must be provided including one possible scheme to

construct the bridge, outlining the sequence of girders and deck placement.

Although significant research on physical testing and advanced analysis of horizontally curved girders

has been underway for the past few decades, the m obilization of many recomm endations has not achieved

practic al em ployment by the engineer ing community nor have standards incorpora ted the findings.

Significant additional research and development are required to accomplish the transition to use the

advanced analytica l techniques in the design and construction processes. Principle features o f this study

include studying the behavior o f the trapezoidal composite bathtub girder bridges during construction

and demonstrating the ability of finite element modeling to assess the stiffness, serviceability performance

and ultimate strength of curved bridges both during construction and service loading phases. The main

objectives o f this thesis are as outlined below:

1. Prepare and summarize a literature review on the experimental and theoretical research work pertain-

ing to curved girder bridges.

2. Review the available specifications and standards pertaining to the curved girder strength.

3. Provide the designer with more insight into the behavio r of the curved bathtub girder bridges. This

will be fulfilled by linear elastic finite element modeling of a curved bathtub girder bridge field test.

The finite element results represented by the strain and the deformations will be compared with mea-

sured field values, as well as a comparison will be made with the two dimensional analytical results.

4. Assess the level o f safety and reliability for the erection and construction sequence o f curved box

girder bridges. This task will be accomplished using nonlinear finite element modeling. The various

modes of failure of every erection step and every construction stage will be examined.

5



23/286


5. Em ploy the advanced analytical techniques such as the finite element method in the analysis and

design process and thus make this method more popular and easy to use by a designer. ANSYS Para-

metric De sign Language (APDL) capabilities are utilized to develop an analysis/design tool for bath-

tub style curved composite steel girder bridges. This tool will be used to evaluate the effects of sev-

eral important design variables on the response and behav ior of the girders during the construction

phase.

The general description of the curved box girder behavior is presented in chapter 2 with emphases on

the behavior during construction. This is followed by a comprehensive review to the research related to

girder erection and construction and the various available static analysis methods. Chapter 3 follows the

simulation of the curved box girder field test using linear elastic threedimensional finite element model-

ing. Comparisons are made between the finite element results, the twodimensional analytical results and

the field test measurements.

The level o f safety and reliability of the curved box girder bridges that would be available at various

stages of the erection and construction processes is evaluated in chapter 4 using nonlinear finite element

modeling. C hapter 5 presents the development o f an analysis/design tool for the bathtub style curved steel

girder bridges during construction utilizing ANSYS Parametric Design Language (APDL). The work is

summarized in Chapter 6 where all important conclusions are presented and recommendations are made

for future work. The analysis/design tool (input file) for twospan continuous curved bathtub girder

bridges during construction is listed in Appendix I.

6



24/286

Chapter 2

Literature Review

2.1 General Behavior of Curved Box Girders

An arbitrary line loading on a simple span box girder (Fig. 2.1a) has bending and torsional

components. Under the bending load (Fig. 2.1b), the section deflects rigidly (longitudinal bending), and

deforms (bending distortion), while the torsional loading Fig. 2.1c rotates the section rigidly (mixed

torsion) and deforms the section (torsional distortion). Curved bridges will develop bending and associated

shear stresses as well as torsional stresses because o f the horizon tal curvature even if they are only

subjected to their own gravitational load.

Longitudinal Bending

According to a survey conducted by ASCE task committee on horizontally curved steel box girder

bridges [47], box girders typically have an average span to depth ratio o f 23 for single spans and 25 for

continuous girder spans. Because o f the large span/depth ratio, transverse load causes significant bending

stresses in the girder.

7



25/286

Chapter 2 Litrature Review

P/2

X -

P/2 P/2

(a) Loading Components.P/2 P/2

(b) Bending Load Actions.

72 P/2

f

P/2

I

+

Long.

Bending

Bending

Distortion

+

Mixed |

Torsion j

J / +

Torsional

Distortion

(c) Torsional Load Actions.

Fig. 2.1. General Behavior of an Open Box Section under Gravity Load Showing Separate Effect.

8



26/286


Assuming elastic behavior and that plane sections remain plane under bending, bending normal

stresses, f, arising from the equ ilibrium o f the cross section (Fig. 2.2a) are calculated by:

Where,

M is the bending moment,

S is the section modulus.

Shear stresses associated with the momen t gradient also occur (Fig. 2.2b) and are calculated by:

fy - X ..............................................................................................(2-2)

Where,

V is the shear force,

I is the m oment o f inertia of the section,

Q is the first momen t of area under consideration,

t is the thickness o f the segment.

Mixed Torsion

Because of the bridge curvature, the transverse loads acting on the girder cause twisting about its

longitudinal axis. Uniform torsion occurs if the rate of change of the angle of twist is constant along the

girder and warping is constant and unrestrained. St. Venant analyzed this problem and found that the St.

Venant shear stresses occur in the cross section (Fig. 2.3). I f there is a va riation o f torque or if warping is

prevented or altered along the girder, longitudinal torsional warping stresses develop.

9



27/286


i

-I !

Ia i

(a) Normal Stress (b) Shear Stress

Fig. 2.2. Normal and Shear Components of Longitudinal Bending Stress.

Fig. 2.3. Shear Flow in Box Girder from Saint-Venant Torsion.

10



28/286


In general, both St.Venant torsion and the warping torsion are developed when thinwalled members

are twisted. Bo x girders are usually dominated by St.Venant torsion because the closed cross section has a

high torsional stiffness. Kollbrunner and Basler [54] indicated that the longitudinal normal stresses

resulting from the nonuniform warping torsion are usually negligible. Box girders have large St.Venant

stiffness, which may be 1001000 times larger than that of a comparable Isection. St.Venant stiffness of

the box section is a function of the shear modulus o f the steel, G, and the torsional constant Kt, which

related to the cross section geometry. Box girders resist the applied torque (in part) by St.Venant torsion

that is given by:

T G K t^ ...................................................................................................... (2.3)

Where,

T is the torque on the cross section o f the member,

t9 is the twist angle of the cross section,

z is the longitudinal axis of the member.

The torsional constant for a single cell box girder is given by:

AA24 Ao

Where,

A0 is the enclosed area o f the box section,

ds is the width o f the given plate in the box,

t is the thickness o f the given plate in the box.

11



29/286


Using Prandtls membrane analogy, the shear stress resulting from pure torsion can be solved [54].

Using Bredts equation, the uniform shear flow (force/length), q, for a single cell box girder can be

determined as follows:

( 2 ' 5 )

Where,

Tis the shear stress, and it is essentially uniform through the thickness o f the plates.

Bending Distortion

Because o f the out o f plane bending of the plates forming the girder, the cross section shape changes.

Bending distortion (spreading) occurs when transverse loads are applied to the open box (without top

bracing). In open box girders, this dis tortion causes outward bending o f the webs, upward bending of the

bottom flange and inplane bending of the top flange (Fig. 2.1b). Ties (Struts) as shown in Fig . 2.4 are

usually placed between top flanges to prevent bending distortion.

Torsional Distortion

Torsional load tends to deform the cross section through bending of the walls (Fig. 2.1c). If the box

girder has no cross frames or diaphragms, the distortion is restrained only by the transverse stiffness of the

pla te elements. Due to the lack of distortional stiffness of the open box girder cross section, cross frames

connecting top and bottom flanges are used to prevent the torsional distortion o f the cross section.

Horizontal bracing can be placed at a small distance below the top flanges to increase the torsional stiffness

of the open box cross section and reduce the twist.

12



30/286


Latera l Ties ,x

(Struts) \

Solid Diaphragm Cross-Frame

Top Bracing System (X-Type)

Top Bracing System (Single Diagonal Type)

Fig. 2.4. Bracing System terminology: Common Configurations.

13



31/286


This chapter summarizes the research conducted to investigate the static behavior of curved steel

girder bridges. The research on curved girders covers a wide range of topics. These can be itemized as

follows:

1. Behavio r of curved girder bridges during construction.

2. Static analysis of curved box girder bridges.

3. Experim ental studies on curved box girder bridges.

4. Ultimate response of curved box girder bridges

5. Design aids for curved box girder bridges

6. Design codes and specifications.

2.2 Behavior o f Curved Girder Bridges during Construction

Despite the fact that the torsional stiffness of a constructed composite box girder is very large, the

quasiclosed section o f a box girder during fabrication, erection and casting concrete stages has a relatively

low torsional stiffness. The bottom flange and webs during the early construction stages are very flexible

elements and high distortion and twist of the cross section can occur. Therefore, top bracing systems are

comm only used to increase the torsional stiffness o f the steel girder. The distortion can be also controlled

by using internal cross fram es and solid diaphragms. Top bracing, internal cross frames and diaphragms act

as secondary and passive bracing members in straight bridges, but in a curved girder, the interaction of

bending and tors ion requires them to be considered as p rim ary load carrying elements. The f lexura l rigidity

o f the steel section of a composite steelconcrete box girder prior to deck hardening is as little as one third

that o f the composite section. Similarly, without top lateral bracing, the torsional rigidity of the resulting

open section before deck hardening may be less than 0.01 to 0.001 that o f the closed section after the deck

has hardened [5]. Curved girders have been erected based generally on experience with straight girders and

this has sometimes led to unexpected problems. Guide specification (AASHTO 93) [39] has some critical

deficiencies and these include lack of fabrication and erection provisions and provisions related to the

14



32/286


construction of curved bridges [41]. The specification (AASHTO 93) commonly does not provide

designers or construction engineers enough information about the behav ior of the girder during

construction to design for bracing. Few experimental and analytical research studies have been carried out

to understand the behavior of curved bridge at various construction stages or enhance the design methods

perta ining to top bracing systems, internal cross frames and diaphragms.

In 1976, McDonald, Chen, Yilmaz and Yen [57] studied two single cell model box girders. The

purpose of thei r study was to investigate the behavior o f such girders and to examine the stresses in a

single cell braced open (quasiclosed) box section under vertical loads eccentric to its longitudinal center

line; the other goal o f their study was to evaluate the stresses in the bracing m embers. The open cross

section of the m odels was rectangular in shape with various arrangements o f longitudinal and transverse

stiffeners in the bottom flange and the webs respectively and with various patterns of top bracing that was

placed at the top flange level o f both models. The specimens had sim ple span with a cant ilever portion.

Both specimens had plate diaphragms at the loading points and the support points. Since the braced open

section of a box girder is loaded primarily during the construction phases and the stresses in the section are

normally within the elastic range, the loads on the model specimens were kept within the elastic range.

Horizontal and vertical deflections of the test box girders were measured w ith dial gages at the supports

and loading points. Afterward, rotations were calculated from the measured deflections. Stresses at various

points of the specimens were estimated using electrical res istance strain gage measurem ents. The open

cross section with top bracing was analyzed using the equivalent thickness concept by Kollbrunner and

Basler [54]. Kollbrunner and Basler [54] considered the bracing system as an equivalent plate and

converted the braced open cross section into an equivale nt closed box section. The thickness, t eq, of the

equivalent plate for different patterns of bracing can be calculated through the consideration of strain

energy results from shear in the equivalent plate and the strain energy of the bracing members. The

equations for the thickness, teq, o f the bracing pa tterns shown in Fig. 2.5 are as follows:

15



33/286


Af

Af

H a

.v

(b>

Ad

(a)

Af

- A d

_

b !

(c)

Af

Ad (d) (e)

Is

/X I f ^ r !Y;

/ 1 \ !

Ad

AsAf

Fig. 2.5. Top Bracing Arrangements.

(f>

16



34/286


abeq

G d3+

2 a

A j 3 A f

. (2 .6a)

eqab

G d 3+ +

A a 4 A c 6A f

. (2 .6b)

*eq Gab

+2 Aa 6Af

. (2 .6c)

eq

E ab

G d 3 b 3

Aj A,

a

6 At

, (2 .6d)

eq G

121 ,+ a

2 b

2 4L

,(2 .6e)

, E ab

eq Q j 3 , 3 3 ^ ^

2A d A s 6A f

Where,

teq is the thickness o f theequivalent top plate,

17



35/286


E is the modulus o f elasticity of steel,

G is the shear modulus o f steel,

a is the spacing between transverse bracing members,

b is the width of the cross section at the bracing level,

d is the length o f diagonal bracing member,

A j is the area of diagonal bracing member,

Af is the area o f the actual top flange plus onequarter o f web,

A s is the area of the strut,

Is is the mom ent o f inertia of the transverse bracing membe r (strut) that is assumed to perform like

a beam member,

If is the moment o f inertia of the actual top flange.

The theory of thinwalled elastic beams has been used by M cDonald et. al. to analyze the equivalent

closed box section. Norm al and shearing stresses at various points in the webs and flanges of the

specimens were analytically determined using the equivalent closed section. The study showed that the

computed normal and shear stresses were in good agreement with the measured stresses. The study

indicated that the measured rotations and deflections of the braced girders were smaller than those

computed for the unbraced members. This indicates the influence of the bracing. Measured and estimated

axial forces in some top bracing members were compared; the results showed that these forces were in

good agreement. The study concluded that using the concept o f an equivalent closed box section was a

good method to approach the calculation o f the forces in the top bracing o f the qua siclosed box girder.

In 1985, Branco and Green [10] investigated the influence of the bracing systems on the behavior of

open box girders. A test program was developed with a onequarter scale model o f an open box girder. The

girder was extensively instrumented at the mid span, three eights, quarter, and oneeight points in such a

way that a complete strain profile o f the section could be obtained. Several tests were conducted on the

girder with concentric and eccentric loading for different types and distributions of bracing systems. The

18



36/286


test results we re compared with analytical results using both the finite strip method and thinwalled curved

beam theory. It was concluded that ties and cross frames were effective in preventing distor tionof the

section. Top truss bracing that is usually placed near the top flange level was found to be effective in

reducing the twisting of the section.

Yoo and Littrell [78] in 1986 used threedimensional finite element models composed entirely o f eight

node brick elements and truss members to analyze curved Igirder steel bridges with varying curvature,

length and cross frame intervals. The analyzed bridges were single span structures horizontally curved in a

circular arc. Bo undary conditions at the ends consisted of pinnedroller bearings radially oriented under the

bottom flanges. The material o f the bridge superst ruc tures was assumed to be linearly elastic and the

deformations were assumed to remain within the limits of small displacement theory. Results from this

param etric study were utilized to develop empirical equations using linear and nonlinear regression.

These equations predict the ratio of the maximum bend ing stress, the maxim um warping stress and the

maximum deck deflection for a curved bridge to corresponding parameters of a straight bridge of equal

length. In addition, an equation suitable for a preliminary estimate o f maxim um bracing spacing was

developed as follows:

Where,

Smax is the maximum bracing spacing,

L is the length o f the outside girder,

R is the radius of the outside girder,

Fws is the ratio o f the maximum warping stress to the maximum bending stress in a curved bridge.

Smax (2.7)

19



37/286


Davidson, Keller and Yoo [30] in 1996 created three dimensional finite element models of a

horizontally curved steel Igirder bridge connected by cross frames. ABAQUS (a Finite Element Program)

was used in the bridge idealization; six shell elements were used to model the webs, while beam elements

were used to model the flanges. All geometry, boundary conditions and loading conditions were modeled

in the cylindrica l coordinate system. As in the previous paper of Yoo et. al. (1986), the material o f the

models was assumed to remain linearly elastic and deformations were assumed to remain within the limits

of small displacement theory. The authors investigated the effects of various parameters (number of

girders, girder spacing, span length, radius of curvature, flange width and cross frame spacing) on the

warpingbending stress ratio, and an equation for the preliminary cross frame spacing was developed from

regression as follows:

Smax = L InZFwsRbfX

V509L2 ;

1.52

. (2 .8)

Where,

bf is the flange width.

In contrast to the previous study by Yoo et. al., the authors o f this study believed th at the flange width

param eter was a major param ete r in relation to reduc tion of the warping stress. Therefore, flange width

appears in the equation for maximum cross frame spacing (Eq. 2.8). The results o f the developed equation

were compared and verified by the finite element model results. Davidson et.al. indicated in their study

that the YooLittrell equation (Eq. 2.7) consistently gave values for the cross frame spacing that were

unconservative compared with the values used in the actual design, while for those spans with a high

degree of curvature, the values generated by Davidson et. al. equation compared more favorably with those

in the AASHTO design [39],

20



38/286


In 1996, Zha ng [79] constructed a three dimensional finite element model for a simple span system of

horizontally curved noncomposite twin box girder bridge using NASTRAN V.68 (a finite element

analysis program). Quadrilateral plate elements were used to model the webs, the top and bottom flanges,

and the internal and external solid diaphragms while beam elements were used to model the cross frame

and lateral bracing members. The author studied the effects of the arrangement of the cross frames, the

lateral bracing and the external solid diaphragms on the box girder system behavior. The equivalent

thickness formula was verified. The study showed that as the number of cross frames increased, the

behavior o f the box girder w ith lateral bracing approached that of the quasic losed bo x girder (equivalent

closed box girder). In addition, the smaller the ratio of span length to radius of curvature was, the closer the

behavior of the box girder wi th lateral bracing wou ld agree with that of the quasi closed box girder.

AASHTO gives the lower bound value of the member size of the lateral bracing as:

A b > 0 .0 3 b ................................................................................................................ (2 .9)

Where,

'yAb is the cross sectional area of lateral bracing mem ber (in ),

b is the bottom flange width between webs (in.).

This formula is based on the ratio of the induced warping stress to the bending stress that is less than

five percent. The results of this study indicated that this low er bound is too low, and thus, inadequate. The

bending mom ent and the torque induced in the box girders were calcu lated by both fin ite elem ent and M/R

method, and the results were compared. It was shown that the M/R method gave reasonable results for the

bending moments and the torques in comparison with the results given by NA STRA N.

In 2000, Helwig and Fan [50] conducted a field study on a curved composite trapezoidal twin box

girder bridge. Girder stresses were measured during erection, construction of the concrete slab and

21



39/286


subsequent live loading. Threedimensional finite element models were developed to simulate the

construction of the bridge. It was shown that the finite element solution and the measured results were in

reasonable agreement. The finite element model was also used to conduct a number o f parametric studies

on the top flange lateral truss and the internal bracing systems for box girders. The results indicated that

current design methods often yield unconservative brace forces and girder stresses in box girders. Based on

equilibrium and compatibility criteria, design equations were developed to evaluate the bending induced

brace forces in the horizontal trusses and the lateral bending stresses o f the top flange . The forces that are

developed in the top flange horizontal truss were shown to be influenced by three parameters: 1) The

torsional mo me nt on the girder. 2) The vertical bending o f the box girder, and 3) The lateral compone nt of

the applied load

Therefore, the total forces in the truss system m embers and the total stresses in the top flange o f the

box girder can be expressed as follows:

S T o t S b e n d + > la t ................................................................................................................................................................( 2 . 1 0 )

M o t M p M + M e n d + M a t ............................................................................... ( 2 1 1 )

fL = fL + f L + f x ............................................................................ (2.12)L Tot ^bend Mat x Top

Where,

Stot is the total forces in the struts,

22



40/286


Sbend the forces in the struts caused by the vertical bending of the box girder,

Sjat is the forces in the struts caused by the lateral component of applied loading,

Dtot is the to tal forces in the d iagonals o f the top flange truss,

Dgpjy[ is the diagonal forces from the torsional moment determined using EPM (Equivalent Plate

Me thod) as shown in Fig. 2.6,

Dbend is foe frce in a diagonal caused by the vertical bending o f the box girder,

Dlat is the force in a diagonal caused by the lateral components o f applied loadings,

fL tot is the total top flange stress,

fx top is the longitudinal stress at the middle of the top flange,

fp iat is the lateral bending stress caused by the lateral component o f applied loadings,

f( bend is the lateral bending s tress caused by the axial forces in the struts tha t result from the

vertical bending o f the box girder.

Two types of top flange truss systems are comm only used: 1) Single Diagonal (SD) and 2) X

diagonals, see Fig. 2.4.

For SDType Trusses:

fv scosax Top_________

(2 .13a)

Sbend ~ be nd S*n a (2 .13b)

23



41/286


1.5s c9 bend'

(bf)tf

. (2.1 3c )

For XType Trusses:

Dbe nd

fT SCOS a'ben d

k ; '. (2 .14a)

Sbend = 2 D b e nds i n a ......................................................................................................( 2 ' 1 4 b )

'bend. (2 .14c)

In which K] and K2 are parameters defined by:

=

_d_

A ,+ (sina) +

A2(bf ) t f

( s ina)^ . (2 .15a)

_ d 2b(sin a). (2 .15b)

24



42/286


Where,

s is the spacing of struts (panel length),

a is the acute angle between the top flange and the diagonal,

bf is th e top flange width,

t f is top flange thickness,

d is leng th o f the diagonal,

b is the distance between centers o f the top flanges,

Ad and A s are the respective cross sectional areas o f the diagonals and the struts.

In a SDtype truss, the struts carry the majo rity o f the lateral component o f the applied loading,

however, the struts and the diagonals in the Xtype share the lateral component. For simplicity, the authors

recommended that each strut should be designed to carry the entire lateral load component since the

diagonal force due to the lateral load component is relatively small compared to the bending and torsional

components. Therefore, the same expressions were recommended for both SDtype and Xtype truss

system:

(2 .16a)

Slat = P s (2 .16b)

25



43/286


2_ p s

fL,.t = (2 J 6 c )f

Where,

p is the entire lateral com ponent of applied loading as shown in Fig. 2.7.

Fictitious

Dsd= qb/sina

(b)DX1= DX25sqb/(2sina)

Fig. 2.6. Diagonal Brace Forces from Torsion According to (EPM) [50].

26



44/286


I

I

ie


45/286


In 2002, Chavel and Earls [17] presented examples of difficulties that commonly happen during

construction o f horizontally curved steel Igirder bridges. Construction difficulties and instabilities can

result during bridge erection from unexpected and excessive out of plane displacements. As part of their

study, the autho rs implemented a nonlinear finite element model o f the Ford city veterans bridge and

analyzed each stage o f the curved span erection sequence. It was shown that the out o f plane displacements

could be minim ized by the use of temporary supports.

In 2002, Jetann, Helwig and Lowery [52] conducted a laboratory test on Permanen t Metal Deck Forms

(PMDF) systems that had been recently used in the bridge construction as a concrete formwork. These

forms are generally not allowed to be considered for bracing in bridge construction. The study proved that

a relatively simple modification to the connection details between the forms and top flange of the girder

enhances the shear strength, stiffness and overall performance of such systems. The shear strength and

stiffness improvements of PMDF systems have encouraged the bridge industry to consider this kind of

formwork as lateral bracing if it is properly detailed.

Chen and Yura in 2003 [19], presented the results of experimental tests conducted on a straight

trapezoidal steel box girder under pure torsion with a permanent metal deck serving as the top flange

lateral bracing. The corrugated metal decking was modeled by an equivalent flat plate that is based on

shear stiffness predictions from the Steel Deck Institute (SDI) Diaphragm Design Manual. The specimen

used in the experimental test was a straight trapezoidal steel box girder measuring 54 in. in depth. Three

bracing configurations that incorporated the metal deck pane ls as the top lateral bracing system were

tested. The three test configurations were 16gauge (0.0598 in.) deck, 20gauge (0.0359 in.) deck and 20

gauge deck with stiffening angles. A three dimensional finite element model was developed using the

commercially available program AB AQUS to analyze the behavior of the test specimen. The experimental

test results demonstrated that permanent metal deck forms could substantially increas

Behavior, Design and Construction of Horizontally Curved Composite Steel Box Girder Bridges

Documents

Transcript of Behavior, Design and Construction of Horizontally Curved Composite Steel Box Girder Bridges