Dynamic and static analyses of continuous curved composite ...

University of Windsor University of Windsor

Scholarship at UWindsor Scholarship at UWindsor

Electronic Theses and Dissertations Theses, Dissertations, and Major Papers

2004

Dynamic and static analyses of continuous curved composite Dynamic and static analyses of continuous curved composite

multiple-box girder bridges. multiple-box girder bridges.

Magdy Said Samaan University of Windsor

Follow this and additional works at: https://scholar.uwindsor.ca/etd

Recommended Citation Recommended Citation Samaan, Magdy Said, "Dynamic and static analyses of continuous curved composite multiple-box girder bridges." (2004). Electronic Theses and Dissertations. 1791. https://scholar.uwindsor.ca/etd/1791

This online database contains the full-text of PhD dissertations and Masters’ theses of University of Windsor students from 1954 forward. These documents are made available for personal study and research purposes only, in accordance with the Canadian Copyright Act and the Creative Commons license—CC BY-NC-ND (Attribution, Non-Commercial, No Derivative Works). Under this license, works must always be attributed to the copyright holder (original author), cannot be used for any commercial purposes, and may not be altered. Any other use would require the permission of the copyright holder. Students may inquire about withdrawing their dissertation and/or thesis from this database. For additional inquiries, please contact the repository administrator via email ([email protected]) or by telephone at 519-253-3000ext. 3208.

https://scholar.uwindsor.ca/

https://scholar.uwindsor.ca/etd

https://scholar.uwindsor.ca/theses-dissertations-major-papers

https://scholar.uwindsor.ca/etd?utm_source=scholar.uwindsor.ca%2Fetd%2F1791&utm_medium=PDF&utm_campaign=PDFCoverPages

https://scholar.uwindsor.ca/etd/1791?utm_source=scholar.uwindsor.ca%2Fetd%2F1791&utm_medium=PDF&utm_campaign=PDFCoverPages

mailto:[email protected]

DYNAMIC AND STATIC ANALYSES OF CONTINUOUS CURVED COMPOSITE MULTIPLE-BOX GIRDER

BRIDGES

BY

MAGDY SAID SAMAAN

A DissertationSubmitted to the Faculty of Graduate Studies and Research through

the Department of Civil and Environmental Engineering in partial fulfillment of the requirements for the

degree of Doctor of Philosophy at the University of Windsor

Windsor, Ontario, Canada 2004

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

1 ^ 1National Library of Canada

Acquisitions and Bibliographic Services

395 Wellington Street Ottawa ON K1A 0N4 Canada

Bibliotheque nationals du Canada

Acquisisitons et services bibliographiques

395, rue Wellington Ottawa ON K1A 0N4 Canada

Your file Votre reference ISBN: 0-612-92551-X Our file Notre reference ISBN: 0-612-92551-X

The author has granted a nonexclusive licence allowing the National Library of Canada to reproduce, loan, distribute or sell copies 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's permission.

L'auteur a accorde une licence non exclusive permettant a la Bibliotheque nationale du Canada de reproduire, preter, distribuer ou vendre 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 celle-ci ne doivent etre imprimes ou aturement reproduits sans son autorisation.

In compliance with the Canadian Privacy Act some supporting forms may have been removed from this dissertation.

Conformement a la loi canadienne sur la protection de la vie privee, quelques formulaires secondaires ont ete enleves de ce manuscrit.

While these forms may be included in the document page count, their removal does not represent any loss of content from the dissertation.

Bien que ces formulaires aient inclus dans la pagination, il n'y aura aucun contenu manquant.

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

Magdy Said SAMAAN©-------------------------------------------------- 2004

All Rights Reserved


I hereby declare that I am the sole author of this document.

I authorize the University of Windsor to lend this document to other institutions or individuals for the purpose of scholarly research.

Magdy Said Samaan

I further authorize the University of Windsor to reproduce the document by photocopying or by other means, in total or part, at the request of other institutions or individuals for the purpose of scholarly research.

Magdy Said Samaan

IV


THE UNIVERSITY OF WINDSOR requires the signatures of all persons using or photocopying this document.

Please sign below, and give address and date.


Abstract

Horizontally curved concrete deck on multiple steel box girder bridges is a

structurally efficient, economic, and aesthetically pleasing method of supporting curved

roadway systems. Modem highway constractions are often in need of bridges with

horizontally curved alignments due to the tight geometry restrictions. Continuous curved

composite box girder bridges allow for the use of longer spans, thus reducing costs of the

substmcture.

Despite all inherent advantages of continuous curved composite box girder

bridges, they do pose challenging problems for engineers in calculating the load

distribution due to moving vehicles across the bridges. Curved bridges are subjected to

high torsional as well as flexural stresses. The interaction between the box girders is also

more complicated in curved bridges than that in straight bridges. North American codes

for bridges have recommended expressions for the load distribution factors only for

straight bridges and not for curved bridges. Impact factors proposed in these codes are

generally restricted also to straight bridges. In addition, simplifled formula to predict the

fundamental frequency of analyzing the bridges is not available. To assist engineers in

dealing with the complexities of continuous curved composite box girder bridges, a

reliable, accurate, and simple method is required to calculate the structure’s response

under self-weight and vehicular loading.

The refined three-dimensional finite-element analysis method is employed to

investigate the static and dynamic responses of the bridge. Two two-equal-span two-box

physical bridge models were constmcted in the laboratory. One of the bridge models was

vi


straight in plan while the other was horizontally curved. The physical models were tested

under several static loading cases to better comprehend their elastic behaviour. Free-

vibration tests were also conducted to obtain the natural frequencies and the

corresponding mode shapes of the bridge models. Both models were loaded up to failure

to examine the collapse mechanism and its correlation with the finite element modeling.

Findings obtained from the two physical bridge models were compared to those predicted

by the analytical models. The agreement between the finite element model and the

experimental model made it possible to use the analytical models to conduct three

parametric studies on several bridges.

Vll


TO MY FAMILY

V lll


Acknowledgements

I would like to express my sincere thanks and deepest gratitude to GOD who

helped me and blessed me all the way of my study.

I would like to express my strongest appreciation to my co-advisor Dr. J.

Kermedy, University Distinguished Professor, for his patience, guidance, and support

throughout the course of this study. I would like to state my genuine thanks to my co

advisor Dr. K. Sennah, Associate Professor, for devoting his time and effort to make this

study a success.

I wish to thank Dr. Madugula, Dr. Hearn, Dr. Ghrib, and Dr. Budkowska for their

help and encouragement.

I wish to acknowledge the financial support provided by the Natural Sciences and

Engineering Research Council of Canada.

I am greatly indebted to my family, parents, and my wife for their strong

encouragement, support, understanding, and patience during the years of this study.

IX


Table of Contents

Abstract................................................................................................................vi

Dedication......................................................................................................... viii

Acknowledgements............................................................................................ix

List of Tables..................................................................................................... xix

List of Figures.................................................................................................... xx

Notation...........................................................................................................xxxv

CHAPTER

I. Introduction...........................................................................................1

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

1.2 The Problem................................................................................................................... 3

1.3 Objectives........................................................................................................................4

1.4 Scope................................................................................................................................ 5

1.5 Outline of the Dissertation........................................................................................... 6

II. Literature Review ................................................................................8

2.1 Introduction.............................................................. 8

2.2 Analytical Methods for Box Girder Bridges.............................................................9

2.2.1 Grillage Analogy Method.......................................................................................9

2.2.2 Orthotropic Plate Theory Method.........................................................................10

2.2.3 Folded Plate Method..............................................................................................10

2.2.4 Finite Strip Method................................................................................................11

X


2.2.5 Finite Element Method......................................................................................... 13

2.2.6 Thin-Walled Beam Theory M ethod.................................................................... 15

2.2.7 M/R-Method.......................................................................................................... 16

2.3 Experimental Elastic Studies......................................................................................16

2.4 Experimental Up-to-CoIIapse Studies....................................................................... 18

2.5 Load Distribution Factors........................................................................................... 19

2.6 Impact Factors..............................................................................................................24

2.7 Fundamental Frequency.............................................................................................29

III. Finite Element Analysis....................................................................32

3.1 Introduction..................................................................................................................32

3.2 Finite Element Technique...........................................................................................33

3.3 Finite Element Program “ABAQUS” .......................................................................36

3.4 Dynamic Analysis........................................................................................................ 38

3.4.1 Natural Frequency Extraction.............................................................................. 38

3.4.2 Transient Modal Dynamic Analysis....................................................................39

3.4.3 Implicit Direct Integration Method...................................................................... 40

3.5 Explicit Dynamic Analysis..........................................................................................41

3.6 Finite Element Modelling of Bridges........................................................................43

3.6.1 Material Modelling.................................................................................................. 443.6.1.1 Modelling of Steel........................................................................................44

3.6.1.2 Modelling of Reinforced Concrete................................................................453.6.1.3 Concrete Model............................................................................................453.6.1.4 Rebar Model................................................................................................ 47

3.6.2 Geometric Modelling............................................................................................ 47

3.6.3 Boundary Conditions............................................................................................ 50

XI


3.7 Finite Element Analysis of Bridge Models..............................................................50

IV. Experimental Study........................................................................... 53

4.1 Introduetion................................................................................................................. 53

4.2 Description of Bridge Models....................................................................................54

4.3 Model Materials...........................................................................................................56

4.3.1 Steel......................................................................................................................... 56

4.3.2 Conerete.................................................................................................................. 57

4.3.3 Steel Wire Reinforeement.....................................................................................58

4.3.4 Shear eonneetors..................................................................................................... 58

4.4 Model Construction.....................................................................................................59

4.4.1 Fabrieation of Open Steel Section........................................................................ 59

4.4.2 Model Supports....................................................................................................... 60

4.4.3 Concrete Form work............................................................................................... 60

4.4.4 Reinforcing Steel W ire.......................................................................................... 61

4.4.5 Casting the Concrete Deck.................................................................................... 62

4.5 Instrumentation........................................................................................................... 62

4.5.1 Strain Gauges...........................................................................................................62

4.5.2 Linear Variable Differential Transducers (LYDTs)............................................ 63

4.5.3 Accelerometers.......................................................................................................64

4.5.4 Load Cells............................................................................................................... 64

4.5.5 Data Acquisition System.......................................................................................65

4.5.6 Hydraulic Jacks.......................................................................................................66

4.6 Test Set-Up................................................................................................................... 66

4.7 T est Procedure............................................................................................................. 674.7.1 Elastic Loading of the Non-Composite Bridge Model..........................................68

4.7.2 Free Vibration.........................................................................................................68

4.7.2.1 Flexural test..................................................................................................69

xii


4.7.2.2 Torsion test.................................................................................................. 70

4.7.3 Elastic Loading the Composite Bridge Models................................................... 70

4.7.4 Loading of Bridge Models Up-to-Collapse..........................................................73

V. Model Validation...............................................................................74

5.1 Introduction................................................................................................................. 74

5.2 Elastic Response of Non-Composite Straight Bridge Model................................75

5.2.1 Loading Case 1 .....................................................................................................75

5.2.2 Loading Case 2 .....................................................................................................76

5.2.3 Loading Case 3 .....................................................................................................77

5.2.4 Loading Case 4 .....................................................................................................78

5.2.5 Loading Case 5 .....................................................................................................79

5.3 Elastic Response of Non-Composite Curved Bridge Model ........................ 80

5.3.1 Loading Case 1 ..................................................................................................... 80

5.3.2 Loading Case 2 ..................................................................................................... 81

5.3.3 Loading Case 3 ..................................................................................................... 82

5.3.4 Loading Case 4 ..................................................................................................... 83

5.3.5 Loading Case 5 ..................................................................................................... 84

5.4 Elastic Response of Composite Straight Bridge Model......................................... 84

5.4.1 Loading Case 1 ..................................................................................................... 84

5.4.2 Loading Case 2 ..................................................................................................... 85

5.4.3 Loading Case 3 ..................................................................................................... 86

5.4.4 Loading Case 4 ..................................................................................................... 87

5.5 Elastic Response of Curved Composite Bridge Model.......................................... 88

5.5.1 Loading Case 1 ..................................................................................................... 88

5.5.2 Loading Case 2 ..................................................................................................... 89

5.5.3 Loading Case 3 .....................................................................................................90

5.5.4 Loading Case 4 .....................................................................................................91

Xlll


5.5.5 Loading Case 5 ...................................................................................................... 92

5.5.6 Loading Case 6 ......................................................................................................92

5.6 Dynamic Characteristics of the Composite Bridge Model...................................94

5.7 Nonlinear Response of the Composite bridge model............................................ 96

5.8 Discrepancies between the Experimental and Theoretical Results......................... 101

5.9 Summary.................................................................................................................... 102

VI. Parametric Studies..........................................................................105

6.1 Introduction...............................................................................................................105

6.2 Description of Bridges Used in the Parametric Studies..................................... 106

6.3 Loading Conditions.................................................................................................. 110

6.3.1 Dead Load.............................................................................................................110

6.3.2 Live Load..............................................................................................................110

6.4 Parametric Study for Load Distribution Factors................................................112

6.4.1 AASHTO Live Loading......................................................................................114

6.4.2 Dead Load.............................................................................................................118

6.5 Parametric Study for Impact Factors....................................................................119

6.6 Parametric Study for the Fundamental Frequency............................................120

VII. Load Distribution Factors............................................................. 121

7.1 Introduction...............................................................................................................121

7.2 Distribution Factors for Tensile Stress..................................................................1227.2.1 Effect of Span Length..........................................................................................122

7.2.2 Effect of Number of Lanes..................................................................................122

7.2.3 Effect of Number of Boxes..................................................................................123

xiv


7.2.4 Effect of Span-to-Radius of Curvature Ratio.................................................... 124

7.3 Distribution Factors for Compressive Stress........................................................124

7.3.1 Effect of Span Length.......................................................................................... 124

7.3.2 EffectofNumber of Lanes.................................................................................. 125

7.3.3 Effect of Number of Boxes................................................................................. 125

7.3.4 Effect of Span-to-Radius of Curvature Ratio....................................................126

7.4 Distribution Factors for Deflection.........................................................................126

7.4.1 Effect of Span Length..........................................................................................126

7.4.2 Effect of Number of Lanes................................................................................. 127

7.4.3 Effect of Number of Boxes................................................................................. 127


7.5 Distribution Factors for S hear................................................................................128


7.5.2 Effect of Number of Lanes................................................................................129

7.5.3 Effect of Number of Boxes................................................................................ 129

7.5.4 Effect of Span-to-Radius of Curvature Ratio.....................................................130

7.6 Distribution Factors for Exterior Support Reaction...........................................130


7.6.2 Effect of Number of Lanes................................................................................ 131



7.7 Distribution Factors for Interior Support Reaction............................................ 132

7.7.1 Effect of Span Length..........................................................................................132




7.8 Distribution Factors for Minimum Reaction........................................................134



XV


7.8.3 Effect of Number of Boxes............................................................... 135


7.9 Empirical Formulas For Load Distribution Factors...........................................136

7.10 Effect of Number of Spans......................................................................................140

7.11 Effeet of Inclined Webs...........................................................................................142

7.12 Effect of Span-to-Depth Ratio............................................................................... 143

7.13 Effect of Cross Bracing...........................................................................................145

7.14 Effect of Different Types of Live Loading........................................................... 147

7.15 Illustrative Design Example....................................................................................149

7.16 Summary................................................................................................................... 152

VIII. Impact Factors.................................................................................. 154

8.1 Introduction.............................................................................................................. 154

8.2 Vehicle Idealization................................................................................................. 155

8.3 Vehicle Loading Positions....................................................................................... 157

8.4 Vehicle Speed............................................................................................................158

8.5 Mode Superposition versus Direct Integration Method................................... 160

8.6 Stability and Accuracy............................................................................................ 162

8.7 Damping Effect.........................................................................................................164

8.8 Dynamic Impact Factor.......................... ......165

8.9 Parametric Study......................................................................................................1668.9.1 Effect of Number of Lanes................................................................................. 1678.9.2 Effect of Number of Boxes................................................................................. 167

8.9.3 Effect of Span Length......................................................................................... 168


xvi


8.10 Expressions for Impact Factor................................................................................169

8.10.1 Impact Factor as a Fimction in Fundamental Frequency................................. 170

8.10.2 Impact Factor as a Function in Bridge Span Length....................................... 172

8.10.3 Impact Factor as a Function in Span-to-Radius of Curvature Ratio............... 174

8.11 Summary................................................................................................................... 175

IX. Fundamental Frequency............................................................... 178

9.1 Introduction..............................................................................................................178

9.2 Effect of Span Length..............................................................................................179

9.3 Effect of Number of Lanes........................................................................................180

9.4 Effect of Number of Boxes........................................................................................180

9.5 Effect of Span-to-Radius of Curvature Ratio..................................................... 181

9.6 Empirical Expressions for Fundamental Frequency...........................................182

9.7 Comparison with Flexural Beam Theory.............................................................. 183

9.8 Effect of Span-to-Depth ratio.................................................................................184

9.9 Effect of End-Diaphragm Thickness..................................................................... 185

9.10 Effect of Cross Bracing........................................................................................... 186

9.11 Effect of Number of Spans......................................................................................186

9.12 Forced-Vibration Analysis......................................................................................187

9.12.1 Effect of Vehicle Speed....................................................................................... 187

9.12.2 Effect of Curvature R atio .................................................................................... 188

9.12.3 Effect of End-Diaphragm Thickness.................................................................. 188

9.12.4 Effect of Number of Cross Braeings.................................................................. 189

9.13 Summary................................................................................................................... 189

xvii


X. Summary and Conclusions............................................................192

10.1 Summary.....................................................................................................................192

10.2 Conclusions..................................................................................................................194

10.3 Recommendation for Further Research......................................... 196

References..........................................................................................................197

Tables .............................................................................................................216

Figures ......................................................................... 223

Appendix A ....................................................................................................... 444

Appendix B ....................................................................................................... 449

Vita Auctoris.................................................................................................... 454

X V lll


List of Tables

TABLE

4.1 Average properties of the eoncrete cylinders for the bridge models........................216

5.1 Natural frequencies and mode shape of tested bridge models..................................216

5.2 Fundamental frequency obtained from the experimental tests.................................217

6.1 Geometries of bridges used in parametrie study for load distribution factor 218

6.2 Geometries of bridges used in parametric study for impact factor and fundamental frequency..................................................................................................................... 219

6.3 Vehicle speed used in parametric study for impact factor........................................ 220

7.1 Comparison between the results obtained from finite element analysis and theproposed method for different codes for load distribution factor............................221

9.1 Comparison between the finite-element results and proposed equations forfundamental frequency............................................................................................... 222

XIX


List of Figures

FIGURE

1.1 Various box girder cross-sections............................................................................. 223

1.2 View of continuous curved composite twin box-girder bridge .............................. 224

1.3 Box girder bridge under construction (US290/IH 35 interchange, Direct connector Z) .................................................................................................................................224

1.4 Typical twin-box girder bridge cross section .......................................................... 225

3.1 Stress-strain relationship for steel adopted in ABAQUS model ............................226

3.2 Tension stiffening model in reinforced concrete..................................................... 226

3.3 Uniaxial stress-strain relationship for plain concrete ............................................. 227

3.4 Concrete failure surfaces in plane stress ................................................................. 227

3.5 Finite element discretization of cross-section of the bridge models.......................228

3.6 Shell element “S4R” used for plate modelling........................................................ 229

3.7 Beam element “B3IH” for beam in space................................................................230

3.8 Boundary condition of the bridges used in the parametric studies......................... 231

3.9 Typical finite element mesh for: a) the non-composite bridge model; and b) the composite bridge m odel............................................................................................. 232

4.1 Plan for the experimental straight bridge model .....................................................233

4.2 Plan for the expereimental curved bridge m odel.....................................................234

4.3 Cross-sectional details of the bridge models ........................................................... 235

4.4 Tension test set-up for steel reinforcement specimen used in the bridge models ..236

4.5 True stress-true strain relationship for structural steel plate...................................236

4.6 View of a tested concrete cylinder after failure.......................................................237

XX


4.7 Stress-strain relationship for concrete cylinders of the curved bridge model ...237

4.8 Stress-strain relationship for the reinforcing s tee l...................................................238

4.9 True stress-true strain relationship for steel shear connectors................................ 238

4.10 View of shear connectors welded to the top flange.................................................239

4.11 View of straight bridge model during fabrication ................................................... 239

4.12 View of the formwork for the curved bridge model.................................................240

4.13 View of the formwork and reinforcing steel bars for the straight bridge model ....240

4.14 View of the formwork and reinforcing steel bars for the curved bridge model ....241

4.15 View of the curved bridge model along with the concrete cylinders during curing ... ...................................................................................................................................... 241

4.16 View of the strain gauges installed along the bottom flange width at the mid-span section...........................................................................................................................242

4.17 Locations of strain gauges on the longitudinal direction of the bridge m odels... 243

4.18 View of the LVDTs in the first span of the bridge m odel.......................................244

4.19 Locations of LVDTs in the cross section of the bridge m odel............................... 245

4.20 View of the accelerometers in the second span of the bridge model .....................246

4.21 Locations of accelerometers in the cross section of the bridge model ....................247

4.22 View of the load eells at the exterior support...........................................................248

4.23 Locations of load cells at support lines of the bridge m odel.................................. 249

4.24 Details of bearings......................................................................................................250

4.25 Data acquisition system cormected to the straight bridge m odel............................251

4.26 Test set-up for the straight bridge m odel..................................................................251

4.27 View of Loading Case 1 applied to the non-composite straight bridge model ....252

4.28 View of the flexural vibration test for straight bridge model................................. 253

4.29 View of the torsional vibration test for curved bridge m odel................................ 254

XXI


4.30 View of straight bridge model imder Loading Case 1 ............................................255

4.31 View of straight bridge model under Loading Case 2 ........................................... 255

4.32 View of straight bridge model under Loading Case 3 ........................................... 256

4.33 View of straight bridge model under Loading Case 4 ............................................256

4.34 View of curved bridge model under Loading Case 1 .............................................. 257

4.35 View of curved bridge model under Loading Case 2 ..............................................257


4.37 View of curved bridge model under Loading Case 4 ............................................. 258



5.1 Cases of loading for non-composite straight bridge m odel................................... 260

5.2 Deflections of the non-composite straight bridge model due to Loading Case 1 ..261

5.3 Longitudinal strains of the non-composite straight bridge model due to Loading C asel ..........................................................................................................................262

5.4 Reactions for the non-composite straight bridge model due to Loading Case 1 ...263


5.6 Longitudinal strain distributions of the non-composite straight bridge model due to loading case 2 .............................................................................................................265



5.9 Longitudinal strains of the non-composite straight bridge model due to Loading Case 3 ..........................................................................................................................268




xxii






5.17 Cases of loading for the non-composite curved bridge model ............................... 276

5.18 Deflections of the non-composite curved bridge model due to Loading Case 1 ...277

5.19 Longitudinal strains of the non-composite curved bridge model due to Loading C asel ..........................................................................................................................278

5.20 Reactions for the non-composite curved bridge model due to Loading Case 1 ....279


5.22 Longitudinal strains of the non-composite curved bridge model due to Loading Case 2 ..........................................................................................................................281











X X lll


5.33 Cases of loading for the composite straight bridge model .................................... 292

5.34 Deflections of the composite straight bridge model due to Loading Case 1 ........293

5.35 Longitudinal strains of the composite straight bridge model due to Loading Case 1 294

5.36 Reactions for the composite straight bridge model due to Loading Case 1 .........295










5.46 Cases of loading for the composite curved bridge model ..................................... 305

5.47 Deflections of the composite curved bridge model due to Loading Case 1 .........306

5.48 Longitudinal strains of the composite curved bridge model due to Loading Case 1 ...................................................................................................................................... 307

5.49 Reactions for the composite curved bridge model due to Loading Case 1 ..........308

5.50 Deflections of the composite curved bridge model due to Loading Case 2 .........309


5.52 Reactions for the composite curved bridge model due to Loading Case 2 .......... 311

XXIV


5.53 Deflections of the composite curved bridge model due to Loading Case 3 ..........312


5.55 Reactions for the composite curved bridge model due to Loading Case 3 ...........314

5.56 Deflections of the composite curved bridge model due to Loading Case 4 ......... 315



5.59 Deflections of the composite curved bridge model due to Loading Case 5 ......... 318



5.62 Deflections of the composite curved bridge model due to Loading Case 6 ..........321


5.64 Reactions for the composite curved bridge model due to Loading Case 6 ........... 323

5.65 Typical acceleration-time history of the straight bridge m odel.............................324

5.66 Typical displacement-time history of the straight bridge m odel........................... 324

5.67 Typical acceleration-time history of the curved bridge m odel.............................. 325

5.68 Typical displacement-time history of the curved bridge m odel............................ 325

5.69 Experimental acceleration frequency response of the straight bridge model in the flexural t e s t ................................................................................................................. 326

5.70 Experimental acceleration frequency response of the straight bridge model in the torsional te s t ................................................................................................................ 326

5.71 Experimental acceleration frequency response of the curved bridge model in the flexural test.................................................................................................................. 327

XXV


5.72 Experimental acceleration frequency response of the curved bridge model in the torsional te s t ................................................................................................................ 327

5.73 First and second mode shapes obtained analytically for the bridge m odels.......... 328

5.74 First and second mode shapes obtained experimentally for the bridge models ....329

5.75 Load-deflection relationship for the straight bridge model at mid-span 1 ............ 330

5.76 Load-strain relationship of the concrete deck for the straight bridge model at midspan 1 ...........................................................................................................................330

5.77 Load-strain relationship of the bottom flange for the straight bridge model at midspan 1 ...........................................................................................................................331

5.78 Load-strain relationship at the top of the web for the straight bridge model at midspan 1 ...........................................................................................................................331

5.79 Load-strain relationship at the bottom of the weh for the straight bridge model at mid-span 1 .................................................................................................................. 332

5.80 Load-deflection relationship for the curved bridge model at mid-span 1 ............. 332

5.81 Load-deflection relationship for the curved bridge model at mid-span 1 .............. 333

5.82 Load-strain relationship of the concrete deck for the curved bridge model at midspan 1............................................................................................................................333

5.83 Load-strain relationship of the concrete deck for the curved bridge model at midspan 1............................................................................................................................334

5.84 Load-strain relationship of the bottom flange for the curved bridge model at midspan 1............................................................................................................................334

5.85 Load-strain relationship of the bottom flange for the curved bridge model at midspan 1............................................................................................................................335

5.86 Load-strain relationship at the top of the web for the curved bridge model at midspan 1 ...........................................................................................................................335

5.87 Load-strain relationship at the bottom of the weh for the curved bridge model at mid-span 1 .................................................................................................................. 336

5.88 Load-strain relationship at the top of the web for the curved bridge model at midspan 1 ...........................................................................................................................336

XXVI


5.89 Load-strain relationship at the bottom of the web for the curved bridge model at mid-span 1 .................................................................................................................. 337

5.90 View of the deflected shape of the straight bridge model at failure........................338

5.91 View of the deflected shape of the curved bridge model at failure............... 338

5.92 Crack pattern of the concrete deck in the straight bridge model at failure....339

5.93 Crack pattern of the concrete deck in the curved bridge model at failure............. 339

5.94 Deformation of the bottom flange in the straight bridge model at failure.............. 340

5.95 Deformation of the bottom flange in the curved bridge model at failure............... 340

6.1 Symbols used for cross-section of four-box girder bridge........................................341

6.2 Cross-section configurations used in the parametric studies................................... 342

6.3 Effect of bottom flange thickness on distribution factor for tensile stress.....343

6.4 Effect of web thickness on distribution factor for tensile stress......................343

6.5 Standard truck and lane loads according to AASHTO Standard Specifications ...344

6.6 AASHTO truck loading cases in the transverse direction of the bridges............... 345

6.7 AASHTO truck loading cases considered in the parametric study for impact factor ...................................................................................................................................... 346

6.8 Idealized four-box bridge ........................................................................................... 347

6.9 AASHTO truck loading cases in the longitudinal direction of the bridges 348

6.10 Typical mode shapes for two-box girder bridge...................................................... 349

6.11 Typical mode shapes of two-equal-span continuous bridges ................................. 350

7.1 Effect of bridge span length on distribution factor for tensile stress for bridges dueto: a) AASHTO live load; and b) dead load .............................................................351

7.2 Effect of number of lanes on distribution factor for tensile stress for bridges due to:a) AASHTO live load; and b) dead load...................................................................352

7.3 Effect of number of boxes on distribution factor for tensile stress for bridges due to:a) AASHTO live load; and b) dead load...................................................................353

xxvii


7.4 Effect of bridge curvature on distribution factor for tensile stress for bridges due to: a) AASHTO live load; and b) dead load...................................................................354

7.5 Effect of bridge span length on distribution factor for compressive stress for bridges due to: a) AASHTO live load; and b) dead load...................................................... 355

7.6 Effect of number of lanes on distribution factor for compressive stress for bridges due to: a) AASHTO live load; and b) dead load...................................................... 356

7.7 Effect of number of boxes on distribution factor for compressive stress for bridges due to: a) AASHTO live load; and b) dead load...................................................... 357

7.8 Effect of bridge curvature on distribution factor for compressive stress for bridges due to: a) AASHTO live load; and b) dead load...................................................... 358

7.9 Effect of bridge span length on distribution factor for deflection for bridges due to: a) AASHTO live load; and b) dead load...................................................................359

7.10 Effect of number of lanes on distribution factor for deflection for bridges due to: a)AASHTO live load; and b) dead load....................................................................... 360

7.11 Effect of number of boxes on distribution factor for deflection for bridges due to: a) AASHTO live load; and b) dead load....................................................................... 361

7.12 Effect of bridge curvature on distribution factor for deflection for bridges due to: a) AASHTO live load; and b) dead load ....................................................................... 362

7.13 Effect of bridge span length on distribution factor for shear force for bridges due to: a) AASHTO live load; and b) dead load...................................................................363

7.14 Effect of number of lanes on distribution factor for shear force for bridges due to: a)AASHTO live load; and b) dead load....................................................................... 364

7.15 Effect of number of boxes on distribution factor for shear force for bridges due to: a) AASHTO live load; and b) dead load...................................................................365

7.16 Effect of bridge curvature on distribution factor for shear force for bridges due to: a) AASHTO live load; and b) dead load...................................................................366

7.17 Effect of bridge span length on distribution factor for exterior support reaction for bridges due to: a) AASHTO live load; and b) dead load..........................................367

7.18 Effect of number of lanes on distribution factor for exterior support reaction for bridges due to: a) AASHTO live load; and b) dead load..........................................368

XXVlll


7.19 Effect of number of boxes on distribution factor for exterior support reaction for bridges due to: a) AASHTO live load; and b) dead load..........................................369

7.20 Effect of bridge curvature on distribution factor for exterior support reaction for bridges due to: a) AASHTO live load; and b) dead load..........................................370

7.21 Effect of bridge span length on distribution factor for interior support reaction for bridges due to: a) AASHTO live load; and b) dead load..........................................371

7.22 Effect of number of lanes on distribution factor for interior support reaction for bridges due to: a) AASHTO live load; and b) dead load..........................................372

7.23 Effect of number of boxes on distribution factor for interior support reaction for bridges due to: a) AASHTO live load; and b) dead load..........................................373

7.24 Effect of bridge curvature on distribution factor for interior support reaction for bridges due to: a) AASHTO live load; and b) dead load..........................................374

7.25 Effect of bridge span length on distribution factor for minimum reaction for bridges due to: a) AASHTO live load; and b) dead load...................................................... 375

7.26 Effect of number of lanes on distribution factor for minimum reaction for bridges due to: a) AASHTO live load; and b) dead load...................................................... 376

7.27 Effect of number of boxes on distribution factor for minimum reaction for bridges due to: a) AASHTO live load; and b) dead load...................................................... 377

7.28 Effect of bridge curvature on distribution factor for minimum reaction for bridges due to: a) AASHTO live load; and b) dead load...................................................... 378

7.29 Effect of number of spans on distribution factor for tensile stress........................ 379

7.30 Effect of number of spans on distribution factor for compressive stress ..............379

7.31 Effect of number of spans on distribution factor for deflection .............................380

7.32 Effect of number of spans on distribution factor for shear force............................380

7.33 Effect of number of spans on distribution factor for exterior support reaction 381

7.34 Effect of number of spans on distribution factor for interior support reaction..... 381

7.35 Effect of number of spans on distribution factor for uplift reaction...................... 382

7.36 Effect of web slope on distribution factor for tensile stress due to AASHTO live load ..............................................................................................................................383

XXIX


7.37 Effect of web slope on distribution factor for compressive stress due to AASHTO live load........................................................................................................................383

7.38 Effect of web slope on distribution factor for deflection due to AASHTO live load ...................................................................................................................................... 384

7.39 Effect of web slope on distribution factor for shear force due to AASHTO live load ...................................................................................................................................... 384

7.40 Effect of web slope on distribution factor for exterior support reaction due to AASHTO live load......................................................................................................385

7.41 Effect of web slope on distribution factor for interior support reaction due to AASHTO live load......................................................................................................385

7.42 Effect of web slope on distribution factor for uplift reaction due to AASHTO live load...............................................................................................................................386

7.43 Effect of span-to-depth ratio on distribution factor for tensile stress .....................387

7.44 Effect of span-to-depth ratio on distribution factor for compressive stress .......... 387

7.45 Effect of span-to-depth ratio on distribution factor for deflection .........................388

7.46 Effect of span-to-depth ratio on distribution factor for shear force........................388

7.47 Effect of span-to-depth ratio on distribution factor for exterior support reaction .389

7.48 Effect of span-to-depth ratio on distribution factor for interior support reaction ..389

7.49 Effect of span-to-depth ratio on distribution factor for uplift reaction...................390

7.50 Effect of number of bracings on distribution factor for tensile stress ....................391

7.51 Effect of number of bracings on distribution factor for compressive stress 391

7.52 Effect of number of bracings on distribution factor for deflection..........................392

7.53 Effect of number of bracings on distribution factor for shear force........................392

7.54 Effect of number of bracings on distribution factor for exterior support reaction .393

7.55 Effect of number of bracings on distribution factor for interior support reaction..393

7.56 Effect of number of bracings on distribution factor for uplift reaction...................394

XXX


7.57 Truck loading considered in AASHTO LRFD and CHBDC codes........................395

7.58 Effect of truck loading specified in different codes on distribution factor for tensile stress.............................................................................................................................396

7.59 Effect of truck loading specified in different codes on distribution factor for compressive stress.......................................................................................................396

7.60 Effect of truck loading specified in different codes on distribution factor for deflection..................................................................................................................... 397

7.61 Effect of truck loading specified in different codes on distribution factor for shear force..............................................................................................................................397

7.62 Effect of truck loading specified in different codes on distribution factor for exteriorsupport reaction...........................................................................................................398

7.63 Effect of truck loading specified in different codes on distribution factor for interiorsupport reaction...........................................................................................................398

7.64 Effect of truck loading specified in different codes on distribution factor for uplift reaction.........................................................................................................................399

8.1 HS20-44 truck loading configuration according to AASHTO Specifications ..... 400

8.2 Vehicle idealization...................................................................................................401

8.3 Loading locations considered in: a) transverse direction; and b) longitudinaldirection...................................................................................................................... 402

8.4 Effect of loading position on tensile stress for 4/-4^»-20 curved bridge ................403

8.5 Effect of loading position on compressive stress for 4/-46-20 curved bridge 403

8.6 Effect of loading position on reaction force for 4/-46-20 curved bridge...............404

8.7 Effect of loading position on shear force for 4l-4b-20 curved b ridge...................404

8.8 Effect of vehicle speed on tensile stress for 4/-66-20 straight bridge ...................405

8.9 Effect of vehicle speed on compressive stress for 4/-6Z>-20 straight bridge..........405

8.10 Effect of vehicle speed on reaction force for 4l-6b-20 straight bridge..................406

8.11 Effect of vehicle speed on shear force for 4l-6b-20 straight bridge ...................... 406

XXXI


8.12 Comparison between direct integration and superposition methods for tensile stress of 2/-2&-20 straight b ridge.........................................................................................407

8.13 Comparison between direct integration and superposition methods for compressive stress of 2/-26-20 straight bridge.............................................................................. 407

8.14 Comparison between direct integration and superposition methods for reaction force of 2/-26-20 straight b ridge.........................................................................................408

8.15 Comparison between direct integration and superposition methods for shear force of 2l-2b-20 straight bridge ............................................................................................. 408

8.16 Effect of time step on tensile stress of 2/-26-20 curved bridge.............................409

8.17 Effect of time step on compressive stress of2l-2b-20 curved bridge .................. 409

8.18 Effect of time step on reaction force of 2/-2Z?-20 curved bridge ...........................410

8.19 Effect of time step on shear force of21-2b-20 curved bridge................................ 410

8.20 Effect of time step on tensile stress of 3l-3b-60 curved bridge.............................411

8.21 Effect of time step on compressive stress of 3l-3b-60 curved bridge ..................411

8.22 Effect of time step on reaction force of 3/-3i-60 curved bridge ...........................412

8.23 Effect of time step on shear force of 3/-36-60 curved bridge................................ 412

8.24 Damping effect on tensile stress of 3/-36-60 curved bridge................................... 413

8.25 Damping effect on reaction force of 3l-3b-60 straight bridge................................ 413

8.26 Effect of number of lanes on impact factor for tensile stress for 46-20 bridges ...414

8.27 Effect of number of boxes on impact factor for tensile stress for 46-20 bridges ..414

8.28 Effect of span length on impact factor for tensile stress for 2Z-26 bridges 415

8.29 Effect of span-to-radius of curvature ratio on impact factor for tensile stress for 41- 26 bridges ................................................................................................................... 415

8.30 Impact factor for tensile stress versus fundamental frequency for: a) straight bridge; and b) curved b ridge.................................................................................................. 416

8.31 Impact factor for compressive stress versus fundamental frequency for: a) straight bridge; and b) curved bridge......................................................................................417

xxxii


8.32 Impact factor for deflection versus fundamental frequency for: a) straight bridge; and b) curved bridge .................................................................................................. 418

8.33 Impact factor for exterior support reaction versus fundamental frequency for: a) straight bridge; and b) eurved bridge........................................................................ 419

8.34 Impact factor for interior support reaction versus fundamental frequency for; a) straight bridge; and b) curved bridge........................................................................ 420

8.35 Impact factor fro uplift reaction versus fundamental frequency for: a) straight bridge; and b) curved bridge......................................................................................421

8.36 Impact factor for shear force at the exterior supports versus fundamental frequency for: a) straight bridge; and b) curved bridge............................................................ 422

8.37 Impact factor for shear force at the interior support versus fundamental frequency for: a) straight bridge; and b) curved bridge............................................................ 423

8.38 Impact factor for tensile stress versus span length for: a) straight bridge; and b) curved bridge ..............................................................................................................424

8.39 Impact factor for compressive stress versus span length for: a) straight bridge; and b) curved bridge..........................................................................................................425

8.40 Impact factor for deflection versus span length for: a) straight bridge; and b) curved bridge ...........................................................................................................................426

8.41 Impact factor for exterior support reaction versus span length for: a) straight bridge; and b) curved bridge .................................................................................................. 427

8.42 Impact factor for interior support reaction versus span length for: a) straight bridge; and b) curved bridge ............................................................................................... ...428

8.43 Impact factor for uplift reaction span length for: a) straight bridge; and b) curved bridge ...........................................................................................................................429

8.44 Impact factor for shear force at the exterior supports versus span length for: a) straight bridge; and b) curved bridge........................................................................ 430

8.45 Impact factor for shear force at the interior support versus span length for: a) straight bridge; and b) curved bridge........................................................................ 431

8.46 Impact factor for tensile stress versus span-to-radius of curvature ratio for curved bridges .........................................................................................................................432

XXXlll


8.47 Impact factor for compressive stress versus span-to-radius of curvature ratio for curved bridges..............................................................................................................432

8.48 Impact factor for deflection versus span-to-radius of curvature ratio for curved bridges..........................................................................................................................433

8.49 Impact factor for exterior support reaction versus span-to-radius of curvature ratio for curved bridges........................................................................................................433

8.50 Impact factor for interior support reaction versus span-to-radius of curvature ratio for curved bridges........................................................................................................434

8.51 Impact factor for uplift reaction versus span-to-radius of curvature ratio for curved bridges..........................................................................................................................434

8.52 Impact factor for shear force at the exterior supports versus span-to-radius of curvature ratio for curved bridges............................................................................. 435

8.53 Impact factor for shear force at the interior support versus span-to-radius of curvature ratio for curved bridges............................................................................. 435

9.1 Effect of span length on: a) fundamental frequency; and b) mode shape ............ 436

9.2 Effect of number of lanes on: a) fundamental frequency; and b) mode shape..... 437

9.3 Effect of number of boxes on: a) fundamental frequency; and b) mode shape 438

9.4 Effect of span-to-radius of curvature ratio on: a) fundamental frequency; and b)mode shape ................................................................................................................. 439

9.5 Effect of span-to-depth ratio on the fundamental frequency..................................440

9.6 Effect of end-diaphragm on the first four natural fundamental frequencies ........ 440

9.7 Effect of number of cross bracings on the first four natural fundamental frequencies ...................................................................................................................................... 441

9.8 Effect of number of spans on the first four natural fundamental frequencies ...... 441

9.9 Effect of vehicle speed on the peak acceleration.................................................... 442

9.10 Effect of span-to-radius of curvature ratio on the peak acceleration.....................442

9.11 Effect of end-diaphragm thickness on the peak acceleration ................................ 443

9.12 Effect of number of bracings on the peak acceleration ..........................................443

xxxiv


Notation

A bridge width

b symbols stands for boxes

B box width

C steel top flange width; damping matrix

d total depth of steel box

De distribution factor for external support reaction under the live load

Di distribution factor for intermediate support reaction under the live load

Dm distribution factor for uplift reaction under the live load; elasticity matrix ofelement m

Dv distribution factor for shear force under the live load

De distribution factor for deflection under the live load

D(jn distribution factor for negative stress imder the live load

Dnp distribution factor for positive stress under the live load

e pavement superelevation

E modulus of elasticity

f coefficient of side friction between truck tire and road surface; fundamentalfrequency

f c fundamental frequency for curved bridges

fs fundamental frequency for straight bridges

F c centrifugal force

Qe distribution factor for external support reaction under the bridge self-weight

9i distribution factor for intermediate reaction under the bridge self-weight

XXXV


9m distribution factor for uplift reaction under the bridge self-weight

Qv distribution factor for shear force under the bridge self-weight

9 5 distribution factor for deflection under the bridge self-weight

9an distribution factor for negative stress under the bridge self-weight

9(jp distribution factor for positive stress under the bridge self-weight

H Total cross section depth including the concrete slab

I impact factor; flexural moment of inertia

Id deflection impact factor

In negative stress impact factor

Ip positive stress impact factor

Ire reaction impact factor at end-support

Iri reaction impact factor at intermediate support

Ise shear impact factor at end-support

Isi shear impact factor at intermediate support

lu uplift reaction impact factor

ki stiffness matrix for element i

K global stiffness matrix

I symbols stands for lanes

L arc length along the centreline of a bridge span

m total mass per unit length

M total mass

Nb number of boxes

Nl number of lanes

P weight of an axle

xxxvi


r radius of the path on which the vehicle centre is traveling

R radius of curvature; global force vector acting in the direction of thedisplacements U

Rb forces per unit volume

Rc concentrated load at nodal C

Rd maximum dynamic response

Re maximum reaction at end-support in three-dimensional bridge obtained fromfmite-element modelling

Rea average reaction at end-support in two-span continuous idealized girder

Rj maximum reaction at intermediate support in three-dimensional bridgeobtained from fmite-element modelling

Ria average reaction at intermediate support in two-span continuous idealizedgirder

Rm minimum reaction support in three-dimensional bridge obtained from finite-clement modelling

Rs maximum static response

Rs forces per unit surface area

t sampling time

T total measuring time

ti thickness of the steel flanges and webs

tj thickness of the concrete deck slab

T„ smallest period of the finite element assemblage with n degrees of freedom

U global modal displacement vector

U virtual global modal displacement vector

V maximum shear force stress in three-dimensional bridge obtained from fmite-element modelling; vehicle speed

xxxvii


Va average shear force stress in two-span continuous idealized girder

Atcr critical time step

5 maximum deflection in three-dimensional bridge obtained from fmite-elementmodelling

5a average deflection in two-span continuous idealized girder

8 virtual strain matrix for element m

s“ failure strain in the concrete model

fraction of critical damping

K = L/R span-to-radius of curvature ratio

p modification factor

On maximum negative stress in three-dimensional bridge obtained from fmite-element modelling

CTna average negative stress in two-span continuous idealized girder

Op maximum positive stress in three-dimensional bridge obtained from finite-element modelling

Opa average positive stress in two-span continuous idealized girder

o “ failure stress in the concrete model

O eigenvectors

CO circular frequency

xxxvni


CHAPTER I

Introduction

1.1 General

Horizontally curved box girder bridges are used extensively in the construction of

highvây systems and interchanges in urban areas vy hen severe restrictions of alignments

and site conditions exist. Box girders are known to have higher flexural and torsional

rigidities, which are required for curved bridges. Because of their closed shape, box

girders are less exposed to the environmental detriments causing corrosion. In addition to

economic considerations, curved box girder bridges provide smooth, aesthetically

pleasing structures. There are different types of curved box girder bridges. They may be

made of reinforced eonerete, prestressed concrete, steel, steel box girders with orthotropic

decks, or steel-concrete composite box girders, i.e. steel box composite with a concrete

deck.

Concrete box girder bridges may be constructed using precast concrete elements,

which are fabricated at a production plant and then delivered to the construction site; or

using cast-in-place concrete, which is formed and cast in its final position using falsework

or a launching frame. In the case of prestressed box girder bridges, there are two types of

prestressing systems: pre-tensioning and post-tensioning. Pre-tensioning systems are

methods in which the strands are tensioned before the concrete is placed and post

tensioning systems are methods in which the tendons are tensioned after concrete has

1


reached a specified strength. High tensile steel or advanced composite-fibre is usually

used as tendons in the concrete. There are also box girder bridges erected using an

orthotropic steel deck. A typical orthotropic deck bridge is formed by welding

longitudinal ribs to the transverse floor system which is supported by the main box

girders. A deck plate is then welded to the ribs.

Curved composite box girder bridges are generally used in moderate- to medium-

span bridges. Horizontal curvature of box girder may be obtained by either heat curving

or cold bending. Heat curving is typically accomplished by fabricating a straight girder in

a conventional manner and then applying thermal stresses and yielding in the top edges of

bottom flanges. Assuming the temperature is high enough, the heated edges will yield

resulting in residual stresses and straining that remains after the flanges cool. Cold

bending may be performed by using either a press or a three-roll bender. The process

must be controlled to prevent the flanges and webs from buckling or twisting out-of

plane. Bottom flanges of the box girders must be cut curved. The top flanges may be

fabricated from a single wide plate, or nested for multiple cutting from a single plate to

minimize the scrap. After preparing the webs and flanges, the webs are then welded to the

flanges and then placed in their location. Vertical camber of the girders should be

provided to allow for dead-load deflections and support rotations about an axis radial to

the girder. Cross bracing, diaphragms, and stiffeners must be provided to prevent any

distortion of the desired shape of the bridge cross section. The deck forms may be either

plywood or steel. Permanent deck forms are highly recommended inside the boxes

because of the difficulty of removing them [62]. Rebars are then placed in their position

and the concrete is cast in approved sequences to form the reinforced concrete deck,


which is connected to the top flange through the shear connectors welded to the top

flanges.

A typical curved steel box girder is a tub girder that consists of independent top

flanges and cast-in-place reinforced concrete decks as previously described. Box girder

bridges have single or multiple boxes as shown in Figure 1.1. A view of a continuous

twin steel box-girder bridge with reinforced concrete deck is shovra in Figure 1.2. A

photo of US290/IH35 interchange, direct connector ‘Z’ is illustrated in Figure 1.3. The

bridge is a continuous curved two-box girder under construction.

1.2 The Problem

A typical cross-section of a composite multiple-box girder bridge, shown in

Figure 1.4, is constructed of a concrete deck slab composite with an open top steel (tub)

girder with either vertical or inclined webs. Continuous curved composite multiple-box

girder bridges are three-dimensional and relatively complex structures. The current design

practice in North America has adopted the concept of load distribution factors to simplify

the analysis of straight multiple-box girder bridges. However, the effect of curvature on

the distribution factors for continuous bridges has not been proposed in any of the current

North American codes. Therefore, a simplified method that accounts for bridge curvature

and continuity is required to design composite box girder bridges.

Transient, wind, or seismic loads on a bridge can cause dynamic deflections due

to bridge oscillations that can be a source of discomfort for pedestrians and motorists,

particularly when the fundamental frequency is mainly torsional. The fundamental


frequency of a bridge is the main characteristic in investigating the effects due to dynamic

loads imposed on the structure. Experience [139] shows that high dynamic response is to

be expected only if bridge resonant frequencies coincide within the peaks of the

fundamental spectrum of the dynamic wheel load. Commercial vehicles exhibit basically

natural frequency between 2 and 5 Hz. Bridges of short to medium span length 10-100 m,

have fundamental longitudinal flexural and torsional frequencies in the range of 1 to 15

Hz. Despite the importance of obtaining the fundamental frequency for bridge structures,

there is no simplified method available in any codes in the literature to evaluate reliably

the fundamental frequency of continuous curved composite box girder bridges.

There is a tendency in most bridge codes to treat loads as static loads avoiding the

use of complicated and difficult dynamic analysis. As a result, impact factors or dynamic

amplification factors are proposed to magnify the maximum straining action exerted by a

moving vehicle to account for the dynamic effects. Impact factors recommended by the

current AASHTO Guide Specifications for Horizontally Curved Highway Bridges, 2003

[5], are based on work done a decade ago by Schelling [120] using the two-dimensional

grid technique to investigate three-dimensional bridge structures. The dynamic load

allowance suggested in the Canadian Highway Bridge Design Code, CHBDC, 2000 [20]

is basically a result of dynamic tests on several bridges other than eontinuous eurved

composite box girder bridges. Thus, the expressions for impact factors for continuous

curved composite box girder bridges based on three-dimensional bridge modeling are as

yet unavailable. Therefore, experimental and theoretical investigations on the dynamic

response o f these latter bridges are required.


1.3 Objectives

Based on the problem diseussed in the previous section, the main objectives of the

conducted research work can be stated as follows:

1. Develop a three-dimensional finite element model capable of predicting

the structural response of continuous curved composite box girder

bridges;

2. Verify and substantiate the analytical model by testing in the laboratory

box-girder bridge models under different loading conditions;

3. Deduce simplified expressions for such bridges in the form of load

distribution factors for stresses, deflection, shear, and reactions;

4. Study the dynamic behaviour of these bridges when subjected to simulated

moving vehicles, and thus propose impact factors for stresses, deflection,

shear force, and reactions; and

5. Provide empirical formulas to estimate the fundamental frequency of

continuous curved composite box girder bridges.


1.4 Scope

To achieve the aforementioned objectives, the scope of this research work

includes the following:

1. Literature review of the analytieal methods, previous experimental and

theoretical research work, and codes of practice for straight and curved

box girder bridges;

2. Develop three-dimensional finite element bridge models using the

commercially available finite element eomputer program “ABAQUS”;

3. Test straight and curved bridge models made of twin-box girders

continuous over two-equal-span, having a cast-in concrete deck;

4. Compare the finite element model predictions with the experimental

findings of the laboratory tested bridge models for various load cases to

verify the finite element model and provide information about the

nonlinear response of box girder bridges;

5. Undertake several parametric studies on the main variables that may affect

the load distribution factors, impact factors, and fundamental frequeneies

of such bridges;


6. Deduce empirical formulas for load distribution and impact factors for

stresses, deflection, shear, and reaction, and fundamental frequencies of

continuous curved concrete deck on multiple steel box girder bridges.

1.5 Outline of the Dissertation

In this dissertation a literature review of the earlier analytical and experimental

work on box girder bridges is presented in Chapter II. The finite element analysis is

described in Chapter III. In that chapter, linear static, free-vibration, and dynamic

analyses, and idealizing and modeling of the bridge components are also incorporated and

explained. Chapter IV includes the details of the experimental work conducted on two

bridge models, including instrumentations, loading systems, and the test procedure. In

Chapter V, the comparison between the experimental results and those predicated by the

finite element modeling is undertaken. Also, the nonlinear structural response is

examined analytically and experimentally. Chapter VI explains the parametric studies

conducted on the prototype bridges. Chapter VII presents the results of the load

distribution factors and the effects of various parameters on these factors. Chapters VIII

and IX deal with the results obtained for the impact factors and fundamental frequency,

respectively. The summary, conclusions and recommendations for further research are

presented in Chapter X.


CHAPTER II

Literature Review

2.1 Introduction

Prior to the design of a curved box-girder bridge, selecting the effective and

suitable analysis method is considered to be paramount. The proper determination of the

structural component of a bridge is highly dependent upon the realistic idealization of the

actual bridges in terms of its geometry, material, boundary conditions, applied loads, as

well as the structural analysis method. The curvilinear nature of continuous curved

concrete deck on steel box girder bridges makes it difficult to accurately predict their

structural response to loading. However, that difficulty in the analysis and design of

continuous curved box girder bridges has been overcome by the use of the digital

computers in the design. Since the overall behaviour of continuous curved box girder

bridges is always elastic under service loads, methods of linear structural analysis, such as

orthotropic plate theory, folded plate and finite strip, may be applied. Engineers have also

been inclined to adopt approximate and conservative methods such as load distribution

factors and impact factors, for static and dynamic analyses. In this chapter, a number of

methods of analysis are reviewed, namely: grillage analogy, orthotrpoic plate theory,

folded plate, finite strip, finite element, and thin-walled beam theory. The approximate

analysis of curved box girders by M/R-method is also described. In addition, a survey of

experimental studies on the elastic response of box girder bridges is undertaken. A brief

review of the ultimate load response of box girder bridges is also given. Moreover, the

8


results of a earlier work concerning load distribution, impact factors and fundamental

frequencies of box girder bridges are presented.

2.2 Analytical Methods for Box Girder Bridges

Several methods are available for the analysis of box girder bridges. In each

analysis method, the three-dimensional bridge structure is usually simplified by means of

assumptions in the geometry, materials and the relationship between its components. The

accuracy of the structural analysis is dependent upon the choice of a particular method

and its assumptions. A review of different analytical methods for concrete box girder

bridges has been published by Scordelis [121] with reference to a large number of

computer programs developed at the University of California, Berkeley. Kirstek [88] has

discussed the theoretical aspect of some of the methods. Also, a comparative study of the

various methods available for the analysis of straight prismatic single-cell box girders has

been presented by Maisel and Roll [93]. A brief review of the aforementioned methods is

presented in the following sections.

2.2.1 Grillage Analogy Method

Grillage analysis has been applied to multiple cell boxes with vertical and sloping

webs and voided slabs. In this method, the bridge deck is idealized as a grid assembly.

The continuous curved bridge is modelled as a system of discrete curved longitudinal

members intersecting orthogonally with transverse grillage members. As a result of the

fall-off in stress at points remote from webs due to shear lag, the slab width is replaced by

a reduced effective width over which the stress is assumed to be uniform. The equivalent


stiffnesses of the continuum are lumped orthogonally along the grillage members. One

problem which arises by using the grillage analogy method is in determining the effective

width of the slab to include the shear lag effects. Another difficulty of this method lies in

estimating the torsional stiffness of the closed cells. Approximate technique may be used

to model the torsional stiffness of closed cells by an equivalent I-beam torsional stiffness.

This technique, established by Evans and Shanmugam [50], provides satisfactory results.

2.2.2 Orthotropic Plate Theory Method

In the orthotropic plate theory method, the interaction between the concrete deck

and the curved girder of a box girder bridge is considered. The stiffness of the diaphragms

is distributed over the girder length. The stiffnesses of the flanges and girders are lumped

into an orthortropie plate of equivalent stiffness. However, the estimation of the flexural

and torsional stiffnesses is considered to be one major problem in this method. Also, the

evaluation of the stresses in the slab and girder presents another difficulty in adopting this

approach. In spite of that Cheung [33] has suggested this method for multiple-girder

eurved bridges with high torsional rigidity. The Canadian Highway Bridge Design Code

[20] has recommended using this method mainly for the analysis of straight box girder

bridges.

2.2.3 Folded Plate Method

A multiple-box girder bridge can be modeled as a folded system which consists of

longitudinal plate elements interconnected at joints along their longitudinal edges and

simply-supported at both ends by diaphragms which are infinitely stiff in their planes and

10


perfectly flexible perpendicular to these planes. Any arbitrary longitudinal joint loading

can be resolved into harmonic component of the loading using Fourier series. Then, a

direct stiffness analysis can be performed for each component. Originally, the folded plate

method is limited to simply supported box girder and no intermediate diaphragms are

assumed. This method produces solutions for linear elastic analysis of a box girder

bridge, within the scope of the assumptions of the elasticity theory. The method has been

used to analyze cellular structures by Al-Rifaie and Evans [2], Evans [49], and Meyer and

Scordelis [95]. Canadian Highway Bridge Design Code [20] restricted the use of this

method to bridges with support conditions closely equivalent to a line support. One of the

major shortcomings of the folded plate method is the large computational effort required

and its complexity.

2.2.4 Finite Strip Method

The finite strip method discretizes the bridge into a longitudinal number of strips,

running from one end support to the other. The strips are connected along their

longitudinal edges by nodal lines. The stiffness matrix is then calculated for each strip

based upon a displacement function in terms of Fourier series, rather than on the theory of

elasticity. Similar to the folded plate method, in the finite strip method the direct stiffness

harmonic analysis is performed. The finite element method is basically different from the

strip method in terms of the assumed displacement interpolation functions. Unlike the

finite element method, the displacement functions for the corresponding finite strip are

assumed as combination o f harmonics varying longitudinally and polynomialy in the

transverse direction. Therefore, the strip method is considered as a transition between the

folded plate method and the finite element method. The method is well suited and is a

11


powerful technique for the analysis of orthotropic and circularly curved plate elements for

which direct application of the theory of elasticity becomes too involved. In 1968,

Cheung [32] introduced this method and then in 1971 Cheung and Cheung [28] applied

the finite strip method for curved box girder bridges. In 1974, Kabir and Scordelis [79]

developed a finite strip computer program to analyze curved continuous span cellular

bridges, with interior radial diaphragms, on supporting planar frame bents. Free vibration

of curved and straight beam-slab and box-girder bridges was conducted by Cheung and

Cheung [34] using the finite strip method. In 1978, the method was adopted by Cheung

and Chan [27] to determine the effective width of the compression flange of straight

multi-spine and multi-cell box girder bridges. Cheung [26] in 1984 used a numerical

technique based on the finite strip method and the force method for the analysis of

continuous curved box girder bridges. In 1989, Ho et al. [71] used the finite strip to

analyze three different types of simply supported highway bridges, slab-on-girder, two

cell box girder, and rectangular voided slab bridges.

The basic advantage of the finite strip method is that it requires small computer

storage and relatively little computation time. Although the finite strip method has

broader applicability as compared to folded plate method, the method is still limited to

simply supported prismatic structures. For multi-span bridges, Canadian Highway Bridge

Design Code [20] restricts the method to those with interior supports closely equivalent to

line supports and isolated columns supports.

12


2.2.5 Finite Element Method

The finite element technique is being extensively applied to complicated

structures and is generally the most powerful and versatile as well as accurate numerical

tool of all the available methods. The finite element method has rapidly become a very

popular technique for the computer solution of a box girder bridge of arbitrary plan

geometry and variable cross section. In the finite element analysis the structure is

modelled using suitable finite elements by subdividing its solution domain into discrete

elements. A large number of elements have been developed for use in the finite element

technique. These finite elements may be one-dimensional beam-type elements, two-

dimensional plate or shell elements or even three-dimensional solid elements.

Since the structure is composed of several finite elements interconnected at nodal

points, the individual element stiffness matrix, which approximates the behaviour in the

continuum, is assembled based on assumed displacement or stress patterns. Then, the

nodal displacements and hence the internal stresses in the finite element are obtained by

the overall equilibrium equations. By using adequate mesh refinement, results obtained

from finite element model usually satisfy compatibility and equilibrium [152].

Aneja and Roll [8, 117] have used the finite element technique for horizontally

curved bridge with a box cross-section using flat plate element with curved boundaries

for discretizing the flanges and flat rectangular elements for the webs. The analytical

results showed poor agreement with the experimental findings, because the elements used

did not have sufficient degrees of freedom at their nodes to account for rotation around all

axes; further the web modelling with flat rectangular element did not seem to be

13


sufficiently accurate. Chu and Pinjarkar [36] in 1971 developed finite element

formulation for curved box girder bridges consisting of horizontal sector plates and

vertical cylindrical shell elements. In 1972, Wiliam and Scordelis [144] analyzed cellular

structures of constant depth with arbitrary plan geometry using quadrilateral element in

the finite element analysis. Bazant and El Nimeiri [11] in 1974 attributed the problems

associated with the neglect of curvilinear bovmdaries in the elements used to model

curved box beams by the loss of continuity at the end cross-section of two adjacent

elements meeting at an angle. Instead of developing curvilinear element boundaries, they

developed the skew-ended finite element with shear deformation using straight elements.

Fam and Turkstra [53, 54] and Fam [52] adopted the finite element method for static and

free vibration analysis of box girders with orthogonal boundaries and arbitrary

combination of straight and horizontally curved sections, the analysis has been shown to

be reliable and efficient. Four-node plate bending annular elements were chosen to

idealize the flange members and conical elements for the inclined web members. In 1995,

Galuta and Cheung [60] combined the boundary element with the conventional finite

element method to analyze box girder bridges. The bending moments and vertical

deflection were found to be in good agreement when compared with the finite strip

solution. Davidson et al. [40] in 1996 utilized the finite element method to develop a

detailed model for horizontally curved steel I-girder bridges. In 1998, Sennah and

Kermedy [128] conducted an extensive parametric study on composite multi-cell box

girder bridges using the finite element analysis. The results obtained from the finite

element method were in good agreement with the experimental findings.

The numerical effectiveness, accuracy as well as the flexibility of the method in

linear, non-linear, static or dynamic analyses has been well established. Therefore, many

14


investigators have been attracted to adopt the finite element method to analyze the

complex mechanics of arbitrary box girder bridges. Canadian Highway Bridge Design

Code has recommended the finite element method for all type of bridges.

2.2.6 Thin-Walled Beam Theory Method

Thin-walled beam theory applicable to box beam has been established by Vlasov

[141] and elaborated by Dabrowski [39] and others. The theory assumes non-distortional

cross-section and, hence, does not account for all warping or bending stress. The

predication of shear lag or the response of deck slabs to local wheel load cannot be

obtained using the theory. In 1966, Kolbrunner and Hajden [84] treated thin-walled beam

structures similar to Vlasov but in more general form by including shear deformation for

closed thin-walled cross sections. The load-deformation response of curved box girder,

which considers bending, torsion and warping deformations, as developed by Vlasov, was

used to predict the behaviour of the cross section assumed to retain their shape xmder

loads [106, 67, 69, 100]. In 1985, Maisel [92] extended Vlasov’s thin-walled beam theory

to account for torsional, distortional, and shear lag effects of straight, thin-walled cellular

box beams. Mavaddat and Mirza [94] implemented formulations into computer programs

to analyze straight concrete box beams with one, two, or three cells and side cantilevers

over a simple span or two spans with symmetric mid-span loading. Li [90] and Razaqpur

and Li [113, 114, 115] developed a box girder finite element, which includes extension,

torsion, distortion, and shear lag analysis of straight, skew, and curved multi-cell box

girders using thin-walled finite element based on Vlasov’s theory. Exact shape functions

were used to eliminate the need for dividing the box into many elements in the

longitudinal direction. The results of the proposed element agreed well with those results

15


obtained from full three-dimensional shell finite element analysis. For both static and

dynamic analyses of multi-cell box girder bridges, Vlasov’s thin-walled beam theory was

cast in a finite element formulation and exact shape function was used by El-Azab [48] to

derive the stiffness matrix.

2.2.7 M/R-Method

The M/R-method provides a means to account for the effect of curvature in curved

box girder bridges. The basic concept behind this method is to load a conjugate beam

with a distributed loading. The load on the conjugated beam is equal to the moment in the

real simple or continuous beam induced by the applied load divided by the radius of

curvature of the girder. The reactions of supports are obtained and thus the shear diagram

can be drawn representing the internal torque diagram of the curved beam. The method

and suggested limitations on its use were discussed by Tung and Fountain [140].

However, the method may be restrictive because the box girder is idealized as a 2-D

beam. The vertical reactions at the interior supports on the concave side of a continuous

span bridge may be significantly underestimated by the M/R-method.

2.3 Experimental Elastic Studies

In order to verify the analytical solutions and computer programs developed,

several experimental studies were conducted on box girder bridges. Occasionally,

experimental studies were reported on field-testing of existing box girders. However, the

majority of experimental tests have taken place in the laboratories on small scale bridge

models.

16


In 1975, Kissane and Beal [83] performed a field test program initiated by the

Engineering Research and Development Bureau of the New York State Department of

Transportation. The program was to evaluate the behaviour of a two-span, continuous,

curved box-girder bridge under dead and static live loads. Yoo et al. [148] in 1976

measured the response of a three-span continuous curved box girder bridge designed for

two-lane traffic. The bridge was tested when the concrete deck was cast and later, when

the construction of the bridge was completed. In 1979, Brennan and Mandel [16]

conducted an experimental study on eight different small-scale horizontally curved

bridges. Six models were 1-section girder bridges and two were three-span two box-girder

bridges. The experimental findings were used to verify a computer program developed at

Syracuse University. In 1982, Buckle and Hood [19] performed an experimental test on a

continuous curved box girder model to validate the finite element method results. In

1987, Xi-jin and De-rong [145] tested a three-span conditions curved box girder bridge.

The main objective of the model was to detect the characteristics of the curved box-girder

under various loading continuous and to further verify the accuracy of the finite strip

method as well as a computer program used in the analytical analysis. In 1988, Siddiqui

and Ng [134] examined two straight plexiglass, single cell, box girder bridge models to

investigate the influence of the transverse diaphragms on the behaviour of the box

section. Ng et al. [101] in 1992 conducted on experimental study of a 1/24 scale

composite concrete deck aluminium four-cell model of the Cyrville Road Bridge overpass

east of Ottawa, under various OHDBC truck loading conditions. The prototype was a

two-span continuous two-lane concrete curved four-cell box girder bridge. Green [61] in

1978, Branco [14] in 1985, and Branco and Green [15] in 1985 conducted experimental

study to examine the effect of construction loadings, as well as the bracing configurations

of simple-span open and quasi-closed cross-section beams. The results from the tests

17


were used to verify those obtained from the analytical study. In 1985, McGill University

[43] conducted an experimental study of a 1/10.45 scale two-span continuous straight

composite concrete deck-steel box girder bridge. In 1997, Ebeido and Kennedy [47]

conducted an experimental study on three continuous skew composite steel concrete

bridge models with two unequal spans. In 1998, Setmab [124] tested five straight and

curved deck-steel three-cell bridge models vmder various static loading conditions and

free vibration tests. Four models were simply-supported and the fifth was a two-equal-

span continuous bridge model. The results obtained from the experimental work were

utilized to verify the finite element model.

2.4 Experimental Up-to-Collapse Studies

Fewer experimental studies have dealt with the up-to-collapse response of straight

and curved box girder bridges. Dogaki et al. [45] in 1979 investigated experimentally the

ultimate behaviour of two horizontally curved steel single-cell box girder bridge models

under two concentrated loads. In 1979, Heins and Humphnay [64] tested up-to-failure a

series of box beam models, composed of top steel flanges, steel webs, steel bottom

flanges and cross-bracings. Some of the beam models had concrete deck, while the rest

did not. An interaction, non-dimensional, equation was developed based on the

experimental findings to predict the load distribution factor of curved steel box girders.

Scordelis et al. [122] tested a two-span, four-cell, reinforced concrete box girder bridge

up-to-collapse. Results obtained from that test agreed well with a three-dimensional

grillage model and a computer program to estimate the non-linear response of multi-cell

reinforced concrete box girder bridges subjected to static loading. In 1985, McGill

University [43] tested a composite deck-steel box girder bridge up to failure. Owens et al.

18


[110] conducted similar experimental study on curved composite concrete-deck steel

multi-spine box girders assembly. In 1994, Soliman [135] performed experimental studies

on straight and curved reinforced concrete single-cell box girder bridge models to

investigate the influence of diaphragms on the bridge behaviour. In 1997, Ebeido and

Kennedy [47] tested three continuous skew composite bridges up to collapse using

simulated truck loads applied on two lanes. Theoretical studies were also undertaken to

better understand the non-linear behaviour as well as the local buckling of individual steel

plates of straight and curved box girder bridges. In 1995, Yabuki et al. [146] developed a

numerical method to estimate the effect of the local buckling in plates and distortional

phenomenon on the non-linear response and ultimate strength of thin-walled curved steel

box girders. In 1998, Sennah [124] employed the finite element method to predict the

non-linear response of composite concrete-deck steel cellular bridges.

2.5 Load Distribution Factors

The distribution of dead load and wheel load on highway bridges is the most

important method in selecting the member size. Engineers can predict the bridge response

by applying the load distribution factor concept. Prior to 1959, design of concrete box-

girder bridges was based on the distribution factor approach in which individual 1-section

were assumed loaded with S/5 wheel lines of a standard H-series vehicle [3], where S is

the spacing (in feet) between centrelines of webs [41]. In 1959, California Design

Engineers [123] suggested to American Association of State Highway Officials

(AASHO) to change this distribution factor to S/7. A computer program was written by

Johnston and Mattock [78] in 1967 to study the lateral distribution of load in simple span

composite box girder bridges without transverse diaphragms or internal stiffeners. In

19


1968, Fountain and Mattock [56] implemented the folded plate method in a computer

program to calculate the lateral distribution of loads in 24 simply supported composite

multiple box girder bridges. The results of the folded plate computer program were

verified by testing one-quarter model of a two-lane, 24.4 m span bridge with three box

girders and one-fifth model of a two-lane, 30.4 m span bridge with two box girders, under

AASHTO truck loadings. The results obtained from the computer program were used to

develop an expression for the live load bending moment distribution factor for each

girder as a function of the roadway width and the number of box girders. The results from

the research program by Fountain and Mattock formed the basis for the lateral

distribution factors for bending moment currently adopted by AASHTO [3] and Ontario

Highway Bridge Design Code in 1983 [108] for multiple box girder bridges. The

application of the deduced expression, however, was limited to bridges having the ratio of

the number of lanes to number of boxes within 0.5-1.5. The results were obtained based

on ratio of relatively limited investigation of a number of bridges considering only the

number of lanes and number of boxes as variables. Most importantly, the curvature and

the continuity effects were not considered in the study. In 1969, Scordelis and Meyer

[123] published an extensive study of wheel load distribution in concrete box girder

bridges and developed formulas that included parameters thought to influence the load

distributions; i.e., span length, number of lanes, cell width, and number of cells.

In 1978, Heins [63] collected the detailed geometry of 82 bridges built until 1975.

Typical composite sections were constructed from these available data. A computer

program developed by Heins and Olenick [67] was utilized to obtain the response o f nine

braced and nine composite sections of horizontally curved multiple box girder bridges. In

1984, Heins and Jin [65] carried out a design-oriented research study on live load

20


distribution of curved composite I-girder bridges. A modification factor to the straight

girder moment distribution developed by Fountain and Mattock [56] was proposed as a

function of the radius of curvature. Bridges with span lengths of 15, 30 and 45 m only

were included. In 1980, Mukherjee and Trikha [99] developed a set of design coefficients

for twin cell curved box girder reinforced concrete bridges using the finite strip method.

These coefficients were for moment, shear, transverse moment, and vertical deflection

under the webs. However, these coefficients were limited only for bridges of two-lanes

with span length between 20 and 40 m, and radius of curvature between 45 and 150 m. In

1988, Nutt et al. [104] developed a set of equations for moment distribution in straight,

reinforced and prestressed concrete, multi-cell box girder bridges as a function of number

of lanes, number of cells, cell width, and span length. In 1989, Ho et al. [71] investigated

straight simply-supported, two-cell box girder and rectangular voided slab bridges using

the finite strip method. As a result of that research, formulas were deduced for the ratio of

the maximum longitudinal bending moment to the equivalent beam moment. However,

the application of the formulas was limited to straight two-cell bridge sections made of

either steel or concrete with span lengths of bridges up to 40 m in case of two-lane, 50 m

in the case of three-lane, and 67 m in the case of four-lane. In 1986, Brockenbrough [18]

derived load distribution factors using the finite element method for curved composite 1-

girder bridges as a function of the span length, radius of curvature, girder spacing, and

cross-bracing spacing. In 1985 and 1992, Bakht and Jaeger [9, 10] presented a particular

case of multi-spine bridges having at least three spines having zero transverse bending

stiffness, with the load transfer between the various spines through transverse shear. They

proposed simplified expressions for the load distribution factors for bending moment and

shear. These expressions formed the basis for the live load distribution factors used by the

OHBDC [109] for multi-spine bridges.

21


In 1992, Zokaie et at. [151] developed moment and shear distribution factors for

moment and shear for reinforced and prestressed concrete box girder bridges. The

proposed expressions were adopted by the current AASHTO LRFD [7] for straight

concrete bridges. In 1994, Noramandin and Massicotte [102] used the finite element

method in determining the distribution patterns for bending moment and shear force in

simply supported straight multi-spine box girder bridges. The effects of internal

diaphragms, external bracings, inclined webs and vehicle loading were studied for such

bridges. They concluded that the internal diaphragms contributed largely to the reduction

of the cross section distortion. However, the external bracing did not significantly

influence the distribution characteristics for bending moments and shear force. In 1995,

Cheung and Foo [29] studied the behaviour of simply supported curved and straight box

girder bridges subjected to OHBDC truck loading. The finite strip element was used in

the parametric study to develop expression for the moment distribution factors of such

bridges as function of span length, number of lanes, box spacing, and radius of curvature.

The effects of the number of boxes and dead load distribution were not included in the

study. Dean [42] in 1994 and Fu and Yang [57] in 1996 investigated the torsional

distribution on multi cellular members. In 1996, Brighton et al. [17] studied the live load

distribution for a new type of concrete double cell box girders proposed for a

prefabricated bridge system for the rapid construction of short-span bridges.

In 1997, Foinquinos et al. [55] studied the influence of intermediate diaphragms

on the live load distribution of straight multiple steel box girder bridges. The results

showed that using only two cross frames sufficed to redistribute the live load stresses and

adding more cross frames did not improve the distribution of live load. In 1998, Mabsout

et al. [91] presented finite element results of a study of the effect of continuity on the

22


wheel load distribution factors for I-girder bridges. Sennab [124] in 1998 and Sennab and

Kennedy [126, 128] in 1999 eonducted an extensive parametric study on curved simply

supported composite concrete deck-steel cellular bridges using the finite element analysis.

Empirical expressions for moment, shear and axial forces in the bracing system were

developed. Nour [103] in 2000 and Sennab et al. [131] in 2003 adopted the finite element

method to deduce empirical formulas for load distribution factors in curved composite

deck-steel multiple-spine box girder bridges. However, the proposed equations were

limited to simply supported box girder bridges. Sennab and Kermedy [129, 130]

presented a comprehensive literature review in analyzing of box-girder bridges.

The superseded version of the Ontario Highway Bridge Design Code [109] and

Canadian Standard for Design of Highway Bridges [21], OHBDC, draft [107], and

CHBDC [20] allow the treatment of a curved bridge as a straight one if the ratio (L^/bR)

is not greater than 1.0, where L is the span length, R is the mean radius of curvature and b

is half-width of the bridge. However, there are no expressions for the more common cases

where the above ratio is greater than 1.0. The superseded 1993 version of the AASHTO

Guide Specifications for Horizontally Curved Highway Bridges [4] ignored the curvature

effect in determining primary bending moments when the subtended angle did not exceed

5°. However, AASHTO 2003 [5] did not propose any other expression for load

distribution factors. Therefore, research work to investigate the live load distribution in

continuous curved concrete deck on multiple steel box girder bridges is required to

provide engineers with simplified method to design such bridges. This is the first

objective for the parametric study.

23


2.6 Impact Factors

The prediction of dynamic behaviour of horizontally curved box girder bridges is

of practical significance in the design of the bridge. As a result of the moving traffic

across the bridge, the stresses in the bridge elements may exceed those obtained

considering only the equivalent static or slow moving vehicle. The effect of the additional

load due to the dynamic response has been reported by numerous investigators since the

1970s. Extensive work has been undertaken on the dynamic analysis of straight bridges.

Comparatively very little work has been conducted to evaluate the impact factors of

continuous curved box girder bridges.

In 1968, Tan and Shore idealized horizontally curved bridge as slender curved

simply supported beams subjected to either a moving force [137] or simulated vehicle

[138]. In their studies, it was concluded that for a vehicle/bridge weight ratio of 0.3 or

less the response of the bridge can, for all practical purposes, be considered to be equal to

that given by solution for a constant moving force solution. In 1975, Rabizadeh and Shore

[1 1 2 ] used the finite element technique for the forced vibration analysis of simply

supported horizontally curved box girder bridges. Their dynamic analysis was conducted

on nine bridges and impact factors were obtained. The results formed the basis for the

impact factor adopted by AASHTO Guide Specifications for Horizontally Curved

Bridges [4]. In that study, the vehicle was simulated by two sets of concentrated forces

having components in the radial and transverse directions and moving with constant

angular velocities on circumferential paths of the bridge.

24


In 1972, Shore and Chaudhuri [133] analyzed a number of horizontally curved 1-

girder bridges under a moving vehicle. Both static and dynamic responses were obtained

and some tentative values for the impact factor for deflections, deck slab stresses, 1-girder

stresses, and support reactions were given. In 1981, Heins and Lee [6 6 ] presented the

experimental results obtained from vehicle-induced dynamie field testing of a two-span

continuous curved composite concrete deck-steel cell bridges located in Seoul, Korea. In

1984, Dey and Balasubramanian [44] studied the dynamic response of horizontally

curved bridge deeks simply supported along the radial edges under the action of a moving

vehicle and using the finite strip method.

In 1984, Billing [13] presented the results of dynamic tests of 27 bridges of

various configurations of steel, timber, and concrete construction and with span length

from about 5 to 122 m to determine the dynamic load allowance. The results from these

tests formed the basis for the dynamic load allowance adopted by Ontario Highway

Bridge Design Code, OHBDC second edition of 1983 [108] and Canadian Standard for

Highway Bridge Design, CAN/CSA-S6 - 8 8 [21]. The dynamic load allowance was plotted

against the first natural frequency of the bridge. However, the dynamic load

allowance/frequency relationship was revised in the third edition of OHBDC, 1992 [109]

as well as in the CHBDC 1997, and 2000 [107, 20] to be a constant value depending on

the number of axles. Akoussahet et al. [1] in 1997 used the three-dimensional finite

element modelling to study the vehicle-bridge interaction and dynamic amplification

factor for simply-supported reinforced concrete bridges. In 1985, Chang et al. [25]

predicted the seismic response of curved composite girder using the Rayleigh-Ritz

method.

25


In 1984, Cantieni [23] published test results on 226 beam- and slab-type highway

bridges conducted from 1958 to 1981. The bridges were tested dynamically through

passages of a single, fully loaded two-axle truck. The gross weight of the vehicle was 160

kN. The dynamic increments were calculated and plotted against the span length of the

investigated bridge. In 1985, O’connor and Pritchard [105] measured the dynamic

response of 137 vehicles on Six Mile Creek Bridge, Brisbane, Australia. Impact values

were calculated and plotted against computed bending moment and gross vehicle weight.

In 1987, Inbanathan and Wieland [77] presented an analytical investigation on the

dynamic response of a simply-supported box girder bridge due to a moving vehicle over a

rough deck. In 1991, Cheung and Megnounif [30] investigated the influence of

diaphragms cross bracings and the bridge aspect ratio on the dynamie response of a

straight twin-box girder bridge of 45 m span. In 1990 and 1992, Kashif and Humar [81]

and Kashif [80], respectively, developed a finite element technique to analyze the

dynamic response of simply-supported multiple box girder bridges considering vehicle-

bridge interaction.

Galdos [58] in 1988, Galdos et al. [59] in 1990 and Schelling et al. [120] in 1992

studied the dynamic response of horizontally curved multi-spine box girder bridges of

different spans. The two-dimensional planar grid analogy was used to model the box

bridges. The vehicle was idealized as a pair of concentrated forces with no mass,

traveling on circumferential paths with constant velocity. Results for the impact factors

formed the basis for those currently used by AASHTO Guide Specification for

Horizontally Curved Highway Bridges [5] for curved multi-spine box girder bridges. In

1992, Paultre et al. [ I l l ] concluded that the dynamic aipplification factors are related to

the fundamental frequency of the bridge. Among many other findings, they established

26


that the peak value of the dynamic amplification factor was not strongly influenced by

vehicle mass. Richardson and Douglas [116] in 1993 conducted a field test on a curved

highway overpass of box girder cross-section using simulated earthquake loads. In 1993,

Huang et al. [73] studied the impact behaviour of multiple vehicles moving across rough

bridge deeks on seven multi-girder concrete bridges with different span lengths. They

concluded that increasing the number of loading lanes increased the impact factors of

short-span bridges. However, the number of loading lanes had little influence on the

maximum impact factors of long-span bridges. In 1995, they investigated [74] the

dynamic response of curved I-girder bridges due to one or two truck loadings (side by

side). It was found that two-truck loading model was better than the one-truck loading

because the two-truck model dominated the maximum static responses at most sections of

the bridge. The one-truck loading might overestimate or underestimate the dynamic load

of the bridge. Moreover, they presented [75] a procedure for obtaining the dynamic

response of thin-walled beam finite-element model. In 1994, Chang and Lee [24]

discussed the effect of the vehicle speed and surface roughness on the impact factors for

simple-span highway girder bridges. They concluded that impact factors increased with

increasing vehicle speed and were almost constant with the bridge span length. In 1995,

Yang et al. [147] developed a new set of impact formulas for simple and continuous

beams subjected to moving vehicle loads. In 1996, Wang et al. [143] investigated the

variation of dynamic loading of nine girder bridges with different girder number and span

length due to several vehicles moving aeross rough bridge decks. In 1996, Wang et al.

[142] studied the free-vibration characteristics and the dynamic response of three-span

continuous and cantilever thin-walled single-cell box girder bridge when subjected to

multi-vehicle load moving across a rough bridge deek.

27


In 1997, Senthilvasan et al. [132] combined the spline finite strip method of

analysis and a horizontally curved folded plate model to investigate the bridge-vehicle

interaction in curved box girder bridges. They established that for curved box girder

bridges, if the mass of the vehicle is less than 35% of the mass of the bridge, the vehicle

can be considered as a moving load rather than a moving mass. Generally, the dynamic

response increased with the speed of the vehicle. Kim and Nowak [82] in 1997 presented

the procedure and results of field tests that were performed on two simply supported steel

1-girder bridges to assess girder distribution and impact factors. In 1998, Fafard et al. [51]

investigated the effect of dynamic loading on the dynamic amplification factors of an

existing continuous bridge. In 1999, Laman et al. [89] evaluated experimentally the

statistics of dynamically induced stress levels in steel through-truss bridge as a function

of bridge component type, component peak static stress, vehicle type, and vehicle speed.

In 2001, Huang [72] analyzed the impact of seven three-span continuous single

box girder bridges with overall span lengths ranging from 76.2 to 213.36 m due to

vehicles moving across rough bridge decks. In 2001, Zhang [149] conducted an extensive

theoretical study to examine dynamie response of simply supported curved composite

concrete deck-steel cellular bridges using the finite element analysis. Expressions for the

dynamic impact factors for moment, reaction, and deflection were deduced. In 2001,

Cheung et al. [35] described recent development in the vibration analysis of girder and

slab girders under action of moving vehicles or trains. Numerical results from analyzing

the entire bridge-track-vehicle system showed that the effect of vibrating track structure

on the dynamic response of the bridge was insignificant.

28


As a result of earlier studies, continuous curved concrete deck on steel multiple

box girder bridges gained very little attention to evaluate their dynamic response. Impact

factors used by the North American Codes were based on examining very limited number

of bridges, majority of which were not of that type. Therefore, expressions for impact

factors for curved bridges are required. This then forms the second objective of this study.

2.7 Fundamental Frequency

The vibration of box girder bridges occurring due to a moving load is a crucial

factor in the study of the dynamic characteristic of the bridge. Most of the previous

studies on the free-vibration analysis have been conducted on simply supported or straight

bridges, with very little information on curved box girder bridges. In 1966, Komatsu and

Nakia [85] studied the free vibration of curved girder bridge with I- or box girder cross-

section using Vlasov’s thin walled beam theory. In 1970, they [8 6 ] conducted a study on

forced vibration of curved single- and twin-box girder bridges using the fundamental

equation of motion. In 1967, Culver [38] established the natural frequencies of a

horizontally curved beam using the closed form solution for the equation of motion. In

1972, Cheung and Cheung [34] determined the natural frequencies and mode shapes of

undamped vibrations of curved or straight single-span beam-slab or box girder bridges

using the finite strip method. Tabba [136] in 1972 and Fam [52] in 1973 conducted free

vibration analysis of curved box girder bridges using the finite element analysis. Results

from testing curved two-cell box girder Plexiglas models were used to verify the method

proposed by them.

29


In 1979, Heins and Sahin [6 8 ] obtained the natural frequency of curved box

girders by utilizing a computer-oriented finite difference scheme. Study was conducted on

nine simply supported, nine two-span, and nine three-span bridges with span lengths

between 15 and 60 m. They used the finite difference technique to solve the differential

equations of motion based on Vlasov’s thin walled beam theory. In 1984, Dey and

Balasubramanian [44] evaluated the natural frequencies of a horizontally curved bridge

simply supported along the radial edges, using the finite strip method.

Cantieni [23] measured the fundamental frequencies of 226 beam- and slab-type

highway bridges in Switzerland. A relationship between the fundamental frequency of a

bridge and its maximum span length was determined through nonlinear regression. In

1985 and 1986, Mirza et al. [98] and Cheung and Mirza [31], respectively, investigated

experimentally and analytically the influence of bracing systems on the fundamental

frequency of composite concrete deck-steel twin-box girder bridge model continuous over

two spans, with varying depth at the intermediate support. In 1992, Kou et al. [87]

presented a theory that incorporates a special treatment of warping in the free-vibration

analysis of continuous curved thin-walled girder bridges. In 1997, Sennah and Kennedy

[125] conducted free and forced vibration analyses of simply supported curved composite

multi-cell bridges. In 1998, they [127] studied the free vibration of composite cellular

bridges continuous over two and three spans using the finite element model. Empirical

formulas for the dominant frequency were deduced for such bridges. In 2001, Zhang

[149] deduced empirical expressions for the fundamental frequency of simply supported

curved composite concrete deck-steel cellular box bridges using the finite element

method. In 2003, Samaan et al. [119] deduced expressions to estimate the fundamental

30


frequency of simply supported curved concrete deck on steel box girder bridges using the

finite element analysis.

Based on the aforementioned review, there seems to be no simplified method to

determine the fundamental frequency for continuous curved concrete deck on multiple

steel box girder bridges. This then is the third objective of the parametric study.

31


CHAPTER III

Finite Element Analysis

3.1 Introduction

Of all the available analysis methods, the finite element method is considered to

be the most powerful, versatile and suitable numerical tool to solve a complex continuum

problem. The method has become an important and frequently indispensable part of

engineering analysis and design. Recent development in computer technology makes it

possible to use finite element computer programs practically in all branches of

engineering. A complex geometry such as that of continuous curved concrete deck on

multiple steel box girder bridges can be readily modelled using the finite element

technique. The method is also capable of dealing with different material properties,

relationships between structural components, boundary conditions, as well as statically or

dynamically applied loads. The linear and nonlinear structural response of such bridges

can be predicted with good accuracy using this method.

In this chapter, the finite element procedure employed to reduce three-dimensional

physical bridges to lumped-parameter numerical models is summarized. The

commercially available finite element program “ABAQUS” was used throughout this

study to determine both linear and nonlinear behaviours of continuous curved concrete

deck on multiple steel box girder bridges subjected to static and dynamic loads as well as

their free vibration response. A brief description of the ABAQUS program as well as the

32


finite element modelling technique for various bridge components is presented in this

chapter.

3.2 Finite Element Technique

The finite element method offers a way to solve a complex continuum

problem by means of subdividing it into a series of simpler interrelated problems.

Essentially, it gives a consistent technique for modelling the whole structure as an

assemblage of discrete parts or finite elements. In other words, in the finite element

analysis, the structure is approximated as an assemblage of discrete finite elements

interconnected at nodal points on the element boundaries. The standard formulation for

the finite element solution of solids is the displacement method. The displacement-based

method of analysis is introduced in detail in many of the finite element literature [1 2 ,

152]. In this section, the method is only briefly presented.

Considering the equilibrium of a three dimensional structure, such as a bridge, the

structure is located in the fixed coordinate system X, Y, Z. The external loads applied to

the structure are considered to be Rb (forces per unit volume), concentrated loads Rc and

Rs (forces per unit surface area). The displacements of the structure from the original

configuration as a result of the externally applied loads measured in the X, Y, Z

coordinate system are denoted as U, where U represents the global nodal displacement

vector. From the stiffness matrices of the individual elements, the global stiffness matrix,

K, o f the complete element assemblage is obtained, where

K = (3.1)

33


where ki is the stiffness matrix for element i, The equilibrium equations for the system

become [1 2 ]

KU = R (3.2)

where R is a vector of forces acting in the direction of the displacements U.

The basis of the displacement-based finite element solution is the principle of

virtual work. The principle states that the equilibrium of a structure requires that for any

compatible small virtual displacements imposed on the body in its state of equilibrium,

the total internal virtual work is equal to the total external virtual work:

jsTdV = |u ,R ,d V + |U ,R ,d S + 2 ;u ;:R i: (33)V V S i

where U are the virtual displacements and e are the corresponding virtual strains.

Equation 3.2 is a statement of the static equilibrium of the element assemblage.

However, if the loadings on the structure are applied rapidly a dynamic problem needs to

be solved. Using d’Alembert’s principle [76], the element inertia forces may be included

parting the body forces. Considering the energy dissipation occurring during vibration,

the equilibrium equations become

M U +C U +K U =R (3.4)

where C is the damping matrix of the structure and M is the mass matrix of the structure.

In the case of linear analysis, the displacement is assumed to be infinitesimally small and

34


the material is linearly elastic. Also, the nature of the boundary conditions remains

unchanged during the application of the loads. Since the displacement is a linear function

of the applied load, the response is obtained directly by applying the loads.

However, in the nonlinear finite element analysis a step-by-step incremental

solution is required to calculate the structural response. While there are several methods

to solve nonlinear problems, ABAQUS applies either the Newton’s method or the BFGS

(Broyden-Fletcher-Goldfarb-Shanno) method as a numerical technique for solving the

nonlinear equilibrium equations. Newton’s method is the most frequently used iteration

scheme for the solution of nonlinear finite element equations. To reduce significantly the

computational cost of generating the stiffness matrix, the alternative form of Newton

methods can be applied. There are several methods known as matrix update methods or

quasi-Newton methods that have been applied for the solution of nonlinear systems of

equations. Among these methods, the BFGS method seems to be the most effective.

These methods provide a secant approximation to the matrix from iteration (i-1) to (i) by

updating the coefficient matrix or its inverse. The BFGS method provides a compromise

between the full-reformation of the stiffness matrix performed in the full Newton method

and the use of a stiffness matrix from a previous configuration as is done in the modified

Newton method. In general, the rate of convergence of the quasi-Newton method is

slower than the quadratic rate of the convergence of Newton’s method, although it is

faster than the linear rate of convergence of the modified Newton method.

An incremental solution strategy based on the iterative methods is considered to

be effective if realistic criteria for the termination of the iteration are used. In a nonlinear

problem it is almost impossible to have a tolerance value of zero. Therefore, a realistic

35


value for the tolerance is required. By default, in ABAQUS this tolerance value is set to

be 0.5% of an average force in the structure, averaged over time. A second convergence

criterion must be satisfied before the program accepts the solution. The last displacement

correction is also checked. For each iteration in a nonlinear analysis the finite element

program forms the model’s stiffness matrix and solves a system of equations which is

equivalent in computational cost to conducting a complete linear analysis for the system.

3.3 Finite Element Program “ABAQUS”

ABAQUS [70] is a powerful engineering simulation program based on the finite

element method that can solve linear and nonlinear problems. The finite element program

contains an extensive library of elements that can model almost any arbitrary structure

geometry. The program has a large list of material models that can simulate the behaviour

of most engineering materials, such as steel and reinforced concrete. The ABAQUS

analysis modules are batch programs; therefore, an input file that describes a problem

must be assembled so that ABAQUS can provide an analysis. An input file for ABAQUS

contains model data and history data. Model data describes a finite element model: the

elements, nodes, element properties, material definitions and any data that define the

model itself. The required model data are the finite element model geometry and the

material definitions. History data define the sequence of events or loadings for which the

model’s response is sought. The required history data that must be included in an input

file are response type, linear or nonlinear, static or dynamic. There are also optional

history data such as loading, boundary conditions, and output control. This history is

divided into sequence of steps in the input file. Each step is a response of a particular

static or dynamic response. Static loading cases might be applied in several steps so that

36


the required output requests can be obtained. When the analysis is completed, several new

files that contain the results and any error or warning messages are created. The basic

coordinate system in ABAQUS is a right-handed, rectangular Cartesian system. However,

the program provides the *TRANSFORM option to choose other local systems for output

of nodal variables and point load or boundary and for material specification the

* ORIENTATION option can be used.

A basic concept in ABAQUS is the division of the problem history into steps. For

each step, an analysis procedure must be selected. This choice defines the type of analysis

to be performed during the step whether static or dynamic stress analysis. ABAQUS

provides solution procedures for analyzing linear or nonlinear response. In nonlinear

problems ABAQUS will increment and iterate as necessary to analyze a step, depending

on the severity of the nonlinearity. In most cases, ABAQUS offers two options for

controlling the solution: automatic time incrementation or user-specified fixed time

incrementation. Automatic incrementation is recommended for most cases. Direet user

control can sometimes save computational cost in cases where the user is familiar with

the problem and knows a suitable incrementation scheme. Direct control can also

occasionally be useful when the automatie control encounters trouble with convergence in

nonlinear problems. In spite of the fact that modified Riks algorithm is assumed to work

well in nonlinear static problems involving collapse behaviour, the algorithm shows

difficulty when dealing with structures containing reinforced concrete elements. In such

cases, cracks in the reinforced concrete element due to the tension forces produce

instability in the structural response and the analysis is automatically terminated before

reaching the ultimate load of the structure as a result of the failure and instability of only

some elements in the model. Thus, in this research work the modified Riks method did

37


not work well in estimating the ultimate load of the prototype bridge models. Instead, a

quasi-static analysis method using ABAQUS/explicit was adopted to predict the

nonlinear response of the bridge models.

3.4 Dynamic Analysis

A dynamic simulation is one that includes the effects of the mass and damping.

ABAQUS offers several methods for dealing with the dynamic analysis of a structure. In

linear problem, model superposition method or direct integration method can be used.

There are two types of direct integration analysis available in ABAQUS, namely; implicit

direct integration method and explicit direct integration dynamic analysis. To extract the

natural frequencies and the corresponding mode shapes, a frequency extraction procedure

can be carried out. In the following sections, a brief description of these methods is

provided.

3.4.1 Natural Frequency Extraction

ABAQUS provides *FREQUENCY option to perform a natural frequency

extraction. The *FREQUENCY procedure applies the eigenvalue techniques to extract

the frequencies of a given structure. The general form of eigenvalue problem for the

natural frequencies of a lumped finite element model is [1 2 ]

( - r a ^ M y + K ' Q t b j = 0 (3 .5)

38


where M*-* is the mass matrix (which is symmetric and positive definite) ; K'-" is the

stiffness matrix; is the eigenvector; and i and j are degrees of freedom; co is the circular

frequency.

ABAQUS offers Lanczos and Subspace iteration eigenvalue extraction methods.

The Lanczos method is generally faster when a larger number of eigenmodes is required

for a structure with many degrees of freedom. The subspace iteration may be faster when

only a few eigenvalues are required. ABAQUS extracts eigenvalues until either the

required number of eigenvalues has been extracted or the last frequency extracted exceeds

the maximum frequency of interest. In extracting the required natural frequencies and the

corresponding mode shapes for bridges, Subspace iteration method was adopted in this

work. For this method, ABAQUS automatically calculates the participation factor and the

effective mass for each mode. The eigenvectors are normalized by-default so that the

largest displacement entry in each vector is unity. In the case of torsional modes where

the displacements may be negligible, the eigenvalues are normalized so that the largest

rotation entry in each vector is unity.

3.4.2 Transient Modal Dynamic Analysis

A modal dynamic analysis is performed in ABAQUS by using the *MODAL

DYNAMIC procedure. This method is used to analyze transient linear dynamic problems

using modal superposition; it can only be performed after a frequency extraction

procedure since it predicts the structure’s dynamic response according to the extracted

natural modes of the problem. The method is a very popular dynamic analysis technique

but it has several important limitations. The method is only valid for linear systems and

39


damping in the system must be proportional. The mode superposition procedure is most

useful when the system response can be accurately estimated by considering only a

relatively small subset of all the vibration modes for the system, which is the case in most

structural systems. Thus, the *MODAL DYNAMIC procedure can be much less

expensive computationally than the dynamic analysis using the direct integration method.

The *MODAL DYNAMIC option provides the time period of the analysis and the time

increment to be used. The *MODAL DAMPING option is often used in conjunction with

a modal dynamic analysis to describe the damping of the system.

Concentrated nodal or distributed pressure or body forces can be applied to the

structure in the modal dynamic analysis. The *AMPLITUDE option can be utilized to

define arbitrary time variation of a load given throughout a step. In order to simulate a

truck moving over a bridge deck, the amplitude of truck load must be specified with time

at each nodal point. The computer program interpolates linearly between these given

values of the time increment in the analysis. The dynamic response of the structure due to

the applied load-time history can be obtained in the form of displacement-, velocity-,

acceleration-, and stress-time histories.

3.4.3 Implicit Direct Integration Method

General linear and nonlinear dynamic responses can be evaluated using the

implicit time integration method. In this method, the equation of motion for a general

system is integrated using a numerical step-by-step procedure. Thus, the system

differential equations are integrated directly in a coupled form, as they exist in the

physical coordinates. Dynamic integration operators are mostly described as implicit or

40


explicit. The implicit direct integration operator used in ABAQUS is called Hiber-

Hughes-Taylor operator; it is an extension of the trapezoidal rule [70]. Implicit schemes

solve for dynamic quantities at time t+At based not only on values at t, but also on these

same quantities at t+At. In the implicit method, the integration operator matrix must be

inverted and a set of simultaneous dynamic equilibrium equations must be solved at each

time increment. The main advantage of Hilber-Hughes-Taylor operator is that it is

imconditionally stable for linear systems. The direct time increment can be automatieally

provided by ABAQUS or speeified by the user. Artificial damping can be introduced by

the ALPHA parameter on the *DYNAMIC option. The parameter values vary from 0,

which gives no artificial damping, to -0.33, which provides the maximum artifieial

damping available for this operator. At the maximum level ALPHA gives a damping ratio

of about 6 % when time increment is 40% of the period of oscillation of the mode being

studied. Therefore, this artifieial damping is never very substantial for realistic time

increments. The moving loaded truck across the bridge deck can be simulated by using

the *AMPLITUDE option or by writing a subroutine *DLOAD to deseribe the load-time

histories at each nodal point on the bridge deek. A typieal input file for the dynamic

analysis using a user-subroutine is given in Appendix A.

3.5 Explicit Dynamic Analysis

The explicit method is well-suited to solving quasi-static analysis with

complicated nonlinear structural behaviour. The explicit dynamic procedure performs a

large number of small-time increments efficiently. The explicit central difference operator

satisfies the dynamic equilibrium equations at the beginning of the increment, t. Then, the

accelerations calculated at time, t, are used to advance the velocity solution to time t+At/ 2

41


and the displacement solution to time t+At. The use of diagonal element mass matrices is

the key to the computational efficiency of the explicit procedure, where the inversion of

the mass matrix required at the beginning of the increment is simple to compute. The

central difference operator is conditionally stable. A small amount of damping is

introduced in the analysis to control high frequency oscillations. The time incrementation

scheme in explicit analysis is fully automatic. The central difference method is required to

be integrated at a time step, At, smaller than a critical time step, Atcr- The critical time

step, Atcr, can be evaluated from the mass and stiffness properties of the complete

structure, where Atcr can be obtained as [70]

At„ = i (3.6)7t

where T„ is the smallest period of the finite element assemblage with n degrees of

freedom. In ABAQUS, the stable time increment is given by [70]

At < + ) (3.7)

where is the fraction of critical damping in the mode with the highest frequency. Hence,

the cost of the analysis may be very expensive in cases where the total analysis time is

high.

This method was adopted in predicting the collapse load of the bridge. While this

method is a dynamic analysis method, only the ultimate static load is sought; the bridge

was loaded slowly enough to eliminate any significant inertia effects. The prototype

42


bridge deck slab was loaded by applying a velocity that increased linearly from 0 to 40

mm/s. Such very low load rate was selected to ensure quasi-static solution. The

computational cost of this solution is relatively high, however, the results obtained from

this method showed good agreement with the experimental findings as described in

Chapter V.

3.6 Finite Element Modelling of Bridge Models

The finite element technique was used to model continuous curved concrete deck-

on multiple steel box girder bridges. Three-dimensional finite element model was

constructed in a way to simulate the actual structural geometry, boundary conditions, and

material properties of the bridge components namely: reinforced concrete deck slab, steel

webs, steel bottom flange, steel top flange, diaphragms, cross bracings, and top chords.

The reinforced concrete slab was fully constrained to the steel top flanges by means of

shear connectors.

In this section, the element types selected for each component, the material

modelling, and the boundary conditions are described. The model presented herein was

verified and substantiated by results gathered from the experimental values from two

continuous composite box girder bridge models tested under several loading cases as

shown in Chapter V.

43


3.6.1 Material Modelling

The material library in ABAQUS allows the modelling of the material used in the

bridges studied. The bridge slab is made of reinforeed eonerete while the rest of the box

girder is made of steel. In the input file, eaeh material definition starts with a

*MATERIAL option. The material option bloeks define the behaviour of a partieular

material throughout the analysis.

3.6.1.1 Modelling of Steel

In the elastie analysis, only the modulus of elasticity, Poisson’s ratio, and the

material density are required to be given in the input file. The density is used in case of

dynamic analysis or to calculate the gravitation loads of the studied bridges. The

*PLASTIC option must be defined, wherever the plastic behaviour of the steel is needed

in the analysis. Since the steel is assumed to be perfectly plastic, only the yield stress

must be given in the *PLASTIC option. This classical metal plasticity model uses von

Mises yield surface with associated plastic flow for isotropic metal behaviour. The von

Mises surface assumes that the metal yielding is independent of the equivalent stress and

it is used to define isotropic yielding. It is defined by giving the value of the uniaxial yield

stress as a function of uniaxial equivalent plastic strain. The true stress-true strain must be

used in defining the plasticity data in ABAQUS. Figure 3.1 presents the elastic-perfectly-

plastie stress-strain relationship assumed for steel in the finite element analysis.

44


3.6.1.2 Modelling of Reinforced Concrete

The reinforced concrete bridge deck slab can be modelled in ABAQUS by

defining the concrete model using * CONCRETE option and the reinforcement in the

concrete by the *REBAR option. The ^TENSION STIFFENING option is required in the

concrete model. This option allows for the effects of reinforcement interaction with

concrete to be simulated in a simple manner where the load is transferred aeross the

concrete cracks through the rebar.

3.6.1.3 Concrete Model

The concrete model in ABAQUS is intended to model plain concrete element or

with the *REBAR LAYER option to model reinforced concrete elements. The most

essential aspect of modelling the concrete behaviour is cracking, which dominate the

concrete model under loading. Once the stress in the concrete reaches a failure surfaee,

called crack detection surface, cracks occur. Cracks in the concrete model are

irrecoverable but they may open and close during the calculation. The concrete model

does not track each individual crack. Instead, the model is considered a smeared crack

model by forming constitutive calculations independently at each integration point of the

finite element model. The presenee of cracks is introduced in the calculations by

considering their effect on the stress and material stiffness associated with the integration

point.

In the case of reinforced concrete model, the strain-softening behaviour for

cracked concrete can be identified by using the setting TYPE parameter equal to STRAIN

on the *TENSION STIFFENING option. In the case of the reinforced concrete model, it

45


is recommended that each element contain a rebar to reduce the mesh sensitivity. The

tension stiffening value must be assumed. It is assumed that strain softening after failure

reduces the stress linearly to zero at a total strain of about 1 0 times the strain at failure.

Figure 3.2 illustrates the simulation of the influence of the rebar in the concrete model

implemented in the finite element model. The failure stress, a “ , occurs at a failure

strain, ef*.

The concrete reveals initially an elastie response when it is loaded in compression.

As a result of increasing the stress in the concrete beyond the elastic region, inelastic

straining takes place and the response of the concrete softens. When the ultimate stress is

reached the material softens such that it can no longer carry any stress. At some point

after inelastistic straining has occurred and the model is unloaded, the reduction in the

model stiffnesses is ignored in the model. If the model is loaded in tension under uniaxial

load, cracks form at a stress corresponding to, typically, 7-10% of the ultimate

compression stress. Figure 3.3 shows that cracking and compression responses of

concrete are integrated in the model by the uniaxial response of a specimen. For the

purpose of developing the model it is assumed that the material loses strength through a

softening mechanism and that this is a dominantly a damage effect in the sense that open

cracks can be represented by loss of elastic stiffness. It is also assumed that cracks are

allowed to close completely if the stress across them becomes compression. In multiaxial

stress states observations can be implemented through the concept of surfaces of failure

and the ultimate strength in the stress space. The computer program defines these surfaces

as shown in Figure 3.4, fitted to experimental data. The *FAILURE RATIOS option can

be used to define the shape of the failure surface. The model is based on the classical

concepts of theory of plasticity.

46


In ABAQUS/explicit, the brittle cracking model is used for the concrete structure.

The *BR1TTLE CRACKING option is used to define the concrete model. The brittle

cracking model allows removal of elements with *BR1TTLE FAILURE option. In this

model, the postfailure behaviour for direct straining across cracks is modeled with the

^BRITTLE CRACKING option.

3.6.1.4 Rebar Model

In ABAQUS, the *REBAR LAYER option is used to define the reinforcement in

the concrete. The rebars are treated in the model as one-dimensional isoparametric

elements. These elements are superposed on the mesh of the plain concrete elements. The

standard metal plasticity model shown in Figure 3.1 is assumed to deseribe the behaviour

of the rebar material. Adopting this model approach, the concrete behaviour is considered

independently of the rebar. This option can model double layers of the rebar in the

longitudinal direction and double layer of the rebar in the transverse direction. The area of

each rebar, the offset of the mid surface in shell element and the spacing can be defined in

the model. In ABAQUS an equivalent smeared orthotropic layer is assumed. The

equivalent thickness of the smeared layer is equal to the area of the rebar divided by the

rebar’s spaeing. The rebar can be also placed in the radial and tangential directions, as

required in the case of curved deck slabs.

3.6.2 Geometric Modelling

A three-dimensional finite element model was created to simulate each bridge

studied. Three-dimensional shell elements were selected to model the reinforced concrete

47


deck slab, steel webs, steel bottom flanges, and steel end-diaphragms. For the steel top

flanges, top chords and cross-bracing, three-dimensional beam elements were chosen in

the finite element model. The connections between the reinforced concrete deck slab and

the steel top flanges were idealized using the *MPC option. Figure 3.5 illustrates a typical

idealized cross-section of a bridge.

ABAQUS offers a wide variety of shell elements for stress/displacement analysis.

A four-node doubly curved general-purpose shell element called S4R was adopted in

modelling the required bridge components. The element can idealize either straight or

curved boundaries depending on the node definition. The element has six active degrees

of freedom at each node; three displacements (Ui, U2, U3) and three rotation ((j)i, ^2 , <j)3).

The general-purpose elements are suitable in all loading conditions for thin and thick

shell elements and provide robust and accurate results. The element allows transverse

shear deformation. They use thick shell theory as its shell thickness increases and become

discrete Kirchhoff thin shell element as the thickness decreases. When a shell element is

made of the same material throughout its thickness, the element is considered a thick

element when the thickness is more than 1/15 of a characteristic length on the surface of

the shell. This characteristic length is the span for a static analysis and the wavelength of

a significant natural mode for the dynamic analysis. This element type uses the reduced

integration to form the element stiffness. However, the matrix and distributed loadings

are still integrated exactly. Reduced integration usually provides accurate results and

significantly reduces the computational cost. The shell element accounts for finite

membrane strains and will allow for change in thickness. Figure 3.6 shows a detailed

description of the shell element S4R.

48


The three-dimensional two-node linear interpolation beam element called B31H

was used to model the steel top flanges, top-chords and cross bracing. The beam element

reduces the problem to one-dimensional problem mathematically and therefore, the

computational time is reduced. The Timoshenko B31H element allows for transverse

shear deformation. The element has two-nodes with six degrees of freedom at each node,

three displacements (Ui, U2, U3) and three rotations ((j)i, (j>2, ^3 ). This hybrid element is

well suited to handle very slender components, where the axial stiffness of the member is

very large compared with its bending stiffness. In this case, the element is considered to

be loaded mainly in tension and compression. The element defines the orientation of the

beam, whether it is straight or curved. Figure 3.7 shows a detailed description of the beam

element chosen for the bridge models.

The *MPC option in ABAQUS allows constraints to be imposed between

different degrees of freedom of the model. The multi-point constraint option was adopted

to simulate the connection between the concrete slab and the steel top flange. This option

is used to ensure full interaction between the concrete deck slab and the steel box girder.

Thus, MFC type is used to model the shear connectors between two nodes. This type is

sorted internally by ABAQUS so that the MFC is imposed by eliminating the degree of

freedom at the first node given. Thus the first node in the MFC option becomes a

dependent node on the last node defined in the option. Therefore, both nodes produce the

same degree of freedom.

49


3.6.3 Boundary Conditions

There were two different boundary constraints considered in modelling the

continuous curved concrete deck on multiple steel box girder bridges: the roller support

and the binged support. The *BOUNDARY option was used in ABAQUS to prescribe

both boundary conditions for the analysis. The roller support was modeled by releasing

the horizontal movements of the node in the required directions. However, the binged

support was constrained from any horizontal movements. All supports were constrained

in the vertical direction, but allowed to rotate around the support line. In the case of

curved bridge models, the tangential, radial, and vertical support arrangements were

adopted. The support conditions were applied at the lower end nodes of each web, at the

outer and internal support lines, as shown in Figure 3.8.

3.7 Finite Element Analysis of Bridge Models

Various finite element meshes were eonstrueted and compared to select the most

suitable mesh for the linear and nonlinear analyses. In idealizing bridges, mesh

convergence was investigated first, by means of several pilot runs. Figure 3.9 illustrates

the final finite element mesh used in the static and dynamic analyses of twin-box girder

bridges. Two elements on each side of the boxes and four elements between the webs

were used in the transverse direction for all the bridges. In the longitudinal direction, the

total number of elements varied depending upon the span length of the bridge. The

number of elements was 72 for 20-m span bridge, 144 for 40-m span bridge, 432 for 60-

m span bridge, 576 for 80-m span bridge, and 720 for 100-m span bridge. The webs and

the end-diaphragms were simulated using six elements in the vertical direction for all

50


bridge models. The mesh proved to be adequate for the static and dynamic analyses. The

chosen mesh was quite adequate to accommodate different truck loading cases, as

described in Chapter VI. The total number of elements used to model the bridges varied

from around 4,000 in the case of bridges with span length of 20 m, to 106,000 in the case

of bridges with span length of 100 m. For curved bridge models, the generation of the

elements were in the radial and tangential directions. The aspect ratios of the element

used for the concrete deck slab and the bottom flanges ranged from 1.0 to 1.9 for all

bridge models. The aspect ratios of the shell element for the webs and end-diaphragms

ranged between 1.2 and 2.1 for all the bridge models. However, for bridges with span

length of 20 m, this aspect ratio was 4.

A sensitivity study was conducted to examine the effect of vertical web stiffeners

on the overall structural behaviour of the bridges. The study showed that these stiffeners

had an insignificant influence on the linear response of the bridge structure. Also, it was

established from the pilot runs that the steel reinforcement in the concrete deck slab had

only marginal effects on the bridge elastie response in the static and dynamic analyses.

However, in the nonlinear analysis the effect of the reinforcement was considerable in

predicting the ultimate load for the bridge models. The cross bracing members were

modeled to connect the points at the comer of the box girders. The connecting gusset

plates were ignored. The top chord members were idealized at the same level as the top

steel flanges. The shear connectors were considered to be at the element nodes over the

top steel flanges. The finite element modelling was conducted using a well-established

technique adopted by Sennah [124]. The finite element model was verified using static

equilibrium checks. In addition, the model was substantiated and validated by results

obtained from testing two continuous concrete deck-steel two-box girder bridge models

51


discussed in Chapter V. The finite element model was then adopted to conduct extensive

parametric studies for static and dynamic responses of continuous curved composite box

girder bridges. Typical input data decks employed in the analysis are given in Appendix

A.

52


CHAPTER IV

Experimental Study

4.1 Introduction

The presence of continuity and curvature in multi-box girder bridges add

considerably to their complex structural behaviour. Due to torsional moments, stresses

and deformations developed in such bridge members are significant. Experimental studies

can provide design engineers and Specification writers with an insight to their response to

loads. A few experimental studies have been undertaken to verify the elastic response of

box girder bridges. However, experimental investigations on continuous curved concrete

deck on steel multiple box girder bridges at construction phase, service and ultimate load

stages as well as under free vibration conditions are yet unavailable. In this research, an

experimental study has been conducted on two continuous, twin-box girder bridge models

to achieve the following main objectives:

(1) To establish accurate experimental data base and compare it with those

obtained from the finite element analysis.

(2) To investigate the elastic response of continuous curved multi-box

girder bridges at the construction phase and under service loading

conditions.

53


(3) To determine the free-vibration response of such bridges.

(4) To use the data gathered experimentally from the inelastic structural

response of the bridge models to validate a finite element model

capable of predicting the structural behaviour of the prototype bridges

up to failure.

(5) To determine the collapse load of such bridges and compare the results

to those predicted by the analytical solution.

The experimental program is described in details in this chapter. Bridge models,

geometry, material properties, instrumentation and loading cases are presented. Views of

the model test equipment and test set-up are also presented.

4.2 Description of Bridge Models

In general, full-scale experimental investigation would entail high

expenditures. The availability of suitable testing equipment and space are the important

consideration in choosing suitable length scale factor. Mirza [97] and others have shown

that models erected to scales as small as 1/6 to 1/8 can be reliable, time-saving and

relatively inexpensive. Based on the previous considerations, a length scale factor 1/8 was

selected. The aim of this experimental study is to compare its results with the ones

obtained from the finite element method for the same model with the same linear scale

length. Thus, for practical reasons a larger linear scale length was used for the thickness

54


of the steel box plates and reinforced concrete deck, in order to avoid difficulties in the

fabrication process.

The concrete deck was constructed without the curbs for simplification and to

match the analytical model used. Care was taken and special procedures were followed to

minimize the amount of possible deformation of the steel box section during fabrication

as a result of welding. The depth of the steel box section was maintained constant at 150

mm throughout for both the straight and curved bridge models.

Two continuous two-box girder bridge models were constructed. The first model

was a straight, while the second one was curved in plan with span-to-radius of curvature

ratio of 1. Figures 4.1 and 4.2 show plans for the straight and curved bridge models,

respectively. The thickness of all steel plates used to construct the steel box sections was

3 mm. Each box girder consisted of a bottom flange 7300-mm long and 270-mm wide,

two top flanges each 7300-mm long and 47-mm wide and two webs each 7300-mm long

and 144-mm deep. At the end support lines and the interior support line only, diaphragms

with access holes were welded inside the boxes, while cross bracing were provided

between the boxes. Also, cross bracings were used inside and between the boxes in both

models at equal intervals between the support lines. Cross bracing members were made of

20 X 3 m m rectangular cross section. It should be noted that these cross members were

installed in the radial direction in the curved model at equal intervals. C-channel section

was used as shear connectors to provide full interaction between the concrete deck and

the steel box section. Shear connectors with a length of 25-mm were placed at 100-mm

intervals, being in the radial direction. Four layers of steel reinforcement wires were

placed in the longitudinal and transverse directions, in the case of the straight bridge

55


model, and in the tangential and radial directions in the curved model. The 3-mm

diameter steel wires were spaced at a distance of 100-mm in both directions with 5-mm

eonerete cover. Typical cross sections are presented in Figure 4.3 for both bridge models.

4.3 Model Materials

4.3.1 Steel

The steel boxes were fabricated using 10 gauge steel plates for the bottom flanges,

webs, top flanges, cross bracing and diaphragms. To obtain the steel material properties,

three tensile coupon tests were fabricated from the steel sheets. Each coupon was tested

under uniaxial tension load up to failure using a 600-kN Tinus Olsen Universal Testing

Machine shovra in Figure 4.4. From the test results, the nominal stress and strain were

obtained. When defining plasticity data needed later on in the finite element software

program, true stress and true strain must be provided. True strain is defined as

^ true (4.1)

where L is the current length and Lo is the original length. The calculated stress that is

conjugate to the true strain is called true stress, atme, defined as

^true Ix Jwhere F is the applied force and A is the current cross sectional area.

56


Nominal strain can be expressed as

L - L o LSno m in a l = “ j— ^ = ~ ~ 1 ( 4 .3 )

L q L q

The relationship between true stress and nominal stress is formed by considering

the following equation:

Lo Ao = L A (4.4)

Carrying out the necessary substitutions provides the relationship between true

strain and nominal strain and true stress and nominal stress, as follows:

^ t ru e ~ n o m in a l ) ( 4 -5 )

^ t r u e “ ^ n o m i n a l ( ^ " ^ ^ n o m i n a l ) ( 4 -6 )

The average relationship between the true stress and true strain for the steel used

to build the box girders is depicted in Figure 4.5.

4.3.2 Concrete

Ready mix concrete was donated by CBM Company for the experimental

program. Smooth limestone 10 mm was used in the concrete mixture with 125-mm

slump. Water-cement ratio and aggregate-cement ratio were selected to obtain a nominal

compressive strength, f c = 40 MPa.

57


Four standard cylinders 100 x 200 mm and four standard cylinders 150 x 300 mm

were sampled concurrently with the casting of the concrete slab. The cylinders were

placed in the same condition as the concrete deck slab. The cylinders and the concrete

slab were cured for at least two weeks under the same room temperature. All cylindrical

specimens were tested under uniaxial compressive load using the 1350 kN Riehle

Compression Testing Machine, as shown in Figure 4.6. Mechanical Dial gauges were

installed on the concrete specimen to measure displacement. The average strength as well

as Young’s moduli of elasticity of the concrete samples are listed in Table 4.1 and a

typical stress-strain relationship is presented in Figure 4.7.

4.3.3 Steel Wire Reinforcement

Three-millimetre diameter steel wires were used as steel reinforcement at the top

and the bottom of the concrete deck in the longitudinal and transverse directions. Three

specimens were tested under uniaxial tension load up to failure. After obtaining the load-

displacement relationship from the laboratory tests, the average true stress-true strain

results were calculated and are shown in Figure 4.8. The yield strength and modulus of

elasticity were found to be 800 MPa and 208 GPa, respectively.

4.3.4 Shear Connectors

To ensure full interaction between the concrete slab deck and the steel section, 25-

mm deep, 10-mm flange width chaimel-sections were used as shear connectors. The

channel section was cut into pieces 25 mm in length. Marks on the top flanges of box

girders at 100 mm were drawn to weld the shear connectors to the top flanges. Three

coupons were cut from the channel section and the average true stress-true strain

58


relationship is presented in Figure 4.9. The shear connectors were positioned on all top

flanges and along the entire length of both bridge models. A view of the shear connectors

welded to the steel top flanges is shown in Figure 4.10.

4.4 Model Construction

4.4.1 Fabrication of Open Steel Section

Steel boxes forming both bridge models were fabricated using the same steel

plates for the bottom flanges, webs, top flanges, cross bracing and diaphragms. Top

flanges and webs were first formed from the flat steel sheets. Then, the webs were

clamped in position with the top flanges and tack welded at discrete locations to reduce

the amount of possible distortion during the welding process. To minimize the heat

generation during the welding process, a medium-heat welding machine was used.

To make it possible to weld the cross bracing and the diaphragm inside the steel

boxes, they were welded to the top flange-web members first before welding the bottom

flange of the box. Then, bottom flanges cut from flat steel sheets were clamped to the

webs in position. Spot welds were used to ensure the right dimension and to stabilize the

cross-section during the welding process. Shear connectors were placed and then welded,

as illustrated in Figure 4.10, on the top flanges and the fabricated open steel box girder

was completed. Continuous welding was provided at the interface of channel shear

connectors. Finally, each box girder was carried and positioned in its testing place on the

supports. A view of the straight bridge model after welding is shown in Figure 4.11. The

two box girders were then connected using the cross bracing members between the boxes.

59


4.4.2 Model Supports

The interior support in both models was placed in the centre of the structural rigid

portal frame available in the structural laboratory. The bridge model support conditions

were designed to simulate the interior bearings as a hinged support and the outer bearings

as roller supports. In the case of the curved bridge model, the outer bearings were released

in the tangential direction. However, at the interior support the bridge model was

prevented from movement in the tangential direction by welding a steel rod just beside

the roller underneath each web. At each support line, the bridge model was tied down

using a tie-down system to prevent the bridge models from uplift movements.

4.4.3 Concrete Formwork

After placing the steel box girders in the testing position under the laboratory

structural frame, the concrete formwork was prepared. Twelve-millimetre plywood sheets

were installed under the steel box girders and supported to the laboratory floor by 100 x

100 mm wooden struts. To form the slab overhang and the slab between the steel box

girders, strips of styrofoam sheets 5 and 10 mm thick were used on the top of the plywood

to form the bottom surface of the concrete deck. One hundred-millimetre of styrofoam

sheets were installed inside the steel boxes with intermediate supports made of small

pieces of these sheets. It should be noted that the styrofoam strips inside the boxes could

not have been removed after easting the eonerete. However, due to their negligible

stiffness, they would not affect the structural performance of the model. Styrofoam

sheets, 40 mm thick, were cut to form the edge of the concrete slab. In the case of the

curved bridge model, small pieces of styrofoam were used to support a very thin flexible

60


wood strip of 40 mm depth, to obtain a well-formed curved edge of the concrete slab. A

view of the formwork for the curved bridge model is presented in Figure 4.12.

4.4.4 Reinforcing Steel Wire

Two meshes of reinforcing steel wires were placed at a distance of 100 mm in

both directions near the top and bottom surface of the concrete slab. To ensure the

stability of the wire mesh during casting the concrete slab, the wires in the transverse

direction were first placed and tied to the bottom of the shear connectors. On the top of

the transverse wires, the longitudinal wires were put on place and held to the transverse

wires. Figures 4.13 and 4.14 show the wire meshes in straight and curved bridge models,

respectively. Due to the flexibility of the steel wire, it was possible to form the steel wires

in the tangential direction in the curved bridge model easily.

Similar procedures were followed to place the top steel wire layer. First, the

transverse wires (the radial in the curved bridge model) were held to the top of the shear

connectors and then the longitudinal wires (the tangential in the curved bridge model)

were placed on top of them. Small pieces of wires were used to ensure a proper cover at

the bottom of the concrete deck and were observed to make sure that they were not

displaced during casting the concrete.

4.4.5 Casting the Concrete Deck

Casting the concrete deck required extreme care in order to prevent the flexible

vsdre meshes or the styrofoam from displacing. Concrete was carried from the mixer to the

end of the bridge and carefully shovelled onto the bridge model. The concrete was

61


compacted manually, where the thickness of the concrete deck was only 40 mm. Then,

the concrete surface was finished with a wooden screed from one end of the bridge and

continued to the other end. To ensure a well-finished surface a final troweling with a steel

screed was made. Later, the concrete was moist-cured for two weeks along with the

concrete cylinders sampled at the time of casting the concrete deck slab, as shown in

Figure 4.15.

4.5 Instrumentation

4.5.1 Strain Gauges

Strain gauges were installed at three cross-sections along the bridge model,

namely: The middle of the first span, the interior support, and the middle of the second

span. Electrical strain gauges type N11-FA-10-350-11 (Showa Co., Ltd) were used at

thirty-two different locations along the bridge model. The length of each strain gauge was

10 mm, with a resistance of 350 + 0.3 % ohms. The gauge faetor varied from 2.14 to 2.15

± 1 %. All the steel strain gauges were placed in the longitudinal direction of the bridge

models. The installed strain gauges on the bottom flange of a steel box at the mid-span

are shown in Figure 4.16.

Eight concrete strain gauges of type N2A-06-20CBW-350 (Showa Co., Ltd) were

used on eaeh bridge model. The length of the concrete strain gauge was 50 mm, with a

resistance of 350. + 0.3% ohms. The gauge faetor was 2.10 ± 0.5% for all concrete strain

gauges used in both models. Three concrete strain gauges were positioned on the concrete

deck at the middle of the first span, three over the interior support and two at the middle

62


of the second span. Figure 4.17 illustrates the distribution of the steel and concrete strain

gauges along the bridge model at the selected cross sections. The readings of the strain

gauges were recorded during the elastic loading tests as well as loading the models up to

failure.

4.5.2 Linear Variable Differential Transducers (LVDTs)

Figure 4.18 shows the linear variable differential transducers, LVDTs, used in the

experimental study to measure deflections. Six LVDTs were arranged in the first span.

Two LVDTs were located at quarter span; two were located at the middle of the span and

two were located at the three-quarter span, as shown in Figure 4.19. Three LVDTs were

placed underneath the outer web and other three underneath the inner web. The purpose

for this arrangement was to capture the vertical displacements of the bridge model to

obtain the model deflections during the static tests and to obtain the frequencies and mode

shapes during the vibration tests. The LVDTs were also installed in a way to capture a

maximum up or down vertical displacement of 30 mm. The readings of the LVDTs were

recorded in all static and vibration load tests. Ten readings per second per sensor during

the static tests and 2200 readings per second during the vibration tests were taken.

4.5.3 Accelerometers

Semiconductor acceleration transducers with built-in amplifiers were installed in

the second span, as shown in Figure 4.20. Six accelerometers were arranged in batteries

of two. The first battery was at one-quarter of the second span, second battery at the

middle of the second span and the third battery at three-quarter of the second span, as

illustrated in Figure 4.21. These accelerometers were glued to the bottom flanges and

63


used only during the vibration tests. The main objective of using the accelerometers was

to obtain another source of data, beside the LVDTs, to obtain the natural frequencies of

the bridge models. However, it should be noted that the sensitivity of the accelerometers

was much higher than the available LVDTs. The accelerometers were adjusted to record

2200 readings per second per sensor.

4.5.4 Load Cells

Six Strainsert Universal Flat Load cell models FL50U-2SGKT and FLIOOU-

2SGKT of 222 and 445 kN capacity, respectively, shown in Figure 4.22 were used to

measure the reactions at different locations. The load cells were arranged in three groups

at the three support lines. Two load cells were positioned at each of the outer supports

and two at the irmer support. Each load cell was placed to measure the reaction

underneath one web. All load cells were installed under only one box girder. For the

curved bridge model, the load cells were under the outer box girder as shown in Figure

4.23. The load cells were calibrated in a Tinius Olsen Universal Testing Machine before

and after each experimental test.

By this arrangement, the load cells were able to measure only the compression

forces. Load cells were cormected to the data acquisition during the static load tests and

during loading the bridges up to failure. They were disconnected during the vibration tests

and their channels were used to connect the accelerometers after making some

modifications regarding the voltage input and the gain factor. This was done in order to

use the maximum available number of channels in the data acquisition. Location and

arrangement of load cells are given in Figures 4.23 and 4.24.

64


4.5.5 Data Acquisition System

The available seven cards installed in the MEG AD AC 3000 series data

acquisition unit were effectively used to obtain as much data as possible from the

experimental investigations. Eaeh card contains eight channels that can be adjusted to a

certain power input and gain factors that depend on the sensors connected to that card.

The first four cards, thirty-two channels, were connected to steel strain gauges and the

fifth one was connected to the concrete strain gauges. One card was used to read the data

from the linear variable differential transducers, LVDTs. The last available card was used

for the load cells in the static load tests (and the accelerometers in the vibration tests).

Test Control Software program (TCS) was installed in the MEGADEC to capture

the data from the sensors and then save and export it in ASCII format for further analysis.

During the static tests, the reading rate of the all sensors was 10 readings per second per

sensor. However, in an attempt to maintain high accuracy in the vibration tests, where the

required readings were for only the sensors connected to the LVDTs and the

accelerometers, it was possible to increase the rate of reading to 2200 readings per second

per sensor. A view of the data acquisition unit beside the straight bridge model along with

electrical cormections to all the sensors is shown in Figure 4.25.

4.5.6 Hydraulic Jacks

Three hydraulic jacks were utilized to perform the static and vibration tests. Two

hydraulic jacks were used in the static load tests where each jack had a capacity of 890

kN. One hydraulic jack was supported on the transverse beam of the structural frame to

allow jack movement in the transverse direction only. The second jack was supported on

65


a longitudinal beam to facilitate its movement in the longitudinal direction. Both

hydraulic jacks were used in the static load tests as well as up-to-failure tests. The third

hydraulic jack with a capacity of 445 kN was positioned underneath the bridge on a beam

fixed to the laboratory floor. This jack was only used in the vibration tests. In the static

load tests, the loads from the jacks were applied manually, however, in the vibration tests,

the excitation was applied electrically.

4.6 Test Set-Up

The straight bridge model was first placed under the rigid portal structural frame.

Then, the steel box girders were supported at two outer support lines and at the interior

support. The six load cells used to measure the reactions in the experimental study were

placed under one box girder, two at each support line. An additional load cell with a

capacity of 890 kN was used to measure the total applied load. Steel strain gauges were

installed and cormected to the data acquisition unit along with the six load cells, the

accelerometers and LVDTs. The strain gauge wires were grouped, bundled and cormected

to cards in the data acquisition unit. A tie-down system was used over each support line

to prevent any uplift movement at the supports to simulate the boundary conditions

assumed in the finite element analysis. LVDTs were supported by a wooden system built

underneath the bridge model and separated from the supports to minimize any electrical

noise transferred from the bridge model to the LVDT during the vibration tests. Similar

test set-up was followed in testing the curved bridge model. For this model, the load cells

and the LVDTs were installed in the bridge radial direction. In order to verify the

performance of the sensors and the reading of the data acquisition unit, a trial test of the

66


bridge was conducted and the readings of the sensors were checked. Any damaged strain

gauges were replaced at this time.

The interior support of the bridge model was placed exactly under the centre line

of the transverse beam carrying the hydraulic jack, while the centre line of the bridge

model was made to coincide with the position of the other jack traveling in the

longitudinal direction. The first jack was used to apply symmetric loads in both spans at

the same time and the second jack was used in the application of load to the first span

only. View of the structural frame and the straight bridge model is shown in Figure 4.26.

4.7 Test procedure

Each model was tested in four stages, namely: elastic loading of the non

composite bridge model, ffee-vibration of the composite bridge model, elastic loading of

the composite bridge model, and loading of the bridge model up-to-collapse. It should be

noted that the tests on the bridge models were conducted in the same sequence as

presented in this section. Each stage is described in detail in the following sections.

4.7.1 Elastic Loading of the Non-Composite Bridge Model

This loading stage was performed to investigate the structural performance of the

steel box girder of the bridge model. The steel section alone should be capable of carrying

its self-weight, the concrete weight before hardening and the weight of construction

equipment. Uniform loading using 222 N steel blocks was applied to the bridge models.

Both bridge models were tested under five loading cases, as shown in Figure 4.27. Pieces

67


of plywood were placed on the top steel flanges between the shear connectors. The

uniform load was then applied on top of these plywood strips,. The load blocks were

distributed equally over the entire loading area. Readings of the sensors were captured

before and after loading the model. Data were recorded and saved to be analyzed and

compared with the results from the finite element method.

4.7.2 Free Vibration

Free vibration tests were first conducted on the composite bridge models. These

tests were necessary to verify the dynamic characteristics of the bridge models. Attempts

were made to utilize the available equipments in the Structures Laboratory to obtain

results as accurately as possible from the experimental test. First, a steel ball was dropped

from a certain height over the concrete deck and the readings from LVDTs and

accelerometers were recorded. However, much electrical noise occurred in the data taken

from the LVDTs and accelerometers due to bouncing of the ball on the concrete deck.

Furthermore, the impact generated was not sufficient to excite the bridge model to the

higher modes. A second attempt was made by suspending a weight by a wire strand from

the bottom flanges of the bridge and then snapping the wire off. However, the space

underneath the bridge was not sufficient enough to apply an adequate weight to excite the

bridge. In addition, electrical noise occurred in the data recorded by the LVDTs and the

accelerometers when the weight dropped on the laboratory floor.

Finally it was decided to vibrate the bridge using a mechanical fuse. A cast iron

round bar was prepared to withstand a certain uniaxial tensile force. The bar was screwed

to a hydraulic jack underneath the bridge model and attached to the floor. The bar was

68


also screwed in a nut welded in the bottom flange of the bridge model. By applying a

tension force to the bar through the hydraulic jack, the cast iron bar first would be pulled

down with the bridge model until the bar suddenly failed, leaving the bridge under initial

displacement and freely vibrating. Two different free vibration tests were conducted by

adopting this method, as described in the following sections.

4.7.2.1 Flexural test

A mechanism welded to the top flanges was built between the steel boxes to hold

a cast iron member without influencing the structural performance of the bridge models.

A horizontal beam supporting a hydraulic jack was fixed to the laboratory floor. The cast

iron bar was screwed to the hydraulic jack and in a mechanism connected to the bridge

model. Before the test, the cast iron bar was tested to evaluate its failure load. Thus, a bar

was chosen to fail at around 25 kN tension force. Six LVDTs and six accelerometers were

used to monitor the response of the bridge model once the cast iron bar broke. LVDTs

and accelerometers readings were captured using the MEGADEC unit, which sampled

the data at 2200 reading per second per sensor. A view of the flexural vibration test is

shovm in Figure 4.28.

4.7.2.2 Torsion test

A nut was welded to the bottom steel flange just underneath the outer web, as

shovm in Figure 4.29. The cast iron bar was screwed in the nut and in the hydraulic jack.

Similar test procedure was followed herein as in the flexural test. However, in this test the

intention was to excite the torsional modes, especially for the straight bridge model.

69


Recording the readings of LVDTs and accelerometers was started just before turning the

hydraulic jack on and stopped after the vibrations of the model died out.

4.7.3 Elastic Loading of the Composite Bridge Models

The objective here of the experimental test was to investigate the elastic structural

behaviour of the bridge models. Since this test was conducted after the free vibration

tests, some cracks were observed over the interior support due to series of attempts to

vibrate the bridge models. Various loading conditions were applied to the bridge model:

four cases in the case of the straight bridge model and six cases in the case of the curved

bridge model. The straight bridge model was subjected to the following loading cases:

(1) Two concentrated loads, one over each web of one box, were

applied over the webs of one box girder at 0.4 of the span from the

outer support line. View of this loading case is shown in Figure

4.30.

(2) Four concentrated loads, one over each web, were applied at 0.4 of

the span from the outer support line. View of this loading case is

shown in Figure 4.31.

(3) Four concentrated loads, two in each span and one over each web

of one box, were applied at 0.6 of the span from the outer support

line. View of this loading case is shown in Figure 4.32.

70


(4) Eight concentrated loads, one over each web, four loads per span,

applied at 0.6 of the span from the outer support line. View of this

loading case is shown in Figure 4.33.

Loadings cases (1) and (2) were intended to simulate partially and fully loaded

lanes to obtain the maximum stresses in the positive moment region. However, loadings

cases (3) and (4) were intended to represent partially and fully loaded lanes to obtain the

maximum stresses in the negative moment region. For all the loading cases, the

concentrated loads were always applied over the webs to prevent any possibility of

punching the 40 mm concrete deck. For all the loading cases, the applied load was

increased slowly until each load reached a value of 15 kN.

Six static tests were conducted on the curved bridge model as follows:

(1) Two concentrated loads, one over each web of the outer box, were



(2) Two concentrated loads, one over each web of the irmer box, were



(3) Four concentrated loads, one over each web of each box, were applied

at 0.4 of the span from the outer support line. View of this loading case

is shown in Figure 4.36.

71


(4) Two concentrated loads in each span, one over each web of the outer

box, were applied at 0.6 of the span from the outer support line. View

of this loading case is shown in Figure 4.37.

(5) Two concentrated loads in each span, one over each web of the inner

box, were applied at a distance of 0.6 the span from the outer support

line. View of this loading case is shown in Figure 4.38.

(6) Four concentrated loads in each span, one over each web, were applied

at 0.6 of the span from the outer support line. View of this loading case

is shown in Figure 4.39.

In all load cases for the curved bridge model, the loads were positioned over the

webs in the radial direction. All LVDTs, strain gauges and load cells readings data were

recorded and analyzed.

4.7.4 Loading of Bridge Models Up-to-CoIlapse

This static load test was conducted to obtain the structural response under

overloads as well as to determine the collapse load for each bridge model. The straight

bridge model was subjected to eight concentrated loads at a distance of 2145 mm from

the outer support line. The curved bridge model was subjected to eight concentrated loads

at a distance of 2150 mm, measured on the central line arc of the bridge model from the

outer support line. Under this loading case, a plastic hinge was expected to develop at the

interior support. Increasing the loads resulted in complete failure of the structure when

72


two other plastic hinges developed near the mid spans. The loads were increased slowly

to have sufficient time to trace the cracks on the concrete deck.

The readings of the LVDTs, strain gauges and load cells were recorded during the

entire loading time history until failure. All loads wee applied over the webs to prevent

any possibility of punching the concrete slab before reaching the overall collapse load of

the bridge model. Significant deflections were observed near the mid spans associated

with severe web bucking of the steel section at the interior support. The test was then

terminated for safety reasons and the load was released slowly.

73


CHAPTER V

Model Validation

5.1 Introduction

Experimental details of the bridge models were presented in the previous ehapter.

Since the main objective of the experimental program was to validate and substantiate the

finite element model used in the static and dynamic analyses of such bridges, comparison

between the experimental results and the results obtained from the finite element solution

is required. Results recorded in the experimental study were compared with those

obtained from finite element analysis using the commercial software “ABAQUS”,

suitably modified. To idealize the continuous composite two-box bridges models, the

modelling techniques presented in Chapter III were employed for both the straight and

curved bridge models. The results from the static load cases presented in this ehapter

include: deflections, longitudinal strains, support reactions, and collapse loads of the

bridge models. The results obtained from the free-vibration tests were also analyzed and

presented.

74


5.2 Elastic Response of Non-Composite Straight Bridge Model

5.2.1 Loading Case 1

The non-composite straight bridge model was loaded with 42 steel blocks, 222 N

each, as shown in Figure 5.1(a). The blocks were placed in the first span and distributed

uniformly over its entire area. The predicted finite-element results and those found

experimentally for the vertical deflection are presented in Figure 5.2. Vertical deflections

were measured at three cross sections in the first span of the bridge underneath the outer

and inner webs. At all cross sections, the experimental results are slightly higher than the

ones computed by finite element method. However, it can be observed that the results

obtained experimentally are generally in good agreement with those predicted by the

finite element model. The maximum deflection in this case obtained from the finite

element model was 1.5 mm while it was 1.8 mm as obtained from experimental testing, a

difference of 16%.

Longitudinal strains at three different cross sections were measured; and they are

compared with the results from the finite element model in Figure 5.3. It should be noted

that the recorded strains are for points at the outer surface of the bottom steel flanges and

webs. As expected, the maximum longitudinal tensile strains in the steel bottom flange

were recorded experimentally at the middle of the first span, while the maximum

longitudinal compression strains at the bottom steel flanges were measured

experimentally at the interior support line. Comparison o f the results obtained

experimentally and those predicted by the finite element mode shows that there is good

agreement between the two set of results for the three cross-sections. Results for support

75


reactions are shown in Figure 5.4. The reactions were measured at six locations, two at

each support line. Similar trends for the distribution can be observed between results of

the support reactions under each box girder obtained theoretically and experimentally.

The values for uplift reactions were not obtained experimentally for lack of suitable

facilities.


For this load case, the bridge model was loaded with 18 steel blocks, 222 N each,

as shown in Figure 5.1(b). The blocks were distributed over the space between the two

boxes in the first span and over the first box in the second span. Similar to loading case 1,

the vertical deflections were measured underneath the outer and inner webs in the first

span at three cross sections. Deflection results obtained from the finite element model and

those from the experiments are compared in Figure 5.5. It can be observed that the

deflections obtained experimentally underneath the outer web are higher than those

estimated by the finite element program. However, deflections at the web far away from

the load location were observed to be smaller than those obtained from the finite element

model. The maximum deflection occurring at the middle of span (1) underneath the outer

web was underestimated by 2%.

Longitudinal strain distributions obtained theoretically and experimentally in

loading case 2 are showm in Figure 5.6. It can be observed that the distribution of the

longitudinal strains at the middle o f spans (1) and (2) are well predicted by finite element

model. The maximum difference between the two sets of results is 20%. The distribution

of the longitudinal strains at the interior support deviated from a straight line due to the

76


localized effect of the boundary conditions. Support reactions due to loading case 2 are

presented in Figure 5.7. It can be observed that the maximum support reaction occurred at

the interior support, as expected. This loading case did not produce uplift at any of the

support points. It can also be observed that total reactions underneath each support line

obtained theoretically and experimentally are in good agreement. Yet, the experimentally

obtained transverse reaction distribution is somewhat different from that estimated by the

finite element model, the maximum difference being under 25%.


Unlike the previous loading cases, there was no distributed load between the

boxes. The loading case 3 was antisymmetric in both spans and only over each box, as

shown in Figure 5.1(c). Deflection plots for this loading case are shown in Figure 5.8.

Results obtained by the finite element model and from tests are compared. As can be

seen, the results predicted by the finite element model agree quite well with those

obtained from the experimental study. The maximum deflection developed under the

outer web in the mid-span 1.

Figure 5.9 compares the longitudinal strains for loading case 3 at three different

cross sections. It can be seen that the experimental results of the distribution of the

longitudinal strains along these cross sections at the mid-span locations can be well

predicted by the finite element model. However, it was no surprise to observe some

fluctuations in the results at the interior support, where the locations o f the strain gauges

were close to the supports. The predicted and measured support reactions for loading case

77


3 are presented in Figure 5.10. It is seen that good agreement exists between the results.

This loading produced, as expected, uplift at the end-support of the outer girder.


In this loading case the non-composite straight bridge model was loaded with 18

steel blocks, 222 N each, only in span 1 distributed over one box girder, as shown in

Figure 5.1(d). Vertical deflection underneath the inner and the outer webs are shown in

Figure 5.11. It can be observed that the results obtained experimentally are in good

agreement with those predicted by the finite element model. The maximum vertical

downward deflection value was also reasonably well estimated by the finite element

model.

Longitudinal strain distributions obtained experimentally and theoretically are

presented in Figure 5.12. The results from the finite element model are in good agreement

with the experimental values at the mid-spans. Again, the longitudinal strain values

obtained experimentally close to the interior support show differ slightly from the finite

element model values. However, in general the structural response was well predicted by

the finite element model. Figure 5.13 presents the results of the support reactions

obtained theoretically and experimentally. It should be noted that this load case produced

uplift at the end-support. Maximum reaction was developed under the outer web of the

interior support. It can be observed that the finite element model can predict reasonably

well the support reactions produced by this loading case.

78



In this loading case, the bridge model was loaded with 18 steel blocks, weighting

222 N each, distributed along the first span between the two boxes. This loading case is

shown in Figure 5.1(e). Vertical deflections underneath the outer and inner webs at three

bridge cross sections were recorded. The theoretical results obtained from the finite

element model were analyzed and compared to the experimental results. The finite

element predictions for the vertical deflection are in good agreement with the

experimental ones, as shown in Figure 5.14. It is also interesting to note that the

maximum vertical deflection due to loading case 4 is more than twice the maximum

deflection due to this loading case.

Longitudinal strains developed at the three cross sections due to this loading case

are presented in Figure 5.15. It can be observed that the finite element model values and

the experimental results are in good agreement. It can also be noted that the transverse

distribution of the longitudinal strains at the three cross-sections for the bottom flange is

relatively uniform. It is observed that the finite element model is capable of predicting the

maximum longitudinal strain at the middle of the first span. Furthermore, the longitudinal

strains obtained from loading case 4 are invariably much higher than those for loading

case 5.

Support reactions due to this loading case are presented in Figure 5.16, with uplift

observed at the support line 2 of the bridge model. Comparison between the results

obtained from the finite element model and experiments shows that the finite element

79


model can well predict the reactions. The maximum support reaction is predicted within

15%.

5.3 Elastic Response of Non-Composite Curved Bridge Model


The bridge model was loaded with 42 steel blocks, weighting 222 N each,

distributed uniformly over the first span, as illustrated in Figure 5.17(a). Results for the

vertical deflections for this loading case are shown in Figure 5.18. It can be observed that

deflections underneath the outer web of span 1 are the highest for any cross-section

location due to the curved geometry of the bridge model. The deflection values

underneath the outer web obtained from the finite element model are in general less than

those recorded experimentally. Nevertheless, the finite element model predicted quite

well the maximum vertical defiection at the first mid span, the difference between the

experimental and analytical values being within 15%. Comparing values due to this

loading case and the same case for the straight bridge model, shown in Figure 5.2, it can

be observed that the maximum deflection at the outer web increased by about 6 folds for

the curved bridge model.

Distributions of longitudinal strains for loading case (1) are presented in Figure

5.19. Comparing the finite-element predictions for the longitudinal strains with those

obtained experimentally, good agreement can be observed at the mid-span locations, but

there is greater percentage differences at the interior support. It is also interesting to note

that the strain values due to this loading case for the curved bridge model are much higher

80


than those for the straight bridge model, shown in Figure 5.3. This is due to the torsional

and curvature effects. Support reactions obtained theoretically and experimentally due to

this loading case are presented and compared in Figure 5.20. Uplift reactions are shown at

the far end-support. Maximum support reaction values at the interior support line are 2.33

and 1.94 kN, predicted by the finite element model and from tests, respeetively. It ean

also be observed that the maximum support reactions in the curved bridge are much

higher than those for the straight bridge model as shown in Figure 5.4.


Each of the outer box girders in the first span and the area between the two boxes

in the second span were loaded with 14 steel blocks, weighting 222 N each, distributed

uniformly, as shown in Figure 5.17(b). Vertical deflections under the outer and irmer

webs in the first span were recorded at the three cross-section locations. Results obtained

theoretically and experimentally are compared in Figure 5.21. It can be observed that the

results obtained from the finite element model are in good agreement with the

experimental values. The finite element model prediction for the maximum vertieal

deflection at the middle span is quite elose to the experimental reading.

The finite-element predictions for the longitudinal strains and those obtained from

experiments are compared in Figure 5.22. In general, the theoretical and experimental

values show good agreement between the two sets of results for this loading ease. It

should be noted that the finite element model underestimates the maximum longitudinal

strain for the bottom flange at the interior support. Support reaction distributions due to

this load case are given in Figure 5.23. The finite element results did not show any uplift

81


at the supports for this loading case. Values obtained from finite element model predicted

well the experimental results. The difference between the maximum support reactions

obtained experimentally and theoretically was within 10%.


As shown in Figure 4.17(c), each of the outer box girder in the first span and the

inner box girder in the second span were loaded with 14 steel blocks, weighting 222 N

each. The blocks were distributed uniformly over the span length. Deflection results for

this loading case are shown in Figure 5.24. As expected, the maximum vertical deflection

occurred underneath the outer web at the mid span location. Theoretical values

underneath the outer web at the three cross-sections in the first span were less than those

recorded during the experimental test. It can be noted also that the deflection results

obtained for the curved bridge model are more than twice those for the straight bridge

model under the same load case, in spite of the fact that load in the case of the curved

bridge model, was less than that applied in the case of straight bridge model.

Longitudinal strains for the three cross sections are presented in Figure 5.25. It is

observed that the results obtained from the finite element model can predict quite well the

longitudinal strain values recorded during the experimental test. Maximum longitudinal

strains under the outer web at the mid span location, predicted by the finite element

model, are in good agreement with the experimental values. Experimental and theoretical

results obtained for the support reactions are presented in Figure 5.26. It can be seen that

the maximum support reaction can be predicted by finite element model. It is interesting

to note that the reaction underneath the outer web at the interior support for the straight

82


bridge model is more than 10 times that obtained for the cnrved bridge model, even

thought, the number of steel block used in the case of the former model is only about 30%

more than those used in the later model.


In this load case, the bridge model was loaded with 14 steel blocks, weighting 222

N each, over the outer box girder in the first span, as shown in Figure 5.17(d). Figure 5.27

compares vertical deflection values under the outer and irmer webs in the first span

obtained by the finite element model and from the experiments. The maximum deflection

was developed, as expected under the outer web at the mid span (1). At this location, the

value obtained from the finite element model underestimates the experimental value of

5.9 mm by 11%. However, the general trend of the deflection values obtained

theoretically and experimentally shows good agreement.

The longitudinal strain distributions are presented in Figure 5.28 for this loading

case. It can be observed that the finite element model generally underestimates the values

obtained experimentally. Maximum longitudinal strains at the bottom flange developed at

mid span (1) location. This is also predicted by the finite element model. It is also

interesting to note that the effect of the load over the outer box girder is transferred to the

inner box girder through the cross-bracings. Results in Figure 5.29 show the predicted

and recorded support reactions at the support lines for this loading case. Comparison

between the values obtained theoretically and experimentally shows that there is good

correspondence between the two sets of results.

83



The bridge model was loaded in the first span between the two boxes. Fourteen

steel blocks, weighting 222 N each, were distributed along the span length as illustrated

in Figure 5.17(e). Figure 5.30 presents the vertical deflection results under the outer and

inner webs obtained from the finite element model and the experiments. It can be

observed that the finite element model can predict the deflection values very close to

those recorded experimentally. However, the result from the finite element model

underestimates the maximum vertical deflection recorded experimentally by 15%.

The longitudinal strain distributions for loading case 5 are shown in Figure 5.31. It

can be observed that longitudinal strain values produced by the finite element model

show good agreement with those recorded experimentally. Results for the support

reactions due to loading case 5 are presented in Figure 5.32. It can be observed that the

maximum support reaction occurs at the interior support under the inner web of the outer

box girder. At that location, the finite element model predicts a reaction of 0.83 kN while

a value of 0.77 kN was obtained experimentally.

5.4 Elastic Response of Composite Straight Bridge Mode!


The bridge model was loaded with two concentrated loads, 15 kN each, above the

webs of the outer box girder. The jacking loads were placed at a distance of 1430 mm

from the support line 1, as shown in Figure 5.33(a). This loading arrangement was

84


applied to investigate the maximum possible torsional and positive moment effects. The

resulting vertical deflections due to this load case are shovm in Figure 5.34. It can be seen

that the finite element model shows slightly lower results than those obtained

experimentally. However, trend of the deflection response predicted by the finite element

model is in good agreement with that obtained from the experiments. The maximum

vertical deflection recorded experimentally under the outer web was 3.6 mm, compared to

2.9 mm obtained from the finite element model.

Figure 5.35 compares the longitudinal strain results due to loading case 1.

Transverse distributions of the longitudinal strains are plotted for the three-instrumented

cross-sections. It can be observed that the elastic strain response of the webs is well

predicted by the finite element model. Also, the compression and tensile longitudinal

strains of the concrete deck estimated analytically are in a good agreement with the

experimental values. Support reactions obtained theoretically and experimentally are

presented in Figure 5.36. Comparing the results from this loading case reveals that the

finite element model can predict well the maximum support reaction, with a 16%

difference between the experimental and theoretical values.


Four coneentrated loads were positioned as shown in Figure 5.33(b) to cause

maximum positive moment and deflection effects in span 1. The resulting vertical

deflections are shown in Figure 5.37 under the inner and outer webs at the three-

instrumented eross-seetions. It can be observed that the theoretieal deflection responses

agree well with those obtained experimentally. The maximum deflection occurred at the

85


same location where the finite element model predicted a value of 4.8 mm while the value

recorded experimentally was 5.3 mm, the percentage difference being 13%.

In Figure 5.38, longitudinal strain distributions for this loading case are presented

for the mid-spans and interior support. The finite element model appears to predict well

the results from the experimental model in terms of the steel and concrete longitudinal

strains. Both results verify the general elastic structural response of the bridge model

under the applied loads in this loading case. Figure 5.39 shows the theoretical and

experimental results of the support reactions. As expected, uplift was observed at support

line 2. Clearly, the analytical and experimental support reaction values correlate well for

this loading case. Comparison of the results with those for loading case 1, Figure 5.36,

reveals that the case of eccentric loading is the one that would produce the maximum

downward reaction at the outer support line.


Unlike the previous loading cases, the two spans were loaded symmetrically in

this case. However, the bridge model in this loading case was subjected to concentrated

loads in such a manner as to produce the maximum negative moment and torsional

effects, as shown in Figure 5.33(c). Deflection results for this loading case are presented

in Figure 5.40. Again the finite element model underestimates the results in terms of

deflection. However, this model estimates correctly the position of the maximum

deflection. The noted percentage differences did not exceed 20%.

86


For the longitudinal strains, the finite element model provides good agreement

with the experimental findings as shown in Figure 5.41. The difference between the

theoretical and experimental results for both steel and concrete longitudinal strains does

not exceed 17% and 15%, respectively. It is interesting to note that the points on the webs

at mid span 1 and the interior support locations appear to predict well the location of their

neutral axes. Support reactions developed due to this loading case are shown in Figure

5.42. Again, the finite element model provides good correspondence with the

experimental values. The finite element model predicts that the maximum reaction to

occur at the same location as obtained experimentally. Difference in values between the

analytical and experimental maximum support reactions does not exceed 8%.


This load case is applied to investigate the elastic structural response of the bridge

model for maximum negative moment. Four concentrated loads were positioned in each

span at a distance of 1430 mm from the interior support, as shown in Figure 5.33(d).

Deflection responses for this loading case are shown in Figure 5.43. From the results

shown it can be observed that the results from the finite element model are in good

correspondence with the experimental data. The maximum deflections occur under the

inner and outer webs, and are underestimated by the finite element model by 5% and

15%, respectively.

The finite element model performs also quite well in terms of predicting the

maximum longitudinal strains in the concrete deck and in the steel sections as shown in

Figure 5.44. Good agreement between the strain values obtained experimentally and from

87


the finite element model can be observed. The differences between the theoretical and the

experimental findings for the maximum concrete and steel longitudinal strain developed

at the interior support are 14% and 7%, respectively. It can be also observed that

longitudinal strain distribution obtained analytically along the webs predicts well the

location of the neutral axes. Analytical and experimental results for the support reactions

are shown in Figure 5.45. Again, the results from the finite element model correlate well

with the experimental results. The maximum support reaction for this loading case

occurred at the interior support. At this location the finite element model underestimates

the maximum support reaction by 5%. It is interesting to note also that the maximum

support reaction due to loading case 3 is much higher than that caused by this loading

case, due mainly to the torsional effects.

5.5 Elastic Response of Curved Composite Bridge Mode!


In this load case two concentrated loads, weighting 15 kN each, were positioned

in the first span as shown in Figure 5.46(a). This loading case was intended to maximize

the torsional and positive moments effect in the first span. Vertical deflections results for

the bottom flange of the box girder obtained experimentally and analytically are

compared in Figure 5.47. Theoretical model predicted lower deflection values at all

points than those obtained experimentally. The maximum experimental deflection

occurred at the same location as predicted by the finite element model, underestimating it

by 10%. It is interesting to note that the maximum deflection due to this loading case is

more than 2 times that of the straight bridge.

88


Comparison of the analytical and experimental longitudinal strains distributions is

shown in Figure 5.48. It is clear that the finite element model can predict reasonably well

the results. It can also be noted that the maximum longitudinal strain due to this loading

case is higher than that for the straight bridge model. Results in Figure 5.49 show the

analytical and experimental support reactions due to loading case 1. Again, good

agreement between the experimental and the theoretical results ean be seen. Maximum

support reaetion values at the interior support are 7.52 kN and 8.63 kN from the finite

element model and experimental model, respectively. It can be observed that maximum

support reaction for the curved model is higher than the one for the straight model.


This load case is similar to the previous load case. However, the inner box girder

was loaded instead of the outer box girder, as shown in Figure 5.46(b). Deflections

calculated from the finite element model and the experimental findings are compared in

Figure 5.50. The results show that the maximum deflection occurs at mid-span 1 under

the outer web. Again, the finite element model underestimates the deflection results at all

points. Maximum deflection values predicted by this model and from the experimental

study are 3.1 mm and 3.2 mm, respectively. Obviously, this loading case produces lower

values in terms of deflection than loading case 1.

Results in Figure 5.51 show the calculated and measured longitudinal strains at

various instrumented cross-sections. It is observed that the transverse longitudinal strain

distributions at all cross-section predicted by the finite element model are in good

agreement with those obtained experimentally. It is also noted that the maximum

89


longitudinal strain at the interior support developed from this loading case is much lower

than those obtained from loading case 1. Support reaction values are compared and

shown in Figure 5.52. Maximum support reaction was observed along support line 1.

Comparison between theoretical and experimental support reactions shows that there is a

good correlation between the two results. Uplift is shown to develop at support line 2.

However, this was not recorded experimentally.


Four concentrated loads were positioned in span 1 as shown in Figure 5.46(c).

Maximum positive moment effects accompanied with torsional effects were produced

due to this loading case. Deflection results are shown for the three instrumented cross-

sections in Figure 5.53. The finite element model predicts lower deflection values at all

points. The maximum deflection occurs at the mid span 1 under the outer web as

predicted by the finite element model. Also, the deflection response for the curved bridge

model is much higher than that for the straight bridge model due to the curvature and

torsional effects.

Analytical and experimental longitudinal strains are compared in Figure 5.54. It is

observed that the finite element model results are in fairly good agreement with the

experimental findings for this loading case. The maximum longitudinal tensile strain at

the bottom flange at the mid span 1 predicted by the finite element model is 627

microstrain, compared to an experimental value of 645 microstrain. The maximum

longitudinal tensile strain at the interior support in the middle of the concrete deck

calculated by the finite element model and from experimental model are 158 and 125

90


microstrain, respectively. Due to the curvature effects, the maximum longitudinal strain

obtained for this loading case is higher than that estimated for the straight bridge model

by about 30% at the mid-span 1 and by 100% at the centre support location. The support

reactions calculated by using the finite element model and those obtained from the

experimental test are shown in Figure 5.55. Fair correlation between the two sets of

results is observed. Maximum measured support reaction was observed at the interior

support experimentally and analytically with values of 12.82 and 11.33 kN respectively.


The model was loaded with four concentrated loads, as shown in Figure 5.46(d),

to investigate the bridge model under maximum negative moment and torsional effects.

For this loading case, the deflection responses are shown in Figure 5.56. Again, the finite

element model underestimates the deflection values at all points. The location of

maximum deflection obtained experimentally is predicted correctly by the finite element

model. At this location, the maximum theoretical and experimental deflection values are

2.7 mm and 3.2 mm, respectively.

Figure 5.57 compares the analytical and experimental results for longitudinal

strain distributions. As expected, the maximum longitudinal strain occurs at the interior

support in both the analytical and tested models. The longitudinal strains in the mid spans

at the bottom flanges and concrete deck are in good agreement for both models. For this

loading case, Figure 5.58 shows the analytical and experimental values for support

reactions. It can be noted that the results from the finite element model agree well with

the experimental findings for the maximum support reaction and its location.

91



This loading case is shown in shown in Figure 5.46(e). Bottom flange deflections

for this loading case are shown in Figure 5.59. Good agreement between the analytical

and experimental results can be seen. The maximum vertical deflection values obtained

from the finite element and the physical models are 1.5 mm and 1.7 mm respectively. It

can be noted that the maximum deflection due to loading case 4 is almost twice as much

as the one developed in this loading case.

Longitudinal strains obtained analytically and experimentally for this loading case

are shown in Figure 5.60. Good correspondence between results recorded experimentally

and calculated analytically is observed. It is not surprising to note that the longitudinal

strain distributions at the interior support location are much lower for this loading case

than those for loading case 4. On the other hand, the results at the mid spans are slightly

higher for this loading case. The finite element model predictions and experimental

findings for support reactions are given in Figure 5.61. Maximum support reaction occurs

at the interior support, as expected. The difference between the maximum support

deflection provided by the physical model and finite element model is about 11%. Fair

agreement is observed between the two sets of results.


To cause the maximum negative effects on the curved bridge model, eight

concentrated loads were positioned as shown in Figure 5.46(f) for this loading case. It can

be observed from Figure 5.62 that maximum deflections occur under the inner and outer

92


webs, and are well predicted by the finite element model. The maximum deflection values

are 4.0 mm and 4.7 mm as obtained from the finite element and experimental models,

respectively. It is obvious from all the previous comparisons that the finite element model

generally provides stiffer response.

Negative longitudinal moment at the interior support is evident from Figure 5.63

by observing the development of tensile strains in the concrete deck and compression

strains in the steel bottom flanges. The results from finite element model are in fair

agreement with those obtained from experimental model. Also, it is clear that there is

very little difference between the maximum longitudinal strains in both bottom flanges at

the interior support from the two models. In addition, the longitudinal strain along the

concrete deck is almost uniformly distributed. It is obvious that the concrete deck and the

presence of cross-bracings provided an excellent distribution for the strains. Comparing

these results with those obtained from the same loading case for the straight bridge shows

that the longitudinal strains at the interior support are much higher in the curved bridge

model. However, at the mid-span, the curved model shows higher values than those

estimated for the straight bridge by about 15%. Figure 5.64 shows the experimental and

theoretical results for the support reactions for this loading case. Maximum support

reaction values are 24.3 kN and 26.7 kN given by the finite element and experimental

models, respectively. Again, the finite element model appears to predict quite well the

experimental model findings in terms of support reactions. It is also interesting to note

that there is no significant change in the maximum support reaction for the straight and

curved bridge models.

93


5.6 Dynamic Characteristics of the Composite Bridge Model

A free-vibration procedure was selected as means of excitation best suited to

obtain natural frequencies and mode shapes of the straight and curved continuous

composite concrete deck-steel two-box girders bridge models. Thus, the recording of the

results of the experimental tests was carried out directly after the end of the excitation.

For the flexural test, the load was applied at the centreline between the two box girders;

whereas, for the torsional test, the load was applied underneath the outer web. Results

were collected and then analyzed using the computer software, DADiSP [46]. Data was

recorded from each sensor at a rate of 2200 reading per second for the total measuring

time.

It is known that the maximum frequency that can be captured is given by [139]

(5.1)

where t is the sampling time. On the other hand, the minimum frequency is given by

4 , .= ] : (5.2)

where T is the total measuring time. The number of data points is therefore given by the

expression

N = 2 . ^ (5.3)

94


Typical acceleration-time history from the accelerometers and displacement-time

history from LVDTs, obtained from the free-vibration tests on the straight bridge model

are shown in Figures 5.65 and 5.66, respectively. Similarly, for the curved bridge model,

results obtained from the accelerometers and LVDTs were recorded and typical

acceleration-time and displacement-time histories are shown in Figures 5.67 and 5.68,

respectively. Signal analysis in the frequency domain was performed on the data captured

using Fast Fourier Transform (FFT) technique. From the FFT analyzer, the spectrum

response of the bridge model in the frequency domain was calculated. Frequency spectra

for the straight bridge model from the flexural and torsional tests are shown in Figures

5.69 and 5.70, respectively. Figure 5.71 shows the frequency spectrum for the curved

bridge from the flexural test; Figure 5.72 shows for the results obtained from the torsional

test. The natural frequencies and their corresponding mode shapes were calculated, where

the peaks in frequency response suggest the locations and the values of the natural

frequencies. The values of the measured and predicted natural frequencies and their

corresponding mode shapes by the finite element model are presented in Table 5.1. It

should be noted that the measured results presented in the table are the average results

obtained from flexural and torsional tests. Table 5.2 shows the values of the fundamental

frequencies obtained experimentally for six LVDTs and six accelerometers from the

flexural and torsional tests for straight and curved bridge models. Comparing the

predictions from the fmite-element model with those obtained experimentally, good

agreement can be observed between the first two natural frequencies in both bridge

models. For the straight bridge model, the fundamental frequency values are 29.9 Hz and

31.2 Hz, obtained from the finite element model and the experimental model,

respectively, with a 4% difference. Also, the corresponding mode shape was purely

flexural in both the theoretical and experimental findings, as shown in Figures 5.73(a)

95


and 5.74(a), respectively. In the case of the curved bridge model, the measured value of

the fundamental frequency was 24.5 Hz and the predicted one by the fmite-element model

was 24.1 Hz, with only 2% difference. The corresponding mode shape for the curved

bridge was a combined flexural and torsional, as given in Figure 5.73(b). This was also

confirmed by the finite element model in Figure 5.74(b). It is also interesting to note from

the these findings, that the fundamental fi-equency of the curved bridge model was lower

than the fundamental frequency of the straight bridge model by almost 29%. This

decrease in the fundamental frequency can be attributed to an increase in the degree of

curvature.

For the straight bridge model, the third and the fourth natural frequency values

were relatively close to each other, which made it difficult to extract them experimentally.

For the curved bridge model, the average third natural frequency value obtained

experimentally was 64.1 Hz. It can be observed that from Table 5.1 that the finite element

model overestimates these values by 22% for the third mode and 14% for the fourth

mode. The mode shapes of the straight and curved bridges obtained analytically and

experimentally were presented in Figures 5.73 and 5.74, respectively.

5.7 Nonlinear Response of the Composite Bridge Model

The straight and curved bridge models were loaded with two sets of concentrated

loads, symmetrically placed. Each set consisted of four equal concentrated loads. One set

was placed at 1430 mm from the interior support in each span. The load was applied at a

low rate and 10 readings per second were taken. For this loading case, the negative

moment capacity of the bridge is expected to control. The data was recorded during the

96


entire loading process until failure. The loading was terminated and the models were

considered failed when they could not carry any further load, or when excessive

deflections were observed. This procedure was followed for safety reasons. Examining

the behaviour of the loaded model at its mid-span and at or near the interior support

would provide an understanding of the collapse mechanism for such bridges. Therefore,

the relationships between the applied load and deflections, steel strain and concrete strain

were investigated.

For the straight bridge model, load-deflection response is shown in Figure 5.75.

The analytical and experimental relationships between the applied load and the deflection

underneath the outer web at the mid-span are compared. It is observed that the behaviour

of the bridge model is elastic up to 250 kN. Cracks were observed on the top of the

concrete deck at about 80 kN which was observed as a kink in the load-deflection

diagram predicted by the finite element model. An increase in deflection can be observed

as the load level increases above 250 kN. However, the changes in load-deflection slope

are not pronounced due to the presence of the two layers of wire mesh in the concrete

deck. At this load level, the initial formation of the plastic hinge at the interior support

was observed. Effective plastic strain at the bottom steel flanges and extensive buckling

of the webs were also taking place. On increasing the load further, the slope of the load-

deflection curve decreased. Failure occurred by crushing of the concrete deck near the

mid-span at an approximate load of 400 kN obtained experimentally. The finite element

model estimated fairly well the load-deflection response of the tested model, predicting

an ultimate load of 365 kN.

97


The results from the finite element model agreed well with the experimental

findings in predicting the longitudinal strain on the top of the concrete slab, as shown in

Figure 5.76. It is interesting to note that a significant decline in the structure stiffness is

observed at a load of 300 kN. This indicates the development of plastic strain within the

element in the maximum positive moment region. Similar observations were made for the

load-strain diagram for the bottom steel flange, as illustrated in Figure 5.77. It is obvious

that the bottom flange started yielding at a load value of about 350 kN in both the finite

and the tested models. Due to material hardening, the tested bridge model continued

carrying further load up to 400 kN when the concrete crushed, leading to failure.

Excessive deformations were also detected.

The development of the plastic hinges though the steel web at the mid span can be

followed by examining Figures 5.78 and 5.79. Analysis indicates that plastic flow started

at the bottom steel flange and propagated through the web from the bottom to the top.

The finite element model predicted the formation of the plastic hinge in the positive

moment region. This was confirmed by the results obtained from the experimental results.

For the curved bridge model, load-displacement diagram for the bottom of the

outer and inner webs are shown in Figures 5.80 and 5.81, respectively. These figures

show the deflection of the bridge model under the same loading case prescribed

previously. As can be noted from these diagrams, the behaviour of the structure model is

mainly elastic up to load of 200 kN. At this load, the concrete slab above the interior

support was extensively cracked and plastic hinges were formed. Subsequently, the bridge

model exhibited pronounced nonlinear behaviour with increasing applied load. The finite

element model corresponded well with the results obtained from the experimental model

98


in terms of the deflection at the bottom of the irmermost web, but it underestimated the

deflection at the bottom of the outermost web. The finite element model estimated the

ultimate load at a value of 320 kN, on the other hand, while the maximum applied load

during the experimental test was 310 kN. The test was terminated due to large deflections

of the model near the mid-span. In addition, the tested model was not able to carry any

further load.

Load-strain relationships of the concrete deck are plotted in Figures 5.82 and 5.83,

respeetively. The finite-element model predicted fairly well the longitudinal strains at the

concrete deck. It can be observed that the finite element model showed greater structural

stiffness than that observed experimentally. Figures 5.84 and 5.85 present the comparison

between the experimental and analytical results for strains at the bottom steel flange

underneath the inner and outer webs, respectively. It can be observed that the bottom

flanges started yielding at almost 250 kN in the finite element model. It is evident from

the results that the longitudinal tensile strain under the innermost web is higher than that

under the outermost web for the same applied load, where the plastic flow near the mid

span started and spread from the bottom flange underneath the innermost web to the

bottom flange underneath outermost web.

The relationships between the applied load and the longitudinal strains for the

innermost web at mid-span are shown in Figures 5.86 and 5.87. Near the top position of

the innermost web, the finite element model results correlate quite well with the results

from the experimental model up to a load of almost 200 kN. Beyond this load, the finite-

element model overestimates the longitudinal tensile strain. Larger longitudinal tensile

strains were obtained experimentally and analytically for the bottom position of the

99


innennost web, where the plastic strains near the mid-span were spreading from the

hottom flange through the weh.

Longitudinal tensile strains are plotted against the applied load for the outermost

web at mid-span in Figures 5.88 and 5.89. Again, results from the fmite-element and

tested models agreed well as long as the general structural behaviour is elastic. For a load

greater than 200 kN, the finite element model appears to be more stiff than the

experimental model. However, both models showed similar trend with respect to the

longitudinal tensile strains for the outermost web. It is also obvious that the plastic strains

started at the bottom flange and propagated through the web upward from bottom.

Figures 5.90 and 5.91 illustrate the deflected shapes of the straight and curved

tested bridge models, respectively. It should be noted that in both cases, the bridge

models exhibited large deflections prior to failure. However, for the curved bridge model,

the deflections were much more pronounced than for the straight bridge model. For the

latter, the test was terminated when excessive deformation of the bridge model was

observed. The development of cracks gives an insight into the progression of failure of

the straight and curved bridge models, as shown in Figures 5.92 and 5.93, respectively.

For the straight bridge model. Figure 5.92, the cracks are close and parallel to the interior

support line. First crack was observed on both side of the interior support at 80 kN. At

almost 200 kN, the concrete slab failed and wide cracks developed at the interior support.

This indicated that a plastic hinge had developed at the interior support.

In the curved bridge model, tracing the cracks was more complicated. The first

crack was observed at a low load of around 40 kN, and was inclined to the support line.

100


Upon increasing the load, cracks progressed on the top concrete slab in the negative

moment region. At a load of 200 kN, wide cracks developed on both side of the interior

support, inclined to the support line, indicating the contribution of the torsional moment

associated with the higher curvature of the bridge model and combined with the flexural

moment. The crack pattern for the curved bridge model is shown in Figure 5.93.

Excessive deformation at the interior support at the bottom flange was observed in

both bridge models, as shown in Figures 5.94 and 5.95, respectively. Severe buckling of

the webs at the interior support was detected. It should be noted that the deformation at

the bottom flange and the buckling in the webs contributed to the development of the

plastic hinge at the interior support. Figures 5.94 and 5.95 show the deformation of the

bottom flange in straight and curved bridge models, respeetively. Cross bracing members

were investigated after terminating the test in both bridge models. No significant

deformation was detected in the straight bridge model. However, excessive deformation

in the cross bracing members was observed in the curved model due to the torsional

moments.

5.8 Discrepancies Between the Experimental and Theoretical Results

The results obtained from the experimental and finite-element models indicate

that there are some differences between the two sets. However, a reasonable agreement

within the range of the experimental and finite-element errors can be concluded. The

experimental errors lie primarily in the sensitivity of equipment measurement and the

simulation of the boundary condition. Also, the tolerance in the rolling of thin sheets used

for the webs of the bridge models may be attributed to the discrepancies between the

101


experimental findings and finite element results. Finite-element errors may be introduced

as a result of structure modelling, boundary condition simulation, shear connectors

simulation, and concentrated loads. In addition, to obtain the collapse load, the explicit

method was adopted to analyze the finite-element model, where the model was subjected

to vertical displacement at the load location points. Material modelling can contribute to

the fmite-element errors, particularly in the study of the nonlinear response of the bridge

models. In the case of the reinforced concrete deck, effects associated with the

rebar/concrete interface, such as bond slip and dowel action, are modeled approximately

by introducing tension stiffening into the concrete modelling to simulate load transfer

across cracks through the rebar. Also, welding between steel plates can be regarded as

constrained points.

5.9 Summary

The structural elastic responses of non-composite and composite continuous two-

box bridge models were examined analytically and experimentally. The inelastic

behaviour as well as the dynamic characteristic was determined for the two composite

bridge models. The finite element commercial program ABAQUS/standard was utilized

to obtain the elastic behaviour and free-vibration analysis. The finite element

ABAQUS/explicit was employed to investigate the structural nonlinear response of such

bridges.

Comparisons between the two sets of results were carried out in terms of vertical

deflection, longitudinal strain, support reaction, natural frequency, mode shape as well as

the collapse load. The comparisons indicate both the reliability of ABAQUS ‘s model in

102


predicting the actual structural responses elastically and inelastically. The physical and

analytical models were in fair to good agreement for all load cases with a reasonably

small percentage of errors. These errors could be attributed to both experimental and

modelling reasons. The experimental errors include: measurements, loading and

unloading, boundary conditions and equipment calibration. Errors in the finite element

modelling can be attributed to: load application, interaction between the concrete and the

steel, boundary conditions and material properties.

Generally the structural stiffness of the fmite-element model is higher than that of

the physical model. As a result, in the elastic loading cases, the finite element model has

underestimated the deflection and overestimated the reactions of the bridge model. It is

also interesting to note that the finite element model estimated the fimdamental frequency

of the bridge models within 5% of the experimental findings. Also, the corresponding

mode shape was accurately predicted. The inelastic responses of the analytical models

were in good correlation with the experimental readings. This validates the use of the

finite element model in predicting the elastic and inelastic behaviour of such bridges. The

finite element model underestimated the collapse load in the case of straight bridge model

by 10% from the experimental results, and overestimated it by 5% for the curved bridge

model.

The reasonable correlation between analytical and experimental findings

presented in this chapter for all loading cases in the static and dynamic tests point to the

following:

103


1. The finite element analysis can reliably predict the static and dynamic

responses as well as the inelastic behaviour of continuous composite box

girder bridges.

2. Loading the outer lane in a continuous curved box girder bridge produces

the maximum vertical deflection, longitudinal strains as well as support

reaction.

3. The vertical deflection, longitudinal strain and support reactions obtained

for a curved bridge are much higher than those for the straight bridge

model.

4. The presence of the concrete deck and the cross bracing inside and

between the boxes enhances the overall structural performance of such

bridges. They are effective in distributing the flexural and torsional

moments throughout the bridge cross-section.

5. The fundamental frequency of such bridges decreases significantly with

increase in bridge curvature.

6. Inelastic structural behaviour as well as the ultimate load of continuous

composite box girder bridges can be well predicted by the finite element

model.

104


CHAPTER VI

Parametric Studies

6.1 Introduction

Parametric studies were performed on continuous curved two-equal-span bridges

having a eoncrete deck on multiple steel box girders. The objectives of the studies were

to: (1) examine the influence of key parameters affecting the structural response; (2)

establish a data base for the various distribution factors, for maximum stresses,

deflection, shear force, and support reaction forces necessary for design; (3) generate

information as yet unavailable for the impact factors for maximum stresses, deflection,

shear force, and support reaction forces; (4) investigate the dynamic behaviour of sueh

bridges; and (5) deduce empirical formulas for load distribution factors, impact factors,

and fundamental frequencies.

The results for the load distribution factors were obtained for traffic live loading

and self-weight of the bridges. AASHTO live loading was mainly used in the parametric

study for the load distribution factors. Subsequently, these load distribution factors were

examined for various truck loading types by applying AASHTO LRFD, CHBDC, and

CHBDC-ONT live loadings. However, only AASHTO truck loading was considered in

the generation of the impact factors. A parametric study was also conducted using free

vibration analysis to obtain the fundamental frequencies of such bridges.

105


6.2 Description of Bridges Used in the Parametric Studies

In the parametric study for load distribution factors, the effects of the following

main parameters that influence the load distribution in the studied bridges: (1) number of

lanes, Nl; (2) number of boxes, Nb; (3) span length, L; (4) span-to-radius of curvature

ratio, K = L/R. The choice of these parameters was based on other studies [9, 63]. The

same variables were also applied in the parametric studies for impact factors and

fundamental frequencies.

Table 6.1 presents the sectional configurations examined for the bridges used in

the parametric study for the load distribution factors, while. Table 6.2 shows the bridges

considered in the parametric studies for impact factors and fundamental frequencies. The

symbols used in Tables 6.1 and 6.2 represent designations of the bridge types considered

in these parametric studies: / stands for lane; b stands for box; and the number in the

middle of the designation embodies the span length of the bridge in meters. For example,

4/-80-6Z> denotes a continuous two-equal-span bridge of four lanes, 6 boxes and each span

being 80 m long. The cross-sectional symbols used in Tables 6.1 and 6.2 are showm in

Figure 6.1.

Five different lengths of 20, 40, 60, 80, and 100 m for each span were considered

in all the parametric studies. Such a range of spans covers medium span bridges. The

number of lanes was taken as 2, 3, and 4 lanes. According to Geometric Design Standards

for Ontario Highways [96], the lane width for two or more lanes should be 3.75 m.

Providing two sidewalks, one on each side of the bridge, of 0.9 m for all bridges, the total

bridge width would be 9.30 m in the case of two-lane, 13.05 m in the case of three-lane

106


and 16.80 m in the case of four-lane bridges. The number of boxes ranged from two to

four in the case of two-lane bridges, two to six in the case of three-lane bridges, and three

to six in the case of four-lane bridges. Figure 6.2 presents the number of boxes along with

the number of lanes considered in the parametric studies.

The curved bridges considered in the parametric studies were assumed to have

constant radii of curvature for both spans. The degree of curvature defined as span-to-

radius of curvature ratio, k = L/R, where the span length of the each bridge span, L, is the

arc length along the centreline of its cross-section and the radius of curvature, R, is the

distance from the origin of the circular arc to the centreline of the cross-section. The L/R

ratios used in the parametric study for the load distribution factors were 0.0, 0.1, 0.2, and

0.4 in the case of span lengths of 20 and 40 m, and 0.0, 0.4, 0.8, and 1.2 in the case of

span lengths of 60, 80, and 100 m. However, in the parametric studies for the impact

factors and fundamental frequencies, the L/R ratios were taken as 0.0, 0.1, 0.2, and 0.4 in

the case of span lengths of 20 and 40 m, and 0.0, 0.4, 0.6, and 1.2 in the case of span

lengths of 60, 80, and 100 m. The values of the radius of curvature were selected to be in

accordance with the Geometric Design Standard for Ontario Highways [96], which

requires that the radius of curvature be no smaller than 45 m.

The practical range of span-to-depth ratio for box girder bridges range from 20 to

30 [62]. For steel girders having a specified minimum yielding stress of 350 MPa or less,

the preferred span-to-depth ratio of the steel girder is not to exceed 25 [62], where the

span to be used in determining this ratio is defined as 0.9 times the arc length for

continuous end spans. In a curved bridge, each box girder in the bridge cross-section is

likely to deflect differently. Increasing the depth, and hence the stiffness, of the girders in

107


curved bridge leads to smaller relative differences in the deflections and smaller cross

bracing forces. Therefore, the span-to-depth ratio was slightly relaxed to 25.

Interior diaphragms were provided inside the box girders at each support in the

radial direetion. The end diaphragm thicknesses were taken to be the same as those of the

webs in all cases. The depth of the diaphragms was taken the same as the depth of the

steel box. A sensitivity study had shown that the aecess holes through the diaphragms had

an insignificant effect on the overall structural response of the bridges. Therefore, solid

end diaphragms were assumed inside the boxes at all supports and bracings were

provided between the boxes at support lines, as shown in Figure 6.1, to resist torsion and

deformation. Intermediate eross braeings were also placed at a spacing of 5 m for all

bridge configurations. Based on a study by Sennah [124] on curved simply supported

bridges, it was shown that this spacing is quite adequate in case of eellular bridges.

Moreover, this spacing is less than the maximum of 7.5 m recommended by AASHTO

Standard [3]. Permanent intermediate external eross braeings between boxes are usually

unnecessary. However, temporary intermediate external bracing may be desirable to

prevent or alleviate the twisting of boxes during casting of the concrete deck slab. In

addition to the cost, the removal of the intermediate external bracing may increase the

stresses in the concrete deek slab. Therefore, in this parametric, intermediate external

bracings were provided at the same spacing as the internal bracings.

Top chords internal to the tub steel girders were placed at the support lines to help

control twist and distortion. A sensitivity study conducted by Sennah [124] revealed that

replacing the area of angle cross-section by a rectangular cross-section for the top chord

showed no effect on the structural response of the bridges. Moreover, changing the

108


stiffness of the rectangular bracing systems or replacing the bracing system by an

equivalent solid plate member of the same volume has also shown no significant effect on

the elastic structural behaviour, irrespective of the degree of curvature. Accordingly, all

X-type bracing members and top chords were chosen to be of 100 x 100 mm rectangular

cross section in all bridges.

The effect of the thickness of the bottom flange and web on the distribution

factors was investigated for 4/-60-36 bridge having L/R = 0.4, where the local buckling of

the steel plates and the plastic behaviour of the materials were not considered in this

study. Figures 6.3 and 6.4 show the influence of the thickness of bottom flange and web,

respectively, on the distribution factor for tensile stress on the AASHTO truck load and

dead load. It should be noted that the change in the thickness of the bottom flange or web

has less than 3% effect, in the case of the live load, and less than 1%, in the case of dead

load. A study conducted by Zhang et al. [150] showed that the bottom flange or web

thickness has no significant effect on the impact factors for horizontally curved composite

bridges. Therefore, the thickness of the bottom flange and web were taken constant

through the parametric studies.

A preliminary sensitivity study regarding the change in flexural stiffness of the

bridge revealed that changing the span-to-depth ratio of the steel section has more effect

on the structural response than changing the concrete slab thickness. Therefore, the

thickness of the concrete deck was kept constant for all bridges considered in the

parametric studies. For all bridges used in the parametric studies, the moduli of elasticity

of concrete and steel were taken as 27 and 200 GPa, respectively. Possion’s ratio was

taken as 0.2 for concrete and 0.3 for steel.

109


6.3 Loading Conditions

The bridges were analyzed under the effect of their self-weight and highway

traffic loadings. The behaviour of curved box girder bridges is unsymmetrical by nature.

As a result, it docs not seem reasonable to distribute the dead loads equally to all box

girders in sueh bridges. Consequently, the assumption made by North American Codes of

Practice [3, 4, 5, 6, 7, 20] that the dead load is considered distributed uniformly between

box girders does not seem to be accurate for curved bridges. Moreover, the traffic loads is

even more unevenly transferred to the box girders. AASHTO live loading was adopted in

the parametric studies to investigate the distribution of live loads. The load distribution

factors were also compared with those obtained by considering AASHTO LRFD,

CHBDC and CHBDC-ONT live loadings. For the study of impact factors, only AASHTO

truck loading, HS20-44, was used in the dynamic analysis.

6.3.1 Dead Load

The dead load due to self-weight of the bridges was considered. Gravity was

specified by the constant, g = 9.81 m/s^. In addition, the material densities were taken as

2400 kg/m^ for concrete and 7800 kg/m^ for steel. Thus, the self-weight of the bridge can

be calculated from these values.

6.3.2 Live Load

AASHTO Standard truck load HS20-44 as well as the equivalent lane load was

considered in the parametric study for the load distribution factors. The truck loading,

110


HS20-44, with a total load of 325 kN is shown in Figure 6.5. The equivalent lane load

consisted of superimposed load of 9.3 kN/m uniformly and centrally distributed within

strip of 3.0 m plus a single concentrated load distributed over 3.0 m width on a line

normal to the centreline of the lane. The single concentrated load was taken as 80 kN to

generate the stress distribution factors and 116 kN to generate the shear distribution

factors. Modification factors of 1, 0.9, and 0.75 for two-, three-, four-lane loading,

respectively, were applied to account for multiple lanes loading in accordance with

AASHTO standard [3].

To obtain the load distribution factors, the above two types of loading were first

applied to a two-equal-span continuous straight girder with a span equal to the span

length of the centreline of one of the two spans of the bridge, to determine which load

type will cause maximum effects. As a result of this investigation, it was established that

the truck loading HS20-44 would be used in the case of bridges with 20 and 40 m span

lengths, and the equivalent lane loading plus the eoncentrated load would be applied for

bridges with 60, 80, and 100 m span lengths.

In order to determine maximum response, two loading cases were applied to each

bridge in the transverse direction using the finite element analysis, viz., full and partial

AASHTO truck loading (or equivalent lane loading) as shown in Figure 6.6. In the partial

loading case, the wheel loads close to the curbs were positioned at a distance of 0.6 m

from the eurb edge of the bridge and the outer lane was loaded to produce the maximum

torsional effects. However, in the parametric study of impact factors, only full AASHTO

truck loading was analyzed in the dynamic analysis for all bridges considered herein, as

shown in Figure 6.7.

I l l


In the AASHTO LRFD [7], the effect of the truck loading or design tandem is

combined with the lane load in the design. Both the design lane load and the truck

loading are the same as those used in the AAHSTO standard. The design tandem consists

of a pair of 110 kN axles spaced 1.2 m apart. Modification factors based on AASHTO

LRFD are 1, 0.85, and 0.65 for two-, three-, four-lane loadings, respectively. A different

vehicular loading is specified in the case of CHBDC of 2000 [20]. The CL-625 truck

loading as well as the lane loading is considered in the design of bridges everywhere in

Canada except in Ontario, where CL-625-ONT truck is applied instead of CL-625 truck.

The lane loading consisted of superimposed load of 9 kN/m uniformly and centrally

distributed within a strip of 3 m width. The truck loading, or 80% of the truck loading

combined with the lane load, whichever produces higher structural responses is applied.

The modification factors applied in accordance with CHBDC of 2000 [20], are: 0.9, 0.8,

and 0.7 for two-, three-, four-lane loading, respectively.

6.4 Parametric Study for Load Distribution Factors

To calculate the load distribution factors for the bridges, an extensive parametric

study was carried out. The parametric study was conducted on two-equal-span continuous

curved concrete deck on multiple steel box girder bridges to achieve all the objectives

stated in section 6.1. The key parameters chosen for this parametric study were also stated

earlier. In addition, the influence of the inclined webs, span-to-depth ratio, and the cross

bracings on the load distribution factors were examined.

The assumptions made in the parametric study were as follows: (1) the reinforced

concrete slab deck had complete composite action with the top flange of the box girders

112


through the presence of the shear connectors; (2) the analysis was conducted assuming

the behaviours of the steel and concrete were elastic and homogenous. Thus the effect of

the plastic deformation or local buckling has not been considered in the analysis; (3) the

effects of the road superelevation, outer-web-slopes, curbs and railings were not taken

into account; (4) solid diaphragms at the support lines were used in the radial directions;

(5) the bridges had constant radii of curvature between support lines; (6) to avoid stresses

due to the effect of supports constrained in plan, bearing A at the inner support line (pier).

Figure 3.8, was constrained in two directions and the innermost bearings on the outer

support lines (abutments) were constrained in the direction perpendicular to the line from

A to that bearings B at the outer support lines (abutments) [118]; all other bearings at the

abutments remained free to move along the horizontal plan of the bridge; and (7) the

truck wheel loads were simulated as concentrated loads.

The load distribution factors for the bridge straining action were calculated by

dividing the maximum straining action determined from the finite element analysis of

three-dimensional bridge by the maximum value for the corresponding straining action

for an idealized girder. The idealized girder was formulated by partitioning the two-span

continuous composite multiple box girder cross section of the bridge to a number of

individual girders, as shown in Figure 6.8. The span lengths for the idealized girder are

exactly the corresponding centreline lengths of the bridges. Each individual girder

consisted of one steel web, steel top flange, ( A / 2 N b ) portion of concrete deck slab, and

( A / 4 N b ) wide steel bottom flange, where A = bridge width and N b = number of boxes.

A form of the function to be fitted to the data collected from the finite element

analysis was specified using several trials. A nonlinear regression analysis was performed

113


to determine the values of parameters for a function that cause the function to best fit the

set of data obtained from the finite element analysis. The goal of this technique is to

evaluate the values of the parameters that minimize the sum of the squared residuals

values for the set of observations. That is known as a Least Square regression fit. The

sum of the squared differences between the actual value of the dependent variables for

each data and the value predicted by the function, using the final parameters, was

calculated. The average deviation over all observations of the absolute value of the

difference between the actual value of the dependent variable and its predicted value was

calculated.

6.4.1 AASHTO Live Loading

Several loading cases were considered in the transverse and longitudinal

directions for each bridge to obtain the maximum straining actions. In the longitudinal

direction, the truck loadings, or the equivalent lane loading plus the corresponding

concentrated load, were placed as follows: (1) AASHTO live loading near the mid-span

to produce the maximum tensile stresses and deflection at the bottom flange, as presented

in Figure 6.9(a); (2) AASHTO live loading in both spans as shown in Figure 6.9(b) to

evaluate the maximum compression stresses in the bottom flange at the interior support;

(3) AASHTO live loading in one span at 1 m distance from the outer support line, as

shown in Figure 6.9(c) to obtain the maximum shear force and the reaction force at the

exterior support; (4) AASHTO live load on both spans to get the maximum shear force at

the interior support, as shown in Figure 6.9(d); and (5) AASHTO live load in each span to

get the maximum reaction at the interior support, as illustrated in Figure 6.9(e).

114


All AASHTO live loading cases were applied twice, fully and partially loaded

lanes. Using the finite element analysis for the three-dimensional bridge, maximum

straining actions were obtained for all the loading cases and then multiplied by the

corresponding modification factors. The minimum reaction force (uplift) was estimated

by considering the minimum reaction at all support points in all loading cases. Figure 6.6

shows the location of live load in the transverse direction of the bridges.

The maximum tensile stress, Op, was obtained in the three-dimensional bridge

using the finite element analysis. The two-equal-span continuous idealized girder was

loaded with the total AASHTO live loading on the bridge divided by the number of

idealized girders to produce the maximum positive moment near the mid-span. Then, the

maximum tensile stress at the bottom fibre near the mid-span, Opa, for the idealized girder

was calculated using the simple beam bending formula. Thus, the distribution factor for

tensile stress, Dnp, in the bridges was calculated from the following formula:

= — (6 .1)

The distribution factor for compressive stress was formulated in the same manner

as that for the distribution factor for tensile stress. From the finite element analysis of the

bridge, the maximum compressive stress, a„, was calculated. Again, the two-equal-span

continuous idealized girder was loaded with the total AASHTO live loading on the bridge

divided by the number of idealized girders to produce the maximum negative moment at

the interior support. The maximum compression stress at the bottom fibre at the interior

support, Ona, for the idealized girder was calculated using the simple beam bending

115


formula. The distribution factor for compressive stress, Den, in the bridges was

determined from:

D a n = — ( 6 . 2 )

According to the North American Codes of Practice for bridges [3, 20], the

limiting value of the live-load deflection is specified in the form of the span length, span-

to-depth ratio, or first flexural frequency. The maximum live-load deflection of each

girder of the bridge cross section is preferably limited to L/800, except for the girders

under the sidewalks for which the live-load deflection is preferably limited to L/1000

[62], where L is taken as the girder arc length between the support lines. Therefore, the

deflection due the live load should be considered in the design of bridges. The deflection

distribution factor was developed by obtaining the maximum deflection under the webs,

6, and the maximum deflection of the idealized girder, 5a. Thus, the deflection

distribution factor, Dg, was calculated as:

D6 = | - (6.3)

In order to determine the shear distribution factor, Dy, the maximum shear forces

in the three-dimensional bridge, V, and the maximum shear force in the corresponding

idealized girder, Va, were first calculated. It must be noted that the maximum shear forces

of the bridges were obtained by considering the absolute maximum values, regardless of

the loading case or the location of the shear force in the box girders. The shear

distribution factor was then calculated from the following relationship:

116


D. = ^ (6.4)V.

Distribution factors for the exterior reaction forces, Dg, were also determined in

the similar way as those for the shear distribution factors. The maximum exterior reaction

considering all loading cases, Rg, was estimated from the three-dimensional finite element

analysis of bridges. Then, the maximum reaction force at the exterior support, Rea, was

calculated for the idealized girder. The exterior reaction distribution factor was calculated

from;

D e = - ^ (6.5)êa

In the same fashion, the distribution factor for the interior reaction force, Dj, was

formulated as:

(6.6)îa

where Rj is the maximum reaction force at the interior support obtained from all the

loading cases considered in the parametric study, and Rja is the maximum reaction force

at the interior support of the corresponding idealized girder.

The minimum reaction force, Rm, was determined at all support lines from all

loading cases considered in the parametric study for each bridge. To obtain the uplift

(minimum reaction) distribution factors. Dm, the minimum reaction force calculated from

117


the finite element analysis of the bridge was divided by the maximum reaction force at

the exterior support for the idealized girder, Rea, thus,

= - ^ (6.7)êa

6.4.2 Dead Load

To obtain load distribution factors for dead load, the maximum tensile and

compressive stresses in the steel bottom flange, Opa and ana, respectively, the maximum

deflection, 6a, the maximum shear force, Va, the maximum exterior and interior reactions,

Rea and Rja, respectively, were first calculated for a two-equal-span continuous idealized

girder, loaded by a uniform load q, where q = total dead load/ 2 Nb, where Nb is the

number of boxes. The moment of inertia used in calculating the maximum tensile and

compressive stresses corresponded to that of the composite section which included both

the steel girders and portion of the concrete deck slab. Thus, to determine the distribution

factors for the tensile, compressive stresses, gcp, g^n, deflection, gs, shear force, gv,

exterior reaction, ge, interior reaction, gj, and minimum reaction, gm, due to the dead load

the maximum tensile and compressive stresses, ap and a„, deflection, 6, shear force V,

exterior reaction. Re, interior reaction force, Ri, and the minimum reaction force, R^,

were calculated from the finite element analysis for each of the bridges subjected only to

its self-weight. Thus, the distribution factors for dead load were formulated as follows:

9 a p = — (6.8)

118


Qan = (6.9)

98 = (6.10)

9v = V(6 . 11)

Req = —^ (6 .12)

9i = - ^3 iiR„

(6.13)

9 m =R„ (6.14)

where the symbols are explained under “Notations”

6.5 Parametric Study for Impact Factors

A sensitivity study was first undertaken to determine the influence of different

parameters such as the vehicle speed on the impact factors. Table 6,3 presents the vehicle

speed considered in the parametric study for impact factors for all bridges. This will be

discussed further in Chapter VIII.

119


In this parametric study, full loaded lanes with AASHTO trucks HS20-44 were

considered. Each truck travelled with a constant speed along the centreline arc of the

loaded lane. The travelling speed was considered in accordance with the maximum safe

allowable highway speed. It must be stated that in the longitudinal direction of each

bridge, two trucks were placed at certain distance to produce the maximum compressive

stresses at the bottom flange at the interior support line. Static and dynamic analyses were

conducted for each bridge subjected exactly to the same loading condition. The maximum

tensile and compressive stresses, deflection, shear force, exterior and interior reactions,

and the minimum reaction were estimated from both static and dynamic analyses. Thus,

the impact factor for each straining action was obtained by dividing the increase in the

straining action due to the dynamic effect by the straining action resulting from the static

analysis.

6.6 Parametric Study for the Fundamental Frequency

This parametric study was undertaken using free vibration analysis to investigate

main parameters that may influence the natural frequencies and mode shapes of

continuous curved concrete deck-on multiple steel box girder bridges. The mode shapes

of two-equal-span curved bridges are generally either symmetric or antisymmetric about

the interior support line. Schematic views of these mode shapes are shown in Figures 6.10

and 6.11. As a result of an extensive parametric study of 180 bridges, empirical

expressions to estimate the fundamental frequency for each bridge were deduced, together

with recommendations for increasing the torsional resistance of continuous curved

bridges.

120


CHAPTER VII

Load Distribution Factors

7.1 Introduction

Despite all the inherent advantages of a curved composite concrete deek-steel

multiple box girder bridge, its behaviour is more complex than that of the I-girder bridge.

Bridge designers in North America strongly prefer a simplified method of analysis to

reduce the complexity involved in the load distribution analysis of a bridge. Accordingly,

in this chapter the results from a rigorous parametric study are presented. Three-

dimensional finite-clement models are used in the analysis, in which 240 two-span

composite multiple box-girder bridges are studied. Various loading conditions are

considered to evaluate the results for the maximum structural responses of such bridges.

The results from the parametric study are based on AASHTO live load and the

bridge self-weight. The following are obtained: (i) the distribution factor for tensile

stress; (ii) the distribution factor for compressive stress; (iii) the distribution factor for

deflection; (ix) the distribution factor for shear force; (x) the distribution factor for

maximum exterior reaction force; (xi) the distribution factor for maximum interior

reaction force; and (xii) the distribution factor for minimum reaction force (uplift). The

effects of bridge span length, number of lanes, number of boxes and span-to-radius of

curvature on the structural responses of the bridges are discussed. Moreover, the effects

of the number of bracing members on the distribution factors are investigated. Empirical

121


expressions for these distribution factors corresponding to the AASHTO live load and

dead load are deduced. Furthermore, the effects of the number of bridge spans on the

distribution factors are examined as well as the influence of inclined webs in the steel box

girder. Sensitivity study on three bridges is conducted to evaluate the effect on the

aforementioned distribution factors due to AASHTO LRFD, CL-625, and CL-625-ONT

truck loadings.

7.2 Distribution Factors for Tensile Stress

7.2.1 Effect of Span Length

Figures 7.1(a) and 7.1(b) show the distribution factors for maximum longitudinal

tensile stress in four-lane three-box girder bridges with L/R = 0.0 and 0.4 due to

AASHTO live load and dead load, respectively. For both loading cases, it can be

observed that the stress distribution factors for straight bridges decrease by about 12%

with increasing span length from 20 m to 60 m. For straight bridge with span length

exceeding 60 m, the distribution factor for tensile stress changes by less than 1%. In the

case of curved bridge, the distribution factors for tensile stress are affected by the span

length, with differences not exceeding 14 %.

7.2.2 Effect of Number of Lanes

The bridge width, represented by the number of lanes, is considered to be one of

the criteria that affect the distribution factor for tensile stress in curved bridges. The

distribution factors for longitudinal tensile stress in curved bridges with four boxes and

122


span lengths of 20 m and 60 m, due to AASHTO live load and dead load are shown in

Figures 7.2(a) and 7.2(b), respectively. It can be observed that the distribution factors

vary almost linearly with the number of lanes varies for both span lengths. The

distribution factor for tensile stress decreases with increasing number of lanes for

AASHTO live load. However, for bridges with span length of 60 m, the changes in the

distribution factor for tensile stress are higher than those obtained for bridges with span

lengths of 20 m. The distribution factor for tensile stress under the self weight of the

bridge increases by almost 10% for short span bridges and becomes almost constant for

bridges with longer span lengths.

7.2.3 Effect of Number of Boxes

The effects of the number of box girders on the distribution factor for tensile

stress for two bridges having different span lengths are illustrated in Figures 7.3(a) and

7.3(b) for AASHTO live load and dead load, respectively. The distribution factor for

tensile stress is almost uniform in case of live load irrespective of the number of boxes

for bridges with span length of 40 m. However, it increases by about 13% with increase in

the number of boxes from 2 to 6 for bridges with span lengths of 100 m. In the case of

dead load. Figure 7.3(b) shows a small increase in the distribution factor for tensile stress

when more boxes are added. The above results reveal that increasing the number of boxes

may not result in a cost effective design in most cases in terms of tensile stresses.

123


7.2.4 Effect of Span-to-Radius of Curvature Ratio

As expected, the span-to-radius of curvature ratio of box girder bridges had a

significant effect on the distribution factor for tensile stress. Figure 7.4(a) shows the

relationship between the distribution factor for tensile stress and the span-to-radius of

curvature ratio for AASHTO live load. It can be noted that for both bridge span lengths,

considered herein, the distribution factor for tensile stress increases significantly with

increase in the span-to-radius of curvature ratio. Similar trends are observed in Figure

7.4(b) for the distribution factors due to the bridge self weight. Interestingly these two

figures show that the longitudinal tensile stress increases by almost 40% for the same

bridge configurations when the L/R is increased from 0 to 1.2.

7.3 Distribution Factors for Compressive Stress


The results for the bridge span length effect on the distribution factors for

compressive stress are presented in Figures 7.5(a) and 7.5(b), for AASHTO live load and

the bridge dead load, respectively. It is observed that the distribution factors for

longitudinal compressive stress decrease when the bridge span length changes from 20 m

to 40 m. Increasing the span length beyond 40 m, increases somewhat the distribution

factors for compressive stress. This observation was noted for bridges under AASHTO

live load for both straight and curved bridges. This may be attributed to the fact that

maximum compressive stress obtained in the case of 20 m and 40 m bridges are based on

AASHTO live load case. For bridges with larger span length, maximum values are

124


obtained from AASHTO lane loading. In the case of bridge self-weight, the distribution

factors for compressive stress increase generally by almost 6% with increasing span

length for both straight and curved bridges from 40 to 100 m.


The relationship between the distribution factors for compressive stress versus

number of lanes in bridges due to AASHTO live load and bridge dead load are plotted in

Figures 7.6(a) and 7.6(b), respectively. It can be observed that the distribution factors for

compressive stress vary almost linearly with the number of lanes in both figures. It is

interesting to note that the distribution factors for compressive stress increase by almost

3% with the number of lanes only for bridges under their own weight. In contrast, these

factors decrease by almost 11% with increasing number of lanes for bridges subjected to

AASHTO live load. The rate of variation in the factors for live load is greater than that in

the case of self-weight.


Distribution factors for compressive stress of three-lane box-girder bridges with

span-to radius of eurvature ratio of 0.4 are plotted against the number of box-girder as

shown in Figures 7.7(a) and 7.7(b), for AASHTO live load and self-weight, respectively.

It can be observed that the live load distribution factors increase by about 10% with

increase in the number of box girders. It is also interesting to observe that the minimum

value of distribution factors can be obtained when the number of boxes is almost the

125


same as the number of lanes. The same observation can be made for self-weight, Figure

7.7(b).


Figures 7.8(a) and 7.8(b) present the influence of the span-to-radius of curvature

ratio on the distribution factors for compressive stress for AASHTO live load and the

bridge self-weight, respectively. The results show that the higher this ratio is, the higher

the distribution factors for compressive stress are under either live or dead loads. It should

be noted that the distribution factors of 100-m span length bridge with L/R value of 1.2

are higher by more than 65% than those obtained for straight bridges with the same span

lengths. Moreover, in some cases the maximum compressive stress is developed when

only one span is loaded, particularly for bridges with larger span length and span-to-

radius of curvature ratio. In this specific case, high longitudinal compressive stresses

occur at the interior support line due to the combined bending and torsional moments.

Occasionally with only one span loaded, unbalanced forces on the two sides of the

interior support line cause larger stresses than those determined by loading

simultaneously the two spans of the bridge.

7.4 Distribution Factors for Deflection


Distribution factors for maximum deflection versus the span length for four-lane

three-box girder bridges are plotted in Figures 7.9(a) and 7.9(b) for AASHTO live load

126


and self-weight, respectively. Clearly, the distribution factors for maximum deflection

occur for bridges with span length of 20 m, decreasing by about 45% as the span length

increases from 20 to 100 m. However, the rate of the decrease in the distribution factors

for deflection levels off with increase in the span length from 40 m to 100 m for bridges.


Similar to the variations in the distribution factors for tensile and compressive

stresses with numbers of lanes, the number of lanes also influences the distribution

factors for deflection in the same marmer. For AASHTO live load. Figure 7.10(a) shows

the reduction in the distribution factors for deflection by about 11% with increase in the

number of lanes from 2 to 4. Figure 7.10(b) shows an increase by almost 11% in the

factors with numbers of lanes for self-weight.


The relationships between the distribution factors for deflection and the number of

boxes are illustrated in figure 7.11(a) and 7.11(b) for bridges subjected to AASHTO live

load and dead load, respectively. The results in both graphs exhibit the same pattern.

Increasing the number of boxes produces a change of about 5% in the distribution factors

for deflection in all cases, irrespective of the span length. The distribution factors for

deflection remain almost unchanged as the number of boxes is increased from 3 to 6 for

bridges subjected to their dead load or to AASHTO live load.

127



Figures 7.12(a) and 7.12(b) show the relationship between the distribution factors

for deflection with the span-to-radius of curvature ratios for bridges subjected to

AASHTO live load and self-weight, respectively. The distribution factors for deflection

increase by almost 46% as the span-to-radius of curvature ratio increases from 0 to 1.2.

Furthermore, the trend of the increase is the same irrespective of the span length.

7.5 Distribution Factors for Shear


Distribution factors for maximum shear force are plotted for 4/-3Z> straight and

curved bridges in Figures 7.13(a) and 7.13(b) for AASHTO live load and dead load,

respectively. Under AASHTO live load, distribution factors for shear are almost the same

for spans between 20 m and 40 m for straight and or curved bridges. For spans greater

than 40 m, the distribution factors for straight bridges approach unity. This means that the

live load will be distributed evenly on all bridge girders. In contrast, the distribution

factors for curved bridges increase by 8% due to an increase in the torsional effects

present in the curved bridges with larger span lengths. On the other hand, distribution

factors for shear due to self-weight load showed almost the same trend for both straight

and curved bridges, with no significant change.

128



The variations in the distribution factors for shear force with number of lanes are

presented in Figure 7.14(a) for AASHTO live load. It can be observed that the highest

distribution factor for shear force occurs for two-lane bridges. These factors decrease with

increasing the number of lanes for both bridges. However, for bridges with larger span

length, the distribution factors remain almost constant when the number of lanes

increases from 3 to 4. It must be noted that the maximum shear forces are obtained from

different loading cases and accordingly various multiple presence factors should be

applied. This can cause an inconsistent trend with increase in the number of lanes. The

distribution factor for shear force due to the bridge self-weight is illustrated in Figure

7.14(b). It can be observed that the changes in the distribution factors are marginal for

both bridges spans.


Number of boxes reflects the number of webs present in the bridge cross-section:

and the webs are the main members in resisting the shear forces in the bridge

superstructure. Therefore, increasing the number of boxes affects the distribution factors

for shear force. However, this effect is too complex to quantify in some cases, given the

fact that the distribution factor for shear developed herein is the envelope of factors

resulting from different loading eases shown in Figure 6.7 in Chapter VI. Figure 7.15(a)

shows the change in the distribution factors for shear with an increase in the number of

boxes. No general trend is observed except that the lowest distribution factor for shear

occurs when the number of boxes is three, which is also equal to the number of lanes. An

129

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

increase in the number of boxes beyond three produces higher values of distribution

factors for shear. Figure 7.15(b) presents the relationship between the distribution factors

for shear force and the number of boxes for self-weight. The variation in these factors

with number of boxes is only nominal.


The span-to-radius of curvature ratio effects on the distribution factors for shear

are shown in Figure 7.16(a) and 7.16(b), for live and self-weight, respectively. It can be

seen that the span-to-radius of curvature ratio, L/R, has a significant influence on the

distribution factors for shear for both load cases. For live load, the distribution factors

increase by about 80 % and 160% for 60-m and 100-m span bridges, respectively. As

expected, this large increase is due to the torsional effects, which give rise to high shear

forces in the webs near the support lines. The same applies to the self-weight case, except

that the rate of change is not that steep.

7.6 Distribution Factors for Exterior Support Reaction


Distribution factors for maximum reaction force at the end supports are plotted

against the bridge span length for straight and curved bridges as shown in Figures 7.17(a)

and 7.17(b) for AASHTO live load and bridge dead load, respectively. It can be noted

that the distribution factors for exterior reaction change by almost 5% with regard to span

length for straight bridge subjected to either AASHTO live or self-weight. However,

130


curved bridges with spans more than 40 m exhibit an increase by about 19% in these

factors as the span length is increased. For example, the distribution factor for exterior

reaction for a bridge with 60 m span length is 20% higher than that obtained for bridge

with 40 m span length. With regard to live loading case, it should be also noted that

measurable change in the distribution factors occurs by increasing the bridge span length

from 40 to 60 m due to the use of different type of live load on the bridges; i.e. truck load

or lane load whichever produces higher distribution values.


Figure 7.18(a) and 7.18(b) show the change in distribution factors for exterior

reaction with increase in the number of lanes for bridges subjected to AASHTO live load

and self-weight, respectively. As shown in Figure 7.18(a), the distribution factors vary by

a maximum of 10% for bridges subjected to AASHTO live load. However, this change is

also dependent on the bridge span length; i.e. on the type of AASHTO live load, truck or

lane load, considered in the study to obtain the maximum exterior reaction force. Figure

7.18(b) shows the variation of these factors with the number of lanes for bridges under

their self-weight. It is observed that in this case the maximum range of fluctuation in the

distribution factors for maximum exterior reaction is no more than 5%.


Distribution factors for exterior reaction are plotted against the number of boxes

in Figure 7.19(a) and 7.19(b) for AASHTO live load and dead load, respectively. The

distribution factors for reaction due to the live load for bridges show only an 11%

131


variation as the number of boxes is increased. For bridges subjected to self-weight, a

change in the number of boxes changes the distribution factor by less than 6%.


The change in the distribution faetors for exterior reaction force with the span-to-

radius of curvature ratios is presented in Figures 7.20(a) and 7.20(b) for AASHTO live

load and the bridge self-weight, respectively. The graphs show very significant changes in

the distribution factors for reaction as the span-to-radius of curvature ratio increases from

0 to 1.2. It is also obvious that the variation rate increases with increase in the bridge span

length. Similar observations can be made for bridges subjected to their own self-weight.

For both loading cases considered in Figure 7.20, the distribution faetors for reaction

increase by about 160% as the span-to-radius of curvature ratio increases from 0 to 1.2 for

bridges considered in this investigation.

7.7 Distribution Factors for Interior Support Reaction


The influence of bridge span on the distribution factors for reaction force at the

interior support is shown in Figures 7.21(a) and 7.21(b) for live load and self-weight,

respectively. From the results shown, it is seen that the distribution factors for reaction

under AASHTO live load decrease by about 8% with increase in the span length for

curved bridges. However, there is no significant variation in these faetors for self-weight.

132



Straight bridges loaded with AASHTO live load show a decrease by about 19% in

the distribution factors for interior reaction with increase in the number of lanes as

illustrated in Figure 7.22(a). For bridges with larger span length, the distribution factors

remain almost constant when the number of lanes exceeds 3. Figure 7.22(b) presents the

effect of number of lanes on the distribution factors for interior reaction for bridges under

their self-weight. It should be noted that the number of lanes has no significant effect on

these factors.


As before, the number of boxes shows no consistent trend in its effects on the

distribution factors, as revealed in Figure 7.23(a) for bridges subjected to AASHTO live

load. It is interesting to observe that as the number of boxes increases from 3 to 4 for 40

m bridge the distribution factor for reaction increases by almost 12%. Distribution factors

for interior reaction are plotted against the number of boxes in Figure 7.23(b) for bridges

subjected to dead load. Obviously, an increase in the number of boxes does not reveal any

general trend in the variation of distribution factors.


For bridges under AASHTO live load, the effect of span-to-radius of curvature

ratio, L/R, on the distribution factors for interior reaction is presented in Figures 7.24(a).

It is clear that increasing this ratio increases the distribution factors for interior reaction

133


by almost 20%. For bridges with 100-m span length, the effect is more pronounced than

for those with 60-m span length for L/R above 0.8. The influence of the span-to-radius of

curvature ratio on the distribution factors is plotted in Figure 7.24(b) for bridges subjected

to their self-weight. The change in the distribution factors follows the same trend as

observed for bridges subjected to AASHTO live load. It is estimated that the distribution

factors increase by almost 30% with increase in L/R ratio from 0.0 to 1.2 for bridges

having 100-m span length. This increase can be attributed to the torsional effects

produced by bridge curvature.

7.8 Distribution Factors for Minimum Reaction


The effect of span length on the distribution factors for minimum reaction (uplift)

for bridges under AASHTO live load is shown in Figure 7.25(a). The behaviour of this

factor seems to follow a certain pattern. The distribution factors are negative signifying

that there is a net uplift. For straight bridges, the absolute value of the factor increases

with increase in the span length from 20 m to 40 m. However, it remains reasonably

constant after that. For curved bridges, the uplift force continues to increase rapidly, as

expected, with increase in the bridge span length due to the torsional moments produced

by the bridge curvature. Figure 7.25(b) shows the factors calculated for bridges subjected

to their own self-weight. Continuous straight multiple box girder bridges do not develop

uplift reactions and the minimum reaction force remains almost constant for all span

lengths. Curved bridges also show no uplift. However, the minimum reactions keep

decreasing almost linearly with increase in the span-to-radius ratio. The minimum

134


reaction force under self-weight for Al-2b 100 m-bridge with L/R value of 0.4 is almost

negligible.


Figure 7.26(a) presents the variation of distribution factor for uplift for straight

bridges having three boxes. It is obvious that no general trend is observed in the variation

of distribution factors with increase in the number of lanes. This may be attributed to the

fact that the uplift forces are produced by several different loading cases while only the

maximum values are considered herein, mostly by loading the outmost lanes in both

bridge spans. In the case of bridges subjected to self-weight, number of lanes does not

have a pronounced effect on the distribution factors, as shown in Figure 7.26(b).


Figure 7.27(a) shows the influence of number of boxes on the distribution factors

for minimum reaction force for bridges subjected to live load. There is a net uplift

reaction force in this case. The absolute value of the factors increases significantly with

increase in the number of boxes. Figure 7.27(b) presents the distribution factors under

self-weight. For 40 m span bridge, the factor remains almost constant as the number of

boxes increases. However, for bridges having 100-m span length, slight decrease in these

factors is observed with increase in the number of boxes. It is interesting to note that the

distribution factors for reaction remain positive, which means that there is no uplift at the

supports for straight bridges.

135



As expected, the span-to-radius of curvature ratio gives rise to uplift forces at the

support lines due to torsional effects. Figures 7.28(a) shows this effect on bridges

subjected to AASHTO live load. The increase in the distribution factors for uplift reaction

is dramatic for 100-m span bridge. It can be observed that under self-weight the minimum

reaction becomes an uplift force at the support lines with increase in bridge curvature as

presented in Figure 7.28(b). This is a very important result since it shows that for curved

bridges uplift force occurs and the self-weight of the bridge may not be able to balance

the uplift forces due to the live load.

7.9 Empirical Formulas For Load Distribution Factors

Based on the results obtained from the parametric study on 240 bridges,

expressions were developed for the distribution factors for maximum tensile, compressive

stresses, deflection, shear, as well as for reactions. The empirical formulas are in terms of

multiplying factors applied to the distribution factors for straight bridges, in order to

account for the effect of the bridge curvature. The empirical formulas for self-weight and

for AASHTO live load are in terms of: (1) span length of the bridge, L (L = one half of

the total centreline span in a continuous two-equal-span bridge in meters); (2) number of

lanes, Nl; (3) number of boxes, Nb; (4) span-to-radius of curvature ratio, k, where k =

L /R . For AASHTO live load, both full and partial loadings were considered. Using a

statistical computer package for best fit based on the Method of Least Squares for

nonlinear data, the following empirical formulas were generated for the various

distribution factors. Typical samples of the finite element results and proposed

136


expressions are given in Appendix B. It should be noted that the impact factor was not

incorporated in these expressions. However, the modification factors for multilane

loading specified by AASHTO (1996) were accounted for in the proposed formulas:

For AASHTO live load:

Distribution Factor for Maximum Tensile Stress in the Bottom Flange along Bridge

Span, Dcrp:

op 1 + ' - Tv i o j v N w(L)'

.0.75 (7.1)

Distribution Factor for Maximum Compressive Stress in the Vicinity of Pier

Section, D<ji,:

/ ( / \V-KT A'* ^1 +

K

vTOyN ,

v N . y(L )- (7.2)

Distribution Factor for Maximum Deflection, Dg:

vN lv

0,2 f1 +

/ -KT / 2 \ 0 .1 5 ^

vN l 7(7.3)

Distribution Factor for Maximum Shear Force along Bridge Span, Dy

137


2N 0.1 A

0 .1 x t O.2

L yL" ‘ NX

/

1 4"1.8 ^

/ Vl i o o j

y

(7 .4)

Distribution Factor for Maximum Exterior Support Reaction, De

^1.9^ rV TV y

X 1 +V V 300 N L y

(7.5)

Distribution Factor for Maximum Interior Support Reaction, Di:

vL“'1 +

V U O y y(7.6)

Distribution Factor for Minimum Reaction Force at the Supports, D„

D„ =\ v 6 /

+ ■0.2 N"-'N«.4 (7.7)

For Self-weight:

Distribution Factor for Maximum Tensile Stress in Bottom Flange along Bridge

Span, 9ap:

(7.8)l . lOx/

1 + f - 1f N j

0.5 ^

VU J I n l J /

138


Distribution Factor for Maximum Compressive Stress in the Vicinity of Pier

Section, go„:

9 a n = 1 . 1 0 X 1 +/ \ l - 5' K '

v2y

/ -KT AN

0.5 ^

(7.9)

Distribution Factor for Maximum Deflection, gs:

Qs =1.10x 1 + (4k)' ^0.5 \

(7.10)

Distribution Factor for Maximum Shear Force along Bridge Span, gv*.

9v =1.10x 1 +v50y

(7.11)

Distribution Factor for Maximum Exterior Support Reaction, ge

9 e = l - 1 0 x 1 -----(TT3NV V L y y(7.12)

Distribution Factor for Maximum Interior Support Reaction, gi;

9i = 1 . 1 0 X 1 +5 0 ,

(7.13)

139


Distribution Factor for Minimum Reaction Force at the Supports, g^:

9m =0.8 X 1 -

10 ^ 0.06 (7.14)J )

7.10 Effect of Number of Spans

In the previous sections, results from a parametric study, using three-dimensional

finite-element simulation of two-equal-span bridges were analyzed and reported.

Empirical expressions for the various distribution factors were deduced taking into

account the variations in the span length, number of lanes, number of boxes and span-to-

radius of curvature ratio. In this section, the effect of number of spans on the distribution

factors are investigated to determine whether the proposed expressions are applicable for

multiple-box girder bridges with various numbers of spans.

Four-lane three-box girder straight and curved bridges with span length of 60 m

were considered in this study. The span-to-radius of curvature ratio of the curved bridge

was taken as 1.2. Bridges with one, two, three and four spans were analyzed. Distribution

factors were calculated in each case through dividing the straining action obtained from

the 3-D finite element model by the corresponding straining action determined from the

idealized girder. It should be emphasized that the number of spans in the idealized girder

must always be the same as in the 3-D fmite-element model of the bridge analyzed.

Figure 7.29 compares the results for the distribution factor for tensile stress for all

cases considered herein. It can be observed that these factors remain almost constant for

140


the various numbers of spans analyzed. The same trend was observed for both straight

and curved bridges. This confirms that distribution factor for tensile stress does not

depend on the number of spans if the distribution factor is calculated using the previously

adopted approach. Distribution factors for compressive stress for two-, three-, and four-

span bridges are plotted in Figure 7.30. It is observed that the fluctuations in the values of

distribution factors for compressive stress are less than 2%, which suggests that the

number of spans has no significant effect on the distribution factors for compressive

stress as presented. Distribution factors for deflection are illustrated for one-, two-, three-,

and four-spans bridges in Figure 7.31. The results in this Figure show that the distribution

factors for deflection for simply or continuously supported straight or curved bridges are

not significantly affected by the number of spans in the bridge.

The effect of number of spans on distribution factors for shear is presented in

Figure 7.32. No variations in the distribution factors for shear are observed for different

number of spans in the case of straight bridges. However, for curved bridges, about 5%

variations in the load distribution factor can be observed. Results for the distribution

factors for exterior reaction versus the number of spans are shown in Figure 7.33. The

trends are similar to the ones discussed for the distribution factors for shear in Figure

7.32. Similar behaviour is presented for the distribution factors for interior support

reaction in Figure 7.34. Because of the complex behaviour of the uplift reaction, some

differences in the distribution factors can be observed with different number of spans, as

shown in Figure 7.35. Such differences may be due to the changes in the location of the

uplift reaction, as governed by the loading position in the longitudinal and transverse

directions. However, it is interesting to note that the value for the distribution factors for

uplift obtained for two-span bridges can be used in estimating the uplift in straight and

141


curved bridges with span length varies from 2 to 4. For curved bridges the proposed

empirical equation for distribution factors for uplift will provide conservative values

when applied to three- or four-span curved bridges.

In conclusion, it can be stated that the number of spans has an insignifieant

influence on the distribution factors. Therefore, applying the proposed empirical

equations to multiple-box girder bridges with 3 or 4 spans will lead to a fair prediction of

the actual straining action in the three dimensional bridge structures.

7.11 Effect of Inclined Webs

As a practical matter, inclined webs are often used in tub girders. The use of

inclined webs reduces the width of the bottom flange to provide for greater efficiency in

the design of bridge substructure. Their use is also aesthetically pleasing. It has been

suggested [3] that the inclination of the webs relative to a plane normal to the bottom

flange should not exceed 1 to 4. For simplicity and to reduce the number of variables in

the parametric study, webs were considered to be normal to the bottom flanges in all

bridge configurations listed in Table 6.1. In this section, the effect of inclined webs on the

load distribution factors is examined. Two bridges were analyzed herein with web slopes

having values of 0, 1/8 and 1/4. Each bridge had three lanes with span-to-radius of

curvature ratio of 0.4. The maximum results for the distribution factors were obtained

considering full and partial loading cases.

Distribution factor for tensile stress versus web slope is shown in Figure 7.36. The

results reveal differences of about 1% in these factors due to the inclination of the webs.

142


It should be noted that the idealized girder consists of one-half the reduced bottom flange

and an inclined web. Figure 7.37 show the effect of web slope on the distribution factors

for compressive stress. The graph indicates that there is no noticeable variation in the

factor for the bridge with two-box girders. In the case of four-box girder bridge, increase

in the distribution factors of less than 5% is observed. The changes are quite small.

Similar trends are noted for the distribution factors for deflection, shear, exterior support

reaction, interior support reaction, and uplift reaction, shown in Figures 7.38, 7.39, 7.40,

7.41, and 7.42, respectively.

In conclusion, web slope has an insignificant effect on the load distribution

factors. However, for web design, the shear force component in the direction of the web

can be calculated by dividing the vertical shear force by the cosine of the slope angle.

7.12 Effect of Span-to-Depth Ratio

The preferred span-to depth ratio of steel box girder is not to exceed 25 [62] for

girder having a specified minimum yield stress of 350 MPa or less. With the adoption of

composite design, the preferred span-to-depth ratio for the steel box girder was relaxed to

30 [62]. The span-to-depth ratio for the composite girder, including the eoncrete deek

slab, was set at 25. Due to the curvature, the span-to-depth ratio of the steel box girder

was increased to be 25, excluding the concrete deck, for all bridges considered in this

parametric study. Accordingly, it was decided to examine the effect of span-to-depth

ration on the distribution factors.

143


Two bridges with span-to-radius of curvature ratio of 0.0 and 1.2 were considered

in this study. Each bridge was a two-equal-span continuous bridge with span length of

100 m. The bridges had four-lane three-box girders each. Span-to-depth ratio of 20, 25

and 30 were considered for each bridge. Various loading cases using AASHTO live load

were applied to determine the maximum structural responses in terms of tensile stress,

compressive stress, deflection, shear force, exterior support reaction, interior support

reaction and uplift force.

Distribution factors for tensile stress are plotted versus span-to-depth ratio for

both bridges in Figure 7.43. It can be observed that these factors remain constant for the

straight bridges, with only minor variation in the results for the curved bridges. It can be

observed that as the span-to-depth ratio increase the distribution factors for tensile stress

decrease. The same trends are seen in Figures 7.44 and 7.45 for the distribution factors

for compressive stress and deflection, respectively. Therefore applying the results derived

herein, which are based on a span-to-depth ratio of 25, to bridges with span-to-depth ratio

> 25, will lead to conservative results.

Distribution factors for shear force versus span-to-depth ratio are presented in

Figure 7.46. It can be observed that for straight bridges this ratio does not influence these

factors. For curved bridges, the distribution factor for shear does increase marginally with

increase in the span-to-depth ratio. Distribution factors for exterior support reaction seem

to have the same trend as that for shear, as shown in Figure 7.47. The results for

distribution factors for interior support reaction versus span-to-depth ratio are presented

in Figure 7.48, showing no significant effect. The results for reaction distribution factors

for straight and curved bridges are given in Figure 7.49. The figure shows a decline in the

144


factors with increase in the span-to-depth ratio. This means that adopting the proposed

equations for interior support reaction and uplift reaction can provide conservative values

in case of curved bridges and fairly accurate predictions for straight bridges.

7.13 Effect of Cross Bracing

Internal cross bracing are necessary to maintain the steel box shape and to reduce

transverse bending and longitudinal warping stresses in the box girder. External bracings

are also required between the boxes, as a good practice for curved bridges, to resist

torsion and distortion. Certainly, at all external brace locations, there must be a brace

placed inside the box girder to assist in the force transfer. To study the effect of cross

bracings on the structural response, a three-lane two box-girder bridge of 60-m span and

span-to-radius of curvature of 0.8 was analyzed for AASHTO truck loading. Internal

bracings, inside the boxes, and external bracings, between the boxes, were examined with

respect to their numbers between the support lines and their combined effect.

The relationship between distribution factor for tensile stress and number of cross

bracing is shown in Figure 7.50. It is evident that providing external bracing in addition to

the internal bracing slightly improves the distribution factor for tensile stress. Maximum

distribution factor for tensile stress occurs when the number of bracing is 5, at a spacing

of 10 m. Providing more cross bracings enhances marginally the distribution factors.

Figure 7.51 presents the change in the distribution factors for compressive stress versus

the niimber of cross bracings. Significant improvement in the distribution factor for

compressive stress is gained with increase in the number of bracing from 1 to 5. No

further enhancement can be realized by adding more cross bracings. Again, the graph

145


shows that the presence of external bracings in addition to internal bracings does not

improve the load distribution significantly. No change is shown in the distribution factors

for deflection with increase in the number of bracings, as illustrated in Figure 7.52. It can

also be observed that external bracings do not help in reducing these factors. Figure 7.53

shows the changes in the distribution factor for shear with the number of cross bracings.

No significant effect is noted with increase of the number in cross bracings or with the

presence of external bracings. Figure 7.54 shows the same trend for the distribution

factors for exterior reaction and number of cross bracings. The influence of number of

bracings on the distribution factors for interior support reaction and distribution factors

for uplift reaction are presented in Figures 7.55 and 7.56, respectively. It can be observed

that for five bracing or more, distribution factors remain unchanged with increase in the

number of bracings. It is interesting to note that providing external bracing in addition to

internal bracing can reduce the distribution factors for uplift reaction by approximately

10%.

Figures 7.50 through 7.56 present also the effect of number of bracings on the

distribution factors for tensile stress, compressive stress, deflection, shear force, exterior

support reaction, interior support reaction, and uplift reaction for bridges under only their

self-weight. It can be observed that the graphs show similar trends to those for bridges

subjected to AASHTO live load. For number of bracings of 5 and more, the distribution

factors continue to be almost unchanged. It should also be noted that providing external

bracing maybe necessary only during casting of the concrete deck slab to prevent twisting

of the steel box girders. Thus, external bracing seems mostly unnecessary as permanent

members. However, it is good practice not to remove them since their removal can be

quite cumbersome and costly.

146


7.14 Effect of Different Type of Live Loading

The American Association of State Highway and Transportation Officials,

AASHTO 1996 [3] specifies HS20 truck semi-trailer combination in their loading. Also,

the equivalent AASHTO lane loading, consisting of a concentrated loading that is 80 kN

for bending moments and 116 kN for shear, together with a uniformly distributed load of

9.34 kN/m for all span lengths. In the parametric study conducted herein, both AASHTO

truck and lane loading were considered in order to arrive at maximum responses. In 1998,

AASHTO published a new LRFD Bridge Design Specifications [7], where the loading

consists of a design truck, or a tandem, coincident with a design lane load. The design

truck is effectively the old AASHTO truck loading. The Canadian Highway Bridge Code,

CHBDC 2000 [20] specifies two types of loading namely; CL-625 truck loading and CL-

625 lane loading. The CL-625 lane load consists of a CL-625 truck, with each axel

reduced to 80% of its normal value, together with a uniformly distributed load of 9 kN/m,

on a lane that is 3.0 m wide. In Ontario, CL-625 truck loading is replaced by CL-625-

ONT truck loading. Figure 7.57 shows the AASHTO LRFD and CHBDC truck loads. To

examine the effect of using any of these vehicular loadings on the distribution factors

derived herein, three bridges. Bridge 1, Bridge 2, Bridge 3, were analyzed using the

fmite-element method and the corresponding distribution factors were computed. The

three bridges were of two-equal-spans, continuous and horizontally curved bridges with

spans of 20, 60 and 100 m, respectively. Bridge 1 had two-lane two-box girder with span-

to-radius of curvature ratio of 0.4. Four-lane three-box girders were considered for Bridge

2 with span-to-radius of curvature ratio of 0.8. Bridge 3 was a four-lane six-box girder

brodge with a value of 1.2 for span-to-radius of curvature ratio. The multiple presence

factors specified in each code were also included in the reported results.

147


Distribution factors for tensile stress are compared in Figure 7.58. It appears that

CL-625 and CL-625-ONT lane loads produce the same distribution factors. It can be

observed that AASHTO truck load gives higher values for the distribution factors for

tensile stress than those calculated on the basis of AASHTO LRFD, CHBDC and

CHBDC-ONT truck loadings. Figure 7.59 shows the variation of distribution factors for

compressive stress for bridges subjected to live loads given by the various codes. Results

based on AASHTO live loads are relatively higher than those obtained by considering the

other loadings. For shorter span length, the graphs show insignificant variation in the

distribution factors for compressive stress for all live load types examined herein.

However, for larger span length, the difference between the distribution factor for stress

based on AASHTO code and those obtained based on the other codes increases. Similar

observation can be made in Figure 7.60 for distribution factors for deflection.

Distribution factors for shear force obtained using the AASHTO, AASHTO

LRFD, and CHBDC truck loadings are compared in Figure 7.61. It can be observed that

these factors fluctuate with changes in the span length. For Bridge 1 and 3, AASHTO

truck loading gives higher results than the results corresponding to the other loadings.

However, in the case of Bridge 2, AASHTO truck loading produce lower results by

almost 5% than those obtained from CL-625-ONT. Figures 7.62 through 6.64 present the

effect of truck loadings on distribution factors for exterior, interior and uplift reactions,

respectively. In almost all cases, AASHTO truck loading provides conservative results for

design when compared to the other loadings considered herein.

148


7.15 Illustrative Design Example

Given a two-lane, two-box, curved two-equal span continuous steel girder bridge,

with a composite concrete deck slab. The box girders are simply supported at the

abutments and continuous over the pier. The bridge, designed to accommodate the

AASHTO live load (1996), has the following details: Length of one span = 45.0 m; deck

width = 9.75 m; k =L/R =0.5; deck slab thickness = 200 mm; bottom flange width =

2.4375 m and its thickness = 24 mm; web depth = 1.80 m and its thickness = 16 mm; top

steel flange width = 360 mm and its thickness = 38 mm; modulus of elasticity of steel =

200 GPa; and modular ratio = 7.4. The wearing surface is assumed to be distributed

uniformly and is = 1.2 kN W . Calculate the maximum design stresses and maximum

deflection.

The moment of inertia of an idealized girder including the concrete deck slab, I =

0.0875 m“, and the distance from the neutral axis to the bottom fibers is y = 1.277 m.

Taking the concrete and steel densities as 2,400 and 7,800 kg/m \ respectively, the total

dead load per meter length for the bridge is 20.8 kN/m. Based on this value and assuming

that the distribution for dead load is uniform, the maximum dead load positive and

negative moment in a two-45 m equal span continuous idealized girder are: 2,948 kN.m,

5,265 kN.m, respectively, and the maximum deflection and shear force are 26 mm and

585 kN, respectively. Exterior and interior support reactions are calculated as: 35land

1170 kN, respectively.

From the simple beam bending formula, the maximum tensile and compressive

stresses in the idealized girder due to the above dead load are 43.0 MPa and 76.9 MPa,

149


respectively. The distribution factors for the continuous two-span bridge are now

calculated. From (7.8) and (7.9) the distribution factors for maximum tensile and

compressive stresses = 1.24 and 1.24, respectively; from (7.10) and (7.11) the distribution

factor for maximum deflection and shear force = 1.50 and 1.12, respectively. The

distribution factor for exterior, interior, and minimum support reactions obtained from

Eqs. (7.12), (7.13) and (7.14) are 1.84, 1.12, and -0.04, respectively. Based on these

distribution factors and the results from the two-span continuous idealized girder under

its own self-weight, the dead load design values for the curved continuous composite box

girder bridge according to (6.8) through (6.14) become: Op = 53.3 MPa; o„ = 95.4 MPa; 6

= 39.0 mm; V = 655 kN, Re = 646 kN; Rj = 1312 kN; R„ = -14.3 kN.

For maximum stresses, deflection, shear, and reactions due to live load on the

bridge, the procedure is to calculate first the maximum moment at the mid-span of a two-

span continuous idealized girder of 45 m span, loaded by: either a line of AASHTO

wheel loads or half the lane loading in additional to half the concentrated load. The

resulting maximum positive moment at mid-span and the negative moment in the vicinity

of the pier support are 1,270 kN.m, 1,530 kN.m, respectively, and the maximum

deflection, 5a, and shear are 13 mm and 188 kN, respectively. Maximum exterior and

interior support reactions are 149 and 320 kN, respectively. From the simple beam-

bending formula, the maximum tensile and compressive stresses, resulting from the

above moments are determined as: Opa = 18.5 MPa, and a„a = 22.3 MPa. The distribution

factors for the continuous two-span bridge are now calculated. From (7.1) and (7.2) the

distribution factors for maximum tensile and compressive stresses = 1.20 and 1.21,

respectively; from (7.3) and (7.4) the distribution factor for maximum deflection and

shear force = 1.26 and 1.36, respectively; from (7.5), (7.6), and (7.7) the distribution

150


factors for exterior, interior, and minimum support reactions = 1.71, 1.34, and -0.54,

respectively. Thus, based on these distribution factors and the above results for maximum

tensile and compressive stresses, deflection, shear, and reactions for the two-span

continuous idealized girder, the design values due to live load for the box girder bridge

according to (6.1) through (6.7), become: Op == 22 MPa; a„ = 27 MPa; 6 = 0.016 m; V =

256 kN; Re = 258 kN; Ri = 429 kN; R„ = -80 kN.

Analyzing this bridge by the finite-element method, the maximum tensile and

compressive stresses obtained due to the dead load are 47 and 86 MPa, respectively, and

the maximum deflection is 34 mm. Also, for AASHTO live load, the maximum tensile

and compressive stresses are 20 and 26 MPa, respectively. The deflection due to the

AASHTO live load obtained from the finite-element analysis model is 16 mm.

Comparing the results from finite-element analysis with the ones obtained above using

the empirical formulas derived herein, it can be observed that they are in fair agreement.

For comparison, the bridge in the illustrated deign example was subjected to

AASHTO LRFD live load (1998) and also to CHBDC live load (2000). Following the

same procedure, the resulting maximum live load positive moment near mid-span and

negative moment at the interior support of a two-span continuous idealized girder were

calculated. The maximum deflection near mid-span and shear force along the span were

obtained. Also, the maximum exterior and interior support reaction forces were

calculated. The results obtained from the finite element analysis and the empirical

formulas were compared in Table 7.1. The comparisons show a fair agreement between

the results predicted by the finite element and the proposed formulas.

151


7.16 Summary

The load distribution factors in continuous curved concrete deck on multiple steel

box girder bridges, subjected to AASHTO live load and dead load, were examined.

Parametric study was conducted to evaluate the maximum tensile, and compressive

stresses, deflection, shear force, exterior support reaction, interior support reaction and

minimum support reaction (uplift) using the calibrated three-dimensional fmite-element

model. Based on results obtained from the parametric study presented in this chapter, the

following conclusions can be drawn:

1. The bridge span, number of lanes, number of boxes and span-to-radius of

curvature ratio are the key parameters affecting the distribution factors.

2. Empirical expressions for the aforementioned distribution factors were deduced

for straight and curved two-span concrete-steel multiple box girder bridges. These

expressions have been shown to be reliable, simple to apply and provide

information unavailable in the current codes. They obviate the need for a rigorous

analysis involving the repeated use of three-dimensional finite element program

for different load conditions.

3. The proposed expressions for the load distribution factors can be used for the

design of equal-span continuous curved multiple box girder bridges with number

of spans varying from 2 to 4.

152


4. Web slope has no significant effect on the distribution factors. Therefore, the

proposed formulas are applicable for bridges with web slope not exceeding 1 to 4.

5. The proposed expressions can be applied with confidence for bridges with span-

to-depth ratio of 25 to 30.

6. Reducing the spacing of cross bracings from 7.5 m does not improve the structural

response for the continuous curved composite multiple box girder bridges in terms

of load distribution factors. The proposed expressions are valid if internal bracings

are provided at equal intervals between the support lines with spacing not

exceeding 7.5 m.

7. The proposed expressions for distribution factors can also be used for the design

of bridges subjected to AASHTO LRFD or CHBDC live loads.

153


CHAPTER VIII

Impact Factors

8.1 Introduction

In this chapter the finite element analysis of dynamic impact factors for

continuous straight and curved composite multiple-box girder bridges is described. The

main eriteria that affeet the dynamic response of curved continuous composite box girder

bridges are investigated. Methodology of the vehicle idealization, loading positions and

the vehicle speed are first presented. Then, forced vibration analysis methods are

described and compared. Stability and accuracy of the numerical methods are discussed.

Consequently, a suitable time step interval for the dynamic analysis method and damping

coefficient are selected.

Sensitivity study is conducted to decide on the unfavourable vehicle loading

conditions and the vehicle speed adapted in this dynamic analysis for such bridges.

Comparison between the mode superposition and direct integration methods is made to

choose the most effective dynamie analysis method in evaluating the impact factors. An

extensive parametric analysis is then undertaken to examine the effects of the various

aspects on the impact factors of such bridges. Results obtained from the parametric study

of 180 continuous curved composite multiple-box girder bridges are analyzed and

compared. The effects of number of lanes, number of boxes, bridge span length and the

span-to-radius of curvature ratio are discussed.

154


Then empirical relationships for the impact factors are proposed for the design of

continuous curved composite multiple-box girder bridges. The proposed equations are

compared with the current expressions specified in the AASHTO 1996, AASHTO 2003,

CHBDC 2000, and AASHTO LRFD 1998 codes [3, 5, 20, 7].

8.2 Vehicle Idealization

The HS20-44 truck of three-axle tractor-trailer type in AASHTO specifications

was considered herein. In this study, the vehicle was idealized as a pair of concentrated

forces moving along the concrete deck in a circumferential path with a constant speed. As

mentioned before, this study was conducted on two-span curved composite multiple steel

box girder bridges. Therefore, the mass of the structure was much larger than the mass of

the design vehicle. In 1968, Tan and Shore [137, 138] have shown that when the mass of

the design vehicle to the mass of the bridge is less than 0.3, the mass of the vehicle can be

neglected in the dynamic analysis and the vehicle can be idealized as concentrated forces.

Similar observation was made also by Senthilvasan et al. in 1997 [132]. The concentrated

forces were considered to act normal to the deck in the radial path and separated by 1.8

m, as shown in Figure 8.1. It was also assumed that the vehicle axles remain parallel to

the bridge surface during motion and the wheels were always in contact with the bridge

deck. The roughness and the superelevation of the bridge surface were neglected in this

study.

According to the AASHTO code, the vehicle weight was assumed to act 2.0 m

above the surface of the bridge. The centrifugal forces, Fc, developed in the case of a

curved bridge by the vehicle traveling on the bridge surface, can be expressed as follows;

155


PV^(8.1)127r

where P = weight of an axle (kN)

V = vehicle speed (km/h)

r = radius of the path on which the vehicle centre is traveling (m)

This centrifugal force was considered to act at the vehicle centre 2.0 m above the

surface of the bridge. The centrifugal forces were then replaced by equivalent horizontal

and vertical forces acting at the mid-depth of the concrete slab on which the magnitude of

the vertical components for the outer and inner wheel load location were represented by

Fo and Fj, respectively, as follows

F ,= ( P /2 + 1.17FJ (8.2)

F i= ( P /2 - 1 .1 7 F J (8.3)

Also, the horizontal forces in the radial direction were considered as Fc/2 at each wheel

load location. It should be noted that the inner vertical force (P/2 -1.17 Fc) was not

allowed to act in the opposite direction as a pull force on the bridge surface. Figure 8.2

shows the vertical and horizontal forces for the idealized vehicle used in evaluating the

impact factors.

156


8.3 Vehicle Loading Positions

The maximum static and dynamic responses for the compressive stresses at the

interior support were produced by applying two vehicles in the longitudinal direction, one

on each side of the interior support. Therefore, in all the cases considered in this study,

two truck loadings were positioned apart at a certain distance in the circumferential path.

This distance was selected in such a way as to produce maximum compressive

longitudinal stresses at the interior support.

In the search for the most critical loading position in the transverse direction to

obtain the impact factors, a four lane-four box curved bridge with a span length of 20 m

and a span-to-radius of curvature value of 0.4 was analyzed. Four vehicle loading

position, as shown in Figure 8.3, were considered as follows:

1. The outer lane was loaded with two vehicles, where the wheel loads were

located at a distance of 0.6 m from the curb.

2. The outer two lanes were loaded with four vehicles, where the wheel loads

for the most-outer lane were at a distance of 0.6 m from the curb.

3. The outer three lanes were loaded with six vehicles, where the most outer

wheel loads were placed near the curb as in the previous two cases.

4. All lanes were loaded with eight vehicles and the loads were positioned in

the centre of the lane.

157


Some typical time histories for the curved four-lane four-box bridge with a span

length of 20 m and a radius of 50 m, as shown in Figures 8.4 to 8.7, were calculated for a

truck speed of 50 km/h. The abscissa in the figures is the distance measured from the left

end of the bridge to the front axle of the front vehicle. These figures present the time

histories of tensile stress, compressive stress, reaction force at the interior supports and

shear force at the interior support. Based on the results in these figures, it is observed that

the ease of fully loaded lanes provides the maximum dynamic response in the bridge. It

should be noted that the multiple lane load reduction factors have not been applied for

this parametric study.

8.4 Vehicle Speed

The maximum safe allowable highway speed of 100 km/h is considered in most

regions in North America. In the case of curved bridges, the maximum allowable highway

speed is restricted by the degree of the superelevation and the coefficient of side friction

force between the truck tire and the road pavement [96], and it is expressed as follows:

V = V l 2 7 r ( e + f ) (8.4)

where e = superelevation, it is considered positive if the slope of the bridge

surface is toward the centre of the curvature.

f = coefficient of side friction force between the vehicle tire and the road

surface.

158


r = radius of the path along which the centre of the truck is travelling (m);

and,

V = design speed of the highway (km/h)

The superelevation of curved bridges is controlled by the design speed and the

weather conditions throughout the year at the bridge site. In severe winter condition

region, the maximum superelevation rate of 0.06 has been recommended [96]. However,

generally in most regions a maximum slope of 0.08 can be applied where high

maintenance prevails and little ice or snow accumulation is anticipated. For the purpose

of this study a value of 0.06 for the superelevation was adopted for all vehicle-loading

cases. It should be noted that applying a high rate of superelevation produees higher

allowable vehicle speed.

The friction between the truck tire and the bridge surface depend mainly upon the

type of bridge pavement, the condition of the bridge surface, the type of vehicle tire and

the vehicle speed. The maximum lateral friction influences significantly the driver’s

comfort. However, maintaining the centrifugal force on the curved bridges through lateral

friction rather than the superelevation causes discomfort for passengers. By increasing the

value of lateral friction, the allowable vehicle speed can be increased. Thus, a maximum

friction coefficient of 0.18 is considered for all cases throughout this study.

To select a suitable vehicle speed for the straight bridge, the effect of vehicle

speed was investigated for a 4l-6b-20 straight bridge. Figure 8.8 shows the tensile stress

history for three different vehicle speeds. It should be noted that with increase in the

159


vehicle speed the maximum tensile stress increases. However, the change in the

maximum tensile stress is not pronounced. Similar effect is presented in Figure 8.9 for

the maximum compressive stress history. It is shown in Figure 8.10 that the increase in

the vehicle speed to 140 km/h produces a slightly higher reaction force at the interior

support. However, as seen from Figure 8.11, an insignificant change in the shear force at

the interior support occurs with increase in the vehicle speed.

In the light of above discussion, it was decided to choose a value of 120 km/h for

the maximum vehicle speed for the straight bridge. Also, an increase in the vehicle speed

with a value of almost 20 % for all eases on the curved bridges was considered. This

increase in the vehicle speed was selected for practical reasons, based on the fact that

drivers usually travel with a speed higher than the posted highway speed. Table 6.3 shows

the vehicle speeds chosen for the dynamie analysis.

8.5 Mode Superposition versus Direct Integration Method

Two types of implicit methods for dynamic analysis were discussed in Chapter 111.

A fully loaded 2l-2b-20 m straight continuous bridge was analyzed using the Mode

Superposition and Direct Integration methods.

The dynamic responses of the bridge were calculated using the case of mode

superposition method by considering 50, 100 and 200 mode shapes. These responses

were then compared to the results obtained from the direct integration method. Figure

8.12 illustrates the maximum tensile stress history for the four dynamic analysis cases.

Obviously, considering only the contribution of 50 modes, the mode superposition

160


method underestimates slightly the maximum tensile stress. However, by increasing the

number of modes to 200 the estimation of the maximum tensile stress becomes very close

to the dynamic response obtained from the direct integration method. In the case of the

maximum compressive stress shown in Figure 8.13, it can be observed that the maximum

compressive stresses are close in all cases. Figures 8.14 and 8.15 depict the reaction force

and shear force history at the interior support. It can be seen that choosing only 50 modes

in the mode superposition method led to inaccurate solution. Thus, the predicted dynamic

responses of the bridge improve significantly by increasing the number of contributing

modes considered in the solution. This may be due to the phenomenon of natural

frequency clustering as reported by Galdos in 1988 [58].

As a result of the aforementioned sensitivity study and in order to obtain good

accuracy in the solution either higher number of the modes must be considered in the

solution using the mode superposition method or the analysis should be conducted using

the direct integration method. It was observed that the required time for the analysis using

mode superposition method considering higher number of modes is not less than that

needed by the direct integration method. That is due to the necessity of conducting a free

vibration analysis first in the case of mode superposition method to calculate the required

natural frequencies. Therefore, it was concluded that the direct integration method was

the most suitable method to achieve acceptable accuracy and reasonable computation

time in the dynamic analysis of the bridges.

161


8.6 Stability and Accuracy

Obtaining a good approximation to the actual dynamic responses of the bridges

under consideration is the goal of the numerical integration of the finite element system

equilibrium equations. To achieve this goal, all system equilibrium equations

significantly contributing to the dynamic responses of the structure must be integrated to

a high precision. This means that the selection of the time step, At, is crucial to the

accuracy and the stability of the solution. Theoretically, the time step At would have to be

almost T/10 [12], where T is the smallest period. However, this assumption is not

required in practice. Generally, the dynamic response of a structure is predicted by

inclusion of only the first few modes. Thus, the dynamic response can be determined

accurately by integrating only the first few effective modes. Thus, the choice of At would

be selected according to the smallest period for the most contributing modes instead of

the smallest period for all modes of the structure.

The implicit method used by ABAQUS/Standard is unconditionally stable. This

means that for any time step At, the estimated solution for any initial condition does not

grow without bound. In addition, the accuracy in the predicted dynamic response of the

structure can be preserved in the unconditionally stable integration method by choosing

the time step At small enough that the response in all modes that contributed significantly

to the total structural response is determined accurately. However, it should be noted that

the other model response components are not calculated precisely but the produced errors

are not critical for the obtained dynamic response.

162


As verification of the previous considerations, two continuous curved bridges,

with L/R = 0.4, were analyzed. Both bridges were fully loaded with truck loading. The

first bridge had a span length of 20 m while the second one had a span of 60 m. Dynamic

responses for both bridges were obtained at various time steps using the direct integration

method. For the 20 m span bridge, the dynamic responses using time steps with values of

0.002, 0.010, 0.015 and 0.020 seconds were predicted. However, in the case of 60 m span

bridge, values of the time steps considered in the dynamic analysis were 0.005, 0.014 and

0.020 seconds.

Tensile and compressive stress histories are plotted in Figures 8.16 and 8.17. It is

clear that the dynamic responses for the tensile and compressive stresses obtained using a

value of 0.002 seconds for At are higher by about 6% than those calculated for larger time

steps. However, the difference in the results is not significant in both responses. The

results of the reaction force and the shear force are presented in Figures 8.18 and 8.19. It

should be noted that carrying out the integration of the equilibrium equations for the

maximum reaction force using a time step with a value of 0.002 seconds is higher by

about 14% than those predicted using a larger time steps. However, this effect of the time

step on the maximum shear force is less than 1%. Similar dynamic responses for the 60

m-span bridge are illustrated in Figures 8.20 to 8.23. It is interesting to note that there is

no marked effect of the time step on the tensile stress history. The dynamic response for

compressive stress, reaction force and shear force are somehow affected by reducing the

time step. However, the differences in estimating the maximum values are less than 5%.

Cleary, reducing the time step would provide more accuracy in the prediction of

the dynamic response of the bridges. However, it will increase dramatically the analysis

163


time and consequently the cost. Therefore, as a result of the previous investigations of the

two bridges the time step with a value of 0.020 seconds was selected in the case of 60 m,

80 m, and 100 m span length. On the other hand, a value of 0.01 seconds was applied in

the dynamic analysis of 20 m and 40 m span length bridges.

8.7 Damping Effect

It is well known that the magnitude of oscillation of a structure decreases until the

oscillations stop due to energy dissipation. This energy dissipation in a structure is called

damping. By applying the general direct integration method provided in

ABAQUS/Standard, called Hilber-Hughes-Taylor operator, an artificial damping is

introduced. This damping is purely numerical. The a-parameter introduces damping that

grows with the ratio of the time step to the period of vibration of a mode. The parameter,

a, can take any value from 0, which gives no artificial damping, to -0.33 which provides

the maximum artificial damping available from this operator.

The dynamic analysis of 3l-3b-60 m straight bridge was conducted. The histories

of tensile stress, compressive stress, reaction force and shear force at the interior support

were calculated for a = 0, -0.05, and -0.33. The damping effect on the tensile stress and

reaction force are shown in Figures 8.24 and 8.25, respectively. It should be noted that the

maximum results of all the dynamic responses were only slightly different for all the

above artificial damping values. Therefore, a value of 0.0 for a was selected in all cases

considered in the parametric study.

164


8.8 Dynamic Impact Factor

Maximum vertical loads caused by a travelling vehicle across a bridge will often

exceed those exercised by an equivalent static vehicle due to the dynamic effects [37].

This increase in the vertical loads has commonly known as Impact Factor, I, or Dynamic

Load Allowance. According to AASHTO Guide Specifications, the dynamic design

values can then be obtained from the following relationship

Maximum dynamic response = Maximum static response (1 + 1) (8.5)

While the impact factor considered in this study is defined as

I (o/o) X 100 (8.6)Rs

where Rd = maximum dynamic response

Rs = maximum static response

Dynamic and static analyses were conducted on each bridge for the same applied

loads in both analyses. Maximum dynamic and static responses for tensile stress near the

mid span, compressive stress at the interior support, deflection near the mid span, reaction

force at the exterior supports, reaction force at the interior support, uplift reaction, shear

force at the exterior support and shear force at the interior support were determined for all

bridges considered herein. Thus, impact factors were calculated by applying Equation 8.6

for the 180 bridges analyzed in this parametric study. In the longitudinal direction of the

165


bridge, two trucks for each lane were applied. The trucks were travelling apart from each

other with various distances depending upon the span length. The distances between the

trucks were chosen in such a way as to produce maximum negative moment on the

interior support for a bridge. The vehicle speeds considered in this study for all bridges

are presented in Table 6.3.

As previously mentioned in section 8.3, the case of fully loaded lanes with truck

loading, in general, produces the maximum straining actions in the structure. Therefore,

this loading case is used for all bridges analyzed in the parametric study to evaluate their

impact factors. It should be mentioned that the time required to obtain the dynamic

response of one bridge on a Solaris 9 Sun Fire 4800 with 4 CPUs and 4 GB memory

using ABAQUS/Standard is more than four days in some cases.

8.9 Parametric Study

The literature suggests that the impact factor of vehicle loads travelling on a

bridge depends upon a wide range of parameters. In this study the following four main

factors were selected to have the most influence on the dynamic behaviour of continuous

curved composite multiple-box girder bridges; namely: 1) Number of bridge lanes; 2)

Number of bridge boxes; 3) Bridge span length; and, 4) Span-to-radius of curvature ratio.

In the parametric study, all these parameters were thoroughly investigated. In the

following section, results are presented in graphs and finally expression for impact factors

are derived.

166



In this study, the number of lanes considered herein was two, three and four. Since

the number of lanes is a clear indication of the variation of the bridge width, this

parameter is considered as a key factor in the bridge geometry and consequently, one that

significantly influences the bridge behaviour under dynamic loads. To investigate the

effect of number of lanes on the impact factors a four-box bridge with a span length of 20

m was analyzed for moving vehicles.

Impact factor for tensile stress is plotted against the number of lanes in Figure

8.26. The graph shows that the values of impact factor do not follow a certain pattern for

bridges with increase in the number of lanes. For curved bridges, it can be observed that

there is no general trend for the impact factors with change in the number of lanes.

Similar observations were made for impact factors for compressive stress, deflection,

shear, and reaction. Based on the above results, it is interesting to note that the number of

lanes, and hence the bridge width, does not have a patterned effect on the impact factors

for continuous curved composite multiple-box girder bridges.


In this section, the effect of the number of boxes on the impact factors for straight

and curved bridges is investigated. Two lanes-20 m bridges with L/R= 0.0, 0.1, 0.2 and

0.4 are considered in this study.

167


As shown in Figure 8.27 the impact factors for tensile stress are plotted against

the number of boxes. It can be observed that the calculated results for the impact factors

for straight bridges increase slightly with increasing the number of boxes. However, in

the case of curved bridges the impact factors do not change in any systemic manner with

the number of boxes. In general, the effect of number of boxes on the impact factors for

continuous curved concrete deck on multiple steel box girder bridges does not follow any

pattern. Basically, the variation of the impact factor values with number of boxes is not

pronounced in most cases.


The effect of the bridge span length on the impact factors for tensile stress is

shown in Figure 8.28. For straight bridges, it is clear from this figure that the impact

factors decrease with increasing span length of the bridge. Similar observations are

obtained for impact factors for compressive stress, deflection, shear, and reaction. In

summary, for straight bridges, the impact factors for tensile, and compressive stresses,

exterior support, and interior support reactions decrease with increase in the span length.


Impact factor values for two selected bridges with span length of 20 m and 40 m,

respectively, were calculated. Each bridge had two lanes and two boxes. The vehicle

speed eonsidered in this examination varied according to the radius of curvature in each

case. The effect of span-to-radius of curvature ratio on the impact factor for tensile stress

is presented in Figure 8.29. While no definite trend is observed in the case of 2/-26-20

168


bridges, bridges with span length of 40 m show a uniform increase in the impact factor.

Similar behaviour is observed for the impact factors for compressive stress, deflection,

shear, and reaction as those obtained for the impact factors for tensile stress for both

bridges. Thus, the effect of the span-to-radius of curvature on the impact factors of curved

or straight bridges does not follow any pattern of behaviour.

8.10 Expressions for Impact Factor

In the previous sections, the effects of several variables on the impact factors in

straight and curved continuous concrete deck on multiple steel box girder bridges were

examined using the finite element computer program ABAQUS/standard. The maximum

selected design speed as shown in Table 6.3 and the full loaded lane condition were

applied for all bridges considered in this parametric study. Impact factor results were

plotted versus the number of lanes, munber of boxes, bridge span length and the span-to-

radius of curvature ratio. In some cases, the impact factors have shown a clear trend with

regard to a specific variable. However, in many cases no pattern was observed. As

expected, impact factors for stresses, deflection, reaction forces and shear forces have

been influenced differently by each of the above variables. This is due to the complicated

structural dynamic behaviour of continuous curved composite multiple-box girder

bridges.

Previous researchers often related the fundamental frequency of a bridge to its

impact factor [108, 22]. Also, bridge span length was considered as the most important

influential factor in estimating impact factors in most codes [3]. Thus, the next section

deals with impact factors expressed as a function of each of the following: the

169


fundamental frequency; and, the bridge span length. In addition, while the span-to-radius

of curvature ratio of a bridge plays a vital role in the overall structural dynamic behaviour

of bridges, expressions for impact factors are also given in terms of the span-to-radius of

curvature ratio.

The main objective of this study is to proposed simplified and practical

expressions for impact factors. Therefore, upper bound values for the impact factors in all

cases were established as follows.

8.10.1 Impact Factor as a Function of the Fundamental Frequency

The impact factor results for all analyzed straight and curved bridges for tensile

stress, compressive stress, deflection, exterior support reaction, interior support reaction,

uplift, shear at the exterior support and shear at the interior support are plotted against the

bridge ftmdamental frequencies as shown in Figures 8.30 though 8.37. It is interesting to

note that there is a difference between straight and curved bridges for low frequencies.

However, results for straight bridges with f =5 Hz fall into the range of values for the

curved bridges with the same frequency. That is mainly due to the fact that small span-to-

radius of curvature ratio was assumed for short span bridges. Impact factors are presented

separately for straight bridges to emphasis at the change in the impact factor values due to

the bridge curvature. Expressions based on the upper bovmd values for the impact factors

for curved bridges were derived as follows:

1) Impact Factor for Tensile Stress (Figure 8.30)

170


Ip = 2 3 -2 .5 f (8.7)

2) Impact Factor for Compressive Stress (Figure 8.31)

I „ = 2 7 - 2 . 4 f (8.8)

3) Impact Factor for Deflection (Figure 8.32)

I d = l l + 0.5f (8.9)

4) Impact Factor for Exterior Support Reaetion (Figure 8.33)

I , e = 2 2 - f (8.10)

5) Impact Factor for Interior Support Reaction (Figure 8.34)

I , i= I8 + 0.5f (8.11)

6) Impact Factor for Uplift Reaction (Figure 8.35)

I u = 1 6 0 (8 .1 2 )

7) Impact Factor for Shear at Exterior Support (Figure 8.36)

I s e = 1 6 - 2 f (8.13)

171


8) Impact Factor for Shear at Interior Support (Figure 8.37)

I s i = 3 7 - 4 f (8.14)

where f is the fundamental frequency in Hz.

8.10.2 Impact Factor as a Function of Bridge Span Length

The values of impact factors are presented as function of the bridge span length in

Figures 8.38 through 8.45. The available expressions for impact factors given in

AASHTO 1996, AASHTO 2003, CHBDC 2000 and AASHTO LRFD 1998 guide

specifications [3, 5, 20, 7] are also presented for comparison. Straight bridge results are

first plotted separately to show the variations in the impact factor values when

considering the results from curved bridges. Upper bound impact factors as a function of

the curved bridge span length were deduced as follows:


Ip =21 (8.15)


I „ = 2 5 (8.16)


172


Id =13 (8.17)

4) Impact Factor for Exterior Support Reaction (Figure 8.41)

h = 2 0 (8.18)

5) Impact Factor for Interior Support Reaction (8.42)

I r i =2 0 (8.19)


lu = 160 (8.20)


I s e = 2 4 ( 8 . 2 1 )


Isi = 34 (8.22)

173


8.10.3 Impact Factor as a Function of Span-to-Radius of Curvature Ratio

In this section, results of the impact factors are presented as functions of span-to-

radius of curvature ratio in Figures 8.46 to 8.53. Considering only this parameter, upper

bound expressions were developed for the impact factor values as follows:


l p = 1 0 ( l + L / R ) (8.23)


1 „ = 5 ( 4 + L/R) (8.24)


Id = 1 4 - 3 (L /R ) (8.25)

4) Impact Factor for Exterior Support Reaction (Figure 8.49)

Ire = 20 (8.26)

5) Impact Factor for Interior Support Reaction (Figure 8.50)

l,i = 2 2 - 1 0 (L/R) (8.27)

174



I„ =190-135(L/R) (8.28)


I,, = 2 9- 13 (L/R) (8.29)


I,i = 1 6 -1 5 (L /R ) (8.30)

Since all values obtained from the three equations for impact factors for any

structural quantity are upper bound, the designer is free to use any of the equations for a

safe design.

8.11 Summary

The direct integration method was applied to calculate the dynamic response of

continuous straight and curved composite multiple-box girder bridges. A suitable time

step At was selected in such a way to maintain low computational cost for the analysis

without jeopardizing the accuracy of the solution. Finite element models of bridges were

subjected to an idealized vehicle loading in all bridge lanes. The vehicle speed was

selected as 120 km/h for the straight bridges. However, for curved bridges the vehicle

speed was reduced according to the span-to-radius of curvature ratios. It should be noted

175


that the chosen vehicle speeds are in all cases higher than the maximum allowable

highway speed for safety reasons. Based on the results of the dynamic analysis carried out

in the parametric study, the following conclusions on the impact factors of these bridges

can be made:

1. No discernible influence of the number of lanes and boxes on the impact

factors was observed.

2. In the case of straight bridges, the impact factors for tensile, and

compressive stresses, deflection, exterior, and interior support reactions,

decreased, in general, with increase in the span length. However, this trend

was not the case for curved bridges.

3. The span-to-radius of curvature ratio has a measurable influence on the

magnitude of the impact factors regardless of the span length of the bridge.

4. Impact factors for tensile and compressive stresses, deflection, exterior,

and interior support reactions for straight bridges increase with higher

fundamental frequencies. However, this trend was not preserved when the

results of impact factor for curved bridges were considered.

5. Formulas given in AASHTO 1996 [3] underestimate, in most cases, the

impact factors for bridges with large span lengths.

176


6. Impact factors for uplift reaction are generally underestimated by all

current North American Codes.

7. Both AASHTO LRFD and AASHTO 2003 [5, 7] overestimate the impact

factors for tensile, and compressive stresses, deflection, reactions, and

shear at the exterior support.

8. Impact factors based on CHBDC 2000 [20] appear to be the closest values

to those predicted by the proposed expressions. However, this Code

underestimates the impact factors for the uplift reaction and shear at the

interior support.

9. Given the scattered nature of most of the impact factors values obtained, it

was necessary to develop the impact factor expressions as upper bound

solutions from all results gathered from the parametric study.

177


CHAPTER IX

Fundamental Frequency

9.1 Introduction

There are several factors, which influence the dynamic characteristics of

continuous curved concrete deck-on-multiple steel box-girder bridges. In order to

consider some of factors for calculating the dynamic response, they need to be assembled

into a form that is convenient. This is normally achieved by characterizing a bridge

mainly in terms of its natural frequency and mode shape. By knowing the value of

fundamental frequency of a bridge, impact factors can be evaluated. Since most heavy

vehicle frequencies occupy a relatively narrow frequency band in practice, 1.5 to 4.5 Hz,

it is preferred to design bridges in such a manner as to avoid this critical range, if at all

possible. In addition, due to bridge vibration, the dynamic deflection can cause

discomfort to pedestrians using the bridge. It has been known that the human body tends

to react more to torsional oscillations than flexural ones.

Accordingly, it is crucial to offer a reliable method to evaluate accurately the

fundamental frequencies of multiple box girder bridges. Therefore, in this chapter, the

results of extensive analytical investigations on the vibration characteristics of continuous

curved concrete deck-on-multiple steel box-girder bridges are generated. The analysis of

the three-dimensional bridges was performed using the finite-element method. Empirical

expressions were then derived from these results to evaluate the fundamental frequencies

178


of such bridges. The influences of bridge spans, number of lanes, number of boxes and

span-to-radius of curvature ratios on the fundamental frequencies were examined.

Furthermore, the effects of number of spans, span-to-depth ratio, end-diaphragm

thickness and number of cross-bracings were also investigated. Forced vibration analysis

was also conducted to examine the peak acceleration due to AASHTO truck loading

passing along the bridge. Effects of truck speed, end-diaphragm thickness and number of

cross-bracings on the peak acceleration were also studied.

9.2 Effect of Span Length

The effect of bridge span length is considered as a major factor affecting the

fundamental frequencies of bridges. Results obtained from 180 continuous curved

concrete deck-on-multiple steel box-girder bridges were plotted in Figure 9.1(a). As

illustrated in the graph, flmdamental frequencies decrease considerably with increasing

bridge span length. It is interesting to observe, however, that for eaeh span length, the

values of the fundamental frequencies vary within a maximum range of 20 %. These

variations are caused by the effects of other parameters, such as number of lanes, number

of boxes or L/R ratios. Mode shapes were also derived for all bridges. For straight

bridges, the first mode shape is always purely flexural, which is not the case for curved

bridges. Figure 9.1(b) shows the mode shapes of 3/-2Z> bridges with L/R ratio of 0.6. It

appears that the contribution of the torsional effects to the first mode shape of continuous

curved multiple box-girder bridges decreases with inerease in the span length. In that case

the flexural effect becomes more predominant.

179


9.3 Effect of Number of Lanes

Four-box 20-m and 60-m bridges with L/R ratio of 0.4 were considered to

illustrate the effect of the number of lanes on the fundamental frequencies, as shown in

Figure 9.2(a). It is observed that the fundamental frequencies of 20-m bridges decrease by

almost 7% when the number of lanes increases from 2 to 4. However, in the case of

bridges with 60-m span length, the fundamental frequencies remain almost unchanged. It

is obvious from the results of for all bridges considered in this study that the number of

lanes has no pronounced influence on the fundamental frequencies. Regardless of the

number of lanes, the first mode shape for a straight bridge is purely flexural. However, for

curved bridges the torsional effects influence the first mode shapes. Figure 9.2(b) shows

that the torsional influence in the first mode increases slightly by increasing the number

of lanes.

9.4 Effect of Number of Boxes

The influence of number of boxes on the fundamental frequencies is presented in

Figure 9.3(a). Three-lane straight bridges with span lengths of 20 and 60 m and having

different number of box girders were analyzed. The results shown revealed that

increasing the number of boxes only slightly enhances the fundamental frequencies of the

bridges. An increase in the fundamental frequency of bridges with 60-m of about 7% was

noted when the number of boxes increased from 2 to 5, with almost 2% in the case of

bridges with 20 m span lengths. It should be noted that the effect is not significant due to

the fact that there is no change in the width of the bottom flange even when the number of

boxes is increased. The increase is in the number of webs. Results not shown herein

180


reveal that the number of boxes had no effect on the first mode shape of straight bridges.

However, it seems that increasing the number of boxes in curved bridges increases the

torsional effect on the first mode shape as shown in Figure 9.3(b). This may be attributed

to a decrease in the torsional stiffness with inerease in the number of boxes for the same

flexural stiffness.

9.5 Effect of Span-to-Radius of Curvature Ratio

The relationship between the fundamental frequency and span-to-radius of

curvature ratio (L/R) of three-lane two-box bridges with span lengths of 60 and 100 m are

presented in Figure 9.4(a). The graph shows that L/R ratio has a signifieant effeet on the

fundamental frequencies. Increasing the L/R ratio significantly decreases the fundamental

frequency. For instance, the fundamental frequency of bridges with span length of 60 m

decreased by about 29% when L/R ratio increased from 0 to 1.2, and by about 32% in the

case of bridges with span length of 100 m. It can be observed that first mode shapes of

curved bridges are also affected considerably by increasing the L/R ratio. For straight

bridges, the first mode shape is purely flexural for all bridge span lengths. However, this

behaviour changes by introducing curvature in the bridge geometry, where the torsional

effects play a significant part. Figure 9.4(b) shows the mode shapes of 3/-2Z>-100 bridges

with span-to-radius ratios of 0 and 1.2, showing the effect of torsion due to the bridge

curvature.

181


9.6 Empirical Expressions for Fundamental Frequency

Based on the results generated from the parametric study of continuous straight

concrete deck-on-multiple steel box-girder bridges, the following simplified empirical

equation was deduced for the fundamental frequency, fj:

94f ^ = - Hz (9.1)

The accuracy of this expression is within 3% for all straight bridges considered in

this parametric study regardless of the number of boxes or lanes. It should be noted that

same expression is valid for simply-supported straight bridges [119].

Since bridge curvature significantly affect the fundamental frequencies, a

modification factor is introduced in Eq. 9.1 to account for the curvature effect. The

modified expression for the fundamental frequency of curved bridges, fc, is given as:

f, = p . f, Hz (9.2)

where q is a modification factor defined as:

q = l - 0 . 0 71.5 f N , . L ] 0.3

u . N , J (9.3)

where L is the centreline arc length of the curved bridge in one span in meters; R is the

radius of curvature of the bridge centreline; Nb is the number of boxes; and Nl is the

182


number of lanes. It should be noted that the number of lanes and number of boxes do

affect to some degree the fundamental frequency of curved bridges. This is reflected in

Eq. 9.3. Equation 9.2 is quite accurate in predicting the fundamental frequency for two-

span curved concrete deck on multiple steel box girder bridges included in the parametric

study with an error not exceeding 3%. It should be noted that Eq. 9.2 applies only to span-

to-depth ratios of 25. For different such ratios, another equation was developed as shown

in a forthcoming section.

9.7 Comparison with Flexural Beam Theory

Flmdamental frequency of a continuous beam girder can be evaluated using the

flexural beam theory, as follows;

J - Hz (9.4)2L V m

where m is the total mass per unit length of the beam girder; I is the flexural moment of

inertia of the beam girder; L is the span length of the beam girder; and E is the modulus

of elasticity of the girder material. In order to include the curvature effect, Heins and

Sahin [68] had evaluated the fundamental frequency of curved box girders using the

differential equations of motion based on Vlasov’s thin-walled beam theory assuming that

the entire cross section at the supports would not twist. In practice, box-girders in bridges

are supported only at the bottom flanges underneath the webs. Accordingly, the end cross-

section of a bridge at the support line becomes more flexible than assumed, particularly

for bridges with small span-to-depth ratios.

183

Repro(juG8(j with permission of the copyright owner. Further reproctuction prohibitect without permission.

In this parametric study, results obtained from the three-dimensional fmite-element

models were normalized by the fundamental frequencies for a similar straight girder. The

section properties of the straight girder were taken as those obtained form the

corresponding bridge with span length equal to the centre line arc length. The

fundamental frequency of the straight girder can be evaluated using Eq. 9.4. Modification

factors were applied to the fundamental frequency of the straight girder to develop the

following empirical expression for the fundamental frequency, f, of two-span curved

concrete deck-on-multiple steel box-girder bridges:

f=0.94.p.fbeam Hz (9.5)

where p and fbeam are evaluated from Eqs. 9.3 and 9.4, respectively. A comparison of the

results using Eq. 9.5 and the finite element solution for two-span curved concrete deck-

on-multiple steel box-girder bridges shows good correlation, with the errors not

exceeding 8% for all the bridges considered in the parametric study.

9.8 Effect of Span-to-Depth ratio

The value of span-to-depth ratio of 25 is preferred when the yielding stress of the

steel does not exceed 350 MPa. For yielding stress beyond 350 MPa, span-to-depth ratio

can be relaxed to 30 [62]. Until now, all bridges considered in this parametric study had a

span-to-depth ratio of 25. However, since the fundamental frequencies of bridges are

affected by this ratio three different values of span-to-depth ratios of: 20, 25 and 30, were

investigated. Four-lane three-box bridges with span length of 100 m were examined in a

sensitivity study. Span-to-radius ratios of 0.0, 0.6 and 1.2 were considered.

184


Figure 9.5 shows the effect of span-to-depth ratio on the fundamental frequency. It

is obvious that the fundamental frequencies decrease almost linearly with increase in the

span-to-depth ratio, irrespective of the degree of curvature considered herein. For

example, in the case of straight bridges, the fundamental frequency decreased by more

than 30% when span-to-depth ratio increased from 20 to 30. Similarly, the fundamental

frequencies of curved bridges declined significantly by almost 28%. Thus, these results

show that changing the span-to-depth ratio affects considerably the fundamental

frequency for continuous curved concrete deck-on-multiple steel box-girder bridges. It

should be noted that the applieation of the empirical Eq. 9.2 is limited to bridges with

span-to-depth ratio of 25. On the other hand, Eq. 9.5 can be applied to bridges with

various span-to-depth ratios. Table 9.1 shows the results obtained from the three-

dimensional finite-element models correlating quite well with the results estimated by

empirical Eq. 9.5 for various span-to-depth ratios.

9.9 Effect of End-Diaphragm Thickness

Figure 9.6 shows the effect of end-diaphragm thickness on the first four natural

frequencies for continuous 2l-2b bridges with span length of 40 m and L/R ratio of 0.4.

From the results shown it is apparent that end-diaphragm thickness has no effect on the

flexural modes. However, providing end-diaphragms over the bridge supports enhances

significantly the torsional effects and accordingly increases the third and fourth natural

frequencies. As the end-diaphragm thickness increases to 10 mm, significant

improvement in the torsional modes is obtained. Nevertheless, any further increase in the

end-diaphragm thickness produces very little enhancement in the torsional modes.

Therefore, providing end-diaphragm at the support line inside the boxes with the required

185


minimum practical thickness specified in the codes is quite adequate to improve the

torsional frequency modes for continuous curved multiple box girder bridges.

9.10 Effect of Cross Bracings

The effeet of internal and external cross bracings on the natural frequencies of 3/-

3b bridges with span length of 60 m and L/R ratio of 0.8 is illustrated in Figure 9.7.

Different numbers of cross bracings were provided in each span and the first four

frequencies are presented. Clearly, the first and second frequencies of the bridges are not

affected by the presence of internal cross bracings or internal and external cross bracings.

This is because the first and second natural modes are basically flexural modes. The

contribution of the torsional effect becomes more dominant in the third and fourth modes.

Thus, it is shown that the presence of cross bracings enhances dramatically the third and

fourth natural frequencies. Providing both internal and external cross bracings improves

even more the torsional modes. It can be observed, also, that a minimum of five cross

bracings, with 10 m spacing (as specified in the bridge code) is necessary to achieve the

maximum enhancement in the torsional frequencies. Increasing the number of cross

bracing beyond 5 does not appear to contribute to the dynamic characteristics of such

bridges.

9.11 Effect of Number of Spans

The relationships between the number of spans and the first four natural

frequencies are plotted in Figure 9.8 for straight and curved bridges with four lanes and

three boxes. The bridges have span length of 60 m with number of spans 2, 3 and 4. It is

186


interesting to observe that the value of the first natural frequency remain constant with

change in the number of spans for both straight and curved bridges. This means that all

previous empirieal expressions obtained for two-span bridges may be applied to three-

and four-span bridges. However, it should be noted that for all the bridges analyses the

span lengths were equal. It is also observed that the third and fourth natural frequency

results decrease with an increase in the number of spans.

9.12 Forced-Vibration Analysis

The forced-vibration analysis was carried out to study the effeets of truck speed

and span-to-radius of eurvature on the peak acceleration at the bridge deck. Two different

transverse positions of AASHTO truck loading were considered in the forced vibration

analysis of 2/-36-60 bridges with L/R ratios of 0, 0.4, 0.6 and 1.2. Furthermore, the

effects of end-diaphragm thickness and number of cross-bracings on the peak acceleration

were also examined in this study. In all cases, full and outer lane truek loadings were

considered.

9.12.1 Effect of Vehicle Speed

The maximum peak accelerations for 2l-3b-60 bridges with span-to-radius of

curvature ratio of 0.6 were plotted against the vehicle speed in Figure 9.9. Various vehicle

speeds, 50, 60, 80 and 120 km/h, were considered in the analysis. It ean be observed that

the accelerations are higher, generally, for fully loaded lanes with two trueks traveling in

the eentre of their lanes. The peak acceleration occurred in both loading cases for a

vehiele speed of 60 km/h. Inereasing the vehicle speed above 80 km/h, the peak

187


acceleration increased. However, for vehicle speed 120 km/h, the peak acceleration was

lower than those obtained for vehicle speed of 60 km/h.


The effect of span-to-radius of curvature ratio of 2/-3&-60 bridges on the peak

acceleration is illustrated in Figure 9.10. It is clear that increasing the bridge L/R ratio

affects significantly the maximum peak acceleration values. The maximum peak

acceleration increases by approximately 150 % when the L/R ratio is increased from 0 to

1.2. Full and partial truck loading cases showed similar trend. However, the full truck

loading case produced higher values of peak acceleration than those calculated for partial

truck loading case.

9.12.3 Effect of End-Diaphragm Thickness

Two-lane three-box bridges with span length of 60 m and span-to-radius ratio of

1.2 were examined in this study. Various end-diaphragm thicknesses were considered. A

vehicle speed of 120 km/h was maintained in all cases. Figure 9.11 presents the

relationship between the maximum peak acceleration and diaphragm thickness. The

results show that providing end-diaphragm reduces the peak acceleration considerably,

particularly for full truck loading case. Also, increasing the diaphragm thickness beyond

10 mm has no significant effect on the peak acceleration for both loading cases.

188


9.12.4 Effect of Number of Cross Bracings

Two types of cross bracings were selected for this study. The first type is made of

internal bracings, inside the boxes, while the second one is made of internal and external

crossing bracing, i.e. inside and between and the boxes. The 2/-3Z>-60 bridges with span-

to-radius of curvature ratio of 1.2 were examined in this investigation. One, five, eight,

and eleven cross bracings, a vehicle speed of 120 km/h and full truck loading were

considered in analysis. Figure 9.12 shows the variation of peak acceleration against the

number of cross bracings. It can be observed that the peak accelerations for both cross

bracing configurations decrease with increasing the number of bracing from 1 to 5. The

reduction in peak acceleration is about 65% for the case of internal bracing only and 14%

for the case of external and internal bracing. Cleary, increasing the number of bracings

beyond 5 has an insignificant effect on the peak acceleration.

9.13 Summary

A free-vibration analysis of continuous curved composite multiple box-girder

bridges was conducted using a three-dimensional finite-element model. A parametric

study was undertaken to scrutinize the effects of various parameters on the natural

frequencies and the corresponding mode shapes. A forced-vibration analysis was then

conducted to examine the effect of some parameters on the peak acceleration values.

Based on this study, the following conclusions can be draAvn:

1. The fundamental frequency of a continuous curved composite multiple box-

girder bridge decreases considerably as the span length is increased.

189


2. The span-to-radius of curvature ratio influences the fundamental frequency of

bridges significantly. The fundamental frequency decreases with increase in the

bridge curvature. The torsional effect in the first mode increases for higher span-

to-radius of eurvature ratios.

3. Empirical expressions for the fundamental frequency for continuous curved

concrete deck on multiple steel box girder bridges are dedueed. These

expressions are based on the frequency obtained from the flexural beam theory,

so that the expressions ean be applied to bridges with different span-to-depth

ratios.

4. The magnitude of the fundamental frequency drops off almost linearly as the

span-to-depth ratio increases for straight and curved bridges.

5. The end-diaphragm thiekness has no influence on the flexural modes. However,

providing an end-diaphragm with a minimum thickness of 10 mm will enhance

the torsional modes.

6. The presenee of cross bracing improves the dynamic response of curved box

girder bridges. Providing both external and internal bracings with a maximum

spacing of 10 m increases measurably the third and fourth natural frequency,

where the torsional effeets predominate.

190


7. Increasing the number of spans from 2 to 4 does not noticeably affect the

fundamental frequency of the bridges having equal span lengths. However, it

may reduce the natural frequency for the second, third, and fourth modes.

8. The peak acceleration of the bridges is greater for two-equal-span bridges with

higher values of span-to-radius of curvature ratio.

9. The presence of end-diaphragm with minimum practical thickness and cross

bracing with maximum spacing of 10 m would considerably reduce the peak

acceleration and accordingly enhance human comfort for pedestrians using the

bridge.

191


CHAPTER X

Summary and Conclusions

10.1 Summary

Extensive analytical and experimental programs were undertaken to establish the

statie and dynamic characteristics of continuous curved concrete deck-on multiple steel

box girder bridges. A detailed literature review was carried out to set up the basis for this

research work. The results of the literature review indicated lack of expressions to

describe the load distribution factors for such bridges. The impact factors included in the

AASHTO Guide Specification for Horizontally Curved Bridges were derived based on

the grillage analogy which does not accurately represent the complex nature of a three-

dimensional bridge structure. Moreover, there was no simplified method to calculate the

fundamental frequency of such bridges which is required in defining the behaviour of the

bridges under dynamic loads. As a result of this lack of information found in the

literature, the research described in this dissertation was conducted with the objective of

providing design engineers and code writers with thoroughly investigated information to

better understand the static and dynamic responses of continuous curved composite box

girder bridges.

The well-established and suited nonlinear finite element technique was adopted to

analyze the continuous curved box girder prototype bridges studied. Two physical bridge

models were fabricated and constructed in the Structures Laboratory to perform static and

192


free-vibration tests. The main goal of the tests on physical bridge models was to validate

and substantiate the finite element modeling of the prototype bridges. Several static load

cases were applied to determine the elastic response of the bridge models. The bridge

models were subjected also to free vibration tests to validate the natural frequencies and

the corresponding mode shapes predicted by the analytical model.

The experimental findings correlated quite well with the predicted results from the

analytical models; such correlation validated the reliability and accuracy of the results

drawn from the finite element program. Three comprehensive parametric studies were

then carried out. The first one was to deduce empirical expression for the load

distribution factors for stresses, deflection, shear, and reactions. The influences of the key

parameters namely: span length, number of lane, number of box, and span-to-radius of

curvature ratio were investigated. In addition, the effects of other parameters, such as the

number of spans, span-to-depth ratio, cross bracings, web inclination, were examined.

Expressions were developed for the dead and vehicular loads using the nonlinear

regression analysis. The second parametric study was conducted to simulate an idealized

moving vehicle across the continuous curved composite box girder bridge. Based on the

results obtained from 180 prototype bridges analyzed statically and dynamically under

exactly the same load conditions, formulas were derived to estimate the impact factors for

stresses, deflection, shear, and reactions. The proposed expressions were developed in

three sets as follows: as a function of the (a) span length, (b) fundamental frequency, and

(c) span-to-radius of curvature ratio, of the bridge. The third parametric study was carried

out with the intention of generating a simplified expression for the fundamental frequency

of such bridges. The influences of span-to-depth ratio, end-diaphragm thickness, cross

bracings, and number of spans on the fundamental frequencies were scrutinized. In

193


addition, the effects of vehicle speed, span-to-radius of curvature ratio, diaphragm

thickness, and cross bracings on the peak acceleration were examined in the forced-

vibration analysis of the prototype bridges. A best-fitting curve technique was adopted to

develop a simplified formula for the fundamental frequency based on the results obtained

from the parametric study. The results from this research provide an insight into the

design of continuous curved composite box girder bridges.

10.2 Conclusions

Based on the theoretical and experimental investigations carried out on continuous

curved concrete deck-on multiple steel box girder bridges, the following conclusions can

be drawn:

1. The theoretical three-dimensional finite element models developed herein,

can predict quite well the elastic behaviour as well as the natural

frequencies and mode shapes of continuous curved composite multiple-

box girder bridges. They can be used also as a valuable tool to predict the

inelastic bridge response with a fair degree of accuracy.

2. Simplified empirical expressions are developed for the load distribution

factors for tensile and compressive stresses, deflection, shear force,

exterior and interior reactions, and uplift reaction under the bridge self

weight, as well as for AASHTO truck loading. These factors are not

affected by using either vertical or inclined webs in the box girder.

194


3. The above expressions can be used in the design of equal-span continuous

bridges with number of spans up to four. They can also be used to design

bridges subjected to AASHTO LRFD or CHBDC truck loadings.

4. The bridge span length, number of lanes, number of boxes, and the span-

to-radius of curvature ratio are the most crucial parameters that affect the

load distribution factors of such bridges.

5. Using cross bracings with a maximum spacing of 7.5 m is a valid

stipulation in the current AASHTO codes.

6. Empirical expressions are deduced for impact factors for positive and

negative stresses, deflection, shear, and reactions. The expressions are

proposed as function of the bridge span length, fundamental frequency,

and span-to-radius of curvature ratio.

7. Impact factors found in AASHTO 2003 overestimate the design stresses,

deflection, reactions, shear force, and underestimate uplift reaction.

8. Empirical expressions are proposed for the fundamental frequency. The

expressions can be applied to bridges with various span-to-depth ratios.

9. The fundamental frequency of the bridge decreases significantly with

increase in the span length. However, it is not influenced by either the

number of lanes or the number of boxes. The fundamental frequencies

195


decrease by about 30% as a result of increasing the bridge span-to-radius

of curvature ratio from 0 to 1.2. For higher ratios, the dominant frequency

tends to be mainly torsional.

10. A forced-vibration study showed that increasing the bridge span-to-radius

of curvature ratio magnifies the peak acceleration of the bridge.

Furthermore, providing end-diaphragms with thickness of not less than 10

mm, in addition to the presence of cross bracing, reduces the torsional

effect and hence the discomfort of pedestrians and/or motorists.

10.3 Recommendation for Further Research

1. Develop expressions to predict the ultimate load carrying capacities of

curved composite box girder bridges

2. Study the fatigue response of curved composite box girder bridges.

3. Study curved box girders with skew support lines.

4. Study buckling of box girder components subjected to combined stresses.

196


References

1- Akoussah, K. E., Fafard, M., Talbot, M., and Beaulieu, D. 1997. Parametric study

of dynamic load amplification factor for simply-supported reinforeed conerete

bridges. Canadian Journal of Civil Engineering, 24(2): 313-322.

2- Al-Rifaie, W. N., and Evans, H. R. 1979. An approximate method for the

analysis of box girder bridges that are eurved in plan. Proceedings of

International Association of Bridges and Structural Engineering, lABSE, 1-21.

3- American Association of State Highway and Transportation Officials, AASHTO.

1996. AASHTO Bridge Design Speeifieations. Washington, D.C.

4- American Association for State Highway and Transportation Officials, AASHTO.

1993. Guide specification for horizontally curved highway bridges.

Washington, D.C.

5- American Association for State Highway and Transportation Officials, AASHTO.

2003. Guide speeification for horizontally curved highway bridges.

Washington, D.C.


1994. AASHTO LRFD Bridge Design Specifications. Washington, D.C.


1998. AASHTO LRFD Bridge Design Speeifieations. Washington, D.C.

197


8- Aneja, I. and Roll, F. 1971. Model analysis of curved box-beam highway bridge.

Journal of the Structural Division, ASCE, Vol. 97: 2861-2878.

9- Bakht, B. and Jaegor, L. G. 1985. Bridge deck simplified. McGraw-Hill, New

York, N.Y.

10- Bakht, B. and Jaegor, L. G. 1992. Simplified methods of bridge analysis for the

third edition of OHBDC. Canadian Journal of Civil Engineering, 19(4): 551-559.

11- Bazant, Z. P. and El Nimeiri, M. 1974. Stiffness method for curved box girders.

ASCE Journal of the Structural Division, lOO(STlO): 2071-2090.

12- Bathe, K. J. 1996. Finite Element Procedures. Prentice Hall, New Jersey, U.S.A.

13- Billing, J. R. 1984. Dynamic loading and testing of bridges in Ontario. Canadian

Journal of Civil Engineering, 11(4): 833-843.

14- Branco, F. A. 1981. Composite box girder bridge behaviour. Thesis presented to

the University of Waterloo, Waterloo, Ontario, Canada, in partial fulfilment of the

requirements for the degree of Master of Science.

15- Branco, F. A. and Green, R. 1985. Composite box girder bridge behaviour

during construction. ASCE Journal of Structural Engineering, 111(3): 577-593.

16- Brennan, P. J. and Mandel, J. A. 1979. Multiple configuration of curved bridge

model studies. ASCE Journal of the Structural Division, 105(ST5): 875-890.

17- Brighton, J., Newell, J., and Scanlon, A. 1996. Live load distribution factors for

double cell box girders. Proceeding of the 1st Structural Specialty Conference,

Alberta, Canada, 419-430.

198


18- Brockenbrough, R. L. 1986. Distribution factors for curved I-girder bridges.

ASCE Journal of Structural Engineering, 112(10); 2200-2215.

19- Buckle, I. G. and Hood, J. A. 1982. Testing and analysis of a curved continuous

box girder model. Department of Civil Engineering, University of Aucland, New

Zealand, Report No. 181.

20- Canadian Standards Association, 2000. Canadian highway bridge design code,

CHBDC. Etobicoke, Ontario, Canada.

21- Canadian Standard Assoeiation. 1988. Design of highway bridges (CAN/CSA-S6-

88). Rexdale, Ontario, Canada.

22- Cantieni, R., 1983. Dynamic load tests on highway bridges in Switzerland - 60

years of experience. Repost 211, Federal Laboratory for Testing of Materials

(EMPA), Duebendorf, Switzerland.

23- Cantieni, R., 1984, Dynamic load testing of highway bridges. lABSE

Proceedings P-75/84; 57-72.

24- Chang, D. and Lee, H. 1994. Impact factors for simple-span highway girder

bridges. Journal of Structural Engineering, ASCE, 120(1): 704-715.

25- Chang, P. C., Heins, C. P., Guohao, L., and Ding, S. 1985. Seismic study of

curved bridges using the Rayleigh-Ritz method. Computers & Struetures, 21(6):

1095-1104.

26- Cheung, M. S. 1984. Analysis of continuous curved hox-girder bridges by the

finite strip method. In Japanese. Japanese Society of Civil Engineers, 1-10.

199


27- Cheung, M. S. and Chan, M. Y. T. 1978. Finite strip evaluation of effective

flange width of bridge girders. Canadian Journal of Civil Engineering, 5(2): 174-

185.

28- Cheung, M. S. and Cheung, Y. K. 1971. Analysis of curved box girder bridges

by the finite-strip method. International Association of Bridges and Structural

Engineering, lABSE, 31(1): 1-8.

29- Cheung, M. S. and Foo, S. H. C. 1995. Design of horizontally curved composite

box girder bridges: a simplified approach. Canadian Journal of Civil

Engineering, 22(1): 93-105.

30- Cheung, M. S. and Magnount, A. 1991. Parametric study of design variation on

the vibration modes of box girder bridges. Canadian Journal of Civil

Engineering, 18(5): 789-798.

31- Cheung, M. S. and Mirza, M. S. 1986. A study of the response of composite

concrete deck-steel box girder bridges. Proceedings of the 3rd International

Conference on Computational and Experimental Measurements, Porto Caras,

Greece, 549-565.

32- Cheung, Y. K. 1968. Finite strip method analysis of elastic slabs. Journal of the

Engineering Mechanics Division, ASCE, 94 EM6: 1365-1378.

33- Cheung, Y. K. 1969. The analysis of cylindrical ortbotropic curved bridge

deck. lABSE Publications, Vol. 29, part II, 41-52.

200


34- Cheung, Y. K. and Cheung, M. S. 1972. Free vibration of curved and straight

beam-slab or box-girder bridges. International Association of Bridges and

Structural Engineering, 32(2): 41-52.

35- Cheung, Y. K. Cheung, Y. S., and Au, F.T.K. 2001. Vibration analysis of bridges

under moving vehicle and trains. Structural Engineering, Mechanics and

Computation, Vol. 1 Elsevier Science: 3-14.

36- Chu, H. and Pinjarkar, S. G. 1971. Analysis of horizontally curved box girder

bridges. ASCE Journal of the Structural Division, 97(ST10): 2481-2501.

37- Colin O’Connor and Peter A. Shaw 2000. Bridge Loads. Spon Press, New York,

NY.

38- Culver, C. G. 1967. Natural frequencies of horizontally curved beams. ASCE

Journal of the Structural Division, 93(2): 189-203.

39- Dabrowski, R. 1968. Curved thin-walled girders, Theory and analysis. Springer,

New York.

40- Davidson, J. S., Keller, M. A., and Yoo, C. H. 1996. Cross-frame spacing and

parametric effects in horizontally curved I-girder bridges. ASCE Journal of


41- Davis, R. E., Bon V. D., and Semans, F. M. 1982. Live load distribution in

concrete box girder bridges. Transportation Research Record, Transportation

Research Board, 871:47-52.

42- Dean, D. L. 1994. Torsion of regular multicellular members. ASCE Journal of


201


43- Department of Civil Engineering and Applied Mechanics McGill University 1985.

An analytical-experimental study of the behaviour of a composite concrete

deck-steel box girder bridge. Montreal, Quebec, Canada.

44- Dey, S.S. and Balasubramanian, N. 1984. Dynamic response of orthotropic

curved bridge decks due to moving loads. Computer and Structures Journal,

18(1): 27-32.

45- Dogaki, M., Harada, M., Yonezawa, H., and Ozawa, K. 1979. Further test of the

curved girder with orthotropic steel plate deck. Tech. Rep. No 20, Kanasi

University, Osaka, Japan.

46- DSP Development Corporation. 1992. Data analysis and display software,

DADiSP. Cambridge, MA.

47- Ebeido, T. and Kennedy, J.B. 1997. Shear and reaetion distributions in

eontinuous skew composite bridges. Journal of Bridge Engineering, ASCE, 1(4):

155-165.

48- El-Azab, M.A. 1999. Static and dynamic analysis of thin walled box girder

bridges by exaet beam finite elements. Ph.D. Thesis, Department of Civil

Engineering, A1 Azhar University, Cairo, Egypt

49- Evans, H. R. 1982. Simplified methods for the analysis and design of bridges of

cellular cross-section. Proceedings of the NATO Advanced Study Institute on

Analysis and Design of Bridges,Cesme, Izmir, Turkey, 95-115.

50- Evans, H. R. and Shanmugam, N. E. 1984. Simplified analysis for cellular

structures. ASCE Journal of the Structural Division. 110(ST3): 531-543.

202


51- Fafard, M. Laflamme, M., Savard, M., and Bennur, M. 1998. Dynamic analysis of

existing continuous bridge. Journal of Bridge Engineering, ASCE, 3(1): 28-37.

52- Fam, A. R. M. 1973. Statie and free-vibration analysis of eurved box bridges.

Structural Dynamic Series No. 73-2, Department of Civil Engineering and Applied

Mechanics, McGill University, Montreal, Quebec, Canada.

53- Fam, A. R. and Turkstra, C. J. 1975. A finite element seheme for box bridge

analysis. Computers and Structures Journal, Pergamon Press, 5: 179-186.

54- Fam, A. R. and Turkstra, C. J. 1976. Model study of horizontally eurved box

girder. ASCE Journal of the Structural Division, 102(ST5): 1097-1108.

55- Foinquinos, R., Kuzmanovic, B., and Vargas, L. 1997. Influence of diaphragms

on live load distribution in straight multiple steel box girder bridges,

Proceedings, 15*'’ Structures Congress, ASCE, Portland, OR, USA, 1 (1): 89-103.

56- Fountain, R. S. and Mattock, R. S. 1968. Composite steel-eoncrete multi-box

girder bridges. Proceedings of the Canadian Structural Engineering Conference,

Toronto, 19-30.

57- Fu, C. C. and Yang, D. 1996. Designs of concrete bridges with multiple box cells

due to torsion using softened truss model. ACI Structural Journal, 93(6): 696-

702.

58- Galdos, N. H. 1988. A theoretical investigation of the dynamie behaviour of

horizontally curved steel box girder bridges under truck loadings. Ph.D.

Thesis, University of Maryland at Collage Park, Md., USA

203


59- Galdos, N. H., Schelling, D. R., and Sahin, M. A. 1990. Methodology for the

determination of dynamic impact factor of horizontally curved steel box girder

bridges. ASCE Jonmal of Stractural Engineering, 119(6): 1917-1934.

60- Galuta, E. M. and Cheung, M. S. 1995. Combined boundary element and finite

element analysis of composite box girder bridges. Computers and Structures

Journal, Pergamon Press, 57(3): 427-437.

61- Green, R. 1978. Composite box girder bridges - The construction phase.

Proceedings of the Canadian Conference on Structural Engineering, CSICC,

Toronto, Canada.

62- Hall, D.H. and Yoo, C.H. 1999. Improved Design Specifications for

Horizontally Curved Steel Girder Highway Bridges, NCHRP Project 12-38,

ESDI, Ltd., Coopersburg, PA.

63- Heins, C. P. 1978. Box girder bridge design-State-of-the-Art. AISC Engineering

Journal, 2: 126-142.

64- Heins, C. P. and Humphrey, R. 1979. Bending and torsion interaction of box

girders. ASCE Journal of Structural Division, 105(ST5): 891-904.

65- Heins, C. P. and Jin, J. O. 1984. Live load distribution on curved I-girders.

ASCE Journal of Structural Engineering, 110(3): 523-530.

66- Heins, C. P. and Lee, W. H. 1981. Curved box-girder bridge: field test. ASCE

Journal of the Structural Division, 107(2): 317-327.

67- Heins, C. P. and Olenick, J. C. 1976. Curved box beam bridge analysis.

Computers and Structures Journal, Pergamon Press, London, 6(2): 65-73.

204


68- Heins, C. P. and Sahin, M. A. 1979. Natural frequency of curved box girder

bridges. ASCE Journal of the Structural Division, 105(12): 2519-2600.

69- Heins, C. P. and Sheu, F. H. 1982. Design/analysis of curved box girder bridges.

Computers and Structures Journal, 15(3): 241-258.

70- Hibbitt, H.D., Karlson, B.I. and Sorenson, E.P. 2002. ABAQUS version 6.2, finite-

element program. Hibbitt, Karlson and Sorenson, Inc., Providence, R.I.

71- Ho, S., Cheung, M. S., Ng, S. P., and Yu, T. 1989. Longitudinal girder moments

in simply supported bridges by the finite strip method. Canadian Journal of

Civil Engineering, 16(5): 698-703.

72- Huang, D. 2001. Dynamic analysis of steel curved box girder bridges. Journal of

Bridge Engineering, ASCE, 6(6): 506-512.

73- Huang, D., Wang, T., and Shahawy, M. 1993. Impact studies of multi-girder

concrete bridges. Journal of Structural Engineering, ASCE, 119(8): 2387-2402.

74- Huang, D., Wang, T., and Shahawy, M. 1995. Dynamic behaviour of horizontally

curved I-girder bridges. Computers & Structures, 57(4): 703-714.

75- Huang, D., Wang, T., and Shahaway, M. 1995. Vibration of thin-walled box-

girder bridges exited by vehicles. ASCE Journal of Structural Engineering,

121(9): 1330-1337.

76- Humar, J. L. 1990. Dynamic of structures. Prentice-Hall Inc.

77- Inbanathan, M. J. and Wieland, M. 1987. Bridge vibration due to vehicle moving

over rough surface. ASCE Journal of Structural Engineering, 113(9): 1994-2008.

205


78- Johanston, S. B. and Mattock, A. H. 1967. Lateral distribution of loads in

composite box girder bridges. Highway Research Record, Highway Research

Board, 167: 25-33.

79- Kabir, A. F. and Scordelis, A. C. 1974. Computer programs for curved bridges

on flexible bents. Structural Engineering and Structural Mechanics Report No.

UC/SESM 74-10, University of California, Berkeley, CA.

80- Kashif, A. M. 1992. Dynamic response of highway bridges to moving vehicles.

Ph.D. dissertation. Department of Civil Engineering, Carleton University, Ottawa,

Canada.

81- Kashif, A. M. and Humar, J. L. 1990. Analysis of the dynamic characteristics of

box girder bridges. Proceedings of the Third International Conference on Short

and Medium Span Bridges, Toronto, Ontario, 2: 367-378.

82- Kim, S. and Nowak, A. 1997. Load distribution and impact factors for I-girder

bridges. Journal of Bridge Engineering, 2(3): 97-104.

83- Kissane, R. and Beal, D. B. 1975. Field testing of horizontally curved steel

girder bridges. Research Report 27, The U.S. Department of Transportation.

84- Kollbrunner, C.F. and Hajdin, N. 1966. Warping torsion of thin-walled beams of

closed section. Verlage Schweizer Stahlbaus Vereinigung, Mitteillung der

Technishen Kommission, Zurich, Heft 32: 177

85- Komatsu, S. and Nakai, H. 1966. Study on free vibrations of curved girder

bridges. Transactions, Japan Society of Civil Engineers, Tokyo, Japan, (136): 35-

60.

206


86- Komatsu, S. and Nakai, H. 1970. Fundamental study on forced vibration of

curved girder bridges. Transactions, Japan Society of Civil Engineers, Tokyo,

Japan, 2(1): 37-42.

87- Kou, C., Benzely, S. E., Haung, J., and Firmage, A. A. 1992. Free-vibration

analysis of curved thin-walled girder bridges. ASCE Journal of Structural

Engineering, 118(10): 2890-2910.

88- Kristek, V. 1979. Theory of box girders. Wiley-interscience Publication, John

Wiley & Sons.

89- Laman, J.A., Pechar, J.S., and Boothby, T.E. 1999. Dynamic load allowance for

through-truss bridges. Journal of Bridge Engineering, ASCE, 4(4): 231-241.

90- Li, H. G. 1992. Thin-walled box beam finite elements for static analysis of

curved haunched and skew multi-cell box girder bridges. Ph.D. Thesis,

Department of Civil Engineering, Carleton University, Ottawa, Ontario, Canada.

91- Mabsout, M.E., Tarhini, K.M., Frederick, G.R., and Kesservan, A. 1998. Effect of

Continuity on Wheel Load Distribution in Steel Girder Bridges. Journal of

Bridge Engineering, ASCE, 3(3): 103-110.

92- Maisel, B. I. 1985. Analysis of concrete box beams using small computer

capacity. Canadian Journal of Civil Engineering, 12(2): 265-278.

93- Maisel, B. I. and Roll, F. 1974. Methods of analysis and design of concrete box

beams with side cantilevers. Technical Report 42.494, Cement and Concrete

Association, London.

207


94- Mavaddat, S. and Mirza, M. S. 1989. Computer analysis of thin walled concrete

box beams. Canadian Journal of Civil Engineering, 16(6); 902-909.

95- Meyer, C. and Scordelis, A. C. 1971. Analysis of curved folded plate structures.

ASCE Journal of the Structural Division, 97(ST10): 2459-2480.

96- Ministry of Transportation and Communications. 1994. Geometric design

standards for Ontario highways. MTO, Downsview, Ontario, Canada.

97- Mirza, M. S., Ferdjani, A., Hadj-arab, A., Joucdar, K., and Khaled, A. 1990. An

experimental study of static and dynamic responses of prestressed eonerete

box girder bridges. Canadian Journal of Civil Engineering, 17(3): 481-493.

98- Mirza, M. S., Manatakos, C. K., Murali, R. D., Igwemezie, J. O., and Wyzykow^ski,

J. 1985. An analytical-experimental study of the behaviour of a composite

concrete deck-steel box girder bridge. Report for Public Works Canada,

Department of Civil Engineering and Applied Mechanics, McGill University,

Montreal, Quebec, Canada.

99- Mukberjee, D. and Trikba, D. N. 1980. Design coefficient for curved box girder

concrete bridges. Indian Concrete Journal, 54(11); 301-306.

100- Nakai, H. and Heins, C. P. 1977. Analysis criteria for curved bridges. ASCE

Journal of Structural Division, 103(ST7):1419-1427.

101- Ng. S. P., Cheung, M. S., and Hacbem, H. M. 1993. Study of a curved continuous

composite box girder bridge. Canadian Journal of Civil Engineering, 20(1): 107-

119.

208


102- Normandin, P. and Massicotte, B. 1994. Load distribution in composite steel box

girder bridges. Proceedings of the 4th International Conference on Short and

Medium Span Bridges. Halifax, N.S., Canada, 165-176.

103- Nour, S. I. 2000. Load distribution in curved composite deck-steel multiple-

spine box girder bridges. M.A.Sc. Thesis, Department of Civil and Environmental

Engineering, University of Windsor, Canada.

104- Nutt, R. V., Schamber, R. A., and Zokaie, T. 1988. Distribution of wheel loads on

highway bridges. Transportation Research Board, National Cooperative Highway

Research Council, Imbsen & Associates Ine., Saeramento, Calif.

105- O’Connor, C. and Pritehard, R.W. 1985. Impact studies on small composite

girder bridge. Journal of the Structural, 111(3): 641-653.

106- Oleinik, J. C. and Heins, C. P. 1975. Diaphragms for curved box beam bridges.

ASCE Journal of the Structural Division, lOl(STlO): 2161:2179.

107- Ontario Ministry of Transportation and Communications, 1997. Canadian

Highway Bridge Design Code, CHBDC, Draft. Downsview, Ontario.

108- Ontario Ministry of Transportation and Communieations, 1983. Ontario Highway

Bridge Design Code, OHBDC. Second edition, Downsview, Ontario.

109- Ontario Ministry of Transportation and Communications, 1992. Ontario Highway

Bridge Design Code, OHBDC. Third edition, Downsview, Ontario

110- Owens, G. W., Dowling, P. J., and Hargreaves, A. C. 1982. Experimental

behaviour of a composite bifurcated box girder bridge. Proceedings of

209


International Conference on Short and Medium Span Bridges, Toronto, Ontario,

Canada, 1: 357-374.

111- Paultre, P., Chaallai, O., and Proulx, J. 1992. Bridge dynamics and dynamic

ampliflcation factors -a review of analytical and experimental flndings.

Canadian Journal for Civil Engineering, 19: 260-278.

112- Rabizadeh, R. O. and Shore, S. 1975. Dynamic analysis of curved box-girder

bridges. ASCE Journal of the Structural Division, 101(9): 1899-1912.

113- Razaqpur, A. G. and Li, H. G. 1990. Analysis of multi-branch multi-cell box

girder bridges. Proceeding of the 3th International Conference on Short and

Medium Span Bridges, Toronto, Ontario, Canada, 2: 153-164.

114- Razaqpur, A. G. and Li, H. G. 1994. Refined analysis of curved thin-walled

multi-cell box girders. Computers and Structures Journal, Pergamon Press, 53(1):

131-142.

115- Razaqpur, A. G. and Li, H. G. 1997. Analysis of curved multi-cell box girder

assemblages. Structural Engineering and Mechanics, 5(1): 33-49.

116- Richardson, J. A. and Douglas, B. N. 1993. Results from field testing a curved

box girder bridge using simulated earthquake loads. Earthquake Engineering &

Structural Dynamics, 22(10): 905-922.

117- Roll, F. and Aneja, I. 1966. Model tests of box-beam highway bridges with

cantilever deck slabs. Paper presented at Annual meeting and Transportation

Engineering Conference, Philadelphia, Pa. (Preprint 395).

210


118- Samaan, M., Sennah, K.M. and Kennedy, J. B. 2002. Positioning of bearings for

curved continuous spread-box girder bridges. Canadian Journal of Civil

Engineering, 29: 641-652.

119- Samaan, M., Sennah, K.M. and Kennedy, J. B. 2003. Vibration of simply-

supported multiple-box girder bridges. Annual conference of the Canadian

society for Civil Engineering, Moncton, New Brunswick, GCF-277-1: 7.

120- Schelling, D. R., Galdos, N. H., and Sahin, M. A. 1992. Evaluation of impact

factors for horizontally curved steel box bridges. ASCE Journal of Structural

Engineering ,118(11): 3203-3221.

121- Scordelis, A. C. 1982. Berkeley computer programs for the analysis of concrete

box girder bridges. Proceedings of the NATO Advanced Study Institute on

Analysis and Design of Bridges, Izmir, Turkey, 119-189.

122- Scordelis, A. C., Bouwkaup, J. C., and Larsen, P. K. 1974. Structural behaviour

of a curved two-span reinforced concrete box girder bridge model; Volumes I,

II, and III. Reports No. US/SESM 74-5, US/SESM 74-6, and US/SESM 74-7,

University of California, Berkeley, CA.

123- Scordelis, A. C. and Meyer, C. 1969. Wheel load distribution in concrete box-

girder bridges. Structural Engineering and Structural Mechanics Report SESM 69-

1, NTIS: PB 183923, University of California, Berkeley, CA.

124- Sennah, K.M. 1998. Load distribution factors and dynamic characteristics of

curved composite concrete deck-steel cellular bridges. Ph.D. Thesis, Department

of Civil Engineering, University of Windsor, Windsor, Ontario, Canada.

211


125- Sennah, K.M. and Kennedy, J. B. 1997 Dynamic characteristics of simply

supported curved composite multi-cell bridges. Canadian Journal of Civil

Engineering, 24(4): 621-636.

126- Sennah, K.M. and Kennedy, J. B. 1998. Shear distribution in simply-supported

curved composite cellular bridges. Journal of Bridge Engineering, ASCE, 3(2):

47 -55.

127- Sennah, K.M. and Kennedy, J. B. 1998. Vibration of horizontally curved

continuous composite cellular bridges. Canadian Journal of Civil Engineering,

25: 139-150.

128- Sennah, K.M. and Kennedy, John B. 1999. Load distribution factors for

composite multicell box girder bridges. Journal of Bridge Engineering, ASCE,

4(1): 71-78.

129- Sennah, K.M. and Kennedy, J. B. 2001. Stat-of-tbe-art in design of curved box-

girder bridges. Journal of Bridge Engineering, ASCE, 6(3): 159-167.

130- Sennah, K. M. and Kennedy, J.B. 2002. Literature review in analyzing of box-

girder bridges. Journal of Bridge Engineering, ASCE, 7(2): 134-43.

131- Sennah, K. M., Kennedy, J.B. and Nour, S. I. 2003. Design for shear in curved

composite multiple-steel box girder bridges. Journal of Bridge Engineering,

ASCE, 8(2): 144-152.

132- Senthilvasan, J., Brameld, G. H., and Thambiratnam, D. P. 1997. Bridge-vebicle

interaction in curved box girder bridges. Microcomputers in Civil Engineering,

12(3): 171-181.

212


133- Shore, S. and Chaudhuri, S. 1972. Free-vibration of horizontally curved beams.

ASCE Journal of the Structural Division, 98(3): 793-796.

134- Siddiqui, A. H. and Ng, S. F. 1988. Effect of diaphragms on stress reduction in

box girder bridge sections. Canadian Journal of Civil Engineering, 15(1):127-135.

135- Soliman, M. I. 1994. Different aspects affecting the torsional stiffness of R.C.

straight and curved box girder bridges. Proceeding of the 4th International

Conference on Short and Medium Span Bridges, Halifax, N.S., Canada, 129-140.

136- Tabba, M. M. 1972. Free-vibration of curved box girder bridges. M.Eng. thesis.

Department of Civil Engineering and Applied Mechanics, McGill University,

Montreal, Quebec, Canada.

137- Tan, C. P. and Shore. 1968. Dynamic response of a horizontally curved bridge.

ASCE Journal of the Structural Division, 94(3): 761-781.

138- Tan, C. P. and Shore, S. 1968. Response of horizontally curved bridge to moving

load. ASCE Journal of the Structural Division, 94(9): 2135-2151.

139- Tilly, G.P. 1986. Dynamic behaviour of concrete structures. Developments in

Civil Engineering, Vol. 13, Report of the Rilem 65MDB Committee. Elsevier, New

York, N.Y.

140- Tung, D.H. and Fountain, R.S. 1995. Approximate torsional analysis of curved

box girder by M/R method. AISC Engineering Journal, 65-74.

141- Vlasov, V. Z. 1965. Thin-walled elastic beams. OTS61-11400, National Science

Foundation, Washington, D. C.

213


142- Wang, T., Huang, D., and Shahawy, M. 1996. Dynamic behaviour of continuous

and cantilever thin-walled box girder bridges. ASCE Journal of Bridge

Engineering, 1(2): 67-75.

143- Wang, T., Huang, D., Shahawy, M., and Huang, K. 1996. Dynamic response of

highway girder bridges. Computer and Structures, Elsevier Science, GB, 60(6):

1021-1027.

144- Wiliam, K. J. and Scordelis, A. C. 1972. Cellular structures of arbitrary plan

geometry. ASCE Journal of the Structural Division, 98(ST7): 1377-1394.

145- Xi-Jiu, Lin and De-Rong, Chen. 1987. Prespex model tests of three span

continuous curved box girders. Research Institute of Highway, China, 1-10.

146- Yabuki, T., Arizumi, Y. and Vinnakota, S. 1995. Strength of thin-walled box

girder bridges curved in plan. ASCE Journal of Structural Engineering, 121(5):

907-914.

147- Yang, Y., Liao, S. and Lin, B. 1995. Impact formulas for vehicles moving over

simple and continuous beams. Journal of Structural Engineering, ASCE, 121(11):

1644-1650.

148- Yoo, C., Buchanan, J., Heins, C. P., and Armstrong, W. L. 1976. Analysis of a

continuous curved box girder bridge. Transportation Research Record, 79: 61-

71.

149- Zhang, X. 2001. Evaluation of impact factors of straight and horizontally

curved composite concrete deck-steel cellular bridges. Thesis presented to the

214


University of Windsor, Windsor, Ontario, Canada, in partial fulfilment of the

requirements for the degree of Master of Science.

150- Zhang, X., Sennah, K.M. and Kennedy, J. B. 2003. Impact factors for

horizontally curved composite box girder bridges. Accepted for possible

publication in ASCE Journal of Structural Engineering.

151- Zokaie, T., Imbsen, R.A., and Osterkamp, T.A. 1991. Distribution of wheel loads

on highway bridges. Transportation Research Record, Transportation Research

Board, 1290: 119-126.

152- Zienkeiwicz, O. C. 1977. The fiuite-elemeut method. McGraw-Hill Book

Company, Third Edition.

215


Tables

Table 4.1. Average properties o f the concrete cylinders for the bridge models

BridgeModel

Ultimate compressive strength (MPa)

fc

Secant modulus of elasticity (GPa)

EeStraight 45 42Curved 41 40

Table 5.1. Natural frequencies and mode shapes of the tested bridge models

BridgeModel

ModeFinite element results Experimental results

number AverageFrequency

(Hz)

Modeshape

Averagefrequency

(Hz)

Modeshape

1 29.9 L A P 31.2 L A PbO-o 2 44.7 L F 39.9 L FUh

3 72.7 T S 72.3-£3bO 4 75.4 T A S

5 102.3 L F

6 110.3 T S

bJO1 24.1 L A F - T S 24.5 L A F - T S

2 40.4 L F - T S 35.1 L F - T SVh 3 81.6 T S 64.1<U&3u

4 86.2 T A S - L F

5 88.8 T A S

6 98.3 T SN ote : Symbols fo r mode shapes are explained in Figure 6.11

216


Table 5.2. Fundamental frequency obtained from the tests

Instrument SensorStraight Bridge

(Hz)Curved Bridge

(Hz)Number Flexural

TestTorsional

TestFlexural

TestTorsional

Test1 31.2 31.0 24.8 —

2 32.0 32.4 23.9 25.3HQ 3 29.0 29.2 23.8 24.3

4 30.2 32.4 24.3 23.65 31.0 31.0 24.7 24.46 31.7 30.4 24.2 24.9

1/3 1 31.5 32.8 24.1 —

(D 2 31.6 29.2 24.3 24.3So 3 31.5 33.0 24.8 24.9

1 3oo

4 31.4 29.0 24.6 24.45 31.5 32.4 25.0 —

< 3 6 32.9 31.4 24.4 24.6

217


Table 6.1. Geometries o f bridges used in parametric study for load distribution factor

B r i d g e N u m b e r o f S p a n ( L ) N u m b e r o f K = C r o s s S e c t i o n D i m e n s i o n s ( m )l a n e s ( N l ) ( m ) b o x e s ( N b ) L / R A B C d H t i ta

( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) ( 6 ) ( 7 ) ( 8 ) ( 9 ) ( 1 0 ) ( 1 1 ) ( 1 2 )2l-20-2b 2 0 2 9 .3 0 2 .3 2 5 0 .3 0 0 .8 0 1 .025 0 .0 1 6 0 .2 2 52l-20-3h 20 3 o 9 .3 0 1.550 0 .3 0 0 .8 0 1.025 0 .0 1 6 0 .2 2 52l-20-4h 2 0 4 2 9 .3 0 1.163 0 .3 0 0 .8 0 1 .025 0 .0 1 6 0 .2 2 52l-40-2b 40 2 9 .3 0 2 .3 2 5 0 .3 0 1.60 1 .825 0 .0 1 6 0 .2 2 52l-40-3b 40 3 o 9 .3 0 1 .550 0 .3 0 1.60 1 .8 2 5 0 .0 1 6 0 .2 2 52l-40-4b 40 4 9 .3 0 1.163 0 .3 0 1.60 1.825 0 .0 1 6 0 .2 2 52l-60-2b <U 60 2 9 .3 0 2 .3 2 5 0 .3 0 2 .4 0 2 .6 2 5 0 .0 1 6 0 .2 2 52l-60-3b 60 3 9 .3 0 1.550 0 .3 0 2 .4 0 2 .6 2 5 0 .0 1 6 0 .2 2 52l-60-4b (N 60 4 cs 9 .3 0 1.163 0 .3 0 2 .4 0 2 .6 2 5 0 .0 1 6 0 .2 2 52l-80-2h 80 2 oc 9 .3 0 2 .3 2 5 0 .3 0 3 .2 0 3 .4 2 5 0 .0 1 6 0 .2 2 52l-80-3b 80 3 o 9 .3 0 1 .550 0 .3 0 3 .2 0 3 .4 2 5 0 .0 1 6 0 .2 2 52l-80-4b 80 4 9 .3 0 1.163 0 .3 0 3 .2 0 3 .4 2 5 0 .0 1 6 0 .2 2 5

2l-100-2b 100 2 o" 9 .3 0 2 .3 2 5 0 .3 0 4 .0 0 4 .2 2 5 0 .0 1 6 0 .2 2 52l-100-3b 100 3 9 .3 0 1 .550 0 .30 4 .0 0 4 .2 2 5 0 .0 1 6 0 .2 2 52l-100-4b 100 4 9 .3 0 1.163 0 .3 0 4 .0 0 4 .2 2 5 0 .0 1 6 0 .2 2 53l-20-2b 2 0 2 13.05 3 .2 6 3 0 .3 0 0 .8 0 1.025 0 .0 1 6 0 .2 2 53l-20-3b 20 3 13.05 2 .1 7 5 0 .3 0 0 .8 0 1 .025 0 .0 1 6 0 .2 2 53l-20-4b 2 0 4 13.05 1.631 0 .3 0 0 .8 0 1.025 0 .0 1 6 0 .2 2 53l-20-5b 2 0 5 o 13 .05 1.305 0 .3 0 0 .8 0 1 .0 2 5 0 .0 1 6 0 .2 2 53l-20-6h 2 0 6 o 13.05 1 .088 0 .3 0 0 .8 0 1.025 0 .0 1 6 0 .2 2 53l-40-2b 40 2 13.05 3 .2 6 3 0 .3 0 1.60 1 .825 0 .0 1 6 0 .2 2 53l-40-3b 4 0 3 o 13 .05 2 .1 7 5 0 .3 0 1.60 1.825 0 .0 1 6 0 .2 2 53l-40-4b 40 4 13.05 1 .631 0 .3 0 1 .60 1 .825 0 .0 1 6 0 .2 2 53l-40-5b 4 0 5 13.05 1.305 0 .3 0 1.60 1.825 0 .0 1 6 0 .2 2 53l-40-6h 40 6 13.05 1 .088 0 .3 0 1 .60 1 .825 0 .0 1 6 0 .2 2 53l-60-2b 60 2 13 .05 3 .2 6 3 0 .3 0 2 .4 0 2 .6 2 5 0 .0 1 6 0 .2 2 53l-60-3b § 60 3 13.05 2 .1 7 5 0 .3 0 2 .4 0 2 .6 2 5 0 .0 1 6 0 .2 2 53l-60-4b S

hJ 60 4 13.05 1.631 0 .3 0 2 .4 0 2 .6 2 5 0 .0 1 6 0 .2 2 53l-60-5b 60 5 13.05 1.305 0 .3 0 2 .4 0 2 .6 2 5 0 .0 1 6 0 .2 2 53l-60-6b 60 6 13.05 1 .088 0 .3 0 2 .4 0 2 .6 2 5 0 .0 1 6 0 .2 2 53l-80-2b 80 2 13.05 3 .2 6 3 0 .3 0 3 .2 0 3 .4 2 5 0 .0 1 6 0 .2 2 53l-80-3b 80 3 OC 13.05 2 .1 7 5 0 .3 0 3 .2 0 3 .4 2 5 0 .0 1 6 0 .2 2 53l-80-4b 80 4 o 13 .05 1 .631 0 .3 0 3 .2 0 3 .4 2 5 0 .0 1 6 0 .2 2 53l-80-5b 80 5 13.05 1.305 0 .3 0 3 .2 0 3 .4 2 5 0 .0 1 6 0 .2 2 53l-80-6b 80 6 o ' 13.05 1 .088 0 .3 0 3 .2 0 3 .4 2 5 0 .0 1 6 0 .2 2 5

3l-100-2b 100 2 13.05 3 .2 6 3 0 .3 0 4 .0 0 4 .2 2 5 0 .0 1 6 0 .2 2 53l-100-3b 100 3 13.05 2 .1 7 5 0 .3 0 4 .0 0 4 .2 2 5 0 .0 1 6 0 .2 2 53l-100-4b 100 4 13.05 1.631 0 .3 0 4 .0 0 4 .2 2 5 0 .0 1 6 0 .2 2 53l-100-5b 100 5 13 .05 1 .305 0 .3 0 4 .0 0 4 .2 2 5 0 .0 1 6 0 .2 2 53l-100-6b 100 6 13 .05 1 .0 8 8 0 .3 0 4 .0 0 4 .2 2 5 0 .0 1 6 0 .2 2 54l-20-3b 20 3 16 .80 2 .8 0 0 0 .3 0 0 .8 0 1 .025 0 .0 1 6 0 .2 2 54l-20-4b 20 4 16 .80 2 .1 0 0 0 .3 0 0 .8 0 1.025 0 .0 1 6 0 .2 2 54l-20-5b 2 0 5 o 16 .80 1.680 0 .30 0 .8 0 1.025 0 .0 1 6 0 .2 2 54l-20-6b 20 6 CN 16 .80 1 .400 0 .3 0 0 .8 0 1 .025 0 .0 1 6 0 .2 2 54l-40-3h 4 0 3 16 .80 2 .8 0 0 0 .3 0 1.60 1.825 0 .0 1 6 0 .2 2 54l-40-4b 40 4 o 16 .80 2 .1 0 0 0 .3 0 1 .60 1.825 0 .0 1 6 0 .2 2 54l-40-Sb 40 5 16 .8 0 1 .6 8 0 0 .3 0 1 .60 1 .825 0 .0 1 6 0 .2 2 54l-40-6b 40 6 16 .80 1 .400 0 .3 0 1.60 1 .825 0 .0 1 6 0 .2 2 54l-60-3b 60 3 16 .80 2 .8 0 0 0 .3 0 2 .4 0 2 .6 2 5 0 .0 1 6 0 .2 2 54l-60-4b i 60 4 16 .80 2 .1 0 0 0 .3 0 2 .4 0 2 .6 2 5 0 .0 1 6 0 .2 2 54l-60-5b J 60 5 16 .80 1 .680 0 .3 0 2 .4 0 2 .6 2 5 0 .0 1 6 0 .2 2 54l-60-6b 60 6 cs 16 .80 1.400 0 .3 0 2 .4 0 2 .6 2 5 0 .0 1 6 0 .2 2 54l-80-3b 80 3 16 .80 2 .8 0 0 0 .3 0 3 .2 0 3 .4 2 5 0 .0 1 6 0 .2 2 54l-80-4b 80 4 00 16 .80 2 .1 0 0 0 .3 0 3 .2 0 3 .4 2 5 0 .0 1 6 0 .2 2 54l-80-Sh 80 5 16.80 1 .680 0 .30 3 .2 0 3 .4 2 5 0 .0 1 6 0 .2 2 54l-80-6b 80 6 o 16 .80 1 .400 0 .3 0 3 .2 0 3 .4 2 5 0 .0 1 6 0 .2 2 5

4l-l00-3b 100 3 o 16.80 2 .8 0 0 0 .3 0 4 .0 0 4 .2 2 5 0 .0 1 6 0 .2 2 54l-100-4h 100 4 16 .80 2 .1 0 0 0 .3 0 4 .0 0 4 .2 2 5 0 .0 1 6 0 .2 2 54l-100-5b 100 5 16 .80 1 .680 0 .3 0 4 .0 0 4 .2 2 5 0 .0 1 6 0 .2 2 54l-100-6b 100 6 16 .80 1 .400 0 .3 0 4 .0 0 4 .2 2 5 0 .0 1 6 0 .2 2 5

Note : Symbols fo r cross-sectional dimensions are explained in Figure 6.1

2 1 8


Table 6.2. Geometries o f bridges used in parametric study for impact factor and

B r i d g e N u m b e r S p a n ( L ) N u m b e r K = C r o s s S e c t i o n D i m e n s i o n s ( m )o t l a n e s

( N l )( m )

o f b o x e s( N b )

L / RA B C d H

0 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) ( 6 ) ( 7 ) ( 8 ) ( 9 ) ( 1 0 ) ( 1 1 ) ( 1 2 )2l-20-2b 20 2 9 .3 0 2 .3 2 5 0 .3 0 0 .8 0 1 .025 0 .0 1 6 0 .2 2 52l-20-3b 2 0 3 O 9 .3 0 1 .550 0 .3 0 0 .8 0 1 .025 0 .0 1 6 0 .2 2 52l-20-4b 20 4 CS 9 .3 0 1.163 0 .3 0 0 .8 0 1.025 0 .0 1 6 0 .2 2 52l-40-2b 40 2 9 .3 0 2 .3 2 5 0 .3 0 1.60 1 .8 2 5 0 .0 1 6 0 .2 2 52l-40-3b 40 3 9 .3 0 1 .550 0 .3 0 1 .60 1 .825 0 .0 1 6 0 .2 2 52l-40-4b 4 0 4 9 .3 0 1.163 0 .3 0 1.60 1.825 0 .0 1 6 0 .2 2 52l-60-2b § 60 2 9 .30 2 .3 2 5 0 .3 0 2 .4 0 2 .6 2 5 0 .0 1 6 0 .2 2 52l-60-3b 60 3 9 .3 0 1 .550 0 .3 0 2 .4 0 2 .6 2 5 0 .0 1 6 0 .2 2 52l-60-4b CS 60 4 cs 9 .3 0 1.163 0 .3 0 2 .4 0 2 .6 2 5 0 .0 1 6 0 .2 2 52l-80-2b 80 2 9 .3 0 2 .3 2 5 0 .3 0 3 .2 0 3 .4 2 5 0 .0 1 6 0 .2 2 52l-80-3b 80 3 o

d9 .3 0 1.550 0 .3 0 3 .2 0 3 .4 2 5 0 .0 1 6 0 .2 2 5

2l-80-4b 80 4 9 .3 0 1 .163 0 .3 0 3 .2 0 3 .4 2 5 0 .0 1 6 0 .2 2 52l-100-2b 100 2 o ' 9 .3 0 2 .3 2 5 0 .3 0 4 .0 0 4 .2 2 5 0 .0 1 6 0 .2 2 521-100-3b 100 3 9 .3 0 1.550 0 .3 0 4 .0 0 4 .2 2 5 0 .0 1 6 0 .2 2 52l-l00-4b 100 4 9 .3 0 1.163 0 .3 0 4 .0 0 4 .2 2 5 0 .0 1 6 0 .2 2 53l-20-2b 2 0 2 13 .05 3 .2 6 3 0 .3 0 0 .8 0 1 .025 0 .0 1 6 0 .2 2 53l-20-4b 20 4 o

cs13 .05 1.631 0 .3 0 0 .8 0 1 .025 0 .0 1 6 0 .2 2 5

3l-20-5b 2 0 5 13 .05 1.305 0 .3 0 0 .8 0 1.025 0 .0 1 6 0 .2 2 53l-40-2b 40 2 —r 13 .05 3 .2 6 3 0 .3 0 1.60 1.825 0 .0 1 6 0 .2 2 53l-40-4b 40 4 o

o13 .05 1.631 0 .3 0 1.60 1 .825 0 .0 1 6 0 .2 2 5

3l-40-5b 40 5 13 .05 1.305 0 .3 0 1.60 1 .825 0 .0 1 6 0 .2 2 53l-60-2b § 60 2 13 .05 3 .2 6 3 0 .3 0 2 .4 0 2 .6 2 5 0 .0 1 6 0 .2 2 53l-60-4b 5 60 4 13.05 1.631 0 .3 0 2 .4 0 2 .6 2 5 0 .0 1 6 0 .2 2 53 l6 0 -5 b CO 60 5 cs 13 .05 1.305 0 .3 0 2 .4 0 2 .6 2 5 0 .0 1 6 0 .2 2 53l-80-2b 80 2 d ' 13 .05 3 .2 6 3 0 .3 0 3 .2 0 3 .4 2 5 0 .0 1 6 0 .2 2 53l-80-4b 80 4 o 13.05 1.631 0 .3 0 3 .2 0 3 .4 2 5 0 .0 1 6 0 .2 2 53l-80-Sb 80 5 13.05 1.305 0 .3 0 3 .2 0 3 .4 2 5 0 .0 1 6 0 .2 2 5

3l-100-2b 100 2 o*' 13 .05 3 .2 6 3 0 .3 0 4 .0 0 4 .2 2 5 0 .0 1 6 0 .2 2 53l-100-4b 100 4 13.05 1.631 0 .3 0 4 .0 0 4 .2 2 5 0 .0 1 6 0 .2 2 53l-100-5b 100 5 13.05 1.305 0 .3 0 4 .0 0 4 .2 2 5 0 .0 1 6 0 .2 2 54l-20-3b 20 3 16 .80 2 .8 0 0 0 .3 0 0 .8 0 1 .025 0 .0 1 6 0 .2 2 54l-20-4b 20 4 o 16 .80 2 .1 0 0 0 .30 0 .8 0 1.025 0 .0 1 6 0 .2 2 54l-20-6b 20 6 16 .80 1 .400 0 .3 0 0 .8 0 1 .025 0 .0 1 6 0 .2 2 54 l40 -3b 40 3 16 .80 2 .8 0 0 0 .3 0 1.60 1 .825 0 .0 1 6 0 .2 2 54l-40-4b 40 4 o

o16 .80 2 .1 0 0 0 .3 0 1.60 1.825 0 .0 1 6 0 .2 2 5

4l-40-6h 40 6 16 .80 1.400 0 .3 0 1.60 1 .825 0 .0 1 6 0 .2 2 54l-60-3b 60 3 16 .80 2 .8 0 0 0 .3 0 2 .4 0 2 .6 2 5 0 .0 1 6 0 .2 2 54l-60-4b J 60 4 16 .80 2 .1 0 0 0 .3 0 2 .4 0 2 .6 2 5 0 .0 1 6 0 .2 2 54l-60-6b Tj- 60 6 cs 16.80 1.400 0 .3 0 2 .4 0 2 .6 2 5 0 .0 1 6 0 .2 2 54l-80-3b 80 3 d" 16 .80 2 .8 0 0 0 .3 0 3 .2 0 3 .4 2 5 0 .0 1 6 0 .2 2 54l-80-4b 80 4 o

d16 .80 2 .1 0 0 0 .3 0 3 .2 0 3 .4 2 5 0 .0 1 6 0 .2 2 5

4l-80-6b 80 6 16 .80 1.400 0 .3 0 3 .2 0 3 .4 2 5 0 .0 1 6 0 .2 2 54l-100-3b 100 3 o" 16 .80 2 .8 0 0 0 .3 0 4 .0 0 4 .2 2 5 0 .0 1 6 0 .2 2 54l-100-4b 100 4 16 .80 2 .1 0 0 0 .3 0 4 .0 0 4 .2 2 5 0 .0 1 6 0 .2 2 54l-W 0-6b 100 6 16.80 1.400 0 .3 0 4 .0 0 4 .2 2 5 0 .0 1 6 0 .2 2 5

N ote : Sym bols fo r cross-sectional dimensions are explained in Figure 6.1

219


Table 6.3. Vehiclespeed used in parametric study for impact factoK =

L/R(1 )

Span length (L) (m)(2 )

Speed (v) (km/h)

(3)

0

20 12040 12060 12080 120too 120

0.120 98

40 120

0.220 70

40 98

0.4

20 5040 7060 8580 98

100 110

0.6

60 7080 80

100 90

1.2

60 5080 57

100 63

220


Table 7.1. Comparison between the results obtained from the finite element analysis

M a x .S t r a i n i n g

D e a d L o a d A A S H T O1 9 9 6

A A S H T O L R F D 1 9 9 8

C H B D C2 0 0 0

A c t i o nF .E .M P r o p o s e d

m e th o d F .E .M P r o p o s e dm e t h o d F .E .M P r o p o s e d

m e t h o d F .E .M P r o p o s e dm e t h o d

a p ( M P a ) 4 6 5 3 2 0 2 2 3 6 4 4 4 2 4 5

a „ ( M P a ) 8 6 9 5 2 6 2 7 4 3 4 5 4 1 3 8

5 ( m ) 0 .0 3 4 0 .0 3 9 0 .0 1 6 0 .0 1 6 0 .0 2 8 0 .0 2 6 0 .0 3 4 0 .0 3 3

V ( k N ) 6 4 3 6 5 5 2 4 8 2 5 6 3 8 7 4 0 6 4 2 7 4 0 8

R e ( k N ) 6 0 2 6 4 6 2 2 0 2 5 5 3 6 7 4 0 5 4 2 5 4 8 8

R i ( k N ) 1 2 8 6 1 3 1 0 4 1 7 4 2 9 6 7 5 7 1 8 6 2 5 6 3 1

Rm ( k N ) 6 0 - 1 4 -8 3 - 8 0 - 1 3 9 - 1 2 8 - 1 7 2 - 1 5 4

221


Table 9.1. Comparison between the fmite-element results and those obtained from proposed equations for fundamental frequency

Bridge L/D N l NbL

(m)L/R

I

(m'*)

M(kN.

s^/m^)

fbeam(Eq.9.4)

h

Frequency (Hz)

Eq.2

Eq.5

F.E.M.

4l-100-3b 2 0 4 3 too 0 .0 4 .4 8 1 5 .3 4 1 .2 0 1 .0 0 0 .9 4 1 .1 4 1 .1 7

4l-100-3b 2 5 4 3 too 0 .0 2 .7 1 1 5 .3 4 0 .9 3 1 .0 0 0 .9 4 0 .8 8 0 .9 5

4l-100-3b 3 0 4 3 too 0 .0 1 .8 1 1 5 .3 4 0 .7 6 1 .0 0 0 .9 4 0 .7 2 0 .7 9

41-100-3 b 2 0 4 3 too 0 .6 4 .4 8 1 5 .3 4 1 .2 0 0 .8 8 0 .8 3 0 .9 9 0 .9 9

4l-100-3b 2 5 4 3 too 0 .6 2 .7 1 1 5 .3 4 0 .9 3 0 .8 8 0 .8 3 0 .7 7 0 .8 3

4l-W0-3b 3 0 4 3 too 0 .6 1 .8 1 1 5 .3 4 0 .7 6 0 .8 8 0 .8 3 0 .6 3 0 .7 1

4l-I00-3b 2 0 4 3 too 1 .2 4 .4 8 1 5 .3 4 1 .2 0 0 .6 6 0 .6 2 0 .7 5 0 .7 2

4l-I00-3b 2 5 4 3 too 1 .2 2 .7 1 1 5 .3 4 0 .9 3 0 .6 6 0 .6 2 0 .5 8 0 .6 2

4l-100-3b 3 0 4 3 too 1 .2 1 .8 1 1 5 .3 4 0 .7 6 0 .6 6 0 .6 2 0 .4 8 0 .5 4

222


CD■DOQ.CoCDQ .

■DCD

(/)Wo'o

oo■oc 5 ‘

Concrete section Composite section Steel section

oCD

cCD

CD■oOQ .Cao= 5

1 7

N>K)

Da) Types of single-box girders

a O ' QTQS«n

CDa. b) Types of multiple box girders

oc■oCD

(/)o'= 3

I 7c) Types of cellular girders

Figure 1.1. Various box girder cross-sections

Figure 1.2. View of eontinuous curved composite twin box-girder bridge

Figure 1.3. Box girder bridge under construction (US290/IH 35 interchange, Direct connector Z)

224


<D

§(UijiVit/1O)HoD00•cVhD•!a00X0

1

cSO‘BnH

(U

.1

225


Strain

Stress

Figure 3.1. Stress-strain relationship for steel adopted in ABAQUS model

Stress, a

Failure point

Tension stiffeningcurve

Strain, sFailure point, s“ =

Uncracked Process zone Crack open

Figure 3.2. Tension stiffening model in reinforced concrete

226


StressFailure point in compression

(peak stress)

Start of inelastic stage

\ldealized (elastic) unload/reload response

StrainSofteningCracking Failure

Plasticstrain

Figure 3.3. Uniaxial stress-strain relationship for plain concrete

UniaxialCrack detection surface \ tension Biaxial

tension

Uniaxialcompression

Biaxialcompression / _________

Figure 3.4. Concrete failure surfaces in plane stress

227


Shell elementBeam element for steel top tlange

Rigid link for top slab

Node

Shell elementBeam elementfor web Shell element

for cross bracing for bottom flange

Figure 3.5. Finite element discretization of cross-section of the bridge models

228


SF2 SF3 SFlSF5 SF4

2(a) Output forces (c) Shell element "S4R"

(in ABAQUS)

SM2 SMISM3 SM3

2

1

(b) Output moments (d) REBAR in a shell

- Four-node element- Degrees of freedom- Output forces- Output moments- S t r e s s c o m p o n e n t s

U1,U2, U3,§l<l2,f3 SFl, SF2, SF3, SF4, SF5 SM1.SM2, SM3 811,822,512

- REBAR Option available in two directions

Figure 3.6. Shell element "S4R" used for plate modelling

229


1 25•

(a) Integration points o f beam in space (for output results)

(b) Element "B31H" in ABAQUS

Two-node element Degrees of freedom Output forces Output moments Stress components

Ul, U2, U3,$l,$2,$3 SFl, SF2, SF3 SMI, SM2, SM3 S11,S12

Figure 3.7. Beam element "B3IH" for beam in space

230


CD■DOQ .CoCDQ .

■DCD

C/)

o '3O

Oo■D

c5'

o3O"ncO’oCD

■DOQ .Caoo■Oo

CDQ .

■DCD

(/)(/)

tow

\ /Key to symbols

\O F i x e d i n t h e t h r e e d i r e c t i o n s

F r e e i n o n e d i r e c t i o n

F r e e i n t w o d i r e c t i o n s

Figure 3.8. Boundary condition of the bridges used in the parametric studies

a)

b)

Figure 3.9. Typical finite element mesh for: a) the non-composite bridge model; and b) the composite bridge model

232


CD“DOQ.CoCDQ .

“DCD

C/)(/)

OO■D(5'

CD■aoQ .cao3T3O

CDQ .

■DCD

C/)C/)

toU)U)

&

Span 1 Span 2 &C/DO)

-3575- -3575

-715 -715 ■715 -715 -715 -715 -715 -715 -715

D ia p h r a g m .

D ia p h r a mD ia p h r a g mC r o s s b r a c in g

A l l d i m e n s i o n s a r e in m m

Figure 4.1. Plan for the experimental straight bridge model

jjoddns

/<uo

(DbOT3•crOT30)

I(UB

•COhX!<u(U’B

)-(

sdcd

CN

U

P-l

234


Cross Section (A-A)

1120-

-2 5 0 -•185- -250- ■250- 185-

O n O

-275- -275-l ia p h r a m

A c c e s s h o le

Cross Section (B-B)

T o p f la n g e

1120-

- 2 5 0 -•185- •250- -250- ■185-

3-1

■275- -275-

B o t t o m f la n g eA ll d im e n s io n s are in m m

Figure 4.3. Cross-sectional details o f the bridge models

235


Figure 4.4. Tension test set-up for steel reinforcement specimen used in the bridge models

300

250

200Ph2

1<U2^ 100

0 ^ - - - - - - - - - - - - - - - - - - - - - - - i - - - - - - - - - - - - - - - - - - - - - - - - - 1- - - - - - - - - - - - - - - - - - - - - - - - - i - - - - - - - - - - - - - - - - - - - - - - - - - i----------- i---------- i-----------1----------- i-----------1----------0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020

T r u e s t r a in

Figure 4.5. True stress-true strain relationship for structural steel plate

236


MAX. LOAD 300,000 LBS.

Figure 4.6. View of a tested concrete cylinder after failure

& 30

0.000 0.001 0.002 0.003 0.004 0.005Strain

Figure 4.7. Stress-strain relationship for concrete cylinders of the curved bridge model

237


1000

8 0 0 -

S ' 6 0 0 -

200 -

0.000 0.002 0 .0 0 4 0 .0 0 6 0 .0 0 8 0.010 0.012 0 .0 1 4 0 .0 1 6 0 .0 1 8 0.020True Strain

Figure 4.8. Stress-strain relationship for the reinforcing steel

400

300 -

100 - <

0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020T r u e s tr a in

Figure 4.9. True stress-true strain relationship for steel shear eonnectors

238


Figure 4.10. View of shear cormectors welded to the top flange

Figure 4.11. View of straight bridge model during fabrication

239


Figure 4.12. View of the formwork for the curved bridge model

Styrofoami m s u l a t i o n c o n

(SOLAWT DC MOUSSE

Figure 4.13. View of the formwork and reinforcing steel bars for the straight bridgemodel

240


Figure 4.14. View of the formwork and reinforcing steel bars for the curved bridge model

Figure 4.15. View of the curved bridge model along with the concrete cylinders during curing

241


Figure 4.16. View of the strain gauges installed along the bottom flange width at themid-span section

242


At mid-span 1

• A •. . 1

E

u .00Tl-1—

X^ X X ^

B------- S------ -

X X Xa ^ / E

Xa E

N ^ ^ /

X ^ X ^ X X

00

At central support

r -------------- 310— - 2 5 0 - - 2 5 0 n - - 3 1 0 -

X

At mid-span 2

• C o n c r e t e s tr a in g a u g e

® S t e e l s tr a in g a u g e

- 3 1 0 - - 2 5 0 - - 5 6 0 -

Figure 4.17. Locations o f strain gauges on the longitudinal direction of the bridge models

243


Figure 4.18. View of the LYDTs in the first span of the bridge model

244


At quarter span 1

-1120-

At mid-span

- 1120 -

\ / •

\ /

X

At three-quarter span 1

- 1120-

Figure 4.19. Locations o f LVDTs in the cross section of the bridge model

245


Figure 4.20. View of the accelerometers in the second span of the bridge model

246


At quarter span 2

-1120-

/ V .

\ /

At mid-span 2

-1120-

■ "'1— ^ \ /

u

At three-quarter span 2

1120-

•N\

> c ^ ^ \ / ^ 'X

^ / \ / \^ 'N,

U 0

Figure 4.21. Locations o f accelerometers in the cross section of the bridge model

247


I

Figure 4.22. View of the load cells at the exterior support

248


At support line 1

-1120-

At central support

- 1120 -

At support line 2

-1120-

Figure 4.23. Locations o f load cells at support lines of the bridge model

249


a) Roller Support

■1120--250-•185* ■250- -250- •185-----

Load cells

PL. 50 X 50 X 10/ PL. 150x 150x20.

Shear connectors

L o a d ce ll

Supporting beam.

C r o s s S e c t i o n ( A - A )

b) Hinged Support

1120--250--185- -250- -250^ -185— -

cells

PL. 50 X 50 X 10 PL. 150 X 150x20.

Shear connectors

Load cell.Supporting beam.

C r o s s S e c t i o n ( B - B )

Figure 4.24. Details o f bearings

250


I

Figure 4.25. Data acquisition system connected to the straight bridge model

Figure 4.26. Test set-up for the straight bridge model

251


Figure 4.27. View of Loading Case 1 applied to the non-composite straight bridgemodel

252


Cast iron barLVDT

Jacking load downward

Figure 4.28. View of the flexural vibration test for straight bridge model

253


Cast iron bar LVDT

Jacking load downward

Figure 4.29. View of the torsional vibration test for curved bridge model

254


Figure 4.30. View of straight bridge model under Loading Case 1

Figure 4.31. View of straight bridge model imder Loading Case 2

255




256


Figure 4.34. View of curved bridge model under Loading Case 1


257


Figure 4.36. View of curved bridge model xmder Loading Case 3


258


i



259


a) Loading Case 15pan 1- 5pan 2 ~T-

S 14 B lo cks ; 14 Blocks!!^14 Blocl

!14 Blocks!

Cross Section

Planb) Loading Case 2

Blocks^ w 8 Blocks^ J 118 Blocks]

Cross Sectioni i i

Plan

c) Loading Case 3

^18 Blocks'V/ / / / / / / / / / / ,

K l8 Blocks; ! ! 1r 1 1 i 1

1 1 1 1 1 f i 1 1 1 1

r . i __ IC ro ss S e c tio n

d) Loading Case 4Plan

V777777777, 18 Blocks /

Vr/%

Cross Section

e) Loading Case 5W77777777m / 1 8 B locksV '/ // / // / / / / /

Cross Section

^ V ///M /////X n Blocks

" " 4 .1 ----- 1 - - - - - - - 1 - - - - - - - 1 -- - - - - - J- - - - - - - - -

Plan

Figure 5.1. Cases of loading for non-composite straight bridge model

260


0.0 0.00.0 0.0

(a) Deflections at L/4 from the outer support in span (1)

0.0 0.00.0 0.0

(b) Deflections at mid span (1) F i n i t e - e l e m e n t r e s u lt s

• E x p e r im e n t a l r e su lts

V a lu e s a r e in m il l im e te r s

0.0 0.00.0 0.0

(c) Deflections at 3L/4 from the outer support in span (1)

Figure 5.2. Deflections of the non-composite straight bridge model due to Loading Case 1

261


52 ~5J

65

(a) Longitudinal strains at mid span (1)

(b) Longitudinal strains at the interior support — F i n i t e - e l e m e n t re su lts • E x p e r im e n t a l r e s u lt s+ Tension

Compression

V a lu e s a r e in m ie r o s tr a in

31'

17

31 3L

17' \ 2 U ' _ " ^2\J.

r31

17

(c) Longitudinal strains at mid span (2)

Figure 5.3. Longitudinal strains o f the non-composite straight bridge model due to Loading Case 1

262


CO■ooQ .coCDQ .

■oCD

C/)

o'o

oo■ov<cq' S u p p o r t l i n e 1 C e n t r a l s u p p o r t S u p p o r t l i n e 2

oCD

Cp.

CD■oOQ .Cao3■oo

Finite-elementresults

o o o

to

Values are in kN

CDQ .

O’o■DCD

Experimentalresults

C/)(/)Figure 5.4. Reactions for the non-composite straight bridge model due to Loading Case 1

0.30.0 0.00.0

0.50.9


0.40.0 0.0

0.7

(b) Deflections at mid span (1)

0.00.4

0.0

0.7



264


-4 4 -------- 7 5 -


(b) Longitudinal strains at the interior support

22

34:

k.49-27 247 :


12

rr2 8

F i n i t e - e l e m e n t r e s u lt s• E x p e r im e n t a l r e su lts+ Tension

Compression

V a lu e s a r e in m ic r o s tr a in

15

M)

11


265


CD■DOQ .CoCDQ .

■DCD

C/)(/)

OO■D<5‘ Support line 1 Central support Support line 2

3.

CD■DOQ .Cao■oo


o (No

toOsOs

Values are in kN

CDQ .

■DCD

Experimentalresults

C/)C/)

Figure 5.7. Reactions for the non-composite straight bridge model due to Loading Case 2

0.40.0 0.00.0

0.6


0.7

0.0 0.0


— F i n i t e - e l e m e n t r e s u lt s • E x p e r im e n t a l r e s u lt s


0.6

0.0 0.0e-.o0.5


Figure 5.8. Deflections o f the non-composite straight bridge model due to Loading Case 3

267


37


54

I------1

(b) Longitudinal strains at the interior support F in i t e - e l e m e n t r e s u lt s• E x p e r im e n t a l r e s u lt s+ Tension

Compression



37


268


73CD■DOQ .CoCDQ .

"DCD

(/)C/)

CD

OO■DS u p p o r t l i n e 1 C e n t r a l s u p p o r t S u p p o r t l i n e 2

CD

CD■DOQ .Cao

■oo

CDQ .

■DCD


toa'O

Values are in kN

Experimentalresults

C/)if)Figure 5.10. Reactions for the non-composite straight bridge model due to Loading Case 3

"o:o0.3

0.00.0 0.00.7


0.0 0.00.00.4

0.0

0.9


— F i n i t e - e l e m e n t r e su lts • E x p e r im e n t a l r e s u lt s


0.00.0 0700.3

0.00.6

(c) Deflections at 3 L/4 from the outer support in span (1)


270


-57


50

32

58

(b) Longitudinal strains at the interior support F i n i t e - e l e m e n t r e s u lt s• E x p e r im e n t a l r e s u lt s+ Tension

Compression


10 IS

11 Ij\ 14" •'I

29

13


Figure 5.12. Longitudinal strains of the non-composite straight bridge model due to Loading Case 4

271


7JCD■DOQ .CoCDQ .

■DCD

(/)(/)

CD

OO■DS u p p o r t l i n e 1 C e n tr a l s u p p o r t S u p p o r t l i n e 2

CD■DOQ .C9-o■oo


to-oto

Values are in kN

CDQ .

■aCD

Experimentalresults

(/)(/)Figure 5.13. Reactions for the non-composite straight bridge model due to Loading Case 4

0.00.5

0.0'0.5

0.00.5

0.00.5


0.0

0.60.0

0.7

0.0

0.7

0.0

0.6


— F i n i t e - e l e m e n t r e s u lt s • E x p e r im e n t a l r e su lts


0.00.4.

0.00.4

0,00.4

0.00.4


Figure 5.14. Deflections o f the non-composite straight bridge model due to Loading Case 5

273


22i

f 38 4 4 ^\ I

k23

28 28

23-i

^ 4 4I / '' 38

28

k22

28

(a) Longitudinal strains at mid span (I)

22 28I 1I I

17 \9 A I S

28

15

I - - - - - - - - - - 1

I_ _ _ _ _ _ I

22

IK19 17 H14

(b) Longitudinal strains at the interior support — F i n i t e - e l e m e n t r e s u lt s • E x p e r im e n t a l r e su lts+ Tension

Compression


13 fl3r


Figure 5.15. Longitudinal strains of the non-composite straight bridge model due to Loading Case 5

274


7JCD-oOQ .coCDQ .

■oCD

I(/)o'oo

oo■oc q ' S u p p o r t l i n e 1 C e n t r a l s u p p o r t S u p p o r t l i n e 2

QCD

Cp.O’oCD■aoQ .cao3■Oo


oK>LTi

Values are in kN

CDQ .

O’oc■oCD

Experimentalresults

C/)C/)Figure 5.16. Reactions for the non-composite straight bridge model due to Loading Case 5

a) Loading Case 1

^ 1 4 Blocks^ / / / / / / / / / . ^14 Blocks''^///////////. V,\A Blocks:

Cross Section

b) Loading Case 2

14 B l o c k s k l 4 Blocksy /y /^ /y /A \/yy/yyy/y.

Cross Section

c) Loading Case 3

Kl4 Blocks; ;14 Blocks ///////////

Cross Section

d) Loading Case 4

p l4 Blockss3'^r

Cross Section

e) Loading Case 57777777777/ 'A A Blocks/

'yyyy/yyyyyy. <ICross Section

Plan

Figure 5.17. Cases of loading for the non-composite curved bridge model

276


0.0 0.00.0 0.0

3.3


0.0 0.00.0 0.0

4.6

7.2

10.0(b) Deflections at mid span (1)

F i n i t e - e l e m e n t r e su lts• E x p e r im e n t a l r e s u lt s


0.0 0.00.0 0.0

4.96.7


Figure 5.18. Deflections of the non-composite curved bridge model due to Loading Case 1

277


R

50

39 J4-■38


114

120’

151

205;116

168 150

123K(b) Longitudinal strains at the interior support F i n i t e - e l e m e n t r e s u lt s

E x p e r im e n t a l r e s u lt s+ Tension

Compression


-73- - -92 116


Figure 5.19. Longitudinal strains o f the non-composite curved bridge model due to Loading Case 1

278


7]CD■oOQ .CoCDQ .

■oCD

C/)

o'13

OoT 3v<c q ' S u p p o r t l i n e I C e n t r a l s u p p o r t S u p p o r t l i n e 2

QCD

Cp.

CD■oOQ .CaoQ■oo


<srs o

to<1

Values are in kN

CDQ .

Oc■oCD

Experimentalresults

(/>o'Q Figure 5.20. Reactions for the non-composite curved bridge model due to Loading Case 1

0.00.3. 0.00.0

0.90.0

2.3


R

0.00.3 (TO'

1.1


0.0

2.03.0

— F in i t e - e l e m e n t r e s u lt s • E x p e r im e n t a l r e su lts

V a lu e s a r e in m il l im e te r sR

0.00 .2* 0.0

0.70.0

(c) Deflections at 3 L/4 from the outer support in span (1)

Figure 5.21. Deflections of the non-composite curved bridge model due toLoading Case 2

280


R

- - 20 -

^ 4 1I / ^

30

^35

'46


104

I _ _ _ _ _ _ 1


R

33

- 8- -

10

F i n i t e - e l e m e n t r e s u lt s• E x p e r im e n t a l r e s u lt s+ Tension

Compression



Figure 5.22. Longitudinal strains of the non-composite curved bridge model due to Loading Case 2

281


CD■DOQ.CoCDQ .

■DCD

C/)(/)

OO■Dc q ' S u p p o r t l i n e 1 C e n t r a l s u p p o r t S u p p o r t l i n e 2

O’QCD■DOQ .Cao■oo


R R

oo o oto00to

Values are in kN

CDQ .

"DCD

Experimentalresults

R R

O O(/)(/)

Figure 5.23. Reactions for the non-composite curved bridge model due to Loading Case 2

R

0.00 .4 , 0.0

1.2

0.00.0

3.0


0.00.5

0.00.0 0.0


F i n i t e - e l e m e n t r e s u lt s• E x p e r im e n t a l r e su lts

V a l u e s a r e in m il l im e te r s

0.00.3 0.00.0 O.tf

2.6(c) Deflections at 3L/4 from the outer support in span (1)


283


R

63 ^3-

2\


109


R

Finite-element results

• Experimental results

+ Tension

Compression

Values are in microstrain

27'

759

PlD. 9- • " - u i

25

13

36



284


7JCD■oOQ.coCDQ .

7DCD

(/)(/)

OO“DS u p p o r t l i n e 1 C e n t r a l s u p p o r t S u p p o r t l i n e 2

3"?CD"DOQ .Cao3■oo

CDQ .

■DCD


co o

to00LhValues are in kN

Experimentalresults

(/)(/)Figure 5.26. Reactions for the non-composite curved bridge model due to Loading Case 3

0.00.5,

0.00.0 0.0

2.6

3.8(a) Deflections at L/4 from the outer support in span (1)

R !

0.0 0.00.0 0.0

2.2

(b) Deflections at mid span (1) 3.7

5.3

R

0.00.6

— F i n i t e - e l e m e n t r e s u lt s • E x p e r im e n t a l r e s u lt s


0.0'

1.5

0:0


0.0

3.5

Figure 5.27. Deflections o f the non-composite curved bridge model due to Loading Case 4

286



105,

(b) Longitudinal strains at the interior support F i n i t e - e l e m e n t r e s u l t• E x p e r im e n t a l r e s u lt s+ Tension

Compression



Figure 5.28. Longitudinal strains of the non-composite curved bridge model due to Loading Case 4

287


7]CD■DOQ .CoCDQ .

■DCD

C/)Wo'oo

oo■Dcq'

3CD

3"OCD■DOQ .Cao3■oo

CDQ.

■DCD


S u p p o r t l i n e 1

R

C e n t r a l s u p p o r t S u p p o r t l i n e 2

R

cso

00fSo o

boooOOValues are in kN

Experimentalresults

C/)C/)

Figure 5.29. Reactions for the non-composite curved bridge model due to Loading Case 4

R

0.00.5 o:o-

1.10.0

2.3


0.00.7

0.00.0 0.0

2.3

3.2(b) Deflections at mid span (1)

R

— Finite-element results • Experimental results

Values are in millimeters

0.00.5

0.00.0 0.0

2.2(c) Deflections at 3 L/4 from the outer support in span (1)


289


R

25

12

' - ' 1

Iii13


r36


R

10

p i ■ 23 ^

13

F i n i t e - e l e m e n t r e s u lt s• E x p e r im e n t a l r e su lts+ Tension

Compression


r38

22



290


CD■DOQ.CoCDQ .

■DCD

C/)Wo'oo

oo■Dc q '

Support line 1 Central support Support line 2

O’oCD■DOQ .CaoQ■oo

CDQ .

"DCD


NJVO

Values are in kl

Experimentalresults

R R

VOo

R

(/)(/)Figure 5.32. Reactions for the non-composite curved bridge model due to Loading Case 5

a) Loading Case 1_l

- 1 4 3 0 --Span 1-

-2145--Span 2 - - 3 5 7 5 —

- J

b) Loading Case 2

(— -1430— r r - - - - ^2145- - 3 5 7 5 -

--------

c) Loading Case 3

- 2 1 4 5 - •1430— r - 1 4 3 0 — r - - - - - ^2145-

--------------^W kN— j—l i k N S ---------------

r

d) Loading pase 4

-2145 - 1 4 3 0 — p l 4 3 0 -

S S E t l ^ v : - :- 2 1 4 5 -

Figure 5.33. Cases of loading for the composite straight bridge model

292


0.0 0.01 .4

0.00.0

2.2


0.0

2.2

0.0

2.6 2 .9

(b) Deflections at mid span (1)— F in ite -e le m e n t resu lts • E x p er im en ta l resu ltsV a lu e s are in m illim e ters

0.00.0 0.01 .3


Figure 5.34. Deflections of the composite straight bridge model due to Loading Case 1

293


1 2 6

1 7 8 2 7 1- - 1 - 8 8 - - 2 5 6 -

200 2102 8 0 ? 2 9 4


6- f^8 2 4 3 2 3 7 .I - - - - - - - - - 1

I _ _ _ _ _ _ I

I - - - - - - - - - 1

2 3 7 1 9 6

(b) Longitudinal strains at the interior support Finite-element results


+ Tension

CotPpTession


2 5 2 8

5 8! , \6 4 ,

56- 5 5 .161

5 4


Figure 5.35. Longitudinal strains of the composite straight bridge model due to Loading Case 1

294


CD■DOQ.CoCDQ .

■ DCD

(/)Wo'o

oo■oc q ’

QCD

CP-O’oCD■oOQ .cao=5

■Oo

CDQ .

Oc■oCD

S u p p o r t l i n e 1 C e n tr a l s u p p o r t S u p p o r t l i n e 2


I C□ X □

> <3 C>

1 r□ □

toVOOl

Experimentalresults

Values are in kN

□ X □ □ X □

o <>

□ X □

oo

oo

(/)o'=3 Figure 5.36. Reactions for the composite straight bridge model due to Loading Case 1

0.0 0.0

3 .6 3 .6

0.00.0

3 .63 .6



0.00.0 0.0

4 .8

— Finite-element results



0.00.0 O.T) 0.0

2.8



296



241 239 239 241

53

(b) Longitudinal strains at the interior support F in ite -e le m e n t resu lts

• E x p e r im e n ta l resu lts+ Tension

Compression

V a lu e s are in m icrostra in

53

.124 124>

112 112112112


Figure 5.38. Longitudinal strains o f the composite straight bridge model due to Loading Case 2

297


7JCD■DOQ.CoCDQ .

■DCD

(/)(/)

CD

OO■oS u p p o r t l i n e 1 C e n t r a l s u p p o r t S u p p o r t l i n e 2

Finite-element

CD

CD■DOQ .Cao

results

ON

□ □

j> c 5

] c□ x: □

) r

K)00

T 3

O

CDQ .

T 3CD

Experimentalresults

□ X □] L

□ X □] C

r-On

□ X □

5ooC/)(/)

Figure 5.39. Reactions for the composite straight bridge model due to Loading Case 2

0.00.7

O D0.8

0.01.0-

0.0

1.2


0.00.00 .9

0.01.2

(b) Deflections at mid span (1)— F in ite -e le m e n t resu lts • E x p er im en ta l resu ltsV a lu e s are in m illim e ters

0.00 . 5 , O.T)

0 .7

- " " - ^

0 .00 .9

0.01.1

( c ) D e f l e c t i o n s a t 3 L / 4 f r o m t h e o u t e r s u p p o r t i n s p a n ( 1 )


299


1 0 5 i n i75~ 1 9 21 1 6

1 2 8 2 0 6 2 2 3


1 7 3

4 8 14 3 4

2 4 6 ,,2 0 4

212 1 7 2 3 4 3 3 8 2



+ Tension

Con^ression


1 1 9 * * 1 2 8

( c ) L o n g i t u d i n a l s t r a i n s a t m i d s p a n ( 2 )


300


7JCD7DOQ.CoCDQ .

“DCD

(/)(/)

CD

OOTD S u p p o r t l i n e 1 C e n t r a l s u p p o r t S u p p o r t l i n e 2

CD

CD"DOQ .Cao


o

□ □ □ X □

cs o o

U)o

■Do

■DCD

Experimentalresults

ooo

oOn

§3

□ X □ □ X □ □ X □oc

C/)(/)Figure 5.42. Reactions for the composite straight bridge model due to Loading Case 3

0.00.0 0.0


b.T) 0.00.0 0.0

2.8

(b) Deflections at mid span (1) — Finite-element results • Experimental results


0.00.0 0.0

1.6


Figure 5.43. Deflections o f the composite straight bridge model due to Loading Case 4

302


1 3 7 1 3 7

2 9 72 9 7

3 3 9 3 3 5 3 3 5 3 3 9(a) Longitudinal strains at mid span (1)

2 4 2 2 4 2686 6866 8 1

1 1 5 p ^ lT - H7- 1 1 5

5 5 3 5 5 4 5 5 4 5 5 3

(b) Longitudinal strains at the interior support F in ite -e le m e n t resu lts• E x p er im en ta l resu lts+ Tension

Compression


3 2 4 • • 3 2 0


3 2 0 3 2 4


303


73CD■DOQ.CoCDQ .

■DCD

C/)C/)

OO■Dc q '

S u p p o r t l i n e 1 C e n t r a l s u p p o r t S u p p o r t l i n e 2

O’QCD■DOQ .Cao


□ > c □ □ X □

o

■ao

CDQ .

■DCD

C/)C/)

Experimentalresults

cS

□ X □

Figure 5.45. Reactions for the composite straight bridge model due to Loading Case 4

a) Loading Case 1

c) Loading Case 3

b) Loading Case 2 ■, \ \

\ y

d) Loading Case

e) Loading Case

f) Loading Case

Figure 5.46. Cases of loading for composite curved bridge model

305


R -►I

0.0

2 .5

O.T) t 0.0

3 .34 .0

(a) Deflections at L/4 from the outer support in span (1) R__________^

0.0 0.0

2.63 .4

(b) Deflections at mid span (1)R

0.00.0

4 .45 .3

— F in ite -e le m e n t resu lts • E x p e r im e n ta l resu ltsV a lu e s are in m illim e ters

0.0 0.00.0

2.6( c ) D e f l e c t i o n s a t 3 L / 4 f r o m t h e o u t e r s u p p o r t i n s p a n ( 1 ) •

Figure 5.47. Deflections of the composite curved bridge model due to Loading Case 1

306


- m - 3 S 9 -

2 6 2 2 0 4


3 7 0 2 3 0



+ Tension

“ Con^ression



Figure 5.48. Longitudinal strains o f the composite curved bridge model due to Loading Case 1

307


CD■oOQ .CoCDQ .

■oCD

(/)o '3

OO■ov<c q '

Q.O’oCD■oOQ .caoQ“Oo

CDQ .

TDCD

S u p p o r t l i n e 1 C e n tr a l s u p p o r t S u p p o r t l i n e 2

Finite-elementresults R R R1 1 1 1 1..... ..................... -

□ □ □ □ □ □

oo <N o

OJO00

Experimentalresults

C/)(/)Figure 5.49. Reactions for the composite curved bridge model due to Loading Case 1

R- ► I

0.00.0 0.0

2 .0. 2.2 2 .3

(a) Deflections at L/4 from the outer su|)port in span (1)

0.00.0 0.0

2 .7 2.9


R— F in ite -e le m e n t resu lts • E x p e r im e n ta l resu ltsValues are in millimeters

0.0 b.T) “ “ 0.00.0

( c ) D e f l e c t i o n s a t 3 L / 4 f r o m t h e o u t e r s u p p o r t i n s p a n ( I )

Figure 5.50. Deflections of the composite curved bridge model due to loading Case 2

309


' - -204 - +236- -

335

66

(a) Longitudinal strains at mid span (I)R

249r -iI I164

123

187 131

34


R


+ Tension

Compression


28

1--

58/1

65 50


Figure 5.51. Longitudinal strains of the composite curved bridge model due to Loading Case 2

310


CD“DOQ.CoCDQ .

“DCD

C/)(/)

OO■D(5' S u p p o r t l i n e 1 C e n t r a l s u p p o r t S u p p o r t l i n e 2

CD■aoQ .cao

Finite-element=! results R R R^ 1 1 1------------------------------------ ----------- 1 1 _

■nc3-

□ □ □ □ □ □

rrou>

■ao

CDQ .

■DCD

(/)(/)

Experimental

'OVOfS

results R R R1 i 1 ...................................... ---------------------------1 1

□ □ □ □ □ □

o

o oo o

Figure 5.52. Reactions for the composite curved bridge model due to Loading Case 2

0.00.0 0.0

■----- '4 .65 .4 - — .

I(a) Deflections at L/4 from the outer support in span (1)

- R J

6.2

0.00.0 0.0

5.3

6.27 .3

(b) Deflections at m id span (1)F inite-elem ent results

R • Experim ental results

V alues are in m illim eters

0.00.0 0.0

3 .2

4 .45 .0

(c) Deflections at 3L/4 from the outer support in span (1) ^


312


- t - 3 7 6 -


2 8 0 5 0 1


• E x p er im en ta l resu lts+ Tension

Conq>i«ssion

V a lu e s are in m icrostrain

,201".1 8 0 1 5 9 ,1 4 2 ,

1 5 9 1 8 0



313


POCD■oOQ.coCDQ .

■oCD

C/)C/)

oo■Dv<cq'

3.

CD■oOQ .cao■Oo

CDQ .

TDCD

C/)C/)

S u p p o r t l i n e 1 Central support S u p p o r t l i n e 2

Finite-element results _

, o

Experimentalresults

Figure 5.55. Reactions for the composite curved bridge m odel due to Loading Case 3

0.00.0


R I- ► i

0.01.1

0.0

1.6

0.00.0

2.22.7


R - ► i



0.01.2 0.0 T 0.0 0.8 1 1,2

0.0

1.6



315


1 1 8 1 4 4

1021 3 0 1 7 92 2 8


2125 7 91532110

3 0 3

2 8 0 2 6 7 4 1 34 8 7


• E x p er im en ta l resu lts+ Tension

Compression


1 1 8 1 4 4

1021 3 0 1 7 92 2 8


Figure 5.57. Longitudinal strains o f the composite curved bridge model due to Loading Case 4

316


CD■DOQ.CoCDQ .

■DCD

C/)(/)

OO■Dc q '

O’QCD■DOQ .Cao


Finite-element results K

■DO

CDQ .

"DCD

(/)(/)

Experimentalresults

o

J L□ □

o oo o


R

0.01.0

0.01.0

0.0l.OL

0.01.0


&-----------------.J '

T

0.0 b.T) “ “ o.ol

1.4

0.01.3


R


Values are in millimeters- ► I

0.00 . 9

Ti

i. r _____1 ______I s

,

0.0 T 0.0 0.8 i 0.7

0.00.6

( c ) D e f l e c t i o n s a t 3 L / 4 f r o m t h e o u t e r s u p p o r t i n s p a n ( 1 )

Figure 5.59. Deflections o f the composite curved bridge model due to Loading Case 5

318



1211 362


R

F in ite -e le m e n t resu lts• E x p er im en ta l resu lts+ Tension

Compression


’ 1 . . w 10 . •" ^ ^^ " V X ^

" i "CM32 - - + - M4- -

251



319


73CD■DOQ.CoCDQ .

■DCD

C/)C/)

OO■Dc q ' S u p p o r t l i n e 1 C e n t r a l s u p p o r t S u p p o r t l i n e 2

Finite-element

O’QCD"DOQ .Cao■oo

CDQ .

■DCD

results

(SU)N)O

Experimentalresults R R

r- 1 1 ■□ □ □ □

w(/)Figure 5.61. Reactions for the composite curved bridge model due to Loading Case 5

0.00.0 0.0

2.2 2 .5

(a) Deflections at ^ from the outer sujpport in span (1)

0.0

2 .7

O.T) t 0.0

3 .64 .0


R

F in ite -e le m e n t resu lts• E x p er im en ta l resu ltsV a lu e s are in m illim e te r s

0.0

1 .5

0:0

1 .7

0.0

1 .9

0.0

2.2



321


236

388. •381


■

557 ! 714


R I

F in ite -e le m e n t resu lts• E x p er im en ta l resu lts+ Tension

Con^tression


2t9-339



322


POCD■oOQ.coCDQ .

■oCD

C/)C/)

oo■ov<c q '

Q.O’QCD■oOQ .CaoQ■oo

CDQ .

TDCD

C/)C/)



ts

U)K)U)

Experimentalresults

O


0wc01 13oo< -2

-4

Flexural test

I l l i i i l W i m (\/WWWWSAAAAAaaa/vwwvvw»

'0 1

Time (sec)

Figure 5.65. Typical acceleration-time history of the straight bridge model

Flexural test

(Dsoo—

C/35

0 2 3 41Time (sec)

Figure 5.66 Typical displacement-time history of the straight bridge model

324


Torsional test

coUt

13oo< -2 -

10 2 3 4Time (sec)

Figure 5.67 Typical acceleration-time history of the curved bridge model

Ss Torsional test

"S

Q29 -

0 2 3 4Time (sec)

Figure 5.68. Typical displacement-time history of the curved bridge model

325


0.8uT33& 0.6edc;.2CS

oo<

0.4

0.2

31.5 Hz

40.9 Hz

___ /

72.4 Fz100.7 Hz

---- -------- 1■ — ■ ■" ...............

0 50 100 150 200 250 300 350 400Frequency, Hz

Figure 5.69. Experimental acceleration frequency response of the straight bridge model in the flexural test

0.10

0.08

I 0-06C01•s 0.04oo<

0.02

0.00

21 .6 Hz

37

1

.9 Hz

J --

50 100 150 200 250 300Frequency, Hz

350 400

Figure 5.70. Experimental acceleration frequency response of the straight bridge model in the torsional test

326


0.20

3 5 . 0 H z

0.16

I 0.12 -C9.

e 0.08 -1 3oo<

0.04 -

- tN>n00

VO

0.000 50 100 150 200 250 300 350 400

Frequency, Hz

Figure 5.71. Experimental acceleration frequency response of the curved bridge model in the flexural test

0.05

3 4 . 7 H z

0.04<D

^ 0.03ados 0.02

1 3oo<

0.01

7 8 . 6 H z(NVO

0.000 50 100 150 200 250 300 350 400

Frequency, Hz

Figure 5.72. Experimental acceleration frequency response of the curved bridge model in the torsional test

327


a) Straight bridge model

LAF mode

b) Curved bridge model

LAF-TS mode

Figure 5.73. First and second mode shapes obtained analytically for the bridge models

328


a) Straight Bridge Model

Second mode shape

b) Curved Bridge Model

Second mode shape

Figure 5.74. First and second mode shapes obtained experimentally for the bridge models

329


450

400

350

300

2 250

o 200

150

100— Finite-element results- - Experimental results

0 -200 -400 -600 -800 -1000 -1200

Strain (microstrain)

Figure 5.75. Load-deflection relationship for the straight bridge model at mid-span 1

400

350 -

300 -

250

-o 200 - cS O J150 -

100 - Finite-element results Experimental results

50 - -

10 150 5 20 25 30 35 40Deflection (mm)

Figure 5.76. Load-strain relationship of the concrete deck for the straight bridge model at mid-span 1

330


450

400 - V

350

300

250

1 200

150

100 Finite-element results- - • Experimental results

10000 2000 3000 4000 5000 6000Strain (microstrain)

Figure 5.77. Load-strain relationship of the bottom flange for the straight bridge modelat mid-span 1

450

400

350

300

250

150

100— Finite-element results- • Experimental results

0 200 400 600 800 1000 1200 1400 1600 1800Strain (mierostrain)

Figure 5.78. Load-strain relationship at the top of the web for the straight bridge model at mid-span 1

331


450

400

350

300

s 250

I 200

150

100

Finite-element results- ■ ■ Experimental results

0 500 1000 1500 2000 2500 3000Strain (microstrain)

Figure 5.79. Load-strain relationship at the bottom of the web for the straight bridge model at mid-span 1

350

300

250

J 150

100

— Finite-element results ■ • Experimental results

0 5 10 15 20 25 30 35 40 45Deflection (mm)

Figure 5.80. Load-deflection relationship for the curved bridge model at mid-span 1

332


350

300

250

z 200

cd^ 150

100

Finite-element results* - • Experimental results

0 10 155 20 25 30 35 40 45 50Deflection (mm)

Figure 5.81. Load-deflection relationship for the curved bridge model at mid-span 1

o-I

350

300

250

200

150

100

Finite-element results • - • Experimental results

50

00 -100 -200 -300 -400 -500 -600 -700 -800 -900

Strain (microstrain)

Figure 5.82. Load-strain relationship of the concrete deck for the curved bridge model at mid-span 1

333


•aISoJ

350

300

250

200

150

100

— Finite-element results • - Experimental results

50

0

0 -100 -200 -300 -400 -500 -600 -700 -800 -900Strain (microstrain)

Figure 5.83. Load-strain relationship of the concrete deck for the curved bridge model at mid-span 1

350

300

250

200

a^ 150

100

— Finite-element results- • Experimental results

0 500 1000 1500 2000 2500Strain (mierostrain)

Figure 5.84. Load-strain relationship of the bottom flange for the curved bridge model at mid-span 1

334


350

300

250

200

^ 150

100

Finite-element results ■ ■ • Experimental results


Figure 5.85. Load-strain relationship of the bottom flange for the curved bridge model at mid-span 1

350

300

250

200

100

Finite-element results‘ - • Experimental results


Figure 5.86. Load-strain relationship at the top of the web for the curved bridge model at mid-span 1

335


350

300

250

100

Finite-element results• - ■ Experimental results

0 200 400 600 800 1000 1200

Strain (mierostrain)

Figure 5.87. Load-strain relationship at the bottom of the web for the curved bridge model at mid-span 1

350

300

250

200

100

Finite-element results~ - ■ Experimental results

0 100 200 300 400 500 600

Strain (mierostrain)

Figure 5.88. Load-strain relationship at the top of the web for the curved bridge model at mid-span 1

336


350

300

250

£200T3

^150

100

Finite-element results- ■ • Experimental results


Figure 5.89. Load-strain relationship at the bottom of the web for the curved bridge model at mid-span 1

337


Figure 5.90. View of the deflected shape of the straight bridge model at failure

Figure 5.91. View of the defleeted shape of the curved bridge model at failure

338


Figure 5.92. Crack pattern of concrete deck in the straight bridge model at failure

Figure 5.93. Crack pattern of the concrete deck in the curved bridge model at failure

339


Figure 5.94. Deformation of the bottom flange in the straight bridge model at failure

Figure 5.95. Deformation of the bottom flange in the curved bridge model at failure

340


H

o.=3o

H

X

CNm

X CQ

X , CQ

X m

x ; om

XCQ

" ' XfN

-23COH

l-H

<L>

T3§

ffiTfo'm<o1/1o

>kHo[L,

o(50■ta

•cX)(U

60XOX5

«4HOc.o

'-Co0t/311/5C/5oIHokH

3oc/3301 00

VO<1

341

pin


t/1(D

;gDI1ttH

xnu60

;g

II<U

H

(/Iu60

t3■c<uI6H

J

11

J

11

J

1

J1

“I 1 1— 1111

J J J— J 1

JU

1 1

J J

c/5•3hac/5O•£<uaKic3&(U

T30)033i03fl

_o

a

ooa_oo0)030303O)-i

u

CN

a

i-iH

342


2.0

c

<2

1.5

1.0tjc.2■3I 0.5

Q

(O............. r^ ....... .. i.

Y— ... )[(' .. N ... / ..... .................................

^ Live Load • -G - Dead Load

0.010 20 30

Bottom flange thickness (mm)Figure 6.3. Effect of bottom flange thickness on distribution factor for tensile stress

40

C/5

c<u

oot+3e_oXI

2.0

1.5

1.0

0.5

0.0

Pi.............r

....... ... nS/ ------7i

......... .......... ........................

~)K Live load - -G - Dead load

10 20 30Web thickness (mm)

Figure 6.4. Effect of web thickness on distribution factor for tensile stress

40

343


7JCD■DOQ .CoCDQ .

■DCD

(/)(/)

OO■D

Design truck HS20-44 17.51 72.5

J l

72.5 WHEEL I LOADS, kN

k - 4.3 m 4 . 3 - 9 m

CD■DOQ .Cao■oo

CDQ.

■DCD

(/)C/)

Design Lane Load

Concentrated load

n n 1 1

116kN for shear

80 kN for moment

9.3 N/mm per lane1 1 i' 1 1 1 i~l I T ' i i~l 1

,5•O

t

Figure 6.5. Standard truck and lane loads according to AASHTO Standard Specifications

7JCD"DOQ .CoCDQ .

■DCD

(/)(/)

OO■D

3 .

CD■DOQ .C2 .5'3■oo

CDQ .

Oc■oCD

Two-laneR

Three-lane R______ J

Four-lane

R______ JJZIL U □ J U l

UJ

a) Partial truck loading

_R______ J R RJ L r—1 ryi 1— 1 ^ u r—1 r—iif—i m ^

b) Full truck loading

Figure 6.6. AASHTO truck loading cases in the transverse direction o f the bridges

(/)o'o

1

o-cdocd

I<L> .

iS

Pi

J .(DI

t2

(DI

IOsS

Pi

J

Pi

J .

c2>%3c/a011c3a,73C3O60.gC3

P4o

oHmc/2

'O

.1tin

346


00

l - n “00

u<a

OT3U_N"c3

OS 2

1)bOt3•cX)Xo

•a.N

OJT3

oo

[In

347


“ I - ; :II "6

r

i f

p

L _

I

i f rE

r

i

r

r

r.aa

oi<Dto

•cX>(L>

C4-1Oa

.2o

.S'O

.2aSia4>

a

M1)MP3O

.s'(J3

O

o£t/3

348

Onv£><D

tL|


CD■DOQ .CoCDQ .

■DCD

C/)Wo'oo

oo"Dcq '

oCD■DOQ .Caoo■Oo

CDQ .

■DCD

(/)(/)

a) Longitudinal flexure, LF b) Symmetrical torsion, TS c) Combined flexure and torsion, LF-TS

r -

4VO d) Combined longitudinal and transverse flexure, LF-TF

e) Antisymmetric torsion, TAS f) Distortion, DS

^

-h-

^ j i , I ;!u

Original shape Deformed shape at mid-span— Deformed shape at quarter-span

Figure 6.10. Typical mode shapes for two-box girder bridge

CD■DOQ .CoCDQ .

■DCD

C/)(/)

OO■Dc q '

O’QCD■DOQ .Cao■oo

CDQ .

■DCD

(/)(/)

U>O a) Antisymmetric flexure, LAP b) Symmetric flexure, LF c) Antisymmetric flexure, LAF

d) Symmetric torsion, TS e) Ajitisymmetric torsion, TAS

Figure 6.11. Typical mode shapes o f two-equal-span continuous bridges

a)

1.8

P 1.6

c/3CCD

u

o-cdd.2px>

1.4

1.2

s .

0.6

■

■ ^ L / R = 0.0 -0- -L/R = 0.4

4l-3b

0 ----- -6)

010 0 (

20 40 60Span Length (m)

80 100

b)

C/3

--------0* ■" —

1.0 -

0.8 -

0.6 480 10020 40 60

Span length (m)

Figure 7.1. Effect of bridge span length on distribution factor for tensile stress for bridges due to: a) AASHTO live load; and b) dead load

351


a)

^ — L = 20 m 0 - -L = 60 m

c/5<L>

co

Q 0.8 -

0.62 3 4

Number of lanes

b)

1.8

H 1-6

1.4

o 1.2

5 1.0a

15 0.8

0.6

L =20m " O ’ ■L=60m

4b,UK=0A

Qh->e

Number of lanes

Figure 7.2. Effect of number of lanes on distribution factor for tensile stress for bridges due to: a) AASHTO live load; and b) dead load

352


a)

<D•4

c2UOIc_o'3XI

1.8

1.6

1.4

1.2

1.0

0.8 -

0.6

■ ^ L = 40 m -O -L= 100m

3/,L/R=0.4

0- - - e -o

b)

h.1Qoao3X3

1.8

1.6

1.4 -

1.2

1.0

0.8

0.6

- ^ L = 40m -G- - L= 100 m

3/,I7R = 0.4

Number of boxes

Number of boxes

Figure 7.3. Effect of number of boxes on distribution factor for tensile stress for bridges due to; a) AASHTO live loading; and b) dead load

353


a)

u

l1o

Q

0.60.0 0.2 0.4 0.6 0.8 1.0 1.2

b)

Span-to-radius of curvature ratio (L/R)

- ^ L = 60 m -© --L = 100mg 1.6 -

O

co"3

S 0.8 -

0.60.0 0.2 0.4 0.6 0.8 1.0 1.2


Figure 7.4. Effect of bridge curvatiu-e on distribution factor for tensile stress for bridges due to: a) AASHTO live load; and b) dead load

354


a)

c/5aS8au

Id.2

I

^ L/R = 0.0 0 - -L/R = 0.4

-<)■e-0.8

0.620 40 60 80 100

Span length (m)

b)

0.8 -

0.620 40 60 80 100

Span length (m)

Figure 7.5. Effect of bridge span length on distribution factor for compressive stress bridges due to: a) AASHTO live load; and b) dead load

355


a)

oo

ooco3:2sc/5

1.8

1.6

1.4

c/5C /5

gl.2

5 1.0

0.8

0.6

— —L=20m *L = 60 m

3b,VR=0

b)

QJ.S

oo

o,c§

1.8

1.6

1.4

Sl.2

§ 1-0

0.8 -

0.6

■ ^ L = 2 0 m -©■ -L =60m

36,L/R=0

()"

Number of lanes

Number of lanes

Figure 7.6. Effect of number of lanes on distribution factor for compressive stress for bridges due to: a) AASHTO live load; and b) dead load

356


a)

X L= 40 m ~ 0 - -L= 100m>

C /3C /3

a.eoo

31, L/R=0.4

so3X)•ccnS 0.8 -

0.62 3 4 5 6

b)

1.8

1.6 -•U&BooucS

1.4

ocSCo

c /3C/5^1-2, C /5

s 1.0 -3x>■£cn

0.8

0.6

Figure 7.7.

Number of boxes

L= 40 m -(>-L=100m

3/,L7R=0.4

(>■•0

Number of boxes

Effect of number of boxes on distribution factor for compressive stress bridges due to: a) AASHTO live load; and b) dead load

357


a)

X L= 60 m -0--L=lO O mcu>

'c /2C /30>aSoo

c/3

O 5^

co3

0.60.0 0.2 0.4 0.6 0,8 1.0 1.2


b)

•X L= 60 m -©-■L=100m

0.8 -

0.60 0.2 0.4 0.6 0.8 1 1.2


Figure 7.8. Effect of bridge curvature on distribution factor for compressive stressbridges due to: a) AASHTO live load; and b) dead load

358


a)

2.6

•2 2.2 O(UCSoT3^ 1.8 u,o

.2 1-4

.oc/3■q 1.0

0.6

- ^ L / R = 0.0 - e - -L /R -0 .4

4l-3b

V. '»s ■V•V,

---- -- J '— — 0

-----------------------^

-..- ............. - .^

.............. —.............. f............. -20 40 60

Span length (m)80 100

b)

2.6- ^ L / R = 0.0 ” <3- -L/R = 0.4

2.2

1.4 -

0.66020 40 80 100

Span length (m)

Figure 7.9. Effect of bridge span length on distribution factor for deflection for bridges due to; a) AASHTO live load; and b) dead load

359


a)

2.6

S 2.2O<D'"O

i01

1.8 -

o 1.43

1.0

0.6

—1— L = 20 m —0 - - L = 60 m

36,L/R=0

■0

Number of lanes

b)

2.6

5 2.2

co

■q

0.62 3 4

Number of lanes

Figure 7.10. Effect of number of lanes on distribution factor for deflection for bridgesdue to: a) AASHTO live load; and b) dead load

360


a)

2.6

S 2.2OCdJTS

otj!=:_o

'SU 5

Q

1.8

1.4

1.0

- ^ L = 40m -©--L=100m

3/,L/R=0

(>-

0.6

-ZTZTzi)

b)

Number of boxes

2.6

.2 2.2ooqi3d>•nkH<2kH01<+Hd_o"SXI

- ) ^ L = 40m -e --L = 1 0 0 m

3/,L/R=0

--------------- .■- ~ ---------- ------------------------------- ______ __________^

1.8

1.4

Q 1.0

0.6

Number of boxes

Figure 7.11. Effect of number of boxes on distribution factor for deflection for bridges due to: a) AASHTO live load; and b) dead load

361


a)

2.6

X L - 60 m - a - L = 1 0 0 m,2 2.2ts0>

<D

OQcoIt o

■q

0.60.0 0.2 0.4 0.6 0.8 1.0 1.2


b)

2.6- X - L = 60 m -© -■L=100m

2.2 -

0.60.0 0.2 0.4 0.6 0.8 1.0 1.2


Figure 7.12. Effect of bridge curvature on distribution factor for deflection for bridges due to: a) AASHTO live load; and b) dead load

362


a)

2.6

2.2.s

g l.8 ÎdoS 1.4X )

1.0

0.6

■ ^ L / R = 0.0 -O -L/R = 0.4

4/-3Z)

O

20

----- 1-----

40 60Span length (m)

80 100

b)

2.6

2.2S(D43C/5

‘r l .

dI 'X )■c

.4

1.0

0.6

- ^ L / R = 0.0 -O -L/R = 0.4

4/-3Z>

20 40 60Span length (m)

80 100

Figure 7.13. Effect of bridge span length on distribution factor for shear force for bridges due to: a) AASHTO live load; and b) dead load

363


a)

2.6

- ^ L = 20m -© - -L = 60m

2.2 -

OCo3•fi■q

0.62 3 4

Number of lanes

b)

2.6

- ^ L = 20m -©■-L = 60m

3b,L/R=0

_____________________________ ^^ ...................... ......................................- ^

c3C/3

iMc2U2

<4-1

d_o"3x>■£

2.2

1.8

1.4

1.0

0.6

Number of lanes

Figure 7,14. Effect of number of lanes on distribution factor for shear force for bridgesdue to; a) AASHTO live load; and b) dead load

364


a)

) ^ L = 40 m 0--L=lO O m

3/,L/R=0

b)

ao3X>

2.6

2.2

1.8

1.4

0.6

40m -©- - L= 100 m

3/,L/R=0

4 5Number of boxes

# ■

Number of boxes

Figure 7.15. Effect of number of boxes on distribution factor for shear force for bridges due to: a) AASHTO live load; and b) dead load

365


a)

2.6

2.2 -u■S

Ico"3,o

1Q

0.60.0 0.2 0.4 0.6 0.8 1.0 1.2


b)

2.6- ) ^ L = 60m -©■ -L - 100m

2.2

0.60 0.2 0.4 0.6 0.8 1.2


Figure 7.16. Effect of bridge span-to-radius of curvature on distribution factor for shear force for bridges due to: a) AASHTO live load; and b) dead load

366


a)

uoaa

.2n><Dt-i<S

Ia.23XI1Q

b)

3.4

3.0

2.6

I 2.2U

1.8

1.4

1.0

>e

20

■ ^ L / R = 0.0 -0- - L/R = 0.4

4/-3 6

0* -O


80 100

3oo-a3C/2

I

3.4

3.0

2.6 -

3_oSXI1Q

s_o

sa2.2

1.8

1.4

1.0

^ 0 " L /R = O.O _ —0 . -L/R —04 -

4/-3h

0 - ---------------------- ^ ----------------------(---------------------------------------------- -

^ ....- ......20 40 10060 80

Span length (m)

Figure 7.17. Effect of bridge span length on distribution factor for exterior support reaction for bridges due to: a) AASHTO live load; and b) dead load

367


a)

3.4

3.0 - -

2.6 -

2 3 4Number of lanes

b)

oaa-;3

'C%X<uIm

o4-1e,g"3X)’H■q

3.4

3.0

2.6

s01 ui-i

1.8 -

1.4

1.0

~"X L = 20 m ~0- ■L = 60m

3b,UR=0

Number of lanes

Figure 7.18. Effect of number of lanes on distribution factor for exterior support reaction for bridges due to: a) AASHTO live load; and b) dead load

368


a)

coa .o.3

X0>

3.4

3.0

2.6

^ -2o pMa_o3X)■£t/i

1.8

1.4

1.0

- )K -L = 40m -© - -L= 100m

3/,L/R-0

___ — <--------------- -^ ------------------^ i - - = r z ------------ ---------------------------------------i 1 ' i ...... ........ ........... ..

b)

Number of boxes

t;

§-O*noX<u

01d

_o"3X)15

3.4

3.0

2.6

2.2

1.8

1.4

1.0

X L= 40 m .- e - -L = 1 0 0 m .

3/,L7R=0

§------ ---------------^ ^ ------------- ---— ^ ^

Number of boxes

Figure 7.19. Effect of number of boxes on distribution factor for exterior support reaction for bridges due to: a) AASHTO live load; and b) dead load

369


a)

3.4

3.0 -D,D.

2.6 -

^ a 2.2 -

0.2 0.40.0 0.6 0.8 1.0 1.2


b)

3.4^ L = 60m ©- -L= 100 m3.0 - -

I "X

c(O .2 2.2 - tH-l *3Vh ^% a42co

0.20.0 0.4 0.6 0.8 1.21.0Span-to-radius of curvature ratio (L/R)

Figure 7.20. Effect of bridge curvature on distribution factor for exterior support reaction for bridges due to: a) AASHTO live load; and b) dead load

370


a)

oaPhpc/5UO*c

b)

1.8

1.6

1.4

U<su>

%fCi

C.9"SXi

a01 1.2

1.0

Q 0.8

0.6

L/R =0.0 L/R = 0.4

20

4/-36


80 100

Uoa.C L3

a;C<2OIHHc.23

X)•c

1.8

1.6

1.4

3_o1<1>

1.2

1.0

0.8

0.6

^ I^ L /R = 0 .0- L/R = 0.4 ---

4/-36

------- ----------- ^

-- ■20 40 80 10060

Span length (m)

Figure 7.21. Effect of bridge span length on distribution factor for interior support reaction for bridges due to; a) AASHTO live load; and b) dead load

371


a)

ImOcxOhoCOuO•c

Vht2

o,cS-1e.9■3

'BCO

s

- ^ L = 20m —©- -L = 60 m

oocd

0.62 3 4

Number of lanes

b)

Uoo.OnoCO

(L>Vmc2Vhoo,cd

C.2'3x>

1.8

1.6

1.4

c.2

1.2

1.0

0.8

0.6

■ ^ L = 20m "©“ - L = 60 m

Figure 7.22.

36,17R=0

■ 0

3 4Number of lanes

Effect of number of lanes on distribution factor for interior support reaction for bridges due to; a) AASHTO live load; and b) dead load

372


a)

X~" L = 40 m - e - - L = 1 0 0 m

3/,L/R=0

co

- 0

0.62 3 4 5 6

b)

Number of boxes

0 - - L = l O O m

3/,L/R=0

y 1.2

-0

Number of boxes

Figure 7.23. Effect of number of boxes on distribution factor for interior support reaction for bridges due to: a) AASHTO live load; and b) dead load

373


a)

X L = 60 m -©■-L^lOOm

.2*cfl

2

0.8 -Q

0.6

0 0.2 0.4 0.6 0.8 1 1.2


b)

~ X L= 60 m -© ■-L=100m

.s

aX I

'BC/5■q

0.8 -

0.60.0 0.2 0.4 0.6 0.8 1.0 1.2


Figure 7.24. Effect of bridge curvature on distribution factor for interior support reaction for bridges due to; a) AASHTO live load; and b) dead load

374


a)

(U

I6’E

o

G.2-2

b)

0.4 C4;

- 1.2

- 1.6

- 2.0

Q -2.4 -

-2.820

-L/R = 0.0-©■ -L/R = 0.4

4 /-3 *

40

“()

60Span length (m)

80 100

E

ISe

<2

GXi■|5

2.8

2.4

2.0

1.6

1.2 -

5 0.8

S 0.4

0.0

X L/R = 0.0 - a - L / R = 0.4

4/-3^»

C>-

20

Figure 7.25.

- X -

-=C)40 60


Effect of bridge span length on distribution factor for minimum reaction for bridges due to; a) AASHTO live load; and b) dead load

375


a)

CdssS

0.0

-0.4

-0.8

O -rt-1.2

aos.£5

offj(L>-1.6

-2.0

-2.4

-2.8

.......... - - . . ..... - . . . . . . . . . . . .............. - ........ ..............

)K' L=20 m - 0 - - L = 60m

3Z),L/R=0- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Number of lanes

b)

c3scSk.1kH2

C_o3

■£!«

2.8

2.4

2.0

1 1.6

<L>^ 1.2

0.8

0.4 4

0.0

■ ^ L = 20m -Q- -L = 60m

3Z),L/R=0

Number of lanes

Figure 7.26. Effect of number of lanes on distribution factor for minimum reaction for bridges due to: a) AASHTO live load; and b) dead load

376


a)

Ssaau,c2Ul

I<4-1c.23

'■§Q

0.0

-0.4 -

-0.8 -

eo

-2.0 - X L= 40m -e - -L = 1 0 0 m-2.4 -

3/,L/R=G-2.8

3 4 62 5Number of boxes

b)

es3

aUi

<2

2.8

2.4

2.0

O .3 1.6

o<4-4c.23

o3 U ^ 1.2

c 0.8

0.4

0.0

- ^ L = 40m -e --L = 1 0 0 m

3/,L/R=0

Number of boxes

Figure 7.27. Effect of number of boxes on distribution factor for minimum reaction for bridges due to: a) AASHTO live load; and b) dead load

377


a)

■2 0.5

e 0.03e!s -0.5 B

-1.0u01 <+-!3O3 -2.0

(5 -2.5

-3.00.0 0.40.2 0.6 0.8 1.0 1.2

b)


0.5o

0.0

§ -0.53B

-1.0

X L = 60 m -© -■L=100m■” -2 5

-3.00.0 0.2 0.4 0.6 0.8 I.O 1.2


Figure 7.28. Effect of bridge curvature on distribution factor for minimum reaction for bridges due to; a) AASHTO live load; and b) dead load

378


2.5

<U2.0

cuI 1.5

%

§1.0

q O.5

■ *-L /R = 0.0 -0- - L/R = 1.2

- e -

0.02 3

Number of spans

Figure 7.29. Effect of number of spans on distribution factor for tensile stress

2.5

2.0

ai<L>DV 5•!>Cl,t l . 5o0

1 1.0

c ^

I 0.5

Q0.0

C)- -0 -

-L/R = 0.0— 0 _ ■L/R = 1.2

-0

Number of spans

Figure 7.30. Effect of number of spans on distribution factor for compressive stress

379


2.5

.2 2.0 ts<u0)*oI 1.50OcfS1 1.0 '3X)

(>-

n 0.5

0.0

-L/R = 0.0- 0 - ■L/R = 1.2

- o

Figure 7.31.

2.5

Number of spans

Effect of number of span on distribution factor for deflection

2.0<Dc/5u

1.oa

.2 1.0 3

.1Q o.5

0.0

-L/R = 0.0—0- ■L/R = 1.2

Number of spans

Figure 7.32. Effect of number of spans on distribution factor for shear force

380


2.5d.9

-ds,&COU|O

<1>

a.9'3

1Q

0

2.0

1.5

1.0

0.5L/R = 0.0

-0 - -L/R= 1.20.0

Number of spans

Figure 7.33. Effect of number of spans on distribution factor for exterior support reaction

OCQd>u*tioaOh

o,ai

2.5

2.0

.9 1.5 -

1.0

do0.5

XI

0.0

L/R = 0.0 -O- -L/R= 1.2

Number of spans

Figure 7.34. Effect of number of spans on distribution factor for interior support reaction

381


0.0

-0.5

a

i<4-1c

.2"3XIID

-1.0

-1.5

-2.0

-2.5

-L/R = 0.0— &- -L/R = 1.2

Number of spans

Figure 7.35. Effect of number of spans on distribution factor for uplift reaction

382


1.6

c/5O1.2

c

c20.8

o

o

I 0.4

0.0

1.6c/3Uhc/3

1.2

o.eoo.2

o.'ciCo

«-Z!r

0.8

sj=

Q

0.4

0.0

-A -2b,L = 40 -O- 4b,L = 40

■ii

1/16 1/8 Web slope

3/16 1/4

Figure 7.36. Effect of web slope on distribution factor for tensile stress due to AASHTO live load

-T ^2b ,L = 40

- G - 4b,L = 40

■o

1/16 1/8 Web sfope

3/16 1/4

Figure 7.37. Effect of web slope on distribution factor for compressive stress due to AASHTO live load

383


1.6

i 1.2u

T3,P

So

0.8

t/)5

0.4

0.0

^ ?s-2b ,L = 40

- -0 - 4b,L = 40

Figure 7.38.

1/16 1/8 Web slope

3/16 1/4

Effect of web slope on distribution factor for deflection due to AASHTO live load

1.6

( 3 -

O

od_o3

/It1.2-

0.8

5 0.4

0.0

<?-

- A - N b = 2,L = 40

- - 0 - Nb = 4,L = 40

<)

1/16 1/8 Web sbpe

3/16 1/4

Figure 7.39. Effect of web slope on distribution factor for shear force due to AASHTO live load

384


1.6co

tia§■o■C(D

( 5 -

Z!r1.2

“ 0.8

o,cdC.2"3X)•£

0.4 -

0.0

-75r-2b,L = 40

-O- 4b,L = 40

<)

1/16 1/8 Web slope

3/16 1/4

Figure 7.40. Effect of web slope on distribution factor for exterior support reaction due to AASHTO live load

„ 1.6 _o1-do 1 2a,3t/5

Z ! r

0.8

o, 3<+H3_o'3JD

0.4 -

t/3

50.0

e-

-75r-2b,L = 40

-0 - 4b,L = 40

o

■tI\

1/16 1/8 Web slope

3/16 1/4

Figure 7.41. Effect of web slope on distribution factor for interior support reaction due to AASHTO live load

385


0.0

B.-0.4

5-1.2

-1.60 1/16 1/8 3/16 1/4

Web slope

Figure 7.42. Effect of web slope on distribution factor for uplift reaction due to AASHTO live load

386


3.0

onS2.5WC/5g 2.0■4-JUi<22 1.5

1.0

t*-!Co

x>1n 0.5

0.0

L/R =0.0 -0- -L/R= 1.2

>0

Figure 7.43.

3.0

20 25Span-to-depth ratio

Effect of span-to-depth ratio on distribution factor for tensile stress

onc/5

u 2.5 .S:*75

c/5

§■2.0B

l u

tw Ci .2 "3

iQ

1.0

0.5

0.0

—)I0--L/R = 0.0_ 0 _ •L/R = 1.2

5 -

20 25

30

i )

30Span-to-depth ratio

Figure 7.44. Effect of span-to-depth ratio on distribution factor for compressive stress

387


3.0

.2 2.5ouC

-O 2.0

ooco

1.5

3 1. JD 1.0

Q 0.5 -

0.020 25

- ^ L / R = 0.0 -0- -L/R= 1.2


Figure 7.45. Effect of span-to-depth ratio on distribution factor for deflection

3.0

2.5

12.0 -O-

43 1.5co3X)c 1.0C/5

Q0.5 -

0.020

e -

25

L/R = 0.0 -0- -L/R= 1.2

Span-to-depth ratio

Figure 7.46. Effect of span-to-depth ratio on distribution factor for shear force

30

388


_o

(Uu<t;oD<O.

3.0

2.5 (>-

2.0.2><w

k.1oo,tdu-ia.2"3

1.5

1.0

0.5 -

0.020

■0


■ ^ L /R = 0.0 -0- -L/R= 1.2

30

Figure 7.47. Effect of span-to-depth ratio on distribution factor for exterior support reaction

c_o

■daat/j

o,c<3

Co

3.0

2.5

2.0

1.5

1.0

2 0.5

0.0

-L/R = 0.0_ 0 _ ■L/R = 1.2


30

Figure 7.48. Effect of span-to-depth ratio on distribution factor for interior supportreaction

389


0.0

-0.5>(-

Cl.3<2 -1-UO'o

1.0 -

« -1.5 -aoZ3

*cc -2.0

-2.5 -

-3.0

■€)

L/R = 0.0 '©■ -L/R= 1.2

20 25 30Span-to-depth ratio

Figure 7.49. Effect of span-to-depth ratio on distribution factor for uplift reaction

390


C3O

o<4HC_o'+33-E00

5

2.0

—♦— Internal & external bracing (live load)- - Internal bracing (live load)—X — Internal & external bracing (dead load) —0 ■ ■ Internal bracing (dead toad)

1.8

1.6

1.4

1.2

1.0

1 3 5 97 IINumber o f cross bracings

Figure 7.50. Effect of number of bracings on distribution factor for tensile stress

t/i

>

&eoO

oC.2

.3

2.0—♦— Intemal & external bracing (live toad)- -A - Intemal bracing (live toad)—X — Intemal & extemal bracing (dead toad) —0 ■ ■ Intemal bracing (dead toad)

1.8

1.6

1.4

1.2

1.0

3 5 7 91 11Number o f cross bracings

Figure 7.51. Effect of number of bracings on distribution factor for compressive stress

391


o(UGa>

<22

e_o"3'200

5

2.0

1.8 -

1.6

L\

1.4

1.2

1.0

—♦— Intemal & extemal bracing (live load)- -A - Intemal bracing (live load)—X — Intemal & extemal bracing (dead load) “ 0 - ■ Intemal bracing (dead load)

'A ’

1 5 7Number of cross bracings

11

Figure 7.52. Effect of number of bracings on distribution factor for deflection

2.0

a 1.8C/3U2

o

ao

1.6

S 1.43X>'B(5

1.2

1.0 4

..♦ ...Intemal & extemal bracing (live load)" -A - Intemal bracing (live load)—X — Intemal & extemal bracing (dead load) —0 ■ ' Intemal bracing (dead load)

-------- 4 ..—.....” .................... & ......................5 ....................... ------A ........... .................... -A

_N/-..... ..........---*5

-------------------- --------------------------------------

---------------- 0

1 3 5 7 9Number of cross bracings

Figure 7.53. Effect of number of bracings on distribution factor for shear force

11

392


.2

■coa.O h

.2'Cu><

................

-♦— Intemal & extemal bracing (live load) -A - Intemal bracing (live load)■X — Intemal & extemal bracing (dead load) ■0 - ■ Intemal bracing (dead load)

5 7Number of cross bracings

9 1 1

Figure 7.54. Effect of number of bracings on distribution factor for exterior support reaction

2.0

C3

ut:oaa.=3c/5UO*CoU4

o,SJc_2"SXi'B

1.8 -

1.6

1.4

1.2

1.0

Figure 7.55.

—♦— Intemal & extemal bracing (live load)- -A - Intemal bracing (live load)—X — Intemal & extemal bracing (dead load) —0 - ■ Intemal bracing (dead load)

^----^/------

..

▲

..................... .

r

-...... ...... . ... . “ ...... ..... ■■5 7

Number of cross bracings11

Effect of number of bracings on distribution factor for interior support reaction

393


-0.4a.3U

3O3

i/j5

- 0.8

- 1.2

- 1.6

- 2.0

Intemal & extemal bracing (live load)- -A ■ Intemal bracing (live load)—X — Intemal & external bracing (dead load) —G - ■ Intemal bracing (dead load)

5 7Number of cross bracings

1 1

Figure 7.56. Effect of number of bracings on distribution factor for uplift stress

394


73CD■oOQ .coCDQ .

■DCD

C/)

o'oo

oo■D AASHTO LRFD17.5 72.5 72.5

HS20-441 ^ 4 , «in

WHEEL LOADS, kN

3CD

CP-

55 55 WHEEL Design Tandem __ , LOADS, kN

1.2m

CD■oOQ .Cao=5

■oo

CDQ.

U)

CL-625 25 62.5 62.5 87.5WHEEL

-— 3.6m — 1.2m . m . Ill

oc■oCD

C/)

o'13

CL-625-ONT 25 70 70 87.5 60

3.6m — .u ii . m . m

WHEEL LOADS, kN

Figure 7.57. Truck loading considered in AASHTO LRFD and CHBDC codes

3.0—©— AASHTO - -A ■ AASHTO LRFD

- CL-625 -X-CL-625-ONT

I 2.5 -

2.0 -C3B

o<4-1

co3bQ 0.5

0.0 H-----Bridge 1 Bridge 2 Bridge 3

Figure 7.58. Effect of truck loading specified in different codes on distribution factor for tensile stress

3.0—9— AASHTO - -A - AASHTO LRFD

CD625 -X -C D 625-O N T

O fl

1.0

Bridge 1 Bridge 2 Bridge 3

Figure 7.59. Effect of truck loading specified in different codes on distribution factor for compressive stress

396


3.0

—e— AASHTO - -A - AASHTO LRFD

- CL-625 -X-CL-625-ONT

-S 2.0

0.5

0.0 ^—


Figure 7.60. Effect of truck loading specified in different codes on distribution factor for deflection

3.0- e — AASHTO - -A - AASHTO LRFD

- CL-625 - X - CL-625-ONT

2.5

oo,sSco

Q0.5

0.0 i -----


Figure 7.61. Effect of truck loading specified in different codes on distribution factor for shear force

397


o

2.0

a—e —AASHTO - -A - AASHTO LRFD

CL-625 -X-CL-625-ONT

3 0.5

0.0 —


Figure 7.62. Effect of truck loading specified in different codes on distribution factor for exterior support reaction

« 3.0Oa 2.5 - -

-e-AASHTO -A - AASHTO LRFD

■CL-625 -X-CL-625-ONT

Bridge 1

Figure 7.63.

Bridge 2 Bridge 3

Effect of truck loading specified in different codes on distribution factor for interior support reaction

398


0.0—e— AASHTO - - A - AASHTO LRFD

CL-625 -X-CL-625-ONT

-0.5 - -

-2.5

-3.0 H-----Bridge 1 Bridge 2 Bridge 3

Figure 7.64. Effect of truck loading specified in different codes on distribution factor for uplift reaction

399


1

35

17,5

2

145

72.5

4.3m 4.3 - 9m

3 Axle number

145 Axle loads, kN

72.5 Wheel loads, kN

3 1 •— 4--------------- i i — E Travel direction S

___________ i:________^____ ~

€

gi-i

Figure 8.1. HS20-44 truck loading configuration according to AASHTO Specifications

400


R I 1 .

Fc

UPj2 I P|2

Figure 8.2. Vehicle idealization

R

(P /2 fl.l7 F c) (P/2+1.17 Fc)

lFc/2

Wheel load and centrifugal force Equivalent forces at centre of the concerte deck

401


a)

R = 5 0 m

One-loaded lane

R = 50 m------------------------►1

................... ^n -

1

Two-loaded lane

R = 50 m

Three-loaded lane

R = 50 m

Four-loaded lane

b)

10m Travel direction, v = 50 km/h

20m 20m

Figure 8.3. Loading locations considered in: a) trasverse direction; and b) longitudinal direction

402


Four-loaded lanes Three-loaded lanes Two-loaded lanes One-loaded lane

Figure 8.4.

10 3020Distance (m)

Effect of loading position on tensile stress for 4l-4b-20 curved bridge

40

40 Four-loaded lanes Three-loaded lanes- ■ ■ - Two-loaded lanes— - - One-loaded lane

II 20

aQ.EoO

10

010 20

Distance (m)30 40

Figure 8.5. Effect of loading position on compressive stress for 4/-46-20 curved bridge

403


F o u r - l o a d e d la n e s T h r e e - l o a d e d la n e s T w o - l o a d e d la n e s O n e - l o a d e d la n e

- A * -I V - - - -Jt I -)ir .250

I 150 Qd

10 20D i s t a n c e ( m )

30 40

Figure 8.6. Effect of load position on reaction force for 4/-4Z>-20 curved bridge

F o u r - l o a d e d la n e s ■ ■ - T h r e e - l o a d e d la n e s

— - - T w o - l o a d e d la n e s — O n e - l o a d e d la n e

5 80V( .

. >V .1'' ,\ It

' a;-:'' ^ft'

10 20D i s t a n c e ( m )

30 40

Figure 8.7. Effect of load position on shear force for 4/-46-20 curved bridge

404


(S

e<u

30

V = 60 km/h V= 100 km/h V= 140 km/h

25

15

10\j

5

0150 10 20 255 30 35 40

Figure 8.8.

Distance (m)

Effect of vehicle speed on tensile stress for 4/-66-20 straight bridge

<ufc

£a.EoU

40 V = 60 km/h- - - -V =100 km/h V =140 km/h

30

20

10

00 105 15 20 25 30 35 40

Distance (m)

Figure 8.9. Effect of vehicle speed on compressive stress for 4l-6b-20 straight bridge

405


240

200 -

160 -I ^ 120 -

80 -

V = 60 km/h V =100 km/h V =140 km/h

40 -

0 10 20 30 40Distance (m)

Figure 8.10. Effect of vehicle speed on reaction force for 4l-6b-20 straight bridge

100

- V= 60 km/h- -V= 100 km/h- V= 140 km/h

§

0 10 20 30 40Distance (m)

Figure 8.11. Effect of vehicle speed on shear force for 4l-6b-20 straight bridge

406


eSOi

U

c<L>H

32

DIRECT 50 MODES 100 MODES 200 MODES

24

16

8

0

80 10 20 30 40

Distance (m)

Figure 8.12. Comparison between direct integration and superposition methods for tensile stress of 2l-2b-20 straight bridge

16 DIRECT 50 MODES 100 MODES 200 MODES12

8

4

0

COCLhS

Ia .6oU

10 20Distance (m)

30 40

Figure 8.13. Comparison between direct integration and superposition methods for compressive stress of 2l-2b-20 straight bridge

407


ac_o1

220— D I R E C T— 5 0 M O D E S- - 1 0 0 M O D E S- - 2 0 0 M O D E S

1 8 0

1 4 0

100

6 0

20

-200 10 20 3 0 4 0

D i s t a n c e ( m )

Figure 8.14. Comparison between direct integration and superposition methods for reaction force of 2l-2b-20 straight bridge

1 4 0— D I R E C T— 5 0 M O D E S- ■ 1 0 0 M O D E S- - 2 0 0 M O D E S

120 -

100 -

8 0 -O

S 6 0 -uJSon

4 0 -

20 -

0 10 20 3 0 4 0D i s t a n c e ( m )

Figure 8.15. Comparison between direct integration and superposition methods for shear force of 2l-2b-20 straight bridge

408


c-20 curved bridge

40

At =0.002 At =0.010 At =0.015 At =0.020

0 10 3 020Distance (m)

Figure 8.17. Effect of time step on compressive stress of 2l-2b-20 curved bridge

4 0

409


400At =0.002 At =0.010 At =0.015 At =0.020

350

300

Uo200

o

100

0 10 20 30 40Distance (m)

Figure 8.18. Effect of time step on reaction force of 2l-2b-20 curved bridge

160At =0.002 At =0.010 At =0.015 At =0.020120 - -

<a80 -

40 -

0 10 20 30 40

Figure 8.19.Distance (m)

Effect of time step on shear force of 2l-2b-20 curved bridge

410


s

c(U

35At =0.005

30At =0.020

25

20

15

10

5

0

Figure 8.20.

20 40 80 10060Distance (m)

Effect of time step on tensile stress of 3l-3b-60 curved bridge

120

BO

cdCL.s

><ua

0

35At =0.005 At =0.014 At =0.020

30

25

15

10

5

020 40 80 10060

Distance (m)

Figure 8.21. Effect of time step on compressive stress of 3l-3b-60 curved bridge

120

411


400

350 -

300 -

250 -o

200 -coS 150 - oi

100 - At = 0.005

50 -= 0.020

0 -H 0 20 40 60 80 100 120

Distance (m)

Figure 8.22. Effect of time step on reaction force of 3/-36-60 curved bridge

lUJ=(/2

0 20 40 8060Distance (m)

Figure 8.23. Effect of time step on shear force of 3l-3b-60 curved bridge

100

250

200

150

100

— At =0.005-• At =0.014- At =0.020

50

0120

412


CO

wdJC<D

30

a = 0.00- - - - a = -0.05 a = -0.3325

15

10

5

02010 30 400

Distance (m)

Figure 8.24. Damping effect on tensile stress of 3l-3b-60 straight bridge

Distance (m)

Figure 8.25. Damping effect on reaction force of 31-3b-60 straight bridge

413

300 a = 0.00- - - - a = -0.05 a = -0.33250

I 200

d>o<s 150cot3

20 3010 400


cflI

eu

o .s

12

— L/R=0.0 --e - L/R=0.1 - - L/R=0.2-A-L/R=0.4

10

8

6

4

232 4

N u m b e r o f la n e s

Figure 8.26. Effect of number of lanes on impact factor for tensile stress for 46-20 bridges

<D

C

Cu

10

8

6

4

-A--— L/R=0.0 - 0 - L/R=0.1 - -X - L/R=0.2 -A-L/R=0.4

2

032 4

N u m b e r o f b o x e s

Figure 8.27. Effect of number of boxes on impact factor for tensile stress for 46-20 bridges

414


(1>

aa>

tjC3£Xe

8

— L/R=0.0 -><- -L/R=0.4

6

4

2

—-------0

20 40 60 80 100Span length (m)

Figure 8.28. Effect of span length on impact factor for tensile stress for 2l-2b bridges

hMcB

Ia.

10— 2l-2b-20 - > e -2l-2b-40

8

6

4

2>e

00.0 0.1 0.2 0 .3 0 . 4


Figure 8.29. Effect of span-to-radius of curvature ratio on impact factor for tensile stress for 2l-2b bridges

415


a)

3 4 60 1 2 5Fundamental frequency (Ffe)

b)

25

^20Vnto

15cj

olO

& 5

----- Proposed Equation

♦

...... ..........>>.

^ Ip = ( 23-2.5 f)♦♦♦

_____ ♦♦

1 ♦ W A . ♦

: vU

i ..................... ........................ i ................

Figure

0 1 2 3 4 5 6Fundamental frequency (Hz)

8.30. Impact factor for tensile stress versus fundamental frequency for: a) straight bridge; and b) curved bridge

416


a)

Cl,

1 2 3 4 5 60Fundamental frequency (Hz)

b)

Proposed Equation

In = ( 27-2.4 f)

Figure 8.31.


Impact factor for compressive stress versus fundamental frequency for: a) straight bridge; and b) curved bridge

417


a)

C3_oo(DG0)

T3

c22o,C3f+H+->OcSD hB

20

15

10

5

01 2 3 40 5 6

Fundamental frequency (Hz)

b)

20

eI 15OCdj

^ 10 VhO•Mo

Qa- ^

00

Figure


Id = ( 11+0.5 f)------- --------- ►

♦*

....V%

.....4 ........

............ * 4 .*

♦♦ * /

F

< 1 ♦

♦


8.32. Impact factor for deflection versus fundamental frequency for: a) straight bridge; and b) curved bridge

418


a)

co20 -

^ 15 - o

10 -oC3as C -


b)

30

25

o20c3

H ,o

0421Q,

15 -

10

-----Propo sed Equation

------------- Iro = (22-f )

..... .............A.

♦ --------------------

▼♦

♦

♦

♦ . ♦

............♦..AI *

1 \t*

i

n...................

♦♦ <

-------------- ' "0 1 2 3 4 5 6


Figure 8.33. Impact factor for exterior support reaction versus fundamental frequency for; a) straight bridge; and b) curved bridge

419


a)

25 -

2 3 40 5 6Fundamental frequency (Hz)

b)

Proposed Equation

Iri = ( 18-0.5 f)

0 1 2 3 4 5 6Fundamental frequency (Hz)

Figure 8.34. Impact factor for interior support reaction versus fundamental frequency for: a) straight bridge; and b) curved bridge

420


a)

200

120 -

63 4 50 1 2

b)

200

^160

IS 120

B^ 80 4-»ocd

40


♦

*

I u - ( 1 6 0 )

♦♦♦♦♦

s

Proposed Equation

- - V r -

i A

0 1 2 3 4 5Fundamental fiequency (Hz)

Figure 8.35. Impact factor for uplift reaction versus fundamental frequency for: a) straight bridge; and b) curved bridge

421


a)

HH

0 1 2 3 4 5 6Fundamental jfrequency (Hz)

b)

30

25s®

1 2 0kN,o

15o

J IOo,S

-----Proposed Equation

...................4..♦% ♦

. .. ^ .... ho ~ ( ][6-2 f )

" ......♦ ♦ ♦

♦

♦ ^ ^

n............ ♦ i - w . , ............ __ ♦ 1 . J r

" - ........... .^ -........... ...... " ......... ............... ................... - ..T“

Figure 8.36.

1 2 3 4 5 6Fundamental frequency (Tfe)

Impact factor for shear force at the exterior support versus fundamental frequency for; a) straight bridge; and b) curved bridge

422


a)

u,aCO

40

30 -

20

0Cjo .1 10 ♦ ♦

t * }t

.............. %

%

2 3 4Fundamental frequency (Hz)

b)

40

ao>C O

Oo,c3

30

20

uc3a.S 10

▲


wt i = (37-41

^

')

♦ ♦♦ ♦

♦.A ♦

♦

^♦♦

♦ . A

_

m ...................

♦ > 4

" 4.....................r '...............1 6

Figure 8.37.

2 3 4 5Fundamental frequency (Hz)

Impact factor for shear force at the interior support versus fundamental frequency for: a) straight bridge; and b) curved bridge

423


a)

I 30-

I 20--Moaaal-H

10-

20 60 80 100 1200 40

b)

60

^ 5 0

u40‘mfl^3 0

J 2 0<4H

o1^10

Span length (m)

------ Proposed Equation- X - CHBDC2000 - -G- AASHTOLRFD -A -A A SH T O 1996 -•-A A S fir O 2 0 0 3

------------ ^-------- i----------------------

6 ------------ i -------------i

'"■'Z

_ _ _ 1 - - I 1 _________ !20 40 60 80


Figure 8.38. Impact factor for tensile stress versus span length for: a) straight bridge; and b) curved bridge

424


a)

^ 50-

40 -1 /1CO

' r 20-

10 -

40 60 80 1000 20 120Span length (m)

b)

COHC l.

oo

Bo

0C31

60

50

40

30

20

10

Proposed Equation- y K - CHBDC2000 - -O - AASHTOIEFD - A -AASHTO 19% - t - AASHTO 2003

r - r i n=( 25)-------------


80 100 120

Figure 8.39. Impact factor for compressive stress versus span length for; a) straight bridge; and b) curved bridge

425


a)

T3

60 800 20 40 100 120

b)

60

40 -

a o*•+3 O(U0>

^ 30i-ioo5 20 o& 10

Span length (m)

------Proposed Equation-)K- CHBDC2000 - O - AASHTOLRFD - A - AASHTO 1996 - • - AASHTO 2003

1 - - - - - ■ ( I" #

! ----- ê --------------) F-------------- ^ --------------k

*------ - _Id = (13)

1 ,--------------------^ ^ I f '

20 40 60 80Span length (m)

100 120

Figure 8.40. Impact factor for deflection versus span length for: a) straight bridge; and b) curved bridge

426


a)

50 -

do

oft's

10 -

200 40 60 80 100 120

Span length (m)

b)

60

50 - -

— ■ Proposed Equation )K- CHBDC2000 O ■ AASHTOLRFD A -AASHTO 1996 • -AASHTO 2003

« 30

Im -(2 0 )


80 100 120

Figure 8.41. Impact factor for exterior support reaction versus span length for: a) straight bridge; and b) curved bridge

427


a)

ao40 -

30 -

20 -

10 -

80 100 1200 20 40 60

Span length (m)

b)

Proposed Equation X - CHBDC2000 -O - AASHTOLRFD A - AASHTO 1996

-AASHTO 2003 •

) r ■ ■ “ )lr ■ ■ ~ ---------------- )KIn = (20)20


80 100 120

Figure 8.42. Impact factor for interior support reaction versus span length for: a) straight bridge; and b) curved bridge

428


a)

200

160 -

80 -

20 40 600 80 100 120

b)

200

^160

120UcSuQi 80

40

Figure 8.43.

Span length (m)

Iu = (160)_♦---------

♦

♦

------ ■ Proposed Equation- X - CHBDC2000- -o ■AASHTOLRFD- A - ■AASHTO 1996

■AASHTO 2003


80 100 120

Impact factor for uplift reaction versus span length for: a) straight bridge; and b) curved bridge

429


a)

g 40

20 40 60 800 100 120

Span length (m)

b)

60

50

| 4 0

01

30

> -

6--

- — ■ Proposed Equation -X- CHBDC2000 -O - AASHTO lEFD -A-AASHTO 1996 -•-AASHTO 2003

‘ ■ (S*.................. "(J)................ (i)

# - ............

Iso = ( 24 )

t ..................

♦

I___ i 1 !20 40 60

Span length (m)80 100 120

Figure 8.44. Impact factor for shear at the exterior support versus span length for: a) straight bridge; and b) curved bridge

430


a)

50 -

^ 30 -u

20 -

10 -

0 20 40 60 80 100 120

Span length (m)

b)

60

50

c3C/a

Proposed Equation-)K - CHBDC2000 - -O - AASHTO U?FD - A -AASHTO 1996 - • -AASHTO 2003

- r ^ . T-. T-. T-. 7^ . r-. - - Isi ( 3 4 )


80 100 120

Figure 8.45. Impact factor for shear force at the interior support versus span length for: a) straight bridge; and b) curved bridge

431


w1/3C /3

atoJi* c oCd>

<2o"S,<soc<3n.B

25

Proposed Equation20

15

10

5

00.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4


Figure 8.46. Impact factor for tensile stress versus span-to-radius of curvature ofcurved bridges

s - /c/3C /3(DLn

30

25

C/3002o.ou

oo

ucs3O.

15

10II

-----Proposed Equation In-5(4HHL/R)— -4-

I

0 + - 0.0

I

t

♦

I♦

i

0.2 0.4 0.6 0.8 1.0

Span-to-radius of curvature ratio (L/R)1.2 1.4

Figure 8.47. Impact factor for compressive stress versus span-to-radius of curvature of curved bridges

432


16

ou<D’OIH

O. a

^ 4 &

Proposed Equation

Id = (14-3iyR)

I♦

I♦

II

II

TI4

0.0 0.2 0.4 0.6 0.8 1.0Span-to-radius of curvature ratio (L/R)

1.2 1.4

Figure 8.48. Impact factor for deflection versus span-to-radius of curvature ratio of curved bridges

a.2csa

, o

o•4-»oaCl,

25

Proposed Equation20

15

10

5

00.80.0 0.2 0.4 0.6 1.0 1.2 1.4


Figure 8.49. Impact factor for exterior support reaction versus span-to-radius of curvature of curved bridges

433


B

Ocd.1a

.2o<sCB

25

Proposed Equation

15

10

5

01.0 1.40.0 0.2 0.4 0.6 0.8 1.2

Span-to-radius of curvature ratio (LTR)

Figure 8.50. Impact factor for interior support reaction versus span-to-radius of curvature of curved bridges

'd-3u,o

o

200

150

100

ocdOdS 50

0

♦♦♦

♦

♦

0.0 0.2

♦♦

i 1 1 ■

Proposed Equation

Iu = (190-135L/R)

i1.2 1.40.4 0.6 0.8 1.0


Figure 8.51. Impact factor for uplift reaction versus span-to-radius of curvature of curved bridges

434


NO

(/)u.2

O<13

D,

30

Proposed Equation25

20

15

10

5

00.0 0.2 1.2 1.40.4 0.6 0.8 1.0


Figure 8.52. Impact factor for shear force at the exterior support versus span-to-radius of curvature of curved bridges

cdOC/5Lh<2

O

oaCl

50

Proposed Equation40

30

20

10

00.4 0.6 0.8 1.0 1.20.0 0.2 1.4


Figure 8.53. Impact factor for shear force at the interior support versus span-to-radius of curvature of curved bridges

435


a)

0 4020 60 80 100 120

b)

S p a n l e n g t h ( m )

3/-2 )-100,L/R = 0.6

Figure 9.1. Effect of bridge span length on: a) fundamental frequency; and b) mode shape

436


a)

&&S=3cr0

1<Ds

§IX

5

4

□ L=20 m

- -A - L=60 m3

4b &L/R = 0.4

2

03 42

Number of lanes

0.774

0.731

2/-3/-80, L/R= 0.4 4l-3b-B0, L/R= 0.4

Figure 9.2. Effect of number of lanes on: a) fundamental frequency; and b) mode shape

437


a)

NE^ 40 ag-< 3

1(D

-a-

-E h - L-20 m

--A- L=60 m

3/ &L/R = 0.0

A-

b)

Number of boxes ^

3/-2Z)-20, L/R = 0.4 3/-5Z)-20, L/R = 0.4

0.365

Figure 9.3. Effect of number of boxes on: a) fundamental frequency; and b) modeshape

438


a)

2.0

■ S — L = 6 0 m 0 - - L = 1 0 0 m

2,1-2b

0 . 4

0.00.0 0.2 0 . 4 0.6 0.8 1.0 1.2

S p a n - t o - r a d i u s o f c u r v a t u r e r a t i o ( r a d )

b)

3 / - 2 6 - 1 0 0 , L / R = 0 . 0 3 / - 2 Z ) - 1 0 0 , L / R = 1 .2

0 . 6 6 3

Figure 9.4. Effect of span-to-radius of curvature ratio on: a) fundamental frequency; and b) mode shape

439


-□-■L/R=0.0

-O-L/R=0.6

-^-iyR=i.2

/= 100m,4/-36


30

Figure 9.5. Effect of span-to-depth ratio on the fundamental frequency

8

oc0O'

*

1 4(Ds•O Tc 33tlH

/=40iu 3/-26,L/R=0.4

" .- .“ ■.T.-J

[3-

)(-

i nTpr—

- ^ n - d -1 2

- A - o f4

E]

■3(

10 20 30 40DiafAragm thickness (mm)

50 60

Figure 9.6. Effect of end-diaphragm thickness on the first four natural frequencies

440


5

ss SB sa

4Internal and external bracing Internal bracing

3

/=60 m,3/-3Z>,L/R=0.8

2

3 5 7 9Number of bracings

Figure 9.7. Effect of number of cross bracings on the first four natural frequencies

8-S - f l -0 -f2

L/R = 0.0 L/R = 1.2

6L = 60m, 4Z-36

^ Ao 4

2

043

Number of spans2

Figure 9.8. Effect of number of spans on the first four natural frequencies

441


X Fully-loaded lanes —B- • Partialy-loaded lanes

0.8

fii 0.6

1

S

0.2

L=60 m,2/-3Z),L/R=0.6

0.090 100 110 12050 70 8060

Vehicle speed (km/h)

Figure 9.9. Effect of vehicle speed on the peak acceleration

X Fully-loaded lanes Partialy-loaded lanes

0.2 L=60m, 21-3b

0.00.2 0.40.0 0.6 0.8 1.0 1.2


Figure 9.10. Effect of span-to-radius of curvature ratio on the peak acceleration

442


2.5X Fully-loaded lanes

—B- - Partialy-loaded lanes2.0

1.5

1.0

0.5L=60m,2/-36,iyR=1.2

0.00 10 155 20 25 30

Diaphragm thickness (mm)

Figure 9.11. Effect of end-diaphragm thickness on the peak acceleration

a_o

o

<uOh

2.5— — Internal and External bracings

Internal bracings2.0

1.5

Q— —-E ]

1.0

0.5

L=60 m ,2/-3i,L/R =1.2

0.0

1 3 5 7 9 I INumber of bracings

Figure 9.12. Effect of number of bracings on the peak acceleration

443


Appendix A

A typical input data deck for the linear analysis of a bridge^ H E A D I N G2 B 0 X C U R V E D C O N T I N U O U S C A S E L = 2 0 , 2 L a n e ( A = 9 . 3 m ) , L / R = 0 . 4 * D A T A C H E C K* P R E P R I N T , E C H O = Y E S , M O D E L = N O , H I S T O R Y = N O ^ R E S T A R T , W R I T E* * * * * * * * * * * * R E F E R E N C E N O D E C O O R D I N A T E S F O R T H E L E F T S I D E * * * * * * * * * * * * * N O D E1 0 0 , 01 1 , 0 0 , - . 1 2 0 51 1 1 , 0 0 , - . 9 0 4 51 0 0 , 5 0 . 3 3 5 9 8 , 2 1 . 2 8 1 7 1 , 01 7 0 0 , 4 1 . 7 7 0 1 2 , 1 7 . 6 6 0 1 2 , 03 1 0 , 4 9 . 2 6 5 2 5 , 2 0 . 8 2 9 0 1 , - . 1 2 0 53 7 0 , 4 9 . 2 6 5 2 5 , 2 0 . 8 2 9 0 1 , - . 9 0 4 51 5 1 0 , 4 2 . 8 4 0 8 5 , 1 8 . 1 1 2 8 2 , - . 1 2 0 51 5 7 0 , 4 2 . 8 4 0 8 5 , 1 8 . 1 1 2 8 2 , - . 9 0 4 5********** R E F E R E N C E N O D E C O O R D I N A T E S F O R T H E R I G H T S I D E * * * * * * * * * * * * *7 2 0 1 0 0 ,7 2 1 7 0 0 ,7 2 0 3 1 0 ,7 2 0 3 7 0 ,7 2 1 5 1 0 ,7 2 1 5 7 0 ,

5 0 . 3 3 5 9 8 ,4 1 . 7 7 0 1 2 ,4 9 . 2 6 5 2 5 ,4 9 . 2 6 5 2 5 , 4 2 . 8 4 0 8 5 , 4 2 . 8 4 0 8 5 ,

- . 1 2 0 5 - . 9 0 4 5 - . 1 2 0 5 - . 9 0 4 5

- 2 1 . 2 8 1 7 1 ,- 1 7 . 6 6 0 1 2 ,- 2 0 . 8 2 9 0 1 ,- 2 0 . 8 2 9 0 1 ,- 1 8 . 1 1 2 8 2 ,- 1 8 . 1 1 2 8 2 ,

N O D E G E N* N G E N , N S E T = S L A B O U T , L I N E = C1 0 0 . 7 2 0 1 0 0 . 1 0 0 0 0 . 1 *NGEN,NSET=SLABIN,LINE=C1 7 0 0 . 7 2 1 7 0 0 . 1 0 0 0 0 . 1 * N F I L L , N S E T = S L A B S L A B O U T , S L A B I N , 1 6 , 1 0 0* * * * * * * * * * * * * * * * * * * * * * * N O D E G E N . F O R A L L T H E W E B S * N G E N , N S E T = W E B 0 U T 1 , L I N E = C3 1 0 . 7 2 0 3 1 0 . 1 0 0 0 0 . 1 1 * N G E N , N S E T = W E B 0 U T 2 , L I N E = C3 7 0 . 7 2 0 3 7 0 . 1 0 0 0 0 . 1 1 1 * N F I L L , N S E T = W E B O U T

W E B 0 U T 1 , W E B 0 U T 2 , 6 , 1 0 * N G E N , N S E T = W E B I N 1 , L I N E = C1 5 1 0 . 7 2 1 5 1 0 . 1 0 0 0 0 . 1 1 * N G E N , N S E T = W E B I N 2 , L I N E = C1 5 7 0 . 7 2 1 5 7 0 . 1 0 0 0 0 . 1 1 1 * N F I L L , N S E T = W E B I N

W E B I N l , W E B I N 2 , 6 , 1 0 * N F I L L , N S E T = W E B

W E B O U T , W E B I N , 3 , 4 0 0

* N F I L L , N S E T = A L L N O D E W E B O U T , W E B I N , 1 2 , 1 0 0 ************************ N F I L L , N S E T = F L A N T

W E B O U T l , W E B I N l , 3 , 4 0 0* * * * * * * * * * * * * * * * * * * * * * n o d e G E N . F O R T H E B O T T O M F L A N G E

F O R T H E S L A B * * * * * * * * * * * * * * * * * * * * *

•k'k'k'k-k-k'k'k'k-k-k'k'k'k-k'k'k'k'k-k

NODE GEN. FOR ALL NODES ■ 'fr'k-k-k-k'k'k'k-k-k'k'k-k-k'k'k'k'k-k

N O D E G E N . F O R T H E T O P F L A N G E

'k'k'k'k'k'k'k'k'k'k'k'k'k'k-k'k

444


* N F I L L , N S E T = F L A N B W E B 0 U T 2 , W E B I N 2 , 1 2 , 1 0 0****'k-k-k-k-k-k*-k-kick'k'k-k-k-k-k-k-k-k-k-k-k-k-k'k'k'k'k-k'k-k-k'k'k'k-k-k-)ck'k-k-k*'k-Jr-^-)ck-^-k-k-k-k-k-k-*r'^-k-k*'k'k'k*-k'k

ELEMENT GEN. FOR TOP SLAB *******************^ E L E M E N T , T Y P E = S 4 R1 0 0 , 1 0 0 , 2 0 0 , 1 0 2 0 0 , 1 0 1 0 0* E L G E N , E L S E T = S L A B1 0 0 . 1 6 . 1 0 0 . 1 0 0 . 7 2 . 1 0 0 0 0 . 1 0 0 0 0* * * * * * * * * * * * * * * * * * * * * * * * * E L E M E N T G E N . F O R T O P F L A N G E * * * * * * * * * * * * * * * * * * * E L E M E N T , T Y P E = B 3 1 H3 1 0 . 1 0 3 1 0 . 3 1 0 * E L G E N , E L S E T = F L A N T3 1 0 , 4 , 4 0 0 , 4 0 0 , 7 2 , 1 0 0 0 0 , 1 0 0 0 0* * * * * * * * * * * * * * * * * * * * * * * * * * * * E L E M E N T G E N . F O R W E B S * * * * * * * * * * * * * * * * * * * *^ E L E M E N T , T Y P E = S 4 R3 2 0 , 3 1 0 , 3 2 0 , 1 0 3 2 0 , 1 0 3 1 0* E L G E N , E L S E T = W E B3 2 0 , 6 , 1 0 , 1 0 , 7 2 , 1 0 0 0 0 , 1 0 0 0 0 , 4 , 4 0 0 , 4 0 0* * * * * * * * * * * * * * * * * * * * * * * E L E M E N T G E N . F O R B O T T O M F L A N G E * * * * * * * * * * * * * * * ** E L E M E N T , T Y P E = S 4 R3 7 1 , 3 7 0 , 4 7 0 , 1 0 4 7 0 , 1 0 3 7 0* E L G E N , E L S E T = F L A N B 13 7 1 . 4 . 1 0 0 . 1 0 0 . 7 2 . 1 0 0 0 0 . 1 0 0 0 0* E L C O P Y , O L D S E T = F L A N B 1 , N E W S E T = F L A N B 2 , S H I F T N O D E S = 8 0 0 , E L E M E N T S H I F T = 8 0 0 * E L S E T , E L S E T = F L A N B F L A N B l , F L A N B 2* * * * * * * * * * * * * * * * * * * * * * * e l e m e n t G N . F O R E N D D I A P H R A G M * * * * * * * * * * * * * * * * ** E L E M E N T , T Y P E = S 4 R3 1 1 , 3 1 0 , 3 2 0 , 4 2 0 , 4 1 0* E L G E N , E L S E T = D I A P L 13 1 1 , 4 , 1 0 0 , 1 0 0 , 6 , 1 0 , 1 0* E L C O P Y , O L D S E T = D I A P L 1 , N E W S E T = D I A P L 2 , S H I F T N 0 D E S = 8 0 0 , E L E M E N T S H I F T = 8 0 0* E L S E T , E L S E T = D I A P L D I A P L l , D I A P L 2* E L C 0 P Y , 0 L D S E T = D I A P L , N E W S E T = D I A P M , S H I F T N O D E S = 3 6 0 0 0 0 , E L E M E N T S H I F T = 3 6 0 0 0 0* E L C O P Y , O L D S E T = D I A P M , N E W S E T = D I A P R , S H I F T N O D E S = 3 6 0 0 0 0 , E L E M E N TS H I F T = 3 6 0 0 0 0* E L S E T , E L S E T = D I A PD I A P L , D I A P M , D I A P R* * * * * * * * * * * * * * * * * * * * * * * * e n d f l a n g e E L E M E N T S * * * * * * * * * * * * * * * * * * * * * * * * * * * E L E M E N T , T Y P E = B 3 1 H3 1 2 . 4 1 0 . 3 1 0 * E L G E N , E L S E T = E F L A N L 13 1 2 , 4 , 1 0 0 , 1 0 0* E L C O P Y , O L D S E T = E F L A N L l , N E W S E T = E F L A N L 2 , S H I F T N 0 D E S = 8 0 0 , E L E M E N T S H I F T = 8 0 0* E L S E T , E L S E T = E F L A N L E F L A N L l , E F L A N L 2* E L C O P Y , O L D S E T = E F L A N L , N E W S E T = E F L A N R , S H I F T N O D E S = 7 2 0 0 0 0 , E L E M E N TS H I F T = 7 2 0 0 0 0*ELSET, ELSET=EFLAN

E F L A N L , E F L A N R* * * * * * * * * * * * * * * * * * * e l e m e n t g e n f o r t h e e n d t r u s s e l e m e n t s * * * * * * * * * * * *

ÊLEMENT, T Y P E = B 3 1 H7 1 1 , 7 1 0 , 1 1 1 0* E L G E N , E L S E T = B R A C E7 1 1 , 2 , 6 0 , 6 0 , 3 , 3 6 0 0 0 0 , 3 6 0 0 0 0’ ^ E L E M E N T , T Y P E = B 3 1 H9 4 1 , 9 4 0 , 7 1 0*ELGEN, ELSET=BRACE9 4 1 , 2 , 2 3 0 , 1 , 3 , 3 6 0 0 0 0 , 3 6 0 0 0 0

445


* E L E M E N T , T Y P E = B 3 1 H9 4 3 , 7 7 0 , 9 4 0*ELGEN,ELSET=BRACE9 4 3 , 2 , 1 7 0 , 1 , 3 , 3 6 0 0 0 0 , 3 6 0 0 0 0* * E L E M E N T G E N F O R T H E I N T E R . T R U S S E L E M E N T S * * * * * * * * * * * * * * E L E M E N T , T Y P E = B 3 1 H 9 0 3 1 1 , 9 0 3 1 0 , 9 0 7 1 0 * E L G E N , E L S E T = B R A C I9 0 3 1 1 . 3 . 4 0 0 . 4 0 0 . 3 , 9 0 0 0 0 , 9 0 0 0 0 ^ E L E M E N T , T Y P E = B 3 1 H , E L S E T = B R A C I 9 0 7 7 1 , 9 0 7 7 0 , 9 1 1 7 0*ELGEN,ELSET=BRACI9 0 7 7 1 . 3 , 9 0 0 0 0 , 9 0 0 0 0 ^ E L E M E N T , T Y P E = B 3 1 H 9 0 5 4 1 , 9 0 5 4 0 , 9 0 3 1 0 * E L G E N , E L S E T = B R A C I9 0 5 4 1 , 2 , 2 3 0 , 1 , 3 , 4 0 0 , 4 0 0 , 3 , 9 0 0 0 0 , 9 0 0 0 0 ^ E L E M E N T , T Y P E = B 3 1 H 9 0 5 4 3 , 9 0 3 7 0 , 9 0 5 4 0 * E L G E N , E L S E T = B R A C I9 0 5 4 3 . 2 . 1 7 0 . 1 . 3 . 4 0 0 . 4 0 0 . 3 , 9 0 0 0 0 , 9 0 0 0 0* E L C O P Y , O L D S E T = B R A C I , N E W S E T = B R A C I , S H I F T N 0 D E S = 3 6 0 0 0 0 , E L E M E N T S H I F T = 3 6 0 0 0 0* * * * * * * * * * * * * * * * * * * * * * * * * M A T E R I A L P R O P E R T I E S * * * * * * * * * * * * * * * * * * * * * * * * * ^ O R I E N T A T I O N , N A M E = L O C A L , S Y S T E M = C Y L I N D R I C A L 0 , 0 , - 1 0 , 0 , 0 , 1 0 3 , 0* S H E L L S E C T I O N , E L S E T = S L A B , M A T E R I A L = C O N , O R I E N T A T I O N = L O C A L

. 2 2 5 , 5^ M A T E R I A L , N A M E = C O N* D E N S I T Y2 . 4 0 0^ E L A S T I C2 7 E 6 , . 2 0* S H E L L S E C T I O N , E L S E T = F L A N B , M A T E R I A L = S T E E L , O R I E N T A T I O N = L O C A L . 0 1 6 , 5

* M A T E R I A L , N A M E = S T E E L* D E N S I T Y7 . 8 0 0^ E L A S T I C2 0 0 E 6 , . 3* S H E L L S E C T I O N , E L S E T = W E B , M A T E R I A L = S T E E L . 0 1 6 , 5

* B E A M S E C T I O N , S E C T I O N = R E C T , E L S E T = F L A N T , M A T E R I A L = S T E E L . 0 1 6 , . 3 0 , 0 , 15 . 5* B E A M S E C T I O N , S E C T I O N = R E C T , E L S E T = E F L A N , M A T E R I A L = S T E E L . 0 1 6 , . 3 0 , 0 , 15 . 5* S H E L L S E C T I O N , E L S E T = D I A P , M A T E R I A L = S T E E L . 0 1 6 , 5

* B E A M S E C T I O N , S E C T I O N = R E C T , E L S E T = B R A C E , M A T E R I A L = S T E E L . 1 , . 1 0 , 0 , 15 . 5*BEAM SECTION,SECTION=RECT,ELSET=BRACI,MATERIAL=STEEL . 1 , . 1 0 , 0 , 1 5 , 5•k-k-k'k'k'k'k'k'k-k'k-kit-k-k-k-k-k-k'k'k'k'k-k'k'k-k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k'k-k'k'k'k'k'k-k'k'k'k'k'k'k'k'k'k'k'k'k'k-k'k-k'k'k-k•k'k-k-k'k'k-k-k-*;-k-k-k-k'k-k-k-k-k-k-k-k'k'k'k'k'k- 'k |V[ULT J POINT CONSTRAINT

446


* N G E N , N S E T = N S L A B O U T , L I N E = C3 0 0 . 7 2 0 3 0 0 . 1 0 0 0 0 . 1 * N G E N , N S E T = N S L A B I N , L I N E = C1 5 0 0 . 7 2 1 5 0 0 . 1 0 0 0 0 . 1 * N F I L L , N S E T = N S L A B

N S L A B O U T , N S L A B I N , 3 , 4 0 0 * N S E T , N S E T = N S L A B E L4 0 0 , 5 0 0 , 6 0 0 , 1 2 0 0 , 1 3 0 0 , 1 4 0 0 * N S E T , N S E T = N S L A B E R7 2 0 4 0 0 , 7 2 0 5 0 0 , 7 2 0 6 0 0 , 7 2 1 2 0 0 , 7 2 1 3 0 0 , 7 2 1 4 0 0 * N S E T , N S E T = N S L A B E M3 6 0 4 0 0 . 3 6 0 5 0 0 . 3 6 0 6 0 0 , 3 6 1 2 0 0 , 3 6 1 3 0 0 , 3 6 1 4 0 0 * N S E T , N S E T = N F L A N E L4 1 0 , 5 1 0 , 6 1 0 , 1 2 1 0 , 1 3 1 0 , 1 4 1 0 * N S E T , N S E T = N F L A N E R7 2 0 4 1 0 , 7 2 0 5 1 0 , 7 2 0 6 1 0 , 7 2 1 2 1 0 , 7 2 1 3 1 0 , 7 2 1 4 1 0 * N S E T , N S E T = N F L A N E M3 6 0 4 1 0 . 3 6 0 5 1 0 . 3 6 0 6 1 0 , 3 6 1 2 1 0 , 3 6 1 3 1 0 , 3 6 1 4 1 0 * M P CB E A M , N S L A B , F L A N T B E A M , N S L A B E L , N F L A N E L B E A M , N S L A B E M , N F L A N E M B E A M , N S L A B E R , N F L A N E R *******■********■*•■*■****************■*■***■*•*■*•*■*•■*•*•*****■*■**■*■*** + ******■*■* + *■*•*** N G E N , N S E T = S U P P 13 7 0 . 1 5 7 0 . 4 0 0 * N G E N , N S E T = S U P P 23 6 0 3 7 0 . 3 6 1 5 7 0 . 4 0 0 * N G E N , N S E T = S U P P 37 2 0 3 7 0 . 7 2 1 5 7 0 . 4 0 0 * N S E T , N S E T = S U P P S U P P l , S U P P 2 , S U P P 3 * T R A N S F O R M , N S E T = S U P P , T Y P E = C 4 6 . 5 1 2 5 , 0 , - 1 0 , 4 6 . 5 1 2 5 , 0 , 1 0 * B O U N D A R YS U P P , 3 S U P P 2 , 2 1 5 7 0 , 23 6 1 5 7 0 . 17 2 1 5 7 0 . 2* E L S E T , E L S E T = X B R A C B R A C E , B R A C I * N G E N , N S E T = M D1 4 0 3 7 0 . 1 4 1 5 7 0 . 1 0 0 * E L G E N , E L S E T = M S1 4 0 3 7 1 . 4 . 1 0 0 . 1 0 03 6 0 3 7 1 . 4 . 1 0 0 . 1 0 0 * E L G E N , E L S E T = M S1 4 1 1 7 1 . 4 . 1 0 0 . 1 0 03 6 1 1 7 1 . 4 . 1 0 0 . 1 0 0

a n a l y s i s ** S T E P

C A S E ( 2 ) : T Y P I C A L T R U C K L O A D I N G C A S E ^ S T A T I C*DLOAD, OP=NEW* N S E T , N S E T = C A S E 21 8 0 4 0 0 , 1 8 0 7 0 0* N S E T , N S E T = C A S E 22 0 4 0 0 , 1 0 0 4 0 0 , 2 0 7 0 0 , 1 0 0 7 0 0* C L O A D , O P = N E W

C A S E 2 , 3 , - 1 8C A S E 2 , 3 , - 7 1

447


* N O D E P R I N T , N S E T = M D U 3* N O D E P R I N T , N S E T = S U P P R F 3* E L P R I N T , P O S I T I O N = A V E R A G E D A T N O D E S , E L S E T = M S 8 2 2* E L P R I N T , E L S E T = X B R A C S F l■k'k-k-k-k-k-k-k-k-k'k-k'k-^-^-k-k'k'k-^-k-k-h'k-k-k-k-k-k-k'k'k-k-k- 'k-k- 'k-k- ' 'k'k'k'k-k-k'k-k'k'k'k-k- 'k-k'k'k'fc'k'k'k'k'k'k-k'k'k'k* E N D S T E P• k ' k ' k ' k ' k - k ' k - k ' k ' k ' k - k ' k ' k ' k ' k ' k ' k - k ' k ' k ' k ' k ' k ' ^ j ^ Q Q v i k ) 2 r 3 . t i o r i* S T E P* F R E Q U E N C Y , E I G E N S O L V E R = S U B S P A C E 4■k'k-k-k-k'k-k'k'k-k'k-k'ick'k'k-k-k-k-k'k'k-k'k'k'k-^-k'k-k'k'k'k'k-k'k'k-k-k'k-k'k^'k-k'k'k-^'k-k'k'k-k-k'k'k-k'k'k'k' ie'k-k'^' if-k-k-k-k-k

^ O U T P U T , F I E L D* N O D E O U T P U T , N S E T = S L A BU' k ' k - k - k - k - k ' k ' k - k ' k ' k ' k ' k - k ' k ' k ' k - k i f i r ' k ' k ' k ' k ' k ' k - k ' k ' k ' k ' k - k i c ' k ' k i r ' k ' k ' k - k ' k - k ' k - k ' k - k - k ' k - k - k ' k ' k - k ' k ' k - k - k - k - k - k - k - k - k - k ' k - k - k ' k - k ' k

* E N D S T E P'k-k'k-k-k-:k'k'k-k-k'k'k'k-k-k-k'k-k-k-k-k-k'k-k'k-k'k-k-k- -k j_ Q SFlSlySiS* S T E P , I N C = 3 0 5 * D Y N A M I C . 0 2 , 6 . 1

* D L O A D S L O A D , P N U S L O A D , B X N U S L O A D , B Y N U-:)r-:k-k'k-k-k-k-k'k'k-kr)ir'k'k'k'k'k-k'k'k'k'k'k'k'k'k'k'k'k'k-k'k'k'k-k'k'k-k'k'k-k-k-k'k'k'k'k'^'k'k'k':)f'k-k-k-k^^'k-k^-k^'k'^-k-k-:k'k'k-k-:)<:

* O U T P U T , H I S T O R Y , F R E Q U E N C Y = 1 * E L E M E N T O U T P U T , E L S E T = A S T S 2 2^ E L E M E N T O U T P U T , E L S E T = S H E S F 3* N O D E O U T P U T , N S E T = D E F U 3* N O D E O U T P U T , N S E T = A R E

R F l , R F 2 , R F 3*************************************************************************** E N D S T E P

q u a s i - s t a t i c a n a l y s i s * * * * * * * * * * * * * * * * * * ** S T E P

C A S E ( 1 ) : M A X I M U M U L T I M A T E L O A D D U E T O N E G A T I V E M O M E N T ^ D Y N A M I C , E X P L I C I T , 1^ B O U N D A R Y , A M P L I T U D E = R A M P , T Y P E = D I S P L A C E M E N T C A S E l , 3 , 3 , - 1 .' k i f i f - k - k ' k ' k ' k i e ' k ' k ' k ' k - k - k - k ' k ' k - k - k - k - k ' k ' k ' k ' k ' k ' k ' k ' k ' k ' k - k ' k - k - k ' k - k ' k - k - k ' k ' k ' k ' k - k ' k ' k - k - k ' k ^ - k - k - k - k ' k ' k - k ' k - k ' k ' k - i e - k - k - ) f - k - k - k

^ O U T P U T , F I E L D , N U M B E R I N T E R V A L = 2 0 ^ E L E M E N T O U T P U T , P O S I T I O N = C E N T R O I D A L E , S^ E L E M E N T O U T P U T , R E B A R = L T O P E , S , R B F O R^ E L E M E N T O U T P U T , R E B A R = L B O T T O M E , S , R B F O R* E L E M E N T O U T P U T , R E B A R = T T O P E , S , R B F O R^ E L E M E N T O U T P U T , R E B A R = T B O T T O M

E , S , R B F O R * N O D E O U T P U T U

*ENDSTEP

448


Appendix B

Samples of the results for the load distribution factors

Bridge ConfigurationDistribution Factor for

Positive StressL L /R N l Nb F.E.M. Eq. 7.120 0 2 2 1.03 1.1540 0 2 2 1.00 1.1560 0 2 2 1.01 1.1580 0 2 2 1.01 1.15too 0 2 2 1.02 1.1520 0 3 4 0.96 1.0640 0 3 4 0.93 1.0660 0 3 4 0.91 1.0680 0 3 4 0.91 1.06too 0 3 4 0.92 1.0620 0 3 5 0.98 1.0640 0 3 5 0.94 1.0660 0 3 5 0.92 1.0680 0 3 5 0.90 1.06too 0 3 5 0.92 1.0620 0 3 6 0.98 1.0640 0 3 6 0.93 1.0660 0 3 6 0.92 1.0680 0 3 6 0.91 1.06too 0 3 6 0.92 1.0620 0.4 2 2 1.10 1.1740 0.4 2 2 1.07 1.1860 0.4 2 2 1.01 1.1980 0.4 2 2 1.19 1.20too 0.4 2 2 1.14 1.2120 0.4 3 4 1.05 1.0840 0.4 3 4 0.98 1.1060 0.4 3 4 1.08 1.1180 0.4 3 4 1.14 1.12too 0.4 3 4 1.09 1.1320 0.4 4 6 1.02 1.0240 0.4 4 6 0.88 1.0460 0.4 4 6 0.93 1.0580 0.4 4 6 0.96 1.06too 0.4 4 6 0.95 1.0820 0.2 2 2 1.07 1.1540 0.2 2 2 1.02 1.1660 1.2 2 2 1.36 1.5180 1.2 2 2 1.66 1.59too 1.2 2 2 1.49 1.6720 0.2 3 4 1.00 1.0640 0.2 3 4 0.94 1.0760 1.2 3 4 1.52 1.5080 1.2 3 4 1.65 1.60too 1.2 3 4 1.54 1.70

449



DeflectionL L/R Nl Nb F.E.M. Eq. 7.320 0 2 2 1.13 1.1540 0 2 2 1.10 1.1560 0 2 2 0.99 1.1580 0 2 2 1.03 1.15100 0 2 2 1.03 1.1520 0 3 4 1.01 1.0640 0 3 4 0.98 1.0660 0 3 4 0.92 1.0680 0 3 4 0.92 1.06100 0 3 4 0.92 1.0620 0 3 5 1.00 1.0640 0 3 5 0.97 1.0660 0 3 5 0.92 1.0680 0 3 5 0.91 1.06100 0 3 5 0.92 1.0620 0 3 6 0.99 1.0640 0 3 6 0.97 1.0660 0 3 6 0.92 1.0680 0 3 6 0.92 1.06100 0 3 6 0.92 1.0620 0.4 2 2 1.30 ^ 1.4740 0.4 2 2 1.27 1.4460 0.4 2 2 1.21 1.4280 0.4 2 2 1.22 1.41100 0.4 2 2 1.23 1.4020 0.4 3 4 1.37 1.4040 0.4 3 4 1.20 1.3760 0.4 3 4 1.16 1.3580 0.4 3 4 1.22 1.34100 0.4 3 4 1.19 1.3320 0.4 4 6 1.39 1.3540 0.4 4 6 1.11 1.3160 0.4 4 6 1.10 1.2980 0.4 4 6 1.08 1.28too 0.4 4 6 1.07 1.2720 0.2 2 2 1.17 1.2640 0.2 2 2 1.16 1.2460 1.2 2 2 2.31 2.7480 1.2 2 2 2.35 2.67100 1.2 2 2 2.50 2.6220 0.2 3 4 1.16 1.1740 0.2 3 4 1.06 1.1660 1.2 3 4 2.38 2.7580 1.2 3 4 2.54 2.68100 1.2 3 ^ 4 2.69 2.63

'2 0 0.2 4 6 1.16 f.ll40 0.2 4 6 0.95 1.1060 1.2 4 6 2.31 2.7080 1.2 4 6 2.37 2.63100 1.2 4 6 2.55 2.57

450



Shear ForceL L /R N l Nb F.E.M. Eq. 7.420 0 2 2 1.32 1.3840 0 2 2 1.23 1.2960 0 2 2 1.13 1.2480 0 2 2 1.12 1.20100 0 2 2 1.11 1.1820 0 3 4 1.41 1.3740 0 3 4 1.30 1.2860 0 3 4 1.03 1.2280 0 3 4 1.05 1.19too 0 3 4 1.05 1.1620 0 3 5 1.38 1.4040 0 3 5 1.31 1.3060 0 3 5 1.08 1.2580 0 3 5 1.09 1.22too 0 3 5 1.10 1.1920 0 3 6 1.29 1.4240 0 3 6 1.25 1.3360 0 3 6 1.11 1.2880 0 3 6 1.13 1.24too 0 3 6 1.13 1.2120 0.4 2 2 1.42 1.4040 0.4 2 2 1.35 1.3460 0.4 2 2 1.25 1.3380 0.4 2 2 1.26 1.36too 0.4 2 2 1.44 M O20 0.4 3 4 1.45 1.3840 0.4 3 4 1.35 1.3260 0.4 3 ^ 4 1.18 1.3280 0.4 3 4 1.28 1.34too 0.4 3 4 1.37 1.3920 0.4 4 6 1.33 1.3640 0.4 4 6 1.36 1.3060 0.4 4 6 1.16 1.3080 0.4 4 6 1.14 1.32too 0.4 4 6 1.22 1.3620 0.2 2 2 1.40 1.3940 0.2 2 2 1.32 1.3060 1.2 2 2 2.05 1.9380 1.2 2 2 2.37 2.32100 1.2 2 2 2.92 2.8120 0.2 3 4 1.41 1.3740 0.2 3 4 1.28 1.2960 1.2 3 4 1.87 1.9080 1.2 3 4 2.21 2.30100 1.2 3 4 2.65 2.7820 0 2 4 6 1.33 1.3540 0.2 4 6 1.30 1.2760 1.2 4 6 1.61 1.8780 1.2 4 6 1.91 2.26100 1.2 4 6 2.41 2.73

451



External Reaction

L L/R Nl Nb F.E.M. Eq. 7.520 0 2 2 1.33 1.4140 0 2 2 1.24 1.3160 0 2 2 1.17 1.2680 0 2 2 1.17 1.23too 0 2 2 1.17 1.2020 0 3 4 1.43 1.4140 0 3 4 1.31 1.3160 0 3 4 1.15 1.2680 0 3 4 1.16 1.23too 0 3 4 1.17 1.2020 0 3 5 1.38 1.4140 0 3 5 1.26 1.3160 0 3 5 1.17 1.2680 0 3 5 1.18 1.23too 0 3 5 1.18 1.2020 0 3 6 1.28 1.4140 0 3 6 1.29 1.3160 0 3 6 1.24 1.2680 0 3 6 1.25 1.23too 0 3 6 1.24 1.2020 0.4 2 2 1.57 1.5040 0.4 2 2 1.41 1.5860 0.4 2 2 1.54 1.7580 0.4 2 2 1.69 1.98too 0.4 2 2 1.96 2.2520 0.4 3 4 1.48 1.4740 0.4 3 4 1.39 1.4960 0.4 3 4 1.57 1.5980 0.4 3 4 1.73 1.73100 0.4 3 4 1.86 1.9020 0.4 4 6 1.43 1.4640 0.4 4 6 1.24 1.4560 0.4 4 6 1.41 1.5180 0.4 4 6 1.53 1.60too 0.4 4 6 1.65 1.7320 0.2 2 2 1.47 1.4540 0.2 r 2 2 1.37 1.4360 1.2 2 2 2.85 3.0980 1.2 2 2 3.64 4.05too 1.2 2 2 4.46 5.1420 0.2 3 4 1.48 1.4440 0.2 3 4 1.40 1.3960 1.2 3 4 2,49 2.4880 1.2 3 4 2.97 3.11100 1.2 3 4 3.60 3.8320 0.2 4 6 1.40 1.4340 0.2 4 6 1.24 1.3760 1.2 4 6 2.14 2.1880 1.2 4 6 2.43 2.64100 1.2 4 6 2.78 3.17

452


Samples of the results for the fundamental frequency

Bridge Configuration Fundamental Frequency (Hz)

L L/R Nl Nb F.E.M. Eq. 9.2 Eq. 9.4 Eq. 9.5

20 0.0 2 2 4.90 4.70 5.15 4.8540 0.0 2 2 2.40 2.35 2.46 2.3160 0.0 2 2 ^ 1.59 1.57 1.62 1.5380 0.0 2 2 1.19 1.18 1.21 1.14too 0.0 2 2 0.95 0.94 0.97 0.9120 0.0 3 4 5.00 4.70 5.15 4.8440 0.0 3 4 2.42 2.35 2.46 2.3260 0.0 3 4 1.60 1.57 1.63 1.5380 0.0 3 4 1.20 1.18 1.21 1.14too 0.0 3 4 0.95 0.94 0.96 0.9120 0.0 4 6 5.03 4.70 5.15 4.8440 0.0 4 6 2.43 2.35 2.47 2.3260 0.0 4 6 1.60 1.57 1.63 1.5380 0.0 4 6 1.19 1.18 1.21 1.14too 0.0 4 6 0.95 0.94 0.96 0.9020 0.4 2 2 4.70 4.50 5.15 4.6340 0.4 2 2 2.30 2.22 2.46 2.1960 0.4 2 2 1.52 1.47 1.62 1.4380 0.4 2 2 1.13 1.10 1.21 1.07too 0.4 2 2 0.89 0.87 0.97 0.8520 0.4 3 4 4.65 4.48 5.15 4.6140 0.4 3 4 2.30 2.21 2.46 2.1860 0.4 3 4 1.51 1.46 1.63 1.4380 0.4 3 4 1.09 1.09 1.21 1.06too 0.4 3 4 0.88 0.87 0.96 0.8420 0.4 4 6 4.49 4.47 5.15 4.6140 0.4 4 6 2.28 2.21 2.47 2.1860 0.4 4 6 1.50 1.46 1.63 1.4280 0.4 4 6 1.11 1.09 1.21 1.05too 0.4 4 6 0.87 0.87 0.96 0.8320 0.2 2 2 4.86 4.63 5.15 4.7740 0.2 2 2 2.38 2.31 ^ 2.46 2.2760 1.2 2 2 1.11 1.07 1.62 1.0580 1.2 2 2 0.79 0.77 1.21 0.75too 1.2 2 2 0.61 0.60 0.97 0.5820 0.2 3 4 4.90 4.62 5.15 4.7640 0.2 3 4 2.38 2.30 2.46 2.2760 1.2 3 4 1.05 1.03 1.63 1.0080 1.2 3 4 0.75 0.74 1.21 0.71too 1.2 3 4 0.57 0.56 0.96 ^ 0.5420 0.2 4 6 4.87 4.62 5.15 4.7640 0.2 4 6 2.39 2.30 2.47 2.2760 1.2 4 6 1.02 1.01 1.63 0.9980 1.2 4 6 0.72 0.72 1.21 0.70too 1.2 4 6 0.55 0.55 0.96 0.53

453


Vita Auctoris

Magdy Said Samaan

1967 Born on the 31 of December in Cairo, Egypt.

1989 Graduated with B.Sc. degree (very good with Honours) in Civil Engineering

from Alexandria University, Alexandria, Egypt.

1990-1996 Worked as a Structural Engineer in several Consulting Engineering

Companies, Egypt.

1998 Graduated with M.Sc. degree in Civil Engineering, Hannover University,

Hannover, Germany.

1999 Worked as a Structural Engineer in an Infrastructure Engineering Company,

Germany.

2000 Enrolled as a full-time graduate student at the University of Windsor in the

Department of Civil and Environmental Engineering pursuing a Ph.D.

degree in Civil Engineering.

454

Dynamic and static analyses of continuous curved composite ...

Documents