STRUCTURAL ENGINEERING, ME CHANIC S AND COMPUTATION

Proceedings of the International Conference on Structural Engineering,

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STRUCTURAL ENGINEERING, ME C HANIC S AND

COMPUTATION

Proceedings of the International Conference on Structural Engineering,

Mechanics and Computation

2- 4 April 2001, Cape Town, South Africa

Edited by

A. Zingoni Department of Civil Engineering, University of Cape Town,

Rondebosch 7701, Cape Town, South Africa

Volume 1

2001

ELSEVIER

A M S T E R D A M - L O N D O N - N E W Y O R K - O X F O R D - P A R I S - S H A N N O N - T O K Y O

ELSEVIER SCIENCE Ltd The Boulevard, Langford Lane Kidlington, Oxford OX5 1GB, UK

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FOREWORD

by Professor Patrick Dowling CBE DL FREng FRS

Vice-Chancellor and Chief Executive University of Surrey, UK

Chairman, Steel Construction Institute, UK

These Proceedings will be regarded in time as seminal in that they record the enormous progress which has been made in Structural Engineering, Mechanics and Computation in the later half of the 20th century, and point the way to the future agenda for research in those areas for the 21 st century.

The two volumes record the collected work of some of the most able researchers working on the world stage at this moment in time, and who are laying the foundations to some exciting new developments in the future. In that respect, the Proceedings should prove essential reading for those newly entering the field.

It is also most appropriate that the SEMC 2001 International Conference be held in South Africa where there is a New Dawn unfolding, and the spirit of cooperation is infusing all sectors of society, not least places of learning such as universities and colleges which are at the very heart of the New Knowledge Economy and Life Long Leaming Agenda.

I salute all of those involved in the organisation, preparation and presentation of the Conference, as well as the delegates for their contribution to the shaping of these Proceedings, and assure all potential readers of the high quality and usefulness of its contents.

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PREFACE

The International Conference on Structural Engineering, Mechanics and Computation was held in Cape Town (South Africa) from 2 to 4 April 2001. Organised by the University of Cape Town, the conference (SEMC 2001) aimed at bringing together from around the world academics, researchers and practitioners in the broad area of structural engineering and allied fields, to review the achievements of the past 50 years in the advancement of structural engineering, structural mechanics and structural computation, share the latest developments in these areas, and address the challenges that the future poses.

The Proceedings contain, in two volumes, a total of 180 papers written by Authors from around 40 countries worldwide. The contributions include 6 Keynote Papers and 12 Special Invited Papers. In line with the aims of the SEMC 2001 International Conference, and as may be seen from the List of Contents, the papers cover a wide range of topics under a variety of themes. There is a healthy balance between papers of a theoretical nature, concerned with various aspects of structural mechanics and computational issues, and those of a more practical, nature, addressing issues of design, safety and construction. As the contributions in these Proceedings show, new and more efficient methods of structural analysis and numerical computation are being explored all the time, while exciting structural materials such as glass have recently come onto the scene. Research interest in the repair and rehabilitation of existing infrastructure continues to grow, particularly in Europe and North America, while the challenges to protect human life and property against the effects of fire, earthquakes and other hazards are being addressed through the development of more appropriate design methods for buildings, bridges and other engineering structures.

I would like to thank all Authors for preparing their work towards this compilation which, on account of the wealth of information it contains in just two volumes, will undoubtedly serve as a useful reference to practitioners, researchers, students and academics in the areas of structural engineering, structural mechanics, computational mechanics, and allied disciplines. Special thanks are due to Members of the International Scientific/Technical Advisory Board of SEMC 2001, who gave their time in advising on the selection of material contained in these Proceedings. The financial support of the Sponsoring Organisations is gratefully acknowledged. I am indebted to my colleagues in the Organising Committee, for the hard work they put into the preparations for the conference. Last but not least, I would like to thank my wife Lydia for the immense contribution she made towards making the SEMC 2001 International Conference a success.

A. Zingoni

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INTERNATIONAL CONFERENCE ON STRUCTURAL ENGINEERING, MECHANICS AND COMPUTATION

Local Organising Committee A. Zingoni, University of Cape Town (Chairman)

M. Latimer, Joint Structural Division of SAICE & IStructE B.D. Reddy, University of Cape Town

J. Retief, University of Stellenbosch F. Scheele, University of Cape Town

A.R. Kemp, University of the Witwatersrand A. Masarira, University of Cape Town

M.G. Alexander, University of Cape Town N.J. Marais, University of Cape Town D. Douglas, University of Cape Town G.N. Nurick, University of Cape Town

International Scientific/Technical Advisory Board

• P.J. Dowling, University of Surrey, UK • Y.K. Cheung, University ofHong Kong, China • P.L. Gould, Washington University, USA • M. Bradford, University of New South Wales,

Australia • M. Papadrakakis, National Technical University

of Athens, Greece • L.A. Clark, University of Birmingham, UK • B.D. Reddy, University of Cape Town, South

Africa • Y. Fujino, University of Tokyo, Japan • D.A. Nethercot, Imperial College of Science,

Technology & Medicine, UK • O. Buyukozturk, Massachusetts Institute of

Technology, USA • A.R. Kemp, University of the Witwatersrand,

South Africa • V. Tvergaard, Technical University of Denmark,

Denmark • M.G. Alexander, University of Cape Town, South

Africa • H. Nooshin, University of Surrey, UK • S.N. Sinha, Indian Institute of Technology at

Delhi, India • D.R.J. Owen, University of Wales at Swansea, UK • G.N. Nurick, University of Cape Town, South

Africa • R. Zandonini, University of Trento, Italy • S. Shrivastava, McGill University, Canada • V. Marshall, University of Pretoria, South Africa

• J.M. Ko, Hong Kong Polytechnic University, China

• J.B. Obrebski, Warsaw University of Technology, Poland

• H.G. Schaeffer, University of Louisville, USA • S. Heyns, University of Pretoria, South Africa • D. Muir Wood, University of Bristol, UK • R. de Borst, Delft University of Technology,

Netherlands • Y. Ballim, University of the Witwatersrand, South

Africa • A. Ghobarah, McMaster University, Canada • A. Nowak, University of Michigan, USA • N.M. Hawkins, University of Illinois at Urbana-

Champaign, USA • U. Schneider, Technical University of Vienna,

Austria • B.W.J. van Rensburg, University of Pretoria,

South Africa • P. Moss, University of Canterbury, New Zealand • J. Retief, University of Stellenbosch, South Africa • P. Dunaiski, University of Stellenbosch, South

Africa • O. Vilnay, Technion Israel Institute of

Technology, Israel • J.J.R. Cheng, University of Alberta, Canada • S.K. Bhattacharyya, University of Durban-

Westville, South Africa • H. Adeli, Ohio State University, USA

Sponsoring Organisations • Joint Structural Division of the South African Institution of Civil Eng&eering (SAICE)

and the UK Institution o f Structural Engineers (IStructE) • The Southern African Institute o f Steel Construction (SAISC)

• The Cement and Concrete Institute (CCI) o f South Africa • The South African Association for Theoretical and Applied Mechanics (SAAM)

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CONTENTS

VOLUME 1

Foreword

Preface vii

Local Organising Committee, International Scientific~Technical Advisory Board Sponsoring Organisations ix

KEYNOTE PAPERS

Y.K. CHEUNG, Y.S. CHENG and F.T.K. AU Vibration analysis of bridges under moving vehicles and trains

D.A. NETHERCOT The importance of combining experimental and numerical study in advancing structural engineering understanding 15

B.D. REDDY and O. EHL Enhanced strain finite elements for Mindlin-Reissner plates 27

P.L. GOULD Recent advances in local-global FE analysis of shells of revolution 39

A.R. KEMP A new mixed flexibility approach for simplifying elastic and inelastic structural analysis 51

P.J. PAHL and M. RUESS Eigenstates of large profiled matrices 63

INVITED PAPERS

F.M. MAZZOLANI and A. MANDARA Advanced metal systems in structural rehabilitation of monumental constructions 75

R. HARTE and W.B. KRATZIG Lifetime-oriented analysis and design of large-scale cooling towers 87

J.T. KATSIKADELIS The BEM for vibration analysis of non-homogeneous bodies 99

xii

J.M. KO, Z.G. SUN and Y.Q. NI A three-stage scheme for damage detection of Kap Shui Mun cable-stayed bridge 111

S.A. SHEIKH Rehabilitation of concrete structures with fibre reinforced polymers 123

D.R.J. OWEN, Y.T. FENG, P.A. KLERCK and J. YU Computational strategies for discrete systems and multi-fracturing materials 135

S.A. TIMASHEV Optimal control of structure integrity and maintenance 147

J.B. OBREBSKI On the mechanics and strength analysis of composite structures 161

H. PASTERNAK, S. SCHILLING and S. KOMANN The steel construction of the new Cargolifter airship hangar 173

T. VROUWENVELDER The fundamentals of structural building codes 183

R.T. SEVERN Earthquake engineering research infrastructures 195

STEEL STRUCTURES: GENERAL CONSIDERATIONS

L.H. TEH and G.J. HANCOCK Beam elements in structural analysis and design of steel frames 213

J.M. DAVIES Second-order elastic-plastic analysis of plane frames 221

J. STUDNICKA Steel structures in Czech Republic 231

A.N. GERGESS and R. SEN Inelastic response of simply supported I-girders subjected to weak axis bending 243

M.M. TAHIR and D. ANDERSON Performance of flush end-plate joints connected to column web 251

R.J. CRAWFORD and M.P. BYFIELD A numerical model for predicting the bending strength of Larssen steel sheet piles 259

A. BURKHARDT Practical use of probabilistic analysis for steel structures 267

J.A. KARCZEWSKI, M. GIZEJOWSKI, S. WIERZBICKI and E. POSTEK Double butt, bolted connections: Influence of prestressing 275

xiii

A. MASARIRA Joint type and the behaviour of frame beams 283

G.J. KRIGE and M.M. KHAN Effect of rock movements on the integrity and performance of mine shaft steelwork 291

J.A. MWAKALI Plasticity enhancement in axially compressed members 301

CONCRETE STRUCTURES: GENERAL CONSIDERATIONS

M.A. MANSUR, K.H. TAN and W. WENG Analysis of reinforced concrete beams with circular openings using strut-and-tie model 311

R.V. JARQUIO True parabolic stress method of analysis in reinforced concrete beams 319

S.H. CHOWDHURY Crack width predictions of reinforced and partially prestressed concrete beams: A unified formula 327

H.Y. LEUNG and C.J. BURGOYNE Analysis of FRP-reinforced concrete beam with aramid spirals as compression confinement 335

R.V. JARQUIO Ultimate strength of CFT circular and square columns 343

H.D. BEUSHAUSEN, R.D. KRATZ and M.G. ALEXANDER The contribution of screed to the structural behaviour of precast prestressed concrete elements 351

L.F. BOSWELL Serviceability criteria for the vibration of post tensioned concrete flat slab floors 359

I. ISKHAKOV Quasi-isotropic ideally elastic-plastic model for calculation of RC elements without empirical coefficients 367

STEEL-CONCRETE COMPOSITE CONSTRUCTION

B. McKINLEY and L.F. BOSWELL Large deformation performance of double skin composite construction using Bi-Steel 377

H.J.C. GALJAARD and J.C. WALRAVEN Behaviour of different types of shear connectors for steel-concrete structures 385

M.P. BYFIELD An analysis of inter shear-stud slip in composite beams 393

xiv

D. LAM and E. E1-LOBODY Finite-element modelling of headed stud shear connectors in steel-concrete composite beam 401

MASONRY, GLASS AND TIMBER STRUCTURES

G. de FELICE Overall elastic properties of brickwork via homogenization 411

E.A. BASOENONDO, R.S. GILES, D.P. THAMBIRATNAM and H. PURNOMO Response of unreinforced brick masonry wall structures to lateral loads 419

H.C. UZOEGBO Lateral loading tests on dry-stack interlocking block walls 427

M.M. ALSHEBANI Cyclic residual strains of brick masonry 437

F.A. VEER, G.J. HOBBELMAN and J.A. van der PLOEG The design of innovative nylon joints to connect glass beams 447

G.J. HOBBELMAN, G.P.A.G. van ZIJL, F.A. VEER and C.N. TING A new structural material by architectural demand 455

N. BOCCHIO, J.W.G. van de KUILEN and P. RONCA The impact strength of timber for guard rails 463

S.J. FICCADENTI, G.C. PARDOEN and R.P. KAZANJY Experimental and analytical studies of diaphragm to shear wall connections 473

PLATE, SHELL AND CONTAINMENT STRUCTURES

N. HASEBE and X.F. WANG Green's functions for the thin plate bending problem under various boundary conditions 483

H. SHIN and D. REDEKOP Nonlinear analysis of a storage tank by the DQM 491

P.D. AUSTIN, D. BUTLER, A.M. NASIR and D.P. THAMBIRATNAM Dynamics of axisymmetric shell structures 499

E.S. MELERSKI Analysis for temperature change effects in circular tanks 507

A. ZINGONI On the possibility of parabolic ogival shells for egg-shaped sludge digesters 515

P.E. TRINCHERO Field testing of column-supported silos and an introduction to the SAISC silo guideline

F. SHALOUF Influence of anti-dynamic tube on reduction of dynamic flow pressure and elimination of pulsation and vibration in grain silo

XV

525

533

LAMINATED COMPOSITE PLATES AND SHELLS

H. MATSUNAGA Vibration of cross-ply laminated composite plates

A. BENJEDDOU and S. LETOMBE Free vibrations of piezoelectric sandwich plates: A two-dimensional closed solution

S.C. SHRIVASTAVA Plastic buckling of spherical sandwich shells under external pressure

P.K. PARHI, S.K. BHATTACHARYYA and P.K. SINHA Hygrothermal effects on the bending behaviour of multiple delaminated composite plates

A. SECU, R. BOAZU and D.P. STEFANESCU New methods to establish the elastic characteristics of the fabric reinforced laminae

541

549

557

565

573

BRIDGES, TOWERS AND MASTS

M. SAMAAN, K.M. SENNAH and J.B. KENNEDY Comparative structural behaviour of multi-cell and multiple-spine box girder bridges

C. GENTILE Full-scale testing and system identification of a steel-trussed bridge

K.M. SENNAH, M.H. MARZOUCK and J.B. KENNEDY Horizontal bracing systems for curved steel I-girder bridges

K.H. RESAN and I. OTHMAN Torsional moments in Y-beam bridge deck under Malaysian abnormal load

X. LIANG, G.J. JUN and J.J. JING Experimental modal analysis of the HuMen suspension bridge

M. IORDANESCU Dedicated software for the structural analysis of guyed antenna towers

B. BEIROW and P. OSTERRIEDER Dynamic investigations of TV towers

583

591

599

607

613

621

629

xvi

FINITE ELEMENT FORMULATIONS

M. BARIK and M. MUKHOPADHYAY A new stiffened plate element for the analysis of arbitrary plates 639

S. GEYER and A.A. GROENWOLD A new 24 d.o.f, assumed stress finite element for orthotropic shells 647

D. SONG, H. WANG, P.K. BANERJEE and D.P. HENRY, Jr Finite element analysis of material and geometry nonlinearities with remeshing 655

A. ZINGONI Subspace formulation for symmetric finite elements 663

G. TABAN-WANI and S.S. TICKODRI-TOGBOA Finite element formulations in the design of underground structures 675

FINITE ELEMENT AND NUMERICAL MODELLING

X.J. YU and D. REDEKOP FEM computation of dynamic properties of a structure using fuzzy set theory 687

M.A. GIZEJOWSKI, J.A. KARCZEWSKI, E. POSTEK and S. WIERZBICKI Development of semi-rigid frame model assisted by testing 695

S.H. LO Analysis of building structures using solid finite elements 703

M.K. APALAK, R. GUNES and E.S. KARAKAS Geometrically non-linear thermal stress analysis of an adhesively bonded tee joint with double support 711

A.T. McBRIDE and F. SCHEELE Validation of discontinuous deformation analysis using a physical model 719

CRACKING AND FRACTURE MECHANICS

G.P.A.G. van ZIJL The time scale in quasi-static fracture of cementitious materials 729

Z. KNESL, L. NAHLIK and Z. KERSNER Calculation of the critical stress in two-phase materials 737

G.P.A.G. van ZIJL A discrete crack modelling strategy for masonry structures 745

xvii

SOIL-STRUCTURE AND FLUID-STRUCTURE INTERACTION

B.B. BUDKOWSKA and A. ELMARAKBI The assessment of shear effect of soil in analysis of laterally loaded models of the piles

B.F. COUSINS and E.S. MELERSKI Numerical analysis of laterally loaded piles under conditions of elasticity

G.A. MOHAMMED and S. BAYOUMI Flexible pipe sewer failure: Numerical analysis

G.A. MOHAMMED Experimental and numerical analyses of multi-storey cracked frames with loss of support

S.K. BHATTACHARYYA and D. MAITY Evaluation of stresses of a flexible structure exposed to fluid considering fluid-structure interaction

S.K. BHATTACHARYYA and O.O. ONYEJEKWE A Green element computational technique applied to a fluid-structure interaction problem

Author Index

755

763

771

779

787

793

S1

Keyword Index $5

VOLUME 2

Foreword

Preface

Local Organising Committee, International Scientific~Technical Advisory Board, Sponsoring Organisations

vii

STABILITY OF THIN-WALLED MEMBERS

P. OSTERRIEDER and J. ZHU Interaction buckling design concepts for thin-walled members

S. SENSOY Degenerate Hopf bifurcation phenomena of a cantilever beam on elastic foundation

N. BJELAJAC Simplified computational procedure for postbuckling equilibrium branches in ideal and imperfect plates

803

811

821

xviii

Q. WANG and Y. LUO Dynamic stability of thin-walled members 829

M. DJAFOUR, A. MEGNOUNIF and D. KERDAL The compound spline finite strip method for the elastic stability of U and C built-up columns 835

VIBRATION AND DYNAMIC ANALYSIS

L.F. YANG, Q.S. LI, J.Z. ZHANG and A.Y.T. LEUNG Stochastic transient variational principle in vibration analysis

J. FARJOODI and A. SOROUSHIAN Efficient automatic selection of tolerances in nonlinear dynamic analysis

T.U. AHMED, L.S. RAMACHANDRA and S.K. BHATTACHARYYA An elasto-plastic free-free beam subjected to pulse load at tip

J. FARJOODI and A. SOROUSHIAN Robust convergence for the dynamic analysis of MDOF elastoplastic systems

N. MUNIRUDRAPPA and V.A. KUMAR Free vibration analysis of slantlegged skew bridge

845

853

861

867

875

VIBRATION CONTROL AND SEISMIC ANALYSIS

A. HENRY, A. RICHARDSON and M. ABDULLAH Placement and elimination of vibration controllers in buildings

Y. RIBAKOV and J. GLUCK Viscous damping system for optimal structural seismic design

N.F. du PLOOY and P.S. HEYNS Reducing vibratory screen structural loading using a vibration absorber

C.F. de ANDRADE, J.C. de ANDRADE FILHO and J.C. de ANDRADE Acceptability vibration criterion for floors with walking occupants

N.A. ALEXANDER, N. GOORVADOO, F.A. NOOR and A.A. CHANERLEY A comparative study of a storey vs. element hysteretic nonlinear model for seismic analysis of buildings

S.T. VASSILEVA Predicting earthquake ground motion descriptions through artificial neural networks for testing the constructions

887

897

905

913

919

927

xix

SEISMIC DESIGN OF STEEL STRUCTURES

G. De MATTEIS, R. LANDOLFO and F.M. MAZZOLANI Contributing effect of cladding panels in the seismic design of MR steel frames

M. MOESTOPO, I. IMRAN, R. RENANSIVA and A. SUDARSONO Ductility formulations of steel structural members

G. DELLA CORTE, G. De MATTEIS, R. LANDOLFO and F.M. MAZZOLANI Seismic analysis of MR steel frames accounting for connection topology

B. FAGGIANO, G. De MATTEIS, R. LANDOLFO and F.M. MAZZOLANI A survey of ductile design of MR steel frames

937

947

955

965

SEISMIC DESIGN OF CONCRETE STRUCTURES

S.S.E. LAM, B. WU, Z.Y. WANG, Y.L. WONG and K.T. CHAU Behavior of rectangular columns with low lateral confinement ratio

H. YIN, P. IRAWAN, T.C. PAN and C.H. LIM Behavior of full-scale lightly reinforced concrete interior beam-column joints under reversed cyclic loading

C.H. HAMILTON, G.C. PARDOEN, R.P. KAZANJY and Y.D. HOSE Experimental and analytical assessment of simple bridge structures subjected to near-fault ground motions

S.M. ELACHACHI, M. BENSAFI and D. NEDJAR Seismic response of reinforced concrete frames using nonlinear macro-element behaviour

E. ATIMTAY and M.E. TUNA Designing the concrete dual system

977

985

993

1001

1009

ANALYSIS AND DESIGN FOR BLAST AND IMPACT

K.H. LOW, K.L. LIM, K.H. HOON, A. YANG and J.K.T. LIM Parametric study on the drop-impact behaviour of mini hi-fi audio products

S.C.K. YUEN and G.N. NURICK Deformation and tearing of uniformly blast-loaded quadrangular stiffened plates

P. BIGNELL, D. THAMBIRATNAM and F. BULLEN Non linear response and energy absorption of vehicle frontal protection structures

F. du TOIT, K. COMN1NOS and P.J. KRUGER Non-linear design of blast/containment reinforced gunite walls for coal mines in SA

1019

1029

1037

1043

XX

R.D. KRATZ A philosophy for blast resistant design 1051

FIRE SAFETY AND FIRE RESISTANCE

P.J. MOSS and G.C. CLIFTON The effect of fire on multi-storey steel frames

W. SHA and N.C. LAU Fire safety design and recent developments in fire engineering

W. SHA, N.C. LAU and T.L. NGU Fire resistance of steel floors constructed with experimental fire resistant steels

K.S. AL-JABRI, I.W. BURGESS and R.J. PLANK The influence of connection characteristics on the behaviour of beams in fire

W. SHA and T.L. NGU Heat transfer in steel structures and their fire resistance

W. SHA and N.C. LAU Temperature development during fire in slim floor beams protected with intumescent coating

1063

1071

1079

1087

1095

1103

STRUCTURAL SAFETY AND RELIABILITY

S. CADDEMI, P. COLAJANNI and G. MUSCOLINO On the non stationary spectral moments and their role in structural safety and reliability

K. RAMACHANDRAN Developments in structuraI reliability bounds

J.O. AFOLAYAN and A. OCHOLI Isosafety parameters for rink-type steel roof trusses

J.O. AFOLAYAN Cost-modelling for the economic appraisal of joint details in steel trusses

1113

1121

1129

1137

STRUCTURAL OPTIMISATION

M. KAHRAMAN and F. ERBATUR A GA approach for simultaneous structural optimization

J. MAHACHI and M. DUNDU Genetic algorithm operators for optimisation of pin jointed structures

K.H. LOW and H.P. SIN Use of a stopper for the stress reduction in beam-block switching systems of audio products

1147

1155

1163

DAMAGE PREDICTION AND DAMAGE ASSESSMENT

M. ZHAO, Y. ZHAO and F. ANSARI Fiber optic assessment of damage in FRP strengthened structures

J.L. HUMAR and M.S. AMIN Structural health monitoring

C.L. MULLEN, P. TULADHAR, B. LeBLANC and S. SHRESTHA 3D seismic damage simulations for an existing bridge substructure using nonlinear FEM calibrated with modal NDT

H.A. RAZAK and F.C. CHOI Damage assessment of corroded reinforced concrete beams using modal testing

M.M. KHAN and G.J. KRIGE Evaluation of the structural integrity of an aging mine shaft

R. MASIH Problems in measuring strains in the remaining parts of partially demolished bridge

xxi

1175

1185

1195

1203

1217

1225

REPAIR, REHABILITATION AND STRENGTHENING

A. GHOBARAH Seismic rehabilitation of beam-column joints

C. ARYA, J.L. CLARKE, E.A. KAY and P.D. O'REGAN TR 55" Design guidance for strengthening concrete structures using fibre composite materials - A review

K.P. GROSSKURTH and W. PERBIX Force transmitting filling of wet and water flled cracks in concrete structures by means of crack injection with newly developed epoxy resins

M.K. RAHMAN Numerical simulation of moisture diffusion in a concrete patch repair

A.I. UNAY Analytical modelling of historical masonry structures for the evaluation of strength capacity of their vulnerable elements

1235

1243

1251

1259

1269

LOADINGS AND CODE DEVELOPMENTS

F. WERNER and P. OSTERRIEDER Actual problems of steel-design- Future of the codes 1279

xxii

T.R. Ter HAAR, J.V. RETIEF .and A.R. KEMP Calibration of load factors for the South African Loading Code 1289

J.V. RETIEF, P.E. DUNAISKI and P.J. de VILLIERS An evaluation of imposed loads for application to codified structural design 1297

A.M. GOLIGER, R.V. MILFORD and J. MAHACHI South African wind loadings: Where to go 1305

J.L. HUMAR and M.A. MAHGOUB Seismic design provisions based on uniform hazard spectrum 1313

P.E. DUNAISKI, H. BARNARD, G. KRIGE and R. MACKENZIE Review of provision of loads to structures supporting overhead travelling cranes 1321

A.M. GOLIGER and J.V. RETIEF Background to wind damage model for disaster management in South Africa 1329

CONCRETE AND CONCRETE MATERIALS

A.S. NGAB Structural engineering and concrete technology in developing countries: An overview 1339

T. YEN, K.S. PANN and Y.L. HUANG Strength development of high-strength high-performance concrete at early ages 1349

H.Y. LEUNG and C.J. BURGOYNE Compressive behaviour of concrete confined by aramid fibre spirals 1357

C.P. LAI, Y. LIN and T. YEN Behavior and estimation of ultrasonic pulse velocity in concrete 1365

C.W. TANG, K.H. CHEN and T. YEN Study on the rheological behavior of medium strength high performance concrete 1373

M.F.M. ZAIN, T.K. SONG, H.B. MAHMUD, Md. SAFIUDDIN and Y. MATSUFUJI Influence of admixtures and quarry dust on the physical properties of freshly mixed high performance concrete 1381

A.S. NGAB and S.P. BINDRA Towards sustainable concrete technology in Africa 1391

S. MOHD, C.K. WAH and P.Y. LIM Development of artificial lightweight aggregates 1399

F. FALADE A comparative study of normal concrete with concretes containing granite and laterite fine aggregates 1407

xxiii

A.R.M. RIDZUAN, A.B.M. DIAH, R. HAMIR and K.B. KAMARULZAMAN The influence of recycled aggregate on the early compressive strength and drying shrinkage of concrete

H.J. CHEN and H.C. CHAN Numerical prediction on the elastic modulus of aggregate

K. RAMACHANDRAN and A. KARIMI Estimation of corrosion time with observed data

1415

1423

1431

CONSTRUCTION TECHNOLOGY AND METHODS

A. KASA, F.H. ALI and N. NASIR Construction and instrumentation of a concrete modular block wall

B.B. BUDKOWSKA and J. YU Analysis of multilayer system with geosynthetic insertion - Sensitivity analysis

L. FLISS The scale pits of Saldanha Steel: An innovative solution to a complex problem

T.A.I. AKEJU and F. FALADE Utilization of bamboo as reinforcement in concrete for low-cost housing

J. KANYEMBA Enhancing housing delivery using a simple precast construction method

I. WEISER Process Chains: A base for effective project management

J.O. AFOLAYAN Analysis of placement errors of bars in reinforced concrete construction

1441

1449

1457

1463

1471

1481

1489

STRUCTURAL ENGINEERING EDUCATION

B.W.J. van RENSBURG Teaching structural analysis: A curriculum for an undergraduate civil engineering degree and learning issues

M.Y. RAFIQ and D.J. EASTERBROOK Interactive use of computers to promote a deeper learning of the structural behaviour

1497

1505

xxiv

LATE PAPERS

STEEL STRUCTURES: GENERAL CONSIDERATIONS

K.F. CHUNG and M.F. WONG Experimental investigation of cold-formed steel beam-column sub-frames: Enhanced performance 1515

CONCRETE STRUCTURES: GENERAL CONSIDERATONS

C.T. MORLEY and S.R. DENTON Modified plasticity theory for reinforced concrete slab structures of limited ductility 1523

MASONRY, GLASS AND TIMBER STRUCTURES

F.S. CROFTS and J.W. LANE Accidental damage on unreinforced masonry structures 1531

LAMINATED COMPOSITE PLATES AND SHELLS

S. MOHAMMADI and A. ASADOLLAHI A contact based dynamic delamination buckling analysis of composites 1539

BRIDGES, TOWERS AND MASTS

S.H. CHENG and D.T. LAU Modelling of cable vibration effects of cable-stayed bridges

S.H. CHENG and D.T. LAU Parametric study of cable vibration effects on the dynamic response of cable-stayed bridges

1551

1559

FINITE ELEMENT AND NUMERICAL MODELLING

T.C.H. LIU, K.F. CHUNG and A.C.H. KO Finite element modelling on Vierendeel mechanism in steel beams with large circular web opening 1567

CRACKING AND FRACTURE MECHANICS

R.A. FODHAIL Loading parameters at cracks and notches 1575

XXV

STABILITY OF THIN-WALLED MEMBERS

E.P. DJELEBOV Investigations on local stability of compressed wall of hollow reinforced concrete bridge pier with rectangular cross section

REPAIR, REHABILITATION AND STRENGTHENING

1583

S.A. EL-REFAIE, A.F. ASHOUR and S.W. GARRITY Strengthening of reinforced concrete continuous beams with CFRP composites 1591

CONCRETE AND CONCRETE MATERIALS

G.C. FANOURAKIS and Y. BALLIM Assessment of a range of design models for predicting creep in concrete

S.J. FICCADENTI Effects of cement type and water to cement ratio on concrete expansion caused by sulfate attack

VIBRATION CONTROL AND SEISMIC ANALYSIS

G.C. PARDOEN, R. VILLAVERDE, R. TAVARES and S. CARNALLA Improved modelling of electrical substation equipment for seismic loads

1599

1607

1615

Author Index S1

Keyword Index S 5

This Page Intentionally Left Blank

KEYNOTE PAPERS

This Page Intentionally Left Blank

Structural Engineering, Mechanics and Computation (Vol. 1) A. Zingoni (Editor) © 2001 Elsevier Science Ltd. All rights reserved.

VIBRATION ANALYSIS OF BRIDGES UNDER MOVING VEHICLES AND TRAINS

Y. K. Cheung, Y. S. Cheng and F. T. K. Au

Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, People's Republic of China

ABSTRACT

The vibration of bridges under moving vehicles and trains is of great theoretical and practical significance in civil engineering. This paper describes some recent developments in the vibration analysis of girder and slab bridges under the action of moving vehicles or trains. A bridge-track- vehicle element is developed for investigating the dynamic interactions among a moving train, and its supporting railway track structure and bridge structure. The effect of track structure on the dynamic response of the bridge structure and the effect of bridge structure on that of the track structure are identified. The proposed bridge-track-vehicle element can be easily degenerated to a vehicle-beam element, which is employed to study the effects of the random road surface roughness and the long- term deflection of concrete deck on the dynamic response of a girder bridge. The plate-vehicle strip for simulating the interaction between a rectangular slab bridge and moving vehicles is also described. Two kinds of plate finite strips, namely the plate-vehicle strip and the conventional plate strip, are employed to model a slab bridge. In the analysis, each moving vehicle is idealised as a one-foot dynamic system with the unsprung mass and sprung mass interconnected by a spring and a dashpot. A train is modelled as a series of moving vehicles at the axle locations. The efficiency and accuracy of the proposed methods are demonstrated by numerical examples.

KEYWORDS

Bridge vibration, moving vehicles, moving trains, finite element method, finite strip method

INTRODUCTION

The vibration of bridges under moving vehicles and trains is of great theoretical and practical significance in civil engineering, and it has attracted much attention during the last three decades. This is in part due to the rapid increase in the proportion of heavy vehicles and high-speed vehicles in the highway and railway traffic, and the trend to use high-performance materials and therefore more slender sections for the bridges. Vehicle-bridge interaction is a complex dynamic phenomenon depending on many parameters. These parameters include the type of bridge and its natural frequencies of vibration, vehicle characteristics, vehicle speed and traversing path, the number of

vehicles and their relative positions on the bridge, roadway surface irregularities, the damping characteristics of bridge and vehicle etc.

The moving force model, moving mass model and moving vehicle model are three essential computational models used to analyse the dynamic responses of bridges due to moving vehicles and trains. The moving force model is the simplest model with which the essential dynamic characteristics of a bridge under the action of moving vehicles can be captured, although the interaction between the vehicles and bridge is ignored. Where the inertia of the vehicle cannot be regarded as small, a moving mass model is often adopted instead. However the moving mass model suffers from its inability to consider the bouncing effect of the moving mass, which is significant in the presence of road surface irregularities or for vehicles running at high speeds. The advent of high-speed digital computer a few decades ago made it possible to analyse the interaction problem with more sophisticated bridge and vehicle models. The vibration of various types of bridges such as girder bridges, slab bridges, cable- stayed bridges and suspension bridges due to moving vehicles and trains can be studied by using a moving vehicle model, in which a vehicle is modelled as a single-axle or multi-axle mass-spring- damper dynamic system.

The analytical methods as described by Fryba (1999) may be used to solve problems involving simple structures. As these analytical methods are often limited to simple moving load problems, many researchers have resorted to various numerical methods such as finite element method (FEM), finite strip method (FSM) and structural impedance method (SIM) to analyse the interaction problems. Among various numerical methods for the class of problems, FEM is no doubt the most versatile and powerful. However, the use of refined meshes often gives rise to large matrix equations with comparatively large bandwidth. In this respect, FSM is particularly suitable for regular slab bridges, and this is especially true for simply supported rectangular slab bridges. Consequently, where applicable, FSM is cheaper than FEM for solutions of comparable accuracy. SIM is also efficient for the analysis of bridge-vehicle interaction problems, as one can treat the modelling of the bridge and the vehicles separately, and therefore changes in one part do not affect those in the other. In addition, modal superposition method does enable further reduction of the problem size.

The vibration analyses of girder bridges under moving vehicles or trains have been extensively investigated. Some of the more recent work includes Olsson (1985), Coussy et al. (1989), Wang et al. (1991), Huang et al. (1992), Yang & Lin (1995), Fryba (1996), Yang & Yau (1997), Henchi et al. (1997), Cheung et al. (1999a), Leet al. (1999), Yang et al. (1999) and Yau et al. (1999). It is noted that, in most of the previous studies on railway bridge vibration, the effects of the track structure have been completely neglected or only partially accounted for.

It is inadequate to model wide slab bridges using beam models, particularly when the vehicle paths are not along the centre-line of the bridge. The vibration of slab bridges modelled as isotropic or orthotropic plates under the action of moving loads has so far received but scant attention. Wu et al.

(1987), Taheri & Ting (1990), Yener and Chompooming (1994), Humar & Kashif (1995) etc. separately used the FEM to analyse the vibration of plates under moving vehicles. Taheri & Ting (1989) used SIM to study the dynamic response of guideways by treating the moving loads and the guideway as components of an integrated structural system. Cheung et al. (1999b) also utilised SIM to solve a similar class of problem, but FSM was employed to obtain the influence function. By using the method of modal analysis, Wang & Lin (1996) analysed the vibration of multi-span Mindlin plates under a moving load.

In this paper, a bridge-track-vehicle element is proposed for investigating the dynamic interactions among a moving train, and its supporting railway track structure and bridge structure. One of its degenerated versions, the vehicle-beam element, is employed to investigate the effects of the random road surface roughness and long-term deflection of concrete deck on the dynamic response of a girder

bridge. A plate-vehicle strip is reported for simulating the interaction between a rectangular slab bridge and the moving vehicles. The efficiency and accuracy of the proposed methods are demonstrated by numerical examples.

THEORY AND FORMULATION

The methods described in this paper are based on the Bemoulli-Euler beam theory and Kirchhoff thin- plate theory where applicable. The bridges under consideration are treated as plane structural systems while each vehicle is modelled as a one-foot dynamic system, in which the unsprung mass and sprung mass are interconnected by a spring and a dashpot. A moving train is modelled as a series of moving vehicles at the axle locations.

The Bridge- Track- Veh&le Element

Figure 1 shows a typical bridge-track-vehicle element with a few vehicles running on it. The upper and lower beam elements modelling the rail and the bridge deck respectively are interconnected by a series of springs and dampers, which reflect the properties of the rail bed. It is assumed that there are Nv moving vehicles in direct contact with the upper beam element and Np spring-damper systems between the upper beam element and the lower beam element. The ith vehicle proceeds with velocity v~(t) and acceleration at(t) in the longitudinal direction. The stiffness of the spring and the damping coefficient of the dashpot of the typical ith vehicle are denoted by k~ and c~ respectively, the unsprung mass is denoted by m~n and the sprung mass is denoted by m,,,2 where i =1,2 ..... Nv. The stiffness and damping coefficients of the typical jth spring-damper system between the upper and lower beam elements are respectively kpj and Cpj where j = 1,2,..., Np.

[ ~ mvi2

mvil

I; x c , _

x

Figure 1: A typica l b r i dge - t r ack -veh i c l e e l emen t

It is assumed that the upward deflections of rail and bridge deck are taken as positive and that they are measured with reference to their respective vertical static equilibrium positions. Let rc(x) denote the top surface irregularities of rail that is defined as the vertically upward departure from the mean horizontal profile. The vector {re} contains the values of the surface irregularities of the rail at the contact points Xc~ (i=1,2 ..... Nv) between the rail and the vehicles. The vertical displacements of the masses mvn and mva are y~l and y~2 respectively, and they are measured vertically upward with reference to their respective vertical static equilibrium positions before coming onto the bridge.

A matrix [Nc] of dimension 4xNv is then defined such that it contains the cubic Hermitian interpolation functions for the beam element evaluated at the contact points xc~ as follows

[N~]=[{N<}, { N c } , ... {Nc},<] (1)

The vector {Nc}, in the matrix [Nc] is given in terms of the vector {N(x)} as

I1 - 3(x lZ) ~ + 2(x/Z) ~ ]1 Jx[,

{Arc }j = {N(x) l ..... = [ 3 ( x l t ) = - 2(x/t) It L xTxlO -(xl t) Jlx -" Xci

i=1,2 ..... N,, (2)

where l is the length of the beam element.

In the case of discrete spring-damper support systems, a matrix [Ncp] of dimension 4xNp, similar to the matrix [Nc], is introduced and it contains the cubic Hermitian interpolation functions for the beam element evaluated at the positions Xcpj (j=1,2 .... ,Np) of the spring-damper systems between the rail and the bridge deck as follows

[N~p]=[{Ncp}~ {N~p}2 ... {N~p}N]=[{N(xcp~) } {N(xcp2) } ... {N(XcpN,)} ] (3)

One may establish the equations of motion for the vehicles, the upper beam element and the lower beam element respectively. These equations are coupled through the unknown contact forces between the moving vehicles and the upper beam element, and the supporting reactions acting through the spring-damper systems between the rail and the bridge deck. Using the constraint equations at the contact points in terms of the displacements, velocities and accelerations at those points, expressing the supporting reactions in terms of the displacements and velocities of upper and lower beam elements at the locations of spring-damper systems and eliminating the degrees of freedom (DOFs) of unsprung masses, one can obtain the equation of motion for a bridge-track-vehicle element with discrete rail supporting systems as follows (Cheng, 2000) using the conventional notations:

[ [ i l ] [0] [0] ]f{//bi } IEcb]+[N~p]Ecp][N~p]T _EN=p][%][N=p]T [o] ]['{Ub}]

[0] [m.lJ[{j~2 [o] [c,,] [c,]J I.{J,~iJ [kb]+[N<p][kp][N<p] T -[N<p][kp][N<p] T [O] ] [ {Ub}] f {Pb} ]

_[Nw][kp][N,p] T [kr]+[ki i ]+[N,p][kp][N,p] T [k,,:l]i {u,/i> = <i {p,/+ {/,/i> (4) [0] [k2,] [k~]Jl.{y2} j [. {£,} J

[roll ] -[N~][m~][N~] T, [c,,] = 2[Nc][mv~][v][Nc].r~ +[Nc][C.][Nc] r (5,6)

[C12 ] = _[Nc ][cv ] , [c21 ] ._. _[cv ][Nc ]T (7,8)

[k,,]=[N~][m~,][a][N~].rx +[N~][m~,][v]2[N~].~ +[Nc][C~][v][Nc] T,~ +[N¢][k.][N~] T (9)

[k,2 ] = -[N c ][k~ ], [k2, ] = -[c, ][v][N c ].r x -[k~ ][N c ]T (10,11)

{fb } = -[N~]({fw } + [k.] {r~} +[c.][v]{rc},~ +[m~l][a]{r~}.x + [m~l ][v] 2 {rc }.~x) (12)

{f~2} = [cv][vl{rc}.x + [kv]{r~}, {fw} = {(my,, + mv,2)g} (13,14)

[mv,]=diag[m,,,, ], [m,,2]=cfiag[mv,2J,[c,,]=diag[c,,,],[k,,]=diag[k,,~] /--1,2 ..... N,, (15-18)

[Cp] = V~ag[cpj ] ,[kp] = diag[kpj ] j= l ,2 ..... Np (19,20)

{Y2 } = {Y,2 }, [a] = diag[a, ], [v] = c~ag[v, ] i = 1,2 ..... Nv (21-23)

Matrices [mb], [Co] and [kb] are the mass, damping and stiffness matrices, respectively, of the lower

beam element representing the bridge deck, the vectors {u b }, {fib } and {//b} are the corresponding

nodal displacement, velocity and acceleration vectors, respectively. Matrices [m r ], [Or] and [k~] are the mass, damping and stiffness matrices of the upper beam element representing the track structure respectively, and vectors {dr} , {dr} and {//,} are their nodal displacement, velocity and acceleration

vectors, respectively. The vector {Pb } is a force vector to account for any external load applied to the

bridge deck apart from the reactions through the spring-damper systems between the rail and the bridge deck. The vector {Pr} is any other external load vector acting on the track structure apart from the reactions and contact forces.

In the case of a rail supporting system with continuous distributed stiffness kp (x) and damping Cp (X),

the terms [Ncp][kp][Ncp] T and [Ncp][Cp][Ncp] T in Eqn. 4 should be replaced by ~ { N } k p ( x ) { N } ~ dx

and a0[t {N}cp(x){N}T dx respectively in which the vector {N(x)} has been defined in Eqn. 2.

A bridge-track-vehicle element can be easily degenerated into a bridge-track element that applies to the rest of the bridge With no moving vehicles on it. It can also be degenerated into special cases of a beam on viscoelastic foundation or simply a beam, with or without the moving vehicles on it. The equation of motion for the vehicle-beam element is obtained by deleting sub-matrices associated with the DOFs of the lower beam element and the rail supporting systems, i.e.

[[mr]~-[mll] [O]l~{i~r}}}~_[[Cr]-~[Cll] [Cl2]l~{~r}~rI[kr]-]-[kll] [k12]lf{~r}~._.~{Pr}-~-{fr}}(24 ) [o] [m~]J[{¢~ [c~,] [c.]][{p~}J [k~,] [k~]Jl{y~}J L {f,,2}

The Plate-Vehicle Strip

A slab bridge is modelled as a rectangular thin plate simply supported at two opposite edges and is subjected to N moving vehicles proceeding with velocity v(t) in the longitudinal direction. The bridge is first split up into a number of rectangular finite strips. The conventional plate finite strips can be employed for those strips not carrying any moving vehicles. However the strips carrying vehicles have to be separately treated.

Figure 2 shows a typical plate-vehicle strip with width b and length l, where the surface irregularity of the bridge is also indicated. It is also assumed that there are Nv moving vehicles in direct contact with the strip. Let w(~,rl, t) denote the vertical deflection at time t of the point (~:,q) of a plate strip (upward positive), which can be expressed as

w(~, r/,t)= £ {N} T {u b }. = {N} v {u o } (25) m=l

where r is the number of terms in the series used in the longitudinal part of the displacement function of a strip, and {N},, and {Ub}m are, respectively, vectors containing the shape functions and nodal line displacement parameters for the mth term of the displacement function. Obviously, {N}m is a function of the position (~:,r/) on the strip and {Ub}m is a function of time t. The vectors {N}m and {Ub}m are given in terms of the typical ith element as

{N},,, = {N~}, {Ub}m={Ub, } /=1,2 ..... sxn~ (26,27)

in which s is the number of nodal lines in a strip and ne is the number of DOFs associated with each nodal line. The vectors {N}m and {Ub}m are given by Cheung & Tham (1998) for various plate finite strips. For a lower order (LO2) mixed plate finite strip, the vectors {N},, and {Ub}m appear respectively as

[[1- 3(~/b) 2 + (~/b)3 ] Y~m (r/)] IWlm (/)] =~[1-2(~/b)+(~/b)Z]Yl,(rl)[ = ~0'm(')~

{g}m [ [3(~:/b) 2 -2(¢/b)3]yzm(17) [ ' {ub} m ]W2m(t)[ [ ~[(~/b)2-(~/b)]Y2m(O) J [O2m(t)]

(28,29)

where Yjm(r/) is the longitudinal displacement function associated with the nodal line j (j= 1,2 for LO2 strip) of the strip for the mth term of the series; wire(t) and 0jm(t) are the deflection and rotation parameters at nodal linesj of the strip for the mth term of the series, respectively.

Let ~:~ = bt and rki = rlc~(t) denote the transverse and longitudinal co-ordinates, respectively, of the ith vehicle. One may then define a matrix [N~s] of dimension (rsne)xNv containing the displacement functions at the positions of the vehicles as follows

i I ii1,, I N = l = ... . . . . . . . . o .

(30)

The sub-matrix [Ncs]q is given in terms of the typical element as

[Nc,]o= {N(2]~j, r/cj ) i=1,2 ..... r ; j=l ,2 ..... N~ (31)

The equation of motion for a plate-vehicle strip can be obtained similarly. Following the sign conventions adopted in the bridge-track-vehicle element and replacing the mass, damping and stiffness matrices of the beam element as well as the matrix [Arc] defined by Eqn. 1 with the counterparts of the plate strip, the equation of motion for a plate-vehicle strip has the same form as that for a vehicle-beam element given by Eqn. 24. The plate-vehicle strip can therefore be taken as the extension of the vehicle-beam element to the two-dimensional case.

t rl, y

rlc,(t)

l~cNv(O

b~ ~ bi

O vehicle i

vehicle Nv 0

O vehicle 1

I~" "~' bN~.._ I S "-'

kvi

mvi2

Cvi

mvil

~,X

Z, W mvl2 ~mv ,2~ . ~mvNv2

L v • v

~,x

Figure 2: A typical plate-vehicle finite strip

NUMERICAL EXAMPLES

A Three-span Continuous Girder Bridge under a Moving Vehicle

The vehicle-bridge element is employed to analyse the dynamic response of a three-span continuous girder bridge with equal span length l of 30m. The material properties of the bridge are, respectively, the mass density p=2400kg/m 3, Young's modulus E=30GPa, cross sectional area A=2.26m 2 and second moment of area of cross section I=0.667m 4. The velocity ratio a is defined as a=vn/coll, where v is the speed of vehicle and col is the fundamental frequency of the bridge, which is 21.068rad/s. The damping effect of the bridge is taken into account by assuming Rayleigh damping of 2%. The moving vehicle is modelled as a one-foot dynamic system for the study, comprising a sprung mass of 31800kg, supported by a spring of stiffness of 9.12x106N/m and a dashpot of damping coefficient of 8.6x104 Ns/m.

The random road surface roughness and long-term deflection of concrete deck are considered in the study. The former is described by a zero-mean stationary Gaussian random process while the latter is represented as a kind of global roadway surface roughness (Cheng, 2000). The Monte Carlo method is used in the simulation. 20 profiles of random road surface roughness are generated using the cut-off spatial frequencies cot = 0.01 cycles/m and cou = 3.0 cycles/m for general good road surface. Statistics are computed for each set of results obtained for a certain vehicle velocity. They include the mean and the standard derivation (SD). For simplicity, the maximum long-term deflection rmax of a span of length l~ is taken as rmax = 0.0016 ls. Four schemes of deck profiles are studied. They include the

2.2

2.0

1.8

• ~ 1.6

.~ 1.4

~ 1.2

"i 1.0 ~ 0.8

perfectly straight and smooth deck (Scheme 1), the deck with long-term deflection only (Scheme 2), the deck with random road surface roughness only (Scheme 3) and the deck with both long-term deflection and random road surface roughness (Scheme 4).

In the analysis, only two kinds of elements are required to simulate the bridge-vehicle system. They are the beam elements and the vehicle-bridge elements. The problem was solved by FEM with 48 elements of equal lengths and 1500 equal time steps for the range of velocity ratio 0.06<a~_0.20. The dynamic magnification factors Dd for displacement and Dm for bending moment at the mid-point of the bridge are shown in Figures 3(a) and 3(b) against the velocity ratio a.

10

0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

1.8

1.6 .

• ~ 1.4

t~ "~ 1.2

~ 0 . 8

0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

Velocity ratio ot Velocity ratio ot

(a) (b)

Figure 3: Three-span continuous girder bridge: dynamic magnification factors (a) Dd for displacement; and (b) Dm for bending moment at middle of Span 2. ~ , Scheme 1" [] , Scheme 2; ~ • Scheme 3 (Mean + SD); X , Scheme 4 (Mean + SD).

It is observed that the effects of random road surface roughness on impact are significant while those of long-term deflection of concrete deck are not as significant. In general, the dynamic magnification factors for displacement are higher than those for bending moment. The effects of the long-term deflection of concrete deck generally become more obvious in the higher range of velocity ratio. However the effects may be adverse or beneficial depending on the velocity ratio.

A Single-Span Simply Supported Railway Bridge under a Moving Train

A single-span simply supported prestressed concrete railway bridge with the two approaches supported on embankments as shown in Figure 4 is considered to investigate the interaction among the train and the supporting track and bridge. The railway track is continuous throughout. The top surface irregularities of rail have been ignored in this example.

The span of the bridge is Lb=20m. For the analysis, the length of track structure taken into account on each approach embankment is assumed to be twice the bridge span, i.e. Lel=Le2=4Om. The flexural rigidity per rail is 4.3x 106Nm 2 while the mass per unit length of one rail is 51.5kg/m. Both the discrete model and the continuous model have been investigated in the modelling of the rail bed. In the discrete model, the spring-damper systems are at a regular spacing of 0.625m. The stiffness and damping coefficient of each spring-damper system underlying one rail are respectively 41125kN/m and 20062.5Ns/m. In the continuous model, the stiffness and damping coefficient of the rail bed for one rail are 65800kN/m 2 and 32100Ns/m 2 respectively. In the analysis, all the rail properties have to be doubled to account for the presence of both rails.

The material properties of the bridge are, respectively, the mass per unit length pA=34088kg/m, Young's modulus E=29430MPa and second moment of area of cross section I=3.81m 4. A train consisting of five cars, as shown in Figure 5, runs over the bridge. Each car can be considered as comprising two vehicles, as shown in Figure 5(b). Each wheel assembly is modelled as an equivalent

11

2-DOF system with unsprung mass ml of 4400kg, sprung mass m2 of 17600kg, spring stiffness k of 9.12x106N/m and damping coefficient c of 8.6xl04Ns/m. The arrangement of wheel assemblies is defined by the parameters Lc=l 8m and Ld=6m.

Figure 4" A typical b r idge- t rack-veh ic le system

(a) Io oHo oHo nHo oHo ol

Figure 5" A moving train and its mathematical models. (a) Plan; (b) Moving vehicle model

The bridge-track-vehicle element and its degenerated versions can be used to simulate the dynamic interaction among a moving train, the track on bridge and the track on embankment. The bridge-track- vehicle elements or the bridge-track elements are used for the bridge as appropriate. The portion of track on embankment is modelled as a beam on viscoelastic foundation with or without vehicles as appropriate. Both the discrete and continuous models for the rail bed have been investigated. To investigate the effect of the vibrating bridge on the dynamic response of the railway track, the problem has also been analysed by replacing the bridge-track elements for the bridge deck by elements for beam on viscoelastic foundation.

-~ 7,0

~ 6 . 0 o

5.0

"~ 4.0

3.0

2.0

~ 1.0

~ 0.0 i i , i i I i i i i i i

0 20 40 60 80 100 120 140 160 180 200 220 240

Train velocity (m/s)

7.0

6.0

5.0 g

"~ 4.0

~ 3.0

2.0 .o I~ 1.0

0.0 | | | | | i | | | , |

0 20 40 60 80 100 120 140 160 180 200 220 240

Train velocity (m/s)

(a) (b) Figure 6: Single-span simply supported bridge under a train modelled as moving vehicles: dynamic magnification factors (a) Dd for displacement; and (b) Dm for bending moment at mid-span. , Track structure ignored; C) , track structure on rail bed with discrete supports;/x , track structure on rail bed with continuous support.

Figures 6(a) and 6(b) show the dynamic magnification factors Dd for displacement and Dm for bending moment respectively at mid-span of the bridge against train velocity. It can be observed that the effect of track structure on the dynamic response of the bridge is insignificant whether the rail bed is represented by a discrete or continuous model. This can be somehow explained by the much smaller

12

flexural rigidity and mass of rails compared to those of the bridge deck. However, the same is not true for the effect of the vibrating bridge on the dynamic response of the rails. Dynamic magnification factors can also be worked out for the railway track. Figures 7(a) and 7(b) show, respectively, the dynamic magnification factors Dd for displacement and Dm for bending moment of the railway track at position A, mid-span of the bridge, against the train velocity. For comparison purposes, the case ignoring bridge vibration is also worked out. Both the discrete and continuous models for the rail bed have been investigated.

"~ 3.5

3.0

2.5 ._~ ~ 2.0

"E 1.5

E 1.0 .u _ 0.5

~ 0.0 I i i I i i i i I i i

0 20 40 60 80 100 120 140 160 180 200 220 240

Train velocity (m/s)

(a)

~1.1

1.0

0.9

0.8 ..~

1~0.7

..- -El

| I , | | | | , | , i

0 20 40 60 80 100 120 140 160 180 200 220 240

Train velocity (m/s)

(b)

Figure 7: Railway track under a train modelled as moving vehicles: dynamic magnification factors (a) Dd for displacement; and (b) Dm for bending moment at A. , Track structure on rail bed with discrete supports and bridge considered; . . . . . , track structure on rail bed with discrete supports and bridge ignored; A , track structure on rail bed with continuous support and bridge considered; [] , track structure on rail bed with continuous support and bridge ignored.

It is noted from Figure 7(a) that the dynamic magnification factor for displacement Dd of the railway track at position A for the entire track-bridge system considered is much higher when the bridge vibration is taken into account than that when the entire track is considered as a beam on viscoelastic foundation. This is particularly noticeable for the velocities around resonant points shown in Figure 6(a). However the effect of bridge vibration on the dynamic magnification factor for bending moment Dm of the railway track at position A is marginal, as shown in Figure 7(b). This may be explained by the fact that the railway track vibrates together with the bridge, giving rise to large deflections with reference to the static equilibrium position of the railway track and therefore higher dynamic displacement magnification factor. In spite of large deflection, the bending curvature has remained low, which explains the relatively low dynamic magnification factor for bending moment.

A Slab Bridge Modelled as an Isotropic Plate under a Moving Vehicle

A simply supported slab bridge modelled as an isotropic rectangular plate with length lb of 20m, width bw of 10m and thickness h of 0.525m is considered. The material properties of the bridge are respectively the mass density p=2450kg/m 3, the Young's modulus E=2.85x 107kN/m 2 and the Poisson's ratio/t=0.15. The fundamental frequency of the bridge without the vehicle is col=12.79rad/s. For comparison with results given by Humar & Kashif (1995), the vehicle is modelled as a single mass supported by a spring only. In other words, the unsprung mass and dashpot in the derivation are ignored. The three dimensionless parameters chosen are the speed parameter a=vx/cOl, the vehicle to bridge mass ratio x=m2/plbwh and the vehicle to bridge frequency ratio ~=COv/CO, where the natural frequency of vehicle COv equals "xl(k/m2).

The bridge-vehicle system was solved by the present method using 10 harmonic terms and 10 strips of equal width with 300 equal time steps for a moving vehicle travelling along the centreline of the bridge

13

o= 1.4 .= ~ 1.3

~ 1.2 .,.

,'o

• -~ 1.0

o 0.9 Z

0.0

i i i i i

0.2 0.4 0.6 0.8 1.0

Vehicle to bridge frequency ratio ¢

Figure 8: A slab bridge modelled as an isotropic plate under a vehicle moving along the longitudinal centreline with h-=0.5 and a=0.15, normalised midpoint deflection. , Present; o , Humar and Ksshif (1995); El, Cheung et al. (1999b).

for the following parameters: a=0.15, x=0.5 and 0.1<~ <1.2. In the presentation of results, the midpoint dynamic deflection is normalised by the corresponding static response at the same point due to the action of a vehicle of mass m2 located at the centre of the bridge. Figure 8 shows the normalised maximum dynamic midpoint deflection plotted against the parameter ~. Good agreement is observed between the present results and those obtained using SIM (Cheung et al. , 1999b). However some discrepancies are observed between the present results and those from Humar & Kashif (1995), which may be due to the use of an unspecified coarse mesh.

CONCLUSIONS

This paper presents the vibration analysis of girder and slab bridges under moving vehicles and trains. A bridge-track-vehicle element has been developed for analysing the vibration of railway bridges under a moving train taking into account the response of the track structure. Numerical results from analysing the entire bridge-track-vehicle system show that the effect of vibrating track structure on the dynamic response of the bridge is insignificant. However, the effect of bridge vibration on the dynamic response of railway track on the bridge is considerable.

The bridge-track-vehicle element can be easily degenerated into a bridge-track element and a beam element on viscoelastic foundation or simply a beam, with or without the vehicles. The vehicle-beam element, one of its degenerated versions, is employed to investigate the effects of the random road surface roughness and long-term deflection of concrete deck on the dynamic response of a girder bridge. The effects of random road surface roughness are considerable while those of the long-term deflection of concrete deck are small to moderate.

A plate-vehicle strip has been proposed for analysing the vibration of slab bridges under moving vehicles. This plate-vehicle strip comprises a conventional plate finite strip with the moving vehicles on it. It can be taken as the extension of the vehicle-beam element to the two-dimensional case. The numerical results obtained from the proposed method agree well with available results.

ACKNOWLEDGEMENT

The financial support of the Hong Kong Research Grants Council is acknowledged.

14

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