Structural Engineering, Mechanics and Computation : SEMC 2001 (2 Volume Set). 9780080541921, 0080541925

Table of contents :
Front Cover
Structural Engineering, Mechanics and Computation
Copyright Page
Foreword
Preface
Local Organising Committee, International Scientific/Technical Advisory Board, Sponsoring Organisations
Contents
PART I: KEYNOTE PAPERS
PART II: INVITED PAPERS
PART III: STEEL STRUCTURES: GENERAL CONSIDERATIONS
PART IV: CONCRETE STRUCTURES: GENERAL CONSIDERATIONS
PART V: STEEL-CONCRETE COMPOSITE CONSTRUCTION
PART VI: MASONRY, GLASS AND TIMBER STRUCTURES
PART VII: PLATE, SHELL AND CONTAINMENT STRUCTURES
PART VIII: LAMINATED COMPOSITE PLATES AND SHELLS
PART IX: BRIDGES, TOWERS AND MASTS.

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STRUCTURAL ENGINEERING, ME CHANIC S AND

COMPUTATION Proceedings of the International Conference on Structural Engineering, Mechanics and Computation

Volume 1

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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 AMSTERDAM

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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

• •

J.M. Ko, Hong Kong Polytechnic University,

•

J.B. Obrebski, Warsaw University of Technology,

China Poland • • • •

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,

• • • • •

• •

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

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

South Africa

Africa • •

A. Ghobarah, McMaster University, Canada A. Nowak, University of Michigan, USA N.M. Hawkins, University of Illinois at UrbanaChampaign, USA U. Schneider, Technical University of Vienna,

Austria

V. Tvergaard, Technical University of Denmark,

Denmark •

Y. Ballim, University of the Witwatersrand, South

Africa

South Africa •

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 •

Africa • •

•

R. Zandonini, University of Trento, Italy S. Shrivastava, McGill University, Canada V. Marshall, 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 DurbanWestville, 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 f o r Theoretical and Applied Mechanics (SAAM)

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CONTENTS VOLUME 1

Foreword Preface

vii

Local Organising Committee, International Scientific~TechnicalAdvisory 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

XV

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

525

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

533

LAMINATED COMPOSITE PLATES AND SHELLS

H. MATSUNAGA Vibration of cross-ply laminated composite plates

541

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

549

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

557

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

565

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

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

583

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

591

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

599

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

607

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

613

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

621

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

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 E L E M E N T AND NUMERICAL M O D E L L I N G

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

755

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

763

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

771

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

779

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

787

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

793

Author Index

S1

Keyword Index

$5

VOLUME 2

Foreword Preface

vii

Local Organising Committee, International Scientific~TechnicalAdvisory Board, Sponsoring Organisations STABILITY OF THIN-WALLED MEMBERS

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

803

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

811

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

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

845

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

853

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

861

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

867

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

875

VIBRATION CONTROL AND SEISMIC ANALYSIS

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

887

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

897

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

905

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

913

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

919

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

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

937

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

947

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

955

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

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

977

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

985

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

993

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

1001

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

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

1019

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

1029

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

1037

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

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

1063

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

1071

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

1079

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

1087

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

1095

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

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

1113

K. RAMACHANDRAN Developments in structuraI reliability bounds

1121

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

1129

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

1137

STRUCTURAL OPTIMISATION

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

1147

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

1155

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

1163

xxi DAMAGE PREDICTION AND DAMAGE ASSESSMENT

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

1175

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

1185

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

1195

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

1203

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

1217

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

1225

REPAIR, REHABILITATION AND STRENGTHENING

A. GHOBARAH Seismic rehabilitation of beam-column joints

1235

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

1243

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

1251

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

1259

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

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

1415

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

1423

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

1431

C O N S T R U C T I O N T E C H N O L O G Y AND M E T H O D S

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

1441

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

1449

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

1457

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

1463

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

1471

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

1481

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

1489

STRUCTURAL ENGINEERING E D U C A T I O N

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

1497

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

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

1551

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

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

1583

REPAIR, REHABILITATION AND STRENGTHENING

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

1599

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

1607

VIBRATION CONTROL AND SEISMIC ANALYSIS

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

1615

Author Index

S1

Keyword Index

S5

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KEYNOTE PAPERS

This Page Intentionally Left Blank

Structural Engineering,Mechanicsand Computation(Vol. 1) A. Zingoni(Editor) © 2001 ElsevierScienceLtd. All rights reserved.

VIBRATION ANALYSIS OF BRIDGES UNDER M O V I N G 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-trackvehicle 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 longterm 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, cablestayed 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-springdamper 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 F O R M U L A T I O N The methods described in this paper are based on the Bemoulli-Euler beam theory and Kirchhoff thinplate 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;

xc,

x

_

F i g u r e 1: A typical b r i d g e - t r a c k - v e h i c l e e l e m e n 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 Id21 > ,,. >-- I kl >- ,.. > Idnl. The matrix D is decomposed into a diagonal matriX E with diagonal elements 1 o r - 1 and a matrix Do > 0. D

= EDo

with

Do>0

(17)

Assume that U T can be decomposed into the product of a left triangular matrix L with diagonal elements 1 and a fight triangular matrix P : UT=

LP

(18)

Equations (17) and (18) are substituted into (16). The matrix product A s is then expressed in terms of an auxiliary matrix Cs: As = UE sC sD sa

(19)

C s .=

DsLDo s

(20)

Cim =

eim \dmo]

(21)

With increasing s, Cs converges to a left triangular matrix L, since [dio I < [dmo 1" D

__.

D1

El ~.I

E= - ~.I

I I

~= - I

D2

E2

L1

1 L21 L22 I

68 lim Cs = L S--~O0

(22)

Tile product U[, is decomposed into an orthogonal matrix H and a fight triangular matrix G. This leads to the following limits of A s in (19)" U[,

= HG

with H T H = I

lim A s = ( U E s U T H ) ( G D s p ) S---~O0 lim I~ s =

=

(23)

lim (~sfis S...-I,O0

U E s U TH

lira £1s = G D s P S---P~

(24)

orthonormal matrix

(25)

right triangular matrix

(26)

In the limit, I~ s = Qs-1 Qs in (12)leads to"

(27)

(~s = W E W T W'=

HTu

with W T W = I

(28)

Similarly, R s R s - 1 in (13) leads to" R s = G D o G -1

(29)

G

(30)

= WL

In the limit, the matrix As = Qs Rs has the following structure" As

= wEWTGDoG

-1 = W E L D o L - l W ' r

As = W D W T

(31)

I

Dl

All A12

WI1 W12

W

=

As =

W21 W22

A21 A22

D2 All

(32)

= ~, (Wll wT1 -- W12 wT2)

A12 = ~.(Wll w T 1 -

(33)

W12wT2 ) -- A~I

(34)

A22 = ~ (W21 wTI -- W22 wT2)

The eigenvalues of A determine the convergence of As as follows : Dl

D1 D2

D1

-kI

kI D2

all values distinct

p-fold value ~,

D2 q-fold value - ~.

If A has a p-fold eigenvalue ~, and a q-fold eigenvalue -X, then the iteration converges to the nondiagonal matrix As in (31). Since the diagonal submatrix of As is orthogonal, it is readily diagonalized by Jacobi rotations.

69 Iteration with interchanges : Assume that Prom becomes zero in the decomposition of U T in (18) and

that consequently ,~im cannot be computed. The eigenvalues d i and dm are then interchanged in D, transforming D to D and U to U. The value of eimin the decomposition of ~T is zero. Since U is regular, interchanges will always yield a decomposable matrix ~T. The matrix Cs for 1~ and U is defined in analogy to (20). If d i and dm have been interchanged, eim--0 so that Cs converges to L even though dio > dmo in (21). The limit value of As is now WI~ W T , so that the eigenvalues after convergence of the iteration are ordered as in I~. If the bounds for the convergence of the iteration are strict, roundoff-errors will force convergence to D instead of D. E x a m p l e : Decomposition of U T with column interchange

Consider a matrix A with eigenvalue matrix D and eigenmatrix U : 1.3

0.5 0.6

O -

U -

0.4

0.5

0.5

0.5

-0.5 -0.5

0.5

0.5

0.5 -0.5

0.5

0.5 -0.5 -0.5

0.5

-0.5

0.1 The matrix U T cannot be decomposed into L P"

2~

3---.

0.5

-0.5 -0.5

0.5

0

~:,,:.!~,~!,.: 1.0 :~ .,,~

-1.0

0

0

0

0

0

1.0

0

0

0

0.5 -0.5 -0.5

1.0

1.0

0

0

0.5 -0.5

lO

0

0.5

0.5 -0.5 -0.5

1.0

0.5

0.5

l O

0.5

0.5 -0.5

0.5

0.5

UT

The diagonal coefficient P22 is zero. The coefficient e32 cannot be computed. The eigenvalues d 2 --0.6 and d 3 = 0.4 as well as the corresponding columns of U are therefore interchanged. The resulting matrix ~T can be decomposed" 0.5

3 -----~

-0.5 -0.5

0.5

0

1.0

1.0

-1.0

0

0

1.0

-1.0

0

0

0

2.0

1.0

0

0

0

0.5 -0.5 -0.5

1.0

1.0

0

0

0.5

1.0

0

0.5 -0.5

0.5 -0.5

1.0

1.0

0.5

0.5

1.0 1.0

1.0

0.5

0.5 -0.5 -0.5

0.5

0.5

b

70 4

ALGORITHM

Iteration : The eigenvalues of A are computed by iteration. In each cycle s, matrix A is decomposed according to section 2. The angle 0 of rotation matrix Sip is stored in location (i, p) of A. Using these angles, A is recombined according to expression (10) in section 2. The convex profile is preserved. If the convergence test after recomposition of A fails, the iteration is continued. Otherwise, A is deflated and iteration continues with the next eigenvalue. Convergence test : Coefficient ann of A s is an eigenvalue of A if the off-diagonal coefficients in row and column n are zero. Since the eigenvalue may be multiple, the diagonal coefficients att ..... ann with equal absolute value are determined. In each of the columns t ..... n of A s the sum of the squares of the elements above row t is computed. If one of these sums exceeds a specified limit, the iteration has not converged. Otherwise, the submatrix in rows and columns t to n is reduced to diagonal form by Jacobi rotations. The iteration has converged. Deflation of A during iteration : Assume that the off-diagonal elements in row and column n of A s are zero and that ann = k. Let the eigenvalue spectrum of A s be shifted by )~. This drives row and column n of A s to zero. The matrix A s is deflated to A by stripping column and row n. If d is an eigenvalue of A s , then d - k is an eigenvalue of A. Convergence of the iteration on A is improved by the shift, since (dio - ~,) / (dmo - )~) < dio / dmo in (21). Partial shifts can be used to accelerate convergence of the iteration.

As

o

--0T

det(A s - dl)

~.

= (d - k) det,~

= 0

(35)

Eigenvectors ofA : The eigenmatrix U is the limit of 0 s in (12). The computational effort to determine Qs Is 0[sb n2], where b is the band-width of A. To reduce this effort, Qs is not accumulated when the eigenvalues are computed in the algorithm. Instead, the eigenvalues corresponding to a q-fold eigenvalue k are computed with the shifted matrix A - k I. This matrix has a q-fold eigenvalue zero. The columns t = n - q + 1 to n of Q in the decomposition Q R o f A - k I contain the eigenvectors k. Therefore only the last q columns of Q in (8) are computed by evaluating the matrix product in (36) from right to left. E =

B :=

S21 ... S~,ll Sn, n_ 1 B

1

n

el 1

(36)

71 The operations for a typical multiplication are" k 13 = S i p B

p --D

i --,

p

i

1

i

p--+

1 1

i-+

1

1

Sip -

fi

^

bpk =

(37)

bpk cos 0 -I- bik sin 0

^

bik

= - bpk sin 0 + bik cos 0

The eigenvector can be used to obtain an improved eigenvalue by computing the Rayleigh quotient. C o m p l e x i t y of the a l g o r i t h m : Consider a matrix A with dimension n and average band-width b. The number of multiplications performed in one cycle of iteration is approximately :

decomposition

A

= QR :

4b 2n

recomposition

~, = R Q •

2b2n

The number of multiplications in s cycles of iteration is thus 6 b 2 ns. The number of multiplications for the computation of an eigenvector with multiplicity 1 is approximately : decomposition of the shifted matrix : 4 b 2 n computation of E in (36)

: 4bn

The number of multiplications for the computation of q distinct eigenvectors is thus approximately 4b2nq. Example : Consider a slightly unsymmetrical arch dam with 548 nodes and 1644 degrees of freedom, whose system stiffness matrix has an average band-width of 114. The iteration method of this paper yields the eigenvalues in the following order"

72

Mode

Value

0 1 2 3 4 5 6 7 8 9

2.5921 3.0182 1.1746 9.4366 2.3417 3.1512 8.3936 3.2573 4.9286 9.8488

Value

Mode

eeeeeee+ e+ e+ e+

13 13 12 12 11 11 00 01 01 01

1.5208 1.6264 2.2936 2.8904 3.3487 4.3323 5.0719 5.4474 6.0141 6.2515

10 11 12 13 14 15 16 17 18 19

e + 02 e + 02 e + 02 e + 02 e + 02 e + 02 e + 02 e + 02 e + 02 e +02

The iteration converges as follows on the eigenmodes" 20 mode

18 16 14 12 10 8 6

I.----(~ ""

4

2

I, 0

2

4

convergence of the iteration

6

8

10

12

14

16

18

20

22

cycle of iteration

The first 20 eigenvalues are obtained with 22 cycles of iteration. The number of multiplications for the computation of these eigenvalues is approximately 6 • 1 1 4 2 , 1 6 4 4 • 22 ~- 3 • 109 . Accounting for loops, index calculations and additions, the time requirement on a workstation is of the order of two minutes. 5

CONCLUSIONS

The proposed method for the solution of the special eigenvalue problem for real symmetric matrices preserves the profile of the matrix and generally yields the eigenvalues in order of ascending magnitude. Shifting and deflation are used to improve convergence of the iteration. The solution effort is proportional to the product of number of unknowns, the square of the average band-width and the number of cycles of iteration. Multiple eigenvalues do not require special treatment. Eigenvalues of equal magnitude but opposite sign require special consideration. Additional research is required to refine the tests for convergence and the strategy for shifting of the eigenspectrum to accelerate convergence. Results of on-going research will be presented at the conference.

INVITED PAPERS

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

75

ADVANCED METAL SYSTEMS IN STRUCTURAL REHABILITATION OF MONUMENTAL CONSTRUCTIONS F. M. Mazzolani I and A. Mandara 2 l Department of Structural Analysis and Design, Engineering Faculty University of Naples Federico II, P.le V. Tecchio 80, Naples (I) 2 Department of Civil Engineering, Engineering Faculty Second University of Naples, Via Roma 29, Aversa (I)

ABSTRACT

The paper outlines the main results of the activity presently in progress in the field of structural rehabilitation of monuments at both universities of Naples under the co-ordination of Federico Mazzolani. The features of the most up-to-dated metal materials and metal-based technologies are shown and discussed, including innovative strategies of seismic protection based on dissipation of seismic input energy. The use of very advanced solutions, relying on shape memory alloys, is addressed in the paper as well. Some relevant applications are also referred to, aiming at illustrating the potential of above materials and technologies in the restoration and preservation of cultural heritage.

KEYWORDS Structural rehabilitation, Monumental constructions, Metal materials, Metal systems, Stainless steels, Aluminium alloys, Shape memory alloys, Seismic protection, Energy dissipation, Special devices.

INTRODUCTION

The modem building industry seems nowadays more oriented to refurbishment and rehabilitation of existing works rather than to new constructions. This trend is mostly to be ascribed to the steady state increase of demand in historical centres for buildings fitted with both structural and functional up-todated features, but also to the need of saving valuable existing constructions from ravages of the time. The last one is a question deeply felt in European countries, notoriously rich in monuments, where the problem is even more delicate, a part of constructions being frequently located in seismic areas. The whole of these problems demands for technological systems able to provide solution not only to specific structural or architectural needs, but also aiming at improving the global performance of the construction, intended as a "system". Similarly, great attention is paid not only to reliability and durability of intervention methods, but also to the possibility to be easily monitored and removed if required. This represents a consequence of a widely shared policy, aiming at the safeguard of monumental works from inappropriate restoration operations.

76 As an answer to these problems, new trends are presently gathering a major importance in structural rehabilitation of existing buildings. The new practice is more and more prone to the use of advanced systems, materials and technologies for both increasing the load bearing capacity of structural elements and for improving, when required, the building seismic performance. At present, the use of advanced metal-based techniques in structural rehabilitation goes on a twofold way. On one side, the use of innovative materials, such as for example, stainless steels, copper, titanium and aluminium alloys, is becoming more and more frequent in rehabilitation interventions. The use of such materials is mainly intended to complement the well known features of constructional mild steel (e.g. high strength and ductility, lightness, ease of transportation and erection, easy market availability, reversibility, etc.), with some special properties, typical of each material, which can be tailored to the specific problem under consideration. On the other side, a corresponding development in the field of seismic protection is presently in progress, allowing new targets to be set in the seismic safeguard of monuments, characterised by unprecedented levels of structural reliability under earthquake actions, even in case of high intensity events. In this field, either active, semi-active or passive systems can be used as well. In particular, dissipative devices based on plastic and/or viscous action have proven to be much suitable to provide existing buildings with a good seismic protection level, from both technical and economical point of view. Most of the them rely on steel or metal alloys, or also on special viscous materials, for achieving the required dissipation capability. In particular, the use of innovative shape memory alloys has been recently introduced for the construction of special dissipative elements or devices for seismic upgrading of damaged monumental buildings. The main aim of this paper is to report the results of a research work carried out at both Universities of Naples under the co-ordination of Federico Mazzolani, where the use of advanced metal systems in the field of structural rehabilitation is pointed out. Particular emphasis is giyen to the investigation of main mechanical features of such materials, at the light of a possible combination with traditional materials, such as masonry, concrete and timber. Similarly, the use of metal materials for special devices intended for seismic protection is also described. To this purpose, the paper also deals with the use of techniques conceived to optimise the seismic performance of monumental buildings and, at the same time, to reduce the effect of thermal changes. For the sake of example, the results of an analysis carried out on single storey constructions are referred to, showing some guidelines on the determination of the optimal value of device properties. Also, suitable criteria are proposed for the evaluation of the most appropriate values for stiffness, nominal capacity and viscous damping features of the devices. Some relevant applications are also referred to, reporting the use of steel and aluminium alloy structures, as well as the implementation of special devices for the seismic upgrading.

ADVANCED M E T A L MATERIALS General

Most of properties provided by special metals make them suitable candidates in the selection of materials for restoration. Sometimes, these features are mistakably disregarded or even ignored by designers, mostly because of the higher initial cost of such materials, often without paying enough attention to the advantages involved over the whole lifetime of the project. The metal materials most liable to be used for structural elements in rehabilitation are: -

-

-

stainless steel; aluminium alloys; titanium alloys; shape memory alloys.

As far the applications in rehabilitation are concerned, the outstanding properties of such materials can be summarised as follows:

77

Low maintenance cost, obtained thanks to a great corrosion resistance. Most of these materials have a very good behaviour in atmospheric and humid environments, whereas some special alloys can even resist chemical attack in acids, alkaline solutions, and chlorine bearing environments. In rehabilitation, this feature involves the possibility to hide the intervention if required, with no risk of future reject due to corrosion. This allows, for example, the possibility to insert tie rods or confinement elements into masonry members. High strength-to-weight ratio, which is an inherent property of all metals and in particular of aluminium and titanium alloys. It allows reinforcing interventions to be made without significant increase of weight and hence with no influence on the mass distribution over the construction. Good ductility, which allows these materials to be used for the fabrication of structural elements or special devices provided with dissipative features. These elements are used to improve the seismic behaviour by means of passive control of input energy, sometimes together with viscous damping devices used to reduce the displacement magnitude. A new chance in the field of energy dissipation is now offered by the so called shape memory alloys, mostly nickel-titanium alloys, able to undergo a transformation of the internal crystalline structure according to temperature and state of stress, which is the main responsible of their hysteretical behaviour under cyclic loads. Ease offabrication and erection. All of these materials can be processed with common modern steelmaking techniques such as cutting, welding, machining, cold and hot forming. In addition, most of them, say aluminium, titanium and copper alloys can be easily extruded, yielding a great freedom in the choice of cross sections of structural elements. Furthermore, in the erection stage, the immediate availability of load bearing capacity involves a reduction of both time and workmanship. Aesthetic appearance, and hence architectural value, provided by the good resistance corrosion of such materials, which makes for an attractive look and a nice harmonisation between new and old materials when used for rehabilitation interventions.

Reversibility, which allows the added elements to be easily removed if necessary. Because of their recyclable character, they can be re-used for different purposes. Thus, when the total lifetime is considered, the choice of such materials can be sometimes less expensive than traditional nonreversible technologies. Product availability, which results in a great number of elements existing on the market. By considering materials altogether, several hundreds of products are commercially available, so to meet any design requirement. The materials

A synopsis of the main mechanical features of above materials is shown in Table 1, where average values of unit weight q(, elastic modulus E, conventional yield strength f0.2 and ultimate strength ft, ultimate tensile strain E:t and linear thermal expansion coefficient et are reported, compared with the corresponding properties of constructional mild steels. It is worth noting that the material choice should be made not only on the basis of the range of mechanical properties like strength, stiffness and ductility, which is quite large as well within any family of materials, but rather by considering other technological factors, such as chemical-physical compatibility with in-situ materials, corrosion resistance, as well as the possibility to obtain particular surface appearances and shapes.

Stainless steel Stainless steels are mostly conceived in order to get adequate corrosion resistance, obtained by adding chromium at 10% or more by weight to low-alloy carbon steel. This involves the development of an

78 invisible stable oxide film (Cr203), which protects the underlying metal surface from corrosion. The addition of molybdenum, nickel and nitrogen improves both corrosion resistance and other important properties. Stainless steel can be martensitic, ferritic, austenitic, duplex and hardened by precipitation. Altogether, more than 60 grades of stainless steel are available. The most commonly adopted alloy is the austenitic one, relying on the addition of both chromium and nickel, which involves the capability to increase strength by work-hardening, together with the best chemical resistance. The duplex grade provides the highest strength within stainless steel alloys, by allowing reduced material thickness compared to conventional grades. The advantage related to the use of stainless steel in rehabilitation practice consists, in first place, of a reduced maintenance cost, made possible by means of corrosion resistance; this aspect also allows to hide the reinforcing elements permanently, with no risk for their performance to be impaired by short or long term corrosion. This feature is particularly important in the case of statues, columns and other stone elements, where any external reinforcing member would be incompatible with the aesthetics of the monument. The good weldability of such materials makes for easy connection between members. A typical consolidation operation based on stainless steel has been proposed for the Main Hall of "Mercati Traianei" (Emperor Traiano's Markets) in Rome (Figure 1). The Hall was interested in past times by several modifications of the structural layout, sometimes for consolidation purposes, which often proved to be neither effective, nor durable. This caused the state of stress in the structure to be much higher than in origin, resulting in a spread damage, mainly consisting of cracks and surface degradation. The intervention of consolidation proposed, consisting of an improved confining system made of stainless steel elements (Figure 2), has been assessed on the basis of a mechanical model of confined masonry developed for this purpose (Mazzolani & Mandara, 1999).

b)

a) Figure 1: Existing confining system in the Main Hall of "Mercati Traianei" in Rome (a) and its damage (b) t .D .

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Figure 2: The new confining system proposed for the Main Hall of "Mercati Traianei" in Rome

79 TABLE 1 SYNOPSIS OF MECHANICAL FEATURES OF SPECIAL METAL MATERIALS COMPARED WITH MILD STEEL MATERIAL

qt E f0.2 (g/cm 3) (kN/mm 2) (N/mm 2) Mild steel 7.85 206 235+365 Stainless steel --- 7.8 ---196 200 + 650 Aluminium alloys --- 2.7 65 + 73 20 + 360 Titanium alloys -- 4.5 --- 106 200 + 1000 SMA Ni-Ti (Nitinol) = 6.5 28 + 75(*) 100 + 560(*) (*) Values referred to martensite and austenite, respectively.

ft (N/mm 2) 360+510 400 + 1000 50 + 410 300 + 1100 750+.'960(*)

x 100 (As) 10+28 10 + 40 2 + 30 8 + 30 15.5

Et

~ x 106 (C °-1) 12+ 15 17 + 19 24 +25 6+7 6.6 +11(*)

Stainless steel can be profitably used also in the construction of special devices adopted to provide an additional seismic protection to the building. This is the case of oleodynamic dampers installed in buildings, where the use of stainless steel for the sliding parts of the device has been chosen in order to avoid any extraordinary or premature maintenance intervention due to corrosion. The devices, whose stiffness varies as a function of the rate of applied load, are intended for both dissipating a part of seismic energy and for optimise the structural response under thermal loads.

Aluminium alloys Aluminium alloys can be considered as innovative materials in civil applications, even though they are well experienced in fields other than building. Best features of aluminium alloys are the reduced weight (about 1/3 of steel) and the good corrosion resistance, which allows to limit the amount of added masses, by reducing at the same time the maintenance problems. Alloys under consideration are obtained by adding to pure aluminium, having poor strength but good ductility properties, elements such as magnesium, silicon, copper, zinc, manganese, etc. The range of mechanical features which can be obtained in such a way is very wide: for instance, it is possible to have work hardening capability and good corrosion resistance in AI-Mg alloys (5000 series), high strength in heat treatment alloys AI-Si-Mg (6000 series) and AI-Zn-Cu (7000 series), even though with a relatively reduced corrosion resistance and ductility (Mazzolani, 1995). In addition, a great number of alloys can be extruded, allowing both solid and hollow profiles to be obtainable. Extrusion turns to be very useful in the production of long components, or items having complex cross sections designed to meet special functional requirements. Contrary to what sometimes erroneously deemed, aluminium alloys are fully suitable to important structural use. In this, their peculiar features in terms of strength-to-weight ratio and corrosion resistance are expressly pointed out. The use of aluminium, in fact, has been adopted or proposed in many important operations of structural rehabilitation, as for example the construction of vertical extensions on the top of existing buildings, where it is possible to take advantage of the great lightness of material, as well as of the good corrosion resistance which limits to a minimum the maintenance interventions. At the same time, aluminium elements can be used for the creation of light internal floors, when a greater surface is required, without any significant increase of dead load. In such cases, the lightness of structural elements also allows a great ease of in-door movement. Within the preservation of the archaeological site of the Mercati Traianei in Rome, the construction of new covered areas, intended for both protecting some of the existing ruins and for improving public facilities, has been based on fully bolted space reticulated structures made of aluminium alloy members (Figure 3). The new structures are based on an innovative type of joint, named GEO System, conceived in such a way to optimise both weight and structural performance, also improving erection features (Mazzolani et al., 2000). The choice of aluminium alloys has been motivated by their properties of lightness, corrosion resistance and nice appearance. In addition, the conception of the new structures makes all elements easily removable, providing the intervention with a fully autonomous character from the architectural point of view. Aluminium has been also used to rebuild the deck of old suspension bridges, so to reduce the magnitude

80 of load acting on the existing piers. Such applications were made in France (Mazzolani & Mele, 1997), where threebridges not far from Lyon (Trevaux, Montmerle, Grosl6e) have been refurbished by replacing the existing deck with aluminium trussed girders (Figure 4a,b,c). In the bridge at Grosl6e, the deck has been completed with a light-aggregate concrete slab. Recently (Figure 4d), an aluminium solution has been adopted in Italy for the restoration of the suspended Real Ferdinando bridge on the Garigliano river (Mazzolani, 1998).

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Titanium alloys Titanium alloys, obtained by adding molybdenum, vanadium, aluminium etc. to base metal, are widely used in aerospace and industrial applications, where they represent an altemative to aluminium alloys, in particular when a greater strength at high temperature is needed. Together with nickel, titanium is also the main component of Shape Memory Alloys. Titanium alloys can be divided into three groups: 1) Alpha alloys are non-heat treatable, have low to medium strength and good notch toughness and ductility; 2) Alpha-Beta alloys are heat treatable with medium to high strength levels, but they do not exhibit the same behaviour at high temperature as Alpha alloys; 3) Beta or near-beta alloys can be easily heat treated, have high strength and good creep resistance at intermediate temperatures. Also, Beta alloys have good combinations of properties in many structural applications such as sheets, heavy sections, fasteners and springs. All titanium alloy grades have very good. resistance to corrosion and, in most cases, good weldability. Furthermore, they can be extruded. In the view of refurbishment applications, titanium alloys offer a very low linear thermal expansion coefficient (6+8 x 10-6 C°l), which is very similar to that of volcanic or metamorphic rocks, such as granite and marble. This allows titanium elements to be used in redundant or prestressed systems with no risk to impair the effectiveness of intervention due to thermal changes. Similarly, no co-active state of stress would be involved. For this reason, titanium reinforcement elements have been used in the restoration of monuments such as the Parthenon in Athens and the Colonna Antonina in Rome, where titanium alloy stirrups have been

81 inserted and hidden into existing stone blocks (Giuffr6 & Martines, 1989). They proved to be far more effective than conventional steel elements used before, which had involved many cracks due to corrosion and excess of thermal dilatation. In the same way, some concrete bridges in Japan have been repaired with titanium rods. In the construction industry, titanium is bonded to glass for exterior use in buildings and is also used as roof panels, window frames, eaves and gables, flashing, curtain walls, railings, ventilators, and interior and exterior appendages.

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Figure 4: The French suspension bridges at Trevaux (a), Montmerle (b), Grosl6e (c) and the one on the Garigliano river (Italy), where aluminium has been widely employed for the re-construction of decks

Shape memory alloys Shape memory alloys (SMA) belong to the class of so called "smart" materials, together with piezoelectric materials, electrorheological fluids, electro- and magnetostrictive materials. They are obtained from alloying Ni-Ti, Cu-A1-Zn and others elements, and are characterised by yield stress and modulus of elasticity strongly increasing as long as temperature increases within a very small range, called transformation temperature range. This quite particular behaviour is due to a solid martensiteaustenite phase transformation, which involves a change of the metal crystallography from the low yield strength martensitic structure to the high yield strength austenitic one. The transformation range is limited by My and Ay, corresponding to the temperatures where only full martensitic or full austenitic structures can exist, respectively. Such transformation, which occurs at temperature between-100 °C and +100 °C depending on the alloy composition, can be due to both temperature change and mechanical stress. This involves the ability to recover large strains due to load (up to 10%) spontaneously or by heating. In the former case, the behaviour is defined superelastic (Figure 5a) and is characterised by the complete strain recover after unloading. This results from an austenite-martensiteaustenite transformation occurring with the loading-unloading process. Since loading and unloading paths are different from each other, a given amount of energy is dissipated over the cycle. In the latter case, complete strain recovery may occur by heating material above Af, in such a way to obtain austenite again. This behaviour is called memory effect (Figure 5b). If the element is constrained, internal stresses

82 arise, which can be very high owing to the great material stiffness in the austenitic phase. Both these features let SMA-based devices very suitable to many consolidation problems. The memory effect can be exploited when a given state of co-active stress has to be applied to the structure. This is the case of confining interventions of masonry members, where SMA elements can be arranged at a temperature lower than My, where pure martensite exists and material can be easily deformed to reach the required strain. Subsequently SMA elements are heated above Af, in order the required degree of co-action to be automatically applied. The superelastic effect, instead, allows some energy to be dissipated in a full loading cycle. This turns to be useful for the construction of seismic protection devices, as shown by the wide experimental activity now in progress at many research centres (Dolce et al., 20.00, Pegon et al., 2000), aiming at investigating the real possibility to implement such materials in structural engineering. On the other hand, some important applications are starting in the field of seismic protection of monuments. The most important of these is the restoration of the Basilica of S. Francesco in Assisi Italy (Figure 5c), which was severely damaged during the earthquake of 1997. SMA devices have been installed between the front tympanum and the roof structure. They supplement a number of oleodynamic devices placed at a lower level, in order to increase energy dissipation (Croci et al., 2000). Another intervention has been carried out on the Bell tower of S. Giorgio in Trignano (Italy), where vertical steel tie-rods in series with SMA devices have been inserted into the tower and prestressed so to increase its flexural strength. They also exhibit energy dissipation under seismic actions, as shown in both theoretical and experimental investigation carried out on the bell tower (Indirli, 2000).

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Using Hamilton's principle we arrive to the following governing equations for an elastically supported membrane Txu, = +2 Txy u, ~y +Ty U,yy - pii = - g fllu+~q=fl3 u(x,y,O)= f ( x , y ) ,

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where q = 0u/On is the derivative of u normal to the boundary, /~ = k(s) is the stiffness of the elastic support and f12 = T the boundary traction that stretches the membrane. Several membranes have been analyzed. The obtained numerical results are compared with those obtained from existing analytical or other the numerical solutions and validate the accuracy and efficiency of the proposed method Example 1. Free vibrations of a non-homogeneous rectangular membrane The free vibrations of a rectangular membrane ( - b / 2 < x < b / 2 , - a / 2 < y < _ a / 2 ) with side ratio a / b = 8.5/12 are studied for the following three cases: (i) homogeneous membrane with p = Pc, Tx = Ty = T -- constant, Txy = 0 (ii) non-homogeneous membrane with p / Pc = exp(-0.1 ~x[ + lY[}) and

Tx

:

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T :constant,

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non-homogeneous

membrane

with

Ty=(xZ-yZ +T)/T,

T x y = 2 x y / T . The

obtained results are presented in Table 1 and Table 2 as compared with those available from other solutions. The exact values are computed from the relation ~"~m,n : ~[ n2 + mz (b / a) 2 ]1/2, while the approximate ones by the Rayleigh method with shape functions the mode shapes of the homogeneous membrane, i.e. urn,,, = sin mnx/ asin nny / b . Example 2. Membrane of arbitrary shape A membrane of arbitrary shape has been studied. Its boundary is defined by the curve r = (5 + sin 0)(1.2sin4 0 + cos 2 0), 0 0

Limit state model Z(X) Probabilistic Models

Full Probabilisfi¢ Design: v

FORM. SORM, System Analysis Monte Carlo Cheek P{ ZO0 a¢ be equal to 2.0. This may be considered as inadequate and so an inspection is planned at some point tinspduring the lifetime. In that case we have: PF(t) = P(F I crack detected at tinsp) P(crack detected at tinsp) + P(F I no crack detected at tinsp) P(no crack detected at tinsp)

(11)

where "F = failure" corresponds to "a(t) > aj' and "crack detected" to "a(t) > ad", where ad represents some (random) threshold value. Let the decision rule be that the structure will be repaired if a crack has been detected. Assuming perfect repair we have P(F I crack detected) = 0. Neglecting also failure before tinsp, the probability of failure including the inspection can be written as: PF(t) = P(F [ no crack detected) P(no crack detected) = P (a(t) > ac lain < ad) P(am < ad)

(12)

This requirement can again be elaborated on the level of probabilistic methods as on the level of partial factors

ae

T

critical level repair or detection level

ad

tinsp

t

time

Figure 4: Fatigue failure in time t occurs if at inspection the crack length is smaller than ad and at time t the crack length is larger than ac RESEARCH NEEDS

In order to have full profit of the possibilities offered by probabilistic methods, either directly or via the Partial Factor Method, still a great deal of research has to be done (Gulvanessian, 1996). First, there is a need for a further development of the probabilistic calculation techniques. It is still difficult to combine advanced structural models (non-linear, time dependent, dynamic) with advanced probabilistic models (including time and space variability on both load and resistance side, the effects of inspections, monitoring and so on).

192 Maybe even more important than having operational computational methods and corresponding software is the availability of adequate data sets. This is partly due to the fact that data is scarce by definition, as for instance statistics on extreme environmental conditions, but it is also a lack of co-ordination between the various parties in the building industry. The collection of data found in material control procedures, laboratory tests, observations during inspections and so on, should be organised in a more effective way. However, in some cases and for some period of time we simply have to accept that a lack of data exists. This means that in applying the theory so called subjective estimates are necessary. The values chosen in these cases should however reflect the common opinion of all experts, rather than the subjective opinion of a single person. For the application of the theory it is therefore important that the probabilistic properties of environmental conditions, material properties and model uncertainties are codified. In this respect, it is worth to mention that the Joint Committee on Structural Safety has recently developed a first complete Model Code for Probabilistic Design (JCSS, 2001 and Vrouwenvelder, 2001). This code offers a general probabilistic design philosophy, and a set of operational probabilistic models for loads (self weight, wind, snow, live load, etc), materials (steel, concrete, etc) and model uncertainties (for beam models, columns, plates, etc). It is aimed for that this code will be improved and extended in the years to come. In particular fatigue and other deterioration mechanisms, for instance for concrete should be added. SUMMARY AND CONCLUSIONS Modem performance based codes are based on limit state concepts and probability based reliability requirements. Such a concept is rational and flexible and offers a maximum of freedom to the designer to find economic and or non-conventional solutions. For standard design solutions codified methods for verification are offered, usually based on the partial factor method. However, for alternative solutions and alternative verification methods (for example design by testing), the essential basis for the verification can remain the same. A frequently used argument against probabilistic methods is that data is too scarce to perform such a type of analysis. To some extend this is true, even if we accept engineering judgement and expert opinions as valuable additional information. On the other hand, Full Probabilistic Methods and Partial Factor Design are connected by the theory of reliability. The Partial Factor Method is in fact to be considered as a well defined simplification of Probabilistic Design. Therefore, if data is scarce for the one, it is scarce for the other. Fortunately, both Partial Factor Methods and Probabilistic Design can be calibrated to the successful design procedures of the past. This enables to have more confidence in both the methods than would be possible on the basis of statistics and expert judgements alone. In the future we may see a development into two directions: one way is to incorporate more safety differentiation into the target reliabilities for the Partial Factor Method, thus enabling Risk Analysis reasoning to be incorporated into the standard design. The other route will be that for special design and assessment cases probabilistic methods will be used directly. For both developments it is necessary that further standardisation of methods and numbers is done and that more user friendly and efficient software becomes available. Given the economic pressure to save money on the building of new structures and the inspection and maintenance of existing ones, these developments may prove to be a vital issue. REFERENCES Ferry Borges, J. and Castanheta, M. (1971), Structural Safety, Laboratorio Nacianal de Engenharia Civil, Lisbon.

193 CEN TC250 SC (1994), Eurocode 1 Basis of Design, ENV 1991 Basis of Design and Actions on Structures: Part 1, CEN-ENV- 1991 Ditlevsen, O. and Madsen, H. (1989), Proposal for a Code for the direct use of Realibility Methods in Structural Desing, ABK-report R-248, Technical University of Denmark, Lyngby, Denmark Dhamarvasan, S., Peers, S.M.C., Faber, M.H., Dijkstra, O.D., Cervetto, D., Manfredi, E. (1994), Reliability Based Inspection scheduling for fixed offshore structures, OMAE, Vol. II, Safety and Reliability ASME. Foeken, F.J. van, Kerstens, J.G.M., Van Manen, S.E., Vrouwenvelder, A.C.W.M. (1994), Probabilistic Design of the Steel Structure of the "Nieuwe Waterweg" Storm Surge Barrier, The Proceedings of the Fourth International Offshore and Polar Engineering Conference, Osaka, Japan, p. 483, Vol. IV. Gulvanessian, H. (1996), Eurocode 1: Part 1: Basis of Design, Introduction, Development and Research Needs, IABSE Conference on Basis of Design and Actions on Structures, Delft. JCSS (1996), Basis of Design Background Documentation, ECCS Publication no 94, Brussels. JCSS (2000), Probabilistic Assessment of Existing Structures, RILEMpublications. JCSS (2001), Probabilistic Model Code, JCSS internet web page ISO/TC98 (1994), General Principles on Reliability for Structures - Revision of IS 2394. Ltichinger, P. (1996), Basis of Design Serviceability Aspects, IABSE Conference on Basis of Design and Actions on Structures, Delft. Madsen, H.O., Krenk, S., and Lind, N.C. (1986), Methods of structural safety, Prentice Hall Inc., Englewood Cliffs, New Jersey. Murzewski, J. (editor) (1993), Combinations of actions to structures, Structural Safety, Vo113, no. 1 & 2. Ostlund. L. (1991), Structural Performance Criteria, IABSE publications. Rackwitz R. (2000), Optimisation - the basis of code making and reliability verification, Structural Safety, 22, pp 27-60. Siemes, T. and Rostam, S. (1996), Durable safety and serviceability - a performance based design format. IABSE Conference on Basis of Design and Actions on Structures, Delft. Vrouwenvelder, A. and Schiessl P (1999), Durability Aspects of Probabilistic ULS Design, HERON, 44, no 1, pp 19-30 Vrouwenvelder A. et al. (2000), Betrouwbaarheidsaspecten Boortunnels (Reliability aspects of boring tunnels), in Dutch, CUR-COB, Gouda, Holland. Vrouwenvelder, A. (2001), The JCSS Probabilistic Model Code, Safety Risk and Reliability, Trends in Engineering, Intemational Conference, Malta. Waarts, P.H.(2000), Structural Reliability using Finite Element Methods, Thesis, Delft University of Technology

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Structural Engineering,Mechanicsand Computation(Vol. 1) A. Zingoni(Editor) © 2001 ElsevierScienceLtd. All rights reserved.

195

EARTHQUAKE ENGINEERING RESEARCH INFRASTRUCTURES R T Severn Coordinator, Earthquake Engineering Round Table University of Bristol, Earthquake Engineering Research Centre, Bristol BS8 1TR, UK

ABSTRACT A summary is given of the development of collaboration between European earthquake engineering laboratories during the period 1991-2000, as a direct consequence of financial support from the European Commission within the various Framework Programmes. Five major shaking tables were involved, together with the ELSA reaction wall facility at JRC Ispra. As a result of providing access to researchers from Member States, a major step forward was made in the fidelity and accuracy with which these six facilities could be used, leading to significantly enhanced performance. This major advance has put European earthquake engineering infrastructures at international level and has opened up several new research areas for experimental study. Collaboration between the infrastructures group and three successive Research Networks has allowed significant progress to be made towards the validation of many aspects of Eurocode 8 and the mitigation of seismic risk. KEYWORDS Large facilities, shaking tables, reaction walls

THE ROLE PLAYED BY EARTHQUAKE ENGINEERING FACILITIES IN EUROPEAN RESEARCH

The Community's Large Installations Plan (LIP) In the 1990 report to the Advisory Committee on LIP, the study panel for Earthquake Engineering made a number of recommendations regarding the utilisation of experimental facilities in the European Union. It took the view that with the completion of the reaction-wall facility at JP,C Ispra and the new shaking table at LNEC Lisbon, together with existing facilities, there would be adequate provision of such facilities for the needs of European researchers. However, the various laboratories had been working in isolation, and what was needed as a matter of urgency, as a component of any coherent European research programme in earthquake engineering, was a study of the true performances of these shaking tables. Clearly, such a programme would be too extensive to carry out on one shaking

196

table, and unless the various tables to be used had guaranteed performance, it would be impossible satisfactorily to compare the results obtained. The immediate result of this recommendation was the award by the Commission of a contract to the Laboratory for Earthquake Engineering (LEE) in Athens and the Earthquake Engineering Research Centre (EERC) in Bristol, for a common study on a well-defined but realistic problem. Such a study was to involve reviews of software, operational procedures, and actual performance of a testpiece, which was to be designed so as to stretch the performance of each table to its limits. Many options were considered for this testpiece before the structure shown in Figure 1 was adopted. It consists of a variable mass supported on 4 steel columns. The mass can be adjusted between 2-10t by adding It steel blocks at various levels, allowing the centre of gravity to be adjusted between 1-3m above the table. The design also allowed the steel blocks to be used as a rigid testpiece. Two such identical testpieces were fabricated in Bristol, one of which was transported to Athens. The Third Framework Programme (FP3, 1993-96)

FP3 overtook the shaking table 'standardisation' study described above, and in it were two sections of the Human Capital and Mobility Programme (HCMP) which were relevant to this paper. The first of these was Access to Large Scale Facilities (LSF), and the second Research Networks. Shaking tables were accepted as one category of LSF, allowing the opportunity to be taken of creating the European Consortium of Earthquake Shaking Tables (ECOEST), membership of which now included LNEC Lisbon and ISMES Seriate as well as Athens and Bristol. With the last of these as coordinating partner, a research contract was obtained from the Commission having the following 5 themes:(i) (ii) (iii) (iv) (v)

Shaking table standardisation research. Access to shaking tables for PREC8 researchers. Access by others in ongoing research at the ECOEST laboratories. Research programmes initiated by users. Training in shaking table procedures.

Theme (ii) above refers to the second of the two relevant sections of the HCMP because an earthquake engineering research network was approved by the Commission having the title Pre-normative Research for Eurocode8 (PREC8), and the ECOEST partners proposed that in view of the importance of Eurocode8 for the European Construction Industry, equal priority should be given to the first two themes in the above list.

Shaking Table Standardisation Research A full description of this research is given in Technical Report 1 (see section 7). The conclusions are given here relating to the first four of the shaking tables described in Table 1. (i)

For testpieces which remained elastic, a satisfactory match between desired and achieved input could be obtained given time, but the success of the matching process depended on skilled operators. (ii) If the measured acceleration of the table was used for matching purposes, its displacement was not controllable. (iii) Small, spurious, rotational motions of the table had a very strong influence on the response, but were difficult to remove. (iv) All four shaking tables operated 'out-of-real-time' and could not be controlled unless the properties of the testpiece remained constant.

197

Table 1. Shaking table specifications NTU EERC ISMES LNEC Athens Bristol Bergamo Lisbon Table size (m) 4x4 3x3 4x4 5.6x4.6 Table mass (T) 10 3 11 40 Max. specimen mass (T) 10 15 30 40 Max. specimen height (m) 11 4 10+ 10+ Controlled degrees of free- 6 6 6 3 dom Translation X, Y, Z X, Y, Z X, Y, Z X,Y,Z Rotation 0x, 0y, 0z 0x, 0y, Oz 0x, 0y, 0z Longitudinal (X) or Lateral

(v)

+_Displacement (mm) 100 + Velocity (mm/s) 1000 + Acceleration (g) -zero 3 payload Vertical (Z) + Displacement (mm) 100 + Velocity (mm/s) 1000 + Acceleration (g) - zero 4 payload Frequency Limit (Hz) 100

CEA Saclay 6x6 25 100 10 6 X,Y,Z 0x, 0y, 0z

150 700 4.5

100 550 3.

175 730 1.8

125 1000 4

150 700 7

100 440 2

175 450 1.1

100 1000 4

100

120

20

80

Taking conclusions (i) and (iv) together, their significance lies in the fact that economically efficient modern seismic design depends on the utilisation of the non-linear properties of the construction materials - the building is designed to deform inelastically in a predictable way, but not to collapse. Tests on shaking tables ought to be able to follow such inelastic behaviour, but they could not do so in a controlled manner. Conclusion (ii) above points to the fact that acceleration was the parameter mostly used for control, whereas what is needed is a control system which utilises both acceleration and displacement in an optimum manner covering the whole operating range of the table. Finally, conclusion (iii) notes that the control systems used were deficient in removing unwanted movements of the cable, particularly rotational components. Although small, they made a significant contribution to the response in some cases.

Shaking Table Studies for the PREC8 Research Network. The PREC8 network involved 15 European research groups working on the following 4 broad themes, chosen because of their importance to the new European Seismic Design Code, Eurocode 8. 1. Reinforced Concrete Highway Bridges (ISMES, Bristol, JRC Ispra). 2. Reinforced Concrete Frames (Athens). 3. Infilled Frames (Athens, ISMES, Bristol, LNEC). 4. Geotechnical structures (Bristol). Full details of the experimental research carried out by ECOEST in collaboration with PREC8 are given in Technical Reports 1-9 (see section 7) and illustrative examples are given in Severn (1998), together with examples of research carried out in collaboration with non-PREC8 colleagues. The laboratories taking part in each of the themes is given above in parentheses; here, the ELSA laboratory at JRC Ispra appears for the first time and its contribution will be dealt with later. It received funding through PREC8 rather than ECOEST.

The Fourth Framework Programme (FP4, 1996-99) In FP4, HCMP became Training and Mobility of Researchers (TMR) but still contained sections devoted to Access to Large-scale Facilities and Research Networks. ECOEST became ECOEST2 with the original 4 members now joined by the shaking table laboratory at CEA Saclay and the reaction-

198

wall facility at JRC Ispra. The research network changed its name to ICONS - Innovative Concepts for New and Existing Structures, with emphasis on the following topics:(i) Seismic Actions. (ii) Assessment, strengthening and repair. (iii). Innovative design concepts. (iv) Composite structures. (v) Shear-wall structures. Figures 8-24 give an indication of the experimental research carried out, either with PREC8 and ICONS colleagues on these 4 themes, or with other European researchers. The JRC Reaction-Wall Facility

In the experimental aspects of earthquake engineering research, shaking tables (Fig. 2) and reactionwalls (Fig. 3) are complementary facilities; they each have positive and negative attributes. Shaking tables must usually test scaled model, whereas reaction walls can often test at full-scale. Shaking tables can input the real seismic motion at the correct timescale to the base of the testpiece, thereby generating inertia forces in every element of mass, whereas the reaction-wall inputs forces to discrete masses at a small number of points on the testpiece itself, at a time-scale which is determined by the speed at which a computer can integrate the equations of motion. This provides a stop-go application of forces leading to the description of the process as 'pseudodynamic'. The reaction wall test process is carried out quasi-statically, using on-line computer calculation, and control, together with experimental measurement of the properties of the testpiece to provide a realistic simulation of the dynamic response. The equations of motion for a discrete parameter model are solved on-line by numerical integration. Inertial and viscous damping forces are modelled analytically, whereas the restoring forces are measured experimentally. The process automatically accounts for hysteric damping due to inelastic deformation and damage. The given acceleration input is applied to the equations of motion, the small time-step integration of which gives displacements at the controlled degrees of freedom. These displacements are then applied to the testpiece by the actuators fixed to the reaction wall (Fig. 3). Load cells on the actuators measure the forces necessary to achieve these displacements, and these restoring forces are returned to the computer for the next time-step calculation. This iterative process does not occur in real-time; in fact, there are 'hold' periods whilst the computer performs the required integration, so that the application of an earthquake input lasting seconds, can take many minutes or even hours. Figures 20-21 give illustrations of research carried out at JRC Ispra during the ECOEST2/ICONS period. The Fifth Framework Programme (FP5, 2000-2003)

Again, this time in the HPRI programme, there are components of FP5 relating to access to large research facilities, and also research networks. The latter now has the title Safety Assessment for Earthquake Risk Reduction (SAFEER), and the former is described as the European Consortium for Large Earthquake and Dynamic Engineering Research (ECOLEADER) with the 4 partners ISMES, JRC Ispra, Bristol and Athens. It will be observed from this title that it is the intention of the group to apply the advances made in earthquake engineering experimental research to other areas concerned with structural dynamics. The collaboration between the two groups has already been planned. The research of the 13 partners of SAFEER will be concentrated on the 4 themes of: (i) Characterisation of seismic hazard. (ii) Assessment and design in low seismicity regions. (iii) Strategies/techniques for risk reduction. (iv) Risk assessment system.

199

NETWORKING AND OTHER ACHIEVEMENTS OF THE CONCERTED ACTION

Research Into the Control of Shaking Tables. The results of the study of how shaking tables actually performed is given above in section 1.3 of this paper, from which it is seen that in 1996 there were serious deficiencies, most notably that all four tables studied operated 'out-of-real-time' and could not follow the non-linear behaviour of the testpiece. Because the utilisation of such energy absorbing mechanisms is an essential feature of economic seismic design, this was regarded as a serious shortcoming. Fortunately, the Commission had introduced into FP4 a programme, described as RTD Projects for Access Improvement, which made it possible to perform research into the performance of Large Scale Facilities, aiming at improvements in that performance. The four partners of ECOEST made a successful RTD proposal to the Commission in 1996 for a basic study of the construction and operation of shaking tables, referred to as CESTADS-Control Enhancement of Shaking Tables from Analogue to Digital systems. A brief description of this research will now be given. Modem shaking tables (Fig. 2) consist of a platform, which can move in all 6 degrees of freedom (DOF) under the control of 8 actuators (see section 6). Its primary function is to replicate in the laboratory the true nature of earthquake input, to a testpiece which is attached to the table. This it does through a control system which is responsible for sending signals to the servo-valves of the 8 hydraulic actuators, the oil pressure in which causes displacement of the rams. The signal sent is a time-history of either acceleration or displacement, but by the time it reaches the testpiece it has been corrupted by various effects in the total system. Thus, what is measured at the testpiece is not the intended input. The error can be measured however, and a correction applied to the input in an iterative manner until what was intended is actually input to the testpiece. But the crucial question here is the time taken to measure the error of the input to the testpiece relative to the time taken for the control system to compute eight, probably different, correction signals. If the latter is greater than the former, the control is said to be 'out-of-real-time', and this, as previously noted, was the situation obtaining at the 4 ECOEST shaking tables in 1996. At that time, to produce the desired testpiece input could take many minutes due to the use of analogue systems and inefficient software and hardware. The practical significance here, is that the stability of convergence of control to the required input requires the properties of the testpiece to remain unchanged, and this clearly precludes any tests during which non-linear behaviour is exhibited. Consideration of this problem led to the discovery that control system for shaking tables had not kept pace with developments in control engineering per se, and particularly that the Minimal Control Synthesis (MCS) algorithm developed by Stoten & Gomez (1999) would probably remove the difficulties produced by existing linear control systems. The reason for such optimism was the adaptive nature of MCS. That is to say, it does not require the dynamic parameters of the tested system to be known at the outset, but adapts itself to deal with changes in these parameters as the test proceeds. This is particularly valuable in earthquake engineering because the testpiece, whose mass will probably be greater than the table itself, changes its characteristic during the test. The CESTADS partners undertook this research, a first step in which was to make a fundamental study of the kinematics of a system having 6DOF controlled by 8 actuators (Gomez, 1999). The CESTADS study addressed the four issues listed in Section 1.3. Taking the first and fourth together, MCS calculation of error signals to the 8 actuators took less than 0.001 sec, and so 'real-time' control was achieved, at the same time producing a much better match between desired and achieved input (Figs. 4a and 4b). Referring to the second issue, research was carded out on a control system which used displacement at low frequencies where LVDT ~measurements are accurate, and acceleration at higher frequencies where accelerometers are accurate, the break-point between the two being obtained by test. Figure 5 shows that this composite filter technique gives appreciable benefits. For the fourth issue, that of spurious motions, Figure 6a refers to a test with intended motion in the X-direction, showing the spurious pitch motion when the table is empty and when it is loaded with the testpiece of Figure 1. Figure 6b shows the MCS adaptive control signal sent to one of the vertical actuators to counteract the spurious pitch. Similar signals were sent to the other three vertical actuators.

200

Other Concerted Actions. The ECOEST group of four laboratories were awarded a Concerted Action contract by the Commission, essentially to perform two tasks - to organise Round Tables for European researchers in earthquake engineering, and to produce a series of major technical reports. These reports are listed as Technical Reports 1-9 in section 7 of this paper, their essential feature being that they contained a more detailed account of the research carried out by ECOEST/PREC8 teams than could be published in technical journals. Each 200-page report was written on a theme, containing contributions from several research groups. Publishing was carried out by LNEC Lisbon, who distributed more than 150 copies of each report to major earthquake engineering laboratories throughout the world. Each report was presented and discussed at a Round Table meeting to which a number of international experts were invited. A similar Concerted Actions contract was awarded in Framework Four to the ECOEST2 group of 6 laboratories together with the 5 coordinating research teams of the ICONS research network. The major reports, again to be produced by LNEC, will appear early in 2001 (listed as Technical Reports 10-18 in section 7 of this paper). For Framework Five, the concerted actions component of our activities have been absorbed within a Thematic Network having 14 partners, which will broaden the scope to include dynamic test facilities outside earthquake engineering. It also contains 7 research programmes which will be carried out in addition to Round Tables, Workshops and a third series of technical reports. A web-site is being created for EU-funded research in earthquake engineering. At the halfway stage of the CESTADS contract it was clear that the MCS adaptive control algorithm would be successful in making a major change in the accuracy with which shaking tables, reaction walls and other dynamic test facilities could be controlled. This would make possible research into several new areas, and a second contract was obtained from the Commission (FUDIDCOEEF Further Developments in Dynamic control of Earthquake Engineering Facilities) to explore the topics which are described in Section 3. By the beginning of 2000, proof-of-concept had been obtained in each topic, and further research and development is proceeding. CURRENT INFRASTRUCTURE DEVELOPMENTS

Controlled Testing In the Non-Linear Range of Material Behaviour. Economic provision of seismic resistance requires that use is made of the energy absorbing characteristics of the construction materials. The structure is designed to follow a progressive deformation caused by failure of individual redundant members, but not to suffer either sudden or complete collapse. The ability to follow such a process has not hitherto been possible in shaking table experiments because the control systems available have required the properties of the testpiece to remain constant. The MCS algorithm provides real-time adaptive control, allowing non-linear stress/strain paths to be followed, and this means that progressive deformation of the structure can now be followed experimentally on a shaking table. To do this, accurate measurements of the stress/strain paths of construction materials will be required, and here again, MCS control is being used for the first time in newly designed materials testing machines.

Substructuring on Shaking Tables. In the majority of cases, the testpiece on a shaking table is a scaled model of the complete real system, such as a building, even though interest may be centred on only certain parts of that building. The scale of the model will be as large as the capacity of the table allows, but some compromises have always to be made in obtaining satisfactory similitude between model and full-scale, and the modelling of joints, for example, is hardly ever satisfactory. But the speed at which the MCS algorithm allows calculations to be made opens up a new approach, in which the greater pfirt of the system is modelled numerically (eg: by finite element methods) in a computer, whilst a smaller part of greater interest is modelled physically and tested on the shaking table. Figure 7 illustrates this substructuring concept, which has the merit of using either full, or large, scale for the testpiece. Of course, the key area is the

201

interface between the numerical and physical models, and it is here that the MCS adaptive control algorithm has been shown to be fast enough to make the necessary links. It is anticipated that the implementation of substructuring will be particularly valuable for structure/foundation interaction studies, which are very difficult to deal with using shaking tables. It is envisaged that the structure itself (or a model) will be physically tested on the table, whilst the foundation material will be modelled computationally. Research in this area is being actively pursued on a worldwide basis; Igarashi et al (2000) and Williams et al (2000) are examples of recent publications.

Multiple Support Input Earthquakes travel at a finite speed (250-800 m/s) and are modified by the ground through which they pass. Large and important structures, such as bridges, therefore receive different input motions at their several support points. Eurocode8, for example, requires all bridges over 600m in length to be designed with this in mind, and although various theoretical approaches have been put forward, none have been subjected to experimental corroboration. What is required is very precise control of the several different inputs, replicating the field situation. This has been achieved by MCS and experimental verification is being carried out on the model bridge shown in Figure 8. A great deal is known about this particular irregular bridge, because it was studied extensively in the ECOEST/PREC8 series of research programmes (Technical Report No. 4, see section 7 of this paper). The Effects on

Response of Spurious Motions.

When shaking table studies are made on items of equipment for 'fitness-for-purpose' qualification tests, the input motion is specified by the certifying organisation. It is normally a single-axis translational input, with a tolerance band on the two other translations. It has always been assumed that rotational inputs (roll, pitch and yaw) were either not present, or, if present added to the rigour of the test and could be considered as an additional factor of safety. Our ECOEST studies of true shaking table performance (Technical Report No. 1, see section 7 of this paper) showed that such unwanted rotations were present, could not be removed by the control systems then available, and did have a significant effect on response. Moreover, they actually served to take energy from the main input, thereby reducing the safety factor. The MCS control has remedied this situation; not only does it allow the spurious motions to be removed, but it also allows a specific value of one, or all, of them to be input to the table. Research is now in progress on the effect of these small rotations on the response of the testpiece. The Effects of Vertical Input Motion

Evidence from earthquake site visits, in Europe and elsewhere, is increasingly providing evidence of the importance of the vertical component of the ground motion in causing damage. To study this experimentally on a shaking table requires that the control system be able to produce a pure vertical motion and also a combined vertical and horizontal motion. The control problem, solved by the MCS algorithm, is similar to that of the previous section dealing with spurious motions. Continuous

Pseudodynamic Testing (CPSD)

Because of its hybrid nature - part physical and part computational, the pseudodynamic test method has been shown by JRC Ispra to be capable of utilising a sub-structuring process as well as multiplesupport input. There were, however, two limitations:(i) the computational model assumed elastic behaviour. (ii) the computed displacement at each time step was imposed in ramp form followed by hold periods, introducing possible spurious strain relaxation in the testpiece. The control period is not fixed and is of the order of some seconds. By use of the MCS adaptive control algorithm on the reaction wall actuators it has become possible to avoid the use of a holding period, and to refer to this new process as Continuous Pseudodynamic

202

Testing (CPSD). This allows a considerable reduction of the test duration and consequently improves the quality of the tests. In the conventional PSD procedure, the actuator motion is stopped when the testpiece reaches the target displacement so that reaction forces can be measured and the next target dis, placement computed. In CPSD, actuators are not stopped; instead the MCS servo-controller moves them in such a way that the testpiece follows the target displacement continuously. The forces are measured at every control sampling period (0.001 see) and the equations of motion integrated in real time. The current test programme shows that there are no difficulties in dealing with non-linearities. This research by JRC Ispra is a major step forward in the use of reaction-wall testing techniques. Progress in the same direction, ie. real-time substructure testing, has been made by the Structural Dynamics Laboratory at the University of Oxford (Williams et al, 2000), although on a much smaller scale. ILLUSTRATION OF RESEARCH TRANSLATED INTO PRACTICE In almost all aspects of construction industry research, the benefits are not usually apparent to the untutored eye. They are to be found in the detailed design, which is obscured by surface finishes, the use of load absorbing materials, and in more economic and safe construction practices. These observations are certainly true in earthquake engineering, where research is often concerned with the theoretical and experimental determination of efficient and predictable load paths through the structure, causing predetermined members to absorb dynamic energy through inelastic deformation. Such members can be allowed to fail, but without causing collapse of the complete building or structure. With proper design and construction the failed members can be replaced and the building made useable within a short time after the earthquake. It is possible, nevertheless, to find illustrations where research has provided visually obvious examples of the diminution of seismic risk. Figure 25 shows an eccentric bracing system used to support the first year at a school in Italy; the brace elements contains a component which is designed to absorb seismic forces, and possibly fail. Figure 26 illustrates the use of a base-isolation system on liquid natural gas storage tanks in Greece. The lower picture shows the distributed arrangement of base isolation devices are designed to allow the structure to displace with the earthquake, returning afterwards to its initial position. Clearly, such devices can only be used for isolated structures. CENTRIFUGES Within the context of research infrastructures for earthquake engineering, centrifuges should be mentioned, since several of international calibre exist in Europe. They are chiefly used for studies of the static behaviour of model-scale geotechnical structures, where the deficiencies of gravity introduced by scaling, are overcome by centrifugal action when the model is placed in a rotating container. When the geotechnical structure is additionally subjected to an earthquake, attempts have been made to reproduce this by causing the model to rotate, not on a flat plane, but on a so-called 'bumpy road', the irregularities in which cause accelerations which are an approximation to the true earthquake input. To overcome the serious inaccuracies of this process, thoughts have been given in Japan to siting a small shaking table within a large centrifuge, but this has not yet been achieved. STEWART VIBRATION PLATFORMS As stated in the paper, almost all modem shaking tables employ 8 actuators to produce motion in 6 degrees of freedom (DOF). Kinematically there are two problems associated with control of the table. First, the direct kinematic problem requires that the position of the table be found, given the length of the 8 actuator rams. Second, the in which a given position of the table must be converted in 8 actuator ram lengths. This second calculation is straightforward, but the first is more difficult, requiring numerical appropriations; satisfactory methods have however been recently devised (Stoten & Gomez, 1999). It is important to note here that shaking table control requires the removal of relatively small errors; the initial approximation is good, and so the iterative process converges rapidly.

203

The question naturally arises as to whether 6DOF could not be controlled by only 6 actuators, thereby apparently simplifying the direct kinematic problem. The Stewart/Gough platform is such a device (Fig. 27), but it has been shown since its inception to be of value principally in vibration problems where the 3 rotations are of at least equal importance to the 3 translations; such problems are to be found in aerospace and land vehicle simulations. Figure 27 shows that the six actuators must have specified directions to achieve this, and that the base occupies a considerably bigger area than the platform. It is required that the 3 rotations shall be large, which means, contrary to expectations, that the direct kinematic problem is not easy to solve and research here is still in progress. In summary, shaking tables and Stewart platforms have been devised for different purposes. For the former, the usual requirement is for precision in one of the directions in which some of the actuators are oriented (X,Y or Z), with motion in all other DOF being essentially errors to be corrected. For the latter, all DOF are significant. It can therefore be argued that, despite the number of actuators exceeding the DOF, the shaking table is somewhat easier to control. MAJOR TECHNICAL REPORTS The following 9 references are the Major Technical Reports written as a result of collaboration between ECOEST and PREC8 partners, with funding through a Concerted Actions contract. Copies can be obtained from the publishers, LNEC, Avenida do Brasil 101, Lisboa, Codex, Portugal. 1. Standardisation of Shaking Tables Editor: Adam Crewe 2. Seismic Behaviour and Design of Foundations and Retaining Structures Editors: Ezio Faccioli and Roberto Paolocci 3. Large Scale Shaking Tests of Geotechnical Structures Editor: Colin A. Taylor 4. Experimental and Numerical Investigations on the Seismic Response of Bridges and Recommendations for Code Provisions Editors: G Michele Calvi and Paolo E. Pinto 5. Pseudodynamic and Shaking Table Tests on R.C. Bridges Editor: Artur V Pinto 6. Experimental and Numerical Investigations on the Seismic Response of R.C. Infilled Frames and Recommendations for Code Provisions Editor: Michael N. Fardis 7. Numerical Investigations on the Seismic Response of R.C. Frames Designed in Accordance with Eurocode 8 Editors: Eduardo C Carvalho and Ema Coelho 8. Shaking Table Tests of R.C. Frames Editor: Panayotis Carydis 9. European Activities for the Development of Eurocode 8 - Summary Report Editors: Roy T. Severn and G Michele Calvi The following 9 references are the Major Technical Reports which are to be written as a resultof collaboration between ECOEST2 and ICONS partners, with funding through a Concerted Actions Contract Publication by LNEC Lisbon is expected early in 2001. 10. 11. 12. 13.

Seismic Actions Editor: E. Faccioli Assessment, Strengthening and Repair Editor: E C Carvalho Innovative Design Concepts Editors: M Fardis and M Calvi Composite (Steel/Concrete) Structures Editor: A Plumier

204

14.Shear Wall Structures Editor: P Reynouard and M Fardis 15.Soil Dynamics and Foundation Structures Editors: C Taylor and D Combuscure 16.Base Isolation- Active and Passive Systems Editor: G Franchioni 17.Special Systems: Cable-stayed Bridges; Irregular Bridges; Asymetric structures Editors: R Severn and C Taylor 18.Developments in Control of Experimental Facilities Editors: D Stoten and G Magonette REFERENCES Gomez E G & Stoten, D.P. 2000, A Comparative Study of the Adaptive MCS Control Algorithm on European Shaking Tables. Paper 2626 Proc 12 WCEE, Auckland, N Zealand Gomez, E.G. 1999, Application of the MCS algorithm to the Control System of the Bristol Shaking Table. PhD thesis, Univ. of Bristol Igarashi A, Iemura, H & Suwa, T. 2000, Development of Substructural Shaking Table Test Method, Paper 1775, Proc. 12 WCEE, Auckland, N. Zealand Severn, R.T. 1998, Earthquake engineering - A European Group of Large-Scale Facilities within the Training and Mobility of Researchers Programme, Eur 18445. European Commission, 1998 Sevem, R.T. 2000, Earthquake Engineering - Research Infrastructures, Conf. On Research Infrastructures, Strasbourg, ISBN 92-894-0028-5. 31 pages. Stewart D. 1965. A Platform with Six Degrees of Freedom, Proc. I. Mech. E, 180(5):371-386 Stoten D.P. & Gomez, E.G. 1999, MCS adaptive control on shaking tables using retrofit strategies. IASTED Conf. On Control, Banff Canada Wagg D J, Severn, R.T. 2001 Multiple Support Excitation of Large Civil Engineering Structures. Submitted for publication to Earthquake Eng. And Struct. Dyn. Wagg, D.J. & Stoten, D.P., 2001, Substructuring of Dynamical Systems via the Adaptive Minimal Control Synthesis Algorithm, Submitted for publication to Earthquake Eng. and Struct. Dyn. Williams M.S. et al, 1998, Development of a real-time hybrid dynamic testing system, Proc. 6th SECED Conf., Oxford, UK, 373-380.

Figure 2 The EERC Bristol 6-axis, 8 actuator shaking table Figure 1 Shaking table performance testpiece

205

Acceleration(m/s a)

V ! I~ I !! ii

06

: 'i

'', ! v', i ~', I !I i l !~ i i !~ ~ i ; ', i .

','

!1

,I

,i

24.7

Figure 3 JRC Ispra reaction wall facility

.

24.8

.

~ f ~(

" f

!~ :I

:i .~

'

24.9

26

; ! i'.

2S.I

Time(s)

Figure 5 Reference and output using MCS with composite filters, target 10Hz l m/s

Power spectral density ofacceleration(W/Hz)

ISE(VZs)

:! 0.?

Output using M C S

°==

.

! ~ : ~I ,,

i

....

,

o.,,

,

,

,,f/"

,

/

o.~+ II I i Output using linear control

,

.....

0.4

8

,.-

MCS

i

~_Wd

(34)

where 19 is the orientation of the angle of the diagonal compressed bars relative to the element axis (in the first quadrant of the trigonometric circle), and 0 t - that of the transverse bars (in the second quadrant). The principal unknown in (34) is being 19, VRd2 can be maximized as per (6) d VRd2 / d 19 = Cos 2 19 + Cot 0t Sin 2 19 = 0 (d 2 VRd2 / d 192 < 0)

(35)

o t = - 2 1 9 ( 0 < 19< 45°), ot=180 ° - 2 1 9 (45 ° < c t < 9 0 °)

(36)

VRd2 -"

fRdbZTan19

(37)

According to [ 1], the experimental values of 19 fall in the interval 22.5 ° _< t9 < 67.5 °

(38)

By (36), at these limits of 19, 0t = 45 °, hence 45 ° 5.77 [Allen (1969)]. Bi-Steel typically ranges from 16.7 < d/t < 100. The position of the neutral axis, measured from the underside of the compression steel plate, is determined in accordance with traditional reinforced concrete theories, modified for the alterations in basic geometry, where the basis is formed from the equilibrium of forces. 1 2

z = - m ( t I + t2) + [m.(t I + t2) 2 - m(t~ - 2tch ~ + t 2 )]2

(1)

The moment of resistance of a DSC element can be determined by taking moments about the action of the concrete compressive force, Feu. A linear distribution of the force throughout the depth of the panel is assumed.

{

M = flbt I 3 +

+ fzbt2

(2)

hc - - 3 + - ~ ,

DSC's behave similarly to an under-reinforced concrete beam. That is, at the limit, yield of the tension steel occurs before crushing of the concrete. This is due to the sufficient available compressive capacity in the top steel alone to balance the tension forces. At the elastic limit, therefore, the stress in the tension plate is equal to the yield stress i.e. j~ = fy, and since the position of the neutral axis is known it is possible to calculate the stress in the compression plate, J), in terms ofj'3. Substituting into Equation 2.

M = f y bt, 3 +

z + t'/~ j

(

z

+ fybt 2 hc --~+

(3)

h c -z+t2~22

Hence, as yield of the tension steel occurs, f¢ < f~u and increasing moments are carried by an upward movement of the neutral axis. The total compression force remains constant. This type of structural mechanism is accompanied by cracking of the concrete, which is usually extensive, with wide cracks appearing in the latter stages before collapse. At yield, as the neutral axis begins to rise, the observed cracking of the concrete will continue to rise towards the compression steel plate. The moment capacity of the panel is reached when the neutral axis moves up to the lower surface of the compression plate (i.e. z = 0) and the stress in the tension plate is at the yield stress. By taking moments about the compression steel plate, the ultimate moment of resistance is, therefore: (4) In the case of the experimental work where the steel plates are of equal strength and thickness (i.e. t~ = t2), the ultimate moment of resistance becomes:

= Lbt(hc + t)

qA Tr..~t . . . . . . . i

--- dr ,~--~Ar = aT ATt dr e-- ds . "~'~ As = ar A Tt ds

t~

I

~t i t

.

.

.

.

.

l .

.

.

.

dO

(d)

i+1

.

M°,

i

"*i(e)

\

'ti

\

aaro

lVli(e)

i+1 (a)

(b)

(c)

(e)

~

Si

Figure 2: Temperature effects in circular plate If an infinitesimal element is free to expand in all directions its radial deformation (Figure 2(c)) due to the temperature difference AT (Figure 2(b)) is similar to that caused by radial bending moments. However, no stresses develop within it. The relative change in slope, dO, shown in this figure is:

dO =

.2As . T ,. . arAT 2arA~-ds = ds t t t

or

dO a r A T ~ = ~ ds t

(3)

Hence, owing to the assumption of small displacements (w), the radial curvature of this element is

ar Kr

--

dO ds -

d O _ d2w _W,,~ dr - -----T dr =

aTA..__~T t

(4)

Consider now a ring element of an idealised plate confined between two adjoining nodal lines i and i+l, and shown in Figure 2(d). If this element is separated from the adjoining parts of the plate and subjected to a uniform temperature difference AT, it deforms freely in such a way that its radial curvature follows Eqn. 4. Figure 2(e)shows the radial cross section of this element subjected to circumferentially uniform edge moments of radial sense and identical magnitudes. It can be shown (Timoshenko & WoinowskyKrieger, 1959) that this radial curvature is given by

KM,=

d2w Mi°(e) . -dr----5-=(l+v)Dip

i= El3 Dp v2-------12(1~_

(5)

where Dpi is the plate's rigidity, with E and v being the modulus of elasticity and Poisson's ratio for the plate; and where ti is the thickness of the plate element i. Assuming now that the edge moments case is combined with the case of the temperature difference and that as a result of this the ring element is re-flattened one can write ~r= Mi~e) tCff e + tCr ( i + v ) Dip

arAT_ 0 ti

(6)

511 Whence

o (I+V)aTAT i Mi(e) = Dp ti The edge

(7)

Mi°(e) (Eqn. 7) can be viewed as Fixed-End Moments (fixed-edge moments) resulting

moments

from the temperature difference AT and are accompanied by uniform radial and circumferential bending moments (Timoshenko & Woinowsky-Krieger, 1959) within the ring element of the same magnitudes. However, no transverse forces (shears) accompany these moments; the ring considered being in a state of pure bending. The fixed-end moments given by Eqn. 7 exert moments on the nodal lines i and i+ 1. Such moments associated with tings/-land i and acting on a radial cross section through the neighbourhood of nodal line i are shown in Figure 3(a). This figure shows also a fictitious clamp imposed on the node that prevents its radial rotation and provides the balancing moment given by ~?T

o o = M(i_l)(e ) - Mi(e) = (1 +

i-1

i /

V)aTAT Dp

(8a)

Dp ti

ti-1

where the symbols used are as already defined. --AT Mi

(a)

MiAT

__. ti-l~t~r~-~~i~Mo_---~

hdo

~

"" (i-1)(e)

(b)

i

i(e)

........ j

Figure 3: Clamped nodal line i and radial moment acting on it Releasing the fictitious clamp allows for rotation under the action of a radial nodal moment, numerically equal to the moment developed by the clamp but of opposite sense (Figure 3(b)). Hence M? r = M(°_l)(e)-

Mi°(e,

=

(1 + v)otTAT

i-I

Dp ti-i

i)

Dp ti

(8b)

Fixed-End Forces in Tank Wall induced by Temperature Changes Figure 4(a) shows a radial section through a circular tank whose wall is subjected to a temperature change field that is uniform in the circumferential direction and varies linearly across the wall thickness. This temperature change field is represented by rises (in °C ) on the wall inside and outside faces; each of the two rises being linear in the vertical direction. Such a temperature field can be represented by the temperature change at the wall middle-surface, denoted by T ° , and the temperature difference between the inside and outside faces of the wall, designated by AT. The vertical variations of these temperature changes are depicted in Figure 4(b) and their top and bottom values are: TOUt TO = -Tti°;- +" "t°P top 2 TOUt Tb°ot = ~ T l ~ i+° n"b°t t ;

2

ATtop = Trio;-- TOUt "top

(9a)

ATbot = Tt~t - T°Ut'bot

(9b)

512 iTemperature Change Inside TOUt o "top Ttop ATtop ,1.........................................................................

!

Tt~'~ --in

~ ~

'.~

NT

n :i~"Wall ~

-o~,

~1 ÷(i)

. ~~,w

,Tout Eft]

,~

[

(~

(~

~..-JI~|

~S,,/|~

o

~

Mx(j)x',a!

[

~

r

~ ......t;i---, ...................N - - l -... .............. o .................... iiij, e i T..°ut -~-J H T~/[f~ ATj~--~

x

Vo

f'_._~V~j)

~

........

m77,

! f:-~I~lJii!!::kt('> --~.,~~ ........ $___~f_'_ ......... ......... ~--. ~ ~ e w ~ ' ~ : i i i ] ' l ~ i ~i l ' 'iw ~~' i ~i ~ . i ::::~ T/-in il! E l e m e n t•~ - ~- .............................................. ] T/° AT/ ,,

.

1i

T~t"

4 ew

w!J

..............That

Z°.bo~t

.

.

.

.

.

.

o -f .......

.

/

-~x,j)

E

o

Mx(i )

...............'~ Tbot

Temperature Change Outside -j

(c)

(a)

(b) Figure 4: Temperature changes in wall and fixed-end forces in element i

The temperature changes T ° and AT associated with the typical element i of Figure 4(a) are: Ti° = Zb~t +

(loo p - Tl~ot o )hi. n

A~ = ATt,ot + (ATtop - ATbot)'--/~

o -- Tb~t o )hi n-t-~.i. Z;-Z~°t+ (Tt°P

ATj = mTbo t + (ATto p - ATbo t )

hi + ~.i H

(lOa) (10b)

For an unrestrained wall, the T°-temperature change component yields no bending, producing only a linear (in the vertical direction) change of the wall radius. Hence, the radial displacements and slopes of the wall at the top and bottom are: T° Utop

T°

= Un

o T° o = aTTtopR w• Ubo .~. o t = u T = C~r_oot__

O~bot= 0 top T°

T°_uT°

-- U n ~

H

--

o o (Ttop-T~ot)aTew

H

(lla) (11

b)

Consider now a typical.element of the wall (say element i of Figure 4(a)) assumed to be entirely restrained at all four edges and whose vertical section is shown in Figure 4(c), together with the fixedend forces caused by the AT-field (also illustrated in this figure). This element under the action of the AT-field retains its original shape if its edges are restrained against rotation. On the other hand, when the edges are allowed to freely rotate, the element deflects and its curvatures are directly proportional to the corresponding AT. The original shape of this element can be restored by applying edge moments such that the edge curvatures induced by them are equal to the negatives of the corresponding curvatures due to AT (similarly to the plate case already considered). Such moments can be regarded as fixed-end moments. The values of these fixed-end moments (meridional, M ° , and circumferential, M~ ) are: Mx(i, o = Me(i) o = Dw (1 + v)a~rAT/ tw

o = - M e (oj ) = - D w (1 + v)a r ATj Mx(j) tw

where Dw represents the wall rigidity defined by equation similar to Eqn. 5.

(12)

513 The fixed-end, transverse, radial forces at the element ends i and j (Figure 4(c) can be found as: Vr°(i, =--Wr°(j, = M°(i) + M°(J) - ( l + v ) c z r O w ~'i twgi

(AT/- ATj)

(13)

As shown by Eqns 12, the circumferential fixed-end moments due to the temperature change AT have the same magnitudes and signs as the corresponding meridional fixed-end moments. The variation of meridional bending moments related to these fixed-end moments is shown in Figure 4(c) (diagram M ° ).

ILLUSTRATIVE EXAMPLE Figure 5(a) shows a radial section of the reinforced concrete storage tank under consideration. The elastic properties of its material are Eb = Ew = Er = 20 GPa and vb = Vw = Vr = 0.2, and the coefficient of thermal linear expansion is a r = 1.1 x 10 - 5 / ° C . The tank rests on a soil modelled as an isotropic elastic half-space with Es = 20 MPa and Vs = 0.3 and is subjected to the following temperature change field: at the top: Tt base -- 10 ° C ; at the bottom:

Base Plate:

Tbbase : 5 ° C ,

Roof Plate: at the top:

Tt r°°f -- 40 ° C ; at the bottom: Tbr°°f = 15° C, 0o T wall at the bottom." inside . Twall "b(inside) "-- 1 C" outside: "b(outsiae) = 30° C,

Wall Cylinder:

at the top:

zwall

zwall

inside: "t(inside) = 1 5 ° C"

outside: "t(outside) "- 4 0 ° C

With the aid of the developed software, the tank is analysed under conditions of the given temperature changes only. Monolithic and hinge-type connections are considered and the plate in-plane deformations are taken into account. Since in tanks resting on elastic half-space media the accuracy of the solutions depends strongly on the number of elements employed to discretise the base plate, 220 uniform-width elements are used here to model this substructure, whilst the roof plate is represented by 110 uniformwidth elements, and only 40 (also uniform-width) elements are used for the wall cylinder. As shown by series of tests, these numbers of elements are entirely adequate to yield accurate results. Figure 5(b) illustrates the radial variation of bending moments in the base plate of the tank while the radial profiles

0.4 m :::::::7! :5::_ -i~.~- ..........

I

'~

-50 -30 -20

I

10.Om i i i i Eb, vb-~

, .......

.

/

i;

0 20

-!

'

I I

/ ii

10

.

(a)

Wall'~i::

-10

,~--,

0.4m

I I

0.5m

' FNdial(Monolit) ' Radial (Hinged) --n- Circumfer(Monolit) --x-- Circurnfer(Hinged)

-40

G , Vw - - /

!_......

BENDING MOMENTS IN BASE [kN-nYm]

r-.:.d

Er, Vr

10.0m

ll.0m

~"

. ._.,~_~~~~-~~

30 0

!

! -~!

Figure 5: (a) Radial section through tank;

1

2

3

4 5 6 7 8 Radial Distance [m]

=i ,,~

9

10 11

(b) (b) Variation of bending moments in base plate

514 of deflections in the base and roof plates, and bending moments in the wall are shown in Figure 6. As expected, the maximum bending moments in the tank with monolithic connections are significantly larger than those in the tank with hinge-type connections. Similar features can be observed in bending moments in the wall, the tank displacements (Figure 6), as well as in the structural response fields not shown here because of paper size constraints. DEFLECTIONS [ m m ] -2 -1.6 -1.2 •-0.8 t~-0.4 =" 0 cn 0.4 t~ m 0.8

-40 -35 -30 -25 -20 _~ -15 a. -10 o ° -5

1.2 1.6 2

0 5 10 0 1 2 3 4 5 6 7 8 9 1 0 1 1

Radial Distance [m]

n-

W A L L BENDING M O M E N T S [kN-m/m]

-110 -100 -90 -80

._..~x~ ~ ..~-~-,r -ad-l~'~" ~=--.

__~..~

-70

-'- ~ r 3 ~ d ~ "

/'/

-40 ~

-30 \ ~- / / -20 -10 0 ~ 10 0

j

Merid. (Monolit) Merid. (Hinged) Hoop(Monolit) - o - Hoop (Hinged) . . . .

/

1

2

3

+ ~

\

4 5 6 7 8 Height from Base [m ]

9

\

10

Figure 6: Radial profile of deflections in base and roof plates and bending moments in wall

CONCLUSIONS A numerical analysis of circular-cylindrical tanks for thermal effects is presented. The adequacy of this analysis is not verified in this work because of paper size limitations. However, appropriate tests have been conducted showing a very high level of accuracy of the obtained results. Accordingly, the results of the included example constitute good representations of the tank structural actions. Although the used tank and half-space properties, and also the temperature changes considered, are arbitrary, they are quite realistic. Therefore, one can notice that the temperature change effects in cylindrical tanks can be large. Since, as demonstrated elsewhere (see Priestley, 1976 or Ghali and Elliott, 1992), the current design practice of providing certain residual compression to prevent concrete cracking due to temperature changes is often inadequate, it is desirable to incorporate an appropriate analysis in the design of tanks. The numerical method presented in this paper is an adequate way of accounting for thermal effects. It is also important that this aspect of analysis is just a part of the computing capabilities of an efficient, available software package for analysis of cylindrical tanks (Melerski, 2000).

REFERENCES Ghali, A. and Elliott, E. (1992). Serviceability of Circular Prestressed Concrete Tanks, ACI Structural Journal, 89:3, May-June, pp. 345-355. Priestley, M.J.N. (1976). Ambient Thermal stresses in Circular Prestressed Concrete Tanks, ACI Journal, Proceedings 73:10, Oct., pp. 553-560. Wood J.H. and Adams. J.R. (1977). Temperature Gradients in Cylindrical Concrete Reservoirs, Proc. of the 6th Australasian Conference on the Mechanics and Structures, Christchurch, No. 2. Melerski, E.S. (1995). An Elastostatic Analysis of Cylindrical Tanks. Part I: Outline of the Approach, Proc. of the 14 th Australasian Conference on the Mechanics of Structures and Materials, Hobart, V. 2, pp. 410-415. Melerski, E.S. (2000). Design Analysis of Beams, Circular Plates and Cylindrical Tanks on Elastic Foundations, A. A. Balkema/Rotterdam.

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

515

ON THE POSSIBILITY OF PARABOLIC OGIVAL SHELLS FOR EGG-SHAPED SLUDGE DIGESTERS A. Zingoni Department of Civil Engineering, University of Cape Town Rondebosch 7701, Cape Town, South Africa

ABSTRACT In this paper is reported the results of a study on the possible adoption of the parabolic ogival shell for egg-shaped sludge digesters, which have become popular in relatively recent times owing to their superior functional performance and lower maintenance costs in comparison with conventional cylindrical digesters. The stress distribution in such shells is expressed in terms of a single governing parameter ~, greatly facilitating the investigation. For various values of ¢ covering the most practical range for egg-shaped digester shells, recommendations are made regarding the positioning of supports. Taking into account maximisation of tank capacity, minimisation of peak stress resultants in the shell, and ease of prestressing, the best range of ~ for parabolic ogival digester shells is identified. The overall conclusion is that from a structural and functional point of view, the parabolic ogival profile is suitable for adoption in the design of egg-shaped concrete sludge-digester shells. KEYWORDS Shell structures, Ogival shells, Egg-shaped sludge digesters, Containment structures, Shell analysis, Membrane hypothesis, Bending theory of shells, Shells of revolution INTRODUCTION Egg-shaped shell-of-revolution sludge digesters offer distinct advantages over conventional digesters of wide cylindrical shape. The smooth curved profile of the egg-shaped digester permits better mixing of the sludge, while the greater volume-to-surface ratio of the egg shape reduces heat losses. The better circulation of sludge in the egg-shaped digester results in a reduced accumulation of deposits at the bottom of the digester, and consequent reductions in maintenance costs. Furthermore, the deposits that do settle to the bottom of the egg shell are easy to remove as they all collect in one relatively small area at the capular or pointed bottom of the digester, and the removal of the deposits may be carried out on a continuous basis. Similarly, the capular or tapered shape of the top of the egg-shaped digester allows the crust that forms on the surface of the sludge to be removed more conveniently than were the crust spread-out over the larger surface area of the conventional wide cylindrical digester. Evidently,

516 the more complex geometry of the egg shell implies higher initial costs of construction, but these are offset in the long term by the lower maintenance costs. For all these reasons, a significant number of egg-shaped sludge digesters have been constructed in the recent past in countries such as the USA, Japan, Taiwan, Germany and Australia (Nojiri 1989; Sutter & Jager 1994; Jager 1997). Despite this fairly widespread adoption of the egg shape for sludge containment, not much information is available in the literature on the detailed analysis and structural behaviour of egg-shaped sludge digesters, which partly explains why these large shell structures are not as common as they should be in other countries around the world. In an effort to increase the analytical data available to the designer of these structures, a study has just been completed on membrane and discontinuity effects in eggshaped sludge digester shells comprising spherical ends and a middle circular ogival portion (Zingoni 2001). Noting the potential of the parabolic ogival shell as a form of egg-shaped sludge digester (the parabolic ogival shell has pointed ends and a bulging middle), the present study evaluates the structural feasibility of this shape on the basis of shell theory. Results of relevant stress resultants are presented in generalised parametric form, and design recommendations are made. EQUATION OF THE MERIDIAN AND PRINCIPAL RADII OF CURVATURE By reference to Figure 1, the shell of revolution in question is formed by rotating a parabola that is symmetrical about the horizontal x axis, about the vertical y axis (which therefore is the axis of revolution of the shell). The shell is therefore symmetrical not only about the vertical y axis of rotation, but also about the horizontal "equatorial" plane containing the x axis. Let the overall height of the shell be H , and the equatorial diameter be D, as shown in Figure 1.

H/2

~

_

H/2

~

,

~

Figure 1: Geometrical parameters of the parabolic ogival shell of revolution With the origin 0 taken at the intersection of the axis of revolution and the equatorial plane, the equation of the generating meridian of the shell of revolution is ---x=ky 2 where k is a constant.

(1)

517 When x = 0, y = +H / 2. From this condition, it follows that 2D k = .~ H

(2)

From Eqn (1), and making use of result (2), we may write

Y =-4~~, 2 -x

=+~

-x

(3)

If we define the angular coordinate ¢~ as the angle measured from the upward direction of the axis of revolution of the shell to the normal to the shell midsurface at the point P in question (refer to Figure 1), it is evident that

tan : - ~+:_+ 2 2 , ~

-x

1-'''

(4)

At the upper pole (x = 0, y = +H / 2 ), if ¢ = #o, then

¢~o = tan-~ ( + - ~ )

(5a)

Similarly, at the lower pole (x = 0, y = - H / 2 ), if # = #', then

(Sb)

From Eqn (4), we have tan s ¢ = ~ -

.-~-- x

(6)

leading to the result x =

4D 2 sin 2 ~ - H 2 cos 2 ¢~ 8Dsin 2 ¢~

(7)

When ~ = 90 ° , Eqn (7) yields x = D / 2, as expected. At any given point P of the shell midsurface, the two principal radii of curvature are denoted by fi and r 2 . The first (r~) is the actual radius of curvature of the parabolic meridian at the point P , while the second (r 2 ) is equal to the distance between P and Q, where Q is the point of intersection of the axis of revolution of the shell, and the normal to the shell midsurface at point P (refer to Figure 1). Thus, from Figure 1, we may write

518

r2

R

X

4D 2 sin 2 #- H 2 cos 2 #

sin #

8D sin3 #

(8)

The other radius of curvature is given by the usual relationship

(9)

d2y dx 2 Now, from Eqn (4),

~x-q =-T-42,~.~.

(10)

-x

Equations (4), (7) and (10) enable the evaluation of r 1 from Eqn (9), yielding

.'

'=

rl--

=

4DH

4Dsin 3 #

(11)

(The negative symbol has been dropped, since r l is positive throughout.)

LOADING COMPONENTS AND SHELL STRESS RESULTANTS Assuming the shell is completely filled with liquid of weight 7" per unit volume, the depth of liquid at the vertical coordinate y is given by H d =---y 2

(12a)

From Eqns (3) and (4),

y=_+~

-x

= -+~

+2 24~tan# =4Dtan#

(12b)

Equation (12a) becomes d .

H . .

2

H2 . . 4D tan #

2HDsin # - H 2 cos# 4D sin #

(12c)

Thus, the loading component Pr normal to the shell midsurface is given by

Pr = 7' d = y

2HDsin # - H 2 cos#] 4Dsin #

(13)

519 The loading component p# in the direction of the tangent to the shell meridian is, of course, zero, since hydrostatic pressure acts purely perpendicular to the shell midsurface (Eqn 13). As is well known, for shells of revolution subjected to distributed loadings that vary smoothly, continuously and "not too rapidly" (Hildebrand 1950) over the surface of the shell, the membrane or "momentless" hypothesis (Novozhilov 1970) accurately predicts the state of stress in the interior of the shell, provided the shell geometry (thickness of shell, slope of the meridian, principal radii of curvature) also exhibits the same smoothness properties. Both the loading and shell geometry of present considerations conform to these requirements, so that the membrane solution should be adequate throughout, except in the lowest zones surrounding the bottom pole, over which the shell is assumed to be supported. Since hydrostatic loading is axisymmetric, the only interior shell stress resultants of relevance in the present problem are N¢ (in the meridional direction) and N o (in the hoop direction); these are forces per unit length of the respective edge of a shell element, considered positive when tensile. General expressions for these are as follows (see, for instance, Fliagge (1973), Zingoni (1997) or Gould'(1998)): N~ = 1--------~[Irlr2(P, COS#- pc sin #)sin # d#

rEsin 2 #

+ C]

(14a)

N o=r 2 p~-

(14b)

where C is a constant of integration. Using expressions (8), (11) and (13) for rE, r1 and pr respectively, and noting that p~ =0, we evaluate the integral in Eqn (14a) which, after some simplifications, leads to the result

/

fl_/s sin # 16D 2 4D 2 sin E¢ - H cos 2 #

1[

D(4D 2 + H E) sin 2 #

•

DH 2

H(4D 2 + H2

2sin 4 #

(cos,

+

sin¢)

+ -H(4D2 +2H2) f c o s ; ) 0 +Esin 2 # ) - H 3 ( cos ~)(3 +4sin 2#+8sin 4¢) + C J 3 ~, sin3 - ~ sin 5

(15)

At the apex (¢ = #o ), N¢ = 0 so that

=

sin 2 0o

2 sin 4 #o

sin 0o

/

H(4D2 +2H2) f c o s , )0 H 3 ( c 0 s ' ) ( 3 + 4 s i n 2 ' +8sin4 ' ) ~,sin3 #o + 2 sin 2 #o ) + - ~ sin 5 #o

(16)

With N~ now known, the hoop stress resultant follows from Eqn (14b) which, after eliminating r~, rE and Pr, may be rewritten as N o = (4D 2 sin 2 # - H 2 cos 2 #

y(2HD sin # - H 2 cos #)

3 - ~ sin a #

-

N,.1

2H 2

(17)

520 The actual stresses in the meridional and hoop directions are calculated in the usual manner:

N¢ Na ere-~ ; o-o - ~ t t

(18)

VOLUME CAPACITY OF TANK Considering elemental horizontal discs of radius x and thickness dy, the volume of the tank is twice the integral summation of such discs between the equatorial plane and the apex, that is H

V = 2 So ~ x 2 dy

(19)

D ky 2 D 2D 2 x . .2 . . . .2 . H 2y

(20)

From Eqns (1) and (2),

Therefore -.~ ( D 2

4

D 2

-~Y

H

HD__7~ / n[_.~ 2D2 3 4D2 51~ 2 ~ D 2 H +4 y4 dy = 2 Y - - ~ - ~ Y +~---H-TY = 1---5

(21)

PARAMETRIC RESULTS Making use of Eqn (5a) to eliminate sin¢~o and cOS~o from expression (16), we obtain, after simplifications C=

D 012D4 +120D=H2 +15H4) 30H 2

(22)

Defining a non-dimensional parameter of the ogival shell H ( =m D

(23)

(that is, ( is the height-to-diameter ratio of the tank) allows Eqn (22) to be recast in the form 9 3

C = ~(112 :~o¢" -

+ 120( 2 + 15(4)

(24)

The constant C may be eliminated from expression (15) on the basis of relation (24); when further use is made of relation (23) to eliminate D from expressions (15) and (17), the following nondimensional form of the results for the stress resultants is finally obtained:

521

g H 2 -1---6 4sin 2 # __(2cos 2#

sin 3

-

sin 2# + 2 sin4 # - ( ( 4 +

0 + 2 sin 2 #)_ ~(3

(cos 3 sinS#)

1

+ 30(2 012 + 120~ 2 + 15( 4

l .sin

7 H 2 - 32(2

x - sin/~¢ +

2 sin 4 #

COS2# -~'(4+

~n~ # \sin #

+

+ 4 sin # + 8 sin 4 ~) +

)]

(25a)

3

sin # ) +

',o.0o~I

o oo I ,/', .-

.

_..

----

.o.oo4o.oo2 o I 0.0020.004 \ It \ ',-O.OOei \

""i

Xl -0.01

,,:

oo, t

"

~.

,,,.

:: . . . . .

I

"O.OOe I \. / -0.01

t \'

,,,,0.

0.002 ....

0.004

: ....

:,,

/'

.o.oo4o.ooz o}, 0.002,0.004 I 1',, -7

/I

....

.se-o7

.4o-o:, -5e-O';

Figure 3: Through-thickness variation of the displacements for the eighth mode of the simplysupported adaptive sandwich plate.

555 Electric potential amplitude

Transverse electric displacement amplitude

-1800.

/

/

/

/'

//

/

// /

/ 1800.

0.008

14ooi

0.004

1200i

o .oo2 /

'J~,

Ioooi 8ooi

-o.oo~i

\~

-0.004 /"

',~,

200

-0.004 -0.002 0

-o.oo4 -o.oo2 0

'\

4oo

::

: . . . : :

/

/,'

"

eooi

/

...

-o.ooe

:.

0.002 0.004

Figure 4: Through-thickness variation of the electric potential and transverse displacement for the eighth mode of the simply-supported adaptive sandwich plate.

0.04 ~

y

o.oo

o.oo 0.08

~

0.1 v

0.04

x

0.08

0.1

Figure 5: Modal shape of the simply-supported adaptive sandwich plate ninth mode Vx displacement amplitude

o.o~

/

/"~

/o.bo~I \

i /

.... ,/...... !\~........ /I...

/

-ooo t'\ / -0.011

'\//

displacement amplitude

/oP'

/

/ \,~ .....'...... _'~

/// . ./

.0.0040.002 0~i: 01002,0.004 ~ /' -o.ooe~ ' '\ /-' I

~ -0.01-~

Vz displacement amplitude

o.ool.

o.o~I

/

-0.0040.002 0~ \ 0.002~.004 / -/

/

Vy

',/ v

0.0008 0.0008

0.0004 o.ooo2

-o~0040~002:o ;:o166z01004

Figure 6: Through-thickness variation of the displacements for the ninth mode of the simply-supported adaptive sandwich plate.

556 Electric potential amplitude -0.004-0.002

0.002

\ . . . . . . . . -2ooooot .. o ...........

\

-4ooooot

\, .ooooooi ',,,

\,

-8ooooo.~

/

• -1'.8e+06.

/ --._

/ /

',,,

Transverse electric displacement amplitude

0.004

/

/

/:~ •~ . . . . . .

-0.004-0.002 0t~

I--J,,,L

0.002 0.004

"1"" "

\

-

Figure 7: Through-thickness variation of the electric potential and transverse displacement for the ninth mode of the simply-supported adaptive sandwich plate.

CONCLUSION A closed form two-dimensional solution for the free-vibrations analysis of simply-supported piezoelectric adaptive sandwich plates has been presented for the first time. It has the originality to consider for each layer a quadratic electric potential and a first-order shear deformation theory. The formulation was entirely derived using M A P L E V symbolic software. Hence, avoiding complexities and numerical difficulties due to explicit exact solution of sixth/eighth-order differential equations or static space approach, often retained for the exact three-dimensional solutions. The present approach has been already used for physical understanding of the electromechanical coupling due to the piezoelectric effect. It is clear then that its results can be used for building more sophisticated approximate and numerical approaches. Currently, the static analysis is being implemented and, the shear actuation/sensing mechanism, see Benjeddou (2000bc), obtained with in-plane polarized piezoelectric materials, is also being investigated using this approach.

REFERENCES Batra R.C. and Liang X.Q. (1997). The vibration of a rectangular laminated elastic plate with embedded piezoelectric sensors and actuators. Computers & Structures 63:2, 203-216. Benjeddou A. (2000a). Advances in piezoelectric finite element modeling of adaptive structural elements: a survey. Computers & Structures 76:1-3, 347-363. Benjeddou A. (2000b). Transverse shear piezoelectric actuation of shells. In M. Papadrakakis, A. Samartin and E. Ofiate (eds.), Fourth International Colloquium on Computation of Shells & Spatial Structures, ISASR-NTUA, Athens, Greece. Benjeddou A. (2000c). Piezoelectric transverse shear actuation of shells of revolution: theoretical formulation and analysis. Presented at the European Congress on Computational Methods in Applied Sciences and Engineering, Barcelona, Spain, September 11-14 (Invited Lecture). Heyliger P.R. and Saravanos D.A. (1995). Exact free-vibrations analysis of laminated plates with embedded piezoelectric layers. Journal of The Acoustical Society of America 98:3, 1547-1557. Pagano N.J. (1970). Exact solutions for rectangular bi-directional composites and sandwich plates. Journal of Composite Materials 4, 20-34. Rahmoune M., Benjeddou A., Ohayon R. and Osmont D. (1998). New thin piezoelectric plate models. Journal of Intelligent Material Systems and Structures 9:12, 1017-1029. Saravanos D.A. and Heyliger P.R. (1999). Mechanics and computational models for laminated piezoelectric beams, plates and shells. Applied Mechanics Reviews 52:10, 305-320. Xu K., Noor A.K. and Tang Y.Y. (1997). Three-dimensional solutions for free-vibrations of initially-stressed thermoelectro-mechanical multilayered plates. Computer Methods in Applied Mechanics and Engineering 141, 125-139.

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

557

PLASTIC BUCKLING OF SPHERICAL SANDWICH SHELLS UNDER EXTERNAL PRESSURE S. C. Shrivastava Department of Civil Engineering and Applied Mechanics McCAll University Montreal, Quebec, Canada H3A 2K6 ABSTRACT The bifurcation buckling of spherical sandwich shells stressed by uniform external pressure beyond the elastic limits of the component materials is presented. The analysis accounts for the transverse shear deformations and is based on the inelastic behaviour according to J2-incremental and J2-deformation theories of plasticity. Results for elastic or plastic buckling of one-material shells follow as special cases. KEYWORDS Bifurcation, Buckling, Plastic Buckling, Sandwich Shells, Spherical Shells, Transverse Shear Effects 1. INTRODUCTION The problem of bifurcation buckling of a thin homogeneous elastic complete spherical shell under uniform external pressure was solved by Zoelly (1915), Schwerin (1922), and Van der Neut (1932), see Timoshenko and Gere (1963). The present paper deals with a generalization of this problem in two respects: (1) the shell is of sandwich variety, and (2) the component materials may undergo plastic straining before buckling. These require consideration of the transverse shear effects and the constitutive relations of a strain-hardening plasticity theory. Bijlaard (1949) was one of the first investgators to deal with the plastic buckling of plates and shells in a rational manner. However, he used only the J2-deformation theory of plasticity and considered only the buckling of sandwich plates in his papers. The present work, dealing with sandwich shells, considers the constitutive relations of both the competing theories of plasticity, namely the J2-deformation and J2incremental theories. The theoretical formulation is similar to the author's previous works (1995). Although the incremental theory of plasticity is considered to be the correct phenomenological theory, it is well known that when applied to bifurcation buckling of plates, it predicts in some cases unrealistically higher results than the experimental data. In contrast, the deformation theory results are in safe-side agreement with experiments. However, for the buckling of cylindrical shells in axial compression, the differences are not as significant, Batterman (1965). The present paper provides this comparison between

558 the two theories for the spherical shell buckling. Additionally, we must note that whereas the classical bifurcation results for shells are rendered useless by the imperfection sensitivity in the elastic range, they become meaningful in the plastic range as shown experimentally by Batterman (1965) and others. The presented analysis is exact and leads to formulas generalizing the classical results. The analytical results are illustrated by plotting the nominal buckling stress for sandwich shells of 24S-T3 aluminum alloy faces and a low modulus elastic core. The wrinkling instability of individual faces is precluded. 2. CONSTITUTIVE RELATIONS Figure 1 shows the spherical sandwich shell and the coordinate system ( ~ - 0 - z)used in the analysis. The middle surface radius is denoted by R, the core thickness by h, and the face thickness by t. The shell thickness is assumed much smaller than the radius. The prebuckling states of stress and strain are assumed to be those induced in a thin shell by a uniform all-round external pressure. For a complete sphere, isotropic material behaviour implies a prebuckling state of uniform strain, ~ = ~ee = constant in the faces and in the core (independent of the coordinates) in elastic as well as plastic range. However, Pole [

•...~. ;>.. O) are taken, see Bijlaard (1949), as (2.1)

a¢¢ = B'e¢¢ + Ceoo, aoo = C'e¢¢ + B%oo, ffij = 2F%ij (i ¢ j) where B' = E()~ + 3 + 3e)/{(1 + )~ - 2u)(2 + 2u + 3e)}

(2.2a, b, c)

C' = E(1 - A + 4u + 3e)/{(1 + A - 2u)(2 + 2u + 3e)}

F' = E l ( 2 + 2u + 3e). The parameters )~ = E / E t and e = ( E / E s ) - 1 are obtained from a uniaxial compression test; Et and E~ being respectively the tangent and secant moduli at stress level a. As written, these relations hold for the J2-deformation theory. They apply to the J2-incremental theory when e - 0, and to the elastic case when e = 0 and )~ = 1 are substituted in Eqns. (2.2). Additionally, we note that C r = B' - 2F'.

3. G O V E R N I N G EQUATIONS To account for the transverse shear deformations, the governing equations are derived from the modified Love-Kirchhoff kinematic hypothesis: a normal to the undeflected middle plane remains straight but not necessarily perpendicular to the deflected middle surface. The ¢ - 0 - r displacement components arising due to buckling are therefore taken as (3.1)

= u ( ¢ , o) - z ~ ( ¢ , o), v = ~ ( 0 , o) - z Z ( ¢ , o), w = ~ ( ¢ , o)

where u - u(¢,0), v - v ( f , 0)are the in-plane (i.e. membrane) displacements, and a - - a ( ~ b , 0 ) , /3 --- ~0(¢, 0), w - w(o~,O) are respectively the rotations of the normal and out-of-plane normal z

displacement. The mid-surface radius R is defined by z - 0. Neglecting ~ terms in comparison to unity (r = R + z ~ R), the matrix [L] of the buckling displacement gradients is

I

ILl =

Ou ,

=

,

z Oc~

Ov

z-k

O~

w

)

1 Ou 0ce 1¢0 : Rsin¢ ( - - - ~ - z - ~ ) -

10w

= -R

u

-k

z/31

1 Ov Off cote loo = Rsind~(ff0 - z-0-~)__+ ~ t ( u - z a ) + 1 Ow v R l , . o - Rsin¢ 00 F

l

cot~ -~(v-

lCr -- -- c~

w lo~ =

-

~

l~ -- 0 (3.2)

560 The equilibrium equations and the boundary conditions, appropriate to the above kinematic assumptions can be obtained by applying the principle of virtual work, Washizu (1975), to the buckled sphere: (a,~6L,j + ~hjLm,6L~j)dV -

{tr([a][6L]) + tr([ ~r][L]T[6L]))dV = 0

(3.3)

where ~ijis the initial (prebuckling) stress field (with ~¢¢ = $00 and other components zero), aij that due to buckling (where only art is assumed zero), and Lmi and 6Lmj are respectively the buckling displacement gradients and their virtual variations. Since the buckling of a complete spherical shell is being considered, only the equilibrium equations (and not the boundary conditions) are needed. We obtain these equations as 0N¢¢sin¢ sin¢0¢ ONoo sine00

ON¢o u sin¢00 ~-cotdpgoo - Q¢ + P¢~{(V 2 - cosec2¢) R

v 0N¢0sin¢ _ cotCN¢0 - Qo + Pc~{(V 2 - cosec2¢) R sin¢0¢

c3Q¢sin¢ sin¢0¢

OQo w sine00 t- N¢¢ + Noo + P~{(V 2 - 2) R

0sinCM¢¢ OM¢o + Rsin¢ 0¢ Rsin¢ 00

20usin¢ Rsin¢0¢

2cot¢ Ov 20w Rsin¢ 00 t- ~- ~--~) = 0 2cot~bOu Rsin¢ 08

20w Rsin¢0O} = 0

2 Ov Rsin¢ 0--0} = 0

cote - - M o o - Q¢ = 0 R

OMoo cOM¢osin¢ ~ Rsin¢ 0-----~+ Rsin¢ 0¢ +

M¢o - Qo = 0

(3.4a, b, c, d, e)

02 (9 02 where V 2 = (~-¢--~+ cot¢~-~ + cosec 2¢0--~)

(3.5)

and P~ is the buckling load parameter (3.6)

Per = 2~rlt + ~r~h - q~R 2

in which qcr is the critical external pressure, and ~f and ~Cdenote the uniform prebuckling compressive stresses in the faces and the core respectively. We accept Shanley's concept of plastic buckling under increasing load, so that there is no unloading from plasticity as the sphere deforms into a buckled shape. The loading relations (2.2)then apply throughout the shell thickness, and the buckling stress resultants, found by integrating through the thickness, are S l Ou

C10v

C1 cote u) + w (B~ + C1),

N¢¢ = (-~-~-~ 4 Rsin¢ 00 t- R

C10u B1 Ov BlCOtCu, Noo = ( - ~ - ~ + Rsin¢ 00 + R F10v 1 Ou N¢o = -~(~-~ - vcot¢ + si--~-~),

(B1 +C1),

(3.7a, b,c)

561 62 0/3 4- 62cot¢ a), Rsin¢ 00 R

M¢¢ = - h2(~--~20a

Moo = - h2( mce--

_•0a

B2 0/3

B2R0tCa),

- ~ + Rsin¢ 00 +

(3.8a, b, c)

- h 2F2 03 - 3 c o t ¢ + ~ -1- Oa -~- (~--~

sin ¢ 00 )'

low Q¢=G(Ro¢

u R

a), Qo = G (

1 Ow Rsin¢OO

v R

(3.9a, b)

fl)'

where tB'f tC'l B1 = B~¢h{l+ ~ c }, C, = C~'h{l+ ~ } , F 1 B'~h

B'l

Be= ~{1+

~f(t)},(?2=

C~ h

Cl

~{1+

tF} -~},

= F'h{l+

~f(t)},F2=

(3.10)

F'~h {1 + F}

~

Fc' f(t)},

G = kF~h and f ( t ) = 6(t/h) + 12(t/h) 2 + 8(t/h) 3.

(3.11) (3.12)

k is a correction factor accounting for the (actual) non-constant variations of shear strain and shear stress contrary to the kinematic assumption of the theory. As is well known k = 5/6 for elastic homogeneous plates and shells, and it can be shown that k ~-, 1 for sandwich cases of moderate facing to core thickness ratios, say t / h < 0.2. Substitution of these integrated relations in the five equilibrium equations then transforms them into the following governing equations for the variables u, v, a, 3, and w: (B1

-Pcr)

R2

B1

-

F1 02u

~72u + R2sin2¢ 002

BI - F1 02V

R2sin¢ 0¢00 B1 - F 1 02u

R2sin¢ 0¢00

2F1

G B1 - P c r ~ u - R----Tu+ --~u + R2sin2¢

G _ p . ) R20¢ Ow + --~ aa = 0

B1 + F~ - 2P~ cot¢~_~ _ 2(B~ - FI+-~ R2sin¢ B1 + F1 + 2P~ cote Ou R2sin¢ 00

F1 - P~T

R2

(3.13)

B1 - F1 02v

~Z2v- R2sin2¢ 002

/71 2El G cosec2¢ G 1 Ow G3 -/t ~(~-4-cOsec2¢ -- -~- + -- Pcr R 2 ) v - 2(B, - F I + -~- +Pcr) RZsin¢ 00 + - R - = o (3.14)

B2.02a R2

Oa

F2

02a

+ cot¢~aq~ - a c°sec2¢) - (R2sin2¢ 002

(/32 - F2) 0 2/3 (/32 + F2) cot¢ O~ R2sin¢ 0¢00 I R2sin¢ O0

G Ow G u h2R 0¢ t- -~-~ = 0

(3.15)

562

(B2 F2) 02a Resin¢ 0¢00

(B2 + F2)cot¢ Oa Resin¢ 00

-

/32

0 2 / ~ 2 F 2 ~ _ G/~

-- (R2-strl2¢ 002 + - ~ "

{ 2(B1 - F1) R2

G

2P~

P~

G..02w

Re ) ( ~

((gefl

cot¢~-~

flcosec 2¢)

G Ow Gv )-- h eRsin¢ O0 ~- ~ = 0

1

Ou sin¢

R 2 } s-~n¢(

~- R 2

+ ( Re

~

F~

R e , 0¢2 +

cgv

0¢

Ow

Oa sine

G

+ 0--0) + Rsin¢ ( . t92w

+ cot¢~-~ + cosec 2¢ ~ )

+

(3.16)

0¢

0~ + ~ )

4B1 - 4F1 - 2P~ Re w = 0.

(3.17)

4. SOLUTION FOR AXISYMMETRIC BUCKLING

Although the governing equations, Eqns. (3.13)to (3.17), are suitable for investigating general buckling, further analysis is restricted to axisymmetric buckling. This mode is found critical for the plastic buckling of cylindrical shells, Batterman (1965), and also for the elastic buckling of spherical shells, Van der Neut

o(...)

(1932). The axisymmetry implies 08 - (V12 - cosece¢ + (Vl2

GR 2 u B2 h2 R

-

2F1 - G u

= v =/~ = 0. The equations governing u, a, w become

Ga

B~ )-R+--~I -(2-

cosec2¢ + 2F2 - GR2/h 2 )a B2

2F1 - G

dw

B~"

) R--de = 0

GR 2 dw = 0 B 2h2 Rd¢

(4.1a, b, c)

4B~ - 4F1 + 2Per w G da sine P~, G ~- = o RsinCd¢ J B~ sinCd¢ ~-( B~* V~ + B---~

(2 - 2F1 - G) du sine B~

in which B~ = B1 - Per and V12(...) - sined de (sine d~.~.)). The structure of

this system of

simultaneous ordinary differential equations indicates a buckling mode of the form u

w

-- K1Pnl(cos¢), oL--- g2p~l~ (cos¢), ~ --

g3Pn(cosq~)

(4.2)

where/(1, K2, K3 are constants, and Pn = Pn (cos¢), P~ " P~ (cos¢) are the Legendre and associated Legendre polynomials respectively, and n corresponds to the number of half-waves in the buckling mode. The condition for a nontrivial solution, i.e. the bifurcation, is 2F1-G B~

G B--~ 2F2 - GR2/h 2

GR2/h 2

(2

B2 2F~-G )#

B~'

# -

B2

(2

-

2F1 G B~' ) -

GR2/h 2

G

B2 ( G - P~)# + 4B~ - 4 F 1 + 2P~

S~'#

B~

= o

(4.3)

563 which is arrived at by using the following properties of the Legendre polynomials: dPn

p1 where # = n ( n + 1).

(V 2 - cosec2~b)P1 = - #P2, V2Pn = - # Pn, de

(4.4a, b, c)

The solution is now complete. The buckling load obtained from Eqn. (4.3) is the following function of #: Pc,. - (B1 - F 1 ) { 4 ( R / h ) 2 F ~ G 2 ( G - 2F~)(#B2 - 2F2)} + # B I G ( # B 2 -G ( R / h ) 2 ( g B 1 - 2F1) + (#B2 - 2F2)(ttSl + G - 2F1)

- 2F2)

(4.5)

where for simplicity we have put B~' = B1 - P c r ~ B l b y presuming Pc,. > 1, we obtain 2F1 # = -~1 + ( R / h )

4F~ (BI - F1) {1 + ( BIB2

)

4F1 (B, - F1) x

}

(4.6)

BIB2

and the minimum buckling load as

2B2 14Fl(B, - F~){1

Pc,.- ( R / h )

BIB2

h i 4 F I ( B , - F1) × B2

- (R)

BIB2

-~

}

(4.7)

where in the expression for Pc,. we have omitted some additional higher order terms not containing the core shear stiffness G. These formulas generalize the classical results of the elastic thin shell buckling to the plastic case (d2-incremental or J2- deformation) by virtue of the stress-dependent moduli and to the sandwich case by addition of the transverse shear terms, i.e. the terms containing G. These terms, both for # and Pc,., are rendered negligible when G is large or h / R is small. The results pertaining to the plastic buckling of one-material (homogeneous) shells of thickness h are obtained by taking t / h = 0 and considering the core to be an elastic/plastic material. The substitution of A = 1, e - O, t / h = O, G --~ oo reduces the Pc,. to the exactly equivalent expression obtained by Zoelly (1915), Schwerin (1922), and Van der Neut (1932) for thin shell elastic buckling.

5. NUMERICAL RESULTS The numerical results pertain to a shell consisting of 24S-T3 aluminum faces and a low modulus core. The alloy elastic moduli are taken as E I = 11,100ksi, uf = 1/3 with a stress-strain (a - e) curve

~-

° ); 11, 100 ~°'°°2(4-~.5

(5.1)

(a in ksi units) valid up to a stress of 45 ksi (310MPa), with plasticity setting around 25ksi (170MPa~ The core is assumed to remain elastic with moduli Ec = 53.2 ksi (367 MPa)and t,c = 0.4.Figure 2 shows plots of the critical nominal buckling stress, at,., defined by

564

e. ac~ = 2t + hE~/EI

=

q.n/:

(5.:)

2t + hE~/Ef

against R / h for t/h = 0.1, 0.15. This stress, calculated by using Eqn. (4.5) and the approximation ?rc = (Ec/E I )~rf, is close to the buckling stress in the aluminum faces. The results are similar to those obtained for the axially compressed cylindrical sandwich shells, Shrivastava (1995b). The incremental theory results are always higher than those for the deformation theory, the differences increase from nearly negligible at the beginning of plasticity to roughly 10 % at the upper end of the validity of the a - e curve. It may be noted that the buckling occurs somewhat earlier, i.e. at a lower face stress, for shells with thicker faces and therefore relatively thinner cores (t/h = 0.15 versus t/h = 0.1).

~

~1 .._

45

Cn

40

" " ~, ~ , ',

:

~

~ "~,q ,~,,~

"~,,,j

. . . . . . . . . :. . . . . . . . . :. . . . . . . . . . . . . . . . . . . . . . . . .

I;. ~ t/h-n ', _..-. . . I_=; . /', :

~ ' ~ ~ _ _ _

:

Full Spherical Nandwich ~, h e l l . . . . ~-,= ,ncrf)mentat r, n e o r y ,, D= D e f o r m a t i o n : T h e o r y :

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.E

=ca

D: t/h=0.1 35 .........

~~.~.~~"'.~,'

,:-. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-: . . . . . . . .

: ..........

¢,,,) 30 . . . . . . . . . . . . . . lO ""r-

;

~II

i

l

O

z

254t5

i 70

l

95

' 120

,

i

J

~ 145

i

1 .170

195

i 220

245

N/h=Radius to Core Thickness Ratio

Fig. 2: Variation of the Nominal Critical Stress with Radius to Core Thickness Ratio.

REFERENCES B a t t e r m a n , S.C. (1965). Plastic Buckling of Axially Compressed Cylindrical Shells.

Journal of the

American Institute of Aeronautics and Astronautics, 3, 316-325. Bijlaard, P.P. (1949). Theory and Tests on the Plastic Stability of Plates and Shells. Journal of the

Aeronautical Sciences, 16:9, 529-541. Schwerin, E. (1922). Zur Stabilit~it der d0nnwandigen Hohlkugel unter gleichm/iBigem AuBendruck.

Zeitschrift fiir angewandte Mathematik undMechanik, 2:2, 81-91. Shrivastava, S. (1995a). Inelastic Buckling of Rectangular Sandwich Plates.

International Journal of

Solids and Structures, 32, 1099-1120. Shrivastava,

S. (1995b). Plastic Buckling of Cylindrical Sandwich Shells in Axial Compression.

Abstract, Society of Engineering Science 32nd Annual Meeting, Oct.29-Nov.2, New Orleans, 547-548. T i m o s h e n k o S. and Gere J.M. (1963). Theory of Elastic Stability, 2nd Ed., McGraw Hill, New York. V a n der Neut, A. (1932). De Elastische Stabiliteit van den Dunwandigen Bol, H. J. Paris, Amsterdam. Washizu, K. (1975). VariationalMethods in Elasticity and Plasticity, Pergamon PreSs, Oxford. Zoelly, R. (1915). Ober em Knickungsproblem am der Kugelschale, Dissertation, ZOrich.

Structural Engineering,Mechanicsand Computation(Vol. 1) A. Zingoni(Editor) © 2001 ElsevierScienceLtd. All rights reserved.

565

H Y G R O T H E R M A L EFFECTS ON THE BENDING BEHAVIOUR OF MULTIPLE DELAMINATED COMPOSITE PLATES P. K. Parhi l, S.K. Bhattacharyya2, P.K. Sinha3 1Civil Engg. Deptt.,C.E.T., Bhubaneswar, Orissa, India 2Civil Engineering Discipline, University of Durban-Westville, S. Africa 3Aerospace Engineering Deptt., IIT Kharagpur, W.B., India

ABSTRACT The laminated composite structural components used in the fields of aircraft and nuclear structures, pressure vessels, automobile and marine structures etc. are more often exposed to high temperature and moisture. The hygrothermal deformation induced by non-uniform moisture/temperature is an important feature in the design of composites. The focus of the present work is on the development of a finite element model for the bending analysis of multiple delaminated composite plates subjected to moisture and temperature. The developed model can take care of any general distribution of moisture and temperature. It has been observed that the moisture content and the temperature play an important role in the evaluation of the deflection of plates with multiple delaminations.

KEYWORDS Composites, finite element method, hygrothermal, delamination, structures, moisture, temperature. INTRODUCTION With the increased use of composites in the aircraft and nuclear structures, pressure vessels, automobile and marine structures etc., study on the performance of composite laminates exposed to adverse environmental conditions are essential. The components used in the above fields are more often exposed to high temperature and moisture. The hygrothermal deformation induced by nonuniform moisture/temperature is an important feature in the design of composites. The varying environmental conditions due to moisture absorption and temperature seem to affect the stiffness and strength of the structures severely. As the matrix is more susceptible to hygrothermal environment than the fibre, the deformation is observed to be more in the transverse direction. The rise in moisture and temperature reduces the elastic moduli of the material and induces internal initial stresses causing adverse effects. Despite many significant attributes, the composite materials are, in general, more brittle than their metallic counterparts. Also, in contrast to their in-plane properties, transverse tensile and interlaminar shear strengths of composite laminates are quite low. As a result, they are susceptible to delamination

566 from a wide variety of sources. Delamination, a major type of matrix dominated failure is one of the most prominent damages, faced by the composite laminates. It takes the form of interlaminar separation of layers in the interface between two adjacent laminae resulting from high interlaminar stresses in the laminating direction. Laminates are prone to delamination during fabrication, assembly, transport or field deployment or during normal use. Common types of delamination are due to high and low energy impact, exposure to hygrothermal environment, high stress concentrations and interlaminar shear failure. As the presence of delamination in composite laminates may adversely affect the safety and durability of structures, a comprehensive understanding of the delamination behaviourr is of fundamental importance in the assessment of structural performance of composites. The deformation and stress analysis of laminated composite plates subjected to hygrothermal environment has been the'subject of research interest in last few years. But to the knowledge of the authors, there have practically been no important contributions dealing with the effects of hygrothermal environment on the behaviour of delaminated composite structures. Whitney and Ashton [1971] used the classical laminate plate theory to study the effect of environment on the stability, vibration and bending behaviour of composite laminates. The Ritz method and a closed form solution were used for the symmetric and unsymmetric laminates, respectively. Wu and Tauchert [1980a, 1980b] presented the closed form solutions for the thermal deformations and stresses in symmetric laminates and anti-symmetric cross-ply and angle-ply laminates. Weinstein et al. [ 1983] presented a finite element formulation for the analysis of the coupled problem of bending and stretching of composite sandwich plates under thermal loading using triangular elements. Sai Ram and Sinha [ 1991 ] developed the finite element formulation to study the effect of moisture and temperature on the bending behaviour of laminated composite plates considering the reduced lamina material properties at elevated moisture and temperature. The focus of the present work is on the development of a finite element formulation for bending analysis of multiple delaminated composite plates subjected to moisture and temperature. The analysis incorporates the first order shear deformation theory considering eight noded isoparametric quadrilateral elements. The development of a simple multiple delamination model is employed for the static analysis of laminates containing delaminations of any size placed at any location. HYGROTHERMAL ANALYSIS A composite laminate of uniform thickness h consisting of n number of thin laminae, each of which may be arbitrarily oriented at an angle O with reference to the x axis of the reference co-ordinate system (Fig 1) is considered. Neglecting the normal stress perpendicular to the plane of the lamina, the stress-strain relations in the presence of moisture and temperature, are represented as:

exx -- t

f88~xx e~ 'Yxy e xy and

(xzt:i44451

l~xz

Xyz

LQ45 Q55 LYyz

(1)

(2)

567 Here, ex~, eyy, exy are the non-mechanical strain components due to moisture and temperature in the x-y reference axes, which are derived from the corresponding values in the fibre axes, l-2 after applying the transformation as given below: {e}= fixxt yy = I m2 n2 xy - 2mn

m2 n2 1{ 13, } ( C - C o ) + 2mn [32

nz

m2

°tl ( T - T o )

-2mn

2mn

°t2

where, ,O,,,6'2 are the moisture expansion coefficients in the longitudinal and transverse direction of any arbitrary lamina; a I ,or2 are the respective thermal expansion coefficients; Co, To are the reference moisture content and temperature and C, T are the elevated moisture content and temperature, respectively. Here, m = cos 8 and n = sin O, 8 being the fibre orientation of the lamina with reference to the x axis. The non-mechanical hygrothermal in-plane force and moment resultants are given by N

N

{Nx~ N~

N /T

Nxy, = £

Zk

~(Q,,j)k{e}kdz

k=l Zk_l Zk

xx Myy

Mxy

-"

ij)k

kzdz,

i, j = 1,2,6

(4)

k=l Zk_l

The force and moment resultants are modified to obtain the constitutive equations for the hygrothermal environment, which may be briefly written as {F}= [D]{8}- {FTM}

(5)

FINITE ELEMENT FORMULATION A laminated composite plate of length a, width b and thickness h consisting of n arbitrary number of bonded anisotropic layers is considered as shown in Fig 1. The displacement field is related to midplane displacements and rotations given by u(x, y) = u° (x, y) + z0 x(x, y) v(x, y) = v ° (x, y) +.z0y (x, y)

(6)

w(x, y) = w 0 (x, y)

where u, v and w are displacements in x, y and z directions, respectively and u°,v°,w ° are the respective mid-plane values. Here, 8 x and 8y are the rotations of the cross sections perpendicular to the y and x axes, respectively. The five degrees of freedom considered at each node of the element are u°, v0, w, 8 x and Ry. The element displacements are expressed in terms of their nodal values by using the eight noded isoparametric quadratic element shape functions. The stiffness matrix, the mass matrix and the nodal load vectors of the element are computed based on the principle of minimum potential energy.

568

/ ~//

y) v °

T

.....

|

t

•

l

hh |

TY

b ;' k k*l

2,

,u °

Fig lo

Compositeplote oxes system

Figlb

T

h/2 '/st

+ h/z

I.

delominotion

Zl

T

ZO

Z

2 nd deIominotion

~.,-

L

r

, Pth delominot~ n-I

£

n Ioyered [ominote configuration.

,I hslZ-

r

I

. . . .

S

_ _ .

J-T hs

zr

.t

. . . . .

k

zn

n-1

T'°I-,,°

°

1_

Fig 20 Muttipte de[ominotion

_2

'

t

U

t__

Fig2b Three orbitrory delominotions

MULTIPLE DELAMINATION MODELLING A simple two dimensional delamination model proposed by Gim [1994] has been extended to the general case of a laminated composite plate having arbitrarily located multiple delaminations. A typical composite plate of uniform thickness h with n layers and p arbitrarily located delaminations is considered for the multiple delamination modelling as shown in Fig 2a The resultant stiffness matrix is the sum of all the sublaminate stiffness.

569 N U M E R I C A L E X A M P L E S AND RESULTS Based on the finite element hygrothermal analysis and multiple delamination modelling stated, computer codes are developed to generate numerical results for studying the bending of delaminated composite plates exposed to moisture and temperature. Moisture concentration is varied from 0 to 1.5 % and temperature from 300 K to 425 K. For computing the numerical results, centrally located single and multiple delaminations are considered. Single delamination of three different sizes like 6.25%, 25% and 56.25% of the total plate area are considered. The mechanical properties for the graphite/epoxy composites and the plate geometry are as follows:

E1 =130 GPa, 22 =E3 =9.5 GPa, G12 =G13=6 GPa, G23 =3 GPa, v12 = 0.25, a = b = 0.5m The lamina material properties at the elevated moisture concentrations and temperatures assumed in the present analysis are shown in Table 1 and Table 2, respectively. TABLE 1 ELASTIC MODULIOF GRAPHITE/EPOXYLAMINAAT DIFFERENTMOISTURECONCENTRATIONS; G13 = G12, G23 = 0.5G12 ,/~ = 0 , / ~ = 0.44 Moisture concentration, C (%) Elastic moduli 0 0.25 0.5 0.75 1 1.25 1.5 (GPa) 130 130 130 130 130 130 130 E1 9.5 9.25 9 8.75 8.5 8.5 8.5 E2 6 6 6 6 6 6 6 Gl2 TABLE 2 ELASTIC MODULIOF GRAPHITE/EPOXYLAMINAAT DIFFERENTTEMPERATURES; Gt3 = G12, G23 =0.4 G12, (XI= -0.3 X 10"6/0K, ~2=28.1 x 10"6/°K

Elastic moduli (GPa) E1 E2 Gl2

300 130 9.5 6

325 130 8.5 6

Temperature, T (K) 350 375 400 130 130 130 8 7.5 7 5.5 5 4.75

425 130 6.75 4.5

The accuracy of the present finite element analysis is compared with the closed form solution [Wu & Tauchert, 1980b]. The results for the deflections and moments at temperature, T=400 K at the points A,B,C and D for a (0/90)2 square simply supported plate are presented in Table 3. The entire plate is modelled with a 8x8 mesh size as shown in the Fig 3.

T 1 o

A "S C

O

Figure.3:8x8 mesh discretization of plate

570 TABLE 3 VERIFICATION OF FEM RESULTS WITH CLOSED FORM SOLUTION AT TEMPERATURE

T = 400 K for (0/90)2 square simply supported plate, a/h=1O0 Bending Characteristics

Solution

w (mm)

Closed form Present FEM Closed form Present FEM Closed form Present FEM

Mx(N mm) My(N mm)

Point

A

B

C

D

0.0000 0.0000 -2.753 -2.6984 2.753 2.6984

0.0085 0.0079 -2.518 -2.4661 2.752 2.681

0.0267 0.0249 -1.869 -1.839 2.657 2.578

0.0337 0.0317 -0.966 -0.950 2.237 2.184

Effect of moisture on single delaminated plate Anti-symmetric cross-ply laminate For anti-symmetric simply supported (0/90)2 cross-ply plate, the deflection w and the moments M~ and

My are computed at different moisture concentrations for various cases of central mid-plane delaminations. In case of clamped boundary, the deflection is zero throughout the laminate irrespective of the magnitude of delamination as well as the percentage of moisture content. The values of moments Mx and My are observed to be M x = - M y , My = - M ~ , where M~ and M~ are the non-mechanical moment resultants due to moisture. The values of Mx and My at different moisture concentrations are shown in Table 4. TABLE 4 STRESS RESULTANTS AT DIFFERENT MOISTURE CONCENTRATIONS FOR

Moment (N mm)

Mx M~

0 0 0

0.25 -0.4480 0.4480

(0/90)2 SQUARE CLAMPED PLATE

Moisture concentration C (%) 0.5 0.75 1.0 1.25 -0.8717 -1.2710 -1.6459 -2.0574 0.8717 1.2710 1.6459 2.0574

1.5 -2.4689 2.4689

Anti-symmetric angle-ply laminate For both the simply supported and clamped boundary conditions, Mx and My are zero throughout the H laminate and M w = - M ~H, where, M xy is the non-mechanical moment resultant due to moisture. Effect of temperature on single delaminated plate

Anti-symmetric cross-ply laminate The deflection w and the moments Mx and My for anti-symmetric simply supported (0/90)2 cross-ply plate at different temperatures for various cases of central mid-plane delaminations are computed. The case of clamped boundary is also considered, where the deflection is observed to be zero throughout the laminate irrespective of the magnitude of delamination and also the percentage of moisture content. The values of Mx and My at different moisture concentrations are shown in Table 5.

571 TABLE 5 STRESS RESULTANTSAT DIFFERENTTEMPERATURESFOR (0/90)2 SQUARECLAMPEDPLATE

Moment (N mm)

M~

My

300 0 0

325 -0.3229 0.3229

Temperature, T (KI 350 375 400 -0.6148 -0.8757 -1.1057 0.6148 0.8757 1.1057

425 -1.3435 1.3435

Anti-symmetric angle-ply laminate Simply supported as well as clamped boundary conditions are considered, where, Mx and My are observed to be zero throughout the laminate and M ~ = - M r , where, M ~r is the non-mechanical moment resultant due to moisture.

Effect of moisture and temperature on plate with multiple delaminations The number of delaminations are increased considering 1,3 and 5 delaminations (each delamination interface of 25% plate area) for a (0/90)10 plate and the deflection at point D (Fig 3) is computed for varying percentages of moisture content as well as temperature. The deflection values for these cases are presented in Table 6. TABLE 6 DEFLECTION(10 .5 M) AT THE POINTD FOP.(0/90)1o SQUAREPLATEWITHCENTRALLYLOCATED MULTIPLE DELAMINATIONSOF 25% DUE TO THE EFFECTOF MOISTUREAND TEMPERATURE,A/H= 100

Multiple delaminations no delamination one three five

Moisture concentration, C 0.5 % 1% 1.5% 2.757 5.207 7.811 3.237j 6.114 9.171 3.513 6.635 9.952 3.887 7.341 11.011

Temperature, 325 375 .840 2.267 .986 2.615 1.071 2.838 1.185 3.140

T (K) 425 3.345 3.927 4.262 4.716

CONCLUSIONS A static analysis of multiple delaminated composite plates exposed to hygrothermal environment is investigated taking into account the changes in thermomechanical properties due to moisture and temperature effect. The conventional finite element formulation is modified to include the hygrothermal effects. The accuracy of the formulation has been verified using sample problems available in the literature. Both simply supported and clamped composite plates are studied. The increase in the deflection and stress resultants is observed due to increase in moisture concentration as well as temperature for a simply supported cross-ply plate. For this case, no deflection is produced along the diagonals. For simply supported angle-ply laminates, no hygrothermal deflection is produced. In the case of clamped plates also, no deflection is noted either due to increase in moisture/temperature or due to increase in area of delamination/number of multiple delaminations.

REFERENCES Cook, R.D., Malkaus, D.S. and Plesha, M.E. (1989), Concepts and Applications of Finite Element Analysis, John Wiley and Sons, New York

572 Gim, C. K., (1994), "Plate Finite Element Modeling of Laminated Plates", Computers and Structures, 52(1), 157-168. Sai Ram, K.S. and Sinha, P.K., (1991) "Hygrothermal Effects on the Bending Characteristics of Laminated Composite Plates", Computers and Structures, 40(4), 1009-1015. Tsai, S.W. and H.T. Hahn, (1980), Introduction to Composite Materials, Technomic Publishing Company, INC, Westport CT, Lancaster. Weinstein, F., Putter, S. and Stavsky, Y., (1983), "Thermoelastic Stress Analysis of Anisotropic Composite Sandwich Plates by Finite Element Method", Computers and Structures, 17, 31-36. Whitney, J.M. and Aston, J.E., (1971), "Effect of Environment on the Elastic Response of Layered Composite Plates", AIAA Journal, 9, 1708-1713. Wu, C.H. and Tauchert, T.R., (1980a), "Thermoelastic Analysis of Laminated Plates. 1: Symmetric Specially Orthotropic Laminates", J. Therm. Stresses 3,247-259. Wu, C.H. and Tauchert, T.R., (1980b), "Thermoelastic Analysis of Laminated Plates. 2: Anti-symmetric Crossply and Angle-ply Laminates", J. Therm. Stresses 3,365-378.

Structural Engineering,Mechanicsand Computation(Vol. 1) A. Zingoni(Editor) © 2001 ElsevierScienceLtd. All rights reserved.

573

NEW M E T H O D S TO ESTABLISH THE ELASTIC C H A R A C T E R I S T I C S OF THE FABRIC R E I N F O R C E D LAMINAE

A. Secu 1, R. Boazu 2 and D. P. Stefanescu 1 1 Department of Civil Engineering & Industrial Constructions, 2 Department of Structural Mechanics, Technical University "Gh. Asachi", Faculty of Civil Engineering & Architecture, D. Mangeron nr. 43, Ia~i, 6600, Romania,

ABSTRACT The modem composites are multilayer made up of uni and bi-directional reinforced laminae. The micro and the macro mechanics of the unidirectional reinforced laminae and the macromechanics of the multilayers made up of laminae are very well known from the theoretical point of view and the results are experimentally confirmed. For the multilayers made up of bi-directional reinforced laminae (with texture) the theoretical methods confirmed by experiment do not exist yet. This work presents: the basic elements of two original methods-the sub cells method and the numerical simulation method, the results obtained by using these two ways for the same lamina, the comparing of the results and the conclusions concerning the applicability field of the methods we have presented. The results emphasise the significance of the numerical simulation method as being the only proceeding that determines all the elastic characteristics for the texture reinforced laminae. KEYWORDS: Composite, elastic characteristics, fabric reinforced laminae, sub cells method, numerical simulation method

INTRODUCTION

The elastic characteristics of fabric-reinforced laminae cannot be well determined through theoretic methods yet, and the use of experimental methods is very expensive. We present the basic elements of two original methods - the sub cell method and the method of the numerical simulation- and the comparing of the results for the same lamina. We emphasize the application of every method in determining the elastic characteristics of the fabric reinforced laminae, and the fact the only method of faithful approximation of the reality is given by the method of the numerical simulation.

574 THE SUB CELL METHOD

The definition of the notions of elementary representative volume (REV) and cell for the matrix-

fabric composite (fig.l)

The elementary representative volume - the REV (fig.1 b) - is the portion of the lamina that is small enough to show the microscopic structure of the composite material and large enough to underline the global behaviour of the matrix- fabric lamina; due to its reduced size, we can admit that on the REV the tensions & the deformations are uniformly distributed.

/

a.

b.

c.

d.

Figure 1" Definition of the notions of REV & cell The cell (fig.1 c) is one fourth of the REV. Depending on the mutual position of the threads there are two types of cell: Cl and C2.

The geometric modelling, the partition of the cell, systems of axes The geometric modelling and the partition of cell are based on its equalization with a sandwich made up of two sub cells (fig. 2).

~

iii 1

/co

I 2~/n /

C a)

~Ce /

~ t /

3 lil

I

~~,"t,,,"

b)

]

af'2/cosot2 ~ . . ~

82

/

/ d)

cell2 ~J

c) 2

2LL"" e)

~ , S u b t._

Figure 2: Geometric modelling and partition of the cell

cell 1 "~ J

575 The values 81 and 82 are obtained from the condition that the volume of thread in every sub cell should stimulate a proportional volume of matrix. That way we obtain:

Afl 61 =

A f2

•L

cos a I

A:~ COSa 1

• L+

. ~,

A:2 COSO~2

82 =

A:~ --.L+

.b

.b

cos gr2

•8

Af2

COS0:1

COSO~2

(1)

.b

Every sub cell has its own system of axes 1L and 2L (1 - the main direction, 2 - the transversal direction, L - the lamina axis, every sub cell being treated like a lamina). Therefore: Ci, l=Ci,2 ~i Cij,l=Cij,2

(2)

where: Ci, l(2) is the C type characteristic corresponding to the 'T' direction of sub cell "1" ("2"); Cij, l(2) is the Cij type characteristic of sub cell "1" ("2"). The modality of partition implies the fact that the sub cells have the same elastic characteristics (the same phases and the same volumetric fractions).

The determination of the elastic constants of the composite lamina o f the matrix-fabric type The REV of the m a t r i x - fabric composite, modelled in sub cell is succesively submitted to axial and shearing efforts in the plane state of tension (fig.3). Axes 1 and 2 are the main axes of the REV. We accept the hypothesis that the specific deformations are constant in its thickness of the multilayered material. L

/

I

J

~'' " ~' ':"~""~'~ "~ ........

'

~ ~ 2L e /

c

y

/ 81"

%

N~

i

1L ~

o

2

12,

I

i

_) -

> 1 -

> 1L~ 1 0=0o

~L

~maL

/ °~........ ~!IIP::~.............]~

~ 2L-1

IL/ 2

0=90 2L

/ ' ::

;2

k" .,_. o

NI,2

I

:~:" ~: ".......

"1;12,

N 12,

/ =,/~

2b

1, ~ ......... ,~ 1,

1

8

, ---1, ? --

N2

N2,1

Figure 3" Submitting to efforts of the REV in a state of plane tension

= 0=90

576 The following steps are taken: The expressions of the efforts proceeded from sub cells by the integration of the tensions on the surface of the sub cell that makes up the section of the REV on which one of the N l, N 2 and N12 efforts is defined (the moments of bending and torsion given by the normal and tangent tensions in every sub cell annul each other) and the efforts on the REV proceeding from the constituent sub cells: 81 2

-

-

NI,I =

,

82 2

N 1,2 =

N1 = 2(Nl: + N1,2)

,

61 2

~2 2

81 2

82 2

m

N2 = 2(N2: + N2,2) _ #__~_l 2

(3)

c~2 2

61 2

82 2

61 2

6z 2

N12 = 2N12: + 2N12,2

The tensions are obtained from the relation:

[

•

g

p

is the matrix of the transformed reduced rigidities,

{So }x,y is the vector of the specific deformations of the median surface. •

We write the efforts on the unity of length of the multilayered material.

g l -- 2 - ~ N , - -

oc.1 • [Ult~ q- U2(t~ 1 - t~2)-at- U3t~']d- s 2 " ( U 4 - U3)t~

(4) N12 = ~bNl2 = Y~°z(U,- U3)6

Relations (5) were obtained by expressing the tensions varying with the transformed reduced rigidities, expressing these rigidities varying with invariants Ui, integration of the expressions (3), the summation given by equations (4) and the reduction on the unity of length of the efforts on the REV. •

We express the specific deformations varying with the efforts:

577

f"t [a a2 01f } 62 = a,2

}"1°2

0

0

a66

• N2

(5)

NI2

a~ = f(U~, d, 8~, 82)

where

•

a22

0

(6)

In relation (5) we express the efforts on unity of length varying with the average tensions in the composite, o-~,o-2, r~2,

f"t LFa;al: 0l f:t 6"2 --

Yl°2

•

a22

0

0

a66

"8'

(7)

O"2

ff12

We obtain the elastic characteristic of the multilayered material, therefore of the m a t r i x - fabric composite, from relation (7), by particularizing. Or Nl¢0, N2=0, N12=0 ~

E~ =)J-

e~' = all. o-~. 8 = o-1

E,

(8)

a~8

Similarly we obtain:

E2 :

1 ~ ,

a228

v12 =

a12 , al 1

V21 : ~ ,a12 a22

G12 --

1

(9)

a66 8

THE METHOD OF THE NUMERICAL SIMULATION The numerical simulation is based on the following reasons: the constitutive phases do not combine chemically, their elastic characteristics being known: the geometry of the fabric is determined by measuring: the thickness of the lamina and the real disposal of the reinforcement in the lamina define the geometry of the phases; the discretization of every phase leads to a good approximation of reality and allows us to catch the phenomenon of working together by defining the common knots of the finite elements; by conditions to the limit we approximate the ideal behaviour of the test piece (model used in analytic methods) and we obtain results that are close to the experimental ones.

The stages of the method •

Establishing, by emphasizing, the real geometry of the section of the reinforcement and its discretization (fig. 4).

578 r----r .............. '.......,,-.................. I ......................'....................' .......... " ............. " ............... ?--TY

0,2685 st,, 0,465 0,489 0,49

0,4891

/

Lwarp=4,403 m

Figure 4: Photograph of the section of a fabric thread and its discretization The modelling and discretization of the REV, made up of two threads of warp and two threads of weft (fig. 5);

~i~%i~i~i~i~i~i!~i~iiii~ii~iii~i~i!i~iiiiii!i#~i , ~..~N~,iii~,~!ii!iiiiiiiiiiiiii:iiiiiiiiii~Miiiiiiiiiiiiii:ili!iiiiii ~"

.~ .~.~iii/!i::iii!i iiiii:,iiiiiiiii::i i;ii::iii::iii::iii~:i::i

Figure 5: Modelling of the geometry of the REV The modelling of the numerical test pieces (fig. 6, 7) specific to every type of experimenting (the research that has been performed [2] has demonstrated that, in the case of obtaining an ideal deformed test piece, the ratio between the length and the breadth of the test piece does not influence the results). Every knot blocking of the displacements in the direction of the force

Rigid yoke driven with 1 uniformly distributed rces

~ Pendulums

Figure 6" Square test piece for determining the longitudinal resilience modulus and Poisson's ratio specific to one direction

579

" _ " ~ ~~~~ ! ! ~ ~.~ .- ~ l J ~ , ~ ~ l"~~~. ~, g~ ~ ~ ~ " ~ / _

z

"

•

I Everykn°tb~cking°fall 1 ~ l ~ g ~ i the degrees of freedom ~ ~ ~

A

STangential forces " ~ [ uniformly distributed ] ) that generate a / umtary antenslty | .

.

.

~ ~]1I~/

Figure 7: Square test piece for determining the transversal resilience modulus •

The modelling of the restraint and load conditions systems that insure the ideal looks to every type of deformed test piece. (fig. 6, 7).

•

The selection of the method and program that allows the numeric transcription of the experiment.

•

The rolling of the program and the reading of the deformations at the level of the weight centre of every quarter of REV (QREV) with the help of the automatic calculating program "DAR".

•

Finding the characteristic value of the composite material (the value with minimum error compared to the average).

The values of the elastic characteristics are obtained with the relations: 1

E1= - - , E1

c2

el2 = - - - ,

El

1

E2 " - - - , ~2

c I.

V21=----, ~2

1

G=--

(10, 11, 12, 13)

Y

CASE STUDY The presented methods have been used to find the elastic characteristics of the composite lamina made of epoxidic resin reinforced with glass fibres, having the following elastic constants: Em=3,5GPa, Ef=73GPa, Gm=l,25GPa, Gf=28,07692308GPa, Vm=0,4, vf=0,3 and geometric constants: Vwarp=0.284785219; Vweft=0.2524226734. Table 1 shows the values of these dimensions. In the method of the numerical simulation (I) there were used the results obtained through experiments carried out on 3 types of test pieces: 1 - square test piece, 2 - rectangular test piece with 3 threads of warp, 3 - rectangular test piece with 3 threads of weft. The sub cell method (II) used for determining the elastic characteristics of the unidirectionally reinforced lamina, the laws of the blends (Jones) for E1 and V12 and the Chamis formulas for E2 and G. Table 1 Comparison of results El

i

(Gpa) 1 37,84395 Jones, Chamis 39,45214

E2

(Gpa) 36,46321 36,24832

Vl2

V21

v12/E1

X10"2

vn/E~ xl0 "2

0,2840

0,2729

0,7506

0,7484

0,1597

0,1468

0,40501

0,4050

G (Gpa) 13,07 5,9586

580 CONCLUSIONS a) The method of the numerical simulation, by accomplishing a faithful geometric modelling of the composite, leads, in the conditions of a fine discretization, to results that are practically real. The statement is also sustained by framing the results we have obtained in an "ideal material" (a material that satisfies all restrictions and consequently exists). b) The sub cell method uses as entry data the elastic characteristics of the unidirectionally reinforced lamina (Jones, Chamis, F6rster, Puck, the parallelipipedic model), which leads to a perpetuation and even amplification of the in exactitudes introduced by the analytical models for this composite. c) The subcell method is an analytic method that takes into consideration the ondulation of the threads, correcting the values of the elastic characteristics of the composite lamina that is unidirectionally reinforced depending on the inclination of the reinforcement. d) The sub cell method has very good results for the longitudinal elasticity moduli (4% error) compared to the method of numerical simulation, table 1. e) The improvement off the results obtained through the sub cell method, at the level of the coefficients of the transversal contraction and the transversal resilience modulus, is possible by using the elastic characteristics of the unidirectionally reinforced composite lamina found through numerical simulation. REFERENCES Gay D. (1991), Mat~riaux composites, III-bme Edition revue et augment& Herm6s, Paris. Boazu R., Secu A., Stefanescu D. P., (2000), Types of test pieces used in the method of numerical simulation to find the elastic characteristics of the matrix-texture composites, Proceedings of the 9th

International Conference on Metal Structures, pg.556-562, Timisoara, Romania. Boazu R., Secu A., Stef~nescu D. P., (2000), The method o f numerical simulation alternative

to the experimental methods in order to determine the elastic characteristics o f the composite laminae, iNDiS 2000, NOVI SAD, 22-24 November 2000. Poterasu V. F., Stefanescu D.P, Secu A., Boazu R., Elastic characteristic evaluation of texture composite laminae based on a numerical simulation, European Congress on Computational Methods

in Applied Sciences and Engineeringr-ECCOMAS 2000, Barcelona, 11-14 September 2000. Secu A., Boazu R., Stefanescu D. P., (2000), Restraint and load conditions systems in numerical simulation to esthabilish the elastic characteristics of the composite materials, Proceedings of the 9th

International Conference on Metal Structures, pg. 620-624, Timisoara, Romania. Stefanescu D. P., Secu A., Boazu R., Programme for automatic reading of the deformations and the displacements on systems solved by using finite element, iNDiS 2000, NOVI SAD, 22-24 novembar

2000. Stellbrink K.K.U., (1996) Micromechanics of Composites, Hanser Publishers.

BRIDGES, TOWERS AND MASTS

This Page Intentionally Left Blank

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

COMPARATIVE

583

STRUCTURAL BEHAVIOUR OF MULTI-CELL

A N D MULTIPLF~SPINE BOX GIRDER BRIDGES M. Samaan ~, K. M. Sennah 2, and J. B. Kennedy ~ ~Civil & Environmental Engineering, University of Windsor, Windsor, Ontario, Canada N9B 3P4 2 Department of Civil Engineering, Ryerson Polytechnic University, Toronto, Ontario, Canada M5B 2K3

ABSTRACT The use of curved box girder bridges in interchanges of modern highway systems has become increasingly popular for economic and aesthetic considerations. Composite concrete deck-steel box girder bridges may take the form of single-cell, multi-cell, and multiple-spines. They provide considerable flexural and torsional strengths to resist the applied loads. In this paper, three types of curved box-girder bridges, of the same material content, are investigated, namely braced multi-cell bridge, multiple-spine bridge with internal bracings (inside boxes) and multiple-spine bridge with internal and external bracings between boxes. The finite-element method, using the commercially available "ABAQUS" software, is used to model the bridges. Shell elements are used to model the concrete deck slab, steel cells, and end diaphragms. While three-dimensional beam elements are used to model top steel flanges and cross-bracings. The material non-linearity option in ABAQUS is used to. obtain the structural response under increasing truck loading, as well as the load-carrying capacity of such bridges. Comparison of the theoretical values of selected structural quantities, such as support reactions, tangential stresses and overall load carrying capacity, are presented for the three types of bridges. A comparison of the dynamic characteristics, natural frequencies and corresponding mode shapes, of such bridges is also presented. It is concluded that curved bridge of cellular cross-section is the most economic section to resist applied loads. KEYWORDS Bridges, box girder, curved, load distribution, ultimate, dynamic, finite-element.

584 INTRODUCTION

Curved box girder bridges provide significantly high torsional rigidity and low material content when compared with I-girder bridges. Box girder bridge cross-sections may take the form of single cell (one box), multiple-spine (separate boxes), or multi-cell with common bottom flange (contiguous boxes or cellular shape). A recent literature review of the available work on straight and curved box girder bridges was presented elsewhere (Sennah and Kennedy, 1999-a and 1999-b). The North American specifications for horizontally curved highway bridges (Guide Specifications for Horizontally Curved Highway Bridges, 1993) pertains to only one type of box-girder configuration, namely the curved composite concrete deck-steel multiple-spine box-girders. The objective of this paper is to provide a comparison between three types of simply-supported composite concrete deck-steel box girder crosssection. These types are multi-cell bridge with internal radial bracings, multiple-spine bridge with internal cross-bracings (inside boxes) and multiple-spine bridge with internal and external bracings (inside and between boxes). The theoretical values of selected structural quantities (support reactions, mid-span tangential stresses in steel bottom flange(s), natural frequencies and mode shapes and the ultimate load carrying capacity) are used in the comparison.

I

rConcretedeck ! 1 [ TOp-ch°rd [

l

G1

G2 ~ross-braclng

I

(a) Multi-spine bridge cross-section

R

0°,

~" G3

J

" ~

0Q dt 0o-,i.

(26)

where dt is the infinitesmall increase in time, y = l/r/; 77 is the coefficient of viscosity, y is the coefficient Of plastic flow; cI)(F) is a monotone-increasing function, Owen & Hinton (1980), defined as

~ ( F ) = e t. F° ) _ 1

(27)

cI)(F)=I[F-F° F o ]l N .

(28)

or

It is assumed that, Owen & Hinton (1980), < qS(F)>= 0,

for F= F ,

for F > 0

(30)

or

with F ° depending on the material type such that for the Mor-Coulomb plastic flow function, F ° = ccos~b, and c and ~bdenoting the coefficient and angle of friction, respectively. Incorporating eqn.24, the stress level inthe elastic region can then be expressed as (31)

dcr!/ = Dij~(deo.-de;f ) . where D:jk~are the elements of the D matrix. Finally, therefore, we can write

deo = D~wdo" o. +7" < O ( F ) >

aQ

0%

dt

(32)

681 or

d~j = ~jk,d% - Dok,y < cI)(F) > &r~j OQ art

(33)

For the elasto-visco-plastic solution, the "stress-strain" relationships takes the form of eqn.31 as for nonlinear problems, which gives, taking into account eqn.21 {do'}

" - "

[D]{de}- [D]{devp --- [O][B]{dld

e

}- [D]{devp

(34)

where

{de}- [e]{d.}, 1

EO - v ) [D]= (1 + v')(1 -- 2v)

v 1-v

[D] - matrix of elastic constants which is defined as

v 1-v 1

0

0

v 1-v

v 1-v

1 1-v v

0 0

1-v

1-2v 2(1-v)

0

0

E [D]= ( 1 - v X l + v )

1 o

where v denotes Poisson's ratio.

for plane strain,

1

i1

0 1 v

for plane stress.

Substituting into equation eqn.6 or eqn.23 the appropriate values of the trial solution eqn.9 for {u(e)} and for stresses eqn.34 and eqn. 12 for the shape functions, we get, after applying the principle of virtual work: ~{d~{/*}--

~B]~){dbg}T{d(y}~ - ~{de~P }~){d~f}l~ - ~N~ke)(~)~dI~T~{dlf }l~ - ~N(ke)(~)~dqn}T~){dlf}dS~-- O.

ff

ff

ff

(35)

sr

Or

6{dg ~ }= ~B]{dcr}-[N~k~)(~)~dt~r~{d~ }dfZ - ~{dd'PI6{d~{ }dfZ - ~)(~)~dqm} r 6{dd }{d%}dS ~ =0 (36)

Ff

FI

Since 8{du e } are not equal to zero in f2 e and S e and do not depend on {da}, therefore

we

can:

we

can

write:

or

~e

Applying eqn.34 to eqn.37 gives:

I[8IDI~] ~ {d. ~ }d~ ~ - {de '~ }: 0

08~

[ge]{dble}-{dR'e}--O,

(39)

.(2e

Eqn.38 can be re-written as:

[K e] is the element stiffness matrix which is given by:

682

[Ke]= I [, ]T[D I ,

}/X~ e "

(40)

Using standard FE procedures (summation over all the elements), we get: N{dff} - {dR'},

(41)

which is the equilibrium equation for the whole system. [K] is the global stiffness matrix,

{dR '} - the incre-

ment of the external fictive forces (flux), {~} - the global displacement vector to be determined.

{dR t}: i[ N ]T {dP }{due }t.Q + I[D ]{de vp }dO + I[N ]r {dq. }{due}Is ~

s

or

{dR'}: I[D ]{devp}dO + dR.

(42)

For the system, [x]:

•

(43)

A solution algorithm for the given elasto-visco-plastic problem is then set up for a numerical solution of the ensuing algebraic equations in incremental form using the 0-method (Euler's method) of numerical integration.

The algorithm is as foUows: (1) (2) (3) (4) (5) (6) (7)

The prescribed load {dR °} is input and an elastic path is followed by calculating the visco-plastic strain rate at the Gauss points using the modification of formula ofeqn.24 when t = t o and At 0 = t 1 - t o = 0. Determination of the increment of the external fictive forces (flux) {dR ~}and the stiffness matrix [K]. The displacements at nodes of the elements are calculated. The incremental of the stresses is obtained from the constitutive equations. The residual force is calculated and the total stresses. Plastic zones are determined at the Gauss Points. Check for convergence. If convergence is not obtained per the time step, then the residual force {dr} is added to the initial loading incremental {dR '} and the process continues. The next time step At, is added and the time iteration continues.

The non-linear problem is linearised using the Newton-Raphson iteration method and a computer code is to be developed in Fortran for its numerical solution.

REFERENCES Hinton E., Owen, D.R.J.(19.80). Finite Elements in Plasticity, Theory and Practice. Pineridge Press, Swansea, U.K. Pande, G.N., Naylor, D.J. Finite Elements in Geoteehnical Engineering. Department of Civil Engineering, University College of Swansea. Pineridge Press, Swansea, U.K. Prager, W.(1961). Introduction to Mechanics of Continua. Ginn & Co., Boston. Taban-Wani, G.W. (1996). Elasto-Plastic and Elasto-Visco-Plastic Investigations of Structures in a Deformable Continuum by the Finite Element Method. Thesis in the Pursuance of the degree of Doctor of Philosophy of Moscow State University of Civil Engineering, Moscow. Zienkiewicz, O.C. (1977). The Finite Element Method. McGraw-Hill, New York. Zienkiewicz, O.C., and Cormeau, I.C.(1973). Visco-Plasticity and Plasticity- an Alternative for Finite Element Solutions of Material Non-Linearities. Procedures Colloque Methodes Calcul., Sci., Tech., 171-199, IRIA, Paris. Zienkiewicz, O.C., Humphension; C., Lewis, R.N.(1975). Associated and Non-Associated Visco-Plasticity and Plasticity in Soil Mechanics. Geotechnique, 25, 671-689.

683 Apendix

1.

Algorithm

T i m e Step 1:

The details of the algorithm oulined in the text is as follows:

t. = t~ and

At~ = t 2 - t ~ .

Time Step 0:

Given values:

t.-t

°°°°° . . . . °°°°°°°,°,°°i . . . . . . . °°°° . . . . . . °°°°°°°°°°°° . . . . ,°°°°,°,°°°°,°°°,°°,°°°°°°,°°°°°°°°

o andAt o =t 1-t o =0

Given values:

R'~ , o l , u ~ a n d el .

T i m e Step n:

R ' ° , o ° , u ° a n d e °.

t, = t , a n d Calculations

At,

= t,+ 1 - t , . R ' n , 6 n , a n a n d e n"

Given values:

1.

~(vp)0

_ . y , ..< ( i ) ( F )

2.

de (vp)° =

~>_ b

o,

Calculations

Ato(1-O)~(vP)lAto

where c (vp)~ = c (vp)O +

0 ~ (vp)0

0or

do

1.

c CVp)" : y -< q ) ( F ) > - b " ,

2.

de (v")" =

where ~ (vp).+~ = ~ (vp). + c3~(vp). do 8o

d R , o = f(BO)r Dde (~P)°df2 + d R o, £2

3.

dR'"=

I(B")rDde a

4.

e" = O - 1 6 "

e ° = D - l o °.

K°=

5.

duO

6

d o ° = (B °)T DduO _ Dde

(vp)0

q_

,

7.

H 'de (~p)° = (B o y D d u o

d e ° = D - l d o ° + D - I H ' d e ( V p ) ° B ° d u °, R ,~ = R '° + d R '° , 10.

u ~ = u ° + d u °,

11.

e ~ =e ° +de °

12.

e (vp)l = e (vp)° + d e (vp)° ,

13.

O 1

14. 15.

The plastic flow function F ° is calculated Check on the condition o f F°:

= o ° + d o °,

F ° = 0, F ° > 0 o r F ° = 0,

de" = D-~do " + D-~H'de(Vp)"B"du",

9.

R '"+~ = R ' " + d R ' " ,

10.

u "+l = u" + du",

11.

e "+l =e" +de"

12.

e (vp)"+~= e (vp)"+ d e (~p)",

13.

o ''+l = o" + d o " ,

14. 15.

The plastic flow function F" is calculated Check on the condition o f F": d ' F " < 0, or F" < 0"

16.

Check on the c o n v e r g e n c e o f the time step iteration process using the strain level and then the viscoplastic strain rate level:

rance?

18.

df

"=

I(B°) x do"d.O

,2

< the giventole

rance?

rance?

If the the answer to 15 and 16 is yes, then a new incremental load is added and the the next circle commences. I f the answer is no, then the continuation is as follows: 17.

(vp)" + ,

8.

is ~ < the given tole

-Dde

F" = 0, F" > 0 o r F" = 0,

C h e c k on the convergence o f the time step iteration process using the stain level and then the viscoplastic strain rate level" is ~'llde - ~ _° I1 < the given tole

do" =(B")rDdu"

H'de(Vp)" = ( B " ) T D d u "

d ' F ° __ 0, or F ° < 0 •

~,,p)0

(vP)"d.Q + d R " ,

/ ( B °)T D B ° d r 2 , f2

16.

At.O-O~(VP)"+'At.

+ dR

",

Cvp),

_< the given tole

rance?

If the the answer to 15 and 16 is yes, then a new incremental load is added and the the next circle commences. If the answer is no, then the continuation is as follows: 17.

df" : j'(B") rd."dx2

+ dR ",

~2

dR ~ =df ° +dR ° 18.

d R "+~ = df" + d R "

The given process continues for the step n + l , n+2, etc.

684 The end of the time iteration is when the value of the ~(vp) becomes less than the given tolorance, for example 0.1 for the convergence of most geotechnical problems and then it is followed by a new load increment. A new time step iteration then follows until there is a stationery conditions in the displacement increment.

aN,(~,) = 1(1_ {2)(2~, + ~,2), c~1 4 0N2(~') = -~,(1-~2), c3N3(~;) = 1(1-~2)(2~

Apendix 2.

Shape Functions of the 8-noded Lagrangian Parabolic Isoparametric Element and their Derivatives

c3~l

-~,2),

2

ONs(~') = 1 0 + {2)(2{, + ~2), c3~i 4 N~ = - 4 1 (1-~,)(1-~2)(~, + ~ +1),

ON~(g,) = -g,(1- g~), 0{,

N2=

(1- ~ ) ( 1 - ~2),

ON7(~,,) = 1(1 + ~2)(2~I-{ ), c~ 4 2

N3=

(1 +~1)(1-~2)(~,--~2-1),

c3Ns(~*) = - l ( 1 -~=2), c3~1 2

N4=

(1- ~)(1 + ~2), z

Ns =

(1 +~,)(1 +~2)(~,

"1-~2-1),

N 6 =-(1-~)(1+~2), N 7 = - 1 ( 1 7 ~,)(1 + ~2)(~,-~2 + 1),

ON,(~,)

1

N8 = 2 1 ( 1 - ~ ) ( 1 - ~ 2 ) ' 0N3(¢') = l(1 + {,)({, - 2~=), c3{2 4

ON,(~,) = _{2(1_ ~,),

0g,

cqN,({,) = 1 (1 + {,)({ + 2{2), 4 l Og, 2 0~,2 ON,(~,) = 1 (1 + ~,, )(~,, - 2~,2),

4

ON~(~,,)

FINITE E L E M E N T AND NUMERICAL MODELLING

This Page Intentionally Left Blank

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

687

FEM COMPUTATION OF DYNAMIC PROPERTIES OF A S T R U C T U R E U S I N G F U Z Z Y S E T T H E O R Y X. J. Yu and D. Redekop Department of Mechanical Engineering, University of Ottawa Ottawa, Ontario, Canada, K1N 6N5

ABSTRACT Results for dynamic properties of structures obtained using the finite element method (FEM) generally differ from experimental results. In this paper a method employing fuzzy mathematics is presented in which FEM models are modified such as to bring computed dynamic properties closer to experimental ones. In the process of modification design variables such as geometrical parameters, material parameters, and constraint conditions are treated as fuzzy parameters, and fuzzy logic is used. An illustration is given for a practical engineering problem in which analyses are carried out with and without fuzzy mathematics modifications. It is demonstrated that with modification the FEM results are closer to the experimental ones.

KEYWORDS:

finite elements, fuzzy mathematics, dynamic properties, model modification. •

.

INTRODUCTION There are always differences between FEM results for dynamic properties of structures and experimental results. The differences take place because of uncertainties in the computational model which arise through the lack of precise information for the design variables. Probability theory can be used to deal with uncertainties of a random nature. As to the uncertainties referred to as fuzziness, one can use fuzzy set theory. The consideration of uncertainties in FEM analysis is the focus of the current study. To increase accuracy in results finite element models can be modified taking into account experimental results and fuzzy mathematics. Since experimental results are also subject to error, the comparison is limited to a finite frequency range, i.e. the objective in the modification is to make the computed dynamic properties of a structure closer to the experimental results within a limited frequency range. In the procedure of modifying the model for a practical engineering problem the frequency and vibration shapes of the model are required to be consistent for the first few modes of the experimental model. However, the number of parameters which can be modified is more than the number of model states to be matched. Thus the solution is not unique, in fact there may even be an infinite number of solutions. It depends on the number of design variables selected and their initial values. Sometimes, it is difficult to predict the influence of design variables on the properties of the model. For

688 example, as the stiffness in a certain direction increases the mass also increases, producing a reaction to the model state in.the opposite sense. Therefore, choice of design variables and prediction of the effect on these design variables of modifications depends significantly on experience. The variables cannot be described quantitatively, but only qualitatively, i.e. they are fuzzy. So design variables and constraint conditions can be treated as fuzzy parameters. Even the optimization procedures can be treatd as a fuzzy processes. Fuzzy set and fuzzy logic theory can be used to modify the finite element model to produce the 'best' solution.

MATHEMATICAL

MODEL FOR FUZZY OPTIMIZATION

Probability statistics optimization Assume that an n dimensional vector {xa} = {Xla,:r,2a,... ,Xna} defines the parameters of the structural model after modification. The vector {Xo} = {Xlo, X2o,...,Xno} gives the initial values. {A~} = {lla, 12a,... ,Area}T and { ~ } = {~1~, ~ 2 a , . . . , Om~}T are m eigenvalues and eigenvectors found by the analysis for the condition {x} = {xa}. {l~} = {11~, A2~,..., Imp} T and {~e} = {q~l~, ~2~,..., ~m~} T are eigenvalues and eigenvectors found by experiment. Define {Ax} = {x~-xo} and A a - Ae } {~y} :

~o_ ~

(~)

If {Ax} and {Ay} are independent of each other and follow the normal distribution, then the combined probability density is 1 ~e~p[-~(ZX~%~Ax P(A~,/Xy) = (2~)(~+m)/:v,~ ~ ~

+ Ay%~Ay)]

(2)

where wx = v ~ "1, Wy = V; 1 and vx, vy are the covariant matrices of {Ax} and {Ay}. Clearly the maximum of the combined probability density is equivalent to the minimum of (x), where (x) is defined by

rain J (x) = AxTwxAx + AyTwyAy

(3)

The constraint condition is that the computed modal mass should be consistent with the experimental

one, i.e.

{~i}T[M]{~i} -- Mie = 0; (i - 1, 2 , . . . , m)

(4)

{Ax} and {Ay} of eqn (3) can be described approximately by the truncated Taylor series

{zxy} = [s]{ZXx} + {~}

(5)

Eqn (5) takes into account the experimental error, and IS] is the sensitivity matrix. Using eqns (3) and (5), noting that {Ax} and {Ay} are independent and have normal distributions, one can derive the relations

{A~} = {A~o} + [K]{Zxy} [K] -- [Wz]-l [,.~]T( [S] [Wx]-l [s] T nt- [Wy]-l) -1

(6)

The variance of the estimated error is given by [W;] -1

--

([I]-

[K][S])[w~] -~

(7)

689 At the outset {Zo} and [Wx] are predicted according to experience, and {Ao}, {(I)o} and [S] can be computed by means of {Zo}. Then combining with the known [%], the sensitivity matrix IS] can be obtained. The change in {Zo} can be determined by examining the difference in {Ay} between the computed and experimental frequencies. Synthesizing, the study ability of the algorithm is embodied by the sensitivity matrix IS] and [Wx], [%] in the gain matrix [K]. When the weight determined by eqn (7) is larger, or the parameter variance is small, then the gain matrix [K] is small also. However when the weight determined by eqn (7) is less or the parameter variance is large, then the amount of modification of {Zo} is large. Increasing the values of some parameters may raise some frequencies, but at the same time there will be a decrease in some other frequencies. After all the changed amount of {Zo} (i.e. {z} -{Zo}) depends on the synthesizing study ability of [K]. M o d e l for fuzzy o p t i m i z a t i o n If the dimension of the vectors of the experimental data is equal to or exceeds the dimension of the vector of the structural model parameters, then the rank of the sensitivity matrix equals the dimension of the estimated parameters. Then there exists a unique solution. The solution obtained is near to the true value, but it could experience small changes depending on the initial values of the weights assigned. If on the other hand the experimental data vector is smaller than the parameter vector the system is indeterminate, and the number of solutions is unlimited. In the practical application of engineering it is always likely that the number of frequencies that are required to be close to the experimental values are few, while the number of the parameters which can be modified are many. So the concept of optimization is fuzzy. There are several key questions which should be addressed, such as (1) to what degree does the solution have to resemble the true solution, so that the calculation can be said to be an optimization? (2) which parameters should be taken as the design variables? (3) how should the weight of the design variables be determined? (4) how should the design variables be constrained? The answers depend largely on experience. The issues can only be described qualitatively, hence the application of fuzzy set theory to the modification of the model is appropriate. Let the fuzzy set be represented by F, and the fuzzy constant set by C. Then from eqn (3) fuzzy optimization mathematics can be used to determine the vector {z}. It is necessary to make F(z) rnin, and satisfy ci(z) v- Ci(z), where rnin means close to the minimum and v- means to belong to basically. The membership objective function and constraint may take the form 1; J(x) 0.95) is taken as the objective, and fuzzy deduction is made with eqn (6). Eqn (6) is used to determine the amount of modification of the parameters. Due to internal nonlinearity many iterations are required. The final results obtained by iteration are given in Table 2. It is seen that the results of the process are very good if #r - 1.

CONCLUSIONS The finite element method is used extensively for solving engineering problems. As many input paramters are imprecise, there are always differences between calculated and experimental results. The FEM model can be modified by treating some input parameters as fuzzy values, and using fuzzy set theory. The modified model leads to results which agree more closely with experimental values. In the modification of the model the geometric and material parameters of sensitive parts should be taken as the initial design variables. The fuzzy set of sensitivity can be obtained by deducing the sensitivity of the design variable for fixed frequency and vibration shape in the range of the frequencies of interest. Finally, rasonable design variables are selected by applying fuzzy logic inference. The approach outlined above in which fuzzy mathematics is used to modify the finite element model has the potential to offer significant benefits in analysis. In comparison with the adaptive finite element approach, it focuses on prior treatment to determine design variables, and later addresses the issue of post-error estimates and mesh refinement. Use of both of these approaches, which have their own respective advantages, can lead to improved results.

BIBLIOGRAPHY

Bathe, K. J. (1982). Finite Element Procedures in Engineering Analysis. Prentice Hall, Englewood Cliffs.

694

Kaufmann, A. and Gupta M. M. (1985). Introduction to Fuzzy Arithmetic: Theory and Application, Van Nostrand Reinhold, New York. Kiler, G. J. (1993). Fuzzy Set, Uncertainty and Information, Prentice-Hall, New Delhi. Li, L. Y. and Bettess, P. (1997). Adaptive finite element methods: a review. ASME Appl. Mech. Rev. 50. Reddy, J. N. (1985). An Introduction to the Finite Element Method, McGraw-Hill, New York. Valliappan, S. (1995). Elasto-plastic finite element analysis with fuzzy parameters. Int. J. Numer. Meth. Engng. 38. Zeng, L. F. (1992). Adaptive h-p procedures for high accuracy finite element analysis. Comput. & Struct. 42. Zienkiewicz, O. C. and Zhu, J. Z. (1989). Effective and practical h-p version adaptive analysis procedure for the finite element method. Int. J. Numer. Meth. Engng. 28. Zienkiewicz, O. C. (1989). The Finite Element Method. 4th Ed., McGraw-Hill, New York. Zimmerman, H. J. (1983). Fuzzy Set Theory and Its Application. Kluwer-Nijhoff, Amsterdam.

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

695

DEVELOPMENT OF SEMI-RIGID FRAME MODEL ASSISTED BY TESTING M. A. Gi~ejowski 1 , J. A. Karczewski 1 , E. Postek 2 and S. Wierzbicki 1 1 Institute of Building Structures, Warsaw University of Technology, Armii Ludowej 16, 00-637 Warsaw, PL 2 Institute of Fundamental Technological Research, ~;wi~tokrzyska 21, 00-049 Warsaw, PL

ABSTRACT A computational FEM model is developed to predict the load-deflection characteristic of portal frames with semi-rigid welded and bolted joints. Frame member and joint plate components are modelled by thin shell finite elements while bolt connectors by springs with the force-deformation characteristic evaluated from test results. The computer software ABAQUS is used. Experimental investigations were designed in order to evaluate frame load-deflection characteristics, joint local rotations and the contribution of end-plates and bolt connectors to the joint rotations. Theoretical results of frame load-deflection characteristics predicted with use of ABAQUS program are compared with the experimental results. The modelling of friction effect between the contact surfaces of bolted joints is verified. The comparative analysis leads to the conclusion that the proposed advanced shell model can be very useful in the verification of simplified models being developed for practical applications.

KEYWORDS Semi-rigid joint, advanced finite element model, thin shell element, contact effect, friction effect, experimental verification.

INTRODUCTION Among several joint types, steel end-plate connections have been widely studied in the literature Krishnamurthy & Graddy (1976), Kuktreti et al. (1987), Rothert et al. (1992), Nemati & Le Houedec (1996). FEM has been proved to be the most suitable tool for conducting an extensive numerical study validating different geometrical and mechanical parameters that influence the joint behaviour. Recent research has mainly concentrated on joints with relatively thick end-plates with comparison to the flange and web components of connecting members. A number of FEM models have been developed and verified by means of experimental studies. They were discussed by Nemati & Le Houedec (1996). It has been concluded that further research is needed to study the end-plate joint behaviour and its mode of failure depending on the plate thickness, the bolt diameter

696 and the connector arrangement. The study presented herein aims to develop an experimentally verified FEM model of semi-rigid portal frames with end-plate beam-to-column connections. The joint behaviour and frame response are investigated taking into account connection local effects of bolts, gaps and friction on the global behaviour of the structure, Gi~ejowski et al. (1998), Gi~ejowski et al. (1999), Gi~ejowski et al. (2000). An extensive experimental study was conducted to create a basis for verification of FEM model developed. The description of experimental work is given by Gizejowski (2000). A summary of this work is briefly presented herein. Theoretical frame load-deflection characteristics are compared with experimental ones. Concluding remarks are drawn with respect to both the FEM modelling and the verification procedure based on experimental evidence.

FINITE E L E M E N T MODEL The details of frame specimens considered in numerical simulations and experimental investigations are given in Fig. 1. The ABAQUS general purpose computer program is used. The shell finite element model is adopted. Since relatively thin plate components of the frame members and beam-to-column connections were used, they are fully discretised with thin shell finite elements what serves the amount of degrees of freedom when compared with solid finite element models. The column section IPE 240, the beam section IPE 120 and the end-plate thickness of 6 mm were applied. The column height (from the level of the load application point to the beam longitudinal axis) was equal to 508 mm, except one specimen for which the said distance was taken as 558 mm. The beam length was 3180 mm for all the specimens. The steel grade corresponding to $235 according to the Eurocode 3 (1992) was used, except for the connectors. The bolts M12 of grade 10.9(10) were used. The welding material was selected to conform with the grade of the parent material of the connecting elements (beams, columns and end-plates). Bolts are modelled by springs with the force-deformation characteristic based on the experimental investigations. The frame nonlinear behaviour is examined by applying a bilinear approach to the modelling of properties of beam section, end-plate components and HS bolt connectors. The kinematic hardening law is considered.

o) 2975

,4o

F

~40

lOO

~ b)

~ _

-~10x100xlO0

IPE 120

..

~d-plates

only ~6x192x80 10x250x130

~l+~

+;~o~

~6xl

~-:;~,

I~.L_~I

I p ',', m a

I I ,,,,, I I

; ,fgl a

[?]I!?1

I

I

Figure 1" Details of the considered flame specimens: a) flame specimen details, b) connection details

~----~ 10 x250 x130

697 To include the friction effect, the contact surfaces are "defined between the mid surface of the column flange and the mid surface of the beam end-plate. The coefficient of friction is taken as for the plain steel surfaces being in contact. Numerical simulations are performed for the frames with welded connections (nominally rigid joint frame) and a range of frames with bolted connections (6 connectors with different placement of the bolt row with reference to the beam tension flange and 8 connectors with bolt rows in the tension zone placed symmetrically with reference to the beam flange).

EXPERIMENTAL INVESTIGATIONS Experimental investigations on semi-rigid joints have mostly been performed on isolated joint specimens. Test results were collected and stored in data banks, such as the SERICON bank used for verification of the M-~ prediction model adopted in the Eurocode 3 (1992). In tests conducted for isolated joints, the following assumptions were usually adopted: 1. Joint specimens were loaded in such a way that forces applied on the joint specimen were increasing monotonically, and changing in a quasi-static manner. 2. Load increments were applied up to the joint failure or they were stopped quite arbitrarily at the point of excessive joint rotation, not necessarily associated with the joint failure. 3. Joint rotations were calculated from the measurements recorded by inductive displacement transducers which were placed on the beam at the distance Lo from the column face. The measurements recorded by transducers were able to reproduce the rotation ~r being a sum of the connection rotation ~ and a contribution from the beam length Lo curvature rotation. In general, such a testing procedure is not an accurate enough procedure in order to model the real connection behaviour in steel frame buildings. A more accurate procedure was therefore suggested for the experimental investigations reported herein: 1. The connection behaviour was examined on the basis of tests conducted upside-down for portal frame specimens, Fig. 1. 2. Frame specimens were loaded in such a way that the force applied on the specimen was increasing incrementally in a quasi-static manner, but at each incremental load step the load cycles were repeated up to the stabilisation of the deflection state. The moment amplitude in the connections was selected as 0.4 times the actual connection moment. 3. The measurement set-up was devised in such a way that the overall joint rotation ~ and the rotation (~r of the joint plus the contribution of the beam length L0 could be examined. 4. Tests were continued up to the point of the local failure of the connection or the global failure of the frame system due to instability. The procedure suggested for the experimental investigations of semi-rigid joint behaviour is a component based procedure used by Bernuzzi et al. (1996). It has however a more general aspect since it includes the testing of frame specimens instead the testing of isolated joints. In this way, the interaction of failure modes could be examined and the influence of joint semi-rigidity on the frame ultimate strength evaluated. With the reference to the joint moment-rotation curve, operational definitions of the rotation of the whole connection ~, the rotations associated with the various connection components (~p, ~ ) and the rotation (~r associated with the connection rotation and the beam length Lo were adopted. The definitions of ~, ~)p and 0b are consistent with Bemuzzi et al. (1996). Measurement set-up The horizontal load was applied to the free end of the left column by means of a hydraulic jack. The power jack, which had a 400 kN capacity and a 160 mm stroke, was attached to a specific frame, designed to accommodate the backward movement. The horizontal force was transferred onto the frame beam through the columns, causing its bending and tension or compression, depending upon

698 the direction of the load applied on the specimen. The amount of bending moment and axial force transferred onto the beam depended upon the properties of the beam-to-column connections. The failure mode of the tested frame specimen depended on the connection ability to transfer the forces from the column to the beam. In the case of a strong connection (such as welded and bolted with 8 bolts), laterally unrestrained specimens were more sensitive to the failure in a global instability mode. In the case of weak connections (bolted with 6 bolts), specimens were more sensitive to a local mode of failure (end-plate fracture or its excessive plastic deformations). The fracture mode resulted from the end-plate rupture in the Heat Affected Zone (HAZ). The excessive plastic deformations resulted from the formation of end-plate yield lines. The experimental set-up allowed for the instability mechanism to be monitored. The failure mechanism could be detected trough the examination of the beam mid-span section lateral displacement and its rotation about the beam longitudinal axis. For tested frame specimens, the above displacements were expected to be relatively small in the initial stage of testing and to increase rather slowly in course of the loading programme. In the final stage, close to the frame specimen ultimate strength, the beam lateral displacement and the twist rotation were expected to remain small in the case of specimens insensitive to lateral torsional instability. This is indicative for the following modes of failure to occur: 1. local joint failure due to: a) a large rotation without fracture (for plastic joints), b) a relatively large rotation with localised fracture (for plastic or semi-plastic joints), c) a relatively small rotation with overall fracture of the end-plate expected above the weld line or in the welding material (for non-plastic or brittle joints), 2. global in-plane failure of the frame due to: a) excessive plastic deformations (for plastic beam sections), b) local instability mechanism (for non-plastic beam sections). In the case of specimens sensitive to lateral torsional instability, an increase in the beam section lateral displacement and twist rotation was expected, indicative for the overall instability mode of failure to occur. The loading history was chosen in such a way that it reproduced an average time-dependent connection behaviour in building steel frameworks, Karczewski et al. (2001).

Evaluation of test results The experimental investigations confirm that restrained frames with semi-rigid connections in the form of welded joint and flush end-plate bolted joints (with 6 bolts two of which are placed below the tension flange) have a very satisfactory ductility. The failure mode of the restrained flame specimen with the welded joints and the beam subjected to tension and bending was due to an excessive plastic deformations of column flanges and associated with the distortion of the beam section web, close to the joint. The full plastic capacity of the beam section was reached at the ultimate strength of the flame specimen. There was no sign of weld fracture or any other local mode of joint failure. The restrained flame specimens with flush end-plate bolted connections mentioned above exhibited an excessive plastic deformation of the end-plates (yield lines), with no plastic deformations in beam sections, according to the connection low stiffness and strength. No cracking of the end-plate was observed but the deflected profile of the end-plate was distinguished by a rigidlike rotation of its cantilever part. The restrained specimens with extended end-plate connections with 8 bolts exhibited a better balance between the stiffness and rotational ductility. They were practically equivalent (in terms of stiffness and strength) to the welded connections. This allowed the same joint ultimate strength to be reached for welded connection specimens and extended endplate connection specimens, provided that there are 8 bolt connectors applied. Although the rotation attained by such bolted connections is much lower than that of welded joints, it was yet sufficient for the plastic stress redistribution to take place in the flame members. The lower rotation ability of

699 this type of joint is associated with the localised fracture occurring in the end-plates, in the area of bolt holes. Detail " A "

Detail "A"

0.0

0.0

6

z

-10.0

Average l o a d - d e f l e c t i o n c h a r a c t e r i s t i c /J for f r a m e s p e c i m e n s n n. 3 and 4 //IJ

-10.0

HI

Partial joint f r a c t u r e

z

n-20.O -o

°-_20.0.

8

o

-6

o

?-30.0

°_30.0._

o 22

.-I-

Thin shell FE model (ABAQUS) -40.0-

m_ _--_ _-! C o n t a c t and friction effects --~ ~ ± C o n t a c t effect neglected

~Maximum deflection due to t e s t i n g machine limit Average l o a d - d e f l e c t i o n c h a r a c t e r i s t i c for f r a m e specimens n ~. 5 and 6 Thin shell FE model

-40.0-

included

= ~ Contact :- --" --" C o n t a c t

-50.0 ......... i ......... i ......... l ......... ~........ -250.0 -200.0 -150.0 -100.0 -50.0

Horizontal

deflection

Horizontal

A [mm]

effects

included

effect neglected

deflection

() 0 •

Z~ [ m m ]

Figure 3: Frame with bolted connections

Detail "A

Detail "A"

A"

0.0

0.0

-10.0

-10.0-

vPu

z

(ABAQUS)

-50.0 ......... i ......... ~. . . . . . . . . i ......... i ........ -250.0 -200.0 -150.0 -100.0 -50.0

() 0 -

Figure 2: Frame with bolted connections

~

and friction

z a-_20.0 -o

°-_20.0

8

-6 .NO--30.0 -1-

Average l o a d - d e f l e c t i o n c h a r a c t e r i s t i c for frame specimens n% 7 and 8

O-3o.o ._

Thin shell FE model (ABAQUS) -40.0

r_ = = C o n t a c t --" --" C o n t a c t

and friction

effects

-40.0-

included

deflection

A [mm]

Figure 4: Frame with bolted connections

8

Beam c l o s e - t o - j o i n t

~ ~Thin

web distortion

shell FE model

(ABAQUS)

Load-deflection characteristic for f r a m e specimen nO.9

effect neglected

-50.0 ......... i ......... i ......... , ......... i ........ -250.0 -200.0 -150.0 -100.0 -50.0 Horizontal

vP,,

T

Complete joint f r a c t u r e

() 0 •

-50.0 ......... i ......... i ......... ,. . . . . . . . . i ........ -250.0 -200.0 -150.0 -100.0 -50.0

Horizontal

deflection

(~ 0 •

A [mm]

Figure 5: Frame with welded connections

Only the end-plate connections with 6 bolts two of which are placed above the beam tension flange showed a very low ductility, insufficient for the plastic moment redistribution to take place in real

700 flame structural systems. These connections in all the tested flame specimens failed in a brittle mode by rupture of the end-plate in the HAZ area, above the tension flange (just above the outer flange weld). This type of failure mode is undesirable, and because of that joints with thin endplates and bolts located only on the outer side of the tension flange must be avoided in practice. If such a joint type needs to be applied, a relatively thick end-plate has to be used in order to ensure that stresses in the end-plate, close to the tension flange welds, are in the elastic region. This increases the joint strength and stiffness, resulting in the plastic moment redistribution to take place in structural members. The experimental results showed that the bolt contribution to the connection rotation ~ can be generally neglected in all the bolted connections tested experimentally. The connection deformability is controlled by the ability of the end-plate to plastic deformations. When the endplate most stressed regions are not associated with HAZ, the yield lines can be developed and the connection exhibits a good ductility. Otherwise, the end-plate fracture (overall or localised) is most likely to occur. As a result of the end-plate flexural deformations, the bolt shanks are subjected to combined tension and bending. Despite of an asymmetrical strain state in the bolts, the average strain, associated with the bolt shank elongation, is a dominant factor. The results from the bolt strain state measurements were used for the development of a connector spring model to be adopted in the presented shell FE model.

V E R I F I C A T I O N OF THE FINITE E L E M E N T MODEL The verification of the thin shell FE model developed is presented for the flame specimens with the beam restrained at distances of 1/3 and 2/3 of its length and loaded in tension and bending. Fig. 2 shows the average experimental load-deflection characteristics obtained for flame specimens with 8 bolts and its comparison with the results obtained for two cases by the ABAQUS code. In the first case the contact and friction effects were considered in the analysis while in the second - the contact effect was neglected. From the comparison it is clear that below the load level corresponding to approximately 2/3 of the ultimate load Pu the experimental load-deflection characteristic is below those obtained from the computer simulations. Above this level the flame equilibrium path based on computer simulations with the contact and friction effects included approaches that obtained from experimental investigations. The experimental ultimate load is however lower from that evaluated numerically. The joint failure due to a partial end-plate fracture below the beam tension flange could not be detected in the computer analysis since the material non-homogeneity and anisotropy in HAZ resulting from the welding process as well as the effects of residual stresses and end-plate lack-of-fit were not considered in the FEM modelling. It is seen that the effect of contact-friction phenomena should not be neglected. It appears that the behaviour of bolt connectors is linear even in the ultimate limit state of the flame. The plastic zones are localised in the end-plates. Fig. 3 shows the similar comparison made for the flame specimens with 6 bolts two of which were placed in the tension zone below the beam flange. The experimental load-deflection characteristic coincides with those obtained by ABAQUS code in the region below the load level of approximately V2 of the ultimate load Pu • The shell FE models underestimate the flame ultimate load but again the results obtained for the case in which the effects of contact and friction were considered are closer to the experimental ones. The flame specimen experimental characteristics ends earlier than those obtained numerically. Excessive plastic deformations of the beam system could not be examined experimentally due the existing limitation of the testing equipment. No joint failure is detected in the computer simulations, however they prove that the end-plate is subjected to large bending deformations causing large permanent end-plate deflections observed during unloading of the tested flame specimens. Fig. 4 shows the results obtained for specimens with 6 bolts two of which are placed in the tension zone but above the beam tension flange. The shell FE model overestimates the flame ultimate load Pu. The shape of the loaddeflection curve obtained numerically for the case in which the contact-friction phenomena was

701 included reproduces more accurately the curve obtained from experimental investigations. The joint failure mode resulting from the end-plate complete fracture could not however be detected in numerical simulations. This is due to the fact that the welding degradation effect on steel ductility properties is not considered and the assumption of homogeneous and isotropic material is used. Thus the shell FE model underestimates frame deflections what can be attributed to the effect of end-plate residual stresses causing early yielding of a large volume of end-plates in the region of beam tension flange welds. Finally the results obtained for the welded joint frame are presented in Fig. 5. The joint was arranged by fillet welding of the beam end section to the inner flange of the frame column. The results of numerical simulations are very close to the experimental ones below the load level of approximately 2/3 of the ultimate load Pu • This is due to the fact that the effect of residual stresses induced by welding is not so important in this case unlike in the case of thin endplate bolted joints. The column flange of 17 mm in thickness is less affected by welding in fillet welded joints than the 6 mm end-plate welded to the beam end section in bolted joints. As a result the residual stresses induced in the column flange of welded joints have a lesser impact on the frame equilibrium path. Their importance become more visible in an inelastic region, close to the frame ultimate limit state. Because the effect of welding residual stresses is not considered in the FE model developed, the experimental ultimate load Pu is lower than that obtained from the ABAQUS code. The experimental evidence showed that there was no joint failure. The ultimate limit state was associated with in an excessive inelastic distortion of the beam web in the region between the joint and the first beam span restraint. The same mode of failure is detected in the computer stability eigen-mode analysis.

C O N C L U D I N G REMARKS The number of geometrical and mechanical parameters that can reasonably be expected to influence the behaviour of frames with semi-rigid joints is significant, so that both the experimental and numerical investigations are needed to provide an efficiency in engineering design practice. In the paper a FE model is presented as part of the wider research program carried out at the Warsaw University of Technology on the quasi-cyclic and quasi-static behaviour of steel semi-rigid joints and frame specimens. Theoretical and experimental investigations were concerned with portal frame specimens composed of I section members and welded or bolted joints. At this stage of research the 6 mm end-plate was used. The thin shell FE model is developed to represent tested physical frame models. Bolt connectors are modelled by springs with the force-deformation characteristic evaluated on the basis of average bolt stress-strain response monitored during testing. The member parent steel constitutive relationship for homogeneous and isotropic material with kinematic hardening is used. No allowance is made for the ultimate stress increase and ductility reduction effects resulting from welding. Experiments were designed so that they reproduced average time-dependent joint behaviour in typical steel building frameworks. The study presented herein was purposefully limited to the connections with a relatively small thickness of the end-plate (if compared with real joints). This allowed to investigate different modes of connection and frame failure. The application of thin end-plates and the bolt rows placed only outside of the tension flange, resulted in a brittle mode of joint failure, even below the load level corresponding to a global failure mode due to the out-of-plane instability of laterally unrestrained frame specimens. It indicates, in particular, that a proper selection of the strength and stiffness of the end-plate relative to the bolt group can enable to optimise the strength, stiffness and ductility of the whole connection and the structural system accordingly. More ductile connections are those in which their end-plates were not subjected to excessive bending deformations in the regions subjected to HAZ effect. Depending on the connection strength, the arrangement of beam in-span lateral restraints and the sign of axial force in the beam, the local joint modes of failure or the global instability modes due to lateral torsional buckling can occur. The verification procedure shows that the accurate modelling of the contact-friction phenomena is highly important in FE modelling of end-plate connections.

702 Any simplification would affect seriously the results. The shape of experimental frame loaddeflection characteristics can only be reproduced by the FE modelling scheme when the above mentioned effect is taken into account. The present investigations show that for relatively thin endplates the effect of welding on mechanical properties of steel in the HAZ regions is significant and it has to be included in the FE modelling. The increase of the values of the yield stress and the ultimate stress as well as the reduction of ductility ratio depend on the steel chemical composition, heat input, cooling rate, etc. An increase in the value of the yield stress has a positive effect slowing down the onset of plastification but the reduction of ductility increase the sensitivity to cracking due to the accumulation of plastic deformations and due to the low cycle fatigue, even in case of quasicyclic loading. Because the presented FE model is based on the conventional constitutive relationship used for parent steel, joint failures due to the end-plate fracture observed during experimental investigations can not be captured in numerical simulations. Because no universal laws governing the effects of material non-homogeneity and anisotropy due to welding have yet been developed, the investigations presented herein create a basis for our fracture study on semirigid joint with welded components, Jemioto and Gi~ejowski (2000). .

REFERENCES

Bernuzzi C., Zandonini R., Zanon P. (1996). Experimental Analysis and Modelling of Semi-Rigid Steel Joints under Cyclic Reversal Loading. J. Construct. Steel Research 38:2. Eurocode 3 (1992). ENV 1993-1-1: Design of Steel Structures, Part 1-1, General Rules and Rules for Buildings. CEN, Brussels, Belgium. Gi~ejowski M., Karczewski J., Postek E. (1998). Computational Model of the Frame with SemiRigid Nodes. Proc. EUROMECH 385 ,,Inelastic Analysis of Structures under Variable Loads: Theory and Engineering Applications", Aachen, Germany. Gi~ejowski M., Postek E. (1999). Modelowanie zachowania sit ram z w~ztami podatnymi. ln£'ynieria i Budownictwo, 11,645-651. Gizejowski M., Karczewski J., Postek E. (2000). Badania ram z w~ztami podatnymi- weryfikacja do~wiadczalna modeli. In~ynieria i Budownictwo, 8, 433-439. Gi~ejowski M. (2000). Modele obliczeniowe stalowych ram pt~skich z w~ziami podatnymi. Prace Naukowe, Budownictwo, Oficyna Wydawnicza Politechniki Warszawskiej, 136, 3-285. Jemioto S., Gi~ejowski M. (2000). Modele konstytutywne ze zmiennymi wewn~trznymi do opisu zachowania sit stali w w~ztach spawanych. Cz.l: Podstawy termodynamiczne, Cz.2: Modele lepkoplastyczno~ci i plastyczno~ci. Proc. International Symposium ,,Semi-Rigid Joints in Metal and Composite Structures" (Eds. M.A. Gi2ejowski & A.M. Barszcz), 24-25 November, Warsaw, Poland, 289-318. Krishnamurthy N., Graddy D. (1976). Correlation between 2D and 3D F.E. Analysis of Steel Bolted End-Plate Connections. Computer & Structures 6, 381-389. Nemati N., Le Houedec D. (1996). A Survey on Finite Element Modelling of Steel End-Plate Connections. Proc. IABSE Colloquium ,,Semi-Rigid Structural Connections", Instanbul, IABSE Reports, 75, 269-278. Karczewski J., Gi~ejowski M., Wierzbicki S., Postek E. (2001). Double butt bolted connections influence of pre-stressing. Proc. International Conference ,,Structural Engineering, Mechanics and Computation", 2-4 April, Cape Town, South Africa. Kuktreti A.R., Murray T.M., Abolmali A. (1987). End-Plate Connection Moment-Rotation Relationship. J. Construct. Steel Research, 8, 137-157. Rothert H., Gebbeken N., Binder B. (1992). Nonlinear 3D Finite Element Contact Analysis of Bolted Connections in Steel Frames. Int. J. Num. Meth. Engng. 30, 303-318.

Structural Engineering,Mechanicsand Computation(Vol. 1) A. Zingoni(Editor) 2001 ElsevierScienceLtd.

703

ANALYSIS OF BUILDING STRUCTURES USING SOLID FINITE ELEMENTS S. H. Lo Department of Civil and Structural Engineering, the University of Hong Kong.

ABSTRACT Robust and efficient hybrid stress hexahedral elements capable of handling beam/plate/shell structures have recently been proposed. A major advantage of using the solid elements for tall building analysis is that rotational degrees of freedom are not required to describe the deformation of the structure. No special effort, if at all possible, would be needed in matching the translational and rotational degrees of freedom when structures are inevitably modelled by a combination of beam, plate and shell elements. In this paper, the feasibility and the effectiveness of applying these hybrid stress hexahedral elements to general three-dimensional spatial frame structures will be investigated. Numerical results as compared to those obtained from the classic beam theory will be presented.

KEYWORDS Structural analysis, three dimensions, hybrid solid hexahedral elements

INTRODUCTION In engineering practice, building structures are usually idealized as spatial frames of columns and beams for the purpose of structural analysis and design. For instance, in the analysis of a coupled shear wall, each of the two walls is regarded as a vertical cantilever composed a series of columns residing at the centroidal axis of the wall. By means of frame analogy, connection beams are modelled by incorporating "rigid arms" into the horizontal beam elements to account for the finite width of the shear walls [1]. Obviously, the omission of the shear deformations and whether a wall can be adequately represented by a series of column elements are subject to serious discussions. Even the length of the rigid arm would vary if different assumptions are made in its determination [2,3]. The result could be improved if plate elements are used to represent the walls and the connecting beams are modelled by standard beam elements. However, the definition of beam rotations and the stress concentration effects at the beam-wall junction remain as interesting topics for engineers and researchers [4]. As for the analysis of core wall structures, wall units have to be divided into shell elements. There are usually five degrees of freedom at a typical node of the shell element, namely, three translations and two rotations about the in-plane axes. However, when two shell elements come together at a sharp angle, say at the junction of two wall units, the in-plane rotations of one element cannot be transformed into those of the neighbouring element since the rotations of the second element

704 refer to different axes. In order that a complete correspondence for the transformation of rotational degrees of freedom is possible, three rotations about three orthogonal axes should be used instead of only two. Unfortunately, the physical rotation normal to the plate/shell element surface turned out to be rather hard to define [5]. Although, in principle, analysis of complex wall-frame structures can be done by using 3D classical solid elements alone, it is well known that conventional displacement elements will "lock" when the element become long and thin. Discretization of the whole structure into solid elements with characteristic size equal to the wall thickness is obviously impractical due to the huge number of degrees of freedom involved. Hence, more efficient and flexible solid elements with good structural behaviours are needed for a practical analysis of real structures. THE 8-NODE HYBRID STRESS HEXAHEDRAL ELEMENT (HSH)

The 8-node HSH element is formulated based on the well-known hybrid stress element (PT1813) developed by Pian and Tong [ 16]. Under usual circumstances or even in incompressible applications, the PT1813 element is highly efficient and not susceptible to element distortions; however, it will encounter severe locking when the element becomes long and thin. The assumed stress field used in the PT1813 element is taken as the starting point for the construction of the HSH element. By means of the selective scaling technique, the parasitic strain components which eventually lead to locking when one or more dimensions of the element become very small are identified and appropriately scaled down. After scaling, however, it is found that the originally robust element is unable to pass the patch test. As a result, the admissible matrix formulation technique is employed to modified the assumed stress modes to restore the robustness of the element [6]. According to the numerical studies in reference [6], the HSH element possesses the following characteristics: • it is insensitive to element distortions and geometrical aspect ratios; • it produces accurate solutions comparable to the most efficient 4-node plate elements to difficult plate bending problems in its flattened form (1:1000); • it can give accurate solution (>99%) in the degenerated form of long and thin strips (length/thickness = 4000) under twisting action; • in modelling a cantilever beam under the action of end shear, it does not lock even in the extreme situation with an aspect ratio of 1000, and when compared to the more expensive 27-node Lagrangian displacement element, it gives more accurate solutions with much fewer degrees of freedom.

STRUCTURAL ANALYSIS OF 3D FRAMES Before the HSH element is put to practical analysis of engineering structures, the single element performance is assessed in the bending test of a cantilever beam under the action of end shear. The cantilever beam is of length 3m, fixed at one end and free at the other end. It is of uniform crosssection 0.5mx0.5m throughout, with modulus of elasticity E=10,000MN/m 2 and shear modulus G=4,000MN/m 2. A point load of 4kN is applied at the free end. For the given geometrical definition and material constants, a reference solution of 0.6912mm is obtained based on simple beam theory. When one HSH element was used to model the cantilever, a deflection of 0.59026mm was obtained which is 85.4% of the reference solution, and the deflection was improved to 96% if two HSH elements were used. However, when the cantilever beam was divided into six HSH elements in the

705 form of unit cubes, the end deflection increased to 0.69248mm which is 100.19% of the reference solution. It is more than 100% because shear deformation is taken into account in the full 3D analysis using the HSH elements, whereas shear deformation is not included in the reference solution based on elementary beam theory. To test convergence, a refined mesh was generated by dividing each HSH element into eight smaller elements, and a even more refined mesh was obtained by further cutting each element in the refined mesh again into eight elements. The end deflections obtained by the refined meshes were 0.69351mm(100.33%) and 0.697mm(100.84%) respectively, showing that accurate converged solution can already be achieved by modelling a structural member with two or more HSH elements as shown in Table 2 - Cantilever.

Figure 1. Two-storey square frame Two-storey f r a m e on a square base

A simple two-storey frame on a 5mx5m square base is the first building structure analyzed by the HSH elements. All members, columns and beams, are of the same cross-section of 0.5mx0.5m, and each floor is 3m in height. The modulus of elasticity and the shear modulus used were E=10,000MN/m 2 and G=4,000MN/m 2 respectively. The frame was loaded by a concentrated force of 8kN at one of its top comers as shown in Figure 1. From the spatial frame analysis, the deflection obtained at the position and in the direction of the load is 0.6714mm which is taken as the reference solution. A very coarse mesh was used as an initial test of the HSH element, in which each structural member was modelled by one single HSH element, and one HSH element was used at the junction where structural members meet. The solution given by this minimum mesh is rather disappointing that the deflection obtained is only 0.07196mm, about 11% of the reference solution. When each structural member was modelled by two HSH elements, the deflection improved to 69.39%. A even better solution of 0.5586mm(83.2%) was obtained by dividing the entire frame into unit cubes of HSH elements with characteristic size equal to the section width of the structural members. A refined mesh was generated by curing each unit cube into eight smaller cubes, the deflection corresponding to this very fine mesh is 0.5775mm(86%). A finer mesh would not bring the deflection closer to the reference solution. Hence, there is a discrepancy of 14% between the finite element solution and the spatial beam solution. A closer look at the problem reveals that the discrepancy may be due to the joint connections which reduce the effective length of the structural members. To make an investigation on the junction effect, rigid arms equal to half of the section width were introduced to the ends of each member, which effectively reduce the flexibility of the structural members. The rigid arms were made fairly stiff to very stiff by gradually increasing its modulus of elasticity from 2E, 3E to 100E. The results indicate that joint effect is very significant for structure with members of large cross-sections.

706 In this example, the deflection could be reduced up to 25% if the joints are considered as infinitely rigid, and the deflection will also be reduced by 10% if the joints are considered as twice as rigid as shown in Table 2 - Two-storey square frame A. In order to further verify the joint effect, the same flame was made more slender by reducing the member sections from 0.5mx0.5m to 0.2mx0.2m. The converged solution is about 98% of the reference beam theory solution, whereas in the previous case it was only 86%. If rigid arms were introduced, the deflection was reduced by 10% and 5% respectively in the cases of infinite stiffness and twice stiffness, showing that a slender frame is less affected by the rigid joint effect as shown in Table 2 - Two-storey square frame B.

Figure 2. Three-storey L frame.

Three-storey L-frame A more complicated spatial flame structure with three storeys on a base of L shape is taken as the second example. All structural members are of the same cross-section of 0.5mx0.5m, the centre to centre distance between columns is 6m in both directions, and each floor is of height also 6m as shown in Figure 2. The same material constants were used such that the modulus of elasticity and the shear modulus were E=I 0,000MN/m z and G=4,000MN/m 2respectively. The frame was loaded by a uniform pressure loaded on one external face of the structure. The maximum deflection of 4.1625mm at the top comer of the frame obtained by spatial frame analysis is taken as the reference solution. When the structural members were each modelled by two HSH elements, the maximum deflection obtained was 65% of the reference solution, and the deflection increased to 87% if cubic HSH elements were used to model the whole structure. Only very slight improvement (88%) was made when the mesh was refined to eight times more elements. In fact, the solution is already very accurate using unit cube HSH elements with further refinement, and the discrepancy is mainly due to the rigid joint effects. Accordingly, rigid arms were incorporated in the spatial frame analysis, which resulted in 9% to 18% reduction in deflection for a rigid arm stiffness from 2E to 100E as shown in Table 2 Three-storey L-frame.

707

Comparison of memberforces The member forces computed based on solid elements model are very close to those obtained from a beam theory model. When no rigid arms are introduced in the junctions of beam connections, the difference is about 5%; however, with rigid arms incorporated, the difference in the maximum reaction forces could be reduced to less than 2% as shown in Table 1. TABLE 1 COMPARISON OF CRITICAL REACTION FORCES

L-frame Shear Force Axial Force Bending Moment Torsion

Square frame Shear Force Axial Force Bending Moment Torsion

Beam Theory(no rigid arm) 3.085kN 7.112kN 10.93kNm 0.286kNm Beam Theory(with rigid arm) 3.091kN 3.928kN 5.77kNm 0.425kNm

Hexahedral elements 3.063kN 7.164kN 10.39kNm 0.309kNm Hexahedral elements 3.093kN 3.926kN 5.87kNm 0.433kNm

CONCLUSIONS AND DISCUSSIONS The 8-node hybrid stress hexahedral element provides an alternative for the analysis of spatial frames and its full potential is yet to be established in the analysis of other practical engineering structures in general, shear-wall, core-wall, frame-tube structures, bridges, dams, etc. With an efficiency even better than the more expensive 27-node displacement element, the HSH element made it possible for the first time that the analysis of a building structure be taken as a full three-dimensional problem. In the spatial frame analysis, however, the deflection based on a coarse mesh of one element per member is rather disappointing, indicating that structural response cannot be adequately captured by the minimum mesh. Much better results could be obtained using two elements for each structural member, about 80% if joint effects are taken into account. Very accurate results can be obtained by dividing the entire structure into unit cubes of HSH elements; nevertheless, this is still a very expensive solution process. From the converged solutions using HSH elements, the rigid joint effect can be significant in spatial frame analysis that deflections could be over-estimated by more that 25% dependent on the slenderness of the structural members. As for the reaction forces, the hybrid solid elements give very good results even in the analysis of very slender spatial frame structures. The difference is about 5% compared to a standard beam model, which can be narrowed down to about 2% if rigid arms are introduced in beam connection junctions.

ACKNOWLEDGEMENT The financial support from the Research Grant Council for the project "Analysis of building structures using hybrid stress hexahedral elements" is highly appreciated.

708 TABLE 2 3D BEAM AND FINITE ELEMENT ANALYSIS OF STRUCTURAL FRAMES (B.T. = BEAM THEORY)

Cantilever NN NE NEQ 8 %

L = 3m

BxD = 0.5mx0.5m

Single element 8 1 36 0.59026 85.40

2 elements 12 2 36 0.66383 96.04

6 elements 28 6 84' 0.69248 100.19

Two-storey square frame A Base = 5mx5m

NN NE NEQ CPU time 8 % Rigid Arm %

Height/floor = 3m

Minimum mesh 80 24 240 0.23 0.07196 10.72 2E 0.5907 87.98

Two hex. Per 144 40 432 0.51 0.4659 69.39 3E 0.56296 83.85

Two-storey square frame B Base = 5mx5m NN NE NEQ CPU time 8 % Rigid Arm %

Minimum mesh 80 24 240 0.23 1.8759 7.25 2E 24.54 94.79

Three-storey L-frame Base = 6mx6m

NN NE NEQ CPU time 8 % Rigid Arm %

Height/floor = 3m Five hex. Per 336 88 1008 1.35 23.983 92.63 3E 24.09 93.05

Height/floor = 6m

Minimum mesh 336 120 1008 3.2 0.21131 5.08 2E 3.794 91.15

E = 1000 Refined mesh 117 48 351 0.69351 100.33

Section = 0.5mx0.5m beam

4E 0.5489 81.75

Unit cubes 464 120 1392 1.81 0.5586 83.20 10E 0.5234 77.96

Section = 0.2mx0.2m beam

4E 23.858 92.15

Unit cubes 1232 312 3696 5.48 25.176 97.24 10E 23.45 90.58

Section = 0.5mx0.5m

Five hex. Per beam 672 204 2016 7.5 2.72369 65.43 3E 4E 3.669 3.6905 88.14 86.71

Point Load = 8 Double refined 625 384 1875 0.697 100.84

B.T. PL3/3EI 1 12 0.6912

Point Load = 8 Refined mesh 2124 960 6372 43.12 0.5775 86.01

B.T. 12(40) 16(44) 72(240) 0.6714

100E

0.5078 75.63

Point Load = 8 Refined mesh 5580 2496 16740 145.54 25.383 98.04

B.T. 12(40) 16(44) 72(240) 25.89

100E

23.197 89.60

Distributed load on one face

Unit cubes Refined mesh 3696 7884 960 3648 11088 23652 76.12 850 3.6325 3.663 87.27 88.00 10E 100E 3.4915 3.4222 83.88 82.22

B.T. 48(204) 84(240) 288(1224) 4.1625

709 REFERENCES

1. Michael D., "The effect of local wall deformations on the elastic interaction of cross walls coupled by beams", Proceedings of Tall Building Symposium, University of Southampton, Pergagom Press, 253-270 (1967) 2. Hall A.S., "Joint deformation in building frames", Civil Engineering Transactions, The Institution of Engineers, Australia, 1969, April, 60-62 3. Bhatt P., "Effect of beam-shear wall junction deformation on the flexibility of the connecting beams", Building Science, Vol. 8, 149-151 (1973) 4. A.K.H. Kwan and Y.K. Cheung, "Finite element analysis of coupled wall structures with local deformation at beam-wall joints allowed", Proceedings of the Fifth East Asia-Pacific Conference on Structural Engineering and Construction, 25-27 July 1995, Gold Coast, Australia 5. O.C. Zienkiewicz and R. L. Taylor, The Finite Element Method, 4 th Edition, Volume 2, McGraw-Hill International Edition 6. K.Y. Sze, A. Ghali, "An hexahedral element for plates, shells and beams by selective scaling", Inter. J. Numer. Methods Engrg., 36, 1519-1540 (1993) 7. K.Y. Sze and S.H. Lo, "A twelve-node hybrid stress brick element for beam~column analysis", submitted to Engineering Computations on 3 April 1998. 8. K.Y. Sze, S. Yi, M.H. Tay, "An explicit hybrid-stabilized eighteen-node solid element for thin shell analysis", Inter. J. Numer. Methods Methods Engrg., 40, 1839-1856 (1997) 9. M.F. Ausserer, S.W. Lee, "An eighteen-node solid element for thin shell analysis", Inter. J. Numer. Methods Engrg., 26, 1345-1364 (1988) 10. J.J. Rhiu, R.M. Russell, S.W.Lee, "Two higher-order shell finite elements with stabilization matrix", AIAA J., 28, 1517-1524 (1990) 11. H.C. Park, C. Cho, S.W. Lee, "An efficient assumed strain element model with six dof per node for geometrically nonlinear shells", Inter. J. Numer. Methods Engrg., 38, 4101-4122 (1995) 12. A. Dorfmann, R.B. Nelson, "Three-dimensional finite element for analysing thin plate~shell structures", Inter. J. Numer. Methods Engrg., 38, 3453-3482 (1995) 13. H. Parisch, "A continuum-based shell theory for non-linear application", Inter. J. Numer. Methods Engrg., 38, 1855-1883 (1995) 14. D. Roehl, E. Ramm, "Large elasto-plastic finite element analysis of solids and shells with the enhanced assumed strain concept", Inter. J. Solids & Structures, 33, 3215-3237 (1996) 15. R. Hauptmann, K. Schweizerhof, "A systematic development of solid-shell element formulations for linear and non-linear analysis employing only displacement degrees of freedom", Inter. J. Numer. Methods Engrg., 42, 49-69 (1998) 16. T.H.H. Pian and P. Tong, "Relations between incompatible displacement model and hybrid stress model", Inter. J. Numer. Methods Engrg., 22, 73-181 (1986)

This Page Intentionally Left Blank

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

711

GEOMETRICALLY NON-LINEAR THERMAL STRESS ANALYSIS OF AN ADHESIVELY BONDED TEE JOINT WITH DOUBLE SUPPORT M. Kemal APALAK', Recep GUNES and E. Sinan KARAKAS Erciyes University, Department of Mechanical Engineering, 38039 Kayseri, Turkey

ABSTRACT: In this study, the geometrically non-linear thermal stress analysis of an adhesively bonded tee joint with double support was carried out using the finite element method. The tee joint was also bonded to a flexible horizontal plate and it was subjected to variable thermal boundary conditions, that is, air flows with different temperature and velocity parallel and perpendicular to outer surfaces of the tee joint. First, the heat transfer analysis of the tee joint was carried out in order to determine the temperature distribution through the joint members. Considerable heat fluxes were observed in the adhesive fillets at the adhesive free ends whereas uniform temperature distributions occurred in the vertical and horizontal plates and supports. Later, the geometrically non-linear stress analysis of the tee joint was carried out considering large displacement and rotation effects. The stress analysis of the tee joint was repeated for two boundary conditions in which the edges of horizontal and vertical plates were fully fixed and partially restrained. In both cases considerable stress concentrations occurred in the adhesive fillets around the adhesive free ends. In addition, the horizontal and vertical plates experienced serious stresses along outer surfaces. However, most serious stresses were observed in case the edges of the horizontal and vertical plates were fully fixed. The effect of support length on the peak thermal stresses were investigated, and an optimum support length was determined based on the peak thermal stresses in the adhesive and plates. Finally, the adhesive joints with simple or complex geometry may experience thermal strains resulting in serious thermal stresses under thermal loads due to thermal-and mechanical mismatch. In addition, even a simple mechanical boundary condition may make these thermal stresses serious. Therefore, thermal boundary conditions should be taken into account in the design of the adhesively bonded steel joints. Keywords: Epoxy adhesive; steel adherends; tee joint; geometrical non-linearity; thermal stress analysis; ANSYS.

INTRODUCTION Different configurations of adhesively bonded joints are used widely in engineering structures [ 1]. The adhesive joints consist of adherends (plates, shells) and adhesive layer having different thermal and mechanical properties. These structures experience thermal strains under thermal loads due to mechanical-thermal mismatch of these materials; therefore, considerable thermal stresses occur in the adhesive joints even they are not subjected to any structural constraint. The thermal deformation-stress problem of adhesively bonded joints has been studied by a limited number of researchers [2-11]. In these studies, the thermal linear elastic stress analysis of a single lap (SLP)joint was carried out assuming that all members of the SLP joint had a uniform temperature variation or prescribing the temperature variations along the boundaries of the SLP joint. Apalak et al [12-13] presented some studies in which the geometrical non-linear thermal stress analyses of a single lap joint and a tubular single lap joint were carried out, and an optimum overlap length was determined. They assumed that the joints had an initial uniform temperature distribution through all members of the joints, and were subjected to air-flows parallel and perpendicular to plate outer surfaces in different velocity and temperature. They considered heat transfer through all joint members by conduction and convection. They found that serious heat fluxes, therefore, high temperature distributions occurred in small joint regions corresponding to the adhesive fillets at the adhesive free ends whereas other joint members ,

Corresponding author E-mail: [email protected]

712 experienced lower temperature distributions. Later, they conducted the thermal stress analyses of the joints for different mechanical boundary conditions considering large displacement and rotation effects and assuming that the adhesive and adherends had elastic properties, as a result they determined considerable thermal stress concentrations in the adhesive fillets around the adhesive free ends. In this study, an adhesive tee joint with double support bonded to an elastic horizontal plate was analyzed for the complicated thermal conditions resulting non-uniform temperature distributions throughout the adhesive joint as a result of conductive and convective heat transfer mechanisms. Serious thermal strains along the adhesive-plate interfaces were observed due to thermal-mechanical mismatches. J O I N T C O N F I G U R A T I O N AND FINITE ELEMENT MODEL The main dimensions of an adhesively bonded tee joint with double support are shown in Figure I. The tee joint consists of a vertical plate, a horizontal plate, supports and adhesive layers. The plates and supports are made of steel and an epoxy adhesive was used to join them.

R3 = 0.1 t5

i

........ \,\ I

! R2=2t o

T Rl=t

~=2a

• ft=2t$ •

\\ ""-..,../

L

'

t----z,

,

J

l

Figure 1. Dimensions of an adhesively bonded tee joint with double support A horizontal and vertical plate length L of 360 mm, plate and support thickness t of 5 mm, plate width w of 25 mm and an adhesive thickness ~5of 0.5 mm were kept constant through the analyses. The support length a of 30 mm was used for the main analysis. In practice, adhesive layer is pressed between plates in order to obtain a suitable bonding; therefore, some adhesive called adhesive fillet is squeezed out and accumulates around the adhesive free ends. Its effect on the adhesive stresses at the adhesive free ends is well known [1,2]. In the analysis, the presence of the adhesive fillet around the adhesive free ends was considered, and its shape was idealized to a triangle to simplify its mesh generation. Its height and length ft were taken as twice the adhesive thickness, i.e. 1 mm. The studies on the stress and deformation analyses of single, double lap joints showed that the adhesive stress concentrations occurred around the sharp adherend comers at the adhesive free ends [1,2]. In practice, the bonding surfaces of adherends are etched, and the adherend comers are rounded, consequently, the sharp adherend comers are unusual. The rounded comers result in lower and smoothly varying adhesive stresses. Therefore, the support comers corresponding to the free ends of adhesive-adherend interfaces were rounded with a radius of 0.1 mm as shown in Figure 1. The finite element method was used to analyze the heat transfer and thermal stresses occurring in the adhesive tee joint. Since the width of the adhesive tee joint is uniform the problem can be reduced to two-dimensional case. A four-noded couple field finite element capable of thermal and structural analyses was used to model the adhesive layers and other metal components. This type of finite element can include the effect of temperature increment on the thermal strains and stresses during an incremental finite dement analysis. Since the stress concentrations occur around the adhesive free ends and propagate along the adhesive layer, mesh areas around the adhesive free ends were refined in order to obtain a reasonable accuracy in the computations as shown in Figure 2. The finite element sottware ANSYS 5.3 was used for the thermal stress analysis [14]. T H E R M A L ANALYSIS The adhesive joints may experience thermal loads in practice. Since the joint members have different mechanical and thermal properties, such as thermal conductivity and thermal expansion, the thermal strains may result in serious thermal stresses. In order to determine the thermal stress distributions in the adhesive tee joint it is necessary to know

713 the temperature distribution with respect to the initial temperature distribution. The thermal stress and deformations in the single and double lap joints made by isotropic and composite materials were investigated assuming a uniform temperature throughout the adhesive joint or its boundaries [3-11]. However, a non-uniform temperature distribution in the adhesive joints arises due to adherend-fluid interaction and different material properties. The effect of nonuniform temperature on the thermal stress and deformation states of the single and tubular lap joints was shown by Apalak et al [12-13]. Therefore, the effect of heat transfer by conduction and convection on the temperature distribution in the adhesive tee joint should be determined. 1~10 V---Int/s T=I20C

, " " ~

.

.

.

.

2

I I ] ]

~

~ V--1ntis ~.V__V T=20C

18

v

I

V=133nu~ T=6flC

Figure 2. Mesh detailsand thermal boundary conditions of an adhesively bonded tee joint with double support

In this study, it was assumed that the adhesively bonded tee joint was subjected to variable thermal conditions along its outer boundaries as shown in Figure 2. The le~ surfaces of vertical plate, lel~ support and horizontal plate experience an air flow with a temperature of 120°C and a velocity of I m/s whereas the right surfaces of vertical plate, right support and horizontal plate experience an air flow with a temperature of 20°C and a velocity of I m/s. In addition, an air flow with a temperature of 60°C and a velocity of 0.5 rn/s along the lower surface of the horizontal plate. The initialtemperature distributionin the tee joint was uniform and taken as 20 °C. In the computation of the averaged heat transfer coefficientsthe air flow was considered as normal to the metal and adhesive surfaces pointed by surface numbers I-2, 2-3, 4-5, 5-6, 7-8, 8-9 and I0- I I, I I-12, 13-14, 14-15, 16-17, 17-18, and as horizontal along the remaining surfaces as shown in Figure 2. The flow direction of the air requires the use of different empirical formulas for the computation of the heat transfercoefficient[22]. Thus, in the case of the verticalair flow the heat transfer coefficientis given as •

De~

where Uoo is air velocity (m/s), D ~ is equivalent diameter (m), v is kinematic viscosity conductivity of the air (w/mOC). In the case of the horizontal air flow it is given as h , = Nu---~-2 0 " 8 3=6(R e y 2 L

P r ~ ) ~ = 0"836

(m2/s) and

(~3

k' is thermal (2)

where Nu, Re and Pr are Nusselt, Reynolds and Prandtl numbers, Cp is specific heat (kcal/kg°C) and Ix is dynamic viscosity (kg/ms). The thermal properties of the air were determined using the averaged value of the wall and air temperature as follows: Too+ To (3)

r:=

2

The thermal properties were given in Table 1. The thermal analysis of the adhesive tee joint was camed out for these thermal boundary conditions using the finite element method. The temperature distribution throughout the adhesive tee joint is shown in Figure 3. The adhesive layer exhibits gradually varying a temperature distribution whereas the temperature distributions in the vertical plate and both supports were uniform.

714 TABLE 1 THERMAL PROPERTIES OF AIR AND HEAT TRANSFER COEFFICIENTS ha (w/m2*C) ALONG SURFACES 20"C 40°C 700C 120°C Surfaces h~ Surfaces 1~ . ~i (w/m'C) 0.0257 0.0271 0.0292 0.0328 1-2, 2-3,4-5,5-6,7-8,8-9 34.4794 12-13 34.4794 k(kcal/m*C) 0.0221 0.0233 0.0251 0.0282 10-11,11-12,13-14 39.411 15-16 34.892 V (m2/s) * 10 6 15.11 16.97 19.92 25.23 14-15,16-17,17-18 39.411 9-21 12.762 Ix (kg/ms) * l0 s 1.82 1.91 2.05 2.27 3-4 72.6898 20-22 5.7573 ev (kcal/kg°C) 0.240 0.241 0.241 0.242 6-7 73.5585 18-19 12.914 A B c D I~ F G

=53. 687 =57.094 =60.5 =63.9116 =67.313 = 7 0 . 7 ig =74.12 6

Figure 3. Temperature distribution in the adhesively bonded tee joint with double support (in °C) The temperature values throughout the joint members vary from 53°C to 81 °C. This is a result of a non-uniform temperature distribution, and a maximum difference of 280C is observed in the temperature distribution. When the initial uniform temperature distribution of 20°C is considered the temperature difference reaches 6 IOc, and it is large so that it results in serious thermal strains in the joint members. In addition, since high temperatures occur around the adhesive free ends this effect would be more evident. Finally, the development and propagation of the thermal strains along the joint members may be expected to become more complicated. GEOMETRICALLY NON-LINEAR STRESS ANALYSIS In this section, the geometrically non-linear stress analysis of the adhesively bonded tee joint with double support was carried out taking into account the effects of large rotation and displacement. The stress analysis was carried with the thermal analysis simultaneously. Thus, the thermal strain increments due to the temperature increments were used to determine the corresponding stress and deformation states in the adhesive tee joint, and this process was repeated until the final thermal load was reached. The adhesive, supports and plates were assumed to have linear elastic properties: the adhesive material was the epoxy with E~3.33 GPa and Va =0.34, and the plate and supports were made by the steel with E=210 GPa and v=0.29. In addition, the adhesive tee joint was analyzed for two different plate end conditions as shown in Figure 4. First, the upper edge of the vertical plate and the fight edge of the horizontal plate were fixed, but the left edge of the horizontal plate was free in the horizontal direction. In the second case, the left and right edges of the horizontal plate

Figure 4. Boundary conditions and corresponding deformed tee joint geometries

715 were fixed but the upper edge of the vertical plate was flee in the horizontal direction. The deformed geometries of the adhesive tee joint are considerably different. In both cases the joint region experiences large rotation and displacements. In addition, the distributions of all stress and strain components were evaluated in the joint region, and it was observed that high stress concentrations occurred around the adhesive free ends and in the horizontal plate. The von Mises stress distributions are shown in Figure 5 for both boundary conditions. Ii i ~~

A B DC

=. 177~+08 =.433E+08 =.689E+08 =. 9~5E+08

A B C D

=. 24flE+08 =. 679~+08 =.111~+0~ =. 154E+09

i

=. I~-OE+Qg =. I~6E+OD =. 171R+Og

]~ F G

=. lg7]E+ID9 =. 2~0R+09 =. 2fl3~+09

I

=. 3 70E+09

a)

=. 223E+09

G ,~x~

~

,~

_..

)

.E_..

.

.~

.

.

F.G

.

~--~--'---~

.

Figure 5. Von Mises stress distributions in the adhesive bonded tee joint for a) BC-I and b) BC-II (Stresses in Pa). Since the adhesive free ends and the regions of the horizontal plates corresponding to the adhesive free ends were subjected to very high stresses, these regions were examined in detail. In the horizontal and vertical adhesive fillets, the peak stresses occurred at the free ends of the adhesive-adherend interfaces. The stress levels around rounded comers were moderate as shown in Figures 6 and 7 whereas the previous studies showed that the adhesive stresses had singularities at the sharp adherend comers [1-2]. The reason of these moderate stress levels can be that the rounded comers remove the geometrical discontinuity.

c)

~

A B C D Pr G H I

=. 1~3 ~.-I-08 =. 165~-+08 =. 18 5~-+oe =.208~.+08 =.229~-+08 =. 2 50~-+08 =.272E+08 =.293~+08 =. 315~+08

A

- . 115E+{]8

'~k

E F.

=.175~.+0fl :.lglE+Q8

/ e/r,I \ "b~l f i ~'--~Jl~,,

"

=.~'-~.

,//~'~

A B C D E ¥ G H I

=. 147E+OB =. 171E+08 =. 194E+0fl =. Z 17E+08 =. 240E+08 =. 2 63E+0B =. ~e 6E+0B =. 30gE+O8 =. 33ZE+08

A B

=" gg7E+07

~

D

--. 1501~'H38

=. ls',,,-+o, =.1.3~+o.

~

n

=. 7_171~+08

~I~

i

e F

d)

=. 116E+Ofl

-.133,-+o,

~Cc

(

~

- ~

~

~

~

~

~.L ~

"

.~ v,

,~

~

A' I

/~

.

B

--.oo4,+o-,

~

F

I~

,~ -.,.45~+oa

G

=. 193~+08 = . 2 19E'H38

I

=. 272P-+08

~I~ I~ ~

I

,~

I P, f ~'~,. r.

C

,,

Figure 6. Von Mises stress distributions in a) left vertical, b) right vertical, c) left horizontal, d) right horizontal adhesive fillets and e) bottom adhesive layer of the adhesive bonded tee joint for BC-I (Stresses in Pa). In the case of BC-I, all adhesive fillets are subjected to high stress levels, and the peak stresses occur at the free ends of the adhesive-adherend interfaces, however stress levels in the left vertical and horizontal adhesive fillets are higher by 50% as shown in Figure 6. Similar stress distributions in the adhesive fillets were observed for the BC-II as shown in Figure 7. The stress concentrations occurred around the rounded adherend comers in the left and fight horizontal adhesive fillets and were twice higher than those in the case of BC-I, whereas the stress levels m the vertical adhesive fillets and in the bottom adhesive layer changed slightly. This is a result of the high deformations in the horizontal plate. In the case of BC-I, the first crack can be expected to initiate at the free ends of the adhesiveadherend interfaces in the horizontal adhesive fillets and at the free end of the cap of the vertical adhesive fillets, and to propagate along the adhesive-adherend interfaces. However, in the case of BC-II the first crack may initiate at a small region close to the rounded adherend comer in the horizontal adhesive fillets, and may propagate towards the free surface of the adhesive fillet, or towards the opposite adhesive-adherend interface.

716 = . 113 P.+OB

=. =. =. =. =. =.

143E+08 165~.+08 18 6E+08 208.+08 23 0~-+08 2 51E+08 =. 2 73 E + Q 8 = . 295E+08 =. 316E+08

C~

N

]i~

i"'t~

=. ,6,-~+o8

FFP~:Lh,~

=.:t46',,.+o8

I / ~

: . zg~,~+oe

IJ P~ F.~',. P' ~ FL /' ~

,,

=.z,-vE-,-oB

I/ t"~--'~

c

=.,'78~+o8 =.Zl,,B+oa

I,/LF,,k~------~

s

=.~4~,,-+o8 r t~ ] b) =.zvvs..oe I =.3:tsE-.,.oe I =.3sss-,-o8 r -.:3geE-,..oe I

=. 322E+08 =. 3 66E+OB = . 411R+Ofl = . 455E+0B =. 500E+08 =. 5 4 5 E + 0 8

~i~i~ !

=.:t,-g~+o,

= 147B,,.+oe I

=..58 gE+OB =. 634E+OB = . 678 E+Dfl

(1)

~..,. ~ E \

I I I I

~I

ri.

I,I" ~1 "~

rl

: . 478,.+o7

=.7g,,,+o7

-.11o,,+0B

,:, =..,,,.,+,:,, "

I~ ' i~ ~ ~ "

=.:tvaE.oe =.zo.,,.~+o8 =.zz6,,_+oa =" ~'~''~+°e

e)

=•558E+08 ~.~---¢-~F =. 5gBE+O8 GD-I FF S

~

F ~',~c--~

Figure 7. Von Mises stress distributions in a) left vertical, b) right vertical, c) leR horizontal, d) right horizontal adhesive fillets and e) bottom adhesive layer of the adhesive bonded tee joint for BC-II (Stresses in Pa). The geometrical modification of the members of the adhesive tee joint, such as increasing the bonding area can have an effect of reducing the peak adhesive stresses. The previous studies related to single- and double-lap joints showed that increasing the overlap length reduced the peak adhesive stresses at the free ends of the adhesive-adherend interfaces, and an optimum overlap length was possible [ 1]. The advancements in the adhesive technology have made the use of rubber-toughened adhesives with high strength in the structural designs possible. Since these adhesives can withstand high plastic strains, these adhesives allow the metal adherends to yield. This means that the metal plates may also have critical regions causing joint failure. For this reason, the critical adherend regions of the adhesive tee joint were examined in detail, and it was found that the regions subjected to the high stress levels were along the outer fibers of the horizontal plate for both boundary conditions. In order to determine the effect of the support length on the peak adhesive and adherend stresses the adhesively bonded tee joint was analyzed for the support lengths a of 15, 20, 30, 40, 60 and 80 mm. The analyses were repeated for both boundary conditions as shown in Figure 4. The von Mises stresses were evaluated at the critical locations in the adhesive layers and in the plates. Whereas the critical adhesive locations were the free end of the horizontal plateleft horizontal adhesive fillet interface, the free end of the cap of the left vertical adhesive fillet and the bottom adhesive region corresponding to the base of the vertical plate, the critical adherend locations were the vertical plate region corresponding to the free end of the vertical plate-the left vertical adhesive fillet interface, a small region of the lower free surface of the horizontal plate near the left horizontal adhesive fillet, and the inner elbow of the left support. The variations of the normalized von Mises stresses in the critical adhesive and adherend locations were shown versus the support length/joint length ratio a/L in Figures 8 and 9, respectively. Increasing the support length resulted in an increase of 13% and a small decrease in the peak stresses in the left horizontal adhesive fillet for the BC-I and II, respectively. However, increases of 8-20% in the peak stresses in the bottom adhesive layer were observed, and increasing the support length had a small effect on reducing the peak adhesive stresses in the vertical adhesive fillet for both boundary conditions as shown in Figure 8. In addition, the peak stresses in the horizontal and vertical plates increased as the support length was increased. Thus, increasing the support length resulted in increases of 6-24% in the peak stresses in the vertical plate, and increases of 5-16% in the peak stresses in the horizontal plate for the boundary conditions I and II, respectively (Figure 9). However, decreases of 10% and 3% in the peak stresses in the support elbow were observed for the BC I and II. The support length has an effect of increasing the adhesive and adherend stresses in their critical locations for the present boundary conditions. The larger support length increases the stiffness of the joint region, and the larger bond area and support length result in larger thermal strains, therefore the thermal stresses become higher especially at the free ends of the adhesive-adherend interfaces along which the thermal and mechanical discontinuities exist. In spite of this, a support length/joint length ratio a/L of 0.083 seems to be optimum.

717 1.00

"~, g

0.95

.,9.0 06

........i...... i........~........~ ......:~/--i ....... i........i-.......i........

0.90

el

........ i . 0.85

o.8o

i

i

~

--dk,

.

i

BC-I

~-

, '

.

.

i

.

.

.

t

~

.

.

.

• O m x = 38.0 M P a

l 0.05

,

.

....................... ]~i))i:i:._.

BC-II •O m x = 78.1 MPa

0.00 1.00

.

~

• .L

i 0.10

.

i . .

.

! !

:!!i~P:!!:!:!::: :!:::!:! !:i::! !i!

,

i

.

i !

.

i !

i

0.20

0.15

0.25

.

i !

'j~

ii!ii

-~ o.~ ........~.......~" ~,-~--......r.......~.......!........~........~--iiQ~i~i;i -

i

i ~..i

i

o., ........i........i........

!

........i.....

-'dk- BC-I "O__

0.90

~

~o.

~

BC-II • Omx = 33.9 MPa l , ~ o

i i

0.05

....... ! .

.

.

.

.

.......!

.

: :l

: :i

: '

0.15

.

i

........

.

.

.

.

.

.

.

i

0.10

.

i

........

.... = 34.4 MPa

0.05

i

.

: i'

0.20

.

.

.

.

0.25

.

ef 0.90

~.~

0~

c) #

* o.ao

0.00

i*

, 0.05

i ~ Support

: ; 0.10

/~"--~ ,' ~

;

BC

I

Om~ x

29 l M P a

BC-II • O'mx= 37.7 MPa

, 0.15

~

i 0.20

,

0.25

length/pint length ratio, a//_

Figure 8. The effect of support length on the normalized von Mises stresses in the critical locations in a) the left horizontal, b) the left vertical adhesive fillets, and c) the bottom adhesive layer of the adhesively bonded tee joint. CONCLUSIONS In this study, the thermal stress analysis of an adhesive tee joint with double support bonded to a flexible horizontal plate was carried out using an incremental finite element method considering the small strain-large displacement theory. The thermal boundary conditions were prescribed so that the convective and conductive heat transfers were allowed throughout the joint. It was observed that the thermal strain distribution in the joint was not uniform for two different plate end conditions. Consequently, non-uniform thermal stresses occurred due to thermal and mechanical discontinuities along the adhesive-adherend interfaces. The considerable thermal stress concentrations occurred at the free ends of the adhesive-adherend interfaces in the adhesive fillets, and in the maximum bending regions of the vertical and horizontal plates. The analyses showed that thermal strain and stress distributions were more complex than those in the case the constant temperature distribution was prescribed inside the joint or along its boundaries. Finally, despite increasing the support length resulted in increaseg of 5-20% in the adhesive and plate stresses, the peak adhesive and plate stresses were reasonable for a support length/joint length ratio a/L of 0.083. REFERENCES 1. R.D. Adams and W.C. Wake, StructuralAdhesive Joints in Engineering, Elsevier Applied Science, London (1984). 2. Apalak, M.K. and Davies, R. (1994), Analysis and design of adhesively bonded comer joints: fillet effect, Int. J. Adhesion andAdhesives, 14:3, 163-173. 3. Ioka, S. Kubo, S. Ohji, K. and Kishimoto, J. (1996), Thermal Residual-Stresses in Bonded Dissimilar Materials and Their Singularities, JSME Int. J. Series A-Mechanics and Material Engineering, 39:2, 197-203. 4. Dechaumphai, P. (1996), Adaptive Finite-Element Technique for Thermal-Stress Analysis of Built-Up Structures, JSME Int. J. Series A-Mechanics and Material Engineering, 39:2, 223-230.

718

"~

0.95

~"

0.90

~

0.85

iiiiiiiii i i i i .

a)

0.80 0.00

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

~

,,

005

.

.

.

.

.

illI

.

BC-II-G,w = 503 MPa

o.,o

0i5

020

025

1.05

"•

g

0.93

i

0.85

!

i,

i

i

!

:

:

1 --~

0.78

....... i .

.

.

.

.

0.70

.

-'~ -~ ;

0.00

0.05

0.10

.

BC-I • ~mx- 218 MPa BC-II"a,,~ = 270 MPa i

i

0.15

i

i

0.20

0.25

i.w

i

0.93

. c) z

0.86 0.05

i

i 0.05

'

-4-

'

'

BCI :G BCII ~ ;

0.10 0.15 Support length/joint length ratio, a/L

;

=2 MPa 223MPa ' 0.~20

;

0.25

Figure 9. The effect of support length on the normalized von Mises stresses in the critical locations in a) the vertical plate, b) the horizontal plate, and e) the left support of the adhesively bonded tee joint with double support. 5. 6.

7. 8. 9. 10. 11. 12. 13. 14.

Kim, Y.G. Lee, S.J. Lee, D.G. and Jeong, K.S. (1997), Strength Analysis of Adhesively-Bonded Tubular Single Lap Steel-Steel Joints Under Axial Loads Considering Residual Thermal-Stresses, d. ofA dhesion, 60:1-4, 125-140. Nakano, Y. Katsuo, M. Kawawaki, M. and Sawa, T. (1998), 2-Dimensional Thermal-Stress Analysis in Adhesive Butt Joints Containing Hole Defects and Rigid Fillers in Adhesive Under Non-uniform Temperature-Field, d. of Adhesion, 65:1-4, 57-80. Humfeld, G.R. and Dillard, D.A. (1998), Residual-Stress Development in Adhesive Joints Subjected to Thermal Cycling, J. of Adhesion, 65: 1-4, 277-306. Unger, W.J. and Hansen, J.S. (1998), Method to Predict the Effect of Thermal Residual-Stresses on the Free-Edge Delamination Behaviour of Fiber-Reinforced Composite Laminates, J. of Composite Materials, 32:5, 431-459. Abedian, A. And Szyszkowski, W. (1999), Effects of Surface Geometry of Composites on Thermal-Stress DistributionA Numerical Study, Composite Science and Technology, 59:1, 41-54. Nakagawa, F. Sawa, T. Nakano, Y. and Katsuo, M. (1999), A 2-Dimensional Finite-Element Thermal-Stress Analysis of Adhesive Butt Joints Containing Some Hole Defects, J. ofAdhesion Sci. and Tech., 13:3, 309-323. Katsuo, M. Nakano, Y. and Sawa, T. (1999), 2-Dimensional Transient Thermal-Stress Analysis of Adhesive Butt Joints; J. of Adhesion, 70:1-2, 75-93. Apalak, M.K. and Gunes, R., "On Non-linear Thermal Stresses in an Adhesively Bonded Single Lap Joint" submitted to Computers & Structures. Apalak, M.K., Gunes, R. and Fidanci L., "Geometrically Non-linear Thermal Stress Analysis of an Adhesively Bonded Tubular Single Lap Joint", submittedto Finite Elements in Analysis and Design. ANSYS®, The general purpose finite element software (5.3), Swanson Analysis Systems, Inc., P.O. Box 65, Houston, Texas.

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

719

VALIDATION OF DISCONTINUOUS D E F O R M A T I O N ANALYSIS USING A PHYSICAL MODEL A. T. MCBride and F. Scheele Department of Civil Engineering, University of Cape Town, South Africa

ABSTRACT Discontinuous Deformation Analysis (DDA) is a relatively new numerical method developed for the analysis of media consisting of isolated blocks. This paper presents a DDA analysis of a physical laboratory model, selected to highlight the applicability of DDA in the field of rock slope engineering. The physical model, with exactly known boundary conditions, block geometry, material properties and initial conditions, provides the data to determine whether the DDA simulation is accurate in terms of the displacement history and failure pattern. A procedure to optimise the selection of a critical analysis control parameter is implemented. The ability of DDA to predict the physical model behaviour is discussed and recommendations made.

KEYWORDS Discontinuous Deformation Analysis, numerical modelling, discontinuous media, physical models

INTRODUCTION

The numerical modeling of discontinuous media, such as fractured rock, has numerous applications in the fields of rock engineering, tunnel excavations and foundation engineering. The stability of a rocky block system is influenced by two factors, namely the strength of the intact rock and the block displacements (Goodman, 1980). If the stresses in certain individual rock blocks exceed tolerable levels, causing the rock blocks to fail, possible rock system failure can be induced. Rock system failure can also be caused by blocks sliding along discontinuities and separation between these discontinuities. A rocky block mass is therefore a highly complex system, containing an infinite number of degrees of freedom. A numerical method must thus make certain simplifying assumptions concerning the behaviour and stability of a rocky block system. The nature of these assumptions affects the feasibility and applicability of the numerical model. Limit Equilibrium Methods were used to model the stability of simple rocky block systems prior to the advent of modem computers. A failure mode is assumed and the factor of safety calculated on the basis of the static equilibrium of forces.

720 Modem computers allowed more complex numerical models for the analysis of continuous media to be developed, in particular the Finite Element Method (FEM) and the Boundary Element Method. These methods are suitable for determining the stress state in individual rock blocks but are unable to accurately simulate the large displacements induced by blocks sliding along discontinuities. The Discrete Element Method (DEM) was developed by Cundall (1971) in the early 1970's. The DEM was revolutionary as it represented the first attempt to model the deformation of a discontinuous system purely in terms of individual block displacements. The DEM is currently the most widely used numerical method for analysing discontinuous media, in particular granular flow. The Discontinuous Deformation Analysis (DDA) method was developed by Shi (1988) and provides an alternative to the DEM. The primary simplifying assumption is that the relative displacement of blocks along discontinuities, as opposed to the failure of individual blocks, governs the stability of a discontinuous system. A certain degree of block deformation is allowed. The original DDA method has been modified and improved by several researches (Cheng, 1998; Amadei et al., 1996; Ke & Bray, 1995; etc.), however relatively few systematic studies have been undertaken to compare the analytical results of DDA with those of experiments. The objective of this paper is to validate the DDA method using a physical laboratory model.

BRIEF OVERVIEW OF DDA DDA Method

The DDA formulation is similar to the FEM formulation, the difference being that the elements are physical blocks isolated by discontinuities. Individual blocks displace and deform due to interaction with other blocks, due to loading and their respective boundary conditions. The block displacements are approximated using a first order linear displacement function. This implies that the stress state across individual blocks is constant. The displacement (u, v) of any point (x, y) in a block can be represented by the following six variables

(Uo,Vo,ro,Cx,cy,C~y)

(1)

where (u0, v0) are the rigid body translations of the block centroid, (r0) the rotation angle at the centroid and (ex, cy, Cxy) the normal and shear strains of the block. The displacement of any point within block i can be approximated as Uo

u},

1 0

v (~,y)

0

1

-(Y-Yo) (X-Xo) (X-Xo)

0

0 (Y-Yo)

1

'

-2(Y_( x6y°I

r0 cx (x.yl cy

1

2'x

vo (2)

If the block system is composed of n blocks then the global system of simultaneous equations can be expressed as

721

K,,

1(12 KI3 ...

K,~

d~

K21

K22

/,:,l .

.

.

K23

..-

K2n

d2

/,:,,

---

K3o

d3

F~ F2 =

r3

(3)

.

k.~,~ K~

K,~ .-- K.o

Each individual block has six degrees of freedom and I Kv ] in Eqn. 3 is therefore a 6 x 6 submatrix. {d, } and {F, } are 6 x 1 submatrices, where {dr} represents the deformation variables for block i and {F,} the loading on block/distributed to each of the six deformation variables. Submatrix [K,,] is dependent of the material properties of block i, whilst submatrix IKv ], where i , j , is dependent on the contacts between blocks i andj. The system of simultaneous equations represented by Eqn. 3 is derived by minimising the total potential energy of the block system and solved to determine the displacements. Block contacts are enforced using the penalty function method whereby numerical springs are applied at contacts to "push" contacting blocks apart. Frictional contact and sliding is modelled using the Mohr-Coulomb law. An iterative procedure is performed during each time-step to prevent block penetration at contacts. This procedure ensures that the block system is in equilibrium at the end of each time-step. The large displacement and deformations that occur in rocky block systems are modelled as the accumulation of small displacements within individual time-steps. The assumption of infinitesimal displacement theory in the derivation of DDA requires that the time-step size be suitably small. A unique implicit time integration scheme is used to guarantee unconditional stability of the solution. DDA models dynamic, quasi-dynamic and static problems using the same formulation. In a dynamic formulation the block deformation velocities are inherited from the previous time-step whilst in a static analysis the block deformation velocities are set to zero at the beginning of each time-step. Quasidynamic formulations may be used to simulate different degrees of kinetic damping to model the effect of energy dissipation in a block system. DDA Version 96

DDA Version 96 (Shi, 1996) is the computer software package used to implement the two-dimensional DDA method. The user must specify four analysis control parameters in order to perform a DDA analysis. These parameters influence the accuracy of the results and the feasibility of the analysis. The selection of the analysis control parameters is problematic. The two parameters that have the most significant influence on the analysis of the selected experimental laboratory model are briefly introduced.

Kinetic damping control parameter (k01) The block deformation velocities at the end of a time-step are multiplied by kO1 and become the initial block deformation velocities for the subsequent time-step. Varying degrees of kinetic damping can therefore be simulated by selecting k01 between 0 and 1. The dissipation of energy in a natural system due to mechanisms such as internal friction in materials, surface friction at contacts, aerodynamic and thermodynamic effects, etc. can thus be simulated.

722

Penaltyfunction (gO) The penalty function is the stiffness of the numerical spring used to enforce contact constraints between blocks. If the value of gO is too large, non-existent energy is added to the block system at contacts. If the value of gO is insufficient to "push" penetrating blocks apart within a time-step, the analysis will terminate. The lowest value of gO that allows the analysis to proceed generally produces the most realistic results. The value of gO can be automatically controlled by the programme or specified by the user. Experience shows that allowing the programme to automatically control gO can result in unrealistic block behaviour due to the initial value of gO being too high for certain problems. Improved results may be obtained if the value of gO is specified. Monitoring the automatically controlled value of gO provides an indication as to the optimum value to be specified in subsequent analyses.

JOINTED R O C K SLOPE MODEL

Physical Model The jointed rock slope model, based on a base friction model investigated by Cundall et al. (1977), is shown in Figure 1. The model contains a continuous joint set dipping at 65 ° into an 80 ° cut slope. A second discontinuous joint set dips out of the slope face at 20 °. The joint sets create a system of 50 blocks consisting of square (20x20mm), rectangular (10×20mm) and triangular blocks with a breadth and height of 20mm. The dimension of each block in the z-direction is 80mm in order to minimise out of plane displacement. The toe block is welded to the base to prevent it from displacing. The face of the block system is initially restrained by hand to prevent the model from displacing under gravity. Once released, the block system fails. The displacement of the blocks was captured using a digital video camera.

Block A

Y

~Toe

Figure 1: Jointed rock slope model The laboratory model was made of structural steel. Steel was chosen due to ease of construction, the availability of preformed shapes, weight, robustness and low contact shear strength. The preformed steel sections were cut to size to form the blocks. The combination of weight and low friction value facilitates block sliding and toppling without the application of loading. The frictional properties of the

723 steel blocks, measured in terms of Coulomb's shear parameters, were determined experimentally. The material properties of the jointed rock slope model are listed in Table 1. TABLE 1 MATERIAL PROPERTIES OF THE JOINTED ROCK SLOPE MODEL

Joint friction angle (~b)

degrees

11.4

Density (p)

kg/m 3

7580

GPa

200

. . . .

Young's modulus (E) Poisson's ratio (v)

0.3

.....

The digitised images were analysed using a computer programme written in Matlab (The Mathworks, Inc., 1999) and the displacement of the centroid of block A, as shown in Figure 1, recorded. The quality of the images decreases as the block velocities increase due to blurring. The frame with the image of the block system prior to any displacement was chosen as the reference frame.

Numerical Modelling The DDA analysis is composed of two parts. The first, a static analysis, is used to determine the initial stress state in the physically restrained block system. The second, the dynamic analysis, computes the block displacements once the restraint is removed. The stress state from the static analysis is incorporated into the dynamic analysis as the initial conditions. In the static analysis, a lateral restraint is placed on the slope face to prevent the block system from displacing. The value of gO is automatically controlled and the stresses in each block recorded. Kinetic damping (k01=0.5) is employed to ensure that static equilibrium is obtained. The stress state can be determined by monitoring either the stresses in selected blocks or the variation in the value of gO. If the value of gO stabilises it is an indication that static equilibrium has been achieved. The selection of the analysis control parameters for the dynamic analysis is more complex due to the large displacements, increased block velocities and continuously changing contact conditions. The dynamic analysis is initially run with the value of kO1 equal to unity and the value of gO automatically controlled. The variation in the value of gO over the time period of interest is shown in Figure 2. Initially the blocks are "pushed" apart due to the value of gO being too high (by definition 40xE). The value of gO decreases rapidly and remains under 1x 108N/m until approximately 0.2s have lapsed. The block behaviour during this period is in close approximation with the physical model. After 0.2s the value of gO begins to increase as the velocity of the displacing blocks increases and a greater spring force is required to prevent block penetration. The increased resolution of gO with time is due to the automatic reduction in the time-step size. New contacts detected after 0.25s generally occur between blocks in free fall outside of the analysis domain and no longer of relevance to this study. One of the weaknesses of the automatically controlled penalty function routine and the penalty function method becomes apparent. The same value for gO is applied to all contacts throughout the analysis. A high-energy collision between blocks can cause the value of gO to increase. This may detrimentally influence contacts where the contact energy is low. Three methods have been proposed in an attempt to overcome this weakness. The incorporation of contact damping based on the relative velocity of the impacting blocks (Cheng, 1998; Ke & Bray, 1995) allows the energy arising from block contacts to be dissipated without affecting the rigid body motion of the blocks. Permitting block penetration to occur (Cheng, 1998) prevents the analysis from terminating due to an inability to satisfy the contact constraint of "no penetration". This modification allows lower values of gOto be specified.

724 Contact damping and the removal of the "no penetration" criterion were successfully implemented in a DDA code for rigid disc elements (MCBride, 2000). Applying a unique value for gO at each contact ensures that all contacts are modelled individually. The Augmented Lagrangian Method (ALM) proposed by Amadei et al. (1996) allows the exact value of the contact force between blocks, and therefore a suitable gO value, to be determined. A simple block contact was modelled applying this routine using a DDA code for polygonal blocks with good results (MCBride, 2000).

2.5 x 108

z ._~1.5

g_

g no. 5

.

.

.

.

0.05

0.1

0.15 0.2 0.25 Time (s) Figure 2: Variation in the value of gO with time Specifying a value for gO less than approximately 1x 108N/m causes the analysis of the jointed rock slope model to be prematurely terminated. The block system behaviour appears realistic, when compared to the physical model, for gO equal to 1× 108N/m. For values of gO greater than 1× 109N/m, all blocks are "pushed" apart, similar to the block system being subjected to blasting. Results

The experimental and DDA block system configuration at selected times is shown in Figure 3. A direct comparison between the DDA graphics and recorded images is not intended, as the selected times at which the block system configuration was recorded do not necessarily correspond. Rather, the general behaviour of the block system should be compared. The horizontal and vertical components of the displacement of the centroid of block A obtained from the DDA analysis, using a value of gO equal to 1× 108N/m, and the physical experiment are shown in Figure 4. The inclusion of the initial stress state from the static analysis in the dynamic analysis improved the results. DDA was capable of determining the displacement of the centroid of block A reasonably accurately. Reasons for the discrepancies can be attributed to two factors. Firstly, the manner in which the blocks were laterally restrained prior to the release was not consistent. A mechanical support and release mechanism will eliminate any irregularities. Secondly, the quality of the images obtained from the digital video camera decrease as the block velocities increase. This makes the identification of the centroid of the measured block relatively subjective. The failure pattem of the jointed rock slope model and the DDA simulation compare very well. A typical rock system failure becomes evident; the blocks slide along a discontinuity plane. As the failure

725 progresses, the sliding mode transforms into a mixed sliding and toppling mechanism. Eventually the blocks above the failure plane end up in free fall beyond the extent of the analysis domain.

Figure 3" Comparison of the experimental jointed rock slope model and the DDA simulation

Horizontal Displacement o.o7~___i "-" " ! E0.05

I........ '-+i -'4"{_-'+--

D6A- ...... ~ ....... Test Series l i Test Series2 I TestSedes3~ .......

Vertical Displacement

i .......... i ---//-;i / / / 1 : / : ~ / ::! i / , f , /

-0.01

~

~

.

.

,

i

:

:

!

:

',

-0.06 --i --¢-- T ~ ~ e s 2 uu/- i-+- Testseriesl _

0.05

0.1

0.15 0.2 Time (s)

0.25

0.3

\\\

~_o.o, i illllill illll _iill ill ill

'=

0

I ....

!. . . .

!

-0.02 t .......... !........................ ! ~ - - ~ i - " \ - - , i - .

_

_

....

l ~

0.1

!~% _~

:

i---

i~

:

\

~\\ \ii

i ili ili

-0.15 0.2 Time (s)

ii

~ k -xx ~

i ........... i ........... i - - ~

_o.o,! ......... .......... ........ i -0.09 0.05

i

\, i

-~

iii

- -,. 0.25

0.3

Figure 4: Predicted and observed displacements of the centroid of block A

CONCLUSIONS Discontinuous Deformation Analysis is a highly successful numerical tool for simulating the behaviour of the experimental jointed rock slope model once the material properties are determined and the analysis control parameters optimised. A certain degree of unavoidable variability is present in the dimensions of the blocks and the measurement of the material properties. This variability does not

726 exist in the numerical model and minor discrepancies in the respective results can therefore be expected. The enforcement of the contact constraints using the penalty function method is computationally efficient and simple to implement. If automatically controlled, the variation in the value of gO indicates the equilibrium state in the block system. However, automatic control of the penalty function may introduce errors into the results and it is advised that a suitable value for gO be specified. Unfortunately, the selection of an appropriate value of gO is a complex exercise. In this study a trial and error procedure was used to select an optimum value for gO. The determination of a unique value for gO in more complex analyses is often not possible. Contact damping, the allowing of block penetration and the automatic determination of a unique value for gO at contacts are features that should be incorporated into future DDA codes.

REFERENCES

Amadei B., Lin C. and Dwyer J. (1996). Recent Extensions to the DDA Method, Proc. 1st Int. Forum on Discontinuous Deformation Analysis and the Simulations of Discontinuous Media, M.R. Salami and D. Banks, eds., TSI Press, Albuquerque, 1-30. Cheng Y. (1998). Advancements and Improvements in Discontinuous Deformation Analysis. Computers and Geotechnics 22:2, 153-163. Cundall P.A. (1971). A Computer Model for Simulating Progressive, Large Scale Movements in Blocky Rock Masses, Proc. Syrup. of Int. Soc. for Rock Mech., Nancy, France, Paper No. II-8. Cundall P.A., Fairhurst C. and Voegele M. (1977). Computerized Design of Rock Slopes Using Interactive Graphics for the Input and Output of Geometrical Data, Proc. 16th US. Syrup. on Rock Mech., ASCE, New York. Goodman R.E. (1980). Introduction to Rock Mechanics, John Wiley and Sons, USA. Ke T. and Bray J.W. (1995). Modeling of Particulate Media Using Discontinuous Deformation Analysis. d. Eng. Mech. 121:11, 1234-1243. The Mathworks, Inc. (1999). Matlab Version 5.3. MCBride A.T. (2000). An Investigation of Discontinuous Deformation Analysis, M.Sc. Thesis (in prep.), Department of Civil Engineering, University of Cape Town. Shi G.-H. (1988). Discontinuous Deformation Analysis, Ph.D. Thesis, University of Califomia at Berkeley. Shi G.-H. (1996). Discontinuous Deformation Analysis Programs Version 96. © Gen-Hua Shi, 5650 Pointsett Ave., E1 Cerrito, CA.

CRACKING AND FRACTURE MECHANICS

This Page Intentionally Left Blank

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

729

THE TIME SCALE IN QUASI-STATIC FRACTURE OF CEMENTITIOUS MATERIALS

G.EA.G. van Zijl Faculty of Architecture Delft University of Technology, Delft, The Netherlands e-mail: g.p.a.g.vanzijl @bk.tudelft.nl

ABSTRACT During the last decades, intense research efforts have been put in the formulation of theories for crack initiation and propagation in cementitious materials. However, little attention has been paid to the time dependence of fracture, despite the evidence of its significance not only in dynamic cases, but also in quasi-static cases. Structural collapse under a creep load is an important example. A visco-elastic-plastic model, which combines continuum plasticity and visco-elasticity, was recently developed. Computational continuum plasticity provides a convenient computational framework for modelling cracking in cementitious materials. The visco-elasticity, which is modelled with an aging Maxwell chain, accounts for the time-dependent bulk response. However, it is postulated that different rate dependence acts in the highly localised fracture process zone. Therefore, a rate-dependent contribution to the cracking resistance is included. In the past such a rate dependence has been added artificially (in quasi-static problems) to regularise the continuum description of localisation. In this paper evidence is given that it is not merely a numerical artifact, but introduces the correct time scale of fracture. Without the crack rate dependence, the time to failure is overestimated by several orders of magnitude. Creep failure experiments have been analysed with the model and good agreement has been obtained with the measured time to failure. KEYWORDS: concrete, masonry, cracking, rate dependence, time scale, finite elements INTRODUCTION The behaviour of cementitious material is highly time- and rate-dependent. It is well-known that for dynamic cases a significant increase in structural resistance occurs upon increase in loading rate. However, during the last decade experimental evidence of rate dependence in the quasi-static loading rate range, where inertia and wave effects are negligible, has been produced by for example Ba2ant and

730 Gettu (1992), Zhou (1992), Ba2ant and Xiang (1997). Much earlier already Riisch (1960) reported an approximate 20% difference in concrete strength between specimens subjected to "normal" testing rates and those tested at (infinitely) low rates. Examples of time-dependent quasi-static behaviour of concrete are illustrated in Figure 1. This time dependence in the quasi-static range may be detrimental or beneficial to structural integrity and serviceability (van Zijl 2000a). Time-dependent crack propagation under sustained load may eventually lead to structural failure, while, on the other hand, cracking may be postponed or avoided completely by the relaxation of peak stresses. To enable computational durability assessment, it is imperative to consider this time and rate dependence and ensure that the time scale is captured accurately by computational models. The key to the understanding of the time dependence lies in knowledge of the micro-structural processes. For example, it is generally agreed that creep is caused by the de-bonding and re-bonding of micro particles, in which process the moisture flux between micro- and macro-pores plays an important role (Powers 1960, Ba~ant and Chern 1985). Furthermore, the atomic processes are thermally activated. This indicates an intricate hygro-thermal-mechanical interdependence, which has inspired fully coupled approaches to model concrete behaviour by Biot (1955), Coussy (1995), Lewis and Schrefler (1998), Carmeliet (1998), Grasberger and Meschke (2000). For computational viability a macroscopic approach must be taken, despite the complication caused bythe averaging/homogenisation of the micro-structural processes over the random and continuously changing micro-structure. Even if a macroscopic approach is taken, coupled modelling is computationally costly. This fact is worsened by the dense finite element meshes required in localisation zones (Askes et al. 1998, de Borst et al. 2000). A promising solution to the latter lies in embedded displacement discontinuities, based on partitions of unity (Wells and Sluys 2000). The extension to embedded pore pressure and thermal discontinuities is currently investigated (van Zijl and Wells 2001). To avoid the computational demand of a fully coupled approach, a model which considers a uni-lateral coupling, has been formulated and is presented here. Hygral and thermal effects are included in the formulation for the mechanical response, but the diffusion processes underlying the mechanisms of hygral and thermal shrinkage, are not influenced by the mechanical response. Apart from the time-dependent shrinkage, it is necessary to consider two other sources of time dependence, namely bulk creep and the fracture process zone (FPZ) rate dependence. Bulk creep is modelled with linear visco-elasticity, justified by the linearity of concrete tensile creep isochrones right up to the peak. In the FPZ a cracking viscosity is employed, justifiable by the significant difference in the crack propagation speed in wet and dry concrete specimens (Tait and Garret 1986). This cracking viscosity contribution has been shown to regularize the continuum description of cracking (van Zijl 2000b). An alternative formulation for the FPZ rate dependence was derived by Wu and Ba~ant (1993) from the rate process theory. Both models are briefly outlined in the computational plasticity framework, which was chosen for computational convenience to model cracking in cementitious materials. The plasticity and visco-elasticity is integrated in a novel, unified setting. To illustrate that this phenomenological model can capture the time scale in quasi-static crack propagation accurately, three-point bending experiments under creep loads, which led to failure of notched concrete beams (Zhou 1992), are analysed. A MODEL FOR TIME-DEPENDENT CRACKING

It is necessary to distinguish the rate process in the FPZ from that in the bulk. This requirement stems from the different mechanism of moisture migration in the FPZ, which increases the drying rate of unsealed specimens (Wang et al. 1997) and causes internal redistribution of moisture in sealed specimens. This motivated the formulation of a crack model, which incorporates bulk creep via visco-elasticity, as well as a cracking viscosity. The cracking stress is given by

731

IIIII

III ll I I [llll II till

]llllllltl

1.4

1.2

IIIII11~ llll Illl

1.0

,if-: ' "

',o'-' '

.... i'o°

...... i"o'

D e f l e c t i o n rate

(l.tm/s)

i) ( ) A ( )

" ....,"o'

~IIIIIIIIIIIIIIIIIIIIII

0.9 Fp 0.9 Fp 0.9 Fp 0.9 Fp to tl t2 tf (b) Crack propagation in concrete specimen under sustained tensile load (van Zijl 2000b). 0.4 Fp

(a) Concrete rate-dependent strength increase in quasi-static loading range (Zhou 1992).

__

0.3 ¢-~ 0.2

]Ill II II111111 Ill I II11LILI~I

()lllllllllll

Time for drop in temperature

(AT=-32"C): .. ~ " ~

0.1

AT =-21 °C ..

,

\~

i

-lO

-20 -30 ('C) a~ :-3z'c (c) Cracking in a masonry wall due to shrinkage restraint by concrete floors. Time-dependent stress-relaxation may postpone, or prevent cracks (van Zijl and Verhoef 2000). AT

Figure 1" Time-dependent cracking in cementitious materials illustrated.

~-~t(Wc)

m

l + f,

,

(1)

where ot(Wc) describes the rate-independent degradation of cracking stress from the virgin strength 3~ governed by the crack width Wc, which is smeared over the crack band width gb- m is the cracking viscosity. Wu and Ba~ant (1993) derived an alternative formulation for the crack velocity from the rate process theory, which reads

o-,~,(Wc) - Wcr sinh ko [ot(Wc) + kl ft]

(2)

for isothermal conditions. Wcr is a constant, low reference crack velocity. The other two model parameters k0, kl have no physical meaning and must be found by inverse analyses. Eq. (2) can be rewritten in

732 terms of the cracking stress

c~ ~t(Wc) l + k 0 s i n h - l ( ~ i '~c ) ] -

Wcr

+ k0 kl j~ sinh -1 (~rcr)" ~i'c

(3)

This three-parameter model has a similar form as the one-parameter model Eq. (1), except for the residual term. It has been argued that the relative compressibility of moisture in comparison with the solid phase in cementitious materials explains the insensitivity of the drying rate to mechanical pressure in the absence of cracks, ruling out the possibility to describe bulk creep by considering the pore pressure (Ba2ant and Chern 1985). This implies that special coupling terms should be introduced also in fully coupled models to correctly introduce the time scale. Grasberger and Meschke (2000) introduce a permeability coefficient, which is dependent on the inelastic pore volume change. In addition, they consider the permeability to be (local) damage-dependent. Although the latter dependence was intended to capture the observed increased permeability caused by cracking (Wang et al.), this effectively introduces a FPZ rate dependence. Whether their formulation successfully captures the time scale in fracture remains to be investigated. The two crack rate models (Eqs. 1,3) have been incorporated in a computational framework, which combines linear visco-elasticity and continuum plasticity. The constitutive law can be expressed in rate form as (~= Dve (E--Ecr--EO)-]-~. (4) where Dve is an equivalent, time-dependent stiffness modulus, Z is a viscous stress term which accounts for the history, (~cr is the cracking strain rate and ~0 represents the dimensional change due to hygral and thermal shrinkage. The latter contribution may be stress dependent. To derive Eq. (4) it has been assumed that the strain rate can be decomposed as follows -~- ~,ve -at- Ecr -~-~,0

(5)

where Eve is the visco-elastic strain. The rate Eq. (4) can be integrated with a linear scheme, which produces the stress increment in the time increment At Aa -- D ve ( z ~ - l ~ c r - 1~0) -1--t~,

(6)

where Dve

:

Eo(t*) + ~_~ 1 - e--~nn En(t*) D (7)

te

=

-~..

l-e

~n

tan.

n=l In this model an aging Maxwell chain can be identified, with time dependent element stiffnesses En(t) and viscosities qn - En~n, ~n being the relaxation time of chain element n. The stress vector/an contains the stress components in chain element n at the end of the previous time step, i.e. at time t. Note that the parameters are assumed to be constant in each time interval and are evaluated at a time t < t* < t + At. D is a dimensionless matrix which is dependent on Poisson's ratio v (van Zijl 2000a). The crack strain increment follows from the plastic flow l~cr- A1£~,

(8)

733 with g -- g (a, 1¢) the plastic potential function and A~¢: Aweigh, the crack strain increment. The stress is limited by the yield function f (if, ~¢) _< 0. (9) An anisotropic Rankine yield function is appropriate for the cohesive material of interest. It can be formulated in plane stress as (Lourenqo 1996)

f = (~x-¢Ytx)+ ((Yy--(Yt7)

_+_

2

2

"

-'1-- a'l;2y

. '

(10)

where ot controls the shear stress contribution to failure. The potential function has been chosen as g--fa=l.

(11)

The cracking stresses in the orthogonal directions are given by -

J•x

J~y ---K

K

C~tx -- fix e gfx

,

(Yty

--

(12)

fty e gfY .

The tensile strengths are denoted by fix and fry, while gfx and gfy are the fracture energies in the orthogonal directions. To add the FPZ rate dependence, the cracking strength function can be supplemented to read jS___Lx¢ _jS>,

(Ytx-(ftxq-m~)e

gfx

, (Yty-(fty+m~)e

gfY

(13)

for the simple cracking viscosity Eq. ( 1) and for the three-parameter model of Eq. (3)

ftx CYtx -- ftx e gfx

1 + ko sinh-I

K:

+ k o k l ftx sinh-l (f~r ) (14)

fty ~ty -- fly e gfY

1 + ko sinh-l

~

+koklftysinh-l(~rr

) •

VERIFICATION: CREEP FAILURE

To verify the model and illustrate the introduction of the correct time scale, the three-point bending creep tests performed by Zhou (1992) have been analysed. The geometry of the 100 mm thick notched concrete beams is shown in Figure 2. The specimens were sealed to avoid the complication of simultaneous shrinkage. Of course, despite such measures the moisture flux can still play an important role in determining the crack velocity through internal redistribution of moisture. The experimental program comprised displacement-controlled tests for the total load-deformation response at 5 pm/s, as well as creep tests at 92%, 85%, 80% and 76% of the peak loads. Separate tests were performed, enabling the characterisation of the model parameters, yielding a Young's modulus 36 k N / m m 2 and the tensile strength fi=2.8 N / m m 2. Relaxation tests were performed on cylindrical, notched tensile specimens, see Figure 3a for the fitted 10-element Maxwell chain model response. Zhou also performed three-point bending tests under displacement control on smaller beams (600 mm long by 50x50 mm 2 section) to determine the fracture energy and rate-dependent strength. By varying the deflection rate from slow (0.05 lam/s - peak load after about 80 minutes) to fast (50 lam/s - peak load after about 5 s) he studied the rate influence on the fracture energy, Figure 3b, and peak strength, Figure 3c.

734

~ FI2 F

symmetric

~

ltl] ] i: ] i

_

~'4 mmwidenotch l

.

E

z(~e400mm

Figure 2: Geometry and FE model schematisation of three-point bending tests on notched concrete beams (Zhou 1992). 40 ~'~ ~.~

~

.....................

, .......

, .........

90 ~- • Measured (Zhou 1 9 9 2 ~ .-.. 70

• Measured (Zhou 1992) ~ x w e l l model fit

~ 50

1.4

~30!

30

g

.~- ,.~-

=

20

/ 7

/

y

1.0

i 0

........................... ' • "A ' i 1.6 i H Measured (Zhou 1992) / o • o - - ~ Numerical: 3 param, modcl / ~ a - - ~ Numerical: I param, modcl//

1000

2000 Time (s)

3000

4000

IO~ 7 10.6 10-5 10.4 10.3 10.2 101 l0 ° 10' 10" Deflection rate (BmA')

102

10 I 10° 101 Deflection rate (lamA)

I 0"

(a) (b) (c) Figure 3" Parameter characterisation made possible by separate tests by Zhou (1992). (a) Relaxation test result with fitted Maxwell chain. Rate effect on (b) fracture energy and (c) peak strength. The finite element mesh employed for the analyses is also shown in Figure 2. It consists of plane-stress, four-noded quadrilateral elements. Symmetry is exploited, enabling one half of the model only to be modelled. Inverse analyses were performed to determine the crack rate model parameters for Eqs. (1,3). For these analyses the smaller beam was modelled with the same mesh shown in Figure 2, scaled to the small beam geometry. The parameters obtained in this way are, for Eq. (1): m = 1500 Ns/mm 2 and for Eq. (3): k0 = 0.05,kl = 0.1,Kr-- 10 -7 s -1. Figure 3c compares the normalised numerical peak strengths with the measured values. Reasonable agreement is found with the three-parameter model, but with the simple one-parameter model it is impossible to fit the strength increase over the entire range of loading rates. A possible remedy is to employ a rate-dependent viscosity m (~:). This has not been attempted. Instead, the three-parameter model has been employed for the subsequent analyses. With regard to the apparent increase in fracture energy with loading rate, it must be noted that this follows from the numerical model although constant fracture energy is prescribed. This value can be estimated by extrapolation to the deflection rate at the reference strain rate/Or, Figure 3a. A constant value of Gf=0.035 N/mm has been used. A crack band width gb = 4 mm has been assumed, which is equal to the notch width. The own weight is compensated for by applying a volume load of mass density 2400 kg/m 3 in an initial step in each analysis. In Figure 4 the experimental and numerical results are compared. Both the displacement-controlled and creep responses are in good agreement. The experimental observation that the displacement-controlled response forms an envelope for failure under a sustained load, is confirmed by the computations. Note that in the creep analyses the force control was continued up to a point where equilibrium could no longer be achieved, indicating failure. To ensure that failure under the sustained load was indeed imminent, the analyses were continued with displacement control at this point, resulting in the shown softening responses. It is imperative that the time scale involved is captured accurately by the model. Figure 5a shows that it is indeed the case. Also shown are the times to failure if the rate dependence in the FPZ is ignored. Then, the time to failure is over-estimated by several orders of magnitude. This is the case even when an extreme bulk relaxation is assumed (creep coefficient 5 after 100 days). Furthermore, Figure 5b shows

735 1000

i000

.p•CMOD, z- 0 Figure 2: Coordinates with its origin at the crack tip For a semi-infinite crack with traction-free faces terminating at the interface and oriented perpendicularly to the interface the stress distribution in the vicinity of the crack tip is given as, e.g. Lin & Mar (1976)

=

(5)

where F,j is a known function of bi-material composite parameters and polar angle and the coefficient H z is a generalized stress intensity factor. In the case of a crack in a homogeneous body H I = KI. The value of the stress intensity exponent 3- is given by the solution of the equation sin (1 - A ~ { (1- 3-)2(- 4a 2 + 4aft)+ 2a 2 -2ctfl + 2ct-fl +1 + ( - 2 a 2 + 2aft - 2ct + 2fl)cos(1- 3-)n:}= 0

(6) The dependence 3- = 3-(E~/E 2) is given in Fig. 3. 0.60

0.55

j

•-~ 0.50

0.45

/

0.40 0.50

J

J 0.75

~

f

J

1.00

1.25

1.50

1.75

2.00

EdE2 Figure 3 The dependence of the stress singularity exponent 3- on the ratio E l / E 2

741 As an example, the expression for the stress component or**in the direction of crack propagation (i.e., for (p = 0 ) is given:

a~,~,(r, (p : O): o'~ (x, y = O)= H2-~(1- 2X2 - 2 + g , ) r -~

(7)

where

g r : O-- 2)--Cos(l-- 2)n:-- ilia + 2(1-- 2)-- 6 + 2a - 4a(1- 2):)cos(1 - 2 ~ ] / d e t ( 2 ) and det(2) = 1+ 2a + 2 a : - 2 a ( 1 + a)cos(1- 2 ~ - 4a2 (1- 2) 2 .

(8)

The value of the generalized stress intensity factor H~ depends on the values of the bi-material composite parameters a and r , applied loading, geometry of the body and boundary conditions and must be estimated numerically. The computational model described in the previous section and the finite element method have been used for the calculation of HI.

FORMULATION OF THE STABILITY CONDITION FOR A CRACK TERMINATING AT THE INTERFACE The fact that the stress singularity for a crack with its tip at the interface between two different materials is no longer r -~/2 means that the linear elastic fracture mechanics arguments and criterion Kzc cannot be applied. The stability condition for a crack with the exponent of singularity different from 1/2 is related to the average stress cYcalculated across a distance d ahead from the crack tip, Knrsl, Knhpek & Bedn~ff (1998). For a normal mode of loading the a~,component of the stress for cp=0has to be considered and

1 de cr = -- } (cr~, (r, cp = O)dr d

(9)

For a crack in a homogeneous material a~,~,(r, (# = O)= K,/ff2--~r and

1a or= d-! K, / ff-2-~rdr = 2K1/ ff-2-~d

_

(lo)

This value is related to the critical fracture stress o-c,,t which is a material constant and can be expressed by means of the fracture toughness K~c

1a acnt = ~- [(a~,(r, (p = 0)dr = 2 K i c / . ~ a . "o _

(11)

The value ~c,~t is related to the average stress -~a.d(d,a, r ) calculated across a distance d in the aggregate.

742 If d = D is a dimension of the particle, then the average stress per the particle -~,,.o (D, a, fl) caused by the crack with its tip at the interface is

O'a,D(O,a,fl)= --~1 °[(o'~, (r, ep = O)dr = H , D - * ( 2 _

- ~, + g , )

(12)

"o

u

The average stress per particle, O-o.o, depends on the dimension of the aggregate D and on the bimaterial composite parameters a, fl, i.e. on the elastic mismatch of the matrix and the aggregate. n

The particle ahead of the crack tip fractures when the value of the mean stress cra.o exceeds its critical value cry, Cra,o > o'~n,

(13)

H, > H m

(14)

and the fracture condition has the form

where H~c is the critical value of the generalized stress intensity factor (the generalized fracture toughness) expressed as

Hm =Km

2D~-1/2 (2-~.+g,)

(15)

The value of the critical stress for intersection of the particle, o-a.~t, depends on the fracture toughness K~c of the particle, on its dimension D and on the elastic mismatch of both materials.

NUMERICAL CALCULATIONS AND EXAMPLE The aim of numerical calculations based on the proposed model is the estimation of the generalized stress intensity factor H~ for a typical structural configuration. The knowledge of the value of H~ is the necessary condition for the application of the stability condition

H,. < H,. c

(16)

and for estimation of the critical fracture stress for failure of particles. The calculations have been performed by the finite element method (system ANSYS). A particle is modelled as a circular region with diameter D in a three point bending specimen, see Fig.1. The values of H, were estimated by comparison of the calculated values of the stress component cry, with the corresponding analytical expression. The calculation of the critical aggregate fracture stress corresponding to the suggested procedure consists of the following steps: 1. Estimate a, fl and the value of the singularity exponent ~ and express the value of the stress component try,,,

743 2. evaluate the generalized stress intensity factor Hz for the given size D of the particle, and 3. apply the stability condition and estimate the critical applied aggregate fracture stress cr~xnt

K,c 2D ~'-l/: °~'°~t = H, (1MPa) (2 - 2 + g~ ) "

(17)

A parametric study is presented which makes it possible to estimate the critical value of the applied stress needed to cause failure, for a wide range of ratios El / E 2 and particle sizes D. In Fig. 3 the dependence ;t = 2(E~ ~E z) is shown. Calculated values of H, = H , ( E 1/Ez,D ) for unit applied stress cr~ppl = 1 MPa and crack length a = 25 mm corresponding to configurations from Fig. 1 are summarized in Table 1. The size of aggregates change from 1 mm (corresponding to fine aggregates) to 16 mm (coarse aggregates). TABLE 1 VALUES OF H~ [MPA.M )"] FOR MATRIX/PARTICLE RATIO E1 / E 2 AND PARTICLE SIZE D

D [ram] 1 2 4 8 16

mortar/ sandstone 0.207 0.219 0.231 0.248 0.271

cement/ sandstone 0.262 0.266 0.273 0.279 0.285

cement/ cement

mortar/ basalt 0.353 0.342 0.323 0.311 0.298

0.296

cement/ basalt 0.388 0.368 0.341 0.320 0.299

cement/ granite 0.367 0.352 0.330 0.315 0.299

mortar/ granite 0.330 0.323 0.311 0.304 0.297

As an example let us take following material data for matrix and aggregates: the values of E 1 (Young's modulus of the matrix) corresponding to a cement paste are E1 = 20 - 40 GPa and for mortar E 1 = 25 - 45 GPa, Kergner Z., Bilek V. & Schmid, P. (1999). Typical value of E 2 and K, c (i.e. Young's modulus and fracture toughness) for aggregates are: E 2 = 50 GPa and 1£,c = 1.8 - 6.3 MPa m l/z for granite, E 2 = 20 GPa and Kzc = 0.28 - 0.52 MPa m v2 for sandstone, A~:tcin (1998) and E 2 = 60 GPa and K, c = 1.80 - 6.35 MPa m v2 for basalt, Basham, Chong & Boresi (1993). The resulting values of the critical stress needed for failure of the particle, o-~.c~t, are summarized in Table 2. TABLE 2 CRITICAL STRESS [MPA] FOR MATRIX]PARTICLERATIO E, / E 2 , PARTICLESIZE D, RANGE OF K,c mortar/ sandstone D [mm] 1 2 4 8 16

K/c

0.28 1.332 1.327 1.325 1.297 1.252

0.52 2.473 2.465 2.460 2.408 2.325

cement/ sandstone

mortar/ basalt

K/c

K/c

0.28 1.067 1.067 1.058 1.053 1.045

0.52 1.8 1.981 5.071 1.982 5.102 1.965 5.250 1.955 5.296 1.941 5.377

6.35 17.89 18.0 18.52 18.68 18.97

cement/ basalt

Kic 1.8 4.581 4.616 4.765 4.842 4.959

6.35 16.16 16.29 16.81 17.08 17.49

cement/ granite

K1c 1.8 4.878 4.910 5.063 5.117 5.214

6.3 17.07 17.17 17.72 17.91 18.25

mortar/ granite

Kic 1.8 5.450 5.476 5.602 5.641 5.692

6.3.... 19.07 19.17 19.61 19.74 19.92

744 CONCLUSIONS Concrete has been considered as a two-phase material and the problem of cracking of coarse aggregate in cement paste has been analysed. The boundary between the fractured matrix and the aggregate particle to be fractured is considered as an interface between two different materials and characterized by the corresponding elastic mismatch Et/E2 of both materials. A fracture model that deals with the influence of aggregate particle dimension and elastic mismatch between matrix and aggregate has been suggested. The crack is postulated to grow and to fracture the particle if the stress averaged over the particle exceeds its critical value. The procedure has been applied to concrete, where the matrix consists of cement paste and the particles correspond to fine and coarse aggregates. Calculations have been performed by a combination of analytical and numerical approaches. A parametric study showing the influence of particles dimension and elastic mismatch between the paste and aggregate is presented.

ACKNOWLEDGEMENTS The authors thank for funding under grants K 1076602 of the Academy of Sciences of the Czech Republic and 103/97/K003 from the Grant Agency of the Czech Republic.

REFERENCES A~tcin P.-C. (1998). High Performance Concrete, E&FN SPON, London and New York.

ANSYS, UsersManual. Version 5.6 (1999). Swanson Analysis System, Inc. Houston, Pensylvania. Basham K.D., Chong K.P. and Boresi A.P. (1993). A new method to compute size independent fracture toughness values for brittle materials. Engineering Fracture Mechanics 16:3, 357-363. Kergner Z., Bilek V. and Schmid, P. (1999). Fracture mechanics study of two aspects reducing mechanical characteristics of concrete. Acta Polytechnica (Prague) 39:2, 25-38. Knesl Z., Knhpek A. and Bedn/t~ K. (1998). Evaluation of the critical stress in bonded materials with a crack perpendicular to the interface, in: Surface Modification Technologqes XI. Eds/: T.S.Sudarshan, M.Jeandin and K.A.Khor, The Institute of Metals, London, 153-159. Li Victor C. and Huang J. (1990). Relation of concrete fracture toughness to its internal structure.

Engineering Fracture Mechanics 35:1-3, 39-46. Lin K.Y. and Mar J.W. (1976). Finite element analysis of stress intensity factors for cracks at a bimaterial interface. International Journal of Fracture 12:4, 521-531. Lipetzky P. and Knrsl Z. (1995). Crack-particle interaction in a two-phase composite. Part II: crack deflection. International Journal of Fracture 73:1, 81-92. Shah S.P. and Ouyang C. (1994). Fracture mechanics for failure of concrete. Annu. Rev. Mater. Sci. 24, 293-320.

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

745

A DISCRETE CRACK M O D E L L I N G STRATEGY FOR M A S O N R Y STRUCTURES

G.P.A.G. van Zijl Faculty of Architecture Delft University of Technology, Delft, The Netherlands e-mail: g.p.a.g.vanzijl @bk.tudelft.nl

ABSTRACT

In many cases masonry lies beyond the scope of a homogeneous, continuum mechanics description. This fact and the knowledge that the joints in masonry are the weak links make a discrete strategy particularly appropriate to model cracking, shear slipping and crushing in masonry. In this paper an interface material model is presented, which captures the main failure modes in masonry joints. Of particular importance is the accurate description of shearing dilatancy, or uplift, as this leads to volume increase, which, when confined, causes wedging and associated stress build-up. It is imperative to incorporate the reduction in the dilatancy coefficient with shearing displacement and normal compression, which mechanisms smoothen the shear surface. Such a formulation has been adopted. It is verified and validated by analysing shear experiments on masonry specimens with different boundary conditions. KEYWORDS: discrete cracking, masonry, dilatancy, finite elements

INTRODUCTION The elegance of continuum modelling tempts the analyst to employ it also for modelling masonry. In yet a further step of approximation, the ignorance of the heterogeneity, the modelling is simplified significantly. Reasonable simulation of masonry behaviour with continuum, homogeneous modelling has been reported. Lourenqo (1996) formulated an anisotropic Rankine-Hill model, based on continuum plasticity. The anisotropic Rankine criterion captures the different strengths parallel and perpendicular to the bed joints in masonry, while the Hill criterion allows different crushing behaviour in the orthogohal directions. In this way, the influence of the meso-structure, i.e. the arrangement of brick units in its mortar matrix, is incorporated in a phenomenological way. By also employing orthotropic elasticity, the different orthogonal stiffnesses prior to cracking can be described. With this model good agreement with large masonry shear wall experiments were obtained (Lourenqo 1996). Homogenization techniques have been employed by Pande et al. (1989) for large scale masonry analyses, but such models fail to capture

746 the inelastic behaviour properly. Continuum, isotropic damage models are also employed for masonry, eg. Papa et al. (2000), for its superiority in the prediction of the response of cementitious materials to cyclic loading. However, there are circumstances under which the ignorance of the discrete nature of cracking in cementitious, quasi-brittle materials is inappropriate. For concrete, which may be considered as a continuum on the macro scale, localisation/crack analysis requires some form of enhancement of the continuum description, which invariably requires dense finite element meshes in the localisation zones (de Borst et al. 2000). To make such analyses viable, adaptive meshing techniques should be employed to avoid a-priory knowledge of crack locations and orientations, a disadvantage in common with discrete modelling. However, even with adaptivity, large numbers of elements are required to capture the localisation properly (Askes et al. 1998). Therefore, due to its superiority in the physical and numerical description of cracking, a displacement discontinuity is the most appropriate final step of an adaptive/remeshing strategy. Such a discontinuity may be an interface, or an embedded jump in the displacement field, which incorporates a constitutive description of cohesive cracking. The latter discontinuity is most promising, because it avoids the need for remeshing (Wells and Sluys 2000). For masonry, continuum modelling is often not only inappropriate, but may lead to large errors. Some examples (van Zijl 2000a,b) are given in Figure 1. In all cases the source of error is the ignorance of the meso-structure. This can be avoided by exploiting the fact that the weak joints in masonry provide the a-priori knowledge of crack positions and orientations, making a discrete modelling strategy particularly appropriate. However, not all joints crack, calling for an adaptive strategy, which acknowledges the geometrical arrangement of bricks in the masonry and inserts discontinuities at joint locations only at critical points. In this way a compromise is made between the computational elegance and speed of homogeneous, continuum modelling and the accuracy of discrete modelling, making the analysis of large scale masonry structures viable. As a first step a constitutive law for the accurate description of cracking, shear-slip and crushing in masonry joints is required. Such a model is presented here. A DISCRETE MODELLING APPROACH In Figure 2 two levels of discrete modelling of masonry are illustrated. A detailed strategy, Figure 2a, is required for especially hygro-mechanical analyses. Then, both the bricks and the mortar are modelled. They are considered to be continua and are discretised with continuous finite elements. The bulk actions like creep, as well as the hygral and thermal shrinkage act in these continua. The interfaces between the two constituents are modelled with interface elements, which capture debonding, shear-slipping and crushing. The rate dependence in the fracture process zone, which introduces the time scale of fracture and the rate-enhanced cracking resistance (van Zijl et al. 2000), is also incorporated in the interface constitutive law. For only mechanical analyses, the simplified strategy of Figure 2b may be followed. Then the mortar is not modelled, but interface elements account for the joint actions. In cases of high strength joints it is necessary to supplement the modelling strategy to include the cracking in the bricks. It usually suffices to model central, vertical brick cracks, by placing interfaces at these locations, Figure 2a,b. INTERFACE CONSTITUTIVE LAW A plane stress interface model formulated by Lourenqo (1996), Figure 3a, has been enhanced to address important aspects of meso-mechanical behaviour, as well as extended to three dimensions (van Zijl 2000a), Figure 3b. The models are based on multi-surface plasticity, comprising a Coulomb friction model combined with a tension cut-off and an elliptical compression cap, the latter not yet included in the three-dimensional model.

747 , 0.6

7~

e

iiiiiiil N iiiiii!!iiiigiit

NNii®

I5

v

Discrete model Continuum model

¢1

0.4

E z 0.2

0.0

0.0

0.1

0.2 0.3 Lengthening (mm)

0.4

0.5

(a) Brick/unit size large relative to structural size. Distinction between the fracture and shear-slip along joints is imperative to capture the global response. The role of dilatancy q~ is shown, which cannot be simulated properly in a continuum approach. 1

--=-7,,

o-~

o8 /,ti 0.6

/ i l

'

0.2

"

°0-~

,0 '-~

,0 '-~ '1o ~-'

,0 °

,0'

,0 ~

Time (days)

(b) Uniform loading; here applied for creep fracture investigation. A stress gradient is produced by the masonry meso-structure in a discrete approach. This enables time-dependent crack propagation under sustained load to be analysed, unlike a homogeneous solution of a continuum approach.

~mb,,

h: 0.6 0.7 0.8 0.8 0.9

h: 0.6 0.7 0.8 0.8 0.9

(c) Hygro-mechanical interaction. Interfaces between mortar and bricks act as sinks, causing discontinuity in moisture content. The different drying rates of mortar and bricks cause large pore humidity (h) gradients, which cause cracking at the interfaces. This can not be captured by a continuum approach. Figure 1" Examples of masonry behaviour where continuum modelling is inappropriate.

748 Mortar

Interface

"Brick"

Joint/interface

iiiiiiiiiiiii~ ' iiiii!il. . . . . '

..... -~

(a)

;!tfft!flblf£ !i£c-k

(b)

Figure 2: A (a) detailed and (b) simplified discrete modelling strategy for masonry. Softening acts in all the modes and is preceded by hardening in the case of the cap mode. The most important enhancement is the incorporation of an accurate simulation of shearing dilatancy in the interface, which will be discussed briefly (see van Zijl 2000(a) for full details of the model). Van der Pluijm (1992, 1998) observed a dilatant normal uplift upon shearing along masonry joints, Figure 4. This dilatancy, if confined externally, or by the surrounding masonry, contributes significantly to the strength and toughness of masonry and may determine the failure mode. However, the strength and toughness may be overestimated greatly if the smoothing effect of high confining pressure and slipping displacement, by which the uplift is arrested, is ignored. The dilatancy is incorporated in the plasticity formulation via the flow rule (here elaborated in 2-D) { t~p

~p

_

where the generalised stress and strain vectors o= e

{o~} =

~

{ u v }r,

(2)

are considered, with o and u the stress and relative displacement respectively in the interface normal direction and z and v the shear stress and relative displacement respectively. The subscript p denotes the plastic component and the rate of the plastic flow amount is given by ~, = Or. A suitable potential function gradient is O0 -

sign(x)

'

(3)

• = tan ~ being the mobilised dilatancy coefficient. Following directly from the flow rule W _ u_£ sign(x). /,p

C ap

/ / l (a)

7"

%'--.Coulomb "~-l~riction

"~S

Ixl

/""'" "'"'frb" ..... ~~.Tension :.~,, Intermediate ~ 1 , ~ Mode ( \ yield surface /( i..~:.:..:.' ,, Initial yield surfac9/ / Residual yield surface/'

(4)

-:,1

ft

(y

(b)

Figure 3" (a) 2-D and (b) 3-D interface limit surfaces (Lourenqo 1996, van Ziji 2000a).

749

s

lll hJ

M~7 gauge )oints

-0 r

I I1~ tl } 11.~ Ill 111~ , l l l

I I 1

Z Measurements in mm

i 1t~ 1"~" ill - tl 1'~' 2

(a)

__

] Experimental (Vander Pluijm 1998) !

0.05

mz2 [

0.04

0.12

._

m2

~

Experimental (Vander Pluijm 1992) ""

O=-0.1 N / m m 2

.__

o=-0.5 N/mm'-

~" 0.02 0.04 0.01 0.00

0.0

0.2

(b)

0.4

0.6

0.8

v,, (ram)

0.00

O=- 1.0 N / m m 2 0.0

0.1

(c)

0.2

0.3

0.4

v,, (ram)

Figure 4: (a) Schematical set-up of masonry shear tests (Van der Pluijm 1992, 1998). Dilatant normal displacement upon shear displacement of (b) clay brick masonry and a (c) calcium silicate masonry. By integration the shear-slip induced normal uplift is found to be:

u !, = f W d[Avpl.

(5)

A description of the normal uplift upon shear-slipping is chosen as 0,

up=

O < Ou

tango ( ~ u u ) 5 1-

(

1-

e_~5

) Vp , o u < o < 0

tango ( e_~)Vp), ~5 1-

(6)

o>0,

which yields after differentiation O < Ou

0~

tango,

Vp, Ou_0.

The dilatancy at zero normal confining stress and shear slip (tan ~o), confining (compressive) stress at which the dilatancy becomes zero (o,) and the dilatancy shear slip degradation coefficient (~5)are model parameters to be obtained by, for instance, a least squares fit of eq. (6) to experimental test data. Note that for tensile stress a stress-independent dilatancy coefficient is assumed. The above relations are cast into the standard plasticity formulation, where also the corner regions are treated consistently via the flow rule (Koiter 1953) c)gl c)g2 I~p = k l - - ~ --}-k2 ~

(8)

750 where the subscripts 1 and 2 refer to the two intersecting criteria. A consistent tangent operator has been derived ~a D ep= ~---~ (9) (van Zijl 2000a) which ensures quadratic convergence of the global solution procedure (Simo and Taylor 1985). V E R I F I C A T I O N AND VALIDATION In this section the discrete modelling strategy and the incorporated dilatancy formulation are verified at the hand of analyses of the micro-shear experiments of van der Pluijm (1992, 1998), Figure 4a. The finite element model is shown in that figure. The model parameters have been obtained by averaging and regression of other micro-experiments by van der Pluijm and Vermeitfoort (1991) and van der Pluijm (1998). Young's moduli E = 17400 and E = 13400 for the clay brick and calcium silicate unit respectively, as well as a Poisson's ratio v = 0.2 for both, have been employed. The other model parameters are given in Table 1. The interface normal and shear stiffnesses k,z, ks have been calculated to match the observed elastic behaviour of the shear specimens, fi is the bonding tensile strength, Gf the fracture energy, Co the adhesion strength and ~o, ~r the initial and residual friction angle respectively. The simplified discrete analysis strategy, Figure 2b, has been followed. In Figure 4b,c it can be seen that the model succeeds in capturing the observed dilatant behaviour. The agreement is not surprising, because the model parameters have been calculated from these experimental responses. Nevertheless, it shows that the chosen form of the dilatancy equation is appropriate. A more severe test of the model is presented by the confinement of the dilatancy and the associated normal stress build-up. Van der Pluijm (1998) modified his experimental set-up to control the displacement normal to the joint. He performed shear tests on two specimens of the clay brick masonry type, Table 1. In the first test he applied an initial normal displacement to cause an average stress of-0.1 N/mm 2. The normal boundaries were then fixed in this position, before the displacement-controlled shearing was commenced, as shown schematically in Figure 5a. The same procedure was followed for the second test, except that an initial average tensile stress, Go = +0.1 N/mm 2 was applied instead of the compression in the first test. In Figure 5b the normal force build-up upon shearing is shown. Van der Pluijm limited the normal force to 27.5 kN to protect the test apparatus, at which point he switched back to force control of the normal boundary to sustain the limit force. This has been simulated numerically, Figure 5b. However, the case of unrestricted normal force build-up has also been analysed. These numerical responses are shown in dotted lines. Due to the smoothing of the interface, a point is reached where no further dilatancy occurs and the normal force is arrested. This point coincides with the pressure at which the dilatancy becomes zero (~u. The measured response for the case of initial compression (cY0 = -0.1 N/mm 2) indicates that a limit point was approached just before the switch to force control was made, which confirms the numerical result. In Figure 5c the numerical and experimental shear force-deformation responses are compared. Reasonable agreement is found. Whereas the agreement of the numerical responses with the measured responses is' reasonable, the in-

Table l'Interface parameters employed for (a) clay brick and (b) calcium silicate masonry.

k,,

ks

N

(a) (b)

N ~

825 438

345 182

ft

Glr

c,,

GI/

N

N"

N

N

mm

m m ~-

0.012 0.005

0.87 0.28

' ~

0.4 0.1

tango

tan~,,

tango

0.006 0.09(~ 0.02 0.03(~

a,

8

N :rim ~

mm

1.1 0.97

0.70 0.75

0.74 0.67

-1.57 -1.22

5.6 17

751 appropriate dilatancy modelling can lead to large errors. This is illustrated in Figure 6. For even a small, constant dilatancy coefficient an unlimited strength is predicted for the shear specimen of Figure 5a. A dilatancy coefficient of zero reproduces the response under force control of the normal boundaries, Figure 4a, in which case the initial confining pressure governs the response. The responses for Go - - 0 . 1 N / m m 2 are shown. S

I I ! I I ~ i i I 1! I I I 1LL1A_IA II~II'~

IlIlIIIl[lll[

~ ~]:7i

............... ~ ! !T!i[i[i[!!:

~~![H!!!![![

(a)

/

50

~Rn

50

t ,.~ i ~".

i~ o'

"\

,

Numerical

I

~ I

.... 0

- E perimental Numerical

(RII unlimited)

•

"

-20

(b)

~...

20

\

,,

Numerical (R.>-27.5kN)

-30

(R unlimited)

30

[

20

10 ~

- - - Experimental (R>-27.5kN) ~ Numerical (R >-27.5kN)

- 10

]l'\ I :ll

10

"\~ 0

0

R (kN)

(C)

/

°

[

0.0

"-'-

'

¢y,=-0.1 N/mm

"

0.2

-22.'22-.22.-2--

0.4

Joint shear displacement

2

0.6

0.8

v (mm)

Figure 5" (a) Schematical set-up of the normal confined masonry shear tests (Van der Pluijm 1998). Comparison of experimental and numerical (b) shear force vs. normal reaction force and (c) shear force vs. joint shear displacement. ~-" 2.0 "~

~._--~-~ ~=0.74

1.5

/

/'

"'--..~0.74 (1-¢yl~,)e . -~'','

0.5