Finite_Element_Method_-_The_Basis_-_Zienkiewicz_and

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The Finite Element Method Fifth edition Volume 1: The Basis

Professor O.C. Zienkiewicz, CBE, FRS, FREng is Professor Emeritus and Director of the Institute for Numerical Methods in Engineering at the University of Wales, Swansea, UK. He holds the UNESCO Chair of Numerical Methods in Engineering at the Technical University of Catalunya, Barcelona, Spain. He was the head of the Civil Engineering Department at the University of Wales Swansea between 1961 and 1989. He established that department as one of the primary centres of ®nite element research. In 1968 he became the Founder Editor of the International Journal for Numerical Methods in Engineering which still remains today the major journal in this ®eld. The recipient of 24 honorary degrees and many medals, Professor Zienkiewicz is also a member of ®ve academies ± an honour he has received for his many contributions to the fundamental developments of the ®nite element method. In 1978, he became a Fellow of the Royal Society and the Royal Academy of Engineering. This was followed by his election as a foreign member to the U.S. Academy of Engineering (1981), the Polish Academy of Science (1985), the Chinese Academy of Sciences (1998), and the National Academy of Science, Italy (Academia dei Lincei) (1999). He published the ®rst edition of this book in 1967 and it remained the only book on the subject until 1971. Professor R.L. Taylor has more than 35 years' experience in the modelling and simu- lation of structures and solid continua including two years in industry. In 1991 he was elected to membership in the U.S. National Academy of Engineering in recognition of his educational and research contributions to the ®eld of computational mechanics. He was appointed as the T.Y. and Margaret Lin Professor of Engineering in 1992 and, in 1994, received the Berkeley Citation, the highest honour awarded by the University of California, Berkeley. In 1997, Professor Taylor was made a Fellow in the U.S. Association for Computational Mechanics and recently he was elected Fellow in the International Association of Computational Mechanics, and was awarded the USACM John von Neumann Medal. Professor Taylor has written sev- eral computer programs for ®nite element analysis of structural and non-structural systems, one of which, FEAP, is used world-wide in education and research environ- ments. FEAP is now incorporated more fully into the book to address non-linear and ®nite deformation problems. Front cover image: A Finite Element Model of the world land speed record (765.035 mph) car THRUST SSC. The analysis was done using the ®nite element method by K. Morgan, O. Hassan and N.P. Weatherill at the Institute for Numerical Methods in Engineering, University of Wales Swansea, UK. (see K. Morgan, O. Hassan and N.P. Weatherill, `Why didn't the supersonic car ¯y?', Mathematics Today, Bulletin of the Institute of Mathematics and Its Applications, Vol. 35, No. 4, 110±114, Aug. 1999).

The Finite Element Method Fifth edition Volume 1: The Basis O.C. Zienkiewicz, CBE, FRS, FREng UNESCO Professor of Numerical Methods in Engineering International Centre for Numerical Methods in Engineering, Barcelona Emeritus Professor of Civil Engineering and Director of the Institute for Numerical Methods in Engineering, University of Wales, Swansea R.L. Taylor Professor in the Graduate School Department of Civil and Environmental Engineering University of California at Berkeley Berkeley, California OXFORD AUCKLAND BOSTON JOHANNESBURG MELBOURNE NEW DELHI

Butterworth-Heinemann Linacre House, Jordan Hill, Oxford OX2 8DP 225 Wildwood Avenue, Woburn, MA 01801-2041 A division of Reed Educational and Professional Publishing Ltd First published in 1967 by McGraw-Hill Fifth edition published by Butterworth-Heinemann 2000 # O.C. Zienkiewicz and R.L. Taylor 2000 All rights reserved. No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England W1P 9HE. Applications for the copyright holder's written permission to reproduce any part of this publication should be addressed to the publishers British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data A catalogue record for this book is available from the Library of Congress ISBN 0 7506 5049 4 Published with the cooperation of CIMNE, the International Centre for Numerical Methods in Engineering, Barcelona, Spain (www.cimne.upc.es) Typeset by Academic & Technical Typesetting, Bristol Printed and bound by MPG Books Ltd

Dedication This book is dedicated to our wives Helen and Mary Lou and our families for their support and patience during the preparation of this book, and also to all of our students and colleagues who over the years have contributed to our knowledge of the ®nite element method. In particular we would like to mention Professor Eugenio OnÄ ate and his group at CIMNE for their help, encouragement and support during the preparation process.

Contents Preface 1. Some preliminaries: the standard discrete system Introduction The structural element and the structural system 1.1 1.2 1.3 Assembly and analysis of a structure 1.4 1.5 1.6 1.7 1.8 The boundary conditions Electrical and ¯uid networks The general pattern The standard discrete system Transformation of coordinates References 2. A direct approach to problems in elasticity Introduction Convergence criteria 2.1 2.2 Direct formulation of ®nite element characteristics 2.3 Generalization to the whole region 2.4 Displacement approach as a minimization of total potential energy 2.5 2.6 Discretization error and convergence rate 2.7 Displacement functions with discontinuity between elements 2.8 2.9 Direct minimization 2.10 An example 2.11 Concluding remarks Bound on strain energy in a displacement formulation References 3. Generalization of the ®nite element concepts. Galerkin-weighted residual Introduction Integral or `weak' statements equivalent to the dierential equations and variational approaches 3.1 3.2 3.3 Approximation to integral formulations 3.4 Virtual work as the `weak form' of equilibrium equations for analysis of solids or ¯uids xv 1 1 4 8 9 10 12 14 15 16 18 18 19 26 29 31 32 33 34 35 35 37 37 39 39 42 46 53

viii Contents Partial discretization Convergence 3.5 3.6 3.7 What are `variational principles'? 3.8 `Natural' variational principles and their relation to governing dierential equations Establishment of natural variational principles for linear, self-adjoint dierential equations 3.9 3.10 Maximum, minimum, or a saddle point? 3.11 Constrained variational principles. Lagrange multipliers and adjoint functions 3.12 Constrained variational principles. Penalty functions and the least square method 3.13 Concluding remarks References 4. Plane stress and plane strain 4.1 4.2 4.3 4.4 4.5 4.6 Introduction Element characteristics Examples ± an assessment of performance Some practical applications Special treatment of plane strain with an incompressible material Concluding remark References 5. Axisymmetric stress analysis Introduction Element characteristics Some illustrative examples Early practical applications 5.1 5.2 5.3 5.4 5.5 Non-symmetrical loading 5.6 Axisymmetry ± plane strain and plane stress References 6. Three-dimensional stress analysis 6.1 6.2 6.3 6.4 Introduction Tetrahedral element characteristics Composite elements with eight nodes Examples and concluding remarks References 7. Steady-state ®eld problems ± heat conduction, electric and magnetic potential, ¯uid ¯ow, etc. 7.1 7.2 7.3 7.4 7.5 7.6 Introduction The general quasi-harmonic equation Finite element discretization Some economic specializations Examples ± an assessment of accuracy Some practical applications 55 58 60 62 66 69 70 76 82 84 87 87 87 97 100 110 111 111 112 112 112 121 123 124 124 126 127 127 128 134 135 139 140 140 141 143 144 146 149

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