The Finite Element Method Fifth edition Volume 2: Solid Mechanics 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 (765.035mph) 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 2: Solid Mechanics 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 5055 9 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 On Ä ate and his group at CIMNE for their help, encouragement and support during the preparation process. Preface to Volume 2 General problems in solid mechanics and non-linearity Introduction Small deformation non-linear solid mechanics problems Non-linear quasi-harmonic field problems Some typical examples of transient non-linear calculations Concluding remarks Solution of non-linear algebraic equations Introduction Iterative techniques Inelastic and non-linear materials Introduction Viscoelasticity - history dependence of deformation Classical time-independent plasticity theory Computation of stress increments Isotropic plasticity models Generalized plasticity - non-associative case Some examples of plastic computation Basic formulation of creep problems Viscoplasticity - a generalization Some special problems of brittle materials Non-uniqueness and localization in elasto-plastic deformations Adaptive refinement and localization (slip-line) capture Non-linear quasi-harmonic field problems Plate bending approximation: thin (Kirchhoff) plates and C1 continuity requirements Introduction The plate problem: thick and thin formulations Rectangular element with corner nodes (12 degrees of freedom) Quadrilateral and parallelograpm elements Triangular element with corner nodes (9 degrees of freedom) Triangular element of the simplest form (6 degrees of freedom) The patch test - an analytical requirement Numerical examples General remarks Singular shape functions for the simple triangular element An 18 degree-of-freedom triangular element with conforming shape functions Compatible quadrilateral elements Quasi-conforming elements Hermitian rectangle shape function The 21 and 18 degree-of-freedom triangle Mixed formulations - general remarks Hybrid plate elements Discrete Kirchhoff constraints Rotation-free elements Inelastic material behaviour Concluding remarks - which elements? ’Thick’ Reissner - Mindlin plates - irreducible and mixed formulations Introduction The irreducible formulation - reduced integration Mixed formulation for thick plates The patch test for plate bending elements Elements with discrete collocation constraints Elements with rotational bubble or enhanced modes Linked interpolation - an improvement of accuracy Discrete ’exact’ thin plate limit Performance of various ’thick’ plate elements - limitations of twin plate theory Forms without rotation parameters Inelastic material behaviour Concluding remarks - adaptive refinement Shells as an assembly of flat elements Introduction Stiffness of a plane element in local coordinates Transformation to global coordinates and assembly of elements Local direction cosines ’Drilling’ rotational stiffness - 6 degree-of-freedom assembly Elements with mid-side slope connections only Choice of element Practical examples Axisymmetric shells Introduction Straight element Curved elements Independent slope - displacement interpolation with penalty functions (thick or thin shell formulations) Shells as a special case of three-dimensional analysis - Reissner-Mindlin assumptions Introduction Shell element with displacement and rotation parameters Special case of axisymmetric, curved, thick shells Special case of thick plates Convergence Inelastic behaviour Some shell examples Concluding remarks Semi-analytical finite element processes - use of orthogonal functions and ’finite strip’ methods Introduction Prismatic bar Thin membrane box structures Plates and boxes and flexure Axisymmetric solids with non-symmetrical load Axisymmetric shells with non-symmetrical load Finite strip method - incomplete decoupling Concluding remarks Geometrically non-linear problems - finite deformation Introduction Governing equations Variational description for finitite deformation A three-field mixed finite deformation forumation A mixed-enhanced finite deformation forumation Forces dependent on deformation - pressure loads Material constitution for finite deformation Contact problems Numerical examples Concluding remarks Non-linear structural problems - large displacement and instability Introduction Large displacement theory of beams Elastic stability - energy interpretation Large displacement theory of thick plates Large displacement theory of thin plates Solution of large deflection problems Shells Concluding remarks Pseudo-rigid and rigid-flexible bodies Introduction Pseudo-rigid motions Rigid motions Connecting a rigid body to a flexible body Multibody coupling by joints Numerical examples Computer procedures for finite element analysis Introduction Description of additional program features Solution of non-linear problems Restart option Solution of example problems Concluding remarks Appendix A [...]... this volume we consider more advanced problems in solid mechanics while in Volume 3 we consider applications in ¯uid dynamics It is our intent that Volume 2 can be used by investigators familiar with the ®nite element method in general terms and will introduce them here to the subject of specialized topics in solid mechanics This volume can thus in many ways stand alone Many of the general ®nite element. .. (where "v is the volume change) as a three-®eld formulation (see Sec 12.4, Volume 1) An alternative three-®eld form is the enhanced strain approach presented in Sec 11.5.3 of Volume 1 The matter of which we use depends on the form of the constitutive equations For situations where changes in volume a€ect only the pressure the two®eld form can be easily used However, for problems in which the response... element procedures available in Volume 1 may not be familiar to a reader introduced to the ®nite element method through di€erent texts We therefore recommend that the present volume be used in conjunction with Volume 1 to which we make frequent reference Two main subject areas in solid mechanics are covered here: 1 Non-linear problems (Chapters 1±3 and 10±12) In these the special problems of solving... implementation of the CBS algorithm Appendix A Non-conservative form of Navier±Stokes equations Appendix B Discontinuous Galerkin methods in the solution of the convection± di€usion equation Appendix C Edge-based ®nite element formulation Appendix D Multi grid methods Appendix E Boundary layer ± inviscid ¯ow coupling Preface to Volume 2 The ®rst volume of this edition covered basic aspects of ®nite element approximation... much stronger link to the full three-dimensional theory 2 General problems in solid mechanics and non-linearity This volume will consider each of the above types of problems and formulations which make practical ®nite element solutions feasible We establish in the present chapter the general formulation for both static and transient problems of a non-linear kind Here we show how the linear problems... is shown that by relating the formulation to the deformed body a result is obtain which is nearly identical to that for the small deformation problem we considered in Volume 1 and which we expand upon in the early chapters of this volume Essential di€erences arise only in the constitutive equations (stress±strain laws) and the addition of a new sti€ness term commonly called the geometric or initial... non-linear material, the plate and shell, and the ®nite deformation problems presented in this volume Here the discussion is directed primarily to the manner in which non-linear problems are solved We also brie¯y discuss the manner in which elements are developed to permit analysis of either quasi-static (no inertia e€ects) or transient applications 1.2 Small deformation non-linear solid mechanics problems... Equations (1.3) and (1.4) hold at all points xi in the domain of the problem  Stress boundary conditions are given by the traction condition " ti ˆ ji nj ˆ ti …1:5† for all points which lie on the part of the boundary denoted as ÿt A variational (weak) form of the equations may be written by using the procedures described in Chapter 3 of Volume 1 and yield the virtual work equations given by1;8;9 … …... we omit direct damping which leads to the term C~ (see Chapter 17, Volume 1) …1:18† 5 6 General problems in solid mechanics and non-linearity and … P…r† ˆ  BT r d …1:19† The term P is often referred to as the stress divergence or stress force term In the case of linear elasticity the stress is immediately given by the stress±strain relations (see Chapter 2, Volume 1) as r ˆ De …1:20† when e€ects... yield the values of n ‡ 1 by simple inversion u of matrix M If the M matrix is diagonalized by any one of the methods which we have discussed in Volume 1, the solution for n ‡ 1 is trivial and the problem can be considered solved u However, such explicit schemes are only conditionally stable as we have shown in Chapter 18 of Volume 1 and may require many time steps to reach a steady state solution Therefore . The Finite Element Method Fifth edition Volume 2: Solid Mechanics Professor O. C. Zienkiewicz, CBE, FRS, FREng is Professor Emeritus and Director of the Institute for Numerical Methods in. 1999). The Finite Element Method Fifth edition Volume 2: Solid Mechanics O. C. Zienkiewicz, CBE, FRS, FREng UNESCO Professor of Numerical Methods in Engineering International Centre for Numerical Methods. 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