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 simulation 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 several computer programs for ®nite element analysis of structural and non-structural systems, one of which, FEAP, is used world-wide in education and research environments 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 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 7506 5055 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 Onate and his group at CIMNE for Ä their help, encouragement and support during the preparation process Preface to Volume 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 - 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 bar Prismatic 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 plates thick 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 option Restart Solution of example problems Concluding remarks Appendix A 332 Geometrically non-linear problems The geometric tangent term is given by " K ˆ G I G where " G ˆ …  N;i ij N ;j dV " …10:112† A solution to Eq (10.109) may be formed by solving the third and second rows as d ~ ˆ Kÿ1 Kpu d~ h u p u h d~ ˆ Kÿ1 Ku d~ ‡ Kÿ1 K d ~ p p p ÿ ÿ1 Á ˆ Kp Ku ‡ Kÿ1 K Kÿ1 Kpu d~ u p p …10:113† and substituting the result into the ®rst row to obtain u u KT d~ ˆ ‰Kuu ‡ Ku Kÿ1 Kpu ‡ Kup Kÿ1 Ku ‡ Kup Kÿ1 K Kÿ1 Kpu Š d~ ˆ f ÿ P p p p p …10:114† This result is obtained by inverting only the symmetric positive de®nite matrix Kp , which we also note is independent of any speci®c constitutive model 10.5 A mixed±enhanced ®nite deformation formulation An alternative method to that just discussed is the fully mixed method in which strain approximations are enhanced The key idea of the mixed±enhanced formulation is the parameterization of the deformation gradient in terms of a mixed and an enhanced deformation gradient from which a consistent formulation is derived This methodology allows for a formulation which has standard-order quadrature and variationally recoverable stresses, hence circumventing diculties which arise in other enhanced strain methods.9ÿ14 There is no need to separate any deformation gradient terms into deviatoric and mean parts as was necessary for the mixed approach discussed in the previous section The mixed±enhanced formulation discussed here uses a three-®eld variational form for ®nite deformation hyperelasticity expressed as … ” ” ” Å ˆ ‰W…FiI † ‡ PiI …FiI ÿ FiI †Š dV ÿ Åext …10:115†  ~ ” where FiI is the deformation gradient, FiI is the mixed deformation gradient, PiI is the mixed ®rst Piola±Kirchho€ stress, W is an objective stored energy function in terms ” of FiI , and Åext is the loading term given by Eq (10.41) The stationary point of Å is obtained by setting to zero the ®rst variation of Eq (10.115) with respect to the three independent ®elds Accordingly,   ! … ÿ Á ”iI ‡ FiI @W ÿ PiI ‡ PiI FiI ÿ FiI dV ÿ Åext ˆ …10:116† ” ” ” ” Å ˆ FiI P ”  @ FiI ” ” where FiI and PiI are mixed variables to be approximated directly The reader will note that we now use the deformation gradient directly instead of the usual CIJ , A mixed±enhanced ®nite deformation formulation EIJ , or bij symmetric forms We often will use constitutive models which are expressed in these symmetric quantities; however, we note that they are also implicitly functions of the deformation gradient through the de®nitions given in Sec 10.2 Once again, at this point we may substitute a ®rst Piola±Kirchho€ stress from any constitutive ” model in place of the derivative of the stored energy function @W=@ FiI in Eq (10.116) Thus, the present form can be used in a general context Finite element approximations to the mixed deformation gradient and ®rst Piola± Kirchho€ stress are constructed directly in terms of local coordinates of the parent element using standard tensor transformation concepts Accordingly, we take ” " " " FiI ˆ FiA JA J I p …n† …10:117† ” "ÿ1 "ÿ1 "ÿ1 PiI ˆ FiA JA J I € …n† …10:118† and where n denotes the natural coordinates ; ; , the Greek subscripts are now associated with the natural coordinates (i.e they are not here the ®nite element node numbers), and € and p are the ®rst Piola±Kirchho€ stress and deformation gradient approximations in the isoparametric coordinate space, respectively.à The " " arrays JA and FiI used above are average quantities over the element volume, e " The average quantity JA is de®ned as … @XA " JA dV and JA ˆ …10:119† JA ˆ e e @a where JA is the standard Jacobian matrix as de®ned in Eq (9.10) of Volume (but " now written for the reference coordinates), and FiI is de®ned as … " FiI ˆ F dV …10:120† e e iI The above form of approximation will ensure direct inclusion of constant states as well as minimize the order of quadrature needed to evaluate the ®nite element arrays and eliminate some sensitivity associated with initially distorted elements The form given in Eqs (10.117) and (10.118) are constructed so that the energy term of the physical and isoparametric pairs are equal Accordingly, we observe that ” ” € p ˆ PiI FiI …10:121† This greatly simpli®es the integrations needed to construct the terms in Eq (10.116) To construct the approximations we note that the tensor transformations for the mixed deformation gradient may be written in matrix form as ” F ˆ Ap …10:122† ” P ˆ Aÿ1€ …10:123† and à Note the resulting transformed arrays are objective under a superposed rigid body motion.4 333 334 Geometrically non-linear problems Table 10.1 Matrix±tensor transformation for the nine-component form Row or column i or I or ... 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:. .. Taylor The Finite Element Method: The Basis, Volume Arnold, London, 5th edition, 2000 S.P Timoshenko and J.N Goodier Theory of Elasticity, McGraw-Hill, New York, 3rd edition, 1969 I.S Sokolniko€, The. .. general ®nite element procedures available in Volume 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