finite element method (5th edition) vol2

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 I EINEMANN 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 -@4 member of the Reed Elsevier plc group First published in 1967 by McGraw-Hill Fifth edition published by Butterworth-Heinemann 2000 0 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 WIP 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 FOR EVERY TnLE THAT WE rmLisn. BL~WORTH.HELYEMAW WU PAY FOR BTCV TO PLANl AND CARE FOR A TREE. 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 finite element method. In particular we would like to mention Professor Eugenio Oiiate and his group at CIMNE for their help, encouragement and support during the preparation process. 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 finite 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 field. The recipient of 24 honorary degrees and many medals, Professor Zienkiewicz is also a member of five academies - an honour he has received for his many contributions to the fundamental developments of the finite 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 first 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 field 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 finite 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 finite deformation problems. Front covcr image: A Finite Element Model of the world land speed record (765.035mph) car THRUST SSC. The analysis was done using the finite element method by K. Morgan, 0. Hassan and N.P. Weatherill at the Institute for Numerical Methods in Engineering, University of Walcs Swansea, UK. (see K. Morgan, 0. Hassan and N.P. Weatherill, ‘Why didn’t the supersonic car fly?’, Mat/ic~ma/ic~ Tocluy, Bulletin of the Institirte of Matliematics uncl Irs Appliccrtioiu. Vol. 35. No. 4. 110-1 14, Aug. 1999). Preface to Volume 2 The first volume of this edition covered basic aspects of finite element approximation in the context of linear problems. Typical examples of two- and three-dimensional elasticity, heat conduction and electromagnetic problems in a steady state and tran- sient state were dealt with and a finite element computer program structure was intro- duced. However, many aspects of formulation had to be relegated to the second and third volumes in which we hope the reader will find the answer to more advanced problems, most of which are of continuing practical and research interest. In this volume we consider more advanced problems in solid mechanics while in Volume 3 we consider applications in fluid dynamics. It is our intent that Volume 2 can be used by investigators familiar with the finite 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 finite element procedures available in Volume 1 may not be familiar to a reader intro- duced to the finite element method through different 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 non-linear equation systems are addressed. In the first part we restrict our attention to non-linear behaviour of materials while retaining the assumptions on small strain used in Volume 1 to study the linear elasticity problem. This serves as a bridge to more advanced studies later in which geometric effects from large displacements and deformations are presented. Indeed, non-linear applications are today of great importance and practical interest in most areas of engineering and physics. By starting our study first using a small strain approach we believe the reader can more easily comprehend the various aspects which need to be understood to master the subject matter. We cover in some detail problems in viscoelasticity, plasticity, and viscoplasticity which should serve as a basis for applications to other material models. In our study of finite deformation problems we present a series of approaches which may be used to solve problems including extensions for treatment of constraints (e.g. near incompressibility and rigid body motions) as well as those for buckling and large rotations. xiv Preface to Volume 2 2. Plates and shells (Chapters 4-9) This section is of course of most interest to those engaged in ‘structural mechanics’ and deals with a specific class of problems in which one dimension of the structure is small compared to the other two. This application is one of the first to which finite elements were directed and which still is a subject of continuing research. Those with interests in other areas of solid mechanics may well omit this part on first reading, though by analogy the methods exposed have quite wide applications outside structural mechanics. Volume 2 concludes with a chapter on Computer Procedures, in which we describe application of the basic program presented in Volume 1 to solve non-linear problems. Clearly the variety of problems presented in the text does not permit a detailed treatment of all subjects discussed, but the ‘skeletal’ format presented and additional information available from the publisher’s web site’ will allow readers to make their own extensions. We would like at this stage to thank once again our collaborators and friends for many helpful comments and suggestions. In this volume our particular gratitude goes to Professor Eric Kasper who made numerous constructive comments as well as contributing the section on the mixed-enhanced method in Chapter 10. We would also like to take this opportunity to thank our friends at CIMNE for providing a stimulating environment in which much of Volume 2 was conceived. OCZ and RLT ’ Complete source code for all programs in the three volumes may be obtained at no cost from the publisher’s web page: http://www.bh.com/companions/fem Contents Preface to Volunie 2 1. General problems in solid mechanics and non-linearity 1.1 Introduction 1.2 1.3 Non-linear quasi-harmonic field problems 1.4 1.5 Concluding remarks Small deformation non-linear solid mechanics problems Some typical examples of transient non-linear calculations References 2. Solution of non-linear algebraic equations 2.1 Introduction 2.2 Iterative techniques References 3. Inelastic and non-linear materials 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 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 References 4. Plate bending approximation: thin (Kirchhoff) plates and C, continuity requirements Xlll 1 1 3 12 14 20 20 22 22 23 36 38 38 39 48 56 61 68 71 75 78 84 88 93 101 104 111 viii Contents 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 Introduction The plate problem: thick and thin formulations Rectangular element with corner nodes (12 degrees of freedom) Quadrilateral and parallelogram 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? References 5. ‘Thick’ Reissner-Mindlin plates - irreducible and mixed formulations 5.1 Introduction 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 Forms without rotation parameters 5.1 1 Inelastic material behaviour 5.12 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 thin plate theory Concluding remarks - adaptive refinement References 6. Shells as an assembly of flat elements 6.1 Introduction 6.2 6.3 6.4 Local direction cosines 6.5 6.6 Stiffness of a plane element in local coordinates Transformation to global coordinates and assembly of elements ‘Drilling’ rotational stiffness - 6 degree-of-freedom assembly Elements with mid-side slope connections only 111 113 124 128 128 133 134 138 145 145 148 149 150 151 153 155 157 158 162 164 166 167 173 173 176 180 183 187 196 199 202 203 208 210 21 1 212 216 216 218 219 22 1 225 230 Contents ix 6.7 Choice of element 6.8 Practical examples References 230 23 1 240 7. Axisymmetric shells 244 7.1 Introduction 244 7.2 Straight element 245 7.3 Curved elements 25 1 7.4 Independent slope-displacement interpolation with penalty functions (thick or thin shell formulations) 26 1 References 264 8. Shells as a special case of three-dimensional analysis - Reissner-Mindlin assumptions 8.1 Introduction 8.2 8.3 8.4 8.5 Convergence 8.6 Inelastic behaviour 8.7 Some shell examples 8.8 Concluding remarks Shell element with displacement and rotation parameters Special case of axisymmetric, curved, thick shells Special case of thick plates References 9. Semi-analytical finite element processes - use of orthogonal functions and ‘finite strip’ methods 9.1 Introduction 9.2 Prismatic bar 9.3 Thin membrane box structures 9.4 9.5 9.6 9.7 9.8 Concluding remarks Plates and boxes with flexure Axisymmetric solids with non-symmetrical load Axisymmetric shells with non-symmetrical load Finite strip method - incomplete decoupling References 10. Geometrically non-linear problems - finite deformation 10.1 Introduction 10.2 Governing equations 10.3 10.4 10.5 10.6 10.7 10.8 Contact problems 10.9 Numerical examples Variational description for finite deformation A three-field mixed finite deformation formulation A mixed-enhanced finite deformation formulation Forces dependent on deformation - pressure loads Material constitution for finite deformation 266 266 266 275 277 278 279 280 285 286 289 289 292 295 296 297 303 305 308 309 3 12 3 12 314 319 328 332 336 338 347 355 x Contents 10.10 Concluding remarks References 359 360 1 1. Non-linear structural problems - large displacement and instability 1 1.1 Introduction 1 1.2 11.3 1 1.4 1 1.5 11.6 11.7 Shells 11.8 Concluding remarks 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 References 12. Pseudo-rigid and rigid-flexible bodies 12.1 Introduction 12.2 Pseudo-rigid motions 12.3 Rigid motions 12.4 12.5 Multibody coupling by joints 12.6 Numerical examples Connecting a rigid body to a flexible body References 13. Computer procedures for finite element analysis 13.1 Introduction 13.2 13.3 Solution of non-linear problems 13.4 Restart option 13.5 Solution of example problems 13.6 Concluding remarks Description of additional program features References Appendix A: Invariants of second-order tensors A. 1 Principal invariants A.2 Moment invariants A.3 Derivatives of invariants Author index Subject index 365 365 365 373 315 38 1 383 386 39 1 392 396 396 396 398 402 404 409 410 413 413 414 41 5 428 429 430 430 432 432 433 434 437 445 [...]... elements and numerical integration - ‘infinite’and ‘singularity’ elements 10 The patch test, reduced integration, and non-conforming elements 1 1 Mixed formulation and constraints - complete field methods 12 Incompressible problems, mixed methods and other procedures of solution 13 Mixed formulation and constraints - incomplete (hybrid) field methods, boundary/Trefftz methods 14 Errors, recovery processes... processes and error estimates 15 Adaptive finite element refinement 16 Point-based approximations; element- free Galerkin - and other meshless methods 17 The time dimension - semi-discretization of field and dynamic problems and analytical solution procedures 18 The time dimension - discrete approximation in time 19 Coupled systems 20 Computer procedures for finite element analysis Appendix A Matrix algebra... with the effects of finite deformation on computing stresses and thus the stress-divergence term and resulting tangent moduli As these aspects involve more advanced concepts we have deferred the treatment of finite strain problems to the latter part of the volume where we will address basic formulations and applications References 1 O.C Zienkiewicz and R.L Taylor The Finite Element Method: The Basis,... Computer implementation of the CBS algorithm Appendix A Non-conservative form of Navier-Stokes equations Appendix B Discontinuous Galerkin methods in the solution of the convectiondiffusion equation Appendix C Edge-based finite element formulation Appendix D Multi grid methods Appendix E Boundary layer - inviscid flow coupling General problems in solid mechanics and non-linearity 1.I Introduction In the... strain-displacement matrix given in Eq (1.15) Similarly, the stresses in each element may be computed by using IS = Id8 + mN,p (1.43) where again 6 are stresses computed as in Eq (1.40) in terms of the strains E Substituting the element stress and strain expressions from Eqs (1.42) and (1.43) into Eq (1.41) we obtain the set of finite element equations P+MU=f Pp - c = 0 p -CTE, + EU = 0 ( 1.44) Small deformation... pressure and volumetric strain approximations are taken locally in each element and N, = N, it is possible to solve the second and third equation of (1.44) in each element individually Noting that the array C is now symmetric positive definite, we may always write these as p = c - 1 P, (1.46) E = C-'Eu = Wu , The mixed strain in each element may now be computed as ( 1.47) where B, = N,W ( 1.48) defines... eight-noded brick isoparametric elements with constant interpolation in each element for one-term approximations to N , and Nl, by unity; and nine-noded quadrilateral or 27-noded brick isoparametric elements with linear interpolation for N, and N, * Accordingly, in two dimensions we use N,, = N , = [1 < 771 or [1 -Y 1.1 77 . and finite deformation problems. Front covcr image: A Finite Element Model of the world land speed record (765.035mph) car THRUST SSC. The analysis was done using the finite element method. ways stand alone. Many of the general finite element procedures available in Volume 1 may not be familiar to a reader intro- duced to the finite element method through different texts. We therefore. field methods, bound- 14. Errors, recovery processes and error estimates 15. Adaptive finite element refinement 16. Point-based approximations; element- free Galerkin - and other meshless methods