# Finite Elements in Plasticity ,Theory and Practice

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FINITE ELEMENTS PLASTICITY:Theory and Practice

D. R. J. OWEN E. HINTONDepartment of Civil Engineering University College of S~vansea,U.K.

Pineridge Press Limited Swansea, U.K.

First published 1980 by Pineridge Press Limited 91 West Cross Lane, West Cross, Swansea U.K. ISBN 0-906674-05-2 Copyright 0 1980 by Pineridge Press Limited. OWEN, D. R. J. Finite Elements in Plasticity 1. Elasticity 2. Plasticity 3. Finite Element Method I. Title 11. HINTON, E. 620.1'123 TA418 ISBN 0-906674-05-2

All rights reserved.No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publishers.

Printed and bound in Great Britain by REDWOOD BURN LIMITED Trowbridge and Esher

Contents

PART I1 Introduction 1.1 1.2 1.3 1.4 Introductory remarks Aims and layout Program structure References

2 One-dimensional nonlinear problems2.1 2.2 2.3 2.4 2.5 2.6 2.7 Introduction Basic numerical solution processes for nonlinear problems Systems governed by a quasi-harmonic equation Nonlinear elastic problems Elasto-plastic problems in one dimension Problems References

3 Structure of computer programs for the solution of nonlinear problemsIntroduction Input data subroutine, DATA Subroutine NONAL Subroutines for equation assembly and solution Output of results Subroutine INITAL Load increment subroutine, INCLOD The master or controlling segment Program for the solution of one-dimensional quasiharmonic problems by direct iteration Program for the solution of one-dimensional quasiharmonic problems by the Newton-Raphson methodv

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CONTENTS

3.1 1 3.12 3.13 3.14

Program for the solution of nonlinear elastic problems Program for the solution of elasto-plastic problems Problems References

4 Viscoplastic problems in one dimension Introduction Basic theory Numerical solution process Limiting time-step length Computational procedure Program structure Element stiffness subroutine STUNVP Subroutine INCVP for the evaluation of end of time-step quantities and equilibrium correction terms Convergence monitoring subroutine, CONVP Subroutine INCLOD The main, master or controlling segment Numerical examples' Problems References

5 Elasto-plastic Timoshenko beam analysis5.1 5.2 5.3 5.4 5.5 5.6 5.7 Introduction The basic assumptions of Timoshenko beam theory Finite element idealisation for linear elastic Timoshenko beams Elasto-plastic nonlayered Timoshenko beams Elasto-plastic layered Timoshenko beams Problems References

PART I1

6 Preliminary theory and standard subroutines for two-dimensional elasto-plastic applications6.1 6.2

6.3

Introduction Virtual work expression for various solid mechanics applications .Tsoparametric finite element representation

FINITE ELEMENTS I N PLASTICITY

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6.4 6.5 6.6 6.7

Standard subroutines for linear elastic finite element analysis Standard subroutines for elasto-plastic finite element analysis Problems References

7 Elasto-plastic problems in two dimensionsIntroduction The mathematical theory of plasticity Matrix formulation Alternative form of the yield criteria for numerical computation Basic expressions for two-dimensional problems Singular points on the yield surface Finite element expressions and program structure Additional program subroutines Numerical examples Problems References

8 Elasto-viscoplastic problems in two dimensionsIntroduction Theory of elasto-viscoplastic solids Selection of the time-step length Computational procedure Evaluation of matrix H Program structure Formulation of the tangential stiffness matrix Subroutine STEPVP for the evaluation of end of time step quantities and equilibrium correction terms Subroutine FLOWVP Subroutine STRESS Subroutine ZERO Subroutine STEADY for monitoring steady state convergence The main, master or controlling segment General comparison of implicit and explicit time integration schemes The overlay method for improved material response Numerical examples Problems References

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CONTENTS

9 Elasto-plastic Mindlin plate bending analysis

Introduction Equilibrium equations Discretisation Solution of nonlinear equations Software for the nonlayered approach Software for the layered approach Examples Problems References

PART 1 1 110 Explicit transient dynamic analysis

Introduction Dynamic equilibrium equations Modelling of nonlinearities Explicit time integration scheme Critical time step Program DYNPAK Examples Problems References1 Implicit-explicit transient dynamic analysis 1

11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8

Introduction Implicit time integration Implicit-explicit algorithm Evaluation of the tangential stiffness matrix Program MIXDYN Examples Problems References

12 Alternative formulations and further applications Introduction List of subroutines Alternative material models Further applications Equation solving techniques Other enhancements in elasto-plastic analysis Concluding remarks References

FINITE ELEMENTS IN PLASTICITY

ix

Appendix I

Instructions for preparing input data for one-dimensional problems

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Appendix I1 Instructions for preparing input data for plane, axisymmetric and plate bending problems Appendix I11 Instructions for preparing input data for dynamic transient problems Appendix IV Sample input data and line printer output for one- and two-dimensional applications Author Index Subject Index

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Preface

The purpose of this text is to present and demonstrate the use of finite element based methods for the solution of problems involving plasticity. As well as the conventional quasi-static incremental theory of plasticity, attention is given to the slow transient phenomenon of elasto-viscoplastic behaviour and also to dynamic transient problems. We make no pretence that the text provides a complete treatment of any of these topics but rather we see it as - an attempt to present numerical solution techniques, which have been well tried and tested, for selected important areas of application. In our earlier books on finite elements we have concentrated on linear applications. Here we attempt the much more daunting task of introducing, in detail, the use of finite elements for solving problems in which plasticity effects are present. To our knowledge it is the first such book. Our main idea is to present the theory and detailed algorithms in the form of modular routines written in FORTRAN which can be linked together to form 13 finite element plasticity programs. Writing this book has been in itself, rather like solving a nonlinear finite element problem. We have gone through many iterations and we hope that we have now converged to a reasonable 'solution'. As in many real engineering situations our convergence criterion has been influenced by a deadline. In our case the deadline was largely self-imposed as we have already been engaged on this project for more than three years. We do not believe our solution to be unique or in any sense optimal. We merely offer it to fill a gap in the existing literature. The text is arranged in three main parts. Part I is devoted to onedimensional problems. These relatively simple applications are possibly the most important in the book; since all the essential features of nonlinear finite element analysis are immediately recognisable without the distractions and complications that are present in general continuum problems. Part I1 deals with the two-dimensional applications of plane stress/strain and axisymmetric continua and plate bending problems. Finally in Part I11 we present some dynamic transient applications and briefly describe some further developments. All of the programs presented in this text have been specially written by the authors. In the development of the subroutines for the solution algorithms described, a conflict inevitably arose between computational efficiency

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PREFACE

and clarity of coding. Whatever sacrifices have been made have been biased towards satisfying the latter condition. However, we believe that the codes presented are both reasonably efficient and flexible and have potential usage in commercial as well as teaching and research environments. A total of 132 subroutines are presented which amount to more than 8,000 statements. The 13 assembled programs comprise approximately 20,000 statements. To aid readers wishing to implement the programs a magnetic tape of the computer codes together with the test input data listed in Appendix IV is available kom the publishers. Although every attempt has been made to verify the programs, no responsibility can be accepted for their performance in practice. A further feature of the book is that each chapter contains several exercises for further study. We are indebted to many people for their direct or indirect assistance in the preparation of this text. This preface would not be complete without an acknowledgment of this debt and a record of our gratitude to the following: To Professor 0.C. Zienkiewicz for his pioneering work and stimulating influence. To Professor G. C. Nayak whose work on numerical analysis of plasticity problems has significantly influenced the present text. To Dr. I. C. Cormeau whose thesis on viscoplasticity has been an invaluable source of information. To Professor K. J. Bathe for permission to use the profile equation solver included in Chapter 11. To N. Bicanic, D. K. Paul, H. H. Abdel Rahman and M. M. Huq for their generous assistance in the preparation of several chapters. To our colleagues and former research workers in the Department of Civil Engineering, University College of Swan

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