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Non-linear Modeling and
Analysis of Solids and
Structures
Steen Krenk
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,
São Paulo, Delhi, Dubai, Tokyo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
Information on this title: 0521830546
© Cambridge University Press 2009
This publication is in copyright. Subject to statutory exception and to the
provision of relevant collective licensing agreements, no reproduction of any part
may take place without the written permission of Cambridge University Press.
First published in print format 2009
ISBN-13 978-0-511-60413-3 eBook (EBL)
ISBN-13 978-0-521-83054-6 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy
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accurate or appropriate.
To Jette
Contents
Preface page ix
1 Introduction 1
A simple non-linear problem 2
Equilibrium 3
Virtual work and potential energy 6
Simple non-linear solution methods 7
Explicit incremental method 8
Newton–Raphson method 9
Modified Newton–Raphson method 13
Summary and outlook 14
Exercises 15
2 Non-linear bar elements 17
Deformation and strain 18
Equilibrium and virtual work 20
Tangent stiffness matrix 24
Use of shape functions 26
Assembly of global stiffness and forces 31
Total or updated Lagrangian formulation 36
Summing up the principles 39
Exercises 43
3 Finite rotations 47
The rotation tensor 49
Rotation of a vector into a specified direction 53
The increment of the rotation variation 55
Parameter representation of an incremental rotation 60
Quaternion parameter repres