Static and Dynamic Analyses of Plates and Shells Theory, Software and Applications

Hou-Cheng Huang

Static and Dynamic Analyses of Plates and Shells Theory, Software and Applications

With 77 Figures

Springer-Verlag london Berlin Heidelberg New York Paris Tokyo

Hou-Cheng Huang, Ph.D Department of Civil Engineering, University College of Swansea, Swansea SA28PP, Wales, UK

Front cover: Diagram of a clamped hyperbolic shell.

ISBN -13: 978-1-4471-1671-4 e-ISBN-13: 978-1-4471-1669-1 DOl: 10.1007/978-1-4471-1669-1

British Library Cataloguing in Publication Data Huang, Hou-Cheng, 1947-Static and dynamic analyses of plates and shells: theory, software and applications 1. Structures. Plates & shells. Stress analyses I. Title 624.1'776 ISBN -13:978-1-4471-1671-4

Library of Congress Cataloging-in-Publication Data Huang, Hou-Cheng, 1947-Static and dynamic analyses of plates and shells: theory, software, and applica­tions/Hou-Cheng Huang. p. cm. Bibliography: p. Includes index. ISBN-13:978-1-4471-1671-4 (U.S.) 1. Plates (Engineering) 2. Shells (Engineering) I. Title. TA660.PGH83 1988 88-31122 624.1'776-dc19

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted llnder the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1989 Softcover reprint of the hardcover 1st edition 1989

The use of registered names, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use.

2128/3916-543210 Printed on acid-free paper

To my father

Preface

Over the past decade or so much has been written on the various attempts to produce efficient, accurate and reliable Mindlin plate finite elements. In the late sixties, a degenerated, Mindlin-type, curved shell element was developed and subsequently many improvements in such elements have been made. Reliability and efficiency in use has always been a major objective.

Degenerated shell elements have enjoyed widespread popularity despite certain potential defects, including shear and membrane lock­ing behaviour and spurious mechanisms. After introducing the basic foundations of Mindlin-type elements, this book describes these defects and also gives the reasons for their occurrence. Furthermore, the author proposes an approach to overcome these defects.

A series of linear benchmark tests are proposed to illustrate the performance of the assumed strain element formulations. The formula­tions and applications for material non-linearity are also presented. Both isotropic and anisotropic material models are included together with the results for both static and transient dynamic analyses. Two associated programs are fully documented and provided on floppy discs with test examples. Source codes for the two associated programs are provided: one is for static analysis and the other for dynamic analysis, and the programs can be compiled and run on either a mini or mainframe coniputer via a terminal.

The author hopes that this book may provide further impetus in the important research area of plate and shell element technology. Technical queries should be sent in writing to the Editorial Depart­ment, Springer-Verlag London Ltd, Springer House, 8 Alexandra Road, Wimbledon, London SW19 7JZ; these will be forwarded to the author where necessary.

Swansea, UK 1988

Hou-Cheng Huang

Acknowledgement I would like to thank Dr E. Hinton and Dr R. D. Wood for their help in the preparation of this book.

Software Use Agreement

By accepting the software accompanying this book you accept also the following condi­tions:

1 The software is for your use only on a microcomputer located at your own premises.

2 You will not disassemble or modify the software or prepare works which derive from the software.

3 You will not copy, beyond the making of one back-up copy, nor permit others to copy the program or any part of it.

4 The software is supplied without any war­ranties of any kind, including any implied warranties of merchantability and fitness for a particular purpose or use.

Contents

Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Changing Trends in Shell Analysis. . . . . . . . . . . . . . . . . . . . . 1 1.2 Reliability of Finite Element Analysis . . . . . . . . . . . . . . . . . . 1 1.3 Shell Element Formulations. . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 Objective and Layout ................................ 3

2 Degenerations of Three-Dimensional Theory ............. 7 2.1 Introduction........................................ 7 2.2 Three-Dimensional Theory..... .. . ......... ... ..... . .. 7 2.3 Plate Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Thin Plate Theory (IGrchhoff hypothesis) . . . . . . . . . . 10 2.3.2 Thick Plate Theory (Mindlin-Reissner Hypothesis) 15

2.4 Three-Dimensional Degenerated Curved Shell........ . .. 19 2.4.1 Definition of Strains. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4.2 Definition of Stresses ........................... 21 2.4.3 The Total Potential Energy ...................... 24

3 Defects of Mindlin Plate and Degenerated Shell Elements 25 3.1 Introduction........................................ 25 3.2 Formulation of Mindlin Plate Bending Elements. . . . . . . . . 25

3.2.1 Formulation of Mindlin Isoparametric Bending Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.2 Finite Element Representations of Boundary Conditions .................................... 30

3.3 Formulation of Degenerated Shell Elements. . . . . . . . . . . . . 30 3.3.1 Coordinate Systems. ....... ......... .... ........ 31 3.3.2 Element Geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3.3 Displacement Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4 Defects of Mindlin Plate and Degenerated Shell Elements 36 3.4.1 Locking Phenomena............................ 36 3.4.2 Reduced and Selective Integration. . . . . . . . . . . . . . . . 38 3.4.3 Alternative Methods Avoiding the Locking Problems 41

4 Assumed Strain Finite Element Plate Formulations. . . . . . . . 45 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

x Contents

4.2 Essence of Shear Locking. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.3 Reinterpretation of Selectively Integrated Elements ...... 47 4.4 Elimination of Shear Locking . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.4.1 Assumed Transverse Shear Strain Fields . . . . . . . . . . . 48 4.4.2 Location of the Sampling Points for the Shear Strains 50 4.4.3 The Evaluation of the Stiffness Matrix ............ 55

S Linear Benchmark Tests for Plate Elements. . . . . . . . . . . . . . 57 5.1 Introduction........................................ 57 5.2 Eigen-Analyses of the Stiffness Matrices. . . . . . . . . . . . . . . . 58 5.3 Patch Tests ......................................... 58

5.3.1 Patch Tests for Bending. . . . . . . . . . . . . . . . . . . . . . . . . 58 5.3.2 Patch Tests for Twisting. . . . . . . . . . . . . . . . . . . . . . . . . 60 5.3.3 Patch Tests for Shear ........................... 60

5.4 Locking Tests ........................ , . . . . . . . . . . . . . . 60 5.5 Long Cantilever . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.6 Convergence Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.6.1 Simply Supported Square Plates under Uniform Load 63 5.6.2 Clamped Square Plate under Uniform Load. . . . . . . . 63 5.6.3 Clamped Circular Plates ........ '. . . . . . . . . . . . . . . . . 64

5.7 Skew Plates ......................... , . . . .... ... ... .. 66 5.7.1 Razzaque's 60° Skew Plate. . . . . . . . . . . . . . . . . . . . . . . 66 5.7.2 Morley's 30° Skew Plate. .... ... . . . ..... ... . .. .. . 69

5.8 Stress Resultant Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6 Assumed Strain Finite Element Shell Formulations. . . . . . . . 79 6.1 Introduction........................................ 79 6.2 Shear Locking, Membrane Locking and Selective Integration 80

6.2.1 Shear Locking in Shells ......................... 80 6.2.2 Essence of Membrane Locking . . . . . . . . . . . . . . . . . . . 80

6.3 Elimination of Shear Locking. . . .. . . ... . . . . .. . .. . .. . .. 81 6.4 Elimination of Membrane Locking. . . . . . . . . . . . . . . . . . . . . 82

6.4.1 Orthogonal Curvilinear Coordinate System (r, s, t or ri) 82 6.4.2 Assumed Membrane Strains. . . . . . . . . . . . . . . . . . . . . . 82

6.5 Assumed Strain Degenerated Shell Elements ... . . . . . . . . . 84 6.5.1 Location of Sampling Points for the Membrane Strains 84 6.5.2 The Evaluation of the Stiffness Matrix ............ 90

6.6 Discussion.......................................... 90

7 Linear Benchmark Tests for Shell Elements ......... , .. .. 93 7.1 Introduction........................................ 93 7.2 Eigen-Analysis of the Stiffness Matrices ................ 94 7.3 Patch Tests ......................................... 94

7.3.1 Bending, Twisting and Shear Patch Tests ..... , .. , . 94 7.3.2 Plane Stress Patch Tests. . . . . . . . . . . . . . . . . . . . . . . . . 94

7.4 Locking Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 7.4.1 Shear Locking Tests ............................ 94 7.4.2 Membrane Locking Tests. .. . .. . .. . . . .... . .. . .. . . 95

7.5 Conclusions......................................... 102

Contents xi

8 Formulations and Applications for Elasto-Plastic Shell Analyses ............................................... 103 8.1 Introduction........................................ 103 8.2 Laminated Plate and Shell ModeL.. . . . . . ... ... . .. ... .. 103 8.3 Shear Correction Factors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 8.4 The Anisotropic Yield Criterion. . . . . . . . . . . . . . . . . . . . . . . 106

8.4.1 Generalised Huber-Mises Yield Criterion.......... 106 8.4.2 Determination of Anisotropic Parameters.. . ....... 107 8.4.3 Relation between Elasto-Plastic Stresses and Strains 108 8.4.4 Tangent Stiffness Matrix. . . . . . . . . . . . . . . . . . . . . . . .. 109

8.5 Numerical Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 8.5.1 Clamped Square Plate........................... 110 8.5.2 Clamped Quadratic Shell. . . . . . . . . . . . . . . . . . . . . . .. 111

9 Formulations and Applications for Elasto-Plastic Dynamic Shell Analyses .......................................... 115 9.1 Introduction ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 115 9.2 Dynamic Equilibrium Equations. . . . . . . . . . . . . . . . . . . . . .. 115 9.3 Modelling of Mass Matrix....... . . . . .... . .. . ... ...... 117

9.3.1 Consistent Mass Matrix.... . . . . . .... .... ..... ... 117 9.3.2 Lumped Mass Matrix. .. .. .. . . . . . . ... . .. ... .. . .. 117

9.4 Newmark's Time Stepping Scheme. . . . . . . . . . . . . . . . . . . .. 118 9.5 Numerical Examples .. '. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 119

9.5.1 Rectangular Plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 119 9.5.2 Simply Supported Square Plate..... . ... . .. . .. .... 119 9.5.3 Clamped Circular Plate ......................... 120 9.5.4 Spherical Shell Caps. . . . . . . . . . . . . . . . . . . . . . . . . . .. 122

Appendix 1 Software Description for Elasto-Plastic Static Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 AU Introduction....................................... 131 A1.2 Glossary of Variable Names...... . . . .... ... . ... .. ... 131

A1.2.1 Main Arrays. . .. .. .. .. .. . . . . . . ... ... ... .. ... 131 A1.2.2 Main Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 132

A1.3 Program Overview ................................. 133 A1.4 File Handling... . .. . . .. .. ...... . . . . . ... . .. . ... ..... 137

A1.4.1 Subroutine INPUT. . . . . . . . . . . . . . . . . . . . . . . . .. 137 A1.4.2 User Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . .. 139 A1.4.3 Files....................................... 143

A1.5 Documented Example - Clamped Quadratic Shell. . . . .. 143 A1.5.1 Input Data File ............................. 144 A1.5.2 Output Data File. . . . . . . . . . . . . . . . . . . . . . . . . . .. 146

A1.6 Stiffness Evaluation Module. . . . . . . . . . . . . . . . . . . . . . . .. 150 A1.6.1 Subroutine for Evaluation of Stiffness Matrix ... 150 A1.6.2 Evaluation of Strain-Displacement Matrix. . . . .. 152 A1.6.3 Subroutine BSAMP. . . . . . . . . . . . . . . . . . . . . . . . .. 155 A1.6.4 Subroutine FUNC . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 A1.6.5 Subroutine FRAME ......................... 158 A1.6.6 Subroutine SFR1 . ... .. . . . . . . . . .. . ... . .. .. . .. 160 A1.6.7 Subroutine SFR3 . . . . . . . . . . . . . . . . . . . . . . . . . . .. 160

xii Contents

A1.6.8 Subroutine TBMAT. . . . . . . . . . . . . . . . . . . . . . . . .. 161 A1.6.9 Subroutine XDIC ........................... 162

Appendix 2 Software Descriptions for Elasto-Plastic Transient Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 163 A2.1 Introduction....................................... 163 A2.2 Glossary of Variable Names. . . . . . . . . . . . . . . . . . . . . . . .. 163

A2.2.1 Main Arrays. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. 163 A2.2.2 Main Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 164

A2.3 Program Overview ................................. 165 A2.4 File Handling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 171

A2.4.1 Subroutine INPUTD... ....... . . . ..... ..... .. 171 A2.4.2 Subroutine INTIME . . . . . . . . . . . . . . . . . . . . . . . .. 173 A2.4.3 User Instructions. . . . . . . . . . . . . . . . . . . . . . . . . . .. 175 A2.4.4 Files....................................... 181

A2.5 Documented Example - Thin Spherical Cap. . . . . . . . . .. 182 A2.5.1 Input Data File. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 182 A2.5.2 Output Data File. . . . . . . . . . . . . . . . . . . . . . . . . . .. 185

Author Index ........................................... 191

Subject Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 193

Notation

E J IJI K Kf

Kb Km Ks Mx, My, Mxy N Nx, Ny, Nxy Qx, Qy Uj(u, v, w)

W

strain-displacement matrix in-plane strain-displacement matrix bending strain-displacement matrix membrane strain-displacement matrix transverse shear strain-displacement matrix assumed in-plane strain-displacement matrix assumed transverse shear strain-displacement matrix elasticity matrix membrane-strain tensor in the orthogonal curvilinear coordinate system assumed membrane strain tensor in the orthogonal curvilinear coordinate system Young's modulus Jacobian matrix determinant of the Jacobian matrix stiffness matrix in-plane stiffness matrix bending stiffness matrix membrane stiffness matrix transverse shear stiffness matrix generalized stress components (moments) shape function generalized stress components (in-plane forces) generalized stress components (shear forces) displacement components potential energy of loads

transverse shear strain components in the Cartesian coordinate system transverse shear strain components in the natural coordinate system

xiv

E

Ej

Eb

Em

Es

Es

OXi' Oyi II

7T

(J'

Notation

assumed transverse shear strain components in the natural coordinate system linear strain tensor in-plane strain tensor bending strain tensor membrane strain tensor transverse shear strain tensor assumed transverse shear strain tensor rotations Poisson's ratio total potential energy stress tensor