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

202 Pages·1989·7.78 MB·English

by  Hou-Cheng Huang Ph.D (auth.)

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 applica
tions/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
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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 lock
ing 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 formula
tions 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 Depart
ment, Springer-Verlag London Ltd, Springer House, 8 Alexandra
Road, Wimbledon, London SW19 7JZ; these will be forwarded to the
author where necessary.
Swansea, UK Hou-Cheng Huang
1988
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 condi
tions:
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 war
ranties 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
Contents
x
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