Stresses in Plates and Shells

336 Pages·1981·7.568 MB·english

by  Ugural

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STRESSES IN PLATES AND SHELLS

STRESSES IN PLATES
AND SHELLS
A. C. Ugural, Ph.D.
Profe."sol' and Chairmall
Ml!l'/wllic:al Eugim"'ring DeplIrtmem
Faideigh Dickillson University
McGraw-Hili Book Company
New York St. Louis San Francisco Auckl~md Bogota Hamburg
Johannesburg London Madrid Mexico Montreal N('w Delhi
Panama Paris Sito Paulo Singapoft: Sydney Tokyo Toronto
This book was set in Times Roman.
The editor was Frank J. Cerra;
the production supervisor was Donna Piligra.
Fairfield Graphics was printer and binder.
STRESSES IN PLATES AND SHELLS
Copyright © 1981 by McGraw~HiIl, Inc. All rights reserved.
Printed ill th!! Ullitt!d States of America. No part of this publication
may be reproduced, stored in a retrieval system, or transmitted, in any
form or by means, electronic, mechanical, photocopying, recording, or
otherwise, with om the prior written permission of the publisher.
1234567890FGFG8987654321
Library of Congress Cataloging in Publication Data
Ugura), A C
Stresses in plates and shells.
Includes bibliographical references and index.
I. Plates (Engineering) 2, Shells (Engineering)
3. Strains and stresses, l. Title.
TA660.P6U39 624.1'776 80-13927
ISBN 0-07-065730-0
CONTENTS
Preface
ix
List of Symbols
xiii
1 Elements of Plate Bending Theory 1
1.1 Introduction I
1.2 General Behavior of Plates 2
1.3 Strain~Cllrvature Relations 4
1.4 Stresses and Stress Resultants 6
1.5 Variation of Stress within a Plate 10
1.6 The Governing Equation for Denection of Plates 12
1.7 Boundary Conditions 14
1.8 Methods for Solution of Plate Deflections 16
1.9 Strain Energy Methods 20
1.10 Mechanical Properties and Behavior of Materials 22
Problems 25
2 Circular Plates
27
2.1 lot roduction 27
2.2 Basic Relations in Polar Coordinates 27
2.3 The AsixYlTImetricai Bending 31
2.4 Uniformly Loaded Circular Plates 32
2.5 Effect of Shear on the Plate Deflection 36
2.6 Circular Plates under a Concentrated Load at Its Center 37
2.7 Annular Plates with Simply Supported Outer Edges 40
2.8 Deflection and Stress by Superposition 41
2.9 The Ritz Method Applied to Circular Plates on Elastic Foundation 42
2.10 Asymmetrical Bending of Circular Plates 44
2.11 Deflection by the Reciprocity Theorem 46
2.12 Circular Plates of Variable Thickness under Nonuniform Load 47
Problems 55
v
\'i CO~TI':--; rs
3 Rectangular Plates 59
3.1 Introduction 59
3.2 Navicr's Solution for Simply Supported Rectangular Plates 59
13 Simply Supported Rectangular Plates under Varioll~ Loadings 62
3.4 Levy's Solution for Rectangular Plates 66
3.5 Levy's Method Applied to Nonuniformly Loaded
Rectangular Pl<ltt~ 74
16 Rectangular Plates under Distributed Edge moments 75
3.7 Method of Superposition Applied to Bending of
Rectangular Plates 77
3.8 The Strip Method 80
3.9 Simply Supported Continuous Rectangular Plates 83
3.10 Rectangular Plates Supported by Intermediate Columns 86
3.11 Rcclangular Plates Oil Elastic Foundation 89
3.12 The Ritl Method Applied to Bending of Rectangular Plates 90
Problems 94
4 Plates of Various Geometrical Forms 96
4.1 Introduction 96
4.2 Method of Images 96
4.3 Equilateral Triangular Plate with Simply Supported Edges 98
4.4 Elliptical Plates 100
4.5 Stress Concentration around Holes in a Plate 102
Problems 105
5 Plate Bending by Numerical Methods 106
5.1 Introduction \06
5.2 Finite Differences 106
5.3 Solution of the Finite Difference Equations III
5A Plates with Curved Boundaries 119
5.5 The Polar Mesh 122
5.6 The Triangular Mesh 123
5.7 Properties of a Finite Element 126
5.8 Formulation of the Finite Elcmt:nt Method 128
5.9 Triangular Finite Element 129
5.10 Rectangular Finite Element 133
Problems 137
6 Orthotropic Plates 140
6.1 Introduction 140
6.2 Basic Relationships 141
6.3 Determination of Rigidities 142
6.4 Rectangular Orthotropic Plates 144
6.5 Elliptic and Circular Orthotropic Plates 147
6.6 Multilayered Plates 148
6.7 The Finite Element Solution 150
Problems 152
CO:\TF0:TS \'jj
7 Plates under Combined Lateral and Direct Loads J 53
7.1 Introduction 153
7.2 Governing Equatio!l for the Deflection Surface 153
7.3 Compression of Plates. Buckling 156
7.4 Application of the Energy Method 159
7.5 The Finite Difference Solution 163
7.6 Plates with Small Initial Curvature 167
7.7 Bending to a Cylindrical Surface 16S
Problems 172
8 Large Deflections of Plates 174
8.1 Introduction 174
8.2 Plate Behavior When Dellcctions Are Large 174
8.3 Comparison of Small- and Large-Dellection Theories 175
SA The Governing Equations for Large Deflections 177
8.5 Deflections by the Ritz Method 179
8.6 The Finite Element Solution 181
Problems 184
9 Thermal Stresses III Plates 185
9.1 Introduction 185
9.2 Stress. Strain, and Displacement Relations 186
9.3 Stress Resultants 187
9.4 The Governing Differential Equations 188
9.5 Simply SupP0ricd Rectangular Plate Subject to an Arbitrary
Temperature Distribution 189
9.6 Simply Supported Rectangular Plate with Temperature
Distribution Varying over the Thickness 190
9.7 Analogy between Thermal and Isothermal Plate Problems 191
9.8 Axisymmctrically Heated Circular Plates 193
Problems 197
10 Membrane Stresses 111 Shells 198
10.1 General Behavior and Common Theories of Shells 198
10.:; Load Resistance Action of a Shell 199
10.3 Geometry of Shells of Revolution 202
10.4 Symmetrically Loaded Shells of Revolution 203
10.5 Some Typical Cases of Shells of Revolution 205
10.6 Axially Symmetric Deformal ion 215
10.7 Asymmetrically Loaded Shells of Revolution 217
10.8 Shells of Revolution under Wind Loading 218
10.9 Cylindrical Shells of General Shape 220
10.10 Breakdown of Elastic Action in Shells 224
Problems 226
11 Bending Stresses 111 Shells 231
11.1 Introduction 231
J 1.2 Shell Stress Resultants 231
,·iii ("():\TE~TS
11.3 Force. Moment, and Displacement Relations 2.D
11.4 Compound Stresses in Shells 235
11.5 Strain Energy in the Bending and Stretching of Shells 236
11.6 Axisymmetrically Loaded Circular Cylindrical Shells 236
11.7 A Typical Case of the Axisymm~trjcally Loaded Cylindrical Shells 239
11.~ Shells of Revolution under Axisymmctrical Loads 243
11.9 Governing Equations for Axisymmetrical Displacements 245
11.10 Compari:;on of Bending and Membrane Stresses 247
11.11 The Finite Elcmcnt Representations of Shells of General Shape 249
11.12 The Finite Element Solution ofAxisymmetrically Loaded Shells 249
Prohlcms 25:!
12 Applications to Pipes, Tanks, and Pressure Vessels 254
12.1 Tn troduction 254
12.2 Pipes Subjected to Edge Forces and Moments 255
12.3 Reinforced Cylinders 258
12.4 Cylindrical Tanks 260
12.5 Thermal Stresses in Cylinders 263
12.6 Thermal Stresses in Composite Cylinders 266
12.7 Discontinuity Stresses in Pressure Vessels 269
12.8 Cylindrical Vessel with Hemispherical Heads 270
12.9 Cylindrical Vessel with Ellipsoidal Heads 272
12.10 Cylindrical Vessel with Flat Heads 273
12.11 Design Formulas for Co~vcntional Pressure Vessels 274
Problems 275
13 Cylindrical Shells under General Loads 278
13.1 Introduction 278
13.2 Differential Equations of Equilibrium 279
13.3 Kinematic Relationships 280
13.4 The Governing Equations for Deflections 282
13.5 A Typical Case of Asymmetrical Loading 284
13.6 lnextensional Deformations 287
13.7 Symmetrical Buckling under Uniform Axial Pressure 290
13.8 Nonsymmetrical Buckling under Uniform Axial Compression 293
Problems 296
Appendixes
A Fourier Series Expansions 297
A.l Single Fourier Series 297
A.2 Hair-Range Expansions 299
A.3 Double Fourier Series 301
B Solution of Simultaneous Linear Equations 303
Rl Introduction 303
B.2 The Gauss Reduction Method 304
References 307
Answers to Selected Problems 309
313
Index