R. Szilard - Theories and Applications of Plate Analysis Classical, Numerical and Engineering Methods

R. Szilard - Theories and Applications of Plate Analysis Classical, Numerical and...

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Theories and Applications of Plate Analysis

Theories and Applications of Plate Analysis: Classical, Numerical and Engineering Methods. R. Szilard Copyright © 2004 John Wiley & Sons, Inc.

Theories and Applications of Plate Analysis Classical, Numerical and Engineering Methods

Rudolph Szilard, Dr.-Ing., P.E.

Professor Emeritus of Structural Mechanics University of Hawaii, United States Retired Chairman, Department of Structural Mechanics University of Dortmund, Germany

This book is printed on acid-free paper. Copyright 2004 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data

Szilard, Rudolph, 1921-

Theories and applications of plate analysis : classical, numerical and engineering methods / by Rudolph Szilard. p. cm.

Includes bibliographical references and index. ISBN 0-471-42989-9 (cloth) 1. Plates (Engineering) I. Title.

Printed in the United States of America. 10 987 654321

To the memory of my father,

Dipl.-Ing. Rudolph Seybold-Szilard, senior, who encouraged and inspired my career in structural mechanics

Contents

Preface xvii Symbols xxi

I Introduction 1

I Historical Background 10

PART I Plate Theories and Analytical Solutions of Static, Linear-Elastic Plate Problems 21

1 Elastic Plate Theories and Their Governing Differential Equations 23

1.1 Classical Small-Deflection Theory of Thin Plates*1 23 1.2 Plate Equation in Cartesian Coordinate System* 26 1.3 Boundary Conditions of Kirchhoff’s Plate Theory* 35 1.4 Differential Equation of Circular Plates* 42 1.5 Refined Theories for Moderately Thick Plates 45

1.6 Three-Dimensional Elasticity Equations for Thick Plates 53

1Asterisks (∗) indicate sections recommended for classroom use. vii viii Contents

2 Exact and Series Solutions of Governing Differential Equations 62

2.1 Rigorous Solution of Plate Equation 62

2.2 Solutions by Double Trigonometric Series (Navier’s Approach)* 69

2.3 Solutions by Single Trigonometric Series (Levy’s Method)* 75

2.4 Further Examples of Series Solutions 83 2.5 Extensions of Navier’s and Levy’s Methods 92 2.6 Method of Images 97 2.7 Plate Strips 9

2.8 Rigorous Solution of Circular Plates Subjected to Rotationally Symmetric Loading* 110

2.9 Solutions of Membrane Problems 116 2.10 Series Solutions of Moderately Thick Plates 120 2.1 Summary* 126 Problems* 127

3 Further Plate Problems and Their Classical Solutions 129

3.1 Plates on Elastic Foundation* 129 3.2 Plates with Variable Flexural Rigidity 139 3.3 Simultaneous Bending and Stretching 147 3.4 Plates of Various Geometrical Forms 150 3.5 Various Types of Circular Plates 156

3.6 Circular Plate Loaded by an Eccentric Concentrated Force 161

3.7 Plates with Edge Moments 165

Contents ix

3.8 Solutions Obtained by Means of Superposition 168 3.9 Continuous Plates 173 3.10 Summary 179 Problems 180

4 Energy and Variational Methods for Solution of Lateral Deflections 181

4.1 Introduction and Basic Concepts* 181 4.2 Ritz’s Method* 187 4.3 Galerkin’s Method and Its Variant by Vlasov* 196 4.4 Further Variational and Energy Procedures 212 4.5 Techniques to Improve Energy Solutions 226

4.6 Application of Energy Methods to Moderately Thick Plates 231

PART I Numerical Methods for Solution of Static, Linear-Elastic Plate Problems 237

5 Finite Difference Methods 247 5.1 Ordinary Finite Difference Methods* 247 5.2 Improved Finite Difference Methods 276

5.3 Finite Difference Analysis of Moderately Thick Plates 303

5.4 Advances in Finite Difference Methods 312 5.5 Summary and Conclusions* 314 Problems* 315

6 Gridwork and Framework Methods 317 6.1 Basic Concepts* 317 xC ontents

6.2 Equivalent Cross-Sectional Properties 320 6.3 Gridwork Cells and Their Stiffness Matrices* 328

6.4 Computational Procedures for Gridworks 336 6.4.1 Procedures Using Commercially Available

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