Table Of ContentENERGY AND FINITE
ELEMENT METHODS
IN STRUCTURAL
MECHANICS
ENERGY AND FINITE
ELEMENT METHODS
IN STRUCTURAL
MECHANICS
IRVING H. SHAMES
State University of New York at Buffalo
CLIVE L. DYM
University of Massachusetts, Amherst
CRC Press
Taylor 8i Francis Group
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
ENERGY AND FINITE ELEMENT METHODS IN STRUCTURAL MECHANICS
Copyright S> 1985 by Hemisphere Publishing Corporation. All rights reserved.
Printed in the United States of America. Except as permitted under the
United States Copyright Act of 1976, no part of this publication may be
reproduced or distributed in any form or by any means, or stored in a data
base or retrieval system, without the prior written permission of the publisher.
This book was set in Press Roman by Hemisphere Publishing Corporation. The editors
were Barbara A. Bodling an Mary Prescott; the designer was Sharon Martin DePass; the
production supervisor was Miriam Gonzalez; and the typesetters were Wayne Hutchins
and Sandra F. Wans. R. R. Donnelley and Sons Company was printer and binder.
Reprinted by Braun-Biumfield, Inc.
Library of Congress Cataloging in Publication Data
Shames, Irving Herman, date.
Energy and finite element methods in structural
mechanics.
Bibliography: p.
Includes index.
1. Structures, Theory of. 2. Finite element method.
3. Calculus of variations. 1. Dym, Clive L.
11. Title.
TA645.S4794 1985 624.1’7 84-22412
ISBN 0-89116-505-3
CONTENTS
Preface xiii
Outline, Course I (Senior-Graduate Level), Finite Elements xvii
Outline, Course II (Senior-Graduate Level), Structural Mechanics XVUl
FOUNDATIONS OF SOLID MECHANICS
AND VARIATIONAL METHODS I
1 THEORY OF LINEAR ELASTICITY 3
1.1 Introduction 3
Part A: Stress 4
1.2 Force Distributions 4
1.3 Stress S
1.4 Equations of Motion 9
1.5 Transformation Equations for Stress 11
1.6 Principal Stresses 13
Part B: Strain 20
1.7 Strain Components 20
1.8 Physical Interpretation of Strain Terms 25
1.9 The Rotation Tensor 29
1.10 Transformation Equations for Strain 32
1.11 Compatibility Equations 34
Part C: General Considerations 37
1.12 Energy Considerations 37
1.13 Hooke’s Law 43
1.14 Boundary-Value Problems for Linear Elasticity 46
1.15 St. Venant’s Principle 48
•*1.16 Uniqueness 49
•indicates that section may be bypassed,
••indicates that section is in fine print.
vi Contenti
Part D: Plane Stress 51
1.17 Introduction SI
1.18 Equations for Plane Stress 51
*•1.19 Problem of the Cantilever Beam 55
1.20 Closure 58
References 58
Reading 59
Problems 59
2 INTRODUCTION TO THE CALCULUS OF VARIATIONS 66
2.1 Introduction 66
2.2 Examples of Simple Functionals 68
2.3 The First Variation 71
2.4 The Delta Operator 75
2.5 First Integrals of the Euler-Lagrange Equation 80
2.6 First Variation with Several Dependent Variables 84
2.7 The Isoperimetric Problem 87
2.8 Function Constraints 92
2.9 A Note on Boundary Conditions 95
2.10 Functionals Involving Higher-Order Derivatives 97
2.11 A Further Extension 102
2.12 Closure 103
References 104
Reading 105
Problems 105
3 VARIATIONAL PRINCIPLES OF ELASTICITY 111
3.1 Introduction 111
Part A: Key Variational Principles 112
3.2 Virtual Work 112
3.3 The Method of Total Potential Energy 120
3.4 Complementary Virtual Work 129
3.5 Principle of Total Complementary Energy 133
•3.6 Stationary Principles; Reissner’s Principle 137
Part B: The Castlgliano Theorems and Structural Mechanics 139
3.7 Preliminary Remarks 139
3.8 The First Castigliano Theorem 140
3.9 The Second Castigliano Theorem 147
3.10 Summary Contents for Parts A and B 151
Part C: Quadratic Functionals 151
3.11 Self-Adjoint and Positive Definite Operators 151
3.12 Quadratic Functionals with Homogeneous Boundary Conditions 158
3.13 Variational Process with Nonhomogeneous Boundary Conditions 162
3.14 The Dirichlet and von Neumann Boundary-Value Problems 164
Part D: Approximate Methods 167
3.15 Introductory Comment 167
Contents vii
3.16 The Ritz Method 167
3.17 Galerkin’s Method 172
3.18 Qosure 174
References 175
Reading 175
Problems 175
STRUCTURAL MECHANICS 183
4 BEAMS 185
4.1 Introduction 185
4.2 Technical Theory of Beams 185
4.3 Deflection Equations for the Technical Theory of Beams 189
4.4 Some Justifications for the Technical Theory of Beams 194
4.5 Timoshenko Beam Theory 197
4.6 Comments on the Ritz Method 204
4.7 The Ritz Method for a Series Solution 208
4.8 Use of the Reissner Principle 214
4.9 Additional Problems in Bending of Beams 216
4.10 Closure 222
Reference 223
Reading 223
Problems 223
5 TORSION 229
5.1 Introduction 229
5.2 Total Potential Energy; Equation for Torsion 230
5.3 The Total Complementary Energy Functional 233
5.4 Approximate Solutions for Linear Elastic Behavior
via the Ritz Method 238
5.5 The Method of Trefftz; Upper Bound for Torsional Rigidity 240
5.6 The Method of Kantorovich 246
5.7 Closure 254
References 254
Reading 255
Problems 255
6 CLASSICAL THEORY OF PLATES 257
6.1 Introduction 257
6.2 Kinematics of the Deformation of Plates 257
6.3 Stress Resultant Intensity Functions
and the Equations of Equilibrium 261
6.4 Minimum Total Potential Energy Approach 266
6.5 Principle of Virtual Work: Rectangular Plates 274
6.6 A Note on the Validity of Classical Plate Theory 278
6.7 Examples from Classical Plate Theory; Simply Supported
Rectangular Plates 283
*6.8 Rectangular Plates: Lévy’s Method 287
6.9 The Damped Rectangular Plate; Approximate Solutions 292
viti Contents
6.10 Elliptic and Circular Plates 301
•6.11 Improved Theory-Axisymmetric Circular Plates 305
6.12 Closure 314
References 314
Reading 315
Problems 315
7 DYNAMICS OF BEAMS AND PLATES 323
7.1 Introduction 323
7.2 Hamilton’s Principle 323
Part A: Beams 326
7.3 Equations of Motion for Vibrating Beams 326
7.4 Free Vibrations of a Simply Supported Beam 329
7.5 Rayleigh’s Method for Beams 334
7.6 Rayleigh-Ritz Method for Beams 338
•7.7 The Timoshenko Beam 342
Part B: Plates 351
7.8 Equations of Motion for Plates 351
7.9 Free Vibrations of a Simply Supported Plate 353
7.10 Rayleigh's Method for Plates 356
7.11 Rayleigh-Ritz Method for Plates 358
•7.12 Transverse Shear and Rotatory Inertia-Reissner-Mindlin
Plate Theory 364
Part C: General Considerations 375
7.13 The Eigenfunction-Eigenvalue Problem Restated 375
7.14 The Rayleigh Quotient in Terms of Operators 377
7.15 Stationary Values of the Rayleigh Quotient 378
7.16 Rayleigh-Ritz Method Re-examined 381
7.17 Maximum-Minimum Principle 384
7.18 JustiHcation of the Estimation from above Assertion
of the Rayleigh-Ritz Method 386
7.19 Closure 387
Reading 388
Problems 388
8 ELASTIC STABILITY 392
8.1 Introduction 392
Part A: Stability of Rigid Bodies 392
8.2 Stability 392
Part B: Some Results from Nonlinear Elasticity Theory 392
8.3 Comment 398
8.4 Large Bending Deflection of Columns 398
8.5 The von Karman Theory of Plates 401
Part C: Elastic Stability of Columns 410
8.6 The Euler Load; Equilibrium Method 410
8.7 Energy Methods 415
8.8 Imperfection Analysis 418
Contents ix
8.9 The Kinetic Method 421
8.10 General Remarks 423
8.11 The Elástica 423
**8.12 An Intermediate Theory 428
Part D: Elastic Stability of Plates 429
8.13 The Buckling Equation for Rectangular Plates 429
8.14 The Equilibrium Method—An Example 433
8.15 The Rectangular Plate via the Energy Method 436
8.16 The Circular Plate via the Energy Method 437
Part E: Approximation Methods 441
8.17 Comment 441
8.18 The Rayleigh Quotient for Beam-Columns 441
8.19 The Rayleigh and the Rayleigh-Ritz Methods Applied to Columns 442
8.20 Rayleigh Quotient for Rectangular Plates 444
8.21 The Kantorovich Method 446
8.22 Closure 449
References 450
Reading 450
Problems 450
FINITE ELEMENTS 455
INTRODUCTION TO FINITE ELEMENTS 457
Part A: JYusses 457
9.1 Finite Elements and Trusses 457
9.2 Stiffness Matrix for an Element 461
Part B: More General Preliminary Considerations 474
9.3 Basic Considerations for Finite Elements 474
9.4 General Theory for the Displacement Method 477
9.5 Local and Global Coordinate Systems 481
9.6 Closure 483
Problems 483
10 FINITE ELEMENTS FOR BEAMS 487
10.1 Introduction 487
10.2 Definitions for Beam Element Deformation and Load 487
10.3 Key Matrices for Beam Element and System 491
10.4 Procedural Details for Global Equations 496
10.5 Closure 505
Problems 506
11 ELEMENTS AND INTERPOLATION FUNCTIONS 510
11.1 Introduction 510
11.2 General Element Considerations 510
11.3 Convergence Requirements for Interpolation Functions 513
Part A: One-Dimensional Elements 514
11.4 Interpolation Functions for One-Dimensional Elements 514
