Clive L. Dym · Irving H. Shames

Solid MechanicsA Variational Approach

Augmented Edition

Solid Mechanics

.

Clive L. Dym • Irving H. Shames

SolidMechanics

A Variational Approach

Augmented Edition

Clive L. DymDepartment of EngineeringHarvey Mudd CollegeClaremont, CAUSA

ISBN 978-1-4614-6033-6 ISBN 978-1-4614-6034-3 (eBook)DOI 10.1007/978-1-4614-6034-3Springer New York Heidelberg Dordrecht London

Library of Congress Control Number: 2013932844

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Foreword to the Augmented Edition

The reappearance of this book represents the culmination of a long-standing dream

of mine. The original was published in 1973. A successor volume, built on much of

the foundation of the first book to incorporate an extensive text of the finite element

method, first appeared in 1985. Notwithstanding the passage of time since these

volumes, or the advances in finite element analysis, both theory and application, the

original book on variational methods in mechanics continues to attract attention

from points flung far around the globe. To this day, I get questions about points in

the book and inquiries about the availability of solutions to its vast array of

problems. Consequently, I have long hoped to see our original work reappear.

Happily, due to the foresight, encouragement and effort of my editor, Michael

Luby, Springer agreed, subject to some modification. I proposed adding

two chapters intended to briefly introduce finite element analysis, set very clearly

in its variational context. While accounting for some of the modern theoretical

developments, the two chapters were intended to follow the style and level of the

original Dym and Shames text: state the general principles, and follow them with

detailed illustrative examples. Needless to say, the writing style is now clearly my

own, and I alone am responsible for errors and omissions.

I thank my good friend, colleague and mentor, the late Irv Shames, for encour-

aging me to write my very first book: I told him I would not do it, starting just a year

or two out of graduate school, without his being co-author. I also thank Irv and

Sheila’s children, Bruce and Lisa, for their approval to proceed with this project.

I have also benefitted greatly from suggestions and support from my Harvey Mudd

colleague, Harry E. Williams, while I was writing the (new) Chaps. 10 and 11. And

finally, many thanks to Michael Luby for his perseverance, and to his very able

assistant, Merry Stuber, for helping to keep me on track.

Claremont, CA Clive L. Dym

v

.

Foreword to the First Edition

In this text we shall employ a number of mathematical techniques and methods. We

shall introduce these techniques and methods at places where it is felt maximum

understanding can be achieved. The physical aspects of the concepts will be

stressed where possible. And, although we shall present most of this material

with the purpose of immediate use in solid and structural mechanics, we shall

also “open-end” the discussions where feasible to other fields of study. Such

discussions are of necessity more mathematical in nature and are generally

asterisked, indicating that they can be deleted with no loss in continuity.

We shall employ Cartesian tensor concepts in parts of this text and the

accompanying notation will be used where it is most meaningful. (It will accord-

ingly not be used exclusively.) For those readers not familiar with Cartesian tensors

(or for those wishing a review), we have presented a self-contained treatment of this

subject in Appendix I. This treatment (it includes exercises as well) will more than

suffice the needs of this text.

Clive L. Dym

Irving H. Shames (deceased)

vii

.

Preface to the First Edition

This text is written for senior and first year graduate students wishing to study

variational methods as applied to solid mechanics. These methods are extremely

useful as means of properly formulating boundary-value problems and also as a

means of finding approximate analytical solutions to these boundary-value problems.

We have endeavored to make this text self-contained. Accordingly, virtually all

the solid and structural mechanics needed in the text is developed as part of the

treatment. Furthermore the variational considerations have been set forth in a rather

general manner so that the reader should be able to apply them in fields other than

solid mechanics.

A brief description of the contents of the text will now be given. For those readers

not familiar with cartesian tensor notation or for those wishing a review, we have

presented in Appendix I a development of this notation plus certain ancillary

mathematical considerations. In Chap. 1, we present a self-contained treatment of

the theory of linear elasticity that will serve our needs in this area throughout the text.

Next, in Chap. 2 comes a study of the calculus of variations wherein we consider the

first variation of functionals under a variety of circumstances. The delta operator is

carefully formulated in this discussion. The results of the first two chapters are then

brought together in Chap. 3 where the key variational principles of elasticity are

undertaken. Thus we consider work and energy principles, including the Reissner

principle, as well as the Castigliano theorems. Serving to illustrate these various

theorems and principles, there is set forth a series of truss problems. These truss

problems serve simultaneously as the beginning of our efforts in structural mechan-

ics. In developing the aforementioned energy principles and theorems, we

proceeded by presenting a functional first and then, by considering a null first

variation, arrived at the desired equations. We next reverse this process by

presenting certain classes of differential equations and then finding the appropriate

functional. This sets the stage for examining the Ritz and the closely relatedGalerkin

approximation methods. In later chapters we shall present other approximation

techniques. In Chap. 4 we continue the study of structural mechanics by applying

the principles and theorems of Chap. 3 to beams, frames and rings. These problems

are characterized by the fact that they involve only one independent variable—they

ix

may thus be called one-dimensional structural problems. In Chap. 5 we consider the

elastic and inelastic torsion of shafts. Use is made of earlier methods in the text but

now, because there are two independent variables, new approximation techniques

are presented—namely themethods of Trefftz and Kantorovich. Chapter 6 dwells on

the classical theory of plates. We set forth the equations of equilibrium and the

appropriate boundary conditions, via variational methods, and then we find approx-

imate solutions to various problems via the techniques presented earlier. In Chap. 7,

free vibrations of beams and plates are covered. With time now as a variable we first

present Hamilton’s Principle and then go on to formulate the equations of motion of

beams and plates. The methods of Ritz and Rayleigh–Ritz are then employed for

generating approximate natural frequencies of free vibration as well as mode shapes.

To put these methods on firm foundation we then examine the eigenvalue–eigen-

function problem in general, and this leads to the Rayleigh quotient that will be used

in the study of stability in Chap. 9. Also developed is the maximum–minimum

principle of the calculus of variations—thus providing a continuation of the varia-

tional calculus in Chap. 2. Up to this point only small deformation has been

considered in our undertakings (the non-linear considerations thus far have been in

the constitutive laws), and so in Chap. 9 we consider large deformation theory. In

particular, the principle of virtual work and the method of total potential energy are

presented. The climax of the chapter is the presentation of the von Karman plate

theory. The closing chapter considers the elastic stability of columns and plates.

Various approaches are set forth including the criterion of Trefftz and the asymptotic

postbuckling theory of Koiter.

Note that we have not included finite element applications despite the impor-

tance of variational methods in this field. We have done this because the finite

element approach has become so broad in its approach that a short treatment would

not be worthwhile. We recommend accordingly that this text serve as a precursor to

a study in depth of the finite element approach.

At the end of each chapter there is a series of problems that either call for

applications of the theory in the chapter or augment the material in the chapter.

Particularly long or difficult problems are starred.

We wish to thank Prof. T. A. Cruse of Carnegie-Mellon University for reading

the entire manuscript and giving us a number of helpful comments. Also our thanks

go to Prof. J. T. Oden of the University of Alabama for his valuable suggestions.

Dr. A. Baker and Dr. A. Frankus helped out on calculations and we thank them for

their valuable assistance. One of the authors (C.L.D.) wishes to pay tribute to

three former teachers—Prof. J. Kempner of Brooklyn Polytechnic Institute and

Profs. N. J. Hoff and J. Mayers of Stanford University—who inspired his interest

in variational methods. The other author (I.H.S.) wishes to thank his colleague

Prof. R. Kaul of State University of New York at Buffalo for many stimulating and

useful conversations concerning several topics in this text. Finally we both wish to

thank Mrs. Gail Huck for her expert typing.

Clive L. Dym

Irving H. Shames (deceased)

x Preface to the First Edition

Contents

1 Theory of Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Part A STRESS

1.2 Force Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Transformation Equations for Stress . . . . . . . . . . . . . . . . . . . . 9

1.6 Principal Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Part B STRAIN

1.7 Strain Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.8 Physical Interpretations of Strain Terms . . . . . . . . . . . . . . . . . 25

1.9 The Rotation Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.10 Transformation Equations for Strain . . . . . . . . . . . . . . . . . . . . 31

1.11 Compatibility Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Part C GENERAL CONSIDERATIONS

1.12 Energy Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.13 Hooke’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1.14 Boundary-value Problems for Linear Elasticity . . . . . . . . . . . . 48

1.15 St. Venant’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

1.16 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Part D PLANE STRESS

1.17 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

1.18 Equations for Plane Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

1.19 Problem of the Cantilever Beam . . . . . . . . . . . . . . . . . . . . . . . 57

1.20 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2 Introduction to the Calculus of Variations . . . . . . . . . . . . . . . . . . . 71

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.2 Examples of Simple Functionals . . . . . . . . . . . . . . . . . . . . . . . 73

2.3 The First Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

xi

2.4 The Delta Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

2.5 First Integrals of the Euler–Lagrange Equation . . . . . . . . . . . . 85

2.6 First Variation with Several Dependent Variables . . . . . . . . . . 89

2.7 The Isoperimetric Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

2.8 Functional Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

2.9 A Note on Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 100

2.10 Functionals Involving Higher-Order Derivatives . . . . . . . . . . . 102

2.11 A Further Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

2.12 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3 Variational Principles of Elasticity . . . . . . . . . . . . . . . . . . . . . . . . 117

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Part A KEY VARIATIONAL PRINCIPLES

3.2 Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

3.3 The Method of Total Potential Energy . . . . . . . . . . . . . . . . . . 124

3.4 Complementary Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . 132

3.5 Principle of Total Complementary Energy . . . . . . . . . . . . . . . 136

3.6 Stationary Principles; Reissner’s Principle . . . . . . . . . . . . . . . 140

Part B THE CASTIGLIANO THEOREMS AND STRUCTURAL

MECHANICS

3.7 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

3.8 The First Castigliano Theorem . . . . . . . . . . . . . . . . . . . . . . . . 143

3.9 The Second Castigliano Theorem . . . . . . . . . . . . . . . . . . . . . . 154

3.10 Summary Contents for Parts A and B . . . . . . . . . . . . . . . . . . . 159

Part C QUADRATIC FUNCTIONALS

3.11 Symmetric and Positive Definite Operators . . . . . . . . . . . . . . . 160

3.12 Quadratic Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Part D APPROXIMATE METHODS

3.13 Introductory Comment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

3.14 The Ritz Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

3.15 Galerkin’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

3.16 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

4 Beams, Frames and Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Part A BEAMS

4.2 Technical Theory of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . 188

4.3 Deflection Equations for the Technical Theory of Beams . . . . . 191

4.4 Some Justifications for the Technical Theory of Beams . . . . . . 196

4.5 Timoshenko Beam Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

xii Contents

4.6 Comments on the Ritz Method . . . . . . . . . . . . . . . . . . . . . . . . 208

4.7 The Ritz Method for a Series Solution . . . . . . . . . . . . . . . . . . 212

4.8 Use of the Reissner Principle . . . . . . . . . . . . . . . . . . . . . . . . . 218

4.9 Additional Problems in Bending of Beams . . . . . . . . . . . . . . . 221

Part B FRAMES AND RINGS

4.10 Open Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

4.11 Closed Frames and Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

4.12 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

5 Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

5.2 Total Potential Energy; Equation for Torsion . . . . . . . . . . . . . . 259

5.3 The Total Complementary Energy Functional . . . . . . . . . . . . . . 261

5.4 Approximate Solutions for Linear Elastic Behavior via the

Ritz Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

5.5 Approximate Solutions for a Nonlinear Elastic

Torsion Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269

5.6 The Method of Trefftz; Upper Bound for Torsional Rigidity . . . 276

5.7 The Method of Kantorovich . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

5.8 Extended Kantorovich Method . . . . . . . . . . . . . . . . . . . . . . . . . 289

5.9 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

6 Classical Theory of Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

6.2 Kinematics of the Deformation of Plates . . . . . . . . . . . . . . . . . 299

6.3 Stress Resultant Intensity Functions and the Equations

of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

6.4 Minimum Total Potential Energy Approach . . . . . . . . . . . . . . 309

6.5 Principle of Virtual Work; Ectangular Plates . . . . . . . . . . . . . . 317

6.6 A Note on the Validity of classical Plate Theory . . . . . . . . . . . 321

6.7 Examples From Classical Plate Theory; Simply-supported

Rectangular Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

6.8 Rectangular plates; Levy’s Method . . . . . . . . . . . . . . . . . . . . . 331

6.9 The Clamped Rectangular Plate; Approximate Solutions . . . . . 336

6.9.1 Ritz Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

6.9.2 Galerkin’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

6.9.3 Kantorovich’s Method . . . . . . . . . . . . . . . . . . . . . . . . 344

6.10 Eliptic and Circular Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

6.11 Skewed Plate Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351

6.12 Improved Theory—Axisymmetric Circular Plates . . . . . . . . . . 354

6.13 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

Contents xiii

7 Dynamics of Beams and Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

7.2 Hamilton’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373

Part A BEAMS

7.3 Equations of Motion for Vibrating Beams . . . . . . . . . . . . . . . . 376

7.4 Free Vibrations of a Simply-Supported Beam . . . . . . . . . . . . . 380

7.5 Rayleigh’s Method for Beams . . . . . . . . . . . . . . . . . . . . . . . . 385

7.6 Rayleigh–Ritz Method for Beams . . . . . . . . . . . . . . . . . . . . . . 389

7.7 The Timoshenko Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

Part B PLATES

7.8 Equations of Motion for Plates . . . . . . . . . . . . . . . . . . . . . . . . 403

7.9 Free Vibrations of a Simply-Supported Plate . . . . . . . . . . . . . . 405

7.10 Rayleigh’s Method for Plates . . . . . . . . . . . . . . . . . . . . . . . . . 408

7.11 Rayleigh–Ritz Method for Plates . . . . . . . . . . . . . . . . . . . . . . 411

7.12 Transverse Shear and Rotatory Inertia—Mindlin

Plate Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

Part C GENERAL CONSIDERATIONS

7.13 The Eigenfunction–Eigenvalue Problem Restated . . . . . . . . . . 429

7.14 The Rayleigh Quotient in Terms of Operators . . . . . . . . . . . . . 431

7.15 Stationary Values of the Rayleigh Quotient . . . . . . . . . . . . . . . 432

7.16 Rayleigh–Ritz Method Re-Examined . . . . . . . . . . . . . . . . . . . 435

7.17 Maximum–Minimum Principle . . . . . . . . . . . . . . . . . . . . . . . . 439

7.18 Justification of the Estimation from Above Assertion of the

Rayleigh–Ritz Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441

7.19 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

8 Nonlinear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449

8.2 Kinematics of Points and Line Segments . . . . . . . . . . . . . . . . 450

8.3 Interpretation of Strain and Rotation Terms . . . . . . . . . . . . . . . 455

8.4 Volume Change During Deformation . . . . . . . . . . . . . . . . . . . 457

8.5 Changes of Area Elements During Large Deformation . . . . . . . 459

8.6 Simplification of Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462

8.7 Stress and The Equations of Equilibrium . . . . . . . . . . . . . . . . 464

8.8 Simplification of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 470

8.9 Principle of Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472

8.10 Total Potential Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

8.11 Von Karman Plate Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 478

8.12 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488

xiv Contents

9 Elastic Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

Part A STABILITY OF RIGID BODY SYSTEMS

9.2 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

9.3 Rigid Body Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

Part B ELASTIC STABILITY OF COLUMNS

9.4 The Euler Load; Equilibrium Method . . . . . . . . . . . . . . . . . . . 500

9.5 Energy Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507

9.6 Imperfection Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

9.7 The Kinetic Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

9.8 General Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515

9.9 The Elastica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516

9.10 An Intermediate Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521

9.11 A Note on Koiter’s Theory of Elastic Stability . . . . . . . . . . . . 523

Part C ELASTIC STABILITY OF PLATES

9.12 The Buckling Equation for Rectangular Plates . . . . . . . . . . . . . 530

9.13 The Equilibrium Method—An Example . . . . . . . . . . . . . . . . . 535

9.14 The Rectangular Plate Via the Energy Method . . . . . . . . . . . . 538

9.15 The Circular Plate Via the Energy Method . . . . . . . . . . . . . . . 538

Part D APPROXIMATION METHODS

9.16 Comment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543

9.17 The Rayleigh Quotient for Beam-Columns . . . . . . . . . . . . . . . 543

9.18 The Rayleigh and the Rayleigh–Ritz Methods

Applied to Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544

9.19 Rayleigh Quotient for Rectangular Plates . . . . . . . . . . . . . . . . 546

9.20 The Kantorovich Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 548

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553

10 Finite Element Analysis: Preliminaries and Overview . . . . . . . . . . 559

10.1 Matrix Forms of the Equations of Elasticity and Key

Variational Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560

10.1.1 Matrix Representations of the Equations of Isotropic

Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560

10.1.2 Matrix Representation of the Key Variational

Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

10.2 Strong and Weak Forms of Elasticity Problems . . . . . . . . . . . . 566

10.2.1 Strong Form to Weak Form . . . . . . . . . . . . . . . . . . . . 568

10.2.2 Weak Form to Strong Form . . . . . . . . . . . . . . . . . . . . 570

10.3 A Formal Statement of Finite Element Analysis

in Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571

10.3.1 Finite Element Analysis of Elastic Bodies

and Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572

10.3.2 The Direct Stiffness Method . . . . . . . . . . . . . . . . . . . 577

Contents xv

10.4 Finite Elements: Degrees of Freedom, Shapes,

and Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578

10.4.1 Polynomial-Based Approximations . . . . . . . . . . . . . . 580

10.4.2 Interpolation Across Finite Elements . . . . . . . . . . . . . 581

10.4.3 Finite Element Shapes . . . . . . . . . . . . . . . . . . . . . . . . 584

10.5 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 588

11 Finite Element Applications: Trusses and Beams . . . . . . . . . . . . . 591

11.1 Truss Forces and Deflections and the Direct Stiffness

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

11.1.1 Analyzing an Indeterminate Truss with Castigliano’s

Second Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592

11.1.2 Element Stiffness Matrix for a Truss Finite

Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596

11.1.3 Truss Element Stiffness in Global Coordinates . . . . . . 599

11.1.4 Assembled Stiffness of a Truss System . . . . . . . . . . . 602

11.1.5 Solving the Assembled (Matrix) Equilibrium

Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605

11.1.6 Determining the Bar Forces in the Truss . . . . . . . . . . 607

11.1.7 Remarks on the Direct Stiffness Methods

for Trusses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609

11.2 Applying FEA to Axially Loaded Bars . . . . . . . . . . . . . . . . . . 611

11.2.1 Total Potential Energy and Element Equilibrium

for Bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612

11.2.2 FEA of a Centrally Loaded Bar . . . . . . . . . . . . . . . . . 613

11.2.3 Solving the FEA Assembled Equilibrium

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

11.3 Finite Element Analysis for Beams in Bending . . . . . . . . . . . . 618

11.3.1 Displacements and Shape Functions for Beams

in Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 618

11.3.2 Total Potential Energy and Element Equilibrium for

Beam Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619

11.3.3 Assembling Beam Elements to Model Beams . . . . . . . 622

11.4 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632

Appendix I Cartesian Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635

I.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635

I.2 Vectors and Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635

I.3 Transformation of Coordinates and Introduction to Index

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637

I.4 Interpretation of the Free Indices: Kronecker Delta . . . . . . . . . 640

I.5 Operations with Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642

I.6 Scalars and Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644

xvi Contents

I.7 Tensors: Symmetry and Skew-symmetry . . . . . . . . . . . . . . . . . 648

I.8 Vector Operations Using Tensor Notation: the Alternating

Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653

I.9 Gauss’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657

I.10 Green’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 660

I.11 Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 662

Appendix II Rotation Tensor for a Deforming Element . . . . . . . . . . . . 667

Appendix III Integration ofR 10xa 1� xg½ �hdx . . . . . . . . . . . . . . . . . . . . 670

Appendix IV To Show that Lagrange Multipliers are Zero for

Development of the Rayleigh–Ritz Method . . . . . . . . . . . 672

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679

Contents xvii

Theory of Linear Elasticity 1

1.1 Introduction

In much of this text we shall be concerned with the study of elastic bodies.

Accordingly, we shall now present a brief treatment of the theory of elasticity.

In developing the theory we shall set forth many concepts that are needed for

understanding the variational techniques soon to be presented.

We will not attempt solutions of problems directly using the full theory of

elasticity; indeed very few such analytical solutions are available. Generally we

will work with special simplifications of the theory wherein a priori assumptions are

made as to

(a) the stress field (plane stress problems for example)

(b) the strain field (plane strain problems for example)

(c) the deformation field (structural mechanics of beams, plates and shells).

The significance of these simplications and the limitations of their use can best be

understood in terms of the general theory.

Specifically in this chapter we will examine the concepts of stress, strain,

constitutive relations, and various forms of energy. This will permit us to present

the equations of linear elasticity and to consider the question of uniqueness of

solutions to these equations. The plane stress simplification will be considered in

this chapter; examples of simplification (c) will be presented in subsequent

chapters.

Although we shall present certain introductory notions pertaining to finite

deformations in this chapter so as to view small deformation in the proper perspec-

tive, we shall defer detailed considerations of finite deformation to Chap. 8 wherein

we consider geometrically nonlinear elasticity.

C.L. Dym, I.H. Shames, Solid Mechanics, DOI 10.1007/978-1-4614-6034-3_1,# Springer Science+Business Media New York 2013

1

Part A

STRESS

1.2 Force Distributions

In the study of continuous media, we are concerned with the manner in which forces

are transmitted through a medium. At this time, we set forth two classes of forces that

will concern us. The first is the so-called body-force distribution distinguished by thefact that it acts directly on the distribution of matter in the domain of specification.

Accordingly, it is represented as a function of position and time and will be denoted

as B(x,y,z,t) or, in index notation, as Bi(x1,x2,x3, t). The body force distribution is an

intensity function and is generally evaluated per unit mass or per unit volume of the

material acted on. (In the study of fluids, the basis of measure is usually per unit mass

while in the study of solids the basis of measure is usually per unit volume.)

In discussing a continuum there may be some apparent physical boundary that

encloses the domain of interest such as, for example, the outer surface of a beam.

On the other hand, we may elect to specify a domain of interest and thereby

generate a “mathematical” boundary. In either case, we will be concerned with

the force distribution that is applied to such boundaries directly from material

outside the domain of interest. We call such force distributions surface tractionsand denote them as T(x,y,z,t) or Ti(x1,x2,x3,t). The surface traction is again an

intensity, given on the basis of per unit area.

Now consider an infinitesimal area element on a boundary (see Fig. 1.1) over

which we have a surface traction distribution T at some time t. The force dfi trans-mitted across this area element can then be given as follows:

dfi ¼ TidA

Note that T need not be normal to the area element and so this vector and the unit

outward normal vector v may have any orientation whatever relative to each other.

We have not brought the unit normal v into consideration thus far, but we will find ituseful, as we proceed, to build into the notation for the surface traction a superscript

referring to the direction of the area element at the point of application of the

surface traction. Thus we will give the traction vector as

TðvÞðx; y; z; tÞ

or as

TðvÞi x1; x2; x3; tð Þ

where (v) is not to be considered as a power. If the area element has the unit

normal in the xj direction, then we would express the traction vector on this element

as T(j) or as TðjÞi .

2 1 Theory of Linear Elasticity

In the following section we shall show how we can use the superscript to good

advantage.

1.3 Stress

Consider now a vanishingly small rectangular parallelepiped taken at some time

t from a continuum. Choose reference x1, x2, x3 so as to be parallel to the edges of

this rectangular parallelepiped as has been shown in Fig. 1.2. We have shown

surface tractions on three rectangular boundary surfaces of the body. Note that we

have employed the superscript to identify the surfaces. The cartesian components of

the vector T(1) are then T1(1), T2

(1), T3(1). We shall now represent these components

by employing τ in place of T and moving the superscript down to be the first

subscript while deleting the enclosing parenthesis. Hence, the components of T(1)

are then given as τ11, τ12, τ13. In general we have

Tð1Þi � τ11; τ12; τ13

Tð2Þi � τ21; τ22; τ23

Tð3Þi � τ31; τ32; τ33

In a more compact manner we have

TðiÞj ¼ τij

where the nine quantities comprising τij are called stresses and are forces per unit

area wherein the first subscript gives the coordinate direction of the normal of the

area element and the second subscript gives the direction of the force intensity

x3n

x1x2

dA

T

Boundary

Fig. 1.1

1.3 Stress 3

itself. These nine force intensities are shown in Fig. 1.3 on three orthogonal faces of

an infinitesimal rectangular parallelepiped with faces parallel to the coordinate

planes of x1, x2, x3.Representing the set τij as an array we have

τ11 τ12 τ13τ21 τ22 τ23τ31 τ32 τ33

0

@

1

A

where the terms in the main diagonal are called normal stresses, since the force

intensities corresponding to these stresses are normal to the surface, while the off-

diagonal terms are the shear stresses.

x3

x1

x2

T(3)

T(1)

T(2)

Fig. 1.2

t33

t31

t32

t23

t22t21t12t11

t13

x1 x2

x3Fig. 1.3

4 1 Theory of Linear Elasticity

We shall employ the following sign convention for stresses. A normal stress

directed outward from the interface is termed a tensile stress and is taken by

definition as positive. A normal stress directed toward the surface is called

a compressive stress (it is exactly the same as pressure on the surface) and is, by

definition, negative. For shear stresses we employ the following convention:

A shear stress is positive if (a) both the stress itself and the unit normal point in positive

coordinate directions (the coordinates need not be the same), or (b) both point in the

negative coordinate directions (again the coordinates need not be the same). A mixture of

signs for coordinate directions corresponding to shear stress and the unit normal indicates a

negative value for this shear stress.1

Note that all the stresses shown in Fig. 1.3 are positive stresses according to the

above sign convention.

Knowing τij for a set of axes, i.e., for three orthogonal interfaces at a point, wecan determine a stress vector T(v) for an interface at the point having any directionwhatever. We shall now demonstrate this. Consider any point P in a continuum

(Fig. 1.4(a)). Form as a free body a tetrahedron with P as a corner and with three

orthogonal faces parallel to the reference planes as has been shown enlarged in

Fig. 1.4(b). The legs of the tetrahedron are given as Δx1, Δx2, and Δx3 as has beenindicated in the diagram, and the inclined face ABC has a normal vector v . The

Stresses on coordinatefaces are average stresses

a b

(T (ν))av(T (ν))av

(T (ν))av

Δx3

Δx2Δx1

21

(T (ν))3

x1

x1

x2

x3

x3

x2

P P

A

B

C

av

Fig. 1.4

1Note that normal stress actually follows this very same convention.

1.3 Stress 5

stresses have been shown for the orthogonal faces as have the stress vector T(v) and

its components for the inclined face. We will denote by h the perpendicular distancefrom P to the inclined face (the “altitude” of the tetrahedron). The stresses and

traction components given on the faces of the tetrahedron are average values over

the surfaces on which they act. Also, the body force vector (B)av (not shown)

represents the average intensity over the tetrahedron. Using these average values as

well as an average density, ρav, we can express Newton’s law for the mass center of

the tetrahedron in the x1 direction as follows

� τ11ð Þav PCB� τ21ð Þav ACP� τ31ð Þav APBþ B1ð Þavρav1

3ABCh

þ �T1ðvÞ�avABC ¼ ρav

1

3ABCha1 (1.1)

where a1 is the acceleration of the mass center in the x1 direction. We can next

replace the average values by values taken at P itself, plus a small increment which

goes to zero as Δxi goes to zero.2 Thus we have

τ11ð Þav ¼ τ11ð ÞPþe1 ðB1Þav ¼ ðB1ÞP þ eB

τ21ð Þav ¼ τ21ð ÞPþe2�TðvÞ1

�av¼ �TðvÞ

1

�Pþ eT

τ31ð Þav ¼ τ31ð ÞPþe3 ρav ¼ ρP þ eρ

(1.2)

where e1, e2, e3, eB, eT and eρ go to zero with Δxi. Thus we have for Eq. (1.1):

�½ðτ11ÞP þ e1� PCB� ½ðτ21ÞP þ e2� ACP� ½ðτ31ÞP þ e3� APB

þ ½ðB1ÞP þ eB�ðρP þ eρÞ 1

3

� �

ðABChÞ þ T1ðvÞ

� �

Pþ eT

h iABC

¼ 1

3ðρP þ eρÞðABCÞha1 (1.3)

Next dividing through by areaABCand noting thatPCB=ABC ¼ v1; ACP=ABC ¼ v2,etc., we have:

� ½ðτ11ÞP þ e1�v1 � ½ðτ21ÞP þ e2�v2 � ½ðτ31ÞP þ e3�v3þ ½ðB1ÞP þ eB�ðρP þ eρÞ h

3þ T1

ðvÞ� �

Pþ eT

h i¼ ðρP þ eρÞ h

3a1

Now go to the limit as Δxi! 0 in such a manner as to keep vi constant. Clearly, thee’s disappear and h! 0. We then get in the limit:

� ðτ11ÞPv1 � ðτ21ÞPv2 � ðτ31ÞPv3 þ T1ðvÞ

� �

P¼ 0

2We are thus tacitly assuming that the quantities in Eq. 1.1 vary in a continuous manner.

6 1 Theory of Linear Elasticity

Since P is any point of the domain we need no longer carry along the subscript.

Rearranging terms we get:

T1ðvÞ ¼ τ11v1 þ τ21v2 þ τ31v3 (1.4)

For any coordinate direction i we accordingly get:

TðvÞi ¼ τ1iv1 þ τ2iv2 þ τ3iv3

And so using the summation convention, the stress vector TðvÞi can be given as

TðvÞi ¼ τjivj

where you will note that vj can be considered to give the direction cosines of the unitnormal of the interface on which the traction force is desired. We will soon show

that the stress terms τij form a symmetric array and, accordingly the above equation

can be put in the following form:

TðvÞi ¼ τijvj (1.5)

Thus, knowing τij we can get the traction vector for any interface at the point.

The above is called Cauchy’s formula and may be used to relate tractions on the

boundary with stresses directly next to the boundary.

1.4 Equations of Motion

Consider an element of the body of mass dm at any point P. Newton’s law requires

that

df ¼ dm _V

where df is the sum of the total traction force on the element and the total body force

on the element. Integrating the above equation over some arbitrary spatial domain

having a volume D and a boundary surface S, we note as a result of Newton’s thirdlaw that only tractions on the bounding surface do not cancel out so that we have,

using indicial notation:

ðð�S

TðvÞi d Aþ

ð ð ð

D

Bi dV ¼ð ð ð

D

_Viρ dV (1.6)

Now employ Eq. (1.5) to replace the stress vector TðvÞi by stresses. Thus:

1.4 Equations of Motion 7

ðð�S

τji vj dV þð ð ð

D

Bi dV ¼ð ð ð

D

_Viρ dV

Next employ Gauss’ theorem for the first integral and collect terms under one

integral. We get:ð ð ð

D

½τji;j þ Bi � _Viρ�dV ¼ 0

Since the domain D is arbitrary we conclude from above that at any point the

following must hold:

τji;j þ Bi ¼ ρ _V (1.7)

This is the desired equation of motion.

Suppose next we consider an integral form of the moment of momentum

equation derivable from Newton’s law, i.e.,M ¼ _H. Thus we may say (considering

Fig. 1.5):

ðð�S

r� TðvÞdAþð ð ð

D

r� B dV ¼ð ð ð

D

r� d

dtðV dmÞ

Considering the continuum to be composed of elements whose mass dm is constant

but whose shape may be changing, we can express the integrand of the last

expression as r � _V dm. Thus we have for the above equation in tensor notation

ðð�S

εijkxjTðvÞk d Aþ

ð ð ð

D

εijkxjBk dV �ð ð ð

D

εijkxj _Vkρ dV ¼ 0

where we replace dm by ρ dv. Now replaceTðvÞk by τlkvl and employ Gauss’ theorem.

Thus we may write the above equation as:

r

r

dA

dV

D

T(ν)(x1,x2,x3)

B(x1,x2,x3)

x3

x1 x2

Fig. 1.5

8 1 Theory of Linear Elasticity

ð ð ð

D

εijk xjτlk� �

;lþ xjBk � ρxj _Vk

h idV ¼ 0

Since the above is true for any domain D, we can set the integrand equal to zero.

Carrying out differentiation of the first expression in the bracket and collecting

terms we then get:

εijkxj½τlk;l þ Bk � ρ _Vk� þ εijkxj; lτlk ¼ 0

Because of Eq. ((1.7) we can set the first expression equal to zero and so we get:

εijkxj; lτlk ¼ 0

Noting that xj,l ¼ δjl we have

εijkδjlτkl ¼ εijkτkj ¼ 0 (1.8)

It should be clear by considering the above equations for each value of the free

index i, that the stresses with reversed indices with respect to each other are equal.

That is:

τkj ¼ τjk (1.9)

The stress components form a symmetric array.3

1.5 Transformation Equations for Stress

We have thus far shown how a stress vector T(v) on an arbitrarily oriented interface

at a point P in a continuum can be related to the set of nine stress components on a

set of orthogonal interfaces at the point. Furthermore, we have found through

consideration of Newton’s law, that the stress terms form a symmetric array. We

shall now show that the stress components for a set of Cartesian axes (i.e., for three

orthogonal interfaces at a point) transform as a second-order tensor.

For this purpose consider Fig. 1.6 showing axes x1, x2, x3 and x01; x02; x

03 rotated

arbitrarily relative to each other. Suppose we know the set of stresses for the

unprimed reference, i.e., for a set of orthogonal interfaces having x1x2x3 as edges,and wish to determine stresses for reference x0y0z0. Accordingly, in Fig. 1.7, we haveshown a vanishingly small rectangular parallelepiped having x01x

02x

03 as edges so as

3We have only considered body force distributions here. If one assumes body-couple distributionsthat may occur as a result possibly of magnetic or electric fields on certain kinds of dielectric and

magnetic materials, we will have in Eq. (1.8) the additional integralÐ Ð Ð

M dv where M is the

couple-moment vector per unit volume. The result is that Eq. (1.8) becomes εijkτkj þ Mi ¼ 0. The

stress tensor is now no longer a symmetric tensor. Such cases are beyond the scope of this text.

1.5 Transformation Equations for Stress 9

to present the set of orthogonal interfaces for the desired stresses. The stress vectors

for these interfaces have been shown as well as the corresponding stresses.

Suppose we wish to evaluate the stress component τ031. We can do this by first

computing the stress vector on the interface corresponding to this stress (this

interface is denoted as (30)). The direction cosines v01; v02 and v03 for this interface

can be given in our familiar notation as a31, a32, and a33 where the first subscript

x'3

x'2

x'1

x3

x2

x1

Fig. 1.6

1´ 2´

3'

t'33

t'31

t'32

x'3

x'2

x'1

x3

x2

x1

T(3')

T(2')

T(1')

Fig. 1.7

10 1 Theory of Linear Elasticity

identifies the normal to interface x03ð Þ and the second subscript identifies the properunprimed axis. Thus we can say, using Eq. (1.5):

Tð30Þi ¼ τija3j (1.10)

You must remember that the stress vector components for surface (30) are given

above as components along the unprimed axes. Hence, to get τ031 we must project

each one of the above components in a direction along the x01 direction. This is

accomplished by taking an inner product using a1i in the above equation.4 Thus

τ031 ¼ Tð30Þi a1i ¼ τija3 j

� �a1i ¼ a3 ja1iτij ¼ a3 ja1iτji

where in we have used the symmetry property of τij in the last expression. For

another component of stress such as τ03k we could change the (1) to free index (k) inthe above equation and we get:

τ03k ¼ a3 jakiτji

And to get a corresponding result on any other interface we need only change the

subscript (3) to free index p. Thus we can say for any stressesτ0pk.

τ0pk ¼ ap jakiτji (1.11)

We have thus shown that stress components transform as a second-order tensor.5

1.6 Principal Stresses

We have shown that given a system of stresses for an orthogonal set of interfaces at

a point, we can associate a stress vector for interfaces having any direction in space

according to the formulation:

TðvÞi ¼ τijvj

4a1i represents the set of direction cosines between the x01 axis and the x1, x2 and x3 axes. It is

therefore a unit vector in the x01 direction.5We could also have arrived at this equation by using one of the quotient rules of Appendix I in

conjunction with Cauchy’s formula

TðvÞi ¼ τijvj

since vj is an arbitrary vector (except for length) and TðvÞi is a vector.

1.6 Principal Stresses 11

We now ask this question. Is there a direction v such that the stress vector is

collinear with v? That is to say, is there an interface having a normal such that there

is only one non-zero stress—the normal stress? We shall call such a stress, if it

exists, a principal stress and we shall denote it as σ(see Fig. 1.8). Thus we can

express the above equation for this case as follows:

σvi ¼ τi jvj (1.12)

Now replace σvi by (σvj)(δij) and rearrange the above equation to form the relation:

ðτij � σδijÞvj ¼ 0 (1.13)

We have here three simultaneous equations with unknowns, σ, v1, v2, and v3.However, we do have a fourth equation involving these direction cosines, namely:

vivi ¼ 1 (1.14)

A non-trivial solution to the set of equations (1.13) requires

τij � σδij��

�� ¼ 0 (1.15)

τ11

σ

τ33

τ32

τ31

τ23

τ22τ21 τ12

τ13

x3

x1

x2

v

Fig. 1.8

12 1 Theory of Linear Elasticity