81086474 washizu variational methods in elasticity and plasticity

VARIATIONAL METHODS INELASTICITY AND PLASTICITY

SECOND EDITION

KYUICHIRO WASHIZUProfessor of Aeronautics and Astronautics, University of Tokyo

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Copyright © 1975All Rights Reserved. No part of this publication may bereproduced, stored in a retrieval system, or transmitted, inany form or by any means: electronic, electrostatic, magnetic tape,mechanical, photocopying, recording or otherwise, without the prwrpermission in writing from the publishers.

First edition 1968Second edition 1975Reprinted 1975

Ubrary of Congress Cataloging in Publicstkui Data

Washizu, Kyuichiro, 1921— -

Variational methods in elasticity and plasticity.(International series of monographs in aeronauticsand astronautics, Division 1: solid and structural mechanks, v. 9)Includes bibliographies.1. Elasticity. 2. Plasticity. 3. Ca'culus of variations. 1. Title.QA93I.W3)1974 620.l'123 74—8861

ISBN 0-08-017653—4

Printed in Great Britain by .4. Wheaton & Company, Exeter

FOREWORDTHE variational principle and its application to many branches of mechanicsincluding elasticity and plasticity has had a long history of development.However, the importance of this principle has been high-lighted in recentyears by developments in the use of finite element methods which have beenwidely employed in structural analysis since the pioneering work by M. I.

ci a!. appeared in Vol. 23, No. 9 issue of the Journal of AeronauticalSciences in 1956. It has been shown repeatedly since that time that thevariational principle provides a powerful tool in the mathematical formula-tion of the finite element approach. Conversely, the rapid development of thefinite element method has given much stimulus to the advancement .of thevariational principle and new forms of the principle have been developedduring the past decade as outlined in Section 1 of Appendix I of thepresentbook.

The first edition of Professor Washizu's book, entitled Variational Methods inElasticity and Plasticity and published in 1968, was well by engineers,teachers and students working in solid and structural mechanics. Its publica-tion was timely, because it coincided with a period of rapid growth of applica-tion of the finite element method. The principle features of the first edition wasthat of providing a systematic way of variational principles inelasticity and plasticity, of transforming one variational principle to anotherand of providing a basis for the mathematical formulation of thefinite element method. The book was widely used and referenced trequentlyin literature related to the finite element method.

Now, Professor Washizu has prepared a revised edition which adds a newAppendix I. The new appendix introduces an outline of variational principleswhich are used frequently as a basis for mathematical formulations in elastic-ity and plasticity including those'new variational principles developed inconnection with the finite element method. As in the case of the first edition,Appendix I is written in the clear, concise and elegant style for which Pro-fessor Washizu is so widely known. The revised edition should. form anextremely valuable addition to the libraries and reference shelves of allwho are interested in solid and structural mechanics.

R. L.National ScienceFoundation,

Washington D.C.

ACKNOWLEDGEMENTS•TUB author feels extremely honored and wishes to express his deepestgratitude to Dr. R. L. Bispliughoff, Deputy Director of National ScienceFoundation, for having given the Foreword to the revised edition of this book.The author would like to express his deepest appreciation to ProfessorT. H. H. Pain of the Massachusetts of Technology and ProfessorIL H. Gallagher of Cornell University for having given valuable commentsto the manuscript for the new appendix. Dr. Oscar Orringer of the Massa-chusetts Institute of Technology collaborated again with the author in

the writing of the manuscript of the new appendix. Moreover, theauthor should remember that he has been given numerous comments,criticisms and encouragements from the reader since the publication of thefirst edition of this book. The author would like to express his sincere ap-predation to all of these people, without whose encouragement and collabora-tion, this revised edition couldn't be realized.

K. WAswzu

CONTENTSFOREWORD

ACKNOWLEDGEMENTS

INTRODUCTION 1

cHAPTER 1. Dis THEORY OF ELAsrrcrrY IN RECTANGULAR C*a-COORDINATss 8

1.1. Presentation of a Problem in SmailDisplacement Theory 81.2. Conditions of Compatibility 11

Stress Functions 121.4. Principle of Virtual Work 13

1.5. Approximate Method of Solution Based on the Principle of Virtual Work 15

1.6. Principle of Complementary Virtual Work 171.7. Approximate Method of Solution Based on the Principle of Complementary

Virtual Work 191.8. Relations between Conditions of Compatibility and Stress Functions 221.9. Some Remarks 24

CHAPTER 2. VARIATIONAL PRINcritas IN nm Smw.E. DIsPzAcn,mrrr orEIAsncrrY 27

2.1. Principle of Minmwn Potential Energy 272.2. Principle of Minimum Complementary Energy 292.3. Generalization of the Principle of Minimum Potential Energy 31

2.4. Derived Variational Principles 342.5. Rayleigh—Ritz Method—(1) 382.6. Variation of the Boundary Conditions and Castigliano's Theorem 402.7. Free Vibrations of an Elastic Body 432.8. Rayleigh-Ritz Method—(2) 462.9. Some Remarks 48

CHAPTER 3. FINim THEORY OP ELASTICTrY IN RECTANGULAR C*st-TESIAN COORDINATss 52

3.1. Analysis of Strain 523.2. Analysis of Stress and Equations of EquIlibrium 563.3. Transformation of the Stress Tensor 583.4. Stress-Strain Relations 593.5. Presentation of a Problem 603.6. Principle of Virtual Work 633.7. Strain Energy Function 643.8. Principle of Stationary Potential Energy 673.9. Generalization of the Principle of Stationary Potential Energy 683.10. Energy Criterion for Stability 693.11. The Euler Method for Stability Problem fl3.12. Some Remarks 74

ix

CONTENTS

CHAPTER 4. THEORY IN CURVILINEAR COORDINATES 764.1. Geometry before Deformation 764.2. Analysis of Strain and Conditions of Compatibility 804.3. Analysis of Stress and Equations of Equilibrium 834.4. Transformation of the Strain and Stress Tensors 844.5. Stress-Strain Relations in Curvilinear Coordinates 874.6. Principle of Virtual Work 884.7. Principle of Stationary Potential Energy and its Generalizations 894.8. Some Specializations to Small Displacement Theory in Orthogonal Curvi-

'inear Coordinates 90

CHAPTER 5. EXTENSIONS OF THE PRINCIPLE OF VIRTUAL WORK AND RELAThDVARIATIONAL PRINCIPLES 93

5.1. Initial Problems 935.2. Stability Problems of a Body with Initial Stresses 965.3. Initial Strain Problems5.4. Thermal Stress Problems 995.5. Quasi-static Problems 101

5.6. Dynamical Hems 1045.7. Dynamical Problems of an Unrestrained Body 107

CHAPTER OF BARS 113

6.1. Saim-Vcnant Theory of Torsion 1136.2. The Principle of Minimum Potential Energy and its Transformation 1166.3. Torsion of a Bar with a Hole 1196.4. Torsion of a Bar with Initial Stresses 121

6.5. Upper and Lower Bounds of Torsional Rigidity 125

CHAPTER 7. BEAMS 132

7.1. Elementary Theory of a Beam 1327.2. Bending of a Beam 1347.3. principle of Minimum Potential Energy and its Transformation 1377.4. Free Lateral Vibration of a Beam 1397.5. Large Deflection of a Beam 1427.6. Buckling of a beam 1447.7. A Beam Theory Including the Effect of Transverse Shear Deformation 1477.8. Some Remarks 149

CHAPTER 8. PLATES 152

8.1. Stretching and Bending of a Plate 1528.2. A Problem of Stretching and Bending of a Plate 1548.3. Principle of Minimum Potential Energy and its Transformation for the

SteLhing of a Plate 1608.4. Principlc of Minimum Energy and its Transformation for the

Bending of a Plate 161

8.5. Large Deflection of a Plate in Stretching and Bending 163

8.6. Ruckling of a Ph. te 165

8.7. Thermal Stresses in a Plate 168

8.8. A Thin Plate Theory Including the Effect of Transverse Shear Deformation 170

8.9. Thin Shallow Shell 173

8.10. Somc Remarks 178

CONTENTS xi

CHAPTER 9. SHELLS 182

9.1. Geometry before Deformation 1829.2. Analysis of Strain 1879.3. Analysis of Strain under the Kirchhoff--Lo%e Hypothesis 1899.4. A Linearized Thin Shell Theory under the Kirchhoff-Love Hypothesis 1919.5. Simplified Formulations 1959.6. A Simplified Linear Theory under the Kirchhoff—Love Hypothesis 1979.7. A Nonlinear Thin Shell Theory under the Kirchhoff—Love Hypothesis 1989.8. A Linearized Thin Shell Theory Including the Effect of Transverse Shear De-

formations 1999.9. Some Remarks 201

CHAPTER 10. Snwcruaas 205

10.1. Finite Redundancy 20510.2. Deformation Characteristics of a Truss Member and Presentation of a Truss

Problem 20610.3. Variational Formulations of the Truss Problem 20910.4. The Force Method Applied to the Truss Problem 21010.5. A Simple Example of a Truss Structure 21310.6. Deformation Characteristics of a Frame Member 21410.7. The Force Method Applied to a Frame Problem 21710.8. Notes on the Force Method Applied to Semi-monocoque Structures 22110.9. Notes on the Stiffness Matrix Method Applied to Semi-monocoque Struc-

tures 225

CHAPTER 11. THE DEFORMATION THEORY OF PLAsricrrY 231

11.1. The Deformation Theory of Plasticity 23111.2. Strain-hardening Material 23311.3. Perfectly Plastic Material 235I 14. A Special Case of Hencky Material 237

CHAPTER 12. THE Fi.ow THEbRY oF 240

12.1. The Flow Theory of Plasticity 24012.2. Strain-hardening Material 24212.3. Perfectly Plastic Material 24412.4. The Prandtl-Reuss Equation 24512.5. The Saint-Venant-Levy-Mises 24712.6. Limit Analysis 25012.7. Some Remarks 253

APPENDIX A. EXTREMUM OF A FuNcnoN wrrii A CONDITION 254APPENDIX B. RELATIONS FOR A THIN PLATE 256APPENDIX C. A BEAM THEORY INCLUDING THE EFFECF or TRANSVERSE SHs*&

FORMATION 258APPENDIX D. A THEORY OF PLATE BENDINP INCLUDING THE Emcr

VERSE SHEAR DEFORMATION 262APPENDIX E. SPECIALIZATIONS TO SEVERAL KINDS OF SHEU.s 265APPENDIX F. A Nom ON THE HAAR-KARMAN PRINCIPLE 269APPENDIX G. VARIATIONAL PRINCIPLES IN ThE THEORY Of 270APPENDIX H. PROBLEMS 272

CONTENTS

APPENDIX I. VARIATIONAL PRINCIPLES AS A BASIS FOR ThEMEThoD 345

I. Introduction 3452. Conventional Variational Principles for the Small Displacement Theory of

Elastostatics 3473. Derivation of Modified Variational Principles from the Principle of Minimum

Potential Energy 3514. Derivation of Modified Variational Principles from the Principle of Minimum

Complementary Energy 3575. Conventional Variational Principles for the Bending of a Thin Plate 3606. Derivation of Modified Variational Principles for Bending of a Thin Plate 3647. Variational Principles for the Small Displacement Theory of Elastodynamics 3728. Finite Displacement Theory of Elastostatics 3789. Two IncremçntaLTheories 384

10. Some Remarks on Discrete Analysis 397

APPENDIX J. Noms ON ThE PRnqcIpLE OF VIR11JAL WORX 405

INDEX 409

INTRODUCTIONTIlE calculus of variations is a branch of mathematics, wherein the stationaryproperty of a function of functions, namely, a functional, is studied. Thus,the object of the calculus of variations is not to find of a functionof a finite number of variables, but to find, among the group of admissiblefunctions, the one which makes the given functional A well-established example is to find, among the admissible curves joining twopoints in the prescribed space, that curve on which the distance betweenthe points is a minimum. The problem of finding a curve which enclosesa given area with minimum peripheral length is another typical example.

The calculus of variations has a wide field of application in mathematicalphysics. This is due to the fact that a physical system often behaves in amanner such that some functional depending on its behavior assumes astationary value. In other words, the equations governing the physicalphenomenon are often found to be stationary conditions of some variationalproblem. Fermat's principle in optics may be mentioned as a typical example.It states that a ray of light travels between points along the path whichrequires the least time. This leads immediately to the conclusion that a rayof light travels in a straight line in any homogeneous medium.

Mechanics is one of the fields of nlathematical physics, wherein thevariational technique has been extensively investigated. We shall take aproblem of a system of particles as an example and review the derivationof its variational formulations4

First, we shall consider thó problem of a system of particles in staticequilibrium under external and internal forces. It is well known that thebasis of variational formulation is the principle of virtual whichmay be stated as follows: Amane that the mechanical system is in equilibriumwider applied forces and prescribed geometrical contraints. Then, the ofall the virtual work, denoted by ö' W, done by the external and internal forcesexisting In the system in any arbitrary Infinitesimal virtual displacementssathfying the prescribed geometrical con,trgmu Is zero:

o,w=o. .

The principle may be stated alternatively in the following manner: If ô' Wvanishes for any arbitrary infinitesimal virtual displacements satisfying the

• t For details of the calculus of variations, see Refs. I through 8 (see pp. 6-7).For details of the variational methods in mechanics, see Rcfs. 2,9, 10 and 11.

tt This principle is also called the principle of virtual displacements.flÔ' W is not a variation of some state function P1, but denotes merely the total virtual

work.

I

2 VARIATIONAL METHODS IN ELASTICITY AND PLASTICITY

prescribed geometrical constraints, the mechanical system is in equilibrium.Thus, the principle of virtual work is equivalent to the equations ofequilibrium of the system. However, the former has a much widerfield of application to the formulation of mechanics problems thanthe latter. When all the external and internal forces are derived from apotential function U, which is a function of the coordinates of the systemof particles,t such that

b'W= —ÔU, (2)

the principle of virtual work leads to the establishment of the principle ofstationary potential energy: Among the set of all admissible configurations,the slate of equilibrium is characterized by the stationary properly of thepotential energy U:

t5U=O. (3)

The above formulation may be extended to the dynamical problem of asystem of particles subject to time-dependent applied fprces and geometricalconstiaints. By the use of d'Alembert's principle which states that thesystem can be considered to be in equilibrium if inertial forces are taken intoaccount, the principle of virtual work of the dynamical problem can bedcrived in a manner similar to the static problem case, except that termsrepresenting the virtual work done by the inertial forces are now included.The principle thus obtained is integrated with respect to time t between twolimits i = t1 and t = t2. Through integration by parts and by the use of theconvention that virtual displacements vanish at the limits, we finally obtainthe following principle of virtual work for the dynamical problem:

ofrdt +fa'Wdt =

where T is the kinetic energy of the system. Since Lagrange's equations ofmotion of the system may be derived from the principle of virtual work thusobtained, it is evident that the principle is extremely useful for obtainingthe equations of motion of a system of particles with geometrical constraints. -

When it is further assured that all the external and internal forces arederived from a potential function U, which is defined in the same manneras Eq. (2) and is a function of coordinates and the time4 we obtain Hamil-ton's principle, which states that among the set of all admissible configurationsof the system, the actual motion makes the quantity

f(T — U) di

t Forces of this category are called conservative forces.If U is time-independent, the forces are called conservative. In Ref. 2, the name

'monogenic" is given to forces derivable from a scatar quantity which is in the mostgenetal case a function of coordinates and velocities of the particles and the time.

3

stationary, provided the configuration of the system is prescribed at the limitst = and t = Hamilton's principle may be stated mathematically asfollows:

(6)

where L = T — U is the Lagrangian function of the system. it is well knownthat Hamilton's principle can be transformed by the use of Legendre'stransformation into a new and equivalent principle, and that Lagrange'sequations of motion are reduced to the so-called canonical equations.Transformations of Hamilton's principle were extensively investigated andan elegant theory known as canonical transformation was established.

The main object of this.book is to derive the principle of virtual work andrelated variational principles in elasticity and plasticity in a systematicway.t We shall formulate these principles in a manner similar to the devel-opment in the problem of a system of particles. The outline is as wedefine a problem involving a solid body in static equilibrium under bodyforces plus mechanical and geometrical boundary conditions prescribedon the surface of the body. To begin with, we derive the principle of virtualwork. This principle is equivalent to the equations of equilibrium and themechanical b undary conditions of the solid body,and is-4erived for smalldisplacement theory as well as finite displacement theory 4 Within the realmof small displacement theory we obtain another principle which will becalled the principle of complementary .virtual work.tt It is worthy of specialmention that the principles of virtual work and complementary virtual workare invariant under coordinate transformations and that they hold inde-pendently of the stress—strain relations of the material of the body. However,the stress—strain relations should be taken into account for the formulationsof variational principles, and the theories of elasticity and plasticity shouldbe treated separately.

The variational method finds one of the most fruitful fields of applicationin the small displacement theory of elasticity. When the existence of a strainenergy function is assured and the external forces are assumed to be keptunchanged during displacement variation, the principle of virtual workleads to the çstablishment of the principle of minimum potential energy. Thevariational principle is generalized by the introduction of Lagrange mul-tipliers to yield a family of variational principles which includes the Hellinger—

t For variational principles in elasticity and plasticity, see Rcfs. 11 through 20.In the small displacement theory, the displacements are assumed so small as to

allow linearizations of all governing equations of the solid body except the stress—strainrelations. Consequently, the equations of equilibrium, the strain-displacement relationsand the boundary conditions are reduced to linearized forms in small displacementtheory.if This principle is also called the principle of virtual stress, the principle of virtual

force or the principle of virtual changes in the state of stress.


Reissner principle, the principle of minimum complementary energy andso forth.

On the other hand, the principle of complementary virtual work leads tothe establishment of the principle of minimum complementary enetgy whenthe stress—strain relations assure the existence of a complementary energyfunction and the geometrical boundary conditions are assumed to be keptunchanged during stress variation. The principle of minimum complemen-tary energy is generalized by the introduction of Lagrange multipliers to yieldthe Hellinger—Reissner principle, the principle of minimum potential energyand so forth. It is seen that these two approaches to the formulation of thevariational principles are reciprocal and equivalent to each other as far asthe small displacement theory of elasticity is concerned.

In the finite displacement theory of elasticity, the principle of virtualwork leads the establishment of the principle of stationary potentialenergy when the existence of a strain energy function of the body materialand potential functions of the external forces is assured. Once the principleof stationary potential energy is thus established, it can be generalizedthrough the use of Lagrange multipliers.

The above technique is extended to dynamical elastic body problems bytaking inertial forces into account. Thus, we iierive the principle of virtualwork for the dynamical problem with the introduction of the concept ofkinetic energy. The principle of virtual work is then transformed into avariational orinciple under the assumption of the existence of a strain energyfunction and potential functions of the external forces. The newlyvariational principle may be thought of as Hamilton's principle extendedto the dynamical elastic body problem, and it can be generalized throughthe use of Lagrange multipliers.

The variational principle of an elasticity problem providcs thc governingequations of the problem as stationary it1 that sense, isequivalent to the governing equations. However, the variational formulationhas several advantages. First, the functional which is subject to variationusually has a definite physical meaning and is invariant under coordinatetransformation. Consequently, once the variational principle has beenformulated in one coordinate system, governing equations expressed inanother coordinate system can be obtained by first writing the invariantquantity in the new coordinate system and then applying variational pro-cedures. For example, once the variational principle has been formulatedin the rectangular Cartesian coordinate governing equations ex-pressed in cylindrical or polar coordinate systems can be obtained throughthe above tethnique. It may be observed that this property makes the varia-tional method extremely powerful for the analysis of structures.

Second, the variational formulation is helpful in carrying out a commonmathematical procedure, namely, the transformation of a given probleminto an equivalent problem that can be solved more easily than the original.

INTRODUCTION

In a variational problem with subsidiary conditions, the transformation isachieved by the Lagrange multiplier method, a very useful and systematictool. Thus, we may derive a family of variational principles which areequivalez,it to each other.

Third, variational principles sometimes lead to formulae for upper orlower bounds of the exact solution of the problem under consideration. Aswill be shown in Chapter 6, upper and lower bound formulae for the tor-sional rigidity of a bar are provided by simultaneous use of two variationalprinciples. Another example is an upper bound formula, derived from theprinciple of stationary potential energy, for the lowest frequency of freevibrations of an elastic body.

Fourth, when a problem of elasticity cannot be solved exactly, the varia-tional method often provides an approximate formulation for the problemwhich yields a solution compatible with the assumed degree of approxima-tion. Here, the variational method provides not only approximate governingequations, but also suggestions on approximate boundary conditions.Since it is almost impossible to obtain the exact solution of an elasticityproblem except in a few special cases, we must be satisfied with approximatesolutions for practical purposes. Theories of beams, plates, shells and multi-component structures are typical examples of such approximate fotmulationsand show the power of the principle of virtual work and related variationalmethods. However, one should take care in relying upon the accuracy ofapproximate solutions thus obtained. Consider, for example, an applica-tion of the Rayleigh—Ritz method combined with the principle of stationarypotential energy. The method may provide a good approximate solutionfor the displacements of a body if admissible functions are chosen properly.However, the accuracy of stress distribution calculated from the approxi-.mate displacements is not as reliable. This is obvious if we remember that,in the governing equations obtained by the approximate method, the exactequations of equilibrium and mechanical boundary conditions nave beenreplaced by their weighted means and that,the accuracy, of an approximatesolution decreases with differentiation. Thus, the equations of equilibriumand mechanical boundary conditions are generally' violated at least locallyin the approximate solution. In understanding approximate solutions thusobtained, the principle of Saint-Venant is sometimes helpful. It states :(14)"If the forces acting on a small portion of the surface of an elastic body arereplaced by another statically equivalent system of forces acting on the sameportion of the surface, this redistribution of loading produces substantialchanges in the stresses locally, but has a negligible effect on the stresses at dis-tances which are large in comparison with the linear dimensions of the surfacepn which she forces are changed."

Due to the author's preference, approximate governing equations ofelasticity problems will be derived very frequently from the principle ofvirtual work rather than from the variational principle, since the former


holds independently of the stress—strain relations of the body and theexistence of potential functions. An approximate method of solution usingthe principle of virtual work will be called the generalized Galerkin's method.tAs far as conservative problems in elasticity are concerned, results obtainedby the combined use of the principle of virtual work and the generalizedGalerkin's method are equivalent to those obtained by the combined use ofthe principle of stationary potential energy and the Rayleigh—Ritz method.

It is quite natural in theories of plasticity to make the principle of virtualwork a basis for the establishment of variational principles. If the problemis confined to the small displacement theory, the principle of complemen-tary virtual work be employed as another basis. Since stress—strainrelations in the theories of plasticity are more complicated than those inthe theory of elasticity, it may be expected that the establishment of avariational principle in plasticity is more difficult. Several variationalprinciples which have been established for the theories of plasticity canbe shown to be formally derivable in a manner similar to those in the theoryof elasticity, although rigorous proofs should follow for showing the validityof the variational principles.

The most successful application of variational formulations in the fi.wtheory of plasticity is the theory of limit analysis for a body consisting ofmaterial which obeys the perfectly plastic Prandtl—Reizss equation. Limitanalysis concerns the determination of an eigenvalue called the collapseload of the body. Two variational principles provide upper and lower boundformulae for locating the collapse load.

Since a great many papers have been written on variational treatment ofproblems in elasticity and plasticity, the bibliography of this book is notintended to be complete. The author is satisfied with citing only a limitednumber of papers for the reader's reference. Literature such as Refs. 22 and23 may be helpful for reviewing recent developments of the topic.

The variational method can, of course, be applied to problems otherthan those mentioned herein.. For example, it has been applied to problemsin fluid mechanics, conduction of heat and so forth. (24-26) As a recentapplication of engineering concern, we may add that problems of the per-formance of flight have been extensively treated in the literature bythe optimization techi.

Bibliogrsphy

1. R. and D. Methods of Mathematical Physics, Vol. 1, Interscicnce,New York, 1953.

2. C. LANCZOS, The Variational Principles of Mechanics, University of Toronto Piess,1949.

t This is also called the method of weighting functions. It is a special case of the approxi-mate method of solution called the method of weighted residuals.'2"

INTRODUCTION

3. 0. BOLZA, Lectures on the Calculus of Variations, The University of Chicago Press,946.

4. 6. A. Buss, Lectures on the Calculus of Variations, The University of Chicago Press,1946.

5. C. Fox, An Introduction to the Calculus of Variations, Oxford University Press, Lon-don, 1950.

6. R. WrINsrocK, Calculus of Variations with Application to Physics and Engineering,McGraw-Hill, 1952.

7. P. M. Moasa and H. FESHBACH, Methods of Theoretical Physics, Vols. 1 and 2,McGraw-Hill, 1953.

8. S. 6. MIKHLIN, Variational Methods in Mathematical Physics, Pergamon Press, 1964.9. H. GOLDSTEIN, Classical Mechanics, Addison-Wesley, 1953.

10. J. L. SYNoC and B. A. Gnirrm, Principles of Mechanics, McGraw-Hill, 1959.11. H. L. LANGHAAR, Energy Methods in Applied Mechanics, John Wiley, 1962.12. C. B. BIEZENO and R. GRAMMEL, Technische Dynamik, Springer, Berlin, 1939.13. R. V. SOUTHWELL, Introduction to the Theory of Elasticity, Clarendon Press, Oxford,

1941.14. S. TIMOSHENKO and J. N. GOODIER, Theory of Elasticity, McGraw-Hill, 1951.15. N. J. HOFF, The Analysis of Structures, John Wiley, 1956.16. C. E. PEARSON, Theoretical Elasticity, Harvard University Press, 1959.17. J. Fl. ARGYRIS and S. KELSEY, Energy Theorems and Structural Analysis, Butterworth,

1960.18. V. V. NovozluLov, Theory of Elasticity, translated by J. K. Lusher, Pergamon Press,

1961.19. J. H. GREENBERG, On the Variational Principles of Plasticity, Brown University, ONR,

NR-041-032, March 1949.20. R. Hiu., Mathematical Theory of Plasticity, Oxford, 1950.21. M. BECKER, The Principles and Applications of Variational Methods, The Massachu-

setts Institute of Technology Press, 1964.22. Applied Mechanics Reviews, published monthly by the American Society of Mechanicq1

Engineers.23. Structural Mechanics in U.S.S.R. 1917—1 957, edited by I. M. Rabinovich. English

translation edited by 6. Herrn-iann was publLhcd by Pergamon Press in 1960.24. J. SERRIN, Mathematical Principles of Classical Fluid Mechanics, Handbuch der

Physik, Band Vll/I. Strömungsmechanik I, pp. 125—265, Springer, 1959.25. M. A. BloT, Lagrangian Thermodynamics of Heat Transfer in Systems including

Fluid Motion. Jdurnal of the Aeronautical Sciences, Vol. 25, No. 5, pp. 568—il, May1962.

26. K. WAsHizu, Variational Principles in Continuum Mechanics, University of Washing-ton, College of Engineering, Department of Aeronauticil Engineering, Report 62—2,June 1962.

27. G. LEIThIANN (Editor), Optimization Techniques with Applications to Aerospace Sys-tems, Academic Press, 1962.

CHAPTER 1

SMALL DISPLACEMENT THEORY OFELASTICITY IN RECTANGULAR

CARTESIAN COORDINATES

1.1. Presentation of a Problem in Small Displacement Theory

In the beginning of his classical work,U) Love states: "The MathematicalTheory of Elasticity is with an attempt to reduce to calculationthe state of strain, or relative displacement, within a solid body which issubject to the action of an equilibrating system of forces, or is in a state ofslight internal relative motion, and with endeavours to obtain results whichshall be practically important in applications to architecture, engineering,and all other useful arts in which the material of construction is solid."This seems to have been a guiding definition of the theory of elasticity.

In the first and second chapters of this book we shall deal with the smalldisplacement theory of elasticity and derive the principle of virtual workand ielated variational principles for the problem of an elastic body instatic equilibrium under body forces and prescribed boundaryRectangular Cartesian coordinates (x, y, z) will be employed for definingthe three-dimensional space containing the body. In the small displacementtheory of elasticity displacement components, u, v, w, of a point of the bodyare assumed so small that we are justified in linearizing equations governing theproblem. The linearized governing equations may be summarized as follows:

(a) Stress. The state of internal force at a point of the body is defined bynine components of stress:

43( xx,t:y, (1.1)

which should satisfy the equations of equilibrium:3;,,—+ )+ +1=0,

ox c1y

(l.2)t

arxz

ax

t Throughout the present book, an overbar indicates that the barred quantity is pre-scribed, unless otherwise stated.

8

SMALL THEORY OF ELASTICITY

and = r,,, r,,, r,, = T,X, (1.3)where 1, 1 and 2 are components of the body forces per unit volume. Weshall eliminate t,,, and r,, by the use of Eqs. (1.3), and specify the stateof stress at a point of the body with six components (a,, a,, T7,,Then, Eqs. (1.2) become:

a, az

+ + + 7=0, (1.4)ax ay 8z

+ +2=0.ôx ôz(b) Strain. The state of strain at a point of the body is defined by six com-

ponents of strain (e,, e,, c,, >',,, Vxx' )'x,).(c) Strain—displacement relations. In small displacement theory the strain—

displacement relations are given as follows:

CX = —, C7 = —, c, =

ow t3v Ou Ow Ov c3u

= + 72X = + 7,, = +

(d) Stress—strain relations. In small displacement theory, the stress—strainrelations are given in linear, homogeneous form:

013 014 015 a16 rexa, a21 a22 a23 024 a25 026 e,

031 a32 a33 a34 035 a36(1.6)

a41 a42 043 a44 a45 a46a52 053 a54 056 Y:xa62 063 a64 a65 a66

The coefficients of these equations are called elastic constants. Among them,

there exist relations of the form:= a,., (r, s = 1, 2, ..., 6). (1.7)

Eqs. (1.6) may be inverted to yield:

b11 b12 b13 b14 b15 b16b21 b22 b23 b24 b25 b26 a,b31 b32 b34 b36

, (1.8)b42 b43 b44 b45 b46

Vzx b51 b52 b53 b54 b55 b56 ;,b62 b63 b64 b65 b66 Tx,

whereb,3 = b,,., (r,s = 1,2, ...,6). (1.9)

10 VARIATIONAL METHODS IN ELASTICiTY AND PLASTICITY

For an isotropic material, the number of the independent elastic constantsreduces to 2, and the stress—strain relations are given by:

2G [ex +1 — 2v

(Ex + C, + = Gy,,,

o,=2G[e,+ (1.1o)t

1 2v(e, + a, + €x)] , TX, =

or, inversely,= — + a3, =

Ee, = a, — v(or, -w = (1.t1)t= — + 0,), r,,.

(e) Boundary conditions. The surface of the body can be divided into twoparts from the viewpoint of the boundary conditions:'the part S1 over whichboundary conditions arc prescribed in terms of external forces and the partS2 over which boundary conditions are prescribed in terms of displacements.Obviously S = S1 + S2. DenQting the components of the prescribed ex-ternal forces per unit area of the boundary surface by 1,, F, and 2,,, themechanical boundary conditions are given by

X,=Z,,, Y,=F,, Z,=Z, on S1, (112)where

-l-r,n, (1.13)

1, m, n being the direction cosines of the unit normalv drawn outwards onthe boundary: 1 = cos (x, v), m cos (y,,) and n cos (z, v). On the otherhand, denoting the components of the prescribed displacements by ü, i)

and the geometrical conditions are given byu=ü, v=1, on S2. (1.14)

Thus, 'we obtain all the governing equations of the elasticity problem insmall displacement theory: the equations of eqUilibrium (1.4), the strain—displacement relations (1.5) and the stress—strain relations (1.6) in the interiorV of the body, and the mechanical and geometrical boundary conditions,(1.12) and (1.14), on the surface S of the body. These conditions show thatwe have 15 unknowns, namely, 6 stress components, 6 strain componentsand 3 displacement components in the 15 equations (1.4), (1.5) and (1.6).

t Young's modulus E, Poisson's ratio r and the modulus of rigidity G are related by theequation E = 2G(l + v). Thus there are only two independent elastic constants.

It is noted that the symbol v is used in the present book to denote the Poisson's ratioas well as the wilt normal drawn outwards on the boundary.

SMALL DISPLACEMENT THEORY OF ELASTICITY 11

Our problem is then to solve these 15 equations under the boundary condi-tions (1.12) and (1.14). Since all the governing equations have linear forms,the law of superposition can be applied in solving the problem. Thus, weobtain linear relationships between the prescribed quantities such as theapplied load on S1 and resulting quantifies such as stress and displacementcaused in the body.

1.2. Conditions of Compatibility

We observe from Eqs. (1.5) that when a continuum deforms, the sixstrain components (es, e,, Viz' Vx,) cannot behave independently,but mu$ be derived from three functions u, v and w as shown. This state-ment can be expressed in a different way as follows: Let tie continuumunder consideration be separated into a large number of infinitesimal rec-tangular parallelepiped elements before deformation. Assume that each ele-ment is given six strain components (es, ..., Yx,) of arbitrary magnitude.Then, trials to reassemble the elements again into a continuum are assumedto be made. In general, such trials cannot be successful. Some relations shouldexist between the magnitude of the strain components for a reassemblageto be successful. Thus, a problem will arise which may be stated as follows:What are the necessary and sufficient conditions for the elements to bs reass-

embled into continuous body?The necessary and sufficient conditions that the six strain components

can be derived from three single-valued function as given in Eqs. (1.5) arecalled the conditions of compatibility. It is shown in Refs. 1 through 5, forexample, that the conditions of compatibility are given in a matrix form as,

[RJ I?, = 0, (1.15)U,UXRZ

wherea2

R—

a)2 az2

D — eX v€Z .1it, — — ,

3z2 c3x2 az 3x

a2R

— Yx,ax2' ay2 — 3xay ' (1.16)

— a+X

— ay ôz 3x k ox+

ôz

— 1 f ôy,z \— az ôx

+ôy —

+ I'= — + ! .!. + '3Vzx —

öx3y ay

12 VARIATIONAL METHODS ELASTICITY ANI) PLASTICITY

The proof that the conditions (1j5) are necessary follows immediatelyfrom Eqs. (1.5) by direct differentiation. The proof that they are sufficientis rather lengthy and is not given here. The interested reader Is advised toread the cited references.

It is noted at the end of this section that there exist identities betweenR,,, ..., U,:

3Rx÷8Uz OU,_0+

11(.7)I3U,

+8(1,

+8R,

— o8y- 8z —

These identities can be proved easily by direct calculations. They show thatthe quantities R,, ..., and U, are not mutually independçnt, and thatthe conditions of compatibility (1.15) can be replaced

= R, R2 0 in V. (1.1$a)

and

onS; (l.18b)

or alternatively

U,=U,=U,=0 mV, (l.19a)

and

onS. (l.19b)

1.3. Stress Functions

We know from Eqs. (1.4) that when the body forces are absent, the equa-tions of equilibrium can be written as:

+ + = 0, (1.20)ax 8y

+

These equations are satisfied identically when stress components are cx-

SMALL DISPLACEMENT ThEORY OP ELASTICITY

pressed in terms of either Maxwell's stress functions and X3 defined by

+ = 82Xa8y2 8z2 ' — dy Oz

a + — 82X2' 8z2 ox2' —

— ozax (1.21)

— 82Xz 82Xi — 82X3

or Morera's stress functions and V'3 defined by

— — — 1 8 f- + +

a— 82, = — I 8

122— .1 — + ( )

— 02, 1 0 8V'3ax — ' — __1_ + — — —OxOy Oz

It is interesting to note that, when these two kinds of stress functions arecombined such that

Ø2%3 0X2'IX = 8y2 +

0z2 Oy 8z' "'' (1

T — I 8 / 0Pi 0V2 Ou3 \Oy Oz

÷2 —

+ +

the expressions (1.16) and (1.23) have similar forms.In a two-dimensional stress problem, where the equations of equilibrium

are

—0 + —0 124Ox+ Oy ' ox 8y '

the so-called Airy stress function defined by

82F 82F 82F'I, =

—(1.25)

satisfies the equations of equilibrium identically.

1.4. PrincIple of Virtual Work

In this section we shall derive the principle of virtual work for the problemdefined in Section 1.1. We consider in equilibrium under prescribedbody forces and boundasy coodiie.s, and denote the stress components

14 VARIATIQNAL METHODS IN ELASTICITY AN!) PLASTICITY

by a,, ..., and Obviously,

mV, (1.26)ox 83' t3z

and

X,—i,=O,...,Z,—2,=OonS1. (1.27)

Now, the body is assumed to execute an arbitrary set of infinitesimal virtualdisplacements ôu, öv and 8w from this equilibrating configuration.

Then, we have

JJf+ + + 4?) ôu +(. .

= 0, (1.28)

=; dx dy dz and;dSarethe elementary votU&ne and the elementaryarea of the.surface of the respectively.

Here, we shall choose the arbitrary set of virtual displacements such thatthe geometrical boundary conditions on S2 are not violated. Namely, theyare so chosen as to satisfy the equations:

3w=0 onS2. (1.29)

Then, bf geometrical relations

dy dz = ± 1 dS, dz dx = ± mdS, dx dy = ±n dS (1.30)

which hold on the boundary, and through integrations by parts such that

(1.31)t

we may transform Eq. (1.28) into

fff + + + T,z 43Y,z + Tzx + dV

— fff(Zau + ?öv + 28w) dV

— ff(X,ou ÷ Y,ÔV + 2,8w)dS = 0 (1.32)

t This is an application of the divergence theorem of Gauss expressed by the equation

Mm+

SMALL DISPLACEMENT THEORY OF ELASTICITY

where t3ôu c3öv c3ôwocx = -i---, Or,

= =aow aou ôOw aov— ÷ 0,/tx = + = —i-- +

-b---.(1.33)

This is the principle of virtual work for the problem defined in Section 1.1.The principle holds for arbitrary infinitesimal virtual displacements satisfyingthe prescrib'ed geometrical boundary conditions.*

Next, we shall consider what kind of relations will be obtained if theprinciple of virtual work is required to hold for any admissible virtualdisplacements. Reversing the above development, we may obtain Eq.(1.28) from Eq. (1.32). Since bu, t3v and are chosen arbitrarily in V andon S1, all the coefficients in Eq. (1.28) are required to vairsh. Thus, wehave another statement of the principle of virtual work: Introduction of thestrain—displacement relations (1.5) and the geometrical boundary conditions(1.14) into the principle of virtual work yields the equations of equilibrium(1.4) and mechanical boundary condizionsjjstrain—displacement relations have beenbrium may be obtained from thespecial mention that the principle of w@k tillmaterial stress—strain relations.

1.5. Approximate Method of Solution Basel on the Principle of Virtual yvorxi

An approximate method of s.lution car( be formulated thelprinciple of virtual This approath will beGalerkin method.t The first step of the method is mplacement components u, v and w can be expressed approximately asfollows: -

- u(x, y, z) = uo(x, y, z) + a.u,(x, y, z),

v(x, y, z) = v0(x, y, z) + b,v,(x, y, z), (1

y, z) = w0(x,y,z) +2j c,w,(x, y, z),

where u0, v0 and w0 are so chosen that= i, v0 = w0 = on S2, (1.35)

* For a physical interpretation ot the principle, see Appendix .1.t This is a generalization of the so-called Galerkin method which requires that approxi-

mate displacements of Eqs. (1.34) are chosen to satisfy not only the geometrical boundaryconditions on S2, but also by substitution of the stress—strain relations the mechanicalboundary conditions on S1. For Galerkin's method, see Refs. 5, 7 through 11, for in-stance.

It is noted that the number of the terms under the three summation signs need notbe equal to each other. In other words, some terms among u,, v, and w, may be missing.

16 VARIATiONAL METHODS IN ELASTICITY AND PLASTICiTY

and u,, v,, w,; = 1, 2, ..., n are linearly independent functions whichsatisfy the conditions

14=0, v,=0, w,=0,(r=1,2,...,n) onS2. (1.36)

The con*tants a,, b, and c, are arbitrary. We then have:a a a

= âa,u, i3v = öb,v,, ow = 2 .5c,w,. (1.37)r.1 i—i

Introducing Eqs. (1.34) into the principle (1.32), we havea

2 [L, Oa, + M, Ob, + N, Oc,) 0, (1.38),—1

where

N, + + — zw,)dv_ffZ,w,ds.

(1.39)

Since k,, Ob, and Oc, are arbitrary, we obtain the following equations:

= 0, M, = 0, N, = 0, (r = 1,2,...,n). (1.40)

note that the expressions (1.39) are transformed via integration by partsInto,

L, +

M, =— fff( + + + v, dY + ff( Y, — F,,) v, dS,

N,=+

5, (1.41)

The second step is to calculate the stress components in terms of Eqs.(1 .34) by the use of Eqs. (1.5) and the stress—strain relations. Here we assumeisotropy of the material to obtain the following stress.-displacemept relations:

b 8w,\11

= •••, .... (142)

SMALL DISPLACEMENT THEORY OF ELASTICITY

Introducing Eq. (1.42) into Eq. (1.40), we have a set of 3n simultaneouslinear equations with respect to the 3n unknowns a,, b, and Cr; r = 1, 2, ... ,n.By solving these equations, values of a,, b, and c, are determined. By sub-stituting the constants thus determined into the expressions (1.34), an ap-proximate solution for the displacement is obtained.

By a proper choice of the functions u0, v0, w0, u,, v,, it,; r = 1, 2, ...,and the number n, it is possible to obtain good approximate solutions forthe deformation of the body. However, the accuracy of the stresses calcul-ated by the use of Eqs.(l.42), employing the values of a,, brand c, thus deter-mined, is in general not as good. This is obvious if we remember that wehave replaced the equilibrium conditions (1.4) and the mechanical boundaryconditions (1.12) the 3n weighted expressions shown in Eqs. (1.41), andthat the accuracy of an approximate solution decreases differentiation.The equations of equilibrium as well as the mechanical boundary conditionsare generally violated, at least locally, in the approximate solution.

The accuracy of the approximate solution may be improved by increasingthe number of terms n. If Eqs. (1.34) represent the set of all admissiblefunctions when n tends to infinity, we may hope that the approximate solu-tion will approach close to the exact solution for a sufficiently large n, andtend to it when the number of terms increases without limit. However,experience and intuition are required if one wishes to obtain an accurateapproximation while retaining only a small number of terms in Eq. (1.34).

Modifications of the above method are frequently employed. For example,we might choose

u(x, y, z) =

In /v(x,y,z) (1.43)

I, ni-O /1

w(x, y, z) wm(x, y)

where m = 0, I, 2, ..., n are prescribed functions of z, while Urn, Vm

and wm are undetermined. Equations governing Urn, Vm and iVm are derivedfrom the principle of virtual work. We shall cite frequent examples of thismethod in Chapters 7, 8 and 9.

1.6. Principle of Complementary Virtual Work

Within the realm of small displacement theory we can formulate anotherprinciple which is complementary to the principle of virtual work in defin-ing the problem presented in Section 1.1. We consider the body in equili-brium under the prescribed body forces and boundary conditions, and denotethe strain and displacement components by ..., and u, v, w, respect-

18 VARIATIONAL MLTHODS IN ELASTICITY AND PLASTICITY

ively. Obviously,

(,u Oh .

in V, (1.44)

u—i=O,..., w—*=O on S2. (1.45)

Now, the body is assumed to take an arbitrary set of infinitesimal virtualvariations of the stress components from this equilibrat-ing configuration. Then we have

P Pr / Ou \ I I Ov \ 1

JJJ — +—

+ + — dY

u)Ox, + (v — b)0Y+(w — *)Oz,]dS = 0, (1.46)

which, via integrations by parts, is tranformed

f/f + ôø, + ....+. + '÷)

..t.. + (...) dI'_ff (UOX, ± vOY, + wOZV)dS

— ff(u .31, + vOY, + =0. (1.47)

Here, we shall choose the arbitrary set of virtual strçsses such that the equa-tions of equilibrium and the mechanical boundary conditions are not violated.

they art so chosen as to satisfy the following equations;

+ + —Ox Oy Oz

(148)Oz

t,OTzx+ +

Ox Oy Oz

in tht interior of the bbdy V and

OX, = + ôTx,lfl + OT,xfl =

o Y, = + k,m + =0, (1.49)

— + + 0,

on S1. Then, Eq. (1.47) reduces to

fff ÷ e,k, + + Yx, dV

—ff (ii OX, + t .3 Y, + .3Z,) dS 0. (1.50)

- S2


The formula (1.50) will be called the principle of complementary virtualwork. The principle holds for arbitrary infinitesimal virtual stress variationssatisfying the equations of equilibrium and prescribed mechanical boundaryconditions. It is seen that the principle of complementary virtual work has aform which is complementary to the principle of virtual work given byEq. (1.32).

Next, we shall consider what conditions result if the principle of com-plementary virtual work is required to hold for an arbitrary set of admis-sible virtual stress variations. For such a formulation the Lagrange multi-plier method provides a systematic tool.t We shall treat Eqs. (1.48) and(1.49) as constraints and employ the displacements u, v and w as the Lag-range multipliers associated with these conditions. Thus, reversing theabove development, we obtain Eq. (1.46) from Eq. (1.50). Since the quanti-ties &,, ôa,, ..., have been independent of each other by intro-duction of Lagrange multipliers, all the coefficients in Eq. (1.46) are requiredto vanish. This leads to another statement of the principle of complemen-tary virtual work: Introduction of the equations of equiibriwn (1.4) and themechanical boundary conditions (1.12) into the principle of complementaryvirtual work yields the strain-displacement relations (1.5) and the geometricalboundary conditions (1.14). Consequently, once the equations of equiE-brium have been derived in the small displacement theory, the strain—displacement relations may be. obtained from the principle of cómplemen-tary virtual work. It is worthy of special mention that the principle ofcomplementary virtual work hçlds irrespective of the material stress—strainrelations.

1.7. Approximate Method of Solution Based on the PriOcipleof Complementary Virtual Work

An approximate method of solution can be formulated by employingthe principle of complementary virtual work. This approach is similar tothe one mentioned in Section 1.5 and may also be called the generalizedGalerkin method. For the sake of simplicity, we shall consider a two-di-mensional elasticity problem of a simply connected body4 The side boun-dary of the body is cylindrical with the generating line parallel to the

•f ForLagrange multiplier method, see Chapter4 of Ref. 12, and Chapters2and5ofRef. 13.

The two-dimensional elasticity problem defined here is a good approximation to theeo.called plane stress problem of a thin isotropic plate with traction-free tipper and Lowerswfaces. In a plane stress problem we assume 0 and obtain Es1 +On the othor hand, this elasticity problem can be shown to be mathema-tically equivalent to a plane strain problem of an isotropic body, by replacing E and v InEqs. (1.51) with E'I E/(l — ,2)rJ md ,'[ — sf(1 — v)Jj respectively, and employing thesuumptionas, 0 and i — +

20 VARiATIONAL METHODS IN ELASTICITY AND PLASTICITY

z-axis, and the deformation of the body is assumed independent ofz. Thestress components and are assumed to vanish. The remainingstress components a, and r, are assumed to be functions of (x, y)only, and related to the stran components as follows:

= — va,, Ee, = —var + o,,, = (1.51)where

Under assumption of absence of body forces, the equations of equilibriumthen reduce to Eqs. (1.24), which suggests the use of the Airy stress Iunctiondefined by Eqs. (1.25).

The boundary conditions on the side surface must be prescribed inde-pendently of z, and arp assumed to be given, for the sake of simplicity, interms of external forces only, namely

I, = 1', = F, (1.53)

on the side boundary C, where

In the above 1 and m are the direction cosines of the outward normal v tothe boundary C. If the contour of the side boundary C is given parame-trically in terms of the arc length s measured along C, such that

x = x(s), y = y(s), (155)we have

I = dy/ds, m = —dx/ds. (1.56)

The arc length s is measured as shown in Fig. 1.1. By introducing the Airystress function and Eqs. (1.56) into Eqs. (1.54), we obtain X, and Y, in

au av= e, = (132)

= a) + (1.54)

y Yp

0

Fio. 1.1. A two-dimensional problem.


terms of' F:• 02F dy 82F dx d f OF \

"= + =02F dy 82F dx d I oF

(1.5'?)=

— ox — = — 'LOX

We shall assU* an expression for the stress function of the following form:

F(x, y) = F0(x, y) + E a,F,.(x, 3?), (1.58)

where F0 Fr are chosen so that

d/4F0\_1—

——

= o, —(aFr) = 0, (r = 1, 2, ..., n) (1.59)

on the C, and a,.; r = 1,2, ..., nare arbitrary constants. The equa-tions (1.5w) that both OF,jOx and v3F,./Oy are constant along C. Sincethe a function ax + by .+ c, a, band c are rbitrary constants, is immaterial as far as the simply connectedbody is concerned, we may

F,=Ø, on C, (r= 1,2,...,n) (1.60)

without of generality.Introduction of Eq. (1.58) inta Eqs. (1.25) results in the following expres-

sions for the stress components:82F 02F0 82F,

= + a,

ÔX2, (1.61)

.32F "82F0 82F,=— ÔxOy — Ox Vy Ox

A set of admissible virtual stress variatiàn a then given by

,_l '.'3?

• ?— 2i vXu3?

Substituting Eq. (1.62) into the principle (130), and remembering that allthe surface boundary conditions are given in terms of forces only, we have

(1.63)


whereL, = Jf(ex + E,

3x2 — dx dy. (1.64)

In Eq. (1.64), the length of the body in the direction of the z-axis is taken asunity, and the integrations extend through the region of the body in the(x, y) plane. Since the variations of the constants, 5a,, are arbitrary, we ob-tain the following equations:

L, = 0, (r = 1, 2, ..., ii). (1.65)

We note that by the use of Eqs. (1.60) and via integrations by parts, theexpression (1.64) is transformed into

L — rn 32e c3€,'I F d d 1 66.. XY.

By the use of Eqs. (1.51) and (1.61), Eqs. (1.65) can be reduced to n simul-

taneous equations with respect to a,; r = 1, 2, ..., n. By solving these equa-tions, values of a, are determined. Substituting the value of a, thus deter-mined into Eqs. (1.61), we obtain an approximate solution for the stresses.By judicious choice ofF0, F1, ..., approximate solutions of considerableaccuracy may be obtained. The factors which govern the accuracy of theapproximate solution are similar to those mentioned at the end of Section1.5.

It is noted here that the strains calculated from, the approximate stresssolution and the stress—strain relations do not satisfy, in general, the con-ditions of compatibility, unless the number n is increased without limit.For example, as the expression (1.66) shows, Eqs. (1.65) are weighted means,and consequently, approximations to the condition of compatibility forthe two-dimensional problem. Although we have taken a two-dimensionalproblem as an example, the extension to three dimensions is straightforward.

1.8. Relations between Conditions of Compatibility and Stress Functionst

We have observed in Section 1.4 that the equations of equilibrium canbe obtained from the principle of virtual work (1.32). In view of the devel-opment in Sectioni .4, we might ask what kind of relations will be obtainedif the conditions of compatibility (1.15), instead of u, vand w, are introducedinto the principle (1.32) by the use of Lagrange multipliers. The body forceswilt be assumed absent throughout the present discussion.

We shall employ Eqs. (I.18a) as the field conditions of compatibility andwrite the principle of virtual work (1.32) as follows:

— Xi — X2 — ÔRJ dV+ (surface terms) = 0, (1.67)

t Refs. 14 through 18.


where Xi' X2 and X3 are the Lagrange multipliers. After some calculation,including partial integrations, Eq. (1.67) is transformed into:

Jfff[ffx — + [o, -+

)jâe,

+ + +1

dV

÷ surface terms = 0. (1.68)

Therefore, since the quantities ôe,••, and öy, are arbitrary, we have

— — 16— öz2 '' — —

thus proving that the Lagrange multipliers X2 and X3 are Maxwell'sstress functions. A similar procedure employing Eqs. (1.19a) as the fieldconditions of compatibility leads to Morera's stress functions. The presentmethod of finding stress functions is applicable to any problem where theprinciple of virtual work and conditions of compatibility have been formu-lated.

On the other hand, we have observed in Section 1.6 that the strain—displacement relations may be obtained from the principle of' compleinen-tary virtual work if the equations of equilibrium have been derived. Now, weshall inquire what conditions result if stress functions are used in place ofthe equations of equilibrium and Lagrange multipliers in conjunction withthe principle of complementary virtual work.

We shall employ as an example Maxwell's stress functions defined byEqs. (1.21). The principle (1.50) can now be written as follows:

-

fJ/ (+ X2) + —

c3x ] dxdy

+ (surface 'terms) = 0. (1.70)

After some calculation, including partial integration, Eq. (1.70) is trans-formed into

rn 1] a2E, Ifiii R+

— äy azi + + 32 — 3z 39 Xz

+ —

(1.71)Since 3Xi. and are arbitrary, we have

(1.72)

and conclude that Eq. (1.71) provides Eqs. (l.18a) as the field conditionsof compatibility. A similar procedure employing Morera's stress functionsleads to the conditions of compatibility given by Eqs. (1.19 a).


The reader has already seen in Section 1.7 that the employment of Airystress function in the principle of complementary virtual work leads to thecondition of compatibility for the two-dimensional problem.

It is noted here that for a multiply connected body, such as a body withseveral holes, formulation via the principle of complementary virtual workcombined with stress functions provides other geometrical conditions, theso-called conditions of compatibility in the 9.20) A simple example ofthese conditions will be illustrated in Section 6.3. In Chapter 10 we shallshow that the conditions of compatibility in the large play an essentialpart in the theory of structures.

19. Some Remarks

We have obsçved in Sections 1.4 and 1.6 that the principles of virtualwork and complementary virtual work arc complementary to each otherin defining the elasticity problem. .Here; we consider extensions of theseprinciples.

It has been assumed in deriving the principle of virtual work that thevirtual displacements are so chosen as to satisfy Eqs. (1.29). This restrictionmay be removed to obtain an extension of the principle of virtual work asfollows:

fff + a, Os, + ••. + dY

—fff(Xtu+FOv+Zdw)dV

— ff (Z, Ou + F, Ov + 2,8w)SI

— ff (X, .3u + Y, tv + Z, Ow) dS =b. (1.73)

On the other band, we have assumed in deriving the principle ofmentary virtual work that the virtual variation of the stress componentsarc so chosen as to satisfy Eqs. (1.48) and (1.49). These restrictions may beremoved to obtain an extension of the principle of complementary virtualwork as follows:

+ a, &i, + + dV

—fff(uOX+vOY+wOZ)dV

— ff (u OX, + v 8)', + w OZ,) dS

.

I'OY. .4: =0,

SMALL DISPLACEMENT THEORY OF ELASTiCITY 25

where OX, OY and OZ are given by

+ax •ay e9z+ —

+ +ax ay

+ + + oz = o.ax az

In view of the above developments, we find that these principles arespecial cases of the following divergence theorem:

fff + a,e, + + dV

= fff (Iv + Lv + Zw) dV

+ (X,u + Y,v + Zw) dSSI

+ ff (X,u + Y,v + Z,w) dS, (1.76)

where a,, ..., r,) are an arbitrary set of stress components whichsatisfy the equations of equilibrium. (1.4), and (X,, Y,, Z,) derived fromthe stress components by the use of Eqs. (1.13), while (u, v, w) are an arbi-trary set of displacement components, and (si, e,, ..., y17) are derived fromthese displacement components by the use Eqs. (1.5). The proof of thetheorem (1.76) is given in a manner similar to those mentioned in Sections 1.4and It should be noted here that the sets (a1' 0,, ..., and (e,, e,,

a, v, w) are independent of each other. Namely, no relations areassumed to exist between these two sets. The divergence theorem has a widefield of application in continuum mechanics. We find that this theorem con-stitutes a basis for the unit displacement method and the unit load methodtwhich play important roles in the analysis of structures." 1)

We note that continuity of stresses as well as displacements is assumed forthe derivation of the divergence theorem. If some discontinuity exists instresses and/or displacements, Eq. (1.76) should Contain additional terms.For example, consider that the ... are continuous,while the displacement components (u, v, w) are discontinuous across aninterface S(12) which divides the body V into two parts V(1) and V(2).

Then, a term

ff + Y,[vJ + Z,[w)) dS (1.77)5(12)

f This method is also called the dummy load

26 VARIATIONAL METHODS IN ELASTICITY ANI) PLASTICITY

should be added to the nghthand side of Eq. (1.76), where (X,, Y,, 4)are on the surface S(12) with unit normal v drawn from V(1) to V(2,,and the square brackets denote the jumps of ii, v and w across the surface:Eu] = — U(2), [v] = V(I) — V(2), [w] W(J) — W(2). A similar care should betaken when the stress components show discontinuity.

Bibliography

1. A. E. H. LOVE, A Treatise on the Mathematical Theory of Elasticity, Cambridge Uni-versity Press, 4th edition, (927.

2. S. TIMOSHENKO and .1. N. Gooozaa, Theory of Elasticity, McGraw-Hill, 1951.3. S. MORIOUTI, Fundamental Theory of Dislocation of Elastic Bodies (in Japanese),

Su.qaku Rikigaku, Vol.!, No.2, pp. 87-90, 1947.4. C. PEARSON, Theoretical Elasticity, Har.'ard University Press, 1959.5. V. V. Novovntov, Theory of Elasticity, Translated by .1. K. Lusher, Press,

1961.6. K. WAsmzu, A Note on tbe Conditions of Compatibilit%, ,Jöurrwi of Mathematics

and Physics, Vol. 36, No. 4, pp. 306—12, January 1958.7. W. I. DUNCAN, Galerkin's Method in Mechanics and Differential Equatioiu, Aero-

nautical Research Committee, Report and Memoranda No. 1798, 1931.8. C. BIEZBNO and R. Technirche Dynamik, Springer-Verlag, 1939.9. L. COLL.ATZ, NwnerLcche &han&wtg von Springer-Verlag,

1951.10. N. J. Horv, The Analysis of Structures, John WIley, 1956.11.1. H. Aaovius and S. KaLsar, Energy Theorems and Structural Analym, Butterwozlh.

1960.12. R. COURANT and D. HILBERT, Methods of Moihematical Physics, VoL 1, Intcrecience,

New York, 1953.13. C. LANczoS, The Variational Prusciples of Mechanics. University of Toronto Press,

1949.14. R. V. SOUTHWELL, Castigliano's Principle of Minimum Strain Energy, Pmceedings

of the Royal Society, VoL 154, No. 881, pp. 4-21, March 1936.15. R. V. Soumww.., Castigliano's Principle of Minimum Strain Energy and Conditions

of Compatibility for Strains, S. Timoshenko 60th Aniversary Volume, pp. 211—17,

1938.16. W. S. D0RN and A. SCHILD, A Converse to the Virtual Work Theorem for Deform-

able Solids, Quarterly of Applied Mathenwàtics, Vol. 14, No. 2, pp. 209—13, July 1956.17. C. TRUESDELL, General Solution for the Stresses in a Curved Membrane, Pwceedh?€s

of the National Academy of Science, Washington, Vol. 43, No. 12, pp.1070-2, Decem-

ber 1957.18. C. TRUESDELL, Invariant and Complete Stress Functions for General Continua,

Archives for Rational Mechanics and Analysts, Vol.4, No.1, pp. 1—29, November 1959.

£9. S. On Castigliano's Theorem in Three-Dimensional Elastostatics (inJapanese). Journal of the Society of Applied Mechanics of Japan, Vol. 1, No. 6, pp.175—80, 1948.

20. Y. C. FUNG, Foundations of Solid M.'chanics, Prentice-Hall Inc., 1965.21. W. PRAGER and P. C. HODGE in., Theory of Perfectly Plastic Solids, John Wiley &

Sons, 1951.

CHAPTER 2

VARIATIONAL PRINCIPLES IN THESMALL DISPLACEMENT THEORY

OF ELASTICITY

2.1. Principle of Minimum Potential Energy

We shall treat variational principles in the small displacement theory ofelasticity in the present chapter. In this section the principle of minimumpotential energy will be derived from the principle of virtual work establishedin Section 1.4.

First, it is observed that wd can derive a state function e,, ...,from the stress—strain relations (1.6), such that

= + + •.. + (2.1)

where

2A = (a1 + a1 2e, + + a1

+...+ + a62e, + + ..

For the stress—strain relations of an isotropic material, namely Eqs.(Iwe have

A = 2(1 + — 2v) + + + +

2kfl,: Yzx '

We shall refer to A as the strain energy function.t From physical considera-tions which will be given in Chapter 3, we may assume the strain energyfunction to be a positive definite function of the strain components. Thisassumption involves some relations of inequality among the elastic con-stants.U) For later convenience we introduce a notation A(u, v, w) to indicatethat the strain energy function is expressed in terms of the displacementcomponents by introduction of the strain—displacement relations (1.5). For

t The quatitity 4 is also cafled the strain energy per unit volume or the strain energydensity.

27

28 VARIATIONAL METHODS IN ELASTICITY 4iND PLASTICITY

example, we have

Ev 8u äv 8w2A(u, v, w)

= 2(1 + v) (1 - 2v) + +1/ôu\2 18v\2 iv3wt2+Gu—j +l—J +1—['aX! \Dy/ \8Z

G118v 8w\2 (*3w 10u ôv\21+—Ii—+——J +—+—J +i—+—J i, (2.4)2 i \ ôz *3)' / \ *3x 0z / k *3y *3x i .j

for an isotropic material.When the existence of the strain energy function is thus assured, the

principle of virtual work (1.32) can be transformed into:

tlfffA(u,v, w)dV — fff(Zou + ?ôv + Zôw)dV

— ff(1,ou + ?,ôv + Z,dw)dS = 0. ,(2.5)St

This expression is useful in application to elasticity problems in which ex-ternal forces are not derivable from potential functions.

Next, we shall assume that the body forces and surface forces are deriv-able from potential function v, w) and !F(u, v, w) such that

—o = lou + FOv + 20w, (2.6)

(2.7)

Then, the principle (2.5) can be transformed into

(2.8)where

if ff f [4(u, v, w) + (P(u, v, wfl dV + ff v, w) dS, (2.9)V

is the total potential energy. The principle (2.8) states that wnong all shedisplacements u, v and w which satisfy the prescribed geometrical

boundary condliions, the actual displacements make the total potential energy

Hereafter, we shall confine our elasticity problem by assuming that thebody forces (1, F, 2), the surface forces (I,, F,, 2,) and the surface dis-placements (ii, t,, are prescribed, and kept unchanged in magnitudes anddirections during variation. Then, potential energy functions ate derivedfor these forces as follows: -

(2.10)

(2.11)

and we have a variational principle called the principle of minimum poten-tial energy: Among oil the admissible dispkfeement functions, the actual

VARIATIONAL PRINCIPLES 29

dLrplacements make the total potential energy

H = fff A(u, v, w) dV— fff (lu + Yv + 2w) dV

— ff (lu ÷ ?,v + Z,w) dS, (2.12)5*

an absolute minfrnum.For the proof of the principle of minimum potential energy, let the dis-

placement components of the actual solution and a set of admissible, ar-chosen displacement components be denoted by u, v, wand

u v w + Ow. We.then have

i7(u*, v, w) = II(u, v, w) + 017 + 6211, (2.13)

where 617 and 6211 are the first and second variations of the total potentialenergy. The first and second variations are respectively linear and quadraticIn du, dv, Ow and their derivatives, namely,

Iii It'x (-i.) + •••+ +

— 1du ÷ ... + 20w)]

— du + + Z, Ow) dS, (2.14)

= fff A(Ou, dv, Ow) dY, (2.15)

where . ..,and are the stress components of the actual solution Sinceou=ov=dw=OonS2,andthe'strcsscolnponcntsbclongtothcactualsolution, we find that the first variation Eq. (2.14),

• (2.16)

Furthermore, since 4 is a positive we must have(217)

where the equality sign holds only strain components whichare derived from du, dv and Ow iilCquently, we obtain

(2 18)

Since no restrictions have been of Ou, dv and Owin the above proof, we conclude energy is made anabsolute minimum for the actual

2.2. PrincIple ol Energy

It will now be shown that another vadational principle can be derivedfrom the principle of complementary vijtual wo!'lç (1.50). We observethat a state function i,, ..., ti,,) may be derived from the stress—strain


relations (1.8), such that

= + s, ôø, + + Yx, (2.19)

where

2B — + + ++...+ .+ + + (2.20)

For the stress—strain re'ations of an isotropic material, namely Eqs. (1.11),we have

B = ((as + o, +

+ 2(1 ± v) + + — — — (2.21)

We shall refer to B as the complementary energy function.t It is obviousthat, the strain energy function A defined by Eq. (2.2) is equal to the com-plementary energy function B defined by Eq. (2.20) and that, if the formeris positive definite, so is the latter. When the existence of the complementaryenergy function is thus assured, the principle of complementary virtual workcan be transformed into:

o fff Cr.,, ... , i.,,) dV— ff (u OX, + 10 Y, + OZ,,) dS = 0. (2.22)

V S2

Employing the assumption that the quantities u, t and are kept unchangedduring variation, we can derive from Eq. (2,22) a variational principlecalled the principle of minimum complementary energy: Among all the setsof admissible stresses Cry, ... , and which satisfy the equations ofequili-brium and the prescribed mechanical boundary conditions on S1. the set ofactual stress components makes the total complementary energy 14 definedby

= fff B(or, Cr,, ..., dV— ff (uX, + t7Y, + dS, (2.23)

-'

an absolute minimum.— For the proof, we denote the stress components of the actual solutionand a set of admissible, arbitrarily chosen stress components by ... ,

and + + Ocr,,= + Ot,,. Then, in a manner similar to the development in the

preceding section, we find that the first variation of the total complementaryenergy vanishes for the actual solution and that, since B is a positive definitefunction, the second variation of the total complementary energy is non-

t The quantity B is also called the complementary energy per unit volume, com-plementary energy density or the stress energy per unit volume.


negative. Thus, we are assured of the validity of minimumcomplementary energy.t

We observe that the arguments of A are strain components, while thoseof B are stress components. For the linear stress—strain relations, Eqs. (1.6)and (1.8), B is equal to A and has the same physical meaning: the strainenergy stored in a unit volume of the elastic body. It should be noted,however, that when stress—strain relations are nonlinear, B defined by Eq.(2.19) is different from A defined by Eq. (2.1). For example, in the simplecase of a bar in tension, we have

= =

The functions A and B are then given by

A B (2.25)

These are illustrated iii Fig. 2.1 by the shaded area OPS and unshaded areaOSR, respectively. It is- seen that A and B are complementary to each otherin respresenting the area OPSR, namely A + B =

FIG. 2.1. Strain and complementary energies in a uniaxial tension

2.3. Generalization of the Principle of Minimum Potential Energy

In the preseilt section we shall consider a generalization of the principleof minimum potential energy. To begin with, we shall summarize the stepsby which the principle of minimum potential energy has been obtained

t For an elasticity problem in which the part S2 of the boundary is held rigidly fixed,namely ü = = i' 0, the functional reduces to

-= fff 'i,, ..., dV

to yield the principle of least work.U)

SR

0

32 VARIATIONAL METhODS IN ELASTICITY AND PLASTICITY

from the principle of virtual work. We have assumed that: (I) it is possibkto derive a positive definite state function e,, ..., y,,) from the givenstress—strain relations; (2) the above strain components satisfy the conditionsof compatibility, that is, they can be derived from u, v and w as in the rela-tionships of Eqs. (1.5); (3) the displacement components u, v and w thusdefined satisfy the geometrical boundary conditions (1.14), and'(4) the bodyforces and surface forces can be derived from potential functions andas given by Eqs. (2.10) and (2.11). The principle of minimum potentialenergy then asserts that, on the basis of the above assumptions, the actualdeformation can be obtained from the iitinimizing conditions of the func-tional 11 defined by Eq. (2.12).

We shall now show that the subsidiary conditions stated in the assump-tions (2) and (3) above can be put into the framework of the variationalexpression by introducing Lagrange multipliers,t and the principle ofminimum potential energy can be By the Introduction ofnine Lagrange multipliers a1, a-,, ..., and p,, p,, p, defined in V andon S2. respectively, the generalized principle can be expressed as follows:The actual solution can be given by the stationary cotufitions. of a frnc:Ionat171 defined astt

= e,, ..., — (lu +Yv + dv

+ - +

I ( .3u ôw\+ r,z + —

—Tzz

+ — — *)Tx,] dV— + ?,v +

— ff((u — ü) Px + (v — O)p, + (w — *)pj dS. (2.26)

The independent quantities subject to variation in the functional (2.26) areeighteen in number, namely, e,, ..., yx,; u, v, w; a,, ..., r,,; Pz' J)yand p, with no subsidiary conditions. On taking variations with respect

f See Chapter 4, § 9 of Ref. 3 for the Lagrange multiplier method and involutory tans-fonnaLions. See also Appendix A.tt It should be noted that once Lagrange multipliers have been employed, the phrase

"miniminng conditions" used in the principle of minimum potential energy must bereplaced by "st tionary conditions".

VARLkflONAL PRINCIPLES 33

t6 these quantities, we have

oH1 1ff[(-u- + ... + — Ti,)

f I t3v Ou\

+ + Pz) Ow] dS, (2.27)

and the stationary conditions are shown to be

= + a12e, + + ... in V, (2.28)

t9v öu= •"' Yx, = + lfl V, (2.29)

in V, (2.30)

= 1,, ..., = 2, on S1. (2.31)

W=W. Ofl (2.32)

Px = X,, ..., p, = Z, Ofl S2. (2.33)

It is seen that Eqs. (2.28) and (2.33) determine physical meanings of theLagrange multipliers ..., p, and p1, and that the relation-Ships for to be stauonary are the equations which define the elasticityproblem stated in Section 1.1. If Eqs. (2.29) and (2.32) are taken asconditions, f11 is reduced again to 11 defir.ed by Eq. (2.12).

We may obtain another expression of the variational principle in whichthe Lagrange multipliers Px, p, and have been eliminated. For this pur-pose, we may require the coefficients of Ott, Ov and Ow in the integral term onS2 of the expression (2.27) to vanish. Thus, by the use of Eqs. (2.33) we may


transform the functional (2.26) into

..., yr,) — (Xu + Fv

+— — — — —

—rxr]

— ff(X1. u + ?,v + Z,w) dS

— ff[(u — i) + (v — 13) Y. + (w — iii) Z,) dS, (2.34)

or, through integrations by parts, into

=— fff + o,€, + + — e,,,

+ (c3clx + ÷ )u+( )w}dV

+ ff [(X, — u + (I', — t' + (Z,. — 2,) WI dS

+ff(x1.u + Y,17 + dS. (2.35)

The independent quantities subject to variation in the functional (2.34) or(2.35) are 15 in number, namely, e,, ..., u, v, w, o,, ..., andwith no subsidiary On taking variations of these 15 quantities,we find that the stationary conditions are given by Eqs. (2.28) through(2.32).

2.4. Derived Variational Principles

It will be shown in the present section that the 1-lellinger—Reissner principleand the principle of minimum complementary energy can be interpreted asspecial cases of the generalized principle (2.26). Let the coefficients of

..., in the expression of 6)1, be required to vanish. This means that... and are no longer independent, but must instead be determined

in a new forinulatzon by the conditions (2.28), namely,

= + ..' +(2.36)

Yx, = + •.- + Tx,.

the use of Eqs. (2.36), the strain components can be eliminated from thefunctional (246) to yield another functional of the principle, as follows:


rrr= + -h--- + ... + +

— 0,, ..., — (Its + Iv + dV

—

ff [(u — u) + (v — 1) p, + — dS, (2.37)f

where the quantity B is defined, as the above derivation shows, byB = + 0,C, + + Tx,Yxy A, (2.38)

in which the strain components are eliminated by the introduction of thestress—strain relationships (2.36). Since we have

öB = + + TxPÔVX), + €X&TX + + — ÔA

= + + p',. (2.39)

with the aid of Eq. (2.1), it is seen that the quantity B defined by Eq. (2.38)is the complementary energy function defined by (2.19). The functional(2.37) is equivalent to those in the Hellinger—Reissner

Because of the elimination of the strain components, the number of theindependent quantities subject to variation in the functional HR is reducedto 12: u, v, w; a,, ..., Px' I',, Pz -with no subsidiary conditions. Ontaking variations of these quantities, we find that the stationary èonditionsare

= + ... +

, (2.40)CU

together with Eqs. (2.30) through (2.33).The functional (2.37) may also be written via integrations by parts in the

following form:

—= fff 0,, ... , ti,) + + + + x) u

\ / \ Ox O' (3z / j

—[(X, — I,) u + (V1 — I,) v + (Z , — Z.) 14']

+ dS, - - (2.41)

t This is a special case of the Legendre transformation the calculus of vanation. Theunique inverse relations of Eqs. (2.28) should exist for the transformation to be justified.


where Eqs. (2.33) have been used for the elimination of Px, p, and p2. Thequantities subjected to variation in the functional (2.41) are u, v, w; a,,a,, ... and with no subsidiary conditions.

We shall now impose further restrictions on the number of independentfunctions in the generalized functional. All the coefficients of ôe,, öe,,

ôu, ôv and 6w in the expression of are required to vanish: thus thestrains and displacements are eliminated by the use of Eqs. (2.28), (2.30),(2.31) and (2.33) to transform the functional into a functional definedby

=— fff B(ar, a,, ..., dV ÷ ff (X,,i + Y,D + Z,*) dS, (2.42)

V 52

where the quantities subject to variation are a,, ... and under thesubsidiary conditions (2.30) and (2.31). Taking into account the positivedefiniteness of the function B, we may state this new principle as follows:Of all the admissible functions a,, ... and which satisfy Eqs. (2.30)and (2.31), the stress componenis of the actual solution make the functional

an absolute maxunwn. observe that the principle (2.42) is equivalentto the principle of minimum complementary energy derived in Section 2.2.In reversing the above development, we find that the functions u, v, w in thefunctional (2.41) play the role of introducing the subsidiary conditions(2.30) and (2.31) into the variational expression.

We have seen that, in the expression for H, admissible functions are chosento satisfy the conditions of compatibility, Eqs. (1.5), and the geometricalboundary conditions on S2, Eqs. (1.14), while in the expression ofadmissible functions are chosen to satisfy the equations of equilibrium,Eqs. (1.4), and the mechanical boundary conditions on S1, Eqs. (1.12). Con-sequently, 11 and are complementary to each other in defining the elas-ticity problem. The transformation of 17 into ft. is known as Friedrichs' trans-formation the actual solution characterized by the minimum propertyof H is also given by the maximum property of 17g.

Thus far, it has been shown that once the principle of minimum potentialenergy has been established from the principle of virtual work, it can begeneralized by the introduction of Lagrange multipliers to yield a family ofvariational principles which include the principle, theprinciple of minimum complementary energy and so forth. The avenue ofthis formulation is shown diagramatically in Table 2.1.

The principle of minimum .complementary energy was derived in Section2.2 from the principle of complementary virtual work. It is easily verifiedthat the principle of minimum potential energy can be derived from theprinciple of minimum complementary energy by reversing the developmentin the present and preceding sections. The equivalence between these twoapproaches is quite obvious as far as the small displacement theory ofelasticity is concerned. However, we shall emphasize the avenue of approach

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which leads from the principle of virtual work to the principle of minimumpotential energy and other related variational principles, because thischoice advantageous for a systematic treatment of problems in solidmechanics.

It is noted here that these variational principles can be applied to an elasticbody consisting of several different materials, if the stress—strain relationsof each material assure the existence of a strain energy or complementaryenergy functiop. For example, if the body Consists of is different materials,and the strain energy function of i-th material is denoted by the prin-ciple of minimum potential energy may be formulated by replacing fffA dV

I, Vwith fffA5 dv. The continuity of displacement on the inter-

1—1

face between the various materials must be satisfied if neither slipping nortearing is assumed. Similar statements can be made concerning the othervariational principles. It is also noted here that several other related varia-tional principles in elasticity have been proposed in Refs. 9, 10 and Ii.

2.5. Rayleigh—RItz Method—(1)

It has been shown that the elasticity problem in small displacement theorycan be formulated by variational methods under the assumption that thethree functions A, cl) and !t' exist. The exact differential equations andboundary conditions defining the problem are then given by the stationaryproperty of the total potential energy and related functionals. However, oneof the greatest advantages of the variational procedure is its usefulness inobtaining approximate solutions. The so-called Rayleigh—Ritz method isthe best established technique for obtaining approximate solutions throughthe use of the variational methodt We shall illustrate the Rayleigh—Ritzmethod with two examples.

Let us first consider the principle of minimum potential energy appliedto the elasticity problem of Section 1.5. Let us assume a set of admisibledisplacement functions u, v and w as given by Eqs. (1.34), (1.35) and (1.36).Introducing Eqs. (1.34) into Eq. (2.12) and carrying out the volume andsurface integrals, we can express fl in terms oia,, b, and C, (r = 1, 2, ..., is).The Rayleigh—Ritz method determines the values of these constants byrequiring Ml = 0, which, in thepresent case, becomes:

8= 0, 0, 0, (r= 1, 2, ..., is). (2.43)

The. Eqs. (2.43) lead to a set of 3n linear algebraic equa-tions in which the 3n unknowns are a,., b, and.c, (r = 1, 2, ... ,n). It is ob-served that the 3n equations thus obtainedare equivalent tç those obtainedin Section 1.5.

t Refs. Z 3 and 12 through 17.


Next, let us consider the principle of minimum complementary energyapplied to the twodimet&sioi,al problem of Section 1.7. Noting that thestresses expressed by Eq. (1.61) constitute a set of admissible functions,shall substitute them into

=[(ni + or,)2 + 2(1 ÷ — dx dy, (2.44)

which, after integration, can be written in terms of a, (r = 1, 2, ..., is). TheRayleigh—lUtz method aeaerts that the stationaiy property of the exact solu-.tion can be satisfied approximately by requiring

(r= 1,2,...,n). (2,45)

The is simultaneous thus obtained determine values of a, (r = I,2, ..., n), which, when sub*dtutcd into eqs. (1.61), provide approximatesolutions for the stress coqipOnents. 'We also observe that the is equationsthus derived are equivalent tçothoee obtained in Section 1.7.

Thus, we see that the Ra jgl—Ritz method leads to formulations equi-valent to those of the approximate methods developed in Sections 1.5 and1.7, as far as the elasticity problem of the small displacement theory isconcerned. ó8ch its 'own advantages and disadvantages inapplications to problems outsjde the elaSticity problem. The approximatemethods are villid the stress—strain relations employed andpotentials of the external but tile proof thatthe approximate solutionsconverge to the exact sotutioá with increasing n is usually difficult. On the.other hand? the stress-etrain relations, body forces and surface forces mustassure the existence of the state functions 4, .8, and !P for the formulationof the variational Rayleigh—Ritz method is to be used.However, the convergence especially when the maxi-mum or minimum propertyof th$ variAtional expressions has been estab-'lished. '

When boundary value probjeqis of elasticity can be solved only approxi-mately, it is desirable to obtain upper and lower bounds of the exactlion. However, this reqñiteinent,is sidom answered, because bounds areusually much more to ób*Ih than approximate solutions. Trefftz..proposed a method of derivlóg *pper and lower bound. formulae for thetorsional rigidity ofa bar byshriultaneous use of the principles oIminimum.potential and complementary enârgy (see Ref. 18 and Section 6.5). Since'his paper was published, marty papers on this and related subjects haveappeared in the field, of Among them, the concept of functionspace dàvised 'by W. Prager and J Synge may be mentioned as a notable

In function space a. sei of stress components related toaset of strain (es, s,,,..., Yx',) by Eqs. (1.6), is considered as a

40 VARIATIONAL METHODS IN ELASTICITY ANt) PLASTICITY

vector. Denoting two arbitrary vectors by N and N and their componentsof stress and strain by (as, ..., ..., ... (er, ...respectively, we define the scalar product of the two vectors in functionspace by

(N, fJf + + .— + dV, (2.46)

the integral being taken throughout the body. Since the strain energy func-tion is a positive definite form, the following relations are obtained imme-diately:

(N, N) � 0, (2.47)

(N, N*) � (N, N)* (N*, N*)+. (2.48)

The function space thus defined enables us to grasp intuitively approximatemethods of solution and their convergence characteristics, and to estimatethe error of approximate Due to the space available, the methodof deriving bound formulae in function space will not be shown here. Theinterested reader is directed to Ref. 21 for details of the concept of functionspace.

2.6., Variation of the Boundary Conditions and Castigliano's Theorem,

Thus far, we have derived the principle of minimum potential energyand its family under the assumption that the boundary conditions on S1and S2 are kept constant during variation. Now, we shall consider variationof the boundary conditions. We assume that the problem defined in Sec-.

1.1 has been solved and that components of the stress and strain as well asthe functions A and B of the solution have been expressed in termsof the prescribed body forces, surface forces on S1 and surface displace-ments on We denote the stress, strain and displacement components ofthe actual solution by a,, ...; e,,, ...; u, v, w, respectively in thepresent section.

We shall consider first the variation of the geometrical boundary condi-tions. The displacement components are given infinitesimal incrementsdu, and on S2, while the body forces as well as the mechanical boun-dary conditions of S1 remain unchanged. We assume that the incrementaldisplacements have yielded a new configuration and denote incremental

caused in the body by du, dv and dw. Th*n we haye

dU = fff (1 du + ? dv + Z dw) dV

+ ff (I, du + 7, dv + 2, dw) dS

+ff.(x.du+ (2.49)

PRINCIPLES 41

whereu = fff A dv, (2.50)

is the strain energy of the elastic body. We have derived Eq. (2.49) in amanner similar to the developments of the divergence theorem, Eq. (1.76),remembering that

dA = de, + + (2.51)

and observing that the stress components ... and the incremental staindes, ... satisfy the equations of equilibrium and the conditions of compati-bility, respectively. We shall see in Chapter 3 that Eq. (2.49) holds for finitedisplacement theory of elasticity as well.

The formula (2.49) is useful in determining the values of 1', and Z,on the boundary S2. As an example, we shall consider the truss structureconsisting of two equal members of uniform cross-section shown in Fig. 2.2.Let the problem be defined such that the displacement at the joint is pre-

scribed and the resulting force P is to be obtained. We denote the lengths ofmembers before and after deformation by and 1, respectively, and thestrain of the members by e. From geometrical considerations we haye j2 = a2

+ (b + cS)2 and = a2 + b2, and we obtain

e = (1 — = (2.52)

where higher order terms are neglected. Consequently, we have

U = ((i) EA010e2] x 2 = (2.53)

where A0 is the cross-sectional area of the member. Applying Eq.obtain

P = = 6. (2.54)

P

Fia. 2.2. A truss structure.

42 VARIATIONAL METHODS IN ELASTICITY ANI) PLASTICITY

Next, we shall consider the variation of the body forces and mechanicalboundary conditions. The body forces and the external forces on aregiven infinitesimal increments dX, dY, dZ and dl,, dY,, dZ. respecthely,while the geometrical boundary conditions on S2 remain unchanged. Weassume that these incremental forces yield a new configuration, and denoteincremental stresses caused in the body by ... and Then,we have

dV_—fff(udl+ vdY+

+ff(udl, +vdY,+wdZ,)dSSi

.+ ff (u ÷ i di, + dS, (2.55)

where

v_—fffBdv (2.56)

is the complementary energy of the elastic body. We have derived Eq.(2.55) in a manner similar to the development of the divergence theoremremembering that

dB = + di; + ... + (2.57)

and observing that the strain components ... and the incremental stressessatisfy the conditions of and the equations of equili-

bnuni, respectively.The formula (2.55) is userul in determining the values of u, v and' w on the

boundary S1. As an example, we consider a body which is held rigidw fixedon the boundary S2, and is subject to n concentrated lOads P1. P2, ...,on the boundary S1. For the sake of simplicity, these loadsare independent. In other words, any of these can be as-sumed' to be given increniints Without interfering with those remaining.Denoting the displacement of the point of application of theload P, in the directiim of the load by we have from Eq. (2.55):

(2.5w)

Since V is a function of the external forces, we have:

= dP.

Combining these two equations, we obtain:


Since the forces are assumed independent, we have

Zig (i 1, 2, ..., '0. (2.61)

The formula (2.55) and its family are called Castigliano's theorem—apowerful tool for analyzing problems in the small displacement theory ofelasticity (see Refs. 2 and 12 through 15, for instance).

2.7. Free Vibrations of an Elastic Body

The variational principles derived so far have been for the boundaryvalue problem of elasticity. In the last two sections of this chapter we shallconsider variational formulations of problem of free vibrations of anelastic body in small displacement theory. The problem is defined by allow-ing the body to be mechanically free on and geometrically fixed on S2.Since the problem is confined to small displacement theory, all the equationsdefining the problem are linear, and displacements and stresses in the bodybehave sinusoidally with respect to time. Consequently if we denote theamplitudes of stress, strain and displacement by as,.. , ... and U, v, w,

respectively, we have for the equations of motion,

+ + + Aeu =

+ + + Ant 0, (2.62)ox (3Y

Ox+ + +

A w2 where w is the natural circular frequency, and is thedensity of the material. The boundary conditions are given by

= 0, 1, = Z, = 0 on SI, (2.63)

and

u=O, v=0, w=0 on S2. (2.64)

Fcom Eqs. (2.62) and (2.63), we have

fff[(Oax + + + ++

dV

+ Jj(x. öu + I', ôv + Z, ow) dS = 0. (2.65)

Here we chose the arbitrary set of virtual displacements ãu and ow suchthat the geometrical boundary conditions are not violated, namely, Ou = Ov

44 VARIATIONAL METHODS iN ELASTICITY AND PLASTICITY

= ow = 0 on S2. Then, we may transform Eq. (2.65) into

fff + a, &, + •• +

— A fff(uOu + vc5v + = 0. (2.66)

This is the principle of virtual work for the free vibration problem.If the relations between the amplitudes Qf stress and strain are given

by(2.67)

where= Gjg = 1, 2, ..., 6),

we are assured of the existence of the strain energy function defined byEq. (2.2). Moreover, the body forces AQv and are derivable from apotential function defined by Eq. (2.6) such that

= + V2 + w2). (2.68)

Consequently, we obtain from Eq. (2.66) the principle of stationary potentialenergy as follows: Among all the admissible displacemeutfunctions u, v and wwhich satisfy théprescribed geometrical boundary conditions, the actual dis-placements make the total potential

17 = fff A(u, v, w) dV—

(u2 + v2 + (2.69)

stationary. In the functional (2.69), the quantities subject to variation areu, v and w under the subsidiary conditions (2.64), while ) is treated as aparameter not subject to variation.

The principle of stationary potential energy can be generalized throughthe use of Lagrange multipliers as follows:

= ... , — v2 + W2)

— + p,v + pew) dS, (230)

where the independent quantities subject to variation are ...; u, ...;'Ix, ... and p,. The stationary conditions are shown to be Eqs.(2.67); Eqs. (2.62), (2.63) and

Px = ..., Px = Z, on S2; (2.7))

0Lcx = ..., Yxy + (2.72)

and Eqs. (2.64).


Several variational principles can be derived from the generalized prir-Here, we shall derive a functional for the principle of stationary

complementary energy. It is shown that elimination of the strain componentsby the use of Eqs. (2.67) and a simple calculation by the use of Eqs. (2.62),(2.63) and (2.71) lead to a transformation of the functional (2.70) as follows:

= ff1 ...,Tx,)dV_kAfff(u2 + v2 + (2.73)

where the quantities subject to variation are u, ...; ... and under thesubsidiary conditions (2.62) and and the stationary conditions areshown to be equivalent to Eqs. (2.64) and (2.72). The functional (2.73) isan expression for the principle of stationary complementary energy of thefree vibration note that another expression of the principleof stationary complementary energy can be obtained by eliminating u, vand w from the functional (2.73) by the usc of Eqs. (2.62), thus expressingthe functional in terms of ... and only.

It was shown in Refs. 23 and 24 that the principle of stationary comple-mentary energy might be extended to cigenvalue problems such as freevibration and stability of elastic bodies. The principle was introduced andproved in Ref. 25 by E. Reissner for a problem in which loagis, stresses anddisplacements are simple harmonic functions of time. The functional (2.73)is equivalent to that introduced by E. Reissner.

It is well established that the principle of stationary potential energy (2.69)is equivalent to finding, among admissible functions u, v and w which satisfythe prescribed geometrical boundary conditions, those which make thequotient

(2.74)

stationary, where

U = fff A(u, v, w) dv, (2.75)

T = fff (u2 + v2 + w2) dv, (2.76)

'nd the stationary values of A provide the iigenvalues of the solution. Forthe proof, we see that

ÔA =— — A ö7), (2.77)

where the variation is taken with respect to u, v and w. Consequently, thecondition that the quotient A is stationary is equivalent to the principle ofstationary potential energy. The expression (2.74) is the Rayleigh quotientfor the free vibration 26) 0

It is also well known that the principle of stationary potential energy(2.69) is equivalent to a problem of finding, among admissible functions


u, v and w which satisfy the geometrical boundary conditions, those whichmake U stationary under a subsidiary condition

T(u, v, w) — 1 = 0. (2.78)

For the proof, we see that this problem is equivalent to obtaining the sta-tionary conditions of a functional defined by

U — — 1), (2.79)

where A plays a role of a Lagrange multiplier and the variation is taken withrespect to u, v, w and A under the subsidiary conditions (2.64).

2.8. Rayleigh—Ritz Method—(2)

We have seen the variational principles established for the free vibrationproblem in the preceding section. When the variational expressions are thusaw dable, the Rayleigh—Ritz method provides a powerful tool for obtainingapproximate values of çigenvalues. We shall consider a free vibration problemof a beam as an example and follow the outline of the method.

We shall take a beam clamped at one end x = 0 and simply supportedat the other end x 1 as shown in Fig. 7.5. The functional for the principleof stationary potential energy for the present problem is given by

I I

H = 4f EI(w")2 dc—

4..Af mw2 dx, (2.80) t

where El, w and m are the bending rigidity, deflection and mass per unitspan of the beam, respectively, and ( )' = d( )/dx. In the functional(2.80), the quantity subject to variation is w under the subsidiary conditions

w(0) = w(1) = w'(O) = 0. (2.81)

We denote the exact eigenvalues by

(i = 1, 2, 3, ...), (2.82)

in ascending order of magnitude such that 0 <A2 <We may transform the functional (2.80) to yield a functional for the

principle of stationary complementary energy:1

tic (2.83)t

where the quantities subject to variation are M and w under the subsidiaryconditions

M" + A,nw = 0, (2.84)

t See Section 7.4 for the derivation of these fenctionals.

VARIATIONAL PRiNCIPLES 47

andM(I) = 0. (2.85)

We shall consider first the Rayleigh—Ritz method applied to the principleof stationary potential energy. The well-known procedure proposed by themethod is followed by choosing a set of n linearly independent admissiblefunctions the so-called coordinate functions, which satisfy Eqs.(2.8!), and assuming was a linear combination of these coordinate functions,namely:

w=

c1w1, (2.86)

where c1(i = 1, 2, ..., n) are arbitrary constants. Substituting (2.86) intoEq. (2.80), and setting

0 (i = 1, 2, ..., n), (2.87)

we obtain a set of n homogeneous equations. The requirement that the deter-minant of the set must vanish for a nontrivial solution provides anotheralgebraic equation, called the characteristic equation of the set of the form:

det (m1, — = 0. (2.88)

If we denote the roots of the characteristic equation (2.88) by(i = 1,2, ...,n) in ascending order of magnitude, i.e. A1 <A2 <As, wehave

2, < A, (i = 1, 2, ..., n). (2.b9)t

Next, we shall consider the method applied to tne principle of stationarycomp(ementary energy. This is sometimes called the modified Rayleigh—Ritz

and its outline isas follows: We choose was given by Eq. (2.86),where the coordinate functions w,(x) are so chosen as to satisfy Eqs. (2.81).We substitute Eq. (2.86) into Eq. (2.84) and perform integrations with theboundary condition Eq. (2.85) to obtain

(I/2)M = c(x —1) (2.90)

where c is an integration constant. Substituting Eqs. (2.86) and (2.90) intothe functional (2.83) and requiring that

= 0, (2.91)and

= 0, i = 1,2, ..., n, (2.92)

we obtain a characteristicequation whicbdetermines approximate eigenvalues.For later convenience, these approximate eigenvalues are denoted by(i = 1, 2, ..., n) in ascending order of magnitude, i.e. <...<

t For the proof, see Refs. 3, 26 and 27.

48 VARIATIONAL METhODS IN ELASTICITY AND PLASTICITY

We see that the inclusion of the term c(x — 1) in Eq. (2.90) and the require-nient of Eq. (2.91) are equivalent to obtaining the exact beam deflection dueto the inertial loading Jn Thus, the method is equivalent to theGrammel's method in which the exact deflection due to the inertial loadingis obtained by the use of the Green's function or the so-called influence func-

It is stated in Ref. 28 that if the same assumed modes (2.86)are employed, we have the following inequality relations:

2, � A7 A,, i 1, 2, ..., n. (2.93)

Thus, the Rayleigh—Ritz method provides an upper bound for each eigen-value. It is well established that the accuracy of approximate eigenvaluesthus obtained is good and sometimes excellent if the coordinate functionsare choscp properly. However, since an approximate method of solutionis applied to a problem whose cxact solution cannot be obtained, we canusually expect to have no information on the exact eigenvalues beforehand.Therefore, formulae providing lOwer bounds are indispensable for locatingthe exact eigeüvalues.

There have been proposed scvefal theorems for locating lower bounds ofeigenvalues. Among them, the Temple—Kato theorem and Weinstein'smethod may be mentioned as typical. The Temple-Kato theorem provides alower bound for the aigenvalue when the value or a lower bound of the

is known.'f This theorem often proves to be an effective toolfor eigenvalue location. On the other hand, Weinstein's method employsas a basis one of Rayleigh's principles that, if the prescribed boundary con-ditions are partly relaxed, all the cigenvalues decrease4 That is, if we denoteëigenvalues of a relaxed or intermediate problem by I — 1, 2, ... inascending order of mag that < X2 then we havq

1, 2, ...). (2.94)

Therefore, if we obtain exact of the intermediate problem, theyprovide lower bounds for the cigciwslues of the original problem.

The Rayleigh-Ritz method fpr the free vibration problem has been illus-trated. It is obvious that the methOd also finds a field of application inother cigenvalue problems. The reader is.directed to Refs. 13, 16 and 26 forfurther details and numerical illustrations of the Rayleigh-Ritz methodapplied to eigenvalue problems.

2.9. Some Remarks

We have derived some extensions of the principles of virtual work andcomplementary virtual work in Section 1.9. It is obvious that the first termsof Eqs. (1.73) and (1.74) may be replaced by dU and ÔV,respectively, for

t Ref's. 30 through 35.Refs. 36 through 38.


elàstitityproblems, and several extensions of the principles (2.5) and (2.22)may be obtained from these equations. For example, we have

ÔV— fff (u aX + v ÔY + w dV = 0, (2.95)

•for an elasticity where the boundary conditions are specified suchthat

I, = P. = = 0 on SI, (2.96)

• on S2, (2.97)

and stress variations are so chosen as to satisfy Eqs. (1.75) and the boundaryconditions

= ô V. = = 0 on S1. (2.98)

Next, a mention is made of the generalized Galerkin's method treated inSection 1.5. The object of this mention is to note that the principle (2.5),which may be used instead of the principle (1.32) for elasticity problems,suggests a modification of the generalized Galerkin's method as follows:Since

ÔU = [(aU/aa,) + (aU/al,,) äb, + ôc,J,. (2.99)

we find that the principle (2.5) leads to an approximate method of solutionin which equations for the determination of the unknown constants a,, b,and c, (r = 1, 2, ..., n) are given by

L, = 0, M, = 0, N. 0, (r = 1, 2, ..., n) (2.100)

where

—fff Pt), dV

— ff F,v, dS, (2.101)

N,= _fffzwr dv

We shall see in Section 5.6 that this approximate method of solution is'equivalent to that employed in deriving Lagrange's equation of motion of

the dynamical problem. It obvious that the principle (2.22) suggests asimilar modIfication of the generalized Galerkin's method treated intion 1.7.

The approximate method of solution above mentioned.can also be appliedto cigenvalue problems of an elastic body in which external forces are not


derivable from potential functions. Such applications are usually based onthe principle of virtual work, Eq. (2.5), as illustrated in Refs. 39 and 40.As an example of applications based on the principle of complementaryvirtual work, we may refer to E. Reissner's work for flutterAn examination of his paper reveals that his method may be considered asan application of the principle (2.95) if aerodynamic and inertial forces aretaken as types. of body forces.

Bibliography

I. A. E. II. Lovr. A Treatise on the Maihematièal Theory of Elasticity, Cambridge Uni-versity Press, 4th edition, 1927.

2. S. TIMOSHENKO and J. N. Theory of Elasticity, McGraw-Hill, 1951.3. B.. COURANT and D. HILBERT, Meihod.c of Mathematical Physics, Vol. 1, Interscience,

New York, 1953.4. K. WASHIZU, On the l'ariationai Principles of Elasticity and Plasticity, Aeroelastic

and Structures Research Laboratory, Massachusetts Institute of Technology, Tech-nical Report 25—18, March 1955.

5. H. C. Hu, On Some Variational Principles in the Theory of Elasticity and Plasticity,Scintia Sinica, Vol. 4, No. 1, pp. 33—54, March 1955.

6. F. 1-IELLINGER, Der ailgemeine Ansatz der Mecbanik der Kontinua, Encyclopä4ie derMarhema:ischen Wissenschaften, Vol. 4, Part 4, PP. 602—94, 1914. -

7. E. REISSNER, On a Variational Theorem in Elasticity, Journal of Mathematics andPhysics, Vol. 29, No. 2, pp. 90—95, July 1950.

8. K. FRIEDRICHS, Em Verfahren der Variationstechnung das Minimum eines Integralsals das Maximum cines andcren Ausdruvkes darzustcllen, Nachrich:en der Academie

der Wissenschaften in Götiingen, pp. 13-20, 1929.9. T. IA!, A Method of Solution of Elastic Problems by the Theorem of Maximum

Energy (in Japanese), Journal of the Society of Aeronautical Science of Nippon, Vol. 10,

No. 96, pp. 276-98, April 1943.10. E. REISSNER, On Variational Principles in Elasticity, Proceedings of Symposia in

Applied Mathe,natks, VoL 8, pp. 1-6, MacGraw-I-IilL, 1958.11. P. M. NAGHLI. On a Variational Theorem in Elasticity and its Application to Shell

Theory, Journal of Applied Mechanics, Vol. 31, No. 4, pp. 647—53, December 1964.

12. C. BIEZENO and B.. GRAMMEL, Technische Dynamik, Springer Verlag, 1939.13. B.. L. BIsPLINOH0FF, H. Asm.p.'y and B.. L. HALFMAN, Aeroelasticily, Addison-Wesley,

1955.14. N. J. HoFF, -The Analysis of Structures, John Wiley, 1956.15. J. H. ARGYRIS and S. KaLSEY, Energy Theorems and Structural Analysis, Butterworth,

1960. -

16. G.TEMPLE and W. Ci. B1c11.EY, Rayleigh's Principle and its Applications to Engineer-

ing, Oxford University Press, 1933.17. S. (3. MIKHLIN, Variational Methods in Mathematical Physics, Pergamon Press, 1964.

18. F. TREFFTZ, Fin Gegenstuck zum Ritzschen Verfahren, Proceedings of the 2nd Inter-national Congress for Applied Mechanics, Zurich, pp. 131 -7. 1926.

19. W. PRAGER and J. L. SYNGE, Approximations in Elasticity Based on the Concept of

Function Space, Quarterly of Applied Mathematics, Vol. 5, No. 3, pp. 241-69, October

1947.20. K. Bounds for Solutions of Bourrdary Value Problems in Elasticity, Journal

of Mathematics and Physici, Vol. 32, No. 2—3, pp. 119—28, July—October 1953.

21. J. L. The Hypercircle in Mathematical Physics, Cambridge University Press.

- 1957.


22. K. WASIHZU, Note on the Principle of Stationary Complementary Energy Appliedto Free Vibration of an Elastic Body, International Journal of Solids and Structures,Vol. 2, No. 1, pp. 27—35, January 1966.

23. S. TIMOSHzNKO, Theory of Elastic Stability, McGraw-Hill, 1936.24. I-I. M. WESTERGAARD, On the Method of Complementary Energy and Its Application

to Structures Stressed Beyond the Proportional Limit, to Buckling and Vibrations, andto Suspension Bridges, Proceedings of American Society of Civil Engineers, Vol. 67,No. 2, pp. 199—227, February 1941.

25. E. REISSNER, Note on the Method of Complementary Energy, Journal of Mathematicsand Physics, Vol. 27, pp. 159—60, 1948.

26. L. COLLATZ, Eigenwertaufgaben mit technischen Anwendungen, Akademische Verlags-gesellschaft, Leipzig, 1949.

27. S. H. GOULD, Variational Methods for Eigenralue Problems, University of TorontoPress, 1957.

28. R. GRAMMEL, Em neues Verfahren zur Losung technischcr Eigcnwertproblerne,Jngenieur Archiv, Vol. 10, pp. 35—46, 1939.

29. A. I. VAN DE V0oREN and .1. 11. GREIDANUS, Complementary Energy Mcthod inVibration Analysis, Reader's Forum, Journal of Aeronautical Sciences, Vol. 17, No. 7,pp. 454—5. July 1950.

30. 0. TEMPLE, The Calculation of Characteristic Numbers and Characteristic Functions,Proceedings of London Mathemqtical Society, Vol. 29, Series 2, No. 1690, pp. 257-80,1929.

33. W: Kom.r, A Note on Weinstein's Variational Method, Physical Review, Vol. 71.No. 12, pp. 902-4, June 1947.

32. T. KATO, On the Upper and Lower Bounds of Eigenvalues, Journal of Physical Societyof Japan, Vol. 4, No. 4-6, pp. 334—9, July—December 1949.

33. 0. TEMPLE, The Accuracy of Rayleigh's Method of Calculating the Natural Frequen-cies of Vibrating Systems, Proceedings of Royal Society, Vol. A211, No. 1105, pp.204—24, February 1950.

34. R. V. SOUTRWELL, Some Extensions of Rayleigh's Principle, Quarterly Journal ofMechanics and Applied Mathematics, Vol. 6, Part 3, pp. 257—72, October 1953.

35. K. Wft.swzu, On the Bounds of Eigenvalues, Quarterly Journal of Mechanics andApplied Mathematics, Vol. 8, Part 3, pp. 311—25, September 1955.

36. WEINSTEIN, Etude des spectres des equations aux derivées partielles de Ia théoricdes plaques élastiques, Memorial des Sciences Mathematiques, Vol. 88, Paris, 1937.

37. N. ARONSZAJN and A. WEINSTEIN, On the Unified Theory of Eigenvalues of Platesand Membranes, American Journal of Mathematics, Vol. 64, No. 4, pp. 623—45,December 1942.

38. J. B. DIAZ, Upper and Lower Bounds for Eigenvalues, Proceedings of Symposia inApplied Mathematics, Vol. 8, pp. 53—78, McGraw-Hill, 1958.

39. R. L. BISPLINOHOff and H. ASHLEY, Principles of Aeroelasticity, John Wiley, 1962.40. V. V. BOLOTIN, Nonconservative Problems of the Theory of Elastic Stability, Translated

by T. K; LUStIER and edited by 0. Herrmann, Pergamon Press, 1963.41. E REISSNER, Complementary Energy Procedure for frlutter Calculations, Reader's

Forum, Journal of Aeronautical Sciences, Vol. 16, 5, pp. 316—17, May 1949.42. F. B.. GANTMACHER, The Theory of Matrices, Chelsea Publishing Company, 1959.43. B. M. FRAELJS DE VEUBEKE, Upper and Lower Bounds in Matrix Structural Analysis,

in Matrix Methods of Structural Analysis, edited by B. M. F. de Veubeke and publishedby Pergamon Press, 1964.

CHAPTER 3

FINITE DISPLACEMENT THEORY OFELASTICITY IN RECTANGULAR

CARTESiAN COORDINATES

3.1. Analysis of Sdnln

In the present chapter we shall treat finite displacement theory of elasti-city in rectangular Cartesian coordir .ites.t The difference between spatialvariables and material variables cannot be overemphasized in the formula-tion of finite displacement theory. Unless otherwise stated, we shall employthe Lagrangian approach, in which the coordinates defining a point of thebody before deformation are employed for locating the point during thesubsequent deformation4

Let the rectangular Cartesian coordinates (x', x2, x3) be fixed in spaceand the position vectot of an arbitrary point p(O) of the body before defor-mation be represented by

= x2, x3), (3.1)

as.shown in Fig. 3.1, where the means that the quantity isreferred to the state before deformation.ff Let us henceforth employ a setof values (x', x2, x3), which the point occupies before deformation,as parameters which specify the material point during deformation.

The base vectors in this coordinate system are given by

== (A = 1, 3), (3.2)fl

where and throughout the present chapter, the notation ( denotesdifferentiation with respect to x1, namely, ( = ô( )/3x". They are unitvectors in the directions of the coordinate axes and are mutually orthogonal:

(3.3)tft

t Refs. I through 6.Or the other hand, coordinates associated with the deformed body are employed

in the Eulerian approach.tt Superscript indices should not be mistaken for exponents.fl A Greek index will be assigned in place of (I, 2, 3) in Chapters 3, 4 and 5.

ttt The notation a . b denotes the scalar product of two vectors a and b.52

FINITE DISPLACEMENT THEORY OF ELASTICITY 53

where the Kronecker symbol defined by

¼=O (?.+,u).

Fin. 3.1. Geometzy of an infinitesimal parallelepiped.(a) before deformation. (b) after deformation.

ci

(3.4)

We shall take a point in the neighborhood of the point p(O) and denotethe coordinates of Q(O) by (x' + dx', x2 + dx2, x3 + dx3). Then the positionvector and the distance between these two points can be expressedas

and= 1a

. = dxi',

(3.5)t

(3.6)

t The summation convention will be employed in Chapters 3, 4 and 5. Therdore aGreek letter index which appears twice in the same term Indicates summation with respect to (1, 3). For example:

= x'l, + X212 + x313.

+a12e,2+0r13e13i—i p—i

+ a21e21 + e122e22 + a23e23 + u31e32 + a32e32 + 133e33.

0

Ii

xl


respectively. For later convenience, an infinitesimal rectangular parallel-epiped enclosed by the following six surfaces:

= constant, + dXA = constant (2 = 1, 2, 3)

will be fixed to the body. It has the line element P(O)Q(O) as one of theprincipal diagonals and and as the vertices adjacent to thepoint p(O)•

The body fs now assumed to be deformed into a strained configuration.The points 5(0) and move to new positions denoted byP, Q, R, S and T, respectively, and the infinitesimal rectangular parallel-

epiped is deformed into a parallelepiped which, in general, is no longer

rectangular. Let us denote the position vector of the point P by

r = r(x', x2, x3),

and introduce the lattice vectors defined by

EA = = TA (2 = 1,2, 3). (3.8)

The sides of the parallelepiped issuing from the point P are then given

by E, dx', E2 dx2 and E3 dx3. Consequently, the position vector andthe distance d.c between P and Q can be expressed by

dr r A = EA dxA, (39)and

(ds)2 = dr . dr E2,4 dx2 df, (3.10)respectively, where

Ea,i = EA = (3.11)

Let us consider the geometrical meaning of The lengths of the in-finitesimal line element before and after deformation are

= (dx1)2 and (ds)2 = E,1(dx1)2,

respectively. Therefore, the rate of elongation of is given by

(c/s — -— SE,, —1. (3.12)

The geometrical meanings of £22 and E33 follow similarly.

Next, consider two infinitesimal line elements andwhich are orthogonal before deformation. After deformation, these twoelements to new PP and PS, the relative positions ofwhich arc given by the vectors E, dx' and E2 dx2, respectively. Jf we

denote the acute angle between PR and PS by — y,2), we have

E1 d.v'•E2dx2 = IEIHEZI dx'


orE12 = 1/E11E22siny12. (3.13)t

This gives the geometrieal meaning of £12. The meanings of E23 and £31follow similarly.

Therefore, we conclude that after deformation an infinitesimal rectangularparallelepiped is transformed into a skew parallelepiped, and the geometryof the deformation can be specified by the set of values of the quantities

(a., 4u = 1, 2, 3). Consequently, we define strains of the parallelepipedby

— = = 1, 2, 3), (3.14)

and employ the nine components under the symmetry conditions =as the quantities which specify the strain of the parallelepiped.

Let us express the position 'vector of the point P as

r = + u, (3.15)

where u is the displacement vector, whose components (u1, u2, u3) are de-fined by

u = u%. (3.16)

From Eqs. (3.8) and (3.15), we have(3.17)

is the Kronecker symbol. By the use of Eqs. (3.14) and (3.17), thestrains can be calculated in terms of the displacement components asfollows:

(3.18)

If u, v, w are used instead of u', u2, u3, respectively, and x, y, z in place of1, 2, 3, or x1, x2, x3, respectively, Eqs. (3.18) can be written as follows:

ii 0u2 t3v2 Ow2]

Ot, 110u2 i3,,2 0w21= ++ .+ j,

oW iii Ou\2 I Ov\2 I= + + +

, (3.19fl

Ow Ov Ou Ou t3v Ov Ow Ow

Oy Oz i3y Oz Oy Oz Oy Oz

Ou Ow Ou Ou Ov Ov Ow Ow

3v Ou Ou Ou i3v Ov O,i' Ow

fCompare with Eqs. (3.5).

56 VARIATIONAL METHODS IN ELASTICITY AND

We note here that, when the strain components are sufficiently small,Eqs. (3.12) and (3.13) may be linearized with respect to the strains to obtainthe following approximate relations:

(d.c = }'l — I

V12 = 2e12.

Similar relations hold for the other strain components.

3.2. AnalysIs of Stress slid df Equilibrium

(3.20)

p.21)

It has been shown in the preceding section that an infinitesimal rectangu-lar parallelepiped is transformed into a skew paraliclepiped after deformation.We shall now consider the equilibrium of the deformed

The forces acting on the deformed parallelepiped arc internal forcesexerted by neighboring parts of the body through the ax side surfaces andbody forces, as shown in Fig. 3.2. Let the internal forces acting on one ofthe side surfaces, the area of which before deformation was d.r2 dx3 and the

FiG. 3.2. Equilibrium of an infinitesimal(a) before deformation. (b) after ddormatiøn.

sides of which after deformation are E2 dx2 and E3 dx3, be representedby — dx3. The quantities and are defined in a similar manner.

—

xi

-

FiNITE DISPLACEMENT THEORY OF ELASTICITY 57

The internal forces acting on the six side surfaces are as follows:

—o'dx2dx3,

dx3 dx3

dx1 dx2)dx3.

The body forces acting in the deformed paraflelepiped will be representedby P dx' dx2 dx3. The force equations of equilibrium of the deformedparallelepiped are then given by

(3.22)

Let us define the components of a' by resolving it in the directions of thelattice vectors:

(3.23)tas shown in Fig. 3.3. Then, the moment equation of equilibrium of thedeformed parallelepiped are gven by(a' dx2 dxa) x E, dx' + dx' dx') x E, dx' + (a3 dx' dx') x E, dx'

• (3.24)t

Where higher order terms arc neglected. By the use of Eq. (3.23), we obtain:frorn Eq. (3.24) the following relations:

(3.25)ff

t The quantity o4 by Eq. 0.23) Is called pseudo-stress or generaliud stress.

However, the familiar nomenclatvre stress will be used instead in subsequent formulations.A notation a a vector jwodiict of two vectors a and b.

ttlthnptedthet.'(a"—a")E, x

.xExE,,the three vectors E, x E,, x E3 and,E, x E, are mutually independent.

Pio. — E,..


We shall employ the nine components on', under the symmetry conditions= o"1, as the quantities which specify the state of stress on the infinitesi-

mal parallelepiped.Equation (3.22) is a vector equation. One way of expressing it in scalar

form is to resolve it in the direction of By defining the components ofthe body force as

we obtain the following scalar equations from Eq. (3.22):

(A =1,2,3).

(3.26)

(3.27)t

We note here that are defined per unit area and P is defined per unitvolume, both with respect to the undefornied state.

3.3. Tran,forniadon of the Stress Tensor

The quantities which define the state of stress at the point P dependon the choice of coordinates. Wt shall now find a law of transformationfor the stress tensor. Form an infinitesimal tetrahedronbounded by three surfaces of the rectangular parallelepiped and an obliquesurface, as shown in Fig. 3.4. If the area of the inclined surface before deform-

Fio. 3.4. Equilibrium of an infinitesimal tetrahedron.(a) before deformation. (b) after deformation.

t Compare with Eqs. (1.2).

0

dZ

-a'


ation be denoted by dE and the internal force acting on the surfaceRST after deformation be represented by F dE, the equilibrium equationof the infinitesimal tetrahedion is'

F = a'(dx2 dx3/2) + o2(dx3 dx'/2) + a3(dx' dx2/2). (3.28)

From the geometry before deformation, we have

dx2dx3 = 2(i1 .v)dE, dx3dx' = 2(i2. v)dE,dx' dx2 = 2(13 v) dE, (3.29)t

where p is the unit normal vector drawn outwards on the inclined surfacebefore deformation. Substituting the relations (3.29) into Eq.

(3.28), we obtain

F = v) Ox. (3.30)

This gives the direction and magnitude of the internal force F acting onthe' oblique surface. By resolving F in the direction of the vectors as

F = FaiA, (3.31)

we obtain Eq. (3.30) in scalar form:FA = + (3.32)

where n,, = v.

3.4. Stress-Strain Relations

In the present chapter we shall assume that the deformation under con-sideration takes place either isothermally or adiabatically, and postulatethç existence of functions which define the stress in terms of the strain such

= e,2, ..., (2, = 1, 2,3),

where the zero stress state corresponds to the zero strain state, namely,0, ..., 0) = 0. We also assume the existence inverse func-

tions which define the strain in terms of the stress:

= a12, ..., a33) (2, = 1, 2, 3).

When the strain components are assumed sufficiently small, we mayexpand Eqs. (3.33) into power series with respect to and neglect higher

t Sec footnote of Eqs. (4.63).Only six equations are physically independent in Eqs. (3.33). However, we may write

them and their inverse relations in nine equations of symmetrical form as given by Eqs.(3.33) and (3.34), respectively.

60 VARIATIONAL METHODS IN AND PLASTICITY

order terms to obtain the following linear stress—strain relations:

= (3.35)f

It is obvious that, due to the symmetry property of the stress and straintensors, there exist the following relations among the coefficients:

(3.35) may be inverted to yield: -

= (3.36)t

where

b =

When the material is isotropic, the numerical values of must beindependent of the coordinate system in which the stress and strain aredefined. This leads to the conclusion that is a fourth-order isotropictensor and is given 5>

(1 + v)(I — 2v) + + (3.37)

and Eqs. (3.35) and (3.36) reduce to

= (1 —2v)+ (3.38)t

and

= (1 —2w)

respectively, where E 2(1 .÷v) G. The quantities and aredeviator stresses and deviator strains defined by — and4, = ea,, — respectively, where u = = (1J3)(orhl + a22 + g33)and e= (l/3)eu=(l/3)(e11 + e32 + e33).

Problem

With the above preliminaries, we now defide abonndary valueproblem in the finite displacement theory of elasticity. Consider an elasticbody subjected to the following boundary conditions and body forces:

(1) Mechanical boundary conditions on S1,

(3.40)

where F is given by Eq. (3.30) with the understanding that the vector v isnow the unit normal drawn outwards on the boundary and .P is the pre-

41

t Compare with Eqs. (1.6), (1.8), (1.10) and (1.11), respectivdy.

FiNITE DISPLACEMENT ThEORY OF ELASTICITY 61

scribed external force. Both F and P are defined per unit area of the unde-formed state. Resolving P in the directions of the base vectors,

P (3.41)

we obtain from Eq. (3.40) the following scalar equations:

P = P (2 = 1, 2, 3). (3.42)t

(2) Geometrical boundary conditions on S2.ua = (1 = 1, 2, 3). (3.43)t

(3) The body forces

(A = 1, 2, 3). (3.44)

Our problem is then to find the �tresses and displacements existing in thedeformed body by employing the stress—strain relations (3.33).

By combining Eqs. (3.18) and (3.33), we can represent in terms ofuk. Introducing thus represented into Eqs. (3.27) and (3.42), we obtainthree simultaneous differential equations and the mechanical boundaryconditions in terms of u's. If these differential equations can be solved underthe boundary conditions on S1 and S2. we can obtain the required equili-brium configuration. Once the displacement components have beenobtained, the state of stress induced in the body may be determined fromEqs. (3.18) and (3.33).

Since the problem is nonlinear, the solution, if obtainable, generallyyields nonlinear relations between the applied external loads and the result-ing defoimations. Some typical exanw és of nonlinear relations are illu-strated in Figs. 3.5, 3.6, 3.7 and 3.8, where the ordinate is the applied ex-ternal load and the abscissa denotes the resulting displacement of the point

Fio. 3.5. A load—deflection curve of a truss structure.

f Compare with Eqs. (1.12) and (1.14), respectively.

p

0

a


of application in the direction of the external In the example illus-trated in Fig. 3.5, the slope of the curve, dF/dã, increases with the increaseof the deflection. This shows that the body is stable under the externalload as long as it behaves elastically. On the other hand, in the exampleillustrated in Fig. 3.6, the slope dM14 decreases with increasing deformation

FIG. 3.6. Flattening instability Ma thin cylindrical tube under bending.

Fio. 3.7. Durchschlag or snap-through of a c*irved beam.

and the load 2 eventually reaches its maximum Mer. This shows that thenody ceases to be stable at beyond which the body fails elastically.'8•The phenomenon illustrated in Fig. 3.7, where the load—deflection curvehas the S-shape characteristic, is called Durchschlag orIf P is a dead load and increases from zero to the deflection jumps from

to ö1 with kinetic energy illustrated by the shaded area of the figure. Inthe case vf unloading, another jump occurs, at If the load—deflectionrelation is as shown in Fig. 3.8, where a point of bifurcation exists atthe system has two states at loads beyond the critical load.The body prefers a stable configuration and changes its deflection suddenly

M

P

FINITE DISPLACEMENT ThEORY OF ELASTICITY 63

from unstable to stable under the stimulus of small external disturbances.The phenomena shown in the last three figures constitute the main partsof the theory. of elastic 'i"

Fio. 3.8. Bifurcation of a rod under compression.

3.6. Prhiclple Virtual Work

In $his section we shall derive the principle of virtual work ofth,e continuousbody under consideration. Assume that the body is in equilibrium underthe body forces, the applied external forces on S1 and the prescribed geo-metrical boundary conditions on S2. the body is assumed to executean infinitesimal virtual displacement on from 'this equilibrating configurationwithout , violating the prescribed boundary conditions on S2. Then, byemploying the equations of equilibrium and the mechanical boundaryconditions (3.40), and remembering that bu = Or, we obtain

. 4

— fff + P) . or dV + ff (F — . Or dS = 0, . (3.45)V •sI -

where dV = dx1 dx2 and 45 are the elementary volume of the parallel-epiped and the elementary area on' the surface of the body before deforma-lion, respectively. By the use of,the geometrical relations,

dx2 dx3 ±nrdS, dx2 = dS, (3.46)t

which hold on the surface of the body, the first term. of (3.45) may betransformed.into

1ff Or dV

ff F OrdSS1+S2 -• V-

t Sec footnote of Eqs. (4.79).

P

64 VARIATiONAL METHODS IN ELASTICITY AND PLASTICITY

Introducing the above into Eq. (3.45), and remembering that = 0 onS2, we obtain

fff — fffPc3r dv — = 0. (3.47)V V SI

The integrand 0a• dV of the first term in Eq. (3.47) may be inter-preted as the virtual work done, during the infinitesimal virtual displace-ment, by the body forces and the internal forces acting On the deformedinfinitesimal parallelepiped, while the second and the third terms representthe virtual work done by the body forces and the external forces on S1.respectively. Combination of Eqs. (3.23), (3.11), (3.25) and (3.14) yields

= (ôr)A

= ÔEA,, =

Introducing the above into Eq. (3.47), we obtain

(3.48)

This is the principle of virtual work for the elasticity problem in finitedisplacement theory. By the use of Eqs. (3.16), (3.26) and (3.41), the principlemay be expressed alternatively as follows:

— — = 0. (3.49)tV Si

Reversing the above development, we obtain from the principle, the equa-tions of equilibrium (3.27) and mechanical boundary conditions (3.42) underthe assumptions of the strain-displacement relations (3.18), the symmetryof stress components (3.25), and the prescribed geometrical boundary con-ditions (3.43). We note that the principle holds regardless of the form ofthe stress—strain relations of the body.

3.7. StraIn Energy Function

Let us consider an element of the body which is a rectangular parallel-epiped and occupies a unit volume before deformation. When the elementis subjected to deformation along a loading path and is brought into astrained state expressed by (e11, ..., e33), we may calculate an integral,

(e11. ...,e,3)

(Q,LO)

(3.50)

t Compare with Eq. (1.32). —

For consistency in tensor notation, it is better to write or instead ofpa óuA where is the symbol and ôu1 5u1'. However, we shall use thesimpler expressions for the sake of brevity, whenever the rectangular Cartesian coordinatesystem is employed.

FINITE DISPLACEMENT THEORY OF EI..ASTICITY 65

along the loading path by the use of the stress-strain relations (3.33). Thevalue of the integral thus obtained generally depends on the loading path.However, if the value does not depend on the loading path, but dependsonly on the final strain, the quantity is called a perfect differentialand the existence of a state function A(e11 e12, ..., e3 is assured suchthat

dii = (3.51)

or equivalently,

(2, = 1, 2,3). (3.52)e,,,

The state function thus defined is the strain energy function in the finitedisplacement theory of elasticity.

The present section is concerned with the conditions under which thequantity a" is a perfect -differential. A detailed discussion of theseconditions may be found in any book on partial differential equations.However, we may summarize them as follows:

if the stress—strain relations (3.33) satIsfy the equations

/= (2, fX, = 1, 2, 3) (3.53)

the quantity proves to be a perfect d(fferential.Consequently, if the stress—strain relations (3.35) satisfy the equations

= (3.54)

we have

A (3•55)t

When the material is isotropic and the stress—strain relations given by Eq.(3.38) may be employed, we have

2(1 —2v)e2 + (3.56)t

So far, conditions for the existence of the strain energy function havebeen studied mathematically. We shall now show from physical considera-tions that such a function really exists when an elastic body deforms eitherisothermally or adiabatically in a reversible process.'2 -

We assume a unit volume of an elastic body again and call it an element.The first law of thermodynamics is applied to the element by taking theawn of energy supplied to the element during the strain's increase by toobtain,

dU0 ci' d'Q + . (3.57)

t Compare with Eqs. (2.2) and (2.3), respectively.


where dU0 and d'Q are the increments of the internal energy and heat energysupplied to the element, respectively. The quantity dU0 is the differentialof the inttrnal energy U0, which is a single-valued function of the tem-perature and the instantaneous state of strain of the element. Physicallyspeaking, the internal energy U0 is a function of the mean position of themolecules under the intermolecular forces and the kinetic energy of themolecules about their mean However, Q is not a state functionand d'Q merely denotes an infinitesimal amount of heat energy supplied.Therefore, the special notation: d' is used here to avoid confusion. The quan-tity is an infinitesimal amount of supplied mechanical energy. Afterthese two kinds of energy have been supplied to the element, there can neverbe any distinction between portions of dU0 contributed by d'Q and

The second Jaw of thermodynamics assures the existence of a state func-tion S, called entropy, such that

(3.58)

where T is absolute temperature of the element. Combining Eqs. (3.57)and (3.58), we obtain,

dU0 = TdS + (3.59)

Equation (3.57) shows that if the deformation takes place adiabatically ina reversible process, we have

dU0 = (3.60)

Consequently, the mechanical energy is stored in the element in the formof internal energy and we obtain

A = U0 + constant. (3.61)

On the other hand, if the deformation takes place isothermally in a re-versible process, we have from Eq. (3.59)

dF0 = deAn, (3.62)

where

F0 U0 — TS (3.63)

is the Helmholtz free energy function. This shows that the mechanicalenergy is stored in the form of Helmholtz free energy and we obtain

A = F0 + constant. (3.64)

Therefore, it may be concluded that the quantity a perfect differ-ential for these two special cases, and the existence of the strain energy

is assured.The differences between the assumptions of adia%atic and isothermal

deformations appear, in mathematical formulations, only as differencesbetween adiabatie and isothermal elastic constants. Generally, speaking,


the differences between these elastic constants have been proved by experi-ments to be negligible. Consequently, the strain energy function is usuallyassumed to exist in the theory of elasticity, although the deformationprocess may be somewhere between adiabatic and isothermal.

We know from experimental evidence that when the strains are sufficientlysmall, an element of the elastic body is stable. This requires that the strainenergy function must be a positive definite function of the strain compo-nents for the small strain. Since we have also found that, when the strainsare small enough, the strain energy function can be expressed by Eq. (3.55),we may conclude that the strain energy function (3.55) is a positive definitefunction of the strains. -

3.8. Principle of Stationary Potential Energy

We have investigated in the preceding section the condition under whichthe strain energy function can exist. When the strain energy functitn isassured to exist, the principle of virtual work (3.49) can be written asfollows:

offfA(u1)dv — — ffFAouAds = 0, (3.65)t

where the :train energy is written in terms of by the useof Eqs. (3.18). The principle (3.65) is very useful in application to elasticityproblems in which external forces are not derivable from potential functions.

Next, we shall assume further that the applied external forces are conser-vativç, namely, they are derivable from potential functions and !t'(u1)such that

= OW = (3.66)

If the applied exl*rnal vary neither in magnitude nor in directionduring the virtual displacements, namely, if they are treated as dead loads,we may have

= W —Pus. (3.67)

Under the assumption of the existence of the strain energy function Aandtwo potential functions and W, the principle of virtual work (3.49) yieldsthe principle of stationary potential energy as follows:

611=0, (3.68)

where

H = fff [A(U2) + dV + ff W(UA) dS (3.69fl?

t Compare with Eq. (2.5).Compare with Eq. (2.9).


is called the total potential energy of the mechanical system under considera-lion and is a functional containing which are the independent variablessubject to variation. The variables should be chosen so that they satisfythe required continuity and differentiability criteria and the boundary con-ditions on S2. The principle (3.68) can be stated as follows: alladmissible displacement functions the actual ones are those which renderthe total potential energy stationary. Retracing the development in the latterhalf of Section 3.6, we can easily show that the conditions which make thetotal potential energy stationary provide the equations of equilibrium andthe mechanical boundary conditions on S1. However, in these equationsand conditions all the stress components are expressed in terms of displace-ment components, and we obtain three simultaneous differential equationsof equilibrium in V and threó boundary condition equations on S1 in termsof u'. The formulation thus obtained can also be reached by direct eliminationof the stress components from the equations of equilibrium and the mechani-cal boundary conditions by the use Qf the relations (3.18) and (3.33), asobserved at the end of Section 3.5.

Generalization of the Principle of Stationary Potefltlal Energy

It is obvious that the principle of stationary potential energy establishedin Section 3.8 can be generalized by the use of Lagrange multipliers. Throughfamiliar procedures, we obtain a generalized functional as follows:

17, = fff + ø(u")

— — + dV

- /1 Pa(Ua — ua)dS, (3.70)t

where the independent quantities to variation a;e U2, a" and p2with no subsidiary conditions. The functional for Reissner's principleU7)can be derived from the functiopal (3.70) through elimination of as

,follows:

HR =fff + +

— B(al + dV

(3.7l)tSi S2

where the independent quantities subject to variation are U2, a" and withno subsidiary conditions.

t çomparewith the functionals (2.26) and (2.37), respectively.


The quantity appearing in Eq. (3.71) is the complementary energyfunction. As the above development shows, it is defined by

(3.72)

The stress—strain relations (3.34) are introduced into Eq. (3.72) to expressB entirely in terms of the stress components. From Eqs. (3.51) and (3.72),we have

dB = (3.73)

or equivalently,

= (2,1u = 1,2,3). (3.74)

We note here that if Eqs. (3.34) satisfy the equations

— 1 2 3' (375—

" — ' '

the function B is assured to exist and may be determined from Eq. (3.73)independently of the function A. Consequently, if Eqs. (3.36) satisfy theequations

= (A, fi = 1, 2, 3), (3.76)we have

B (3.77)t

When Eq. (3.39) is employed, we obtain

B = 3(1 — 2,') 2 _L 3 78

for the isotropic material. -

Under the assumptions of small displacement theory, the principle ofminimum complementary energy can be expressed in terms of the stresscomponents only, as shown in Section 2.2. However, coupling of the dis-placements with the stress components in finite displacement problems com-plicates the derivation of the principle of stationary complementary energyfrom the principle can nb longer be expressed purely in terms of stresscomponents.

3.10. Energy Criterion for Stability

We know that, as long as the applied external loads are sufficiently small,we obtain linear relations between the loads and resulting deformations.However, the deformation characteristics gradually deviate from the linearrelations with increaSing loads. This tendency is usually pronounced in

t Compare with Eqs. (2.20) and (2.21), respectively.

70 VARIATIONAL METHODS IN ELASTICITY AND PLASTICITy

slender or thin bodies, and a point is finally reached beyond which thebodies cease to be stable for some loading conditions, as shown in theexamples illustrated in Figs. 3.6, 3.7 and 3.8. In the present section, weshall consider the energy criterion for determining the stability and criticalload of equilibrium configurations of elastic bodies under conservativeexternal fbrces.t

We assume that the equilibrium configuration of the body has been ob-tained for the problem defined in Section 3.5, and call it the original configura-tion. Then, let the original configuration be given small virtual displace-ments without violating the geometrical boundary conditions, thus obtain-ing a new configuration. If the virtual work done by the external forcesdoes not exceed the increase of the stored strain energy, the body is consi-dered stable. If this condition is not met for some virtual displacements,then the excess energy will appear as kinetic energy. This indicates an in-stability of the original configuration fortbe virtual displacements.

The above considerations lead to the following mathematical formula-tion. Let the displacement components of the original equilibrium configura-tion and the new configuration be denoted by u1 and uA + ôuA respectively.Denoting the total potential energy of the original configuration and thenew configuration by I1(ua) + &') we have

IT(UA + â/) = + MI + ö2H + 63H + ..., (3.79)

where Ml, ö217, c53H, ..., are the first, second, third, ... variations of thetotal potential energy. They are linear, quadratic, ... with respect to 5uAand their derivatives, and their coefficients contain the displacement com-ponents of the original configuration as parameters3 Since the originalconfiguration is in equilibrium, we haveS

= 0. (3.80)

With these preliminaries, we may now conclude that the stability of theoriginal configuration in its neighborhood can be determined by the signof the second variation Ô211 as follows:

(1) The configuration Lr stable t1ô211> 0 holds for all adñtissible virtual

t Refs. I and 18 through 22.When all the applied external forces are dead loads and their potential functions are

given by Eqs. (3.67), we have

oIH fff + dv,

where are the stress components of the original configuration and

= + + (6 +

tt This statement, combined with the principle of minimum potential energy derived inSection 2.1, assures that no unstable configurations exist as far as problems-of the smalldisplacement theory of elasticity are concerned.


(2) 21w configuration is unstable (5211 < 0 holds fof at least one ad,nis-sible set of virtual displacements.

Following we shall consider the determination of the lowestcritical load beyond which the body ceases to be stable for the first timeduring the loading process. We have seen that the original configurationis stable as long as (5211> 0 holds for all admissible virtualThis criterion will now be expressed in a differentmanner. We introduce aproperly chosen functional N which is positive definite and quadratic withrespect to and their derivatives,t and seek, among admissible virtualdisplacements which satisfy

c5u2 = 0 on S2, (3.81)

those which make the quotientA = â211/N (3.82)

a minimum. The criterion then states: the ininiinwn value of the quotient.is found to be positive, the original configuration is stable.

We know that since= j((52JJffq) = J((5217 — AN)/N, (3.83)

where indicates that the variation is taken with respect to ôuA, the stat onarycondition of the quotient is givàn by

— AN) = 0. (3.84)

Equation (3.84) yields differential equations and mechanical boundaryconditions, which, together with the geometrical Uoundary conditions(3.81), determine the stationary values of the quotient as eigenvalues.Consequent'y, the stability criterion may be expressed as follows if theminimum of the eigenvalues is foi,nd to be positive, the original configurationis stable.

The above consideration leads to a conclusion for the determination ofthe lowest critical load: the external load actiiçg on the original configurationis considered critical when the minimum of the eigenvalues reaches the valuezero; the variational equation

= 0 (3.85)

under the subsidiary conditions (3.81) then yields governing equations whichdetermine the lowest critical loa4. We note here that the governing equationsdetermipe all the critical configurations which possess at least one eigenvalueof the value zero in the eigenvalue problem derived from Eq. (3.84), andthe configuration corresponding to the lowest critical load is one of thecritical configurations. The lowest critical load thus determined is frequently

t The functional N, which is introduced heràfor the normalization of the virtual dis-placements, has no effect on the final result, namely, Eq. (3.85).


followed by unstable states upon further increase of the external load, butthe sign of the higher-order variation must be considered for a sufficiencyproof.

3.11. The Euler Method for Stability Problqm

Another method of determining the critical load concerns the problemof whether there exists at least one additional, distinct equilibrium configu-ration in the very close neighborhood of the original configuration. If suchan adjacent equilibrium configuration exists, the body may change suddenlyfrom one equilibrium configuration to the other under the stimulus of smallexternal disturbances. We shall formulate the stability problem by thisapproach, which is sometimes called the Euler method, for a body undernonconservative external

We assume the existencébf a critical original configuration and developa linearized theory which determines the adjacent configuration. We denotestresses, strains, displacements and external forces of the original and ad-jacent equilibrium configuration by

uA, p'pA

and

respectively, where

= 0 on S2, (3.86)

because the geometrical boundary conditions are the same for both con-figurations. We may derivefrom Eq. (3.49) the principle of virtual work of theadjacent equilibrium configuration by replacing ... by +

+ z4,..., respectively:

+ + +

• (3.87)

where

• = (u1 + ui),,. + ++ (u's + + (3.88)

and the variation is taken with respect to The equation (3.87) is theprinciple of virtual work for an incremental theory of elasticity

t It is emphasized in Ref. 23 that the stability problcm of a nonconservative systemshould be investigated not only by the Euler method which deals with the static instability,but also by the dynamic method deals with the dynamic instability of small oscilla-tions of the system about the original equilibrium configuration.


Since we are interested in a linearized theory, we may assume that pt'.and are linear functions of and their derivatives. Remembering

that the original configuration is in equilibrium, we may reduce the principle(3.87) into the following form:

fff o4 + dV -

(3.89)V. SI

where

24 = + + (Ô + t4j (3.90)

and higher order terms are neglected. The equation (3.89) yields differential ,i.equations in V and mechanical boundary conditions on S1:

+ + + = 0,

+ + (3.92)

Consequently, if the relations between the incremental stresses and strainsare given in a linearized form by

= (3.93)

we have all the governing equations which determine the critical load andthe adjacent equilibrium configuration.

When Eqs. (3.93) satisfy the symmetry relations:(3.94)

the principle can be. written as follows:

ö iff + dVj

— fff bid dv— ff oz4 dS 0, (3.95)

V Si /

where Eqs. (3.90) have been substituted to express in terms ofWhen the body is elastic and the external forces are cotiservative, we find

that the principle (3.89) reduces to the principle (3.85).The above formulations show that the critical load depends upon the

relations between the incremental stresss and incremental strain measuredfrom the original configuration, rather than the previousrelations. This suggests that the critical load problem qan be treated moregenerally as an instability of a body with initial stresses and deformations.A stability problem of a body with initial stresses will be treated in Section5.2 under an assumption that changes in the geometrical configuration ofthe body remain negligible until the instability occurs.


3.12. Some Remarks

Thus far, the principle of virtual work and related variational principleshave been written fbr the elasticity problem in finite displacement theory.It is noted here that the approximate methods of solution such as the genera-lized Galerkin's method mentioned in Section 1.4 and the Rayleigh—Ritzmethod m.entioned in Section 2.5 can similarly be applied to the elasticityproblem in the finite displacement theory. It is also noted that for a variationof the geometrical boundary conditions on S2 by we have

dU = fff P & dV + ff P dua dS + ff P dü' dS,V SI S2

which is an extension of Eq. (2.49) to the elasticity problem in the finitedisplacement theory.

We have derived in Section 1.2 the conditions of dbmpatibility for thesmall displacement theory in rectangular Cartsian coordinates. The samekind of conditions may be derived of the finitc displacement theory for thestrains es,,, to be derivable from scalar functions However, we shall notshow them in the present chapter, but shall be satisfied with formulatingthe conditions for finitedisplacement theory in general curvilinear coordin-ates later in Section 4.2.'

Mention is made of the principles of complementary virtual workand minimum complementary energy. We have obscrvcd that these prin-ciples play important roles in the small displacement theory of elasticity.However, extensions of these principles to the finite displacement theory ofelasticity are not found successful, since the displacements couple with thestress components as mentioned in Section 3.9.

Bibliography

1. R. lCAppus, Zur ElastizitAtstheode endlicher Zeiisd,rlft für Au-Mathemank Vol.19, pp.271-85, October 1939 and pp. 344-61,

December 1939.2. V. V. NovozlllLov, of Elasticity Pergamon Press, 1961.3. A. E. (jaEN and W. Zws*, Thev.wetlcal Ejaitlcisy, Oxford University Press, 1934.4.1. S. Theory of Elasticity, McGraw-Hill, 1936.5. C. E. Theoretical Elasticity, Harvard University Press, 1959.6. C. ThuEsDEu., editor, Prot4euiu of Ná-lbsear Elasticity, Gordon and Breath, Science

Publishers, 1965.7. C. and R. GRAMMEL, Tedudeche Dynamik, SprInger, 1939.8. L. 0. BRAzmt, The Flexure of Thin Cylindrical Shells and Other Thin Sections,

R. & M. No. 1081, British Aeronsutical Research Council, 1926.9. L 0. BRAZIER., On the Flexiue of Thin Cylindrical Shells and Other Thin Sections,

of the Royal Society of Lofldon, Series A, Vol.116 pp. 104-14,1927.

10. C. B. Biam4o, Das Durchscblagea ames schwach gekrUmmten Stabes. Zeitschraftfür Angewandte Matheniatik sod Mecisanik, Vol. 18, No.1, pp. 21-30. February1938.


11. S. TJMOSHENK0, Theory of Elastic Stability, McGrew-Hil, 1936.12. II. L. General Theory of Buckling, Applied Mechanic.s Reviews, Vol. 11,

No. 11, pp. 585—8, November 1958.13. M. Yosinici et a!., Handbook of Elastic Stability (in Japanese), edited by Column

Research Committee of Japan, Corona Publishing Co. Tokyo, revised edition 1960.14. A. E. H:. LovE, A Treatise on the Mathematical Theory of Fiasticity, Cambridge Uni-

versity Press, 4th edition 1927.15. Y. C. FuNo, Foundations of Solid Mechanic.c, Prentice-Hall, 1965.16. R. L. BISPUNOHOFF, Some Structural and Aeroelastic Considerations of High Speed

Flight, Journal of the Aeronautical Sciences, Vol. 23, No.4, pp. 289—327, April 1956.17. E. REISSNER, Qn a Variational Theorem for Finite Elastic Deformations, Journal of

Mathematics and Physics, Vol. 32, No. 2—3, pp. 129-35, July-October, 1953.18. B. (Jber die Ableitung der StabilitAts-Kriterien des elastischen Gleichge-

wichtes aus des ElastizitAtstheorie endlicher Deformationen, Proceedings of the 3rdInternational Congress for Applied Mechanics, pp. 44-50, Stockholm 1930.

19. K. M4utGuEaRE, Die Behandlung von StabiitAtsproblcznen mit Hilfe des energetiscbenMethoden, Zei:schrzfz für Angewandte Mathemasik und MecJ.anik, Vol. 18, No.1,pp. 57—73, February 1938.

20. K. MARGUERItE, Uber die .Anwendung des energetischen Methode auf Stabilitits-probleme, Jahrbuch der Deutschen Luf:falsrtforsciusng, Flugwerk, pp. 433-43, 1938.

21. V. V. Novozim.ov, Foundations of the Nonlinear Theory of Elasticity, Graylock Press,1953.

22. H. L. LANOHAAR, Energy Methods In Applied Mechardes, John Wiley, 1962.,23. V. V. Boi.o'im, Nonconservative Problems of the Theory of Elastic Stability, translated

by T. K. Lushes, Pergainon Press, 1963.

CHAPTER 4

THEORY OF ELASTICITY INCURVILINEAR COORDINATES

4.1. Geometry Deformation

We shall devote this chapter to the theory of elasticity expressed in generalcurvilinear coordinates.t Let the space coordinates be defined by threeparameters (x', a3) before deformation. We shall employ a set of values

tx3) which locate an arbitrary point p(O) of the body before defor-mation as parameters which specify the point during the deformation.Therefore, the position vector of the point P'°' before deformation is'givenby

= (4.1)

The relations between the rectangular Cartesian coordinates (x1, x2, x3)and the curvilinear coordinates (&, x2, are usually written

A simple example of a curvilinear coordinates, system is cylindricalcoordinates, in which the relations (4.2) are

= r cos 0, x2 = r sin 0, x3 = z, (4.3)

where = r, = 0, = z. By introducing the unit vector associatedwith the axis in the rectangular Cartesian coordinate system, we can writeEq. (4.1) for the present example as follows:

= r cos 0 + Z 13. (4.4)

We shall summarize geometrical relations which are useful in subsequentformulations. For details of their derivations, the reader is advised to referto books on the tensor calculus and differential geometry 4 First, we definethe covariant base vectors associated with the point by

= = (4.5)

as shown in Fig. 4.1, where and throughout the present chapter, the nota-lion ( denotes differentiation with respect to namely, (

t Ref's. I through 6.Refs. 7 through Ii.

76

THEORY OF ELASTIQTY IN CURVILINEAR COORDINATES 77

= By the use of the covariant base vectors, we define themetric tensor by

— = ga,the contravariant metric tensor by

(4.6)

(4.7)

where is the Kronecker symbol, and the contravariant base vector gAby

From these relations, we obtain

(4.8)

=(4.9)

(4.10)

Next, we shall consider the derivative of the covariant base vector g,, withrespect to Since the derivative is again a vector, we may write

ii. i= (4.11)

a

Fio. 4.1. Geometry of an infinitesimal parallelepiped.(a) before (b) aftez deformation.


where the notation {,) is related to the magnitude of the components ofgm,, in the direction of Since

—— S. — —

we obtain

(4.12)(/4VJ LV/SJ

Differentiating both sides of = g, with respect to we have

Substituting (4.11) into the above, we obtain

tel tel —+ — (a,

Interchanging x with v in (a), we have

—

Another interchange of x with in (a) leads to

C

Subtraction of (a) from the sum of(b) and (c) then yields

+

By multiplying both sides of the above equation by and summing withrespect to ,, we finally obtain

Ifl 1= — g,a,.x). (4.13)

The quantity is called the Christoffel three-index symbol of the secondkind. The Christoffel symbols are a measure of the curvature of the curvi-linear coordinate axes, and play an important role in tensor calculus. FromEqs. (4.8) and (4.11), we obtain the derivative of thern contravariant basevector with respect to as follows:

— (4.14)

Let us consider next a vector field in space and denote the vector bycr3). We define components of the vector u by resolving it in the

directions of at the point p(O) as follows:u = (4.15)

THEORY OF ELASTICITY IN CURVILINEAR COORDINATES

where vA is called the contravanant component of the vector u. Differentiat-ing u with respect to x', and using Eq. (4.11), we obtain

a,, = (v2gJ,, = += (4.16)

where is called the covariant denvative vA and is given by

= vA.,+

(4.17)

/ Components of the vector a may be expressed alternatively by resolving itin the directions of gA at the point

U = (4.18)

where v1 is called the covariant component of the vector u and is given fromEqs. (4.9), (4.15) and (4.18) by

(4.19)

Differentiating a with respect to we obtain

a. = (vd)., = Vi: (4.20)

where V1;, is called the covariant derivative of and is given by

Vi:, V1,. (4.21)

It is defined in the theory of tensor calculus that the covariant derivativeof a tensor with respect to is

— +Pi p.:' — th-,'..' p

S I— Ti 1. (4.22)Pi P II/SiP

An application of this relation to ectors has been shown in Eqs. (4.17) and(4.21). As another application) we may show that the covariant derivativesof the covariant and contra var*nt tensors and v,anish:

= 0, 0. (4.23)

It is also added that the followiag forrpula holds for the covanant derivativeof a tensor product of two tensors and T::::

= + (4.24)

/ Two more geometrical relations are noted here before proceeding to thenext section. If we take a point Q(O) in the of the point plO)

and denote the coordinates of by + &, x2 + + dôu3), the


distance between p(O) and Q(O) can be expressed by= . = (g4

= d&'. (4.25)

Next, if we take an infinitesimal parallelepiped enclosed by six surfaces,= constant, + & constant = 1,2, 3), its volume is given by

dV (4.26)

where

(g2 x g3) = g2 . (g3 x g1) = g3 . (g1 x g2), (4.27)

and we have

g11

g = g21 g22 g23 - (4.28)

g32 g32 g33

With these geometrical preliminaries before deformation, we shall proceedto the analysis of stress and strain.

4.2. of Strain and of CompatthUity

After deformation, the point moves to a new position F, the positionvector to which will be denoted by

r = '. (4.29)

We shall define the covariant base vector after deformation by(4.30)

and the covariant and contravariant metric tensors after deformation by

Ga = (4.31)

and= (4.32)

respectively, where is the Kronecker symbol. Differentiation of G, withrespect to yields

= (4.33)

in a manner similar to Eq. (4.11), where

= + (4.34)

THEORY OF ELASTICiTY IN CURVILINEAR COORDINATES 81

The distance between the two points P and Q after deformation is given by

= Jr. Jr = (G1 dzxA). (G,,,

(4.35)

By using the relations (4.25) and (4.35#'we may define the components ofthe strain tensor in general curvilinear coordinates as follows:

= — = (4.36)

Equation (4.36) is a natural extension to curvilinear coordinates of thedefinition (3.14). The specify the state of strain of the infinitesi-mal parallelepiped which was bounded by the six surfaces, = constantand + = constant, before deformation.

Let us consider the strain—displacement relations. Defining the displace-ment vector u(x1, x3) by

(4.37)

and its components byu = vAgi, (4.38)t

we have.= (oX + (4.39)

Substituting Eq: (4.39) into Eq. (4.36), we obtain expressions of the strainin terms of the displacement components as follows:

f,4. + + (4.40)

By the usó of Eqs. (4.19), (4.23) and (4.24), the above relations may be writtenalternatively as follows:

+ + (4.41)

Next, we shall consider the conditions of compatibility in the curvilinearcoordinate system, namely, the necessary and sufficient conditions thatstrain componcnts are derivable from a single-valued vector function

It is known in the theory of tensor calculus that the conditions ofcompatibility are given by

= 0, (4.42)

where is the Riemann—ChristofleI curvature tensor defined by

If H 11

— 0lIpvlf xliii All Jixilli A—

________

— flxvfj ll,svIJ llXC')(4.43)

t We note here that the relation (4.38) is not the only way of defining the componentsof u. For example, may be resolved into the directions of as expressed by the equation(3.16), or in the duections of as expressed by the equation (4.18). Whatever the defini-tions of the components may be, the definition of the strain (4.36) remains unchanged.


The proof that the conditions (4.42) are necessary is easily given by usingthe relations

8 (3G,4 \ — 8 / \ 4 44— — —'

'together with the relations (4.33). However, the proof that they are suffi-cient is rather lengthy, and will not be given here. The reader interested inthe proof is advised to refer to books on tensor calculus or advancedelasticity. It is noted in this connection that the covariant curvature tensor

defined by

= (4.45)

is frequently used instead of and the conditions of compatibility(4.42) can be written alternatively as

(4.46)

where it may be shown that

— 11 82G1I, 826,.—

ko&'+

——

+ G4(({ "}j ffa}) — H,J1 (4.47)

The Greek letters in Eq. (4.42) are assigned in place of (1, 2, 3). Therefore,it appears that relations totaling 34 = 81 in number are contained there.However, there exist relations among the components of such that

= —

and it can be shown that the number of the independent conditions ofcompatibility for three-dimensional space reduces to 6. Moreover, we canprove the existence of Bianchi's identities

14,;. + lrb.:p + = 0 (4.48)

among the Riemann-Christoffel curvature tensors.When the above-mentioned relations are linearized and applied to the

small displacement theory in rectangular Cartesian coordinates we find•that

= —R2323, R, = —R3131, =

= —R1231, U, = —R2312,- U, = —R3123, (4.49)

and the conditions of compatibility (4.46) and Bianchi's identities (4.48)are reduced to (1.15) and (1.17), ftspecfively.

THEORY OF ELASTICITY IN CURVILINEAR COORDINATES 83

- 4.3. of Stress and Equations of Equilibrium

\Ve shall consider the equilibrium of the infinitesimal(see Fig. 4.2) before deformation, was bounded by the six surfaces:

= constant and 4 dd = constant. Let the internal forces acting onone of the surfaces, the sides of are G2 and G3 after

-'

tion, be denoted by —14 do', where g is defined by Eq. (4.28). Thequantities 14 an& 14 are defined in a similar manner. By resolving 14 suchthat

we have the following equations of equilibrium for the infinitesimal parallel-epiped after deformation:

(4.51)

(4.52)

4ere P is the body force vector defined per unit volume of the body beforedeformation (see Section 3.2 for the similar development in rectangularCartesian coordinates).

Equation (4.51) is a vector equathin; one way of expressing it in scalIrform is to resolve it in the directions By the use of Eqs. (4.11) and

we obtain

+ + + v°,,) j

+ = 0 = 1, 2, 3), (4.53)

p

Fio. 4.2. Equilibrium of an paraflelepiped.


where

(4.54)

and the suffix ( ), denotes that the components are taken in the directionsof the base vectors of the generalized coordinate system.

4.4. Transformation of the Strain and Stress Tensors

Let us assume a local rectangular Cartesian coordinate system (y1, y3, y3)issuing from the point p(O) before deformation as shown in Fig. 4.3, anddenote the unit vector in the direction of the y4-axis by $4. Since

we have

and

• )/ôcr" = )/ôyi= E8( )I&*i

FIn. 4.3. A local Cartesian coordinate system (y1, y2, y3)•

8411 = =

= g,.

Scalar multiplication of Eq. (4.55) by $4 yields

Similarly, scalar multiplication of Eq. (4.56) by g,, yields

where the indices and /4 have been interchanged.

(4.55)

(4.56)

(4.57)

(4.58)

y3

1'


We may now formulate the transformation law for strain. Let the straintensor defined with respect to the local rectangular Cartesian coordinatesby denoted by en,, namely:

Dr—

—(4.59)

Then, we have from Eq. (4.36),

Dr Dr—

_f Dr Dr'°>lat Dy'Dy' I

=2ea&'

Consequently, the transformation law between f,4, and e,,, may be writtenoyNoyQ

(4.60)

or converselyos" (4.61)

These relations show and e,4, are covariant tensors of order two.Next, we shall formulate the transformation law for stress. Let us isolate

an tetrahedron which is defined by the th?eesides g1 dx1, g2 dcs2 and g3 issuing from the point before deforma-tion, and consider its equilibrium after deformation, as shown in Fig. 4.4.Let the internal forces acting on the oblique surface of the tetrahedron bedenoted by F LE, where dl' is the area of the oblique surface before deform-ation. The condition of equilibriupi of thà internal forces acting on the tetra-hedronis

F ill' + r2 +

From the geometry of the infinitesimal tetrahedron before deformation, wehave !4.

• v) dcx' = .v) dZ,

lljdcx3 dcx' 2(, . ,) (4.63)t

t In FIg. 4.4, we have .

dl' x — (*2 & — g, a1) x (g3 & —

Combininithis equation with Eqs. (4.27)tno*surnmcd

we obt*in the relations (4.63).


where a' is the unit normal vector drawn on the oblique surface before defor-mation. Substituting the relations (4.63) into Eq. (4.62), we obtain

F =

Fio. 4.4. Equilibrium of an infinitesimal tetrahedron.(a) before deformation. (b) after deformation.

(4.64)

if we an arbitrary set of local rectangular Cartesian coordinates(y1, y2, y3) issuing from the point before deformation as a special caseof the curvilinear coordinates and denote the stress vector defined withrespect to the local Cartesian coordinates by we have, instead of Eq.(4.64),

F (4.65)

Since F is a physical quantity, its magnitude and direction do not dependupon the choice of coordinate system. Therefore, combination of Eqs.(4.64) and (4.65) yie!ds:

(iA = a')

By taking the direction v coincident with the f-axis, we obtain:ar äy'1

which, after an interchange of indices, leads to

0_ha

(4.66)

(4.67)

x3

FdE

xl

THEORY OF ELASTICITY IN CURViLINEAR COORDINATES 87

or conversely

TA" = — (4.68)orThese relations show that and TA" thus defined are contravarjant tensorsof order two.

For later convenience, we note that the resolution of F in the directibnsof the base vectors reads:

F = + VAR) g), (4.69)where is defined by

(4.70)

4.5. Stress—Strain Relations jn Curvilinear Coordinates.

Following the formulations in Section 3.4, we assume stress—strain rela-tions in the local rectangular Cartesian coordinate system to be given by

= (4.71)

and, when the strain components are sufficiently small, by

(4.72)

When a problem of elasticity must be solved in curvilinear coordinates,the stress—strain relations must be as

tAP = (4.73)or, in linearized {orm as

TAP = (4.74)

Since the transformation laws for stress and strain have already been devel-oped (Eqs. (4.60), (4.61), (4.67) and (4.68)), it is rather easy to derive therequired relations. For example, Eqs. (4.72) can be written as,

— Oy'8

or, after multiplication, summation and interchange of indices as,

TAP - (475)Oy" Oy'

which shows that&tA= — — (4.76)Oy' jJye

When the material is isotropic, Eq. (3.37) may be employed for the Stress—strain relations in the rectangular Cartesian coordinate system and Eq.(4.76) takes the form

= (1 + v)(l — 2v)g4 + (3(githlgP$ + (4.77)


4.6. Principle of Virtual Work

The problem which will be treated in the present section is the same asstated in Section 3.5, namely, to find the equilibrium configuration of abody subjected to prescribed mechanical boundary conditions F = P onS1 and prescribed geometrical boundary conditions u = i on S2, togetherwith the imposition of body forces P = P. The object of the present sectionis to derive for this problem the principle of virtual workbpressed in generalcurvilinear coordinates.

From Eq. (4.51) and the mechanical boundary conditions, we have

÷ff(F — P).ördS= 0.V SI

(4.78)By the use of geometrical relations

= ±v1dS, = ±v2dS,

= ±v3 dS, (4.79)t

which hold on the boundary, and followil)g a development similar toSection 3.6, we may transform Eq. (4.78) into

fff — P . or) dv— ff P. Or dS = 0 (4.80)

t In Fig. 4.5, we have

dS = x = (g1 + x (*2 + g3 dc4.)

= g1 x *2 h' — *2 x dc4 — g3 x *i dos',

and consequently

where v is unit normal drawn on the oblique surface. Other relations of Eqs. (4.79) canbe derived similarly.

FiG. 4.5. An area element on the boundary.


where dV is given by Eq. (4.26). This is the principle of virtual work ex-pressed in a general curvilinear coordinate system. By the of Eqs.(4.18), (4.54) and

(4.8 1)

the principle can be expressed alternatively as follows:

ff f — ovA) dV— ff dS = 0. (4.82)

V - SI

From Eqs. (4.60) and (4.68), wn havetAhi = (4.83)

Substituting Eq. (4.83) into Eq. (4.80), we obtain another expression forthe principle of virtual work:

— P.r3r)dV — ffP.ords = 0. . (4.84)

it may appear that the principle (4.84) is identical with the principle (3.48).However, the physical meaning is different, because the quantities and

appearing in Eq. (4.84) are defined with respect to a local rectangularCartesian coordinate system, while and appearing in Eq. (3.48) aredefined with respect to the fixed Cartesian coordinates.

Thus far, the principle of virtual work has been written with respect tothe curvilinear coordinate system. The approximate method of solution,mentioned in Section 1.5, and the technique of finding stress functions withknowledge of the conditions of compatibility, mentioned in Section 1.8,can be applied similarly to the present problem in curvilinear coordinates,once the principle of virtual work has been established. As will be shownin subsequent chapters, the principle plays a very important role in formula-tions of elasticity problems where curvilinear coordinates can be favorably

4.7. Prlraciple of Stationary Potential Energy and Its Generalizationa

We shall assume, as in Section 3.8, the existence of the functions A, GThese are state functions and do not depend on the choice of coordi-

nate system employed, i.e. the functional defined by Eq. (3.69) is invariant.Consequently, once the principle of stationary potential energy has beenestablished in rectangular Cartesian coordinates, the principle in a generalcurvilinear coordinate system can be in terms of by the use ofthe transformation law for strain (4.61), the strain—displacement relations(4.40), and the transfonnation law for displacement components

=where

u = uAIA = (4.86)


It is obvious that the principle can be expressed in terms of vA by the use ofEq. (4.19).

The generalization of the principle :of stationary potential energy israther straightforward. We shall be satisfied with writing one of its generali-zations which is given as follows:

17,= +

— TAM[lAp l(VA;p + Vp;A + dV

+ ff dS— ff — dS, (4.87)

SI S2

where is the prescribed displacement components defined byu = on S2. (4.88)

In the (4.87), the independent quantities subject to variationare 4,, and whilc is dependent of t'., by -

gAMy (4.89)

The stationary conditions are shown to be the governing equations of theproblem, together with

pA = pA;11),(4.90)

which determined the Lagrange multiplier on S2.

4.8. Some Specializations to Small Theory in OrthogonalCurvilinear

In the conclusion of this chapter, some the resu)ts thus far obtainedwill be applied to an orthogonal coordinate system, where therelation (4.25) reduces to

= g1 + g,2(dx2)2 + (4.91)

The summation convention will not be employed in the present section,although Roman letters will beused in place of the numbers 1, 2 and 3.First, we note that Eq. (4.28) reduces to

g = V (4.92)

and the Christoffel symbols given by Eq. (4.13) may be reduced

I. — 1 ö)/glii

= .'.j = 2— , (4.93)

ElfV

ff1= 0, fori,j, k all different.


We shall confine our problem to the small displacement theory and derivethe equations of equilibrium from the principle of virtual work. Since themagnitudes of the displacement components are assumed small, we mayobtain the following linear strain—displacement relations from Eqs. (4.40):

t3v'= +

j. (4.94)

Next, we shall consider a set of local rectangular Cartesian coordinatesy' (i = 1, 2, 3) coincident with the direction of g1 (I = 1, 2, 3) at the point

F"", and denote the unit vector in the direction of g, by We then have

g1 = (4.95)

From Eqs. (4.57), (4.58) and (4.95), we obtain

1— - 1= ô,,, (4.96)

where is the Kronecker symbol. The components of theand body may be. alternatively defined with respect to the localCartesian coordinates by

u = u'j,, (4.97)i_i

3

P ?1j1. (4.98)s_i

Denoting the stresses and strains defined with respect to the local rectangularCartesian coordinates by au, ..., o" and fu' ..., wemay Eqs. (4.38), (4.61), (4.67) and (4.94) to obtain

= 1

•I,j, (4.100)

= Jtg,,g.,jr", (4.101)and

1 ô U

= gjji

(4.102)


When Eqs. (4.97), (4.98) and (4.102) are substituted into the principle ofvirtual work (4.84), we obtain the following equations of equilibrium:"

221181'i 33.r—a ,g33 ,g22

I 12 rT\ 31 i/\

___________

£ g11,g33,+ 2 + 3 + I rg11g22g33 = U,

yg11(4.103)

the other two equations being obtained by cylic permutations of the indices.It is obvious that Eqs. (4.103) arc obtainable alternatively by specializingEqs. (4.53) to small displacement theory in the orthogonal curvilinear coordi-nate system.

1. A. E. and W. ZERNA, Theoretical Oxford University Press, 1959.2. V. V. Novozim.ov. Theory of Fiojlicity, Pergainon Press, 1961.3. Y. C. FUNG, Foundations of Solid Mechanics, 1965.4. E. Kon'a, Methoden der nicbtlinearen Elastizititstheorie mit Anwendungen auf die

dt%nne PlAtte endlicher DUrchbiegung, Zeltschrlft für Angewandte Mathematik mmdMechardk, Vol. 36, No.11/12, pp. 455-62, November/December 1956.

5. K. Geometry of Elastic Deformation and Incompatibility, and A Theory ofStresses aM Stress Densities, Memoirs of the UnifyIng Study of the Problems in

Sciences by Means of Geometry, Vol. 1, pp. 361—73 and 374-91, GakujutsuBunkcn Pukyu-kai, Tokyo, 1955.

6. Y. Meta-theory of Mechanics of Corninqa Subject to Deformation ofArbitrary Ma.qni:udes, Aeronautical Research Institute, University of Tokyo, ReportNo. 343, May 1959.

7. L Bww, Vector and. Tensor AvialyiLs, John Wiley, 1947.8. S. L Smoa and A. Scmw, Tensor Calculus, University of Toronto Press, 1949.9. 1). 3. Smuix, Lectures on Classical Differential Geometry, Addison-Wtslcy, 1930.

10.1. S. Tensor John Wiley, 1951.11. H. D. BLOcK, Introduction to Temawr AnalysLs, Charles E. Merrill,12. P. and H. Methods of Theoretical Physics, Paris I and II, McGraw-

Hill, 1953.13. C. BIEzENo and R. Dynamik, Springer Verlag, 1939.

S. D. Skew Plates and Structures, Pergamon Press,

CHAPTER 5

EXTENSIONS OF THEQF VIRTUAL WORK AND RELATED

VARIATIONAL PRINCIPLES

5.1. Initial Stress Problems

We have derived the principle of virtual work and related variationalprinciples of the boundary value problem in Chapter 3. We shall extendthese principles to other problems of elasticity in the present chapter.tWe shall formulate each problem in the finite displacement theory, specializ-ing to the small displacement theory whenever necessary. The rectangularCartesian coordinate system will be employed for describing the behaviorof the elastic body. However, due to the invariant character observed inChapter 4, expressions of the principles in the general curvilinear coordinatesystem are obtainable through coordinate transformation.

The Lagrangian approach will be employed throughout thechapter. The set of values (x', x2, x3) which locate an arbitrary point ofthe body in a reference state will be used for specification during the sub-sequent behavior. Determination of the reference state depends upon thespecific problem under consideration. Unless otherwise stated, the displace-ment is measured from the reference state.

We shall first consider an initial stress problem.U. 2) By initial stress, wemean those stresses which have existed in a body in the initial state, that is,before the start of a deformation of interest. We choose the initial stateas the reference state of an initial stress problem.

Let a rectangular Cartesian coordinate system (x1, x2, x3) be fixed inspace. Form an infinitesimal .rectangular parallelepiped enclosed by the sixsurfaces: = constant and x1 + dx1 = constant (j( = 1, 2, 3). Denotingthe initial internal forces per unit area acting on the surface = constantby — we define components of the initial stress as

= (5.1)

fit is noted that some of the principles which'will be derived in this chapter may havefields of application outside of elasticity problems.

93

94 VARIATIONAL METHODS IN ELASTICITY AND PLAST?CITY

where I,, is the unit vector in the direction of the x"-axis. Initial body forcesand surface tractions are denoted by P"" and respectively, and theircomponents by

P(O)%, = (5.2)

For the sake of simplicity, we assume that these initial stresses and forcesform a self-equilibrating system, i.e.

+ = 0 (5.3)

in the interior of the body and= (5.4)

on the surface of the body, where ( = ö(We define a boundary value problem of the body with initial stresses by

prescribing additional body forces P*, additional surface forces P on S1 andsurface displacements ua on S2, where the displacements are measured fromthe initial state.

Consider the equilibrium of the infinitesimal parallelepiped after deforma-tion in a manner similar to the development in Section 3.2, and denote theinternal forces acting on the surface with the sides E2 and E3 dx3 by

+ a1") E,, dx2 d*3. Those acting on the other surfaces arc definedin a similar manner. The quantities thus defined will be called incrementalstresses. Then, we find that the

equations for the initial stress problem are derivablefrom Eqs. (3.27) and (3.42) by replacing and P with +p(o)a+ and + Fa, respectively. Consequently following the developmentin Section 3.6, we have the principle of virtual work for the initial stressproblem as fojlows:

f/f + — + P') 6u9 dV

— + P)bu2dS = 0, (5.5)SI

where= + + U"a (5.6)

and ôu1 is required to vanish on S2. When the initial stresses are in self-equilibrium, we employ Eqs. (5.3) and (5.4) to transform Eq. (5.5)into:

f/f + — ôu2) dV

_ffPau2ds0. (5.7)SI

Next, a formulation of the principle of stationary potential energy andrelated variational principles will be considered. First, relations between

VIRTUAL WORK AND RELATED VARIATIONAL PRINCIPLES 95

the incremental stresses and strains are assumed to exist such that

= (5.8)1or conversely

= (5.9)t

where the initial stresses may be contained as parameters. Second Eqs.(5.8) are assumed to satisfy Eqs. (3.53). •Then, the existence of the strainenergy function defined by

(5.10)

is assured. Then Eq. (5.7) may be transformed into

fff [A(uA; + dV

(5.11)

where A(ua; is obtained from a(O))44) by writing e341 in ternis ofu' by the use of Eqs. (5.6), and the variation is taken with respect to u1.

If the existence of the two potential functions defined by Eqs. (3.66) is

also assured, we have a functional for the principle of stationary potentialenergy, which is then generalized by the use of Lagrange multipliers. Here,we record only the expression for H,:

H, = fff + +

— — + + dv

+ ff !P(uA) dS— ff — dS, (5.12)

Si S2

where the independent quantities subjict to variation are and p'with no subsidiary conditions. The stationary shown to bethe governing equations of the initial stress problem, together with

pA + + (5.13)

which determines the Lagrange multijlier on S2. The expression (5.12)suggests that the complementary energy function B(o"; correspond-ing to is given by

(5.14)

where Eq. (5.9) is used to express strains in terms of stresses.We shall derive in this connection a linearized formulation of the initial

stress problem, assuming that the displacements are of infinitesimal magni-.tude, i.e. = O(e)t and the initial stresses are of finite magnitude, i.e.

t We assume throughout the present chapter that stress-strain relations have uniqueinverse relation, unless otherwise

The notation 0(e) stands for "order ole", where e denotes an infinitesimal


= 0(1). This assumption leads to the reduction of the principle ofvirtual work (5.7), and to the linearization of the strain—displacementrelations (5.6) and the stress—strain relations (5.8) as follows:

fff + — PAóui dV

- (5.15)Si

= + u's), (5.16)(T4 = (5.17)

The principle (5.15) yIelds the equation of equilibrium

+ + PA = 0, (5.18)

and the mechanical boundary conditions

+ = P on S2. (5.19)

By combining Eqs. (5.16), (5.17), (5.18), (5.19) and the geometrical boundaryconditions

uA = on S2, (5.20)

we may obtain the governing equations for the desired linearization. Relatedvariational principles for the linearized problem car be derived in the usualmanner.

5.2. Stability Problems of a Body with Initial Stresses

Consider a body with initial stresses under body forces inV and surface tractions kF(0)A on S, where k is a monotonically increasingfactor of proportionality. The quantities p(o)A and F(0)A are assumedto be prescribed. When k is sufficiently small, the equilibrating configurationwill be stable. However, with increase of k, a critical condition may bereached, beyond which the body ceases to be stable. The present sectionwill be with finding the critical nitial stress distribution underthe assumption that, in spite of the increase of k, changes in the geometricalconfiguration of the body remain negligible until the instabilityWe shall confine our problem by assuming that the prescribed body forcesas well as the surface forces on S1 vary neither their magnitudes nor theirdirections, while the body is rigidly fixed on S2. during the period of insta-bility. It is observed that this instability problem may be considered as aspecial case of the formulation developed in Section 3.11.

shall employ as a criterion of instability the existence of an adjacentequilibrium configuration introduced in Section 3.11. It is then obviousthat the linearized formulation in the preceding section yields the governingequations of this instability problem. By replacing by and

VIRTUAL WORK AND RELATEI) VARIATIONAL PRINCIPLES 97

requiring that the incremental body forces surface forces P and displace-ments üÂ vanish in Eqs. (5.15), (5.18), (5.19) and (5.20), obtain

fff + dV = 0, (5.21)

+ 0, (5.22)

+ = 0 on S1. (5.23)

= 0 on S.,. (5.24)

The equations (5.16), (5.17), (5.22), (5.23) and (5.24) describe completelythe instability problem under consideration. The solution of these equationsreduces to an eigenvalue problem, where the critical value of the parameterk is determined as an eigenvalue and the adjacent equilibrium configurationas the corresponding eigenfunction.

When the elastic constants in Eq. (5.17) satisfy the symmetry relation= the principle (5.21) is tranIformed into the principle of station-

ary potential energy, of which the functional is given by

/7 = fJf [A(uA; g(O)1P) + dV, (5.25)

where A(uA; is obtained froma(o)AP) = (5.26)

by writing in terms of ua by the use of Eq. (5.16). in the functional(5.25), the variation is taken with respect to under the subsidiary conditions(5.24), while k is treated as a parameter not subject to variation.

Once thà principle of stationary potential energy is thus deriveçl, it canbe generalized through the use of Lagrange mukipliers. Only the expressionof 11, is shown below:

•= f/f +

— — + dV— ff pAuAdS, (5.27)

where the quantities subject to variation are uA and p2 with no sub-sidiary conditions. The stationary conditions are shown to be the governingequations of the instability problem, together with

= + (5.28)

which determines the Lagrange multipliers pt on S2.Retracing the development in Section it is observed that the principle

(5.25) is equivalent to finding, among admissible displacements u2, thosewhich make the quotient

A(u2) dV

k = — (5.29)4 fjj dV

•98 VARIATIONAL METHODS IN ELASTICITY AND PLASTICITY

stationary. When the smallest positive and largest negative values of k arefound to be K4 and K-, respectively, the body is stable as long as the valueof k lies within the region bounded by the two extremities, i.e. K' <k < K4.The expression (5.29) is closely related to the Rayleigh quotient in eigen-value 6)

5.3. Initial Strain Problems

Assume that there exists a body in a reference state with neither stressesnor strains. Let this body be cut into a number of infinitesimal rectangularparailciepipeds and let each piece be given strains of arbitrary magnitude.'These strains are called initial strains and will be denoted by in thefollowing foTmulation. Then, let the pieces be reassembled and brougbtagain into a continuous body.

A set of strains must be added to the initial' strains to reproducecontinuous body, because of the incompatibility the arbitrary initialstrains. These incremental strains cause internal Stresses, even if neitherexternal forces nor displacements are applied. We shall generalize thepresent problem further by prescribing, together with the initial strains,body forces P. surface forces P on S1, and surface displacements onS3, where the displacements are measured from the reference state.- The set of incremental strains required to produce theifinal configurationis denoted by In general, neither the initial strains nor, the incrementalstrains satisfy the conditions -of compatibility. But their/sums,

+ (5.30)

must satisfy the conditions of compatibility, namely, They should be deriv-able from U', measured from the reference state, such that

= + + ',' (5.31)

Then, retracing the development in Section 3.6, we find that the principleof virtual work for the initial strain problem is also given by Eq. (3.48)through the use of the strain—displacement relatiuns (5.31).

Next, we shall derive the principle of stationary potential energy andrelated variational principles. First, we assume that stress—strain relationsare given by

= ei), (5.32)or conversely

= 4), (5.33)

where the initial strains appear as parameters and = 0. Second,Eqs. (5.32) are assugied to satisfy Eqs7'(3.53), thus assuring the existence ofa strain energy function defined by

dA (5.34)

VIRTUAL WORK AND RELATED VARiATIONAL PRiNCIPLES 99

Third, the existence of the twopotential functions and !P(u') is assumed.We can now derive the principle of stationary potential energy for theinitial strain problem. The principle can be generalized by following familiarprocedures. For example H1 may be shown to be

H1 =fff +

— — + + dV

+ ff dS — If — dS, (5.35)St St

where the independent quantities subject to variation are on', uA andwith no subsidiary conditions. The expression (5.35) suggests that thecomplementary energy function corresponding to is

(5.36)

Eqs. (5.33) are used to.cxpress in terms of -

We may then conclude that the principle of virtual w&k and relatedvariational principles arc derived in the same forms as those in Chapter 3,except for the difference in the expressions for A and B. Similar statementscan be made for initial strain problems of the small displacement theory.

5.4. Thermal Siress Problems

Consider an elastic body in a reference state with neither stresses norstrains and of uniform absolute temperature T0. Then, let adistribution T(x2, x2, x3) be given to the body, while body forces andsurface boundary conditions are applied as prescribed in Section 3.5. Ourproblem is to find the stress distribution thus created in the body.t

Since we know that the coupling between the elastic deformation and theheat transfer is very weak and can usually be neglected, we shall assume thatthe temperature distribution is prescribed and the stress—strain relationsarc given as

= 0), (5.37)where -

0) 0 and 0 T -- T0.

Once the above assumption is employed, the equations which govern thethermal stress problem arc found to be the same as those of the problemin Chapter 3, .except that Ens. (3.33) are now replaced by Eqs. (5.37), inwhich the temperature 0 appears as a parameter. Thus, the principle ofvirtual work for the thermal stress problem is also given by Eq. (3.48).

Rçtracing the development of Section 3.7 and keeping in mind that thetemperature distribution is prescribed, we find that the stran energy func-

f Refs. 7 through 14.


tion exists for each elqment of the elastic body in the thermoelastic problemand is equal to the Helmholtz free energy by Eq. (3.63). Consequently,we need only the existence of the two state functions and W forthe establishment of the principle of stationary potential energy, of whichthe functional is shown to be

II = fff 0) + dV + ff dS, (5.38)V SI

where the quantities subject to variation are under the geometrical bound-ary conditions on 52, while the temperature 0 is treated as prescribed andnot subject to variation. Since the strain energy function appearing in thefunctional (5.38) is equal to the free energy function, the principle of station-ary potential energy for the' thermoelastic problem is frequently called theprinciple of stationary free Once the variational principle is thusestablished, it can be generalized by the use of Lagrange multipliers in amanner similar to the development in Section 3.9.

We may then conclude that the principle of virtual work and relatedvariational principles for the therinoelastic problem are expressed in thesame forms as those in Chapter 3, except for the difference in the expressionsfor A and B. Similar statements can be made for thermoelastic problemaof the small displacement theory.

Mention is made of the stress—strain relations and the expressions forA and B. From the free energy function defined in Eq. (3.63), we may derivethe relation:

dF0 = dU0 — TdS — S dl'. (5.39)

Combining the above with Eq, (3.59), we obtain(5.40)

which yields:

= S — (5.41 a, b)

This means that once the free energy function is given 'explicitly in termsof and T, the stress—strain relations (5.37) are derived functions.

We shall look for linear stress—strain relations for the thermoclasticproblem by assuming

F0 = a0 + + (5.42)

where a0, and are functions of Tand = a. Then, from Eqs.(5.41 a) and (5.42), we have

(5.43)

If we denote the thermal strain by we have= (5.44)

because it is required that = 0 for = The inverse relations ofEq. (5.43) may be obtained as

= + (5.45)

VIRTUAL WORK AND RELATED VARIATIONAL PRINCIPLES

Equations (5.45) show that for a linear relationship, the thermal Strains maybe treated as initial strains. The expressions of the twO functions defined by

dA = dB = (5.46)

are derived from Eqs. (5.43) and (5.45):

A — (5.47)

B = + (5.48)

When the material is isotropic both elastically and thermally, we maychoose

= (5.49)

and derive the following relations:

= (1 — 2v)e +

— (1 —(5.50)

(1 — 2v)+ + (5.51)

2(1 — 2v)e2 +

— (1 — 2v)et'e, (5.52)

B = 3(1

2Eg2 + + 3e°a, (5.53)

where and & are the Kror1ecker symbols. If a linear relation is postulatedbetween the thermal strain e° and the temperature difference 0, may

itee°=xO, (5.54)

where is the coefficient of thermal expansion.Thermal stress is closely associated with high speed flight and has been

one primary problems in the design of flight vehicles in recent years.A great number of papers have been written on the subject, some of whichare listed in the bibliography for the reader's reference.

5.5. Quasi-statIc Problems

Consider a body in a reference state with neither stresses nor strains.Let this body be subject to time-dependent body forces P(x', x2, x3, :),surface forces P(x', x2, x3, :) on S1 and surface displacements (x', x2,x3, t and are measured from the reference state. Ourproblem is to find the deform4tion and stress distribution of the bodydue to the motion.

In the present section, we shall consider a quasi-static formulation of thedynamical problem presented above. By quasi-static, we mean that the

102 VARIATIONAL METHODS IN ELASTICITY AND PLASTICFFY

time rate of change of the prescribed body forces, surface forces and dis-placements is so gradual that inertial terms can be neglected in the equationsof motion. It is then obvious that the principle of virtual work and relatedvariational principles can be formulated in the same manner as in Chapter 3,except that the time i now appears as a parameter. Consequently, weshall be rather interested in the quasi-static problem expressed in terms ofrate as follows: given the distribution of stresses and displacements uain the body at the generic time, find the time rates of change of the stresses,

and displacements, induced in the body, where a dot denotes differen-1iation with respect to time.

Since the equations of equilibrium and boundary conditions are writtenin terms of rate as

in V, (5.55)

= on S1. - (5.56)

on (5.57)where

P + + (5.58)we have

+ + F}ouAdV

+ ff(P - P) aS =0. (5.59)

After some calculation, we may reduce Eq. (5.59) to.

+ -

—ffPou1ds=0, (5.60)

where öê,,, denotes the variation of with respectto u"pnly,

Equation (5.60) is the principle of virtual work for the quasi-static problem.Next, we shall consider variational principles of the quasi-static problem.

First, we assume relations between the stress-rate and strain-rate to begiven by,

= e4), (5.63)or, conversely,

= ed), (5.64)


where and e4 may be contained as parameters. Second, the relations(5.63) arc assumed to satisfy the equations

&r"= , (5.65)

4which assure the existence of a state function e,1,) defined by

= (5.66)

Third, two state functions and !P*(üa) defined by

= d!P* = (5.67)

are assumed to exist. Then, we may obtain the principle of stationary poten-tial energy for the quasi-static problem from. Eq. (5.60). The principle thusobtained can be generalized by the use of Lagrange multipliers; the general-ized expression may be shown to be

17, = fff + dV

+ — — i31)dS, (5.68)Si 'S3

where thà independent quantities subject to variation are 6", dA and ,)A

with no subsidiary conditions, while the and treatedas parameters not subject to variation. The stationary conditions are shownto be the governing equations of the quasi-static problem, together with

(5.69)

which determine the Lagrange multipliers fr' on S2, The functional (5.68)is equivalentto that formulated by SanderS, McComb and Schlechte.""

The expression (5.68) suggests the function ak', e,,) correspondingto to be

(570)

where are expressed in terms of by the use of Eqs. (5.64).When the q)iasi-static problem is confined to the small displacement

theory, the governing equations corresponding to Eqs. (5.55), (5.56), (5.57)and (5.61) arc

+ I = 0 in V, (5.71)

= 1', on (5.72)

= on S2, (5.73)

24, ÷ lfl V, (5.74)


respectively. Here, components of the stress, strain and displacemcnt aredenoted respectively by and white the quantities X's, and u1are respectively components of the prescribed body forces, surface forcesand displacements. From the above relations, we obtain the principle ofvirtual work for the quasi-static problem of the small displacement theoryas,

ffj — X, dV— ff P dS = 0, (5.75)

where Eqs. (5.74) have been substituted. Since Eqs. (5.11) through (5.74)hold without the dot notation, we have the principle in mixed forms as

fff — X1 dV— ff bu,dS = 0, (5.76)

'fff — dV ff dS = 0. (5.77)V SI

It is obvious that we may also obtain the principle of complementary virtualwork corresponding to Eq. (5.75) as

fJf ad,1 dV— ff dS = 0. (5.78)

V S2

The expressions which correspond to Eqs. (5.76) and (5.77) may be derivedin a similar manner. When the relations letween stress or stress-rate andstrain or strain-rate assure quantities such as to be perfect differen-tials, the above principles may lead to variational formulations.

5.6. Dynamical Problems

We shall now consider the dynamical problem defined in Section 5.5without Tequiring the motion of the body to be quasi-static. The equationsof motion for the dynamical problem are obtained from Eqs. (3.22) and(3.25) by replacing P with P —

&A + — e 0, (179)

(5.80)

where is the density of the body per unit volume in the reference state.Consequently, Eq. (3.48) holds for the dynamical problem if the abovereplacement is made. By integrating the equation thus replaced with respectto time between t = t and employing convention thatvalues of r at t = and I = 12 are prescribed such that 5r(x', x2, x3, t,)= cSr(x', x2, x3, 12) 0, we finally obtain the principle of virtual work forthe dynamical problem as follows:

(5.81)


wheree2,4 = + (5.82)

and T is the kinetic energy of the body defined by

T = dV = dv. (5.83)

If the existence of the strain energy function defined by Eq. (3.51) isassured, we find that the principle (5.81) becomes

f — MI + fff P. or dV +f . Or dSJ dt = 0, (5.84)

where U is the strain energy of the elastic body:

U = fff A(UA)dV. (5.85)

The principle (5.84) is useful in application to dynamical problems of elasticbodies in which external forccs are not derivable from potential functions.

If the existence of the two potentials and W defined by Eqs. (3.66) isalso assured, the above principle reduces to

of[T_ U—fffbdV—ffWdSJdt=0, (5.86)

where the variation is taken with respect to u2. Equaiion (5.86) is Hamilton'sprinciple applied to the dynamical problem of the elastic body. It statesthat among all admissible displacements which satisfy the prescribed geo-metrical boundary conditions on S2 and the prescribed conditions at the limitst = 11 and t = t2, the actual solution makes the functional

[T_stationary.

It is an extension of the principle of stationary potential energy (3.68)to the dynamical problem. Its generalization can be formulated in a mannersimilar to the development in Section 3.9.

We shall employ Eq. (5.84) in subsequent formulations, in order to accountfor forces not derivable from a potential function, and we shall consideran approximate solution of the problem under the assumption that thedisplaced components of the body can be expressed in terms of a discretenumber of generalized coordinates (r = 1, 2, ..., n) as follows:

= uA(xl, x2, x3; q1, q2, ..., t), (5.87)

where the generalized coordinates are functions of time. The expressions(5.87) are so chosen as to satisfy the prescribed geometrical boundary con-


ditions on S2, irrespectively of the values of the generalized coordinates.From Eqs. (5.87), we obtain

(5.88),.i

N

ovA = -i--- Oq,. (5.89)uIJ,

Introducng Eqs. (5.87) and (5.88) into Eqs. (p.83) and (5.85), we can expressthe Lagrangian function

L=T—U (5.90)

in terms of q, and q,. With these preliminaries, the first two terms on theleft hand side of Eq. (5.84) are transformed into

t2 *2*3L ØL

v4, jSi I,.

*2a ÔL '2 ,4 R d / .3L OL 1- I

r—1 :i J ,—1 L Ut \ vq,, uq,. jSi

= —Oq, di, (5.91)

where the convention Oq,(:1) = Oq,(t2) = 0 (r = 1, 2, ..., n) is employed.tIntroducing Eq. (5.89) and remembering that Or = Ou, we find that theremaining terms of Eq. (5.84) become

(5.92)

where

Qy =jjf (r= 1,2, ...,n) (5.93)

is called the generalized force. Introducing Eqs. (5.91) and (5.92) into Eq.(5.84), we have

(5.94)

$ t1

Since the tq, are independent, the above equation leads to nequations:

t3L

— -r— Q1, (r = 1,2, ..., n). (5.95)

•t This corresponds toour earlier assumption that

x2, x3, - ?w(xt, X3, 12) = 0.


arc Lagrange's equations of motion for the elastic body. For applica-tions of these cquatioms to motions of elastic bodies, the reader is directedto Refs. 16, 17 and 18. The above formulations have been made in the finitedisplacement theory. However, they may be specialized to the small dis-placement theory through the familiar procedure, i.e. linearization of thestrain—displacement relations (5.82).

5.7. DynamIcal Problems of an Unresfrslned Body

In the last section of this chapter we shall consider a dynamical problemof an unrestrained body.U8. 19, 20) Let a rettangular Cartesian coordinatesystem (11, X2, X3) be fixed in space and let the unit vector in the directionof the Xk.axis be denoted by as shown in Fig. 5.1. Let another rectangularCartesian coordinate system (x', x2, x3), called the body axes, be fixed tothe body in a reference state with neither stresses nor strains. The Lagrangianapproach will be employed, i.e. the set of the values (x', x2, x3) which locatean arbitrary point of the body with respect to the body axes in the referencestate will be used for specification during the mOtion. The position vectorof the point of the body at the generic time t is given by

Fzo. 5.1. Fixed and moving coordinate systems.

xt

(5.96)r = rG +

r u.

K,


Here, rG is the position vector from the origin of the space-fixed coordinatesto the origin of the body axes, is the position vector from the origin ofthe body axes to a point of the body in the reference state and u is the defor-mation vector. Components of the last two vectors are defined withto the body axes as follows:

= u = u2k1, (5.97)

where k2 is the unit vector in the direction of the Since

= w x (5.98)

where= pk1 + qk2 + rk3 (5.99)

is the angular velocity vector of the moving coordinate system, we have

diw x (5.100).

du d*u= + w x ii, (5.101)

where d*( )/dt denotes a partial differentiation, the unit vectors ka(A = 1, 2 3) being held constant. For example, we have d*u/dt =

We shall define the orientation of the body axes with respect to the space-fixed coordinate system by the Eulerian angles 4, 6 and tp, as shownin Fig. 5.2 to obtain the following geometrical and kinematical rela-

22. 23)

k1 cos 6 cos tp, cos 0 sin —sin 0

.

k2 = . . . . . s1n4cos0 I

I

sin sin I

k3 . . . cos4'cosOL

(5.102)1 0 —sinO

q = 0 cos4) sin4'cosO . (5.103)

r 0 —sin 4) cos 4) cos 0

Combining Eqs. (5.96) and (5.100), we may express the kinetic energy ofthe body as follows:

I (drG )2f/f dV+ x

+ dV+

x w) .ffjr(0) o dV

tu.f[f(r0 x (5.104)

WRTUAL WORK AND RELATED VARIATIONAL PRINCIPLES 109

The vector rG is a function of time only. Consequintly, we have+ u)/axA and we find that the strain—displacement relations are

given by(5.105)

Combination of Eqs. (5.96), (5.100) and (5.101) yields the following resultfor the virtual displacement vector or:

where

+ (—öOsin# +

Fia. Si. The Eulerian angles.f.

(5.106)

(5.107)

ii,

t The transfon X2, X3)to the moving axes (x1, x2,x3) is defined bythree sucoesane angles of rulalion. the (Xl, 12, X3) axes are rotatedaround tbel'4xi$ by angle ilto N3) axes. Second, the(,1, ,2axes are.rotated around the vj3-anls by the 0 to obtain (x', 172, axes. Finally, the

(x1, 173)axarsiotalsd xt.uis'by theangle 4, to obtain (x', x2, x3) axes.The three anglel 4,, 0 called the specify the orientation Qf the(x1, x2, x3) axes uniqudy.

XI

XI

xl


We shall define a dynamical problem of the unrestrained body by prescrib-ing, in addition to the body forces P(x', x2, x3, t), the external forcesP(x1, x2, x3, t) applied on the surface S of the body. We then find that theprinciple of virtual work for the dynamical problem of the unrestrainedbody is given by

= 0, (5.108)

which, when the strain energy function exists, becomes

f[5T — 6U + fff P. or dV + ff P. Or dc} di =0. (5.109)

Since dynamical problems are beyond the intended scope of this book,we shall mention briefly only two special cases. The first example is themotion of a rigid body, where the deformation vector a vanishes throughoutthe body. We shall take the origin of the body coordinates in coincidencewith the center of gravity of the body. Equations (5.104), (5.106) and (5.109)then reduce to

r = dV + x dv, (5.110)

Or = OrG + O'ø x (5.111)and

(5.112)

respectively. Since OrG, 04, 06 and O*p are arbitrary, we obtain from theprinciple (5.112) six of motion of the rigid body which are writtenin vector forms as follows:

(5.113)

and

= x + Jf(rc0 x, F)dS, (5.114)t

t Since offf3(w H,5q+H5ôr andV

—4' ôO cosO, ..., we have through integrations by parts a f x dv] at

?I . N di. where N ÷ qHg — rH,) k, + (H, + — pH1) k2

+ (H1 ÷ pH, — k3 =

VI*TUAL WORK AND RELATED VARIATIONAL PRINCIPLES Ill

In the above equations, M and H are the total mass and the total angularmomentum around the centre of mass of the rigid body, respectively,defined by the following relations:

M=fffedV. (5.115)

H + H,k2 + .(5.116)

p

H, = I, q , (5.117)

r

fff (y2 + 1, = fff (z2 + I = f/f (x2 +

—fffxzedV, (5.118)

where notations x, y, z are used in places of x', x2, x3.The second example is a small disturbed motion of an elastic body. We

assume that the body is in rectIlinear flight with a constant velocity beforeit is subjected to small external disturbances. The steady flight state takenas a reference state, in which the origin of the axes is located at thecenter of gravity of the body. The x'-axis is taken in coincidence with thedirection of the constant velocity. The Eulerian angles are measured froma system of reference axes, thç directions of which are coincident with thoseof the body axes in rectilinear flight.

Now, thebody is assumed subject to small external disturbances. Follow-ing Ref. .5.18, we may express the elastic deformation in terms of the normalmodes of the unrestrained elastic body as

ii = (5.119)

where is the i-th natural mode. Then, we obtain the equations of motionof the unrestrained body from the principle of virtual work (5.109), wherethe independent quantities are ôrG, ö4, ôO, and (1 = 1, 2, ...). Afterneglecting higher order terms, we may finally reduce these equations tolinearized form. The reader is directed to Ref. 18 for further detail.

Thus far, we have derived the principle of virtual work and related varia-tional principles several elasticity problems. In the five chapters whichfollow, these principles will 1,e applied to spócial problems such as torsionof bars, beams, plates, shells and structures. In these applications, thematerial is assumed isotropic and homogeneous, and problems are treatedin the small displacement theory, unless otherwise stated. Moreover, weshall employ conventional notations in these problems. For example, u, vand w will be used instead of u1in Chapters 7, 8 and 9, while the symbolsii, v and w will be reserved cor expressing displacement components

112 VARIATIONAL METHODS TN ELASTICITY AND

centroid locus of the beam or of the middle surface of plates and shells.As a second example, we note that a,, ... and will be employed evenin the finite displacement theory with an understanding that these symbolsnow represent defined in Chapters 3 and 5.

1. E. Tnnwrz, Zur Theorie der StabibtAt des elastischen Gkichgewichts, Zeitschr:ft für4ngewandte Mathematik wzd Mecisanik, Vol. 13, No. 2, PP. 160-5, April 1933.

2. V. V. Novozim.ov, Foundations of the Nonlinear Theory of Elasticity, Graylock, 1953.3. W. PRAGER, The General Variational Principle of the Thcoiy of Structural Stability,

Quarterly of Applied Mathen,aiks, Vol.4, No.4, pp. 378-84, January 1947.4.3. N. G000IER and H. J. PLASS, Energy Theorems and Critical Load Approximations

in the General Theory of Elastic Stability. Quarterly of Applied Mathematics, Vol. IX,No.4, pp. 371-40, 1952.

5. L. COLLATZ, E rail technlschen Anwendungen, Akademische Ver-lagsgesdllschaft, Leipzig, 1949.

6. G. Tasil'tn and W. Q. BICELEY, Rayleigh's Principle and its Application to Engineering,Oxford University Press, London, 1933.

7. S. and i. N. Goooma, Theory ofElasticity, McGraw-Hill, 1951.8. W. S. HEMP, Fundamental Principles and Methods of Thermal Elasticity, Aircraft

Vol. 26, No. 302, pp. 126-7, April 1954.9. W. S. Fundamental Principles and Theorems of Thermoelasticity, Aeronautical

Quarterly, Vol. 7 Part 3, pp. 184—92, August 1956.10. W. S. HEMP, Methods for the Theoretical Analysis of Aircraft Structures, AGARD

Lecture Course, April 1957.11. B. E. GAmwooD, Thermal Stresses, McGraw-Hill, 1957.12. B. A. Botay andi. H. Theory of Thermal Stresses, John Wiley, 1960.13. R. L. BIsPuNoaoff, Some Sfructural and Aeroelastic Considerations of High Speed

Flight, Journal of the Aeronautical Sciences, VoL 23, No.4, pp. 289-327, April 1956.14. High Temperature Effects in Aircraft Structures, edited by N. J. Hoff, AGAkDograph

28, Pergamon Press, 1958.15. J. L. SANDERS, JR., H. G. McCorm, IL, and F. R. Sciu.acwrn, A Variational Theorem

for Creep with Applier,) Ions to Plates and Columns, NACA TN 4003.16. Y. C. FuNo, Iqtroduction to the Theory ofAeroelasticity, John Wiley, 1955.17. R.. L. H. AmIzY and R. L. HALPMAN, Aeroelasilcity, Addison-Wesley,

1955.18. R. L. BISPLINOHOFF and H. ASHLEY, Principles ofAeroelasticity, John Wiley, 1962.19. H. GOLusTEIN, Classical Mechanics, Addison-Wesley, 1959.20. J. L. and B. A. Giw'vrm, Principles ofMechanics, McGraw-Hill, 1959.21. R. A. FPJZER, W. 3. DUNCAN and A. R. COU.AR, Elementary Matrices, Cambridge

University Press, 1938.22. M. I. Anzuo, Applications of Matrix Opórators to the of Airplane Mo-

tion, Journal of Aeronautical Sciences, Vol. 23, No. pp. 679—84, July 1956.23. B. BriaN, Dynamics ofFlight, John Wiley, 1959. -

24. B. L. B'v'xuu and E. R. GRADY, General Air-frame Dynamics of a Guided Missile,Journal of Aeronautical Sciences, Vol. 22, No. 8, pp. 534-40, August 1955.

CHAPTER 6

TORSION OF BARS

6.1. Theory of Torsion

In the present section, the Saint-Venant theory of torsion of a cylindricalbar is treated. Unless otherwise stated, the cross section of the bar, denotedby the area S, is assumed to be simply connected. Let the z-axis be takenin the direction of the generating line of the cylinder, and the x- and y-axesin the sectional plane, as shown in Fig. 6.1. Torsion of a bar is defined asthe application of twisting moments at ends of bar, while keepingthe side surface of the bar traction free. Consequently, the mechanicalboundary conditions at the ends, z 0 and z = 1, are given as

andA', = — Y. = — z, = 0, (6.1)

z.=o,

113

(6.2)

x

FiG. 6.1. Torsion of a bar.


respectively, to produce the twisting couple

= — (6.3)

The displacement components u, r and w for a cylindrical bar in torsionare assumed to 2)

u = — ily. v = Ox, w w(x, y, z). (6.4)

where = 0(z) is the twist angle of the cross section thea function of z. The relation (6.4) assures us that the only

ing strain components are and which are given by

aw dO

E'=—a-—,(6.5)t

It is assumed in the Saint-Venant theory of torsion that the deformation ofthe bar takes place independently of z. This means that w(x, y, z) and di9/dzare independent of z. Therefore, we may write

u = — Oyz, v = Oxz, w = w(x, y), (6.6)and

ow= 0, = — yO, = + xO, (6.7)

where 0 dO/dz is the rate of twist. Accordingly, the only non-vanishingstress components are and which are related to and by

= r3.1 = (6.8)f

With the above preliminaries the principle of virtual work for the Saint-Venant theory of torsion can be expressed as follows:

— + + xoo)jdxdy — = 0,

where the length of the cylinder is taken as unity, due to the uniformity ofthe deformation in the direction of the z-axis. After some cakulation theprinciple (6.9) is transformed into:

— ff( +Ow dx dy +fftxz/ +

+—

I and m are the direction cosines of the normal v drawn outward onthe boundary C. If the contour of the boundary C is given by x = x(s)

t Notations and will be preferably employed instead of and in Chapters 6,7 and 8.

Eq. (1.32).

in s,

1 d

N' =

TORSION OF BARS 115

and y = y(s), where s is measured along the contour as shown in Fig. 6.2,we have

1=dy/ds, m——dxfds. (6.11)

Since âw and óO are arbitrary, we have, from Eq. (6.10), the equation ofequilibrium and boundary conditions as follows:

+ =•o in S. (6.12)

+ = 0 on C, (6.13)and

A? = ff (r,2x — rxzy) dx dy. (6.14)

V

Fio. 6.2. Directions of s andy.

One of obtaining the solution is to eliminate and fromEqs. (6.7), (6.8), (6.12) and (6.13). Using Eqs. (6.11), the finallyleads to -

(6.15)

C, (6.16)

(6.17)and

(6.18)

Thus, the function q, called the warping function of the cross section, is aplane harmonic function which satisfies the boundary condition (6.16).Once the solution has been obtained, we have from Eq. (6.14),

(6.19)

s—O

l!6 VARIATIONAL METHODS IN ELASTICITY ANIk PLASTICITY

which means that the torsional rigidity of the cross section is given by G.J,where

+y2}dxdy. (6.20)

Many studies of the Saint-Venant torsion problem have been made.The problem has been solved for various cross sectional forms such as thecircle, ellipse, square, rectangle and so forth. The interested reader is advisedto reads for example, Refs. 1, 2, and 3.

6.2. The Principle of Minimum Potential Energy and its Transformation

it observed that the functional (2.12), when combined with Eqs. (6.1),(6.2), (6.6) and (6.7), provides the total potential energy for the Saint-Venanttrsion problem:

17 = —o,)2

+÷ dx dy — OA?, (6.21)t

where the length of the cylinder is taken as unIty4 and the absolute minimumproperty of the total potential cnergy for the actual solution can be proved.Following a development sinhilar to that in Chapter 2, we may generalizethe expression (6.21) as (ollows:

1— — rx;

—— — ox)T4 dx dy — OM, (6.22)

where the independent functions and scalar quantity subject to variationsare w and 0. Since the fitst variation of is given 1*

oil,= ff — öy7. + —

aN' /' FJw— (Yxz — + 0Y)t5Tx: — — -- —

- - d,J

t This functional is also obtainable frorp the principle of virtual work (6.9) combinedwith Eqs. f6.7) and (6.8).

TORSION OF BARS

it is easily shown that all the conditions which define the torsion problemunder consideration can be obtained from the requirement that bestationary.

Now, let us employ the following stationary conditions as subsidiaryconditions:

= (6.24)

+ = 0 in S, (6.25)

+ = 0 on C, (6.26)

which mean elimination of Yy: and w from Since Eq. (6.25) is auto-matically satisfied through the introduction of a stress function 4)(x, y)defined by

= (6.27)

we shall use 4' in place of Eq. (6.25) in subsequent Then, byusing the relations (6.11), we may write Eq. (6.26) as

ôyds 9xds dsor equivalently

4) = c0, (6.28)

on the boundary C, where c0 is integration constant. Since the crosssection of the bar is assumed to betGnply connected, we may put

on C, (6.29)

without loss of generality. Thus, elimination of and y, through Eqs.(6.24) and introduction of the stress function 4) defined by Eq. (6.27) hans-form H, into U1, defined by

fill f[{1+ (a4))j

— 204)} dx dy

wds — Of(xl + ym) 4) d.c — (6.30)

Imposition of the condition (6.29) further simplifies Eq. (6.30) to

17,1! fJ{L+ (84')2J

284'} dx dy — 02, (6.31)

which is the final result obtained from fl, through the elimination of Yxx,Y,.z and w by the use of the relations (6.24), (6.25) and (6.26).

'Tjn derived above, the independent function and scalar subject tovariation are 4' and 0, respectively, where 4' satisfies the condition (629).


Further reduction of Eq. (6.31) leads to the following stationary conditions:

0241 0241+ + 2G0 = 0 in S, (6.32)

2 = 2ff (6.33)

where Eq. (6.29) is taken as the subsidiary condition. The equation (6.32)thus obtained is the condition of compatibility for the torsion problem.A direct proof will be given as follows: By the use of Eqs. (6.7), (6.8) and(6.27), we have

Ow Owdw = -3—dx + -5--dy

= + Oy) dx + — Ox) dy

1041= + dx

— (-b-+ Ox) 4. (6.34)

By integrating the above relation along an arbitrary closed path withinthe region S, we have

I 041 1 041 +Ox)dy

=—

+ + (6.35)

where the btacket notation indicates the increase of the value of theenclosed function with respect to one complete circuit of the closed path.The notation is the integral around the closed path, and the area integralis defined with respect to the region enclosed by the closed path. Thus,Eq. (6.32) assures that no dislocation is allowed for the displacement w.

Furthermore, if the condition (6.33) is taken as one more subsidiary con-dition, 11111 can be transformed into written as follows:

ffI [(841)2 + (84))2J

dx 4, (6.36)

where the independent function subjectto v nation is 41 under the subsidiaryconditions (6.29) and (6.33). It is verified that among admissiblefunctions 41, the actual solution makes the functional III, a maximum andthat the variational principle thus obtained is equivalent to the principleof minimum complementary energy. If we backward from to H,,,,it is easily seen that the scalar quantity 0 appearing in'Eq. (6.31) plays therole of a Lagrange multiplier via which the subsidiary condition (6.33) isintroduced into the framework of the variational expression,., .4

TORSION OF BARS 119

Finally, it is noted that in the Saint-Venant theory of torsion of a bar,the strain energy and complementary energy stored per unit length of thebar are given by

and

— )2+ + x)2] dxdy = , (6,37)ax

S

1 rr2GJJ

S

dxdy= 2bjM2, (6.38)

respectively, where M is the twisting stress couple at the cross section.

6.3. Torsion of a Bar with a Hole

In the present section, we shall derive variational formulation. thetorsion problem of a cylindrical bar with a hole, as shown in Fig. 6.3. Letthe outer and inner boundaries of the cross section be dónoted by C0 and C1,

respectively. The assumption of Saint-Venant torsion .theory asserts thatthe equations defining the problem are the same as stated in Section 6.1,eicept for one additional condition on the boundary C1:

+ = 0, (6.39)

where / and m are direction cosines of the normal drawn on the inner bound-ary from the interior of the bar S into the hole. By the use of the stress func-tion defined by Eq. (6,27), the condition (6.39) can be written as

c1 on the boundary C,, (6.40)

Fto. 6.3. A bar with a hole.

120- VARIATIONAL METHODS IN ELASTICITY ANt) PLASTICITY

where cg is an arbitrary integration constant. We have put the integrationconstant c0 given in Eq. (6.28) equal to zero before. the samedisposition cannot be applied to the integration constant c,. A formula fordetermining the value of c1 will be given in the following.

With the above preliminaries, it is easily shown .that expression of thetotal potential energy His the same as (6.21), if the integration is taken overthe region between the two curves C0 and C,. However, care should be takenin generalizing the principle. The generalization is the same as (6.22),except for the region of integration. But in the avenue leading from tofl11, we should notice that we now have two boundaries. Thus, we have,

=— 20c6J dxdy —02

•

wds — Of(xI + ym)çbds, (6.41)

where the directions of s and v on the C, are shown in Fig. 6.3. By use of therelations (6.29) and (6.40), can be reduced to

'7"= -41 ((012+ 20Q5J —02

— csof(xI + ym) (is. (6.42)

The quantities subject to independent variation in the above are and 0,and the stationary conditions thereof are the same aS obtained in Section6.2, except for one additional condition that

+ GOf(xI + ym)d.c = 0, (6.43)

whiéh is obtained by requiring the coefficient of ôc, in oil,,, to vanish. Notingthat the direction s is defined clockwise on the boundary C,, we have

f (xl + ym) ds = f (x dy — y dx) = — 2A,, (6.44)I Cl C1

where A, is the area enclosed by the curve C,. Consequently, Eq. (6.43) isieduced

(6.45)

We have shown that, for a bar consisting of a simply connected region,Eq. (6.32) gives the condition of compatibility. For a bar with a hole, Eq.

TORSION OF BARS 121

(6.45) gives an additional condition of compatibility, which is use4 todetermine the value of cg. For the proof, we shall again use the(6.34), which, when integrated along an arbitrary path between two arbitrarypoints F and Q, provides

w(Q) — w(P) =—

ds + of€v dx — zdy), (6.46)

where the directions of s and v on the path FQ are shown in Fig. 6.3. Sinceno dislocation is allowed for the displacement w, it is required that

+ GO (x dy — y dx) = (6.47)

for any arbitrary closed path of integration within the region enclosed byC0 and If the closed path of integration is taken in coincidence with theinner boundary C1, Eq. (6.47) may be shown to reduce to Eq. (6.45). Thus,Eq. (6.45) is an additional condition of compatibility defined along the innerboundary. By using the relations derived above, we can prove that Eq.(6.47) holds for any arbitrary closed path on the cross section, providedthat the relations (6.32) and (6.45) are assumed to hold. Equation (6.45)is called the condition of compatibility in the large for the of a barwithahole. . 0

*

So far, only Saint-Venant torsion problems have been treated. Namely,the strain of the bar is assumed to be independent of z. It is obvious that forthe complete realization of Saint-Venant torsion, the mechanical boundaryconditions at the two ends, Eqs. (6.1) and (6.2), must be prescribed in amanner such that they are exactly coincident with the stress distributiongiven by the Saint-Venant solution. When a bar of finite length is subjectedto twisting moments applied on both ends in an arbitrary manner, stressdistribution induced ifl the bar may be different from that obtained fromthe Saint-Venant torsion theory. However, it is assured from the Saint-Venant principle mentioned in the Introduction of this book that the end

disturb the stress distributions derived from Saint-Venant theoryonly locally. The spread of the disturbed regions in the z-direction is of theorder of magnitude of the lateral dimensions of the bar, and the Saint-Venant theory of torsion can apply quite well to regions away from the endsof the bar. Approximate solutions have been obtained by other authorsthrough variational methods for end constraint problems in the torsion ofa

6.4. Torsion of a Bar with Initial Stresses

Next, we shall consider the problem of torsion of a bar with initial stresses.For the sake of simplicity, the initial stresses are assumed to consist ofonly, which is a function of(x,y) and independent ofz. The governing equations


for this torsion problem will be derived through the principle of virtualwork for the initial stress problem, Eq. (5.5).

Following Ref. 5, we assume the displacement components to be given by

u = — x(l — cos 8) — y sin 8,v = xsinO — y(l — cosO), (6.48)tw = 0ø(x,y)

where 8(z) is the twist angle and 0 = dt9/dz is the rate of twist. The deforma-tipn is assumed to take place independently of z. This assumption makes0 and constant and allows us to take the length of the cylinder as unity.The quantities to be determined are 4)(x, y) and the two constants 0 and,

Froth Eqs. (6.48), we obtain

811 øu . OU

-i---= —I + cos 8, = —sin 8, = —(x sin 8 + y cos 6)0,

8v= sin 8, = —1 + cos 6, = (x cos 0 — y sin 8)0,

0w Ow==

0, = (6.49)

Consequently, Eqs. (3.19) provide:

\

=

1 1

= .y (x2 + y2)02 + . (6.50)

These strain components are assumed to be related to incremental stresses,denoted by a,, ... and by Eqs. (3.38).

We shall be interested in formulating a linearized theory of the problemof pure torsion and we shall follow the development in the latter half ofSection 5.1. Since the constant e0 proves to make no contribution to finalresults of ipterest as far as the linearized theory is concerned, we allowto vanish in. the following formulation. To begin with, the principle ofvirtual wo-k, Eq. (5.5), Is written for the present problem. After neglectinghigher order terms, we find that the contribution from the volume integralof the principle becomes:

ff + + dx dy,

f Compare with Eqs. (6.4) and (6.17).

TORSION OF BARS 123

wherej60 60= — Y)

=+ x) 0, (6.52)

and= Mx2 + y2) 02. (6.53)

We define components of the external force P by

P Fj1 + F,12 + (6.54)

and prescribe them as follows:

F, = — P. = (6.55)

on the end at z = 0, and

F, sinO + (6.56)

on the other end at z = 1. The side boundary is traction-free. Here,and 7. arc the external forces applied at both ends to produce the twistingmoment 11? given by Eq. (6.3). Since the virtual displacements are obtainedfrom Eqs. (6.48) as

ôu=O,ôv=O,ôw=060+0ö0 (6.57)

on the end at z = 0, and

öu =—xôOsinO—yö0cos0,= x 60 cos 0 — yôO sin 0, (6.58)

ôw=0ô0+060on the othci end at z = I, the contribution from the surface integral of theprinciple reduces to

—260. . (6.59)

By the use of the relations (6.51) and (6.59), we obtain the principle ofvirtual work for the present problem. Through partial uflegration, thóprinciple becomes

—+ 60 dx dy + + Tam) 6 ds

30 34)

+ 0 Jf (x2 + yi) dx dy—

2}o0 = 0, (6.60)

124 VARIATIONAL METHODS IN ELASTICITY AND PLASTICiTY

which yields following equations:

in S, (6.61)

+ = 0 on C, (6.62)

= If — dx dy + 0 ff (x2 + y2) dx dy. (6.63)

To complete the formulation the strcss—strain relations (3.38) are linearized,and we obtain

= = (6.64)

Substituting Eqs. (6.52) and (6.64) into Eqs. (6.61) and (6.62), we findthat the function thus determined is equivalent to the warping function

defined in Section 6.1. Consequently, Eq. (6.63) may be written as

= [GJ + ff(x2 + 12)a(o)dxdylo, (6.65)

to yie'd the effective torsional rigidity,

GJerr = GJ + ff(x2 + (6.66)

where J is defined by Eq. (6.20).The last term in Eq. (6.66) shows the effect of the initial normal stresson the torsional rigidity. The effect may be explained as follows: Since

the position of an arbitrary point of the bar after deformation is

r = (x + u) + (y ÷ v) + (z + w) 13, (6.67)we have

+ (xcos 0 — ysinifr)0i1 + (I + eo)i3, (6.68)

with the aid of Eq. (6.49). Therefore, the stress produces a twistingmoment

[kxcosO + u) + (xsin0 + ycos0)(y += (x2 + y2) F) (6.69)

around the z-axis, as shown in Fig. 6.4.It is seen that the resultant of the stress in the sectional plane

may not vanish, but ipstead, may produce bending of the bar, unless theaxis of rotation is chosen properly. The effective torsional rigidity dependson the location of the axis of rotation through the term (x2 + y2) involvedin the surface integral in Eq. (6.66). However, if the initial stress isgiven such that

ff dx dy = ff dx dy = ff dx dy = 0, (6.70)

TORSION OF BARS 125

the torsion does not produce bending and any location of the axis of rotationmay be used for the computation of the effective torsional

We note that the governing equations for a torsion problem of a cylinderwith initial stresses a mannersimilar to the above development. When these initial stresses are functionsof (x, y) only and in self-equilibrium in the cylinder with the side boundarysurface traction-free, we obtain, in place of Eq;(6.61), the following equa-tion:

(6.71)

while Eqs. (6.52), (6.62). (6.63) and (6.64) remain unchanged.

It is well known that the presence of axial tensile or compressive stresscan cause an increase or decrease in the torsional rigidity of aIn recent years, thermal stresses induced in structural members of high speedflight vehicles due to aerodynamic heating have been one of the greatestengineering concerns. Among the difficulties caused by the thermal stressesis the loss of torsional rigidity of the lifting surfaces of flightThis loss is responsible for the reduction of safety margins for static anddynamic aeroelastic phenomena in high speed flight.

6.5. Upper and Lower Bounds of Torsional Rigidityt

The topic of the last section of this chapter is to show that formulaeproviding upper and lower bounds for torsional rigidity are derivable bythe simultaneous use of the principles of minimum potential and comple-

t Refs. 10 through 15.

—8(xsin

Fio. 6.4. Components of ar/az.


mentary energy. For the sake of simplicity, the bar is assumed to be simplyconnected and the torsion problem is defined in the same manner as inSection 6.1.

First, let us derive a lower bound formula employing the principle 'ofminimum complementaryenergy. Let #and represent the stress functionand the total complementary energy corresponding tà the exact soiption,and let and 17 represent the corresponding quantities of an admissiblefunction which satisfies the subsidiary conditions:

A? = 2fJ4*dxdy,. (6.72)

and =0 on . (6.73)

Then, the principle of minimum complementary energy assures that:

where (6.74)

I $2 $2 M2TIC = + (ij;) } 2GJ' (6.75)

and S

I $*2 84*2= + bi) (6.76)

Now, we shall assume 4i as a linear combination of 4,(x, y) (I 1,2, ..., m)such that

= (6.77)i—i

and consider the minimum value of 11. The functions 4,(x, y) are chosento satisfy thc required continuity and differentiability conditions in thedomain S and the boundary conditions 4,(x,y) =0 on C, and a, are arbitraryconstants. The function 4* thus expressed must satisfy the subsidiary con-dition (6.72). Since the subsidiary condition can be into thevariational expression through the use of a Lagrange multiplier 2, the mini-mum value is given by the extreme value of

fffl[(Ø4*)2 + ($*)2]

— 22#*j dxa!y (6.78)

where the quantities to variation are and a, (1 = 1,2, ..., in).After some calculation, we obtain the stationary conditions of the expression(6.78) with respect to these quantities as follows:

+ * = 262ff4edxdy.

(6.79)

TORSION OF BARS 127

and

A? =

By solving these equations, we may express Og and A in terms of M. Substitu-tion of aj and 2 thus obtained into Eq. (6.76) yields:

A?2H.-__ -— 2GJ'where

2IA. (6.82)

Combining Eqs. (6.74), (6.75) and (6.81), we obtain:

GJ GI, (6.83)

which shows that G1 thus obtained provides a lower bound to the torsionalrigidity.

Second, we may derive an upper bound formula via the principle ofminimum potential energy. Let w, 0 and 11 represent displacement, twistangle per unit length and the total potential energy corresponding to theexact solution, and let w', and represent the correspoiding quanti-ties of an admissible function. The principle of minimum potential energythen assures that:

(6.84)where

17 =— oy)2

+ (4 + Ox) Jdxdy — OR ,(6.85)

and

= Gff[(— )2 +

+ 0**x)21 dx dy — O"R.S (6.86)

Now, let us assume as a linear combination of w,(x, y) (1 1,2,..., n),such that

y), (6.87)

and consider the minimum value of 17'. The functions w.(x, y) may bechosen arbitrarily, except that they must satisfy the required differentiabilityand continuity conditions in the domain 5, and b1 are arbitrary constants.Substituting Eq. (6.87) into Eq. (6.86), and taking variations with respectto b1 and &*, we obtain:

+

(1, 1, 2, ..., n) (6.88)'I.

128 VARiATIONAL METHODS IN AND PLASTICITY

M — + o**ff(x2 + y2)dxdy1.

(6.89)

By solving these equations, we may express the quantities and mterms of 2. Substitution of b1 and O** thus obtained into Eq. (6.86) thenyields:

J7** =— 2GJ**' (6.90)

where= M/O**. (6.91)

Combining Eqs. (6.84), (6.85) and (6.90), we obtain the following formulaproviding an upper bound to the torsional rigidity of the bar:

GJ GJ**. (6.92)

So far, no conditions have been prescribed for the admissible functionswg. However, since the exact solution w should satisfy Laplace's equation,it will be more convenient to choose w, so that they satisfy Laplace'sequation

p32w,+ = 0 (1 1,2, ..., ii). (6.93)

Then, the surface integrals on the left band side of Eq. (6.88) can be replacedby line integrals as follows: -

+(6.6)

where the line integrals are taken along the contour of the bar, and is inthe direction of the outward normal to the contour.

Thus, combining Eqs. (6.83) and (6.92), we finally obtain the upper andlower bound formulae for the torsional rigidity of the bar:

(6.95)

The accuracy of the bounds obtained in this way can be improved by in-creasing the number of admissible functions.

an example of the procedure outlined above, bounds for the torsionalrigidity of the prismatic bar with a square cross section, as shown in Fig. 6.5,will be calculated following Ref. 10. To begin with, we shaH obtain a lowerbound by choosing

41(x, y) = a2(x2 — a2) (y2 a2),

y) = (x2 + y2) (x2 — a2) (y2 — a2), (6.96)

TORSION OF BARS 129

thus assuming = a1,b1 + a242. Then Ejs. (6.79) and (6.80) bewritten, alter several integrations and cancellation of a common factor,as:

[26880 92161[a11 (GA\f 16800[9216 11264j[a1j 6720

(6.97)

and

M = + 2a2), (6.98)

y

txFio. 6.5. A square section.

which yield:a1 = (3885/ 6648) (GA/a4),

a2 = (1575/13296) (GA/a'), (6.99)

2 = (5600/ 2493) Ga'A.

Thus, we obtain for the lower bound:

(5600/2493) Ga' � GJ. (6.100)

Next, we proceed to obtain an upper bound by choosing

w1 = x3y — xy3,. (6.101)

and expressing in terms of w1 only, namely, =b1w1. It is easilyseenthatw1istheimaginarypartof(x + = J/-l,whichensuresthat w1 is a plane harmonic fuiLction. further calculation, Eqs. (6.88)and (6.89) can be written as follows:

(96/35)am.b1 = (16/15)a68**, (6.102)

2 = G[—(16/l5)a'b1 + (8/3)a40j, (6.103)

which yield:= (7/18) (O**/a2),

I = (304/135) (6.104)

130 VARIATJOM&. METhODS IN ELASTICITY AND PLASTICITY

Thus, we obtain for the upper bound:

6) (304/135) Ga'. (6.105)

Combining Eqs. (6.100) and (6.105), we have the following upper and lowerbounds for the torsional rigidity:

(5600/2493) Ga' GJ � (304/135) Ga', (6.106)or

2.24628 Ga4 < GJ 2.25186Ga'. (6.107)

The exact value of torsional rigidity is:

GJ = 2.2496 Ga', (6.108)

as shown in on the theory of elasticity. It is seen that the accuracy ofthe bounds is excellent. It should be noted, however, that they do not guaran-teethe same accuracy for displacements or stresses thus determined approxi-mately. More complex techniques are necessary for obtaining pointwisebound formulae for displacements or stresses at an arbitrary point of thebar.'1 618)

Bibliography

.1. A. E. H. Lova,. Mathematical Theory of Elasticity, Cambridge University Press,4th edition, 1927.

2. S. Tno wuco and J. N. GOOD1ER, Theory of Elasticity, McGraw-Hill, 1951.3. 1. S. Mathematical Theory of Elasticity, McGraw-Hill, 1956.4. E. On Non-uniform Torsion of Cylindrical Rods, Journal of Mathematics

and Physics, Vol. 31, No. 2, pp. 214-21, July 1952. Note on Torsion with VariableTwist, Journal of Applied Mechanics, Vol. 23, No.2, p. 315, June 1956. On Torsionwith Variable Twist, Osterreichisches Irigenieur-Archiv, Vol. 9, No. 2—s, pp. 218—24,1955.

5. R. KAPPUS, Zur Elastizitãtstheorie endlicher Verschiebungen, Zeltschrzf: für Ange-wandse Mashematik mid Mechanik, Vol. 19, No. 5, pp. 344-61, December 939.

6. ft. L. BISPLINGHOFP et al., Aerodynamic Heating of Aircraft Structures in HighFlight, Notes for a Special Summer Program, Department of Aeronauticaljug, Massachusetts Institute of Technology, June 25—July 6, 1956.

7. H. Verrfrehung und Knickung von offene,, Profilen, 25th Anniversary Publi-cation, Technische Hochschulc Danzig. 1904-29, pp. 329-44, Druck und Verlagvon A. W. Kafemann GinbH, Danzig, 1929. Translated in NACA TM 807, October

-1936.8. F. Bwcis and H. BLEZCH, Buckling Strength of Metal Structures, McGraw-Hill, l952.9. B. and I. MAYERS, Influence of Aerodynamic Heating on the Effective

Torsional Stiffness of Thin Wings, Journal of Aéror4uakoJ Sciences, VoL 23, No. 12,pp. 1O81—93, December 1956.

10. E. Tpzpprz, Em Gegenstuck zum Ritznchen Verfabren, Proceedings of the 2nd inter-nationqsl Congress for Applied Mechanics, ZUrich, pp. 131-7, 1926.

11. N. M. BASU, On an Application of the New Methodi of Calculus of Variations toSome Problems in the Theory of Elasticity, Philosophical Magazine, Vol. 10, No.pp. 886-904, November 1930.I. B. and A. The Torsional Rigidity and Variational Methods,American of Mathematics, Vol. 10, No 1, pp. 107-16, January 1948.

TORSION OF BARS 131

13. A. WEINSTEIN, New Methods for the Estimat;on of Torsional Rigidity, Proceedingsin Applied Mathematics, Vol. 3, pp. 141-61, McGraw-Hill, 1950.

14. LIN Huijo-Sui', On Variational Methods in the Problem of Torsion for Multiply-connected Cross Sections, Acta Sci-Sinica, Vol.3, pp. 1954.

15. S. CL MHCmJN, Variational Methods ui Mathematical Physics, Pergamon Press, 1964.16. H. 3. GREENBERG, The Determination of Upper and Lower Bounds for Solution of

the Dirichiet Problem, Jourmft of Mathematics and Physics, Vol. 27, No. 3, 161-82,October 1948.

17. J. L. SYNGE, The Diiichlet Problem: Bound at a Point for the Solution and its Deriva-tives, Quarterly of Applied Mathematics, Vol.8, No.3, pp. 213-28, October 1950.

18. K. WAsluzu, Bounds for Solution of Boundary Value Problems in Elasticity, Journalof Mathematics and Physics, Vol. 32, No. 2—3, pp. 117—28, July—October 1953.

19. S. TIMOSHENKO, Theory of Bending, Torsion and of Thin-walled Membersof Open Cross Section, .louriral of the Franklin Institute, Vol. 239, No. 3, pp. 201—19,March 1945.

CHAPTER 7

BEAMS

7.1. Elementary Theory of a Beam

We shall treat slender beams in the present chapter. It will be assumedthat the locus of the centroid of the cross sçction of the beam is a straightline and that the envelopes of the principal axes through the centroid aretwo flat planes perpendicular each other. We shall take the x-axis in thedirection of the centroid locus and the y- and z-axes parallel to the principaldirections. Thus, the x-, y- and z-axes form a right-handed rectangula'Cartesian coordinate system (see Fig. 7.1).

13

a

Saint-Venant has formulated a method of solution for bending of acylindrical cantilever beam of constanj cross section by a terminal 2)

Solutions of the problem have been obtained for beams having circular,elliptic, rectangular and several other cross sections. These results show thatboth bending and torsional deformations occur due to the terminal load.Consequently, it is considered convenient to define the center of shear of

t Only the projections onto the (x. z) plane are shown in Fig. 7.1.

132

4

z4

Fio. 7.1. Geometrical mlations.f

BEAMS 133

the cross section as a point through which a shear force can be appliedwithout producing torsion, thus realizing torsion-free bending. It is knownfrom the above definition that once the shearing stress distribution over thecross section due to the torsion-free bending has been obtained, the centerof shear is determined as the point of application of the resultant of theshearing force.t When the cross section has an axis of symmetry, the centerof shear lies on that axis. When the beam has a doubly symmetric cross sec-tion, the center of shear coincides with the centroid of the cross section.Exact general solutions for bending of a beam with arbitrary cross sectionalong the span under arbitrary external loads have not been obtained.

The present chapter will treat, unless otherwise stated, the elementarytheory of a beam under the tacit assumptions that variation of the geometryof the cross section along the x-axis is gradual and that torsion-free bendinghas been rçalizcd in the (x, z)-planc by proper application of external loads.Since the longitudinal dimension of a slender beam is much larger than itslateral dimensions, it is a common practice in the elementary theory toemploy the following two assumptions. First,. we assume that the stresscomponents or,, a, and;, may be neglected in comparison with the otherstress components and may be set

(7.1)

Then Eqs. (1.10) and Eqs (3.38) reduce to

= rjc, , , (1.2a, b, c)and f7.3a,b,c)

respectively. Second, we employ the Bernoulli—Euler hypothesis that thecross sections which are perpendicular to' the centroid locus before bending

plane and perpendicular to the deformed locus and, suffer no strainsIn their planes. '

We shall show that expressions for the displacements are" greatly. simplifiedby the introduétion orthe hypothesis. We consider an arbitrary point of .ibeam having the coordinates (x, y, z) before denote itsposition vectors before and. after deformation by atid r, respectivelywhich are related to the displacement vector u by

r = + a, (7.4)twhere = x11 + yi2 + z13, and are the unit vectors in thedirections of. the x-, y-, z.axes, respectively. Similarly, we denote position

t The analytical determination of center of shear depends upon the definition of"torsion-free bending", and there exist several different definitions (see RcA. 2 through 6).A detailed and luad discussion on the center of shear, center' of twist, elastic etc..is given in Ref. 7.

The superscript and subscript used in Chapters 7,8 and 9 mean that the quantityis referred to the state before deformation and to the centroid locus cc the n$* sccface,respectively.


vectors of a point (x, 0,0) of the centroid locus before and after deformationby 40) and respectively, which are related to the displacement vector

'°Y r0 =4! +

where = xi1. We define components of a and a0 as follows:

U = + v12 + wi3, (7.6)(7.7)

where u and w are functions of x only. It is seen that the hypothesis allowsus to express r as

r=r0+zn+y12, (7.8)

where n is the unit normal to the deformed locus and is given by

a x 12/VOl.

In Eq. (7.9) and throughout the present chapter, the prime differen-tiation with respect to x, namely, ( )' = )fdx. Since

r0 = (x + u)l1 + 1443, (7.10)

we may express ii in terms of uand w:

— —W'i1 + (I + u')13(7 11)I'

From Eqs. (7.4), (7.5) and (7.8), we obtain

u=Uo+z(n—i3). (7.12)

This is the expression for the displacements of a beam under the Bernoulli—Euler hypothesis. It is observed that the degree of freedom of the beamdeformation implied by Eq. (7.12) is two, namely u(x) and w(x).

When a beam problem is confined to small displacement theory, Eq.(7.12) may be linearized with respect to the displacement components toyield

u=u—zw', v=O, w=w. (7.13)

Then, we find that in the elementary theory of bending, the only non-vanish-ing strain component is

(7.14)

which is related to by Eq. (7.2 a).

7.2. B'uudhig of a Beam

As a simple example of bending of a beam, let us consider a problemshown in Fig. 7.2: a beam of span us clamped at one end x = 0 and issubjected to a disributed lateral load per unit span acting in the direc-

$ See footnotç to p. 133.

BEAMS 135

tion of the z-axis. At x = 1, it is subjected to end forces P,, and indirections of the x- and z-axes, respectively and to an external end momentA?. We may write the principle of virtual work for the present problem asfollows:

_fpOwdx

e3u(l) — öw(I) + Mâw'(!) = 0, (7.15)t

where Eq. (7.14) has been substituted, and ôu and ow should satisfy thegeometrical boundary conditions:

u(0) = 0, (7.16)and

w(O) = w'(O) = 0, (7.17)

—p.

respectively. 1ntegration of the first term in Eq. (7.15) with respect to y andz leads to

— MOw") dx

— + A? Ow'(l) = 0, (7.18)where we define

'\\\\ dz, (7.19)

(7.20)

integrations being takeji 4tion S of the beam. The quantitiesN and M are axial force and of the cross section as shownin Fig. 7.3.

Now, we may proceed to derive approximatj\equations of equilibriumimplied by Eq. (7.18). By the use of the and (7.17) and

t See Eq. (1.32).

Fia. 7.2. A cantilever beam.


partial integrations, we have

—J [N' öu + (M" + dx + (N — öu(I)

+ (M' — öw(I) — (M — A?) ôw'(I) = 0, (7.21)from which, we obtain

N' = 0, 0 x 1, (7.22)tN = at x 1, (7.23)

andM"+p=O, (7.24)t

M'=P,, M=A? at x=1. (7.25)

It"P1 M+dM

Q

L.dxFio. 7.3. Positive directions of N, Q and M.

In order to solve the problem, we must use the stress—strain relation (7.2 a),which, when combined with Eqs. (7.14), (7.19) and (7.20), provide N andM in terms of u and w as follows:

N = E40u', (7.26)'

—EJv/', (7.27)where

Ao=ffdydz. I=ffzzdydz (7.28)

are the area and the moment of inertia of the crois section, respectively.

t As Is well known, these equations can be obtained alternatively by writing theequilibrium conditions with respect to forces and moments of the beam element shown inFig. 7.3 as

N'-O, M'—Q....O

and then eliminating Q, where Q is the shearing force of the cross

A

U

BEAMS 137

Using the relations above obtained, we have the governing differentialequations and the boundary conditions for the beam problem. CombiningEqs. (7.16), (7.22), (7.23) and (7.26), we obtain a differential equation andboundary conditions which determine the stretching of the beam. Alter-natively, combination of Eqs. (7.17), (7.24), (7.25)and (7.27) yields a differ-ential equation and boundary conditions which determine the bending ofthe beam. Thus, in the small displacement theory of a beam where the dis-placement components are assumed to be of the form (7.13), the stretchingand bending do not couple with each other and can be treated separately.It is observed from the above relations that in the elementary theory ofbending of a beam, the stress a,, and the strain energy U are given by

N M,(7.29)

110 .1

andU = fff dx dy dz

El(w")9 (7.30)

respectively.Before leaving the present section, we note that if the distributed load

p(x) is discontinuous at some point along the span, care should be taken inderiving Eq. (7.21). For example, if the beam is subjected to a concentratedload P acting at x = in the direction of the z-axis, Eq. (7.15) is appendedwith a. term — P and we have

dx =f Mow" dx +fMow" dx

_—fM"ôwdx+ fM"owdx+Mt5w' —M'Owo 5+0 0

+ — 0) — + 0)] Ow'(E)

— — 0) — + (7.31)

Consequently, the principle of virtual work yields the connection conditionsat x as follows:

lSlO_AW50—v,

= 0, M' ÷ F = 0. (7.32)

7.3. Principle of Minimum Potential Energy and Its Transformation

We shall consider in the present section variational principles of a beamproblem shown in Fig. a beam is clamped at one end and is subjectedto a distributed lateral iqad p(x), while at the other end, it is supported and


is subjected to an end moment M. Since no external forces are applied inthe x-direction, we may take

U = —zw', v = 0, w = w, (7.33)Lx (7.34)

(7.35)

The functional for the principle of minimum potential energy of this problemis given, as suggested by the relations derived in the preceding section, by

II = EJ(w")2 dx —fpw dx + (7.36)

where w must satisfy the geometrical boundary conditions.

w(O) = w'(O) = w(I) = 0. (7.37)

FiG. 7.4. A beam with clamped supported ends.

Next, let us consider transformations of the principle of minimum poten-tial energy. By the introduction of an auxiliary function defined by

= w" (7.38)

and with the of Lagrange multipliers M(x), Q* and R*, the functional(7.36) is generalized as follows:

H, = E1x2 dx — J'pw dx + Mw'(I)

+ f — w") Mdx + p*w(o) + Q*w(o) + R*w(f), (7.39)

where the quantities subject to variation are x, w, M, P*, Q* and R* with nosubsidiary After some calculation including partial integrations,the first variation is shown to be

all, =J((M + — (M" +p)ôw + — w")aMldx

+ [F* — M'(o)] 6w(o) + (Q* + M(o)1 bw'(o)+ [R* + i%f'(l)I ôw(1) — — M'J àw'(/)

+ w(o) + ;v'(o) ãQ* + w(1) àR*. (7.40)

BEAMS

Consequently, the stationary conditions are shown to be the governingequations which define the problem, together with

= M'(o), Q* — M(o), R* = — M'(l) (7.41)

which determine the Lagrange multipliers Q* and R*.Familiar techniques lead to specializations of the generalized expression

(7.39). For example, by requiring that the coefficients of and ow' inEq. (7.40) vanish, thus eliminating x and w, we obtain the functional for theprinciple of minimum complementary energy as follows:

(7.42)

where the function subject to variation is M(x) under the subsidiary condi-tions

M" + p = 0, (7.43)and

M(1) = 2. (7.44)

We note that the principle of minimum complementary energy for thebeam problem is directly obtainable from the principle (2.23), assumingthat the stress component is given by Eq. (7.35) and all the other stresscomponents make negligible contributions in establishing the complementaryenergy function (see also Appendix C).

7.4. Free Lateral Vibration of a Beamt

Let us consider a free lateral vibration of a beam which is clamped atx = 0 and pimply supported at x = 1, as shown in Fig. 7.5. Following thedevelopment in Section 2.7, we may express the total potential energy forthe free lateral vibration problem as

H = EI(w")2 d; — mw2 dx, (7.45)*

where 2 = co2 andm(x)=ffedydz (7.46)

t Refs. 8 through 11.In the derivation of the last term of Eq. (7.45) by the use of the functional (2.69) and

Eq. (7.33), the term ff pz2(w')2 dydz is neglected in comparison with the term ff dy dz

due to the slenderness of the beam—a1common practice in the elementary theory of thelateral vibration of a beam.


is the mass per unit span. In the functional (7.45), the function subject tovariation is w under the subsidiary boundary conditions

w(o) = w'(o) = w(1) = 0, (7.47)

while A is treated as a parameter and not subject to variation. The stationaryconditions of the functional (7.45) are shown to be the equations of motion,

(EJw")" — 2mw = U, (7.48)

and the boundary conditionE!W" = 0 at x = I. (7.49)

Therefore, the problem reduces to an eigenvalue problem, where the requirednatural modes and frequencies are determined as eigenfunctions and eigen-values of the differential equation (7.48) under the boundary conditions

14FIG. 7.5. A beam with clamped and simply supported ends.

(7.47) and (7.49). It is observed that this eigenvalue problem is equivalentto finding, among admissible functions w, those which make the quotient

f EI(w")2 dx

2= ° (7.50)

0mw dx

statiQnary.U 2)

NeXt, let us consider a generalization of the principle of stationary poten-tial energy."3' Through the usual procedure, the functional (7.45) may begeneralized as follows:

ii, = dx — +)fmw2 dx

— w") Mdx + + Q*w'(o) +

where the quantities subject to variation are ,c, w ,M, P, Q* and withno subsidiary conditions. The stationary conditions with respect to x and

BEAMS 141

w are shown to be+ M = 0, (7.52)

M" + Amw = 0, (7.53)

— M'(o) = 0, Q* + M(o) = 0, R + M'(I) = 0, (7.54)

M(l) = 0. (7.55)

We eliminate x, Q* and by the use of Eqs. (7.52), (7.54) and (7.55),and with the aid of Eq. (7.53), to transform the functional (7.51) into:

nc 1/M2dx — -}Afmw2 dx, (7.56)

where it is assumed that A # 0. In the functional (7.56), the quantitiessubject to variation are M and w under the subsidiary conditions (7.53) and(7.55). The expression (7.56) is a functional for the principle of stationarycomplementary energy of the free vibration problem.

As mentioned in Section 2.8, the Rayleigh—Ritz method be appliedfor obtaining approximate eigenvalues of the free vibration problem, oncethe variational expressions have been established. When the method isapplied to the principle of stationary potential energy (7.45), we may assume

w = c,w1 + c2w2 (7.57)where

w1 = x2(x — 1), w2 = x3(x — 1) (7.58)

are coordinate functions which satisfy Eqs. (7.47). Substituting Eq. (7.57)into the functional and requiring that

--=0, i= 1,2 (7.59)

we obtain approximateNumerical results have been obtained for a beam with constant El and

m, as shown in Table 7.1.

TABLE 7.1. ExAcr AND APPRO IMATE

= jIEI/m14

Exacteigenvalues

Approximate eigenvalues

Rayleigh—Ritz method ap-plied to the functional (7.45)

Rayleigh—Ritzmethod applied to the

tional (7.56)

k1

k2

15.42

49.96

15.45

75.33

15.42

51.93 -—_____________________ —


Next, shall apply the modified Rayleigh—Ritz method to the principleof stationary complementary energy (7.56). We choose w as given by Eq.(7.57). As the derivation of the functional (7.56) shows, it is not necessaryfor the coordinate functions w1(x) and w2(x) to satisfy Eqs. (7.47) for theestablishment of the principle. However, this imposition is desirable forimproving the accuracy of approximate eigenvalues and is essential forobtaining the inequality relations (2.93). We substitute Eq. (7.57) intoEq. (7.53) and perform integrations with the boundary condition (7.55)to obtain

2 Ii(1/).) M = c(x — I) —

c is an integration constant. Substituting Eqs. (7.57) and (7.60) intothe principle (7.56) and requiring that

= 0 (7.61)and

= 0, i 1, 2,

we obtain approximate eigenvalues. Numerical results have been obtainedfor a beam with constant El and in, and are shown in Table 7.1. It is observedthat the inequality relations (2.93) hold between them. See Refs. 11, 12, 14and 15 for other numerical examples of the Rayleigh—Ritz andmodified Rayleigh—Ritz method applied to free vibration probleths;

7.5. Large Deflection of a Beam

We shall consider large doflection of an elastic beam in this section andtake as an example the beam problem treated in Section 7.2. It is obviousthat since the are given by Eq. (7.12) and the strains canbe calculated in terms of u and w by the use of Eqs. (3.19), a finite displace-ment theory of the beam under the Bernoulli—Euler hypothesis may befoi'mulated by the principle of virtual work (3.49). However, we shall besatisfied with confining the problem by assuming that although the deflectionof the beam is small in comparison with the height of the beam, itis still small in comparison with the longitudinal dimension of the beam andemploy the following expressions for displacements and strain—displacementrelations:

u = u zw', v = 0, w = w, (7.63)tu' + — zw". (7.64)t

t These equations may be derived from Eqs. (7.12) and (3.19) by assuming that—. (w')2 1, and terms containing z2 may be neglected in view of the hypothesis and

slenderness bI the beam, where the notation stands for "same order of magnitude".The first assumption states that the square of the slope and the strain of the centroid locusare very small compared to unity.

BEAMS 143

Then, the principle of virtual work for the present problem may be writtenas

_0fpc5wdx

— óu(l) — Ôw(1) + A? öw'(/) = 0, (7.65) t.where Eq. (7.64) has been substituted. By introducing the stress resultantsdefined by Eqs. (7.39) and (7.20), we may transform the principle (7.65)into

f [N(öu' + w' 6w') — M 6w" — p 6w) dx

— ôu(1) — P2 ãw(I) + öw'(l) = 0, (7.66)

where the independent variables are ôu and 6w under the subsidiary condi-tions (7.16) and (7.17). After some calculation, we obtain from Eq. (7.66),the governing differential equations

N' = 0, M" ÷ (Nw')' + p = 0 (7.67, 7.68)

and the mechanical boundary conditions at x = 1:

N = M = M, Nw' + M' = (7.69, 7.70, 7.71)

Comparing Eqs. (7.68), (7.70) and (7.71) with Eqs. (7.24) apd (7.25), wefind that when the deflection of the beam becomes large, the axial, forceN,, has a contribution to the equations of equilibrium in the direction of thez-axis due to the inclination of the centroid locus. From Eqs. (7.67) and.(7.69), we have

N(x) = = constant. (7.72)

Combining Eqs. (7.3a), (7.19), (7.20) and (7.64), we obtain the stress resul-tant--displacement relations as follows:

N = EA0[u' + Mw')2), (7.73)M = (7.74)

The equations (7.68), (7.72), (7.73) and (7.74), together with the boundaryconditions (7.16), (7.17), (7.70) and (7.7fl, constitute the governing equationsfor the large deflection problem. is observed that in the large deflectiontheory of a beam, the stretching and bending couple with each other andmust be treated simultaneously.

We note that, in the large deflection theory of a beam, the stress isgiven by Eq. (7.29) and the strain energy U by

u =

= +f (EA0(u' + Mw')212 + EI(w")2}dx. (7.75)

t See Eq. (3.49).


7.6. Suckling of a Beamt

Next, let us consider the buckling problem of a beam, as shown inFig. 7.6. The beam is fixed at one end, while at the other end it is simplysupported and under a compressive load P applied in the negative directionof the x-axis. When load reaches a critical value, denoted by Ps,, thecolumn may buckle. We shall treat the column as a body with initial stress

the magnitude of which is given from the equilibrium conditions as= 0, = (7.76)

where = It. is assumed that the force changes itsmagnitude nor its' direction during the buckling. Following the derivationin Section 5.1, we may write the principle of virtual work for the presentproblem as

fff + dx dy dz + bu(l) = 0,

where is incremental stress and is given by Eq. (7.64). The subsidiaryboundary conditions of the displacements are

u(o) = 0 (7.78)and

w(o) = w'(o) = w(!) = 0. (7.79)

Since we are interested in the determination of the critical load, we assumethat = 0(1) and U, H' = 0(e) to neglect terms higher than 0(e2) inthe principle of virtual work (see Sections 5.1 a similar devel-opment). Thus, by introducing stress resultants defined by Eqs. (7.19) and(7.20), we have

öu' + Nb,,' + ow' — MOw") dx

+ Ou(I) = 0. (7.80)

By the use of Eqs. (7.76) and (7.78), we find that the terms relating to Ou inEq. (7.80) reduce to

—f N' âu dx + N(1) Ou(1). (7.81)

17.

See Eq. (5.5).

Fio. 7.6. A beam under a critical axial load.

BEAMS 145

Consequently, we have N'(x) = 0 and N(I) = 0, and we conclude thatN(x) = 0 throughout the beam. Thus, the principle (7.80) is simplified into

f(Môw" + 6w') dx = 0, (7.82)

or after some calculation, to

1w" — +

— (M' = 0. (7.83)1' 0

Consequently, taking accoqit of Eqs. (7.79), we have from Eq. (7.83) theequation of' equilibrium

M" — Pcrw" = 0 (7.84)and a boundary conditioà

M(I) = 0. (7.85)

Combining Eqs. (7.20) and (7.64), obtain the stress, resultant—displacement relation a4 follows:

M = —EIw". (7.86)

The equations (7.84) aijd (7.86), together with the boundary conditions (7.79)and (7.85), consxitute the governing equations for the buckling problem.

When combined with Eq. (7.*6), the principle of virtual work (7.82) maybe transformed into the principle of stationary potential energy, of whichthe functional is given by

dx — dx, (7.87)

where the function subject to'variation is w under thesubsidiary conditionsIt is observed that the principle (7.87) to finding,

among functions w, those which make the quotient

ffEI(w'1)zdx= 0

(7.88)

0

Neat,, let us consider a generalization of the principk of stationary poten-tial Through the usual procedure, the functiona} (7.87) may begeneralized as

If, = if E1x2 dx — (w')2 dx

— w") Mdx + P*w(o) + Qw'(o) + RwQ), (7.89)


where the quantities subject variation are w, M, Q* and R* withno subsidiary conditions. The, stationary conditions with respect to x andw are shown to be

EIx + M = 0, (7.90)

— Paw" = 0, (7.91)

PcrW'(O) — M'(o) = 0, Q* + M(o) = 0,

— PcrW'(l) + M'(l) = 0, (7.92)

M(I) = 0. (7.93)

We eliminate x, P*, Q* and R by the use of Eqs. (7.90), (7.92) and (7.93),and with the aid of Eq. (7.91), to transform the functional (7.89) into:

=dx — Pcrf (w')2 dx, (7.94)

where it is assumed that 0. In the functional (7.94), the quantitiessubject to variation are M and w under the subsidiary conditions (7.91)and The expression (7.94) is a functional for the principle of station-ary complementary energy for the buckling problem.

7.2. EXACT AND APPROXIMATE LIGENVALUES

Exacteigenvalues

.

Approximate eigenvalues

Rayleigh—Ritz methodapplied to the func-

tional (7.87)

Modified Rayleigh-Ritzmethod applied to the

functional (7.94)

20.19 20.92 20.30

59.69 107.1 67.70

Once the variational principles have thus been established, we can applyRayleigh—Ritz method and the modified Rayleigh—Ritz method for

obtaining approximate eigenvalues. A numerical example is shown by tak-ing w as given by Eqs. (7.57) for a beam with constant El. Numericalresults are listed in Table 7.2 and compared with the exact eigenvalues.See Refs. 16 and 17 for other examples of the Rayleigh—Ritzmethod and modified Rayleigh—Ritz method applied to buckling problems.We note that nonconservative problems of the stability of elastic beams havebeen extensively treated in Ref. 19.

I3EAMS 147

7.7. A Beam Theory Including the Effect of Transverse Shear Deformation

The elementary beam theory which has been considered in the preceedingSections is based on the Bernoulli—Euler hypothesis, in which no transverseshear deformation is allowed to occur. We shall consider in the presentsection an approximate formulation for a dynamical beam problem takingaccount of the effect of the shear deformation. A dynamical problemdefined in a manner similar to the presentation in Section 7.2 will be takenas an example, except that the external forces are now time-dependent. Theprinciple of virtual work is an avenue which leads to an approximate for-mulation.

Since the displacement vector u is a function of (x, y, z), we-may expandit into a Taylor series about z = 0:

y, z) = u(x, o) + (-p-) z + z2 + . (7.95)

Therefore, tone of the simplest expressions for displacements to include theeffect of transverse shear deformation may be given by retaining the first twoterms only:

u = u0 + zu1, (7.96)

where components of u1 are defined by

= u1i1 + w1i3, (7.97)

and u1 and w1 are functions of x only. The degree of freedom implied byEq. (7.96) is four, namely, u, w, u1 and w1. However, if we continue to usethe assumption (7.1), and employ Eqs. (7.3) as the stress—strain relations, wemay take = + (1 + w1)2 — 1 = 0 (7.98)

as ap additional geometrical constraint to reduce the degree of freedom tothree. Equations (7.96) and (7.98) state that the cross sections perpendicularto the undeformed locus remain plane and suffer no strains planesalthough they are no longer perpendicular to the deformed locus.

We shall confine our problem to small displacement theory. Then, Eq.(7.98) is linearized with respect to the displacements to yield

w1 — 0. (7.99)

Consequently, the simplest expression for displacements to include thetransverse shear deformation is to assume that

jj=u+zu1 v=0, w=w, (7.100)

which provide the following nonvanishing strain cOmponents

= U' + Vxx = W' + U1. (7.101)t

f It is. seen that the hypothesis imposis the constraint conditionUI = —W'.


Then, the principle of 'Virtual work for the present dynamical problem iswritten as

+ — +[U2 +

dx — P ôw(I) — A? ôu1 } di = 0, (7.102)t

(7.100) and (7.101) have been substituted. Here, we introducenew quantities defined by

Q (7.103)

Jf9z2 dy dz, (7.104)

in addition to the stress resultants defined by Eqs. (7.19) and (7.20). Thequantity defined by Eq. (7.103) is the shearing force of the cross sectionas shown in Fig. 7.3, while the quantity defined by Eq. (7.104) is the massmoment of inertia of the cross section. By the use of these quantities, thefirst two terms of Eq. (7.102) may be written as follows:

t2{l+ Môu +Q(ôw' +ôu1)ldx

— ô/ + g'2) + dx} di. (7.105)

Consequently, after some calculation including integrations by parts, theprinciple (7.102) is transformed into

ti I

f f{(rnu — N')ãu + (rnw — Q' —p)ãw0

+ — M' + Q)e3u1}dx

+ [(N — ôu ÷ (Q — ow + (M -,

— [NOu + QOw + 0, (7.106)

from which we obtain the equations of motion

mu = N', (7.lOm

= Q' +ImÜt = M' — Q,

t See Eq. (5.81).Compare with Eqs. (7.22) and (7.24).

BEAMS 149

and the mechanical boundary conditions

at x=l, (7.110)

while it is suggested that the geometrical 1!oundary conditions may bespecified approximately as

u=0, w—0, u1=0 at (7.111)

Combining Eqs. (7.2a, b), (7.19), (7.20), (7.101) and (7.103), we have thefollowing stress resultant—displacement relations:

N=EA0u' (7.112)

= (7.113)

Q = GkA0(w' + u1), (7.114)

where k = I. The factor k in Eq. (7.114) is appended to take account of thenonuniformity of Yxz over the cross section and the effect of y,j. Anmate method of determining the value of k for a beam in static equilibriumis shown in Appendix C, where the minimum complementary energy methodis employed. Another method may be to determine the value of k so thatsome results obtained from the above approximate equations may be coin-cident with those obtained from the exact theory of vibrations or wave

(see Refs. 20 and 21).. Substituting Eqs. (7.112), (7.113) and(7.114) into Eqs. (7.107), (7.108) and (7.109), we obtain

mu = (EA 0u')', (7.115)

mw [GkA0(w' + u1)]' + p. (7.116)

1mÜi = — GkA0(w' + u1). (7.117)

These equations constitute the governing equations for the dynamic.al beamproblem including the effect of transverse shear deformations, the so-calledTimoshenko beam From the above relations, it is observed thatthe strain energy of the beam is given by

U = (EA0(u')2 + + GkA0(w' + u1)2j dx. (7.118)

Effects of shear flexibility and rotary inertia play a very important role intheories of beam vibration and dynamical behavior under impulsive loading(see Refs. 21 through 25).

7.8. Some Remarks

The elementary theory of the beam. formulated in Section 7.1 is based onthe assumption (7.1) and the Bernoulli—Euler hypothesis. However, wehave t, = = 0 from Eqs. (7.13), and we find that theof the assumption (7.1) and the hypothesis does not satisfy the stress-

150 VARIATiONAL METHODS IN ELASTICITY AND PLASTICITY

relations (1 .10), and ill not lead to correct results. The same kind of con-tradiction exists in the formulations of Sections 7.5 and 7.7. We have triedto remove this difficulty approximately by setting a, = = = 0 in thethree-dimensional stress—strain relations and then eliminating e, andFor a complete removal of the inconsistency and an improvement of theaccuracy of the beam theory, we may assume

(1 2] VmZ", V 2] 1'mn(X)m-O.n-O

w = 2] ymz", (7.119)m-O. n-U

where the numbers of terms should be chosen properly. Equatiod governingUrni,, Urn,, and w,,,,, are obtainable by the use of the principle of virtual work. Itis noted here that a theory has been developed in Ref. 26 for expressions ofdisplacements and strain—displacement relations applied to rod problems.

A naturally curved and twisted slender beam presents a classical problemin The principle of virtual work may provide an avenue to anapproximate formulation of the problem, where a curvilinear coordinatesystem may be conveniently employed for describing the curved centroidlocus and the two curved surfaces constituted by the envelopes of the prin-cipal axes through the Variational forniulations have beenproposed for the problem, and Ref. 29 is among recent contributions inthis field.

Bibliography

1. A. E. H. LovE, Mathematical Theory of Elasticity, Cambñdge University Press, 4thedition, 1927.

2. S. TIMOSHENKO and J. N. Theory of Elasticity, McGraw-Hill, 1951.3. E. TREEFIZ, Ober den Schubmittelpunkt in einem durch einc Einzellast gebogenen

Balken. Zeirschrift fur Angewandre Mathematik und Mechanik, Vol. 15, No. 4, pp.220—5, July 1935.

4. A. WEINSThN, The Center of Shear and' the Center of Twist, Quarterly of AppliedMathematics, Vol. 5. No. 1, pp. 97—9, 1947.

5. A. 0. STEVENSON, Flexure with Shear and Associated Torsion in Prisms of Uni-axialand Asymmetric Cross Section, Philosophical Transaction of Royal Society, Vol. A237,No. 2, PP. 161—229, 1938. -

6. J. N. GOODIER, A Theorem on the Shearing Stress in Beams with Applicatioas toMulticellular Sections, Journal of the Aeronautical Sciences, Vol. 11, No. 3, pp. 272—80,July 1944.

7. Y. C. FUNG, An introduction to the Theory of ,4eroelasticity, John Wiley, 1955.8. Loar RAYLEIGH, Theory of Sound, Macmillan, 1877.

3. P. DEN HARTOG, Mechanical Vibrations, McGraw-Hill, 1934.10. S. TIMOSHENKO, Vibration Problems In Engineering, D. van Nostrand, 1928.Ii. R. L. BISPLINGHOFF, H. ASHLEY and R. L. HALPMAN, Aeroelasticity, Addison-Wesley,

4955.12. L. COLLATZ, Eigenwertaufgaben mit technischen Anwendungen, Academische Verlags-

gesellschaft, 1949.

BEAMS 151

13. K. WA.sIuzu, Note on the Principle of Stationary Complementary Energy Appliedto Free Lateral Vibration of An Elastic Body, I,uernauonai Journal of Solids andStructures, Vol. 2, No. 1, pp. 27—35, January 1966.

14. P. A. LIBBY and R. C. SAUER, Comparison of the Rayleigh—Ritz and ComplementaryEnergy Methods in Vibration Analysis, Reader's Forum, Journal of AeronauticalSciences, Vol. 16, No. ii, pp. 700—2, November 1949.

15. S. H. CRANDALL, Engineering Analysis, McGraw-1-Iill, 1956.16. S. TIMOSHENK0, Theory of Elastic Stability, McGraw-Hill, 1936.17. N. J. Horr, The Analysis of Structures, John Wiley, 1956.lB. K. WASHIZU, Note on. the Principle of Stationary Complementary Energy Applied

to Buckling of a Column, Transactions of Japan Society for Aeronautical and SpaceSciences, Vol. 7, No. 12, pp. 18—22, 1965.

19. V. V. BOLOTIN, Nonconservative Problems of the Theory of Elastic Translatedby T. K. Lusher and edited by 0. Herrmann, Pergamon Press, 1963.

20. R. D. MINDUN and G. A. HERRMANM, A One-dimensional Theory of CompressionalWaves in an Elastic Rod, Proceedings of the is: National Congress for Applied Mecha-nics, Chicago, pp. 187—91, 1951.

21. Y. C. FliNG, Foundations of Solid Mechanics, Prentice-Hall, 1965.22. R. W. and A. R. COLLAR, Effects of Shear Flexibility and Rotary Inertia

on the Bending Vibrations of Beams, Quarterly Journal of Mechanics and AppliedMathematics, Vol. 6, No. 2, pp. 186—222, June 1953.

23. H. N. ABRAMSON, F!. 3. PLkss and E. A. RIPPERGE, Stress Wave Propagation in Rodsand Beams, Advances in Applied Mechanics, Vol. 5, pp. 111—94, Academic Press,1958.

24. R. W. LEONARD and B. BUDIANSKY, On Travdlin,g Waves in Beams, NACA Report1173, 1954. -

25. R. W. LEONARD, On Solutions for the Transient Response of Beams, NASA, TechnicalReport R-21, 1959.

26. V. V. NovozHILov, Foundations of:he Nonlinear Theory of Elasticity, Graylock Press,1953.

27. K. WASHIZU, Some Considerations on a Naturally Curved and Twisted SlenderBeam, Journal of Mathematics and Physics, Vol. 43, No. 2, pp. 111—16, June 1964.

28. K. WASHIZU, Some Considerations on the Center of Shear, Transactions of JapanSociety for Aeronautical and Space Sciences, Vol. 9, No. 15, pp. 77—83, 1966.

29. E. REISSNER, Variational Considerations for Elastic Beams and Shells, Journalof theEngineering Mechanics Division, Proceedings of the American Society of Civil Engi-neers, Vol. 88, No. EM!, pp. 23—57, February 1962.

30. R. KAPPUS, Drillknicken zentrisch gedrückter Stãbe mit offenem Profil im elastischenBereich, Luftfahrtforschung, Vol. 14, pp. 444—57, 1937.

31.3. N. GOODIER, Torsion and Flexural Buckling of a Bai of Thin-walled Open Sectionunder Compression and Bending Loads, Journal of Applied Mechanics, Vol. 9, No. 3,pp. A-103-A-107, September 1942.

32. S. TIMOSHENKO, Theory of Bending, Torsion and Buckling of Thin-walledof Open Cross Section, Journal of the Franklin Institute, Vol.239, No. 3, pp. 201—19,March 1945; 239, No.4, pp. 249-68, April 1945; Vol. 239, No. 5, pp. 343—61,May 1945.

33. F. BLEICH and H. BLEICH, Buckling Strength of Metal Structures, McGraw-Hill, 1952.34. K. MARGUERRE, Die Durchschlagskraft eines schwach gekrummten Sitzungs-

fcrichie der Berliner Mathematischen Gesellschaft, pp. 22—40, 1938.

CHAPTER 8

PLATES

8.1. Stretching and Bending of a Plate

Let us consider in the present chapter the stretching and bending of athin plate, the middle surface of which is assumed to be flat. Concerningthe coordinate system employed, the x- and are taken in. coincidencewith the middle surface and the z-axis in the direction of the normal to themiddle surface, so that the x-, y- and z-axes constitute a right handed rect-angular Cartesian coordinate system. The plate is assumed to be simply-connected, and its side boundary surfaces to be cylindrical, i.e. parallel

to the z-axis, as shown in Fig. 8.1. We shall denote the region and peripherywhich constitute the middle surface of the plate by Sm and C, respectively.The direction cosines of' the normal v, drawn outwardly on the boundaryC, are denoted by (1, m, o) namely, 1 = cos(x, v) and m = cos(y, ;'). Acoordinate s is taken along the boundary C, such that v, s and z form aright handed system.

152

Fio. 8.1. Coordinate system for a plate.

PLAThS

Th formulating an approximate theory of,the thin plate in stretching andbending, we shall employ the following assumptions based on the thinnessof the plate. First, we assume that the transverse normal stress may beneglected in comparison with the other stress components and may be set

= 0. (8.1)

Then, as shown in Appendix 13, we have the following stress—strain relationsfor linear theories of the thin plate:

ax = (1 v2)(e + ye,),

= (1 — v2)(vex + i,),

= = = (8.2)

For nonlinear theories, we may have

E E= (1 —

+ a,= (1 — -v2)

+ es,),

= = = 2Ge,1. (8.3)

Second, we shall employ the Kirchhoff hypothesis that the linear filamentsof the plate initially perpendicular to the middle surface remain straightand perpendicular to the deformed middle surface and suffer no exten-

2)f

We shall derive expresions for the displacements underthis hypothesis.We consider an arbitrary point of a plate having the coordinates (x, y, z)before deformation, and deiiote its position vectors before and after defor-mation by and r, respectively, which are related to the displacementvector ii by

r + u, (8.4)

where xi1 + yi2 +- zi3, and i1, 12, i3 are the unit vectors in the direc-tions of the x-, y-, z-axes, respectively. Similarly, we denote position vectorsof a point (x, y, 0) of the middle surface before and after deformation by

and r0, respectively, which are related to the displacement vector n0 byr0 = if + u0, (8.5)

where = xi1 ± v12. We define conponents of u and u0 as follows:

u = Ui1 + v12 + wi3, (8.6)

= Ui1 + V12 + (8.7)

where u, v and w are (x, y) only. It is setn that the hypothesisallows us to express r as r=r0+zn,

(8.8)

t The Kirchhoff hypothesis is usually understood to include the first assumption 0as well as the second assumption. However, only the second assumption will be called the

hypothesis in this book.


where ii is a unit normal to the deformed middle surface and is given by

= / x '89ax 3;'! 3x 3ySince

(x + u)i1 + (y + v)i2 + Wi3, (8.10)

we may express n in terms of u, v and w as follows:— Li1 + Mi2.+Ni3

8111/L2+M2+N2' (.where

3w 3v3w i3v3wL= —-—+——--.———--,3x3y 3y3x3w 0u 3w 3u i3wM= ——+-—--——--————, (8.12)0y t3yDx 3xôy

3u 3v 3u3vN= 1+—+—--+—.-—-——---—.3x ôy ôx3y 3y3x

From Eqs. (8.4), (8.5) and (8.8), we obtainu — + z(n — i3). (8.13)

This is the expression for the displacements of a plate under the Kirch-hoff hypothesis. It is observed that the degree -of freedom of the plate defor-mation implied by Eq. (8.13) is three, namely u(x, y), v(x, y) and w(x, y).When a plate problem is confined to small displacement theory, Eq. (8.13)may be linearized with respect to the displacements to obtain

0w Owu=u—z—, v=v—z—, w=w. (8.14)Ox

Consequently, the strain components are given by

Ou Ov(8.15)

= )'xz = Viz = 0,which are related to the stress components by Eqs. (8.2).

8.2. A Problem of Stretching and Bending of a Plate

We consider a problem of a plate stated as follows: Let the plate besubject to a distributed lateral load p(x, y) per unit area of the middle sur-face in the direction of the :-axis. The lateral load may consist of bodyforces as well as external forces on the upper and lower surfaces of the plate.On part of the side boundary, denoted by S1, external forces are prescribed.They are defined per unit area of the side boundary, and their compoiients

PLATES 155

in the of the x-, y- und z-axes are denoted by F, andrespectively. On the remaining part of the side boundary, denoted by S2,geometrical boundary conditions are prescribed.

The principle of virtual work for the present problem can be written asfollows:

+ — Ifpôwdxdy'V Sm

(8.16)t

where Eqs. (8.14) and (8.15) have been substituted. Here, we define thefollowing stress resultants:

*/2 */2 */2

= f dz, N, = f dz, = f—*12 -*12 —h/2

*/2 h/2 *12

= f a,z dz, M, = f dz, (8.17)-kJ2 -*12 -k!2

and */2

fFxdz, N,,= fF,,dz,

f—*/2

and perform integrations with respect to z in Eq. (8.16), where h(x, y) isthe thickness the plate. Then, through the use of an integration by parts,

f [Mi, âw, + M,, dè— ff + ow,,] dx dy, (8.

C1+C2 Sm

and the geometrical conditions

= I — m = m + I (8.20)

which hold on the boundary C, we may transform Eq. (8.16) into

—ff + + + N,,,,)Ov + + + p)Ow]dxdy

+Cl

— (M, — 2,) Ow,, — — Ow,] is

+ f Ou + N,,, Ov + Ow — M, Ow,, — M,., Ow,,] (is = 0, (8.21)C2

t See Eq. (1.32).Notations ( = d( )/ax, ( ),, = a( )/dy, ( = O( and ( ),, ø(

will be used for the sake of brevity whenever convenient.

156 VARIATIONAL IN ELASTICITY AND PLASTICITY

where C1 and C2 are parts of the boundary Cwhich correspond to and S2.respectively, and it is defined that

—

_____

+ a'= NJ + N,, = NJ+ N,in, (8.23)

= + .MX).m, M,, =M, = ÷ M,,jn M,12 + + M,rn2, (8.24)

M,2 = — + M,,l = — — M,) Im + — m2),

+ (8.25)

A?, = + M,,,n, + M,J. (8.26)

The quantities defined by Eqs. (8.17) are stress resultants and moments perunite length of the lines x and y of the middle surface as shown in Fig. 8.2.

zd

ax

FIG. 8.2. Stress resultants and moments.

The quantities N, and are in-plane stress resultants, while M,and are bending and twisting moments. The quantities and definedby Eqs. (8.22) are proved equal to shearing forces and Q, per unit lengthof the lines x and y of the middle surface by considering the equilibriumconditions of the infinitesimal rectangular parallelepiped in the figure withrespect to moments around the axes parallel to the y- and x-axes, respective-ly.t The quantities defined by Eqs. (8.18) and (8.26) are prescribed external

t See the footnote of Section 7.2 for a similar development.

PLATES 157

forces and moments per unit length along the boundary. It is seen thatis the shearing force acting in the direction of the z-axis, while M, and 2,,are bending and twisting moments as shown in Fig. 8.3.

Returning back to Eq. (8.21), we find that some of the line integral termsmust be transformed through integrations by parts. For example, we have

f [(Vi — ãw — — M,5) ow, ,) ds

= —(M,3 — ii?,5)Ow + f [(Vi + M,,,) — (V2 + M,,,5)]Owdc (8.27)Cl Ci

where the notation ( indicates that the difference in the values at theends of C1 is taken. The above equation shows that under the Kirchhoffhypothesis, the action of the twisting moments M,, and M,, distributed

along the boundary is replaced by that of the shearing forces V2 and V2,

respectively, while M,, and 2,, at the ends of C1 remain as concentratedforces in the ±z-directions, respectively.U. 2) A similar transformation isapplied to the line integral on C3, and it is suggested that the geometricalboundary conditions on S2 can be specified approximately as,

owu=ü, v—D, on C2. (8.28a,b,c,d)

Consequently, we may put Ou = Ov = Ow = Obw/Ov 0 on C2.In view of the above development, the principle of virtual wprk, (8.21),

is finally reduced to

— ff + Ni,,,) öu + (N1,,1 + N,,,) Ov

+ +,,, + p)Ow]dxdy +f{(N1. — + (N,, — N,,)Ov

[(V2 + M,3, ÷ Ow — (M, — 2,) Ow, ,j ds = 0. (8.29)

Fzo. 8.3. Resultant forces and couples on the boundary.


Since ôu, ãv, ãw and are arbitrary in S,, and on C1, we obtain the equa-tions of equilibrium,

+ aNt,— o

oN,— o

Ox c3y — ' Ox

32M 82Max2

+ 2Ox Oy

+0y2

+ p = 0, (8.30a, b, c)

and the mechanical boundary conditions,

N,, = Ni,, N,, = N,,, +OM,.

= V: + Mr = A?, Ofl C1.

(8.31a, b, c, d)

We now seek the stress resultant—displacement relations. Combining Eqs.(8.2), (8.15) and (8.17), we find

Eh IOu Eh I OuN,,

= (1 — v2) N,= (1 — ,2)

IOu Ov\= Gh

+(8.32)

and02w 02w .32w 02wM,=

= — D(1 — v)Ox

(8.33)

where D = Eh3/12(1 — v2) is the bending rigidity of the plate.Combining Eqs. (8.28a, b), (8.30a, b), (8.31a, b) and (8.32), we obtain

two simultaneous differential equations and boundary conditions in termsof u and v. By solving this boundary value problem, we can determine thestretching of the plate. Alternatively, combination of Eqs. (8.28c, d) (8.30c),(8.31 c, d) and (8.33) yields a differential equation and boundary conditionsin terms of w, which determine the the plate. When the plateis of uniform bending rigidity, they take the formW

D4LIw = p. (8.34)

—.(4w) + (1 — v) ._

+ aA?,5

102w f.32w I Ow\1(8.35)

Ow Oww w, = on C2, (8.36)

where4( ) = )/0x2 ± 02( )/0y2 is the two-dimensional Laplace operator.The quantity in Eq. (8.35) is the local radius of curvature of the periphery

PLATES 159

C1 defined by = d?9fds, where is the angle between the tangent to theperiphery and the x-axis as shown in Fig. 8.4. Thus, in small displacementtheory of a plate, where the displacement components are assumed to beof the form (8.14), the stretching and bending do not couple with eachother and can be treated separately.

The stress—strain relations (8.2) ensure the existence of the strain energyfunction as shown in Appendix B. Consequently, with the aid of Eq. (8.15),we have the expression for the strain energy of the plate as follows:

I rr Eh iau ôv\22 +Gh (—.4-—

2 j j (1 —i' ) \ 9x 3y / \S."

1 rr iia2w

It is observed that the two terms on the right-hand side of Eq. (8.37) corre-spond to the straifl çnergies due to stretching and bending, respectively.

Before leaving the present section, we note that, if the slope 9of the con-tour or the quantity A?,, is discontinuous at some points on the boundaryC1, care should be taken in deriving Eq. (8.27). For example, if Al,, is dis-continuous at a point s = we should have

f (— + )Q,bw,)ds = £i,,öwCl Cl

— [Mpc(S* + 0) — A?,s(s* — 0)] ôw(s*)— f (V, ÷ ow ds. (8.38)

Ct

Similar care should be taken in the transformation of the line integral onC2. Howe*r, in subsequent sections, we shall assume that such singularpoints do not exist on the boundary.

a2w'2 r/ \2+ ÷ 2(1 — 3') [(oX oy) —.

02w 132w I1dxdy.

(8.37)

Fia. 8.4. and

160 VARIATIONAL METHODS IN ELASTICITY AND PLASTiCITY

8.3. Principle of Minimum Potential Energy and its Transformation forthe Stretching of a Plate

The formulation in the preceding section suggests that the expression forthe total potential energy of the plate in stretching is given by

I rr i Eh I 3u &' dxdy2 j (I — ) oy / cxji ax ay

+ (8.39)

where the independent quantities subject to variation are u and v under thesubsidiary conditions (8.28a, b). By the introduction of three auxiliaryfunctions defined by

au au &'(8.40)

the functional (8.39) is generalized intoEh Gh_v2) + E,o)2

——

——

N,

— ——

Nxy] dx dy + N,,v) ds

f [(u — i) + (v — U) ds. (8.41)C2

If we eliminate 1x0, )'xyO, u and v through the use of the stationaryconditions:

Eh Eh

= (1 — v2)+ N,

= v2)+ =

(8.42')

+ Nx., = 0, + N,, = 0 in (8.43)

Ri,., N,, = on C1, (8.44)

the functional (8.41) reduces to

= + N)2 + — NXNy)ldxdy

+ UN,,,) ds, (8.45)

where the functions subject to variation are N, and under the subsid-iary conditions (8.43) and (8.44). If the Airy stress function F(x, y) defined by

—8 46= Ny — ax öy

PLATES 161

is employed, the functional (8.45) may bewritten as

I j32F 2 2 32F f32F

d d 'F—

ds, (8.47)

where the independent function subject to variation is F under the subsidiaryboundary condition (8.44), namely,

d/aF\ - diêF\=

—= on C1.

The stationary conditions of the functional (8.47) are an equation in S,and boundary conditions on C2. The equations in Sm comprises the condi-tion of compatibility between the strain components andWhen the plate is of uniform thickness, the equation becomes

ZJAF 0 Hi Sm, (8.49)

where A is the two-dimensional Laplace operator. It is obvious ihat theboundary conditions.on C2 are equivalent to Eqs. (8.28 a, b). It is noted herethat 'the functional (8.45) can be obtained directly from the functional(2.23) by assuming that

ax = a, = 4., = (8.50)

and all the other stress components vanish.The problem of a plate in stretching has been extensively investigated,

and a great number of papers have been written on the subject (see Refs. 3through 6, for example). Variaiional principles combined with the Rayleigh—Ritz method have been employed to obtain approximate solutions for theanalysis of plates in stretching Refs. 3, 7 and 8, for example).

8.4. Principle of Minimum Potential Energy and Its Trangformation for• theBendingofaPlate

Asis observed in Section 8.2, the total potential energy for the plate inbending is given by

1 432H' t32w 2 ô2w 2 32w11=

++2(1 -

—(8.51)


where the independent function subject to variation is w under the sub-sidiary condition (8.28c, d). By the introduction of 'hree auxiliary functionsdefined by

= (8.52)

the functional (841) can be generalized as follows:

iii= ff [(xi + +' 2(1 — v) —

+ — + —

+ 2 Mx,;P:} dx dy

—

F* — (w — dr, (8.53)

where M,, and Q are Lagrange multipliers. Eliminating x,,and w t(hough the use of following stationary conditions:

= — + vx,), M, = — + ,c,),= —D(l — (8.54)

+ + Mi,,, + p = 0, (8.55)M, = + = + on C1 (8.56)

and = M,, = + M,,3 on C2, (8.57)

the generalized functional yields the following functional for the principleof minimum complementary energy:

=[(Ma, + + 2(1 + v) (Me, — M1M,)1 dx dy

(858)

where the quantities subject to variation are M, and under thesubsidiary conditions (8.55) and (8.56). The stationary conditions of thefunctional (8.58) are shown to be the conditions of compatibility, which areequivalent to Eqs. (8.52), and the geometrical boundary conditions (8.28c,d).It is noted that the first term of the functional (8.58) can be obtained fromthe first term of the functional (2.23) by assuming that

M, z z

= (1*2/6) (1*12)'

(1*12)' = (1*2/6) (1*12)(8.59)

and aJI the other stress components vanish (see also Appendix D).

PLATES 163

The variational principles derived above can be applied to the solutionof problems of plates in bending. The principle of minimum potentialenergy (8.51), combined with the Rayleigh—Ritz method, has been success-fully employed for obtaining approximate solutions for the deflection ofplates in bending (see Refs. 2, 9 and 10, for example).

8.5. Large Deflection of a Plate in Stretching and Bending

We shall consider a large deflection theory of a plate proposed by T. vprescribing the plate problem in the same manner as in Section 8.2.

It is assumed that, although the deflecffirn of the plate is no longer small incomparison with the thickness of plate, it is still small in comparisonwith the lateral dimensions of the plate and the following expressions maybe employed for the displacements u, wand for the strain-.displacernent.relations: w=w

ôu I fôw\2+ — z

1

+—

.9u ôv e3w t3w a2w

i3j' 3x ox Oy OxDy

higher order terms being neglected.Since we are dealing with the large deflection theory, we must employ

Eq. (3.49) for establishing the principle of virtual work for the presentproblem, and we have

fff + a, be,, + 2T,, be,,) dr dy dz — ff p 6w dx dy

(8.62)Si

where Eqs. (8.60) and (8.61) have been substituted. After some calculationand introduction of the quantities defined in Section we find that Eq.(8.62) reduces to an equation which may also be obtained from Eq. (8.29)by making the following replacements:

by + +by + + N,w,,.

These replacements mean that when the deflection of the plate becomeslarge, the in-plane stress resultants N,, N, and N,, have contributions to

t SeeRefs.2, 11 and 12.These equations may be derived from Eqs. (8.13) and (3.19) by assuming that

u — u , v — v , (w, ,)2 1 and -ternn containing z2 may be neglectctl.

The first assumption states that the quantities (w,,,)2, the strain of the middle sur-face as well as the rotation of the plate around the 2-axis are very small compared tounity." 2)


the equation of equilibrium in the direttion of the z-axis due to the inclina-tion of the middle surface. Thus we obtain the equations of equilibrium

aNx,+0N,oOx — Ox

32M,, O2MX, 82M, 8 / Ow Ow2 +2 + 2Ox Ox Oy Ox

(8.64)

and the mechanical boundary conditioris on C1,= jc.,,, , = ic11,,

Ow Ow÷ Q,m + N,,, + N,,, -y- + = Pz + a

M. = P,. (8.65)

Combining Eqs. (8.3), (8.17) and (8.61), we obtain the in-plane stress resul-tant—displacement relations as follows:

Eh EhN,, = (1 v2)

(e,,,,0 + ye,,,,0), = (1 — w2)(ye,,,,0 + e,,0).

N,,,, = 2Ghe,r,o, (8.66)

where Ou 1/Ow \2 Op 1/Ow+ , C,,0 = +

Ou Oi' Ow Ow- 2e,,,0 = + + -i-— (8.67)

while the bending moment—curvature relations are still given by Eqs. (8.33).The equations thus obtained, together with the geometrical boundaryconditions (8.28 a, b, c, d), formulate the problem of the flat plate in largedeflection. It is observed that the stretching and bending couple with eachother in the large deflectiop theory and cannot be treated independently.

Next, let us consider variational formulations of the problem. Followingthe development similar to that in small displacement theory, we can writethe principle of stationary potential energy, from which we obtain thefollowing generalized form, III:

=ffI2(1_v2) [(e,,,,0 + + 2(1

+ [(ac,, + + 2(1 — —

I iOu 1 1 1 I.3v 1 /Ow\2\l -- [exxo - + ) JN,, - - + )j

N,

Ou Ov Ow

— (T + + -b--)N,,, + — M

+ (terms on C1 and C2). (8.68)

PLATES 165

We shall eliminate the strain components and by the use of thestationary conditions with respect to these quantities. while Eqs. (8.52) will be

substituted to eliminate M1, and These having beeneliminated, introduction of the Airy stress function defined by Eqs. (8.46) thenallows us to transform the expression (8.68) into,

* cr I I I2 t32F 2

H =jj ÷ + 2(1 + — aX2

- D ô2w ê2w 2 a2w 2

aw 1

adx dy

+ (integrals on C1 C2), (8.69)

where the functions subject to variation are Fand w. Assuming, for the sakeof simplicity, that the thickness h is constant, we obtain as the stationaryconditions of the functional fl* the following two equations in S_:

02F '82w '02F .32w 02F 82waxt3y'

(8.70)

and

M h02w 02w 02w

F— EOx2 0y2

' (8.71)

where LI is the two-dimensional Laplace operator. It is observed that Eq.(8.70) is the equation, of equilibrium in the direction of the z-axis, whileEq. (8.71) comprises the condition of compatibility between the straincomponents e,,0 and Some of papers related to the large deflectiontheory of flat plates are listed in the bibliography (Refs. 13 through 18).

8.6. Buckling of a Plate

We shall now formulate a buckling problem for the flat It isassumed that before buckling occurs, the plate is subject to a system oftwo-dimensional stresses k a monotonicallYincreasing factor of proportionality and the distribution of the stresses

and is prescribed. The stress system will be treated as initialstresses which satisfy the following equations of equilibrium andboundary conditions: - -

± 0, + = 0 in S_, (8.laaJay

= = on C, b)


where— — —

= + =

We shall measure the displacement components u, v and w from the statejust .prior to the occurrence of buckling, and assume that they are given byEqs. (8.60). We also assume that the external forces and onC1 vary neither in magnitude nor in direction during buckling, and that theplate is fixed on C2, requiring that

u = 0, v = 0, w 0, = 0 on (8.73a, b, c, d)

Since we are dealing with an initial stress prQblem, we must employ thenonjinear expressions (8.61) for the strain components in the principle ofvirtual work, which is written for the present problem as follows:

ff f + + + cl),) ôe,, + + ri,) tie,,] dx dy dzV r

—f 3u + ds = 0, (8.74)t

where a, and are incremental stresses. Since we are interested onlyin the configuration and critical load for the buckling, higher order termsmust be neglected in the principle (8.74). The strains are linearized withrespect to the displacement components in the incremental stress—strainrelations for the same reason. In the present problem we assume that theincremental stress—strain relations are given by Eqs. (8.2). Then, employingthe stress resultants defined by Eqs. (8.17), we find that the incrementalstress resultant—displacement relations are given by Eqs. (8.32) and (8.33).

Returning to the principle (8.74)and employing Eqs. (8.72a, b) and (8.73 a,-b), we find that contributions from the ãu and öv terms in Eq. (8.74) provide

+ Ni,., = 0, + N,, = 0 in S,,,, (8.75ajand

Ni,, = 0, N,, = 0 on C1. (8.75b)

Combining' these with Eqs. (8.32) and (8.73 a, b), we concludethat = N, = = 0 throughout the plate. Consequently, we mayreduce the principle to the form:

32ôw ö2ôw3x2

+ + 2Mg, dx dy

+ kff + + + dxdy =0,(8.76)

t See Eq. (5.5).

PLATES 167

which, through a familiar process, provides the equations of equilibriumin Sm,

02M 432M 02MwX÷2Ox(3y

+3y2

+ k (N(0) + Ow+ = 0, (8.77)+

and the mechanical boundary conditions on C1,

+ k1N°' Ow+ +

3M,3= 0, M, = 0. (8.78)

I Os

Equations (8.77) and (8.78), together with Eqs. (8.33) and (8.73c, d), formu-late the buckling problem under consideration. In the case of uniformbending rigidity these equations may be written as follows:

Dzlzlw = kI + + +Ow'1

(8.79)

+ k(JV(0)OW

0—D {-L-(/lw) + (1 —F 3 1t3w'1

F02w I Ow\1D + v

+0 on C1, (8.80)

andOww=0, —=0 on C2. (8.81)(33,

Consequently, the buckling configurations and critical loads can be deter-mined by solving the differential equation (8.79) under the boundary con-ditions (8.80) and (8.81).

When the principle (8.76) is combined with Eqs. (8.33), we obtain theprinciple of stationary potential for the buckling problem as follows:

* (8.82)where

H ,! (ID j(32w 2 r/ .32w (32k, 32w

2jj L\3x2s,n

+ k ff[N0 Ow \2+

(Ow)2+

Ow Owdx dy, (8.83)

5,,'

and the independent function sujiject to variation is w under the subsidiaryconditions (8.73c, d). By employing the auxiliary functions defined by Eqs.(8.52), we may generalize the, principle (8.82) in a manner similar to theusual development, obtaining the principle of stationary complementaryenergy. Due to the limited space available, the derivation will not be shown•here.


It is noted here that the principle (8.82) is equivalent to finding, amongadmissible w, that function which makes the quotient defined by

+2(1 — v)

____

—dx dv

k——- 4..ff [N? (PjL) + N?(f-) + dxdy

S... (8.84)stationary.

8.7. Thermal Stresses in a Platet

We shall nàw consider a problem of thermal stresses in a flat plate subjectto a temperature distribution O(x, z). The temperature U is measured from areference state of uniform temperature in which the plate hasneither stresses nor strains. Confining the problem to the small displacementtheory of elasticity and introducing results from Appendix B. we mayemploy the fbllowing stress—strain relations:

E Ee8

= (1 — v2)+ ye,)

— (1 — v) '

Ee°

= (1 — v2)(vex + t,)

— (1 — )(8.85)

Tx, = = =where e denotes the thermal strain. We assume that the displacementponents measured from the reference state may be expressed by Eqs. (8.14).For the sake of simplicity, the boundary S is prescribed to be fixed, namely

t3w- u = 0, v = 0, w = 0, = 0 on C, (8.86a, b, c, d)

while the surfaces z = ±h/2 are assumed to be traction free.The derivation of governing equations for the thermal stress problem

proceeds in a manner similar to the development in Section 8.2, the effect'of thermal expansion having been accounted for by including e° in the stress—strain relations. Combining Eqs. (8.85) with Eqs. (8.17), we have

Eh au— (1 — +

—

Eh N.3-

— (1 — r)(8.87)

f Ref. 20. -

PLATES 169

and

M _,D(ôw 82w\ MT

/ 82w\ M(

= —D(l — v)82w

wherehI2 *12

= f Ee° dz, MT = f Ee°z dz. (8.89)-h/2 -472

These relations show that thermal stretching and bending are decoupled insmaH displacement theory in which the displacement components are givenby Eqs. (8.14).

Let us now consider variational formulations of the problem. With theaid of Eqs. (8.15) and some results from Appendix B, we may express thestrain energy of the plate as follows:

I fri Eh /öu av\2 f/0u .9v\2 öu öv2 +Ghii—+——-i —4——2,, t(1 — v) 'ox êy, Oy

2NT Du Ov 1 82w 82w2—

I 82w 2 82w 82w\1 2MT /8W ô2w\l+ 2(1 - - + (I + V)ldxdy.(8.90)

Consequently, the total potential energy for thermal stretching of the plateisgivenby -

ff1 Eh i Gh II \2 8u 8v11= il 1— + —, + —u— + —I —4—-—jj 12(1 — ,2) t3y/ 2 fly Ox Oy

Sm

— +dz4y, (8.91)

where the functions subject to variation are ii and v under the subsidiaryconditions (8.86 a, b).

The functional (8.91) is generalized in a manner similar to the developmentin Section 8.3, and we obtain the following functional for the principle ofstationary complementary energy:

[IC =+ N,)2 + 2(1 + v) — N1N,)]

+ 2NT(NX + N,)) dx dy,

170 VARIATIONAL METhODS IN ELASTICITY AND PLASTICITY.

where the functions subject to variation are N, and under thesubsidiary conditions (8.43). Introduction of the Airy stress function definedby Eqs. (8.46) reduces the functional further to -

I t32F Ô2F 2 2

++ 2(1 +

a2F\l.+ dx dy, (8.93)

where the only function subject to is F(x, y), upon which no sub-sidiary conditions are imposed.

• Thus, far, the variational formulations have been made for The thermal• stretching. Formulations can also be made for the thermal bending in amanner similar to Section 8.4, by employing the latter half of the handside of (8.90). Extensions of the above formulations to thermal stressproblems of plates in Jarge deflection may be made in a manner similar tothe developments in Section 8.5. These variational principles have beenused for obtaining approximate solutions in combination with the Rayleigh—Ritz method.t2 • 22) Thermal stresses in a plate are responsible pheno-mena such as thermal buckling or variation of stiffeness and -vibrationfrequencies of the plate.t23 24)

A Thin Plate Theory Including the Effect of Transverse Shear Deformation

So far, theories of a thin plate have been established on the Kirchboffhypothesis. In this section, we shall consider a small displacement theoryof a thin plate including the effect of transverse shear deformation. In makingthis extension, we are forced to abandon the hypothesis; it&hlternative mustbe chosen judiciously. V

Since the displacement vector u is a function of (x, y, z), we may expandit in power series of z:

'8u' 1 'ö2u' z2+.... (8.94)g.0 2. \ ('Z /s-O

Therefore, one of the simplest expressions for displacements to include theeffect of transverse shear deformation may be given by retaining the first -

two terms only: u=u0+zu1, (8.95)

where components of u1 are defined byu1 u111 + v1i2 -I, w1i3, (8.96)

and ut, v1, w1 are functions of (x, y) only. The degree of freedom impliedby Eq. (8.95) is six, namely, u, v, w, u1, VL and w1. However, if we continuetu use the assumption (8.1), and employ Eqs. (8.3) as the stress—strainrelations, we may take

+ + (1 + — I = 0 (8.97)

PLATES 171

as an additional geometrical constraint to reduce the degree of freedom tofive. Equations (8.95) and (8.97) state that the linear filaments perpendicularto the undeformed middle surface remain straight and suffer no strainsalthough they ak no longer perpendicularto the deformed middle surface.

Since we are interested in a small displacement theory,t Eq. (8.97) islinearized with respect to the displacements to yield

w1 = 0. (8.98)

Consequently, we observe that the most natural and simplest expressionto include the effect of transverse shear deformation is to assume that

U = u + zu1, zv1, w = w. (8.99)

In a manner similar to the development in Section 8.2, it can be shown thatthe functions u and v are related to the stretching of the plate, while thefunctions u1, v1 and w are relaCed to the bending of the plate, and thesetwo problems can be treated separately. Therefore,- we confine our sub-sequent interest to bending only by assuming that

u=zu1, v=zv1, w=w, (8.100)and we obtain

3v1

8w

(8.101)

It is seen from Eqs. (8.15) and (8. 101) that the Kirchhoff hypothesis imposesthe constraint conditions, I

8w 8wu1 =

— —(8.102)

We shall consider a dynamical problem defined in a manner similar tothe presentation in Section 8.2, except that the external forces and geo-metrical boundary conditions are flow time-dependent. The form of theprinciple of virtual work for this dynamical problem is suggested by Eq.(5.81) to be

• {fff + o, 6€, 8Yx, + + dx dy dz

• offf3(i12 V2 + w2) — ffpawdxdy

(8.103)

t Ref. 12 for a finite 8isplacement theory in which Eq. (8.95) is employed asan expression of displacements.


where Eqs. (8.100) and (8.101) have been substituted. Here we shall introducenew resultants defined as follows:

• *12

= f dz, Q, = f dz, (8.104)—*12 -*12

hJ2 *12

m = fedz. 4, = foz2dz. (8.105)—5/2 —5/2

The quantities defined by Eqs. (8.104) are shearing forces per unit lengthacting in the direction of the z-axis.t The quantities defined by Eqs. (8.105)are the mass and mass moment of inertia per unit area of the middle surface.With these preliminaries and some calculation including integrations byparts, Eq. (8.rn3) is finally reduced to

J2 I [(1mÜi — — Mi,., +

+ — — M,, + Q,)öv,

+(miv— — Q,,, —p)öw]dxdy

+ f [(Mi, — ôu1 + (M,,, — 2,,) öv1 + + Q,m V) owl ds

[Mi, Ou1 + 0v1 + (QJ + Q,m) Ow] dc} di = 0. (8.106)

Thus, the principle provides the equatiops of motion,

8Mg,= + — (8.107)vXj

ÔM,4,v1 = + —Q,, (8.108)

(8.109)

and the mechanical boundary conditions on C1,

= Ms,, M,, A?,,, QJ ÷ Q,,n = P, (8.110)

while it suggests that the geometrical boundary conditions on C2 can bespecified approximately as

= v1 = w (8.111)

f Thus, the shearing forces Q and Q, appear u jndepaident quantities in the thinplate theory including the effect of absas ddetwation. Compare EqL (L104),(8.107) and (8.108) with Eqs. (8.22).

PLAThS

The stress resultant-displacement relations are obtained from Eqs. (8.2),(8.17), (8.101) and (8.104) as

/ au, I t3u1 8v1

= -} D(l —+

and= Gkh(_/L + Q, = + (8.113)

where k = 1. The factor k in Eqs. (8.113) has been included to account forthe nonuniformity of the shearing strains over the cross section. In Appen-dix D, a theory of a thin plate based on the principle of minimum comple-mentary energy is introduced following paperst and the valueof k is found to be 5/6 for the isotropic plate. On the other hand, from theresult obtained in a vibrational problem of a thin plate, suggeststhat k = which is. very close to 5/6 obtained from the formulationbased on the complementary energy principle. Introducing Eqs.(8.112) and

113) into Eqs. (8.107) through (8.109), we obtain three simultaneous dif-férential equations in terms of u1, and w. Consequently, the dynamicalproblem is reduced to solving these differential equations under the bouudaryconditions (8.110) and (8.Ilql).

it is seen from the formulation that we have three mechanicalboundary conditions (8.116) on C1 and three geometrical bpundary con-ditions (8.111) on C2 in the thin plate theory including the eflbct of trans-verse shear deformation. have replaced through integrations by partsthe action of ic?,, and M,, by that of P. and respectively in the thin platetheory under the Kirchhoff hypothesis; However, such replacements areno longer necessary in the thin plate theory including the effect of transverseshear deformation.

/

8.9. Tha Shallow Shell'\

In the present section we shall consider a nonlinear theory of thin, shallowshells proposed by K. Let the rectangular Cartesian coordi-nates fixed in space be (x, y, z) and let the middle surface of the thin, shallowshell be represented by

z = z(x,y) (8.114)

as shown in Fig. 8.5. The position vector of an arbitrary point F:°' in theundeformed middle surface is given by

. (8.115)

t R.efs. 25 through 28.


where and 13 are unit vectors in the of the x-, y- and z-axes,respectively. Then; the position vector oIan arbitrary point p(O) outside themiddle surface before deformation may be given by

= ÷ (8.116)

where is a unit vector drawn perpendicular to the undeformed middlesurface and is calculated by

-ax ay/

Iax

and where is the distance from the middle surface to the pQint. Equation(8.116) suggests that an arbitrary point in the shell can be specified by thecoordinates (x, y, which form a curvilinear coordinate system. Conse-quently, by taking x, = = we may apply the formulationsdeveloped in Chapter 4.

The shell is now assumed to be subject to deformation and the positionvectors of the two points and F after deformation are represented by

=

r

r

V

x

Fio. 8.5. A plate with small initial deflection.

PLATES 175

respectively, where u0 and u are displacement vectors, their componentsbeing defined by

U0 = U)1 + vi2 + wi3 (8.120)and

u = UI1 + vi2 + Wi3 (8.121)

respectively, where u, v and w are functions of x and y only.In subsequent formulations, we shall employ the Kirchhoff hypothesis,

under which the position vector r is related to r0 byr = r0 + (8.122)

where n is the unit normal to the deformed middle surface and is given by- .3r0 .3r0 I 0r0n=_x_/_x_. (8.123)a, ox .3y

Combining Eqs. (8.116), (8.118), (8.119) apd (8.122), we obtain

u = U0 + C(n — (8.124)

Then, the strain tensors can be obtained in terms of the displacementsu, v and w by the use of Eqs. (4.36), (8.116) and (8.122).

It is assumed hereafter that the shell is shallow and thin to the extentthat

/ / 8z .3z

-r—T 4 1, (8.125)

<<1, (8.126)

and terms containing be neglected. Then, we have

= — — + '3, (8.127)

and we observe that the (x, y, system can be taken approximate-ly to be locally rectangular Cartesian. In addition to the above assumption,restrictions on the orders of magnitude of the displacements of the middlesurface are introduced. It is assumed that the initial deflection z(x, y) andthe displacement w(x, y) are of the same order of magnitude. Then, we havethe following approximate expressions:t

i.3z IOz Ow\I =— + — + + (8.128)

u = + + (8.129)

111 = 122 = ê,Yo —

f12 = — (8.130)

f These equations may be derived by assuming that u,. (w,

— I and terms containing C2 may be neglected.'3"

176 VARIAflONAL METHODS IN ELASTICITY AND PLASTICITY

where- Di, e3z8w I

- Dv DzDw lIDw\2+ r -y- + (8.131)

Dii Dv Dz Dw Dz Dw ow OwDy Dx Dx Dy Dy Ox Ox Dy

and02w 02w= ,C, = = (8.132)

higher order terms being neglected.We may define the stress tensor with respect to the (x, y, C) coordinates

in a manner similar to the definition in Chapter 4, and employ Eqs. (4.74)and (4.77) as the stress—strain relations. However, since the shell is thin andshallow, the-transverse normal stress uc may be neglected and the (x, y, C)coordinate system can be considered approximately locally rectangularCartesian. Consequently, we may have

= (1 ,2) + '122)' r22 = (1(vf11 +122),

T12 = 2Gf12. (8.133)

A problem of a thin shallow shell is stated as follows. External forcesare prescribed per unit area of the (x, y) plane and their components in thedirections of the x-, y- and z-axes are denoted by 1, F and 2, respectively.The side boundary generated by the envelope of: normals drawn perpendicu-lar to the middle surface divides into two S1 and S2. External forcesper unit area are prescribed on S1, with components F, and andgeometrical boundary conditions arc prescribed on S2.

With the above preliminaries, we have the foftbwing expression for theprinciple of virtual work:

fffrruioij, + r228f22 +

_ffixou÷

—— + !', (ôv — C + on] dr dC =0. (8.1 34)t

t See Eqs. (4.80).

PLATES 177

Here we introduce stress resultants defined as follows:

M,=fT22CdC, (8.135)

(8.136)

M,3 = —(Mi — M,) bn + M,(12 — mi), (8.137)

Nxr=ffxdC,

M=fF,1dc, (8.138)

(8.139)

where integrations extend through the thickness of the shell. Following theusual procedure, we find that the principle (8.134) provides the equations ofequilibrium,

+1=0,Ox ay

X + 282MX,

+ +1+ +

Ow' IOz Ow'+ +

+ Z= 0, (8.140)I

and the mechanical boundary conditions of C1,

= Na,, N,, = (oMx + + +aMa)

Ox Oy) s3y

/ Oz Ow' / Oz Ow'—

DRIPS M, =a

(8.141)

while it suggests that the geometrical boundary conditions on C2 are givenby

Ow• u=ü, v=V, —=—. (8.142)

Or Or

Combining Eqs. (8.130), (8.133) and (8.135), we obtain the following rela-tions between the stress resultants and strains:

Eh Eh= (1 — v2)

+ ye,,0), N,(1

— ,2) + e,,0),

= (8.143)

and= — + v,c,),M, = —

D( I — v) (8.144)

178 VARIATIONAL METHODS iN ELASTiCITY AND PLASTICITY

These equations formulate the nonlinear theory of the thin, shallowIt is observed that the total potential energy for the present problem isgiven by

I rn Eh11= ki v2)+ e,,0)2 + 2(1 — —

+ + x,)2 + 2(1 — v) — dx dy

— ff(xu + Yv + 2w) dx dy

+ + + — (8.145)

where Eqs. (8.131) and (8.132) have been substituted. Generalizations andtransformations of the functional (8.145) can be formulated in the usualmanner.

So far, the nonlinear theory of the thin shallow shell has been derived.It is noted in this connection that a linear theory can be obtained by lineariz-ing the strain—displacement explessions as

au ezaw ô2w111 = + —

av az Ow 02wf2'2 = + r —

(8.146)

Ou Oz Oiv Oz Ow2112 =—+-—+——+—-——2C——,ay ay Ox Ox Oy

and deriving equations in a manner similar to the development for thenonlinear theory. Some related papers are listed in the

8.10. Some Remarks

The theories of thin plates developed in this chapter are based on theassumption that the transverse normal stress may be neglected in the stress—strain relations. Rigorously speaking, the transverse stress

a in Appendix D showsthat unless the surface forces are highly concentrated, the stress is ingeneral of smaller order of magnitude than and 0,. Consequently, theterms containing are usually neglected in the stress—strain relations. Onthe other hand, it is seen from Eqs. (8.15) that we have = 0 under theKirchhoff hypothesis. The three dimensional stress—strain relations, Eqs.(1.10), show that a theory which includes = 0 as well as 0 would

f See Refs. 36.

PLATES 179

fail to produce correct results. The same kind of exsts in theformulations of Sections 8.5 and 8.8. We have tried to avoid this difficultyby putting = 0 in the three-dimensional stress—strain relations and theneliminating To remove the inconsistency completely, it would be neces-sary to employ

w(x, y, z) w(x, y) + y) z + w2(x, y) z2, (8.147)

instead of the last equatibn in Eqs. (8.14) or (8.99). However, these addi-tional linear and quadratic terms are usually found small in comparisonwith the leading far as the small displacement theory of thin plat6sis concerned and may be omitted in a first theory. The th&riesof thin plates developed in this chapter have been based on the above con-siderations, which are due mainly to Ref. 37. The accuracy of these platetheories may be improved by assuming the displacertient compoflents as

U =. y) zm, v =Evm(x, y) w w,,,(x, y) z", (8.148)

thus adding terms of high powers with respect to z. We note here that atheory of thin plates including the effect of transverse shear deformationhas been derived by Yu with the use of generalized Hamilton's principle inwhich the variation is taken with respect to displacements, strains andstresses.t38'

Variational formulations can, of course, be made for fite vibrationalproblems of elastic plates, although no mention has been made of thistopic in this chapter.t39' 40. 41) Ap application of the variational methodhas been made to the problem of free vibration of non-isotropic, rectangular,AT-cut quartz We also note that the self-excited or forced vibra-tion of plates due to aerodynamic forces has been one of the central pro-blems in the theory of aeroelasticity.t43 44)

Bibliography—

1. A. E. H. LoVE, Mathematical Theory of Elasticity, Cambridge University Press, 4thedition, 1927.

2. S. and S. WOINOWSKY-KRIEGER, Theory of Plates and Shells, McGraw-Hill,1959.

3. S. TIMOSHENKO and 3. N. GOODIER, Theory of Edasticity, McGraw-Hill, 1951.4. I. S. ixorv, Mathematical Theory of Elasticity, McGraw-Hill, 1956.5. N. I. MUSCHELISVILI, Practische Losung der fundamentalen Randwertaufgaben der

Elastizitäts-Theorie in der Ebene für cinige Berandungsformen, Zeitschr:ft für An-gewandie Mathematik und Mecht. ilk, Vol. 13, No. 4, PP. 264—82, August 1933.

6. S. Moiuoun, Theory of Two-Dimensional Elaslicity (in Japanese), Series on ModernApplied Mathematics, Iwanami Book Publishing Co., 1957.

7. E. REISSNER, Least Work Solutions of Shear Lag Problems, Journal of Me AeronauticalSciences, Vol. 8, No. 7, pp. 284—91, May 1941.

8. E. REISSNER, Analysis of Shear Lag in Box Beams by the Principle of Minimum Poten-tial Energy, Quarterly of Applied Mathematics, Vol. 4, No. 3, p0. October1946.


9. E. and M. STEIN, Torsion and Transverse Bending of Cantilever Plates,NACA TN 2369, June 1951.

10. R. L H. Asiuiy and R. L. HALFMAN, Aeroelassicity, Addison-Wesley,1955.

11. T.v. KAIU.tAN, Festigkeitsprobleme im Maschinenbau, Encyklopddie derMathema:ischenWissenichaften, Vol. IV, pp. 314—85, 1910.

12. V. .V. NovozlnLov, Foundation of the Nonlinear Theory of Elasticity, Graylock, 1953.13. K. MARGUERRE, Die über die Ausbeulgrenze belastete Platte, Zeitschr,f: für Ange-

wandte Mathematik und Mechanik, Vol. 16, No. 6, pp. 353—5, December 1936.14. K. MARGUERRE and E. Twrrz, Uber die Tragfahigkeit eines Iãngsbelasteten Platten-

streifens nach Uberschreiten der Beullast, für Angewandte Mathematik undMechanik, Vol. 17, No. 2, pp. 85-100, April 1937.

15. A. FROMM and K. MARGUERRE, Verhaken elites von Schub- und Druckkrãften be-anspruchten Plattenstreifens oberhaib dcc Beulgrenzc, Lzifsfahrtforschw*g, Vol. 14,No. 12, pp. 627—39, December 1937.

16. C. T. WANG, Principle and Application of Complementary Energy Method for ThinHomogeneous and Sandwich Plates and Shells with FInite Deflecuons. NACA TN2620, 1952.

17. E. issiisa, Finite Twisting and Bending of Thin Rectangular Elastic Plates, Journalof Applied Mechanics, Vol. 24, No. 3, pp. 391-6, September 1957..

18. R. L. BISPLINOHOFF, The lInite Twisting and Bending of Heated Elastic Ljfting Sur-faces. Mitteilung Nr. 4 aus dem Institut für Flugzeugstatik und Leichtbau, E. T. H.,Zurich, 1957.

19. S. Theory of Stability, McGraw-Hill, 1936.20. B. A. and J. H. WEIpgp, Theory of Therniai Stresses, John Wiley, 1960.21. R. R. HE t1Ifl.S and W. t4 Experimental and Theoretical Determination

of Thermal Siresses in a Flat Plate, NACA TN 2769, 1952.22. M. L. 0065MW, P. W. M. RonaRis, Thermal Buckling of Plates, NACA TN

2771, 1952.23. R. L. BISPLINGHOFP et a!., Heating of Aircraft Structures in High-speed

Flight, Notes for a Special Summer Program, Department of Aeronautical Engineer-ing, Massachusetts Institute of Technology, June 25-July 6, 1956. -

24. N. J. Hose, Editor, High Temperature Effeqs in Aircraft Structures, AGARDograph28, Pergamon Press, 1958.

25. E. On the Theory of Bending of Elastic Plates, Journal of Mathematics andPhysics, Vol. 23, No. 4, pp. 184—91, November 1944.

26. E. REISSNER, The Effect of Transverse-Shear Deformation on the Bending of ElasticPlates, Journal of Applied Mechanics, Vol. 12, No. 2, pp. 69-77, June 1945.

27. E. REISSNER, On Bending of Elastic Plates, of Applied Mathematics, VoL. 5,No. 1, pp. 55—68, April 1947.

28. E. REISSNER. On a Variational Theorem in Elasticity, Journal of Mathematics andPhysics, Vol. XXIX, No. 2, pp. 90—5, July 1950.

29. R. D. MINDLIN, Thickness-Shear and Flexural Vibrations of Crystal Plates, Journalof Applied Physics, Vol. 23, No.3, pp. 316-23, March 1951. -

30. K. M*aovsaar., Zur Theorie der gelcrummtcn Platte grotler Formãnderung, Pro-ceedings of she 5th International Congress for Applied Mechanics, pp. 93—10%, 1938.

31. E. REISSNER, On Some Aspects of the Theory of Thin Elastic Shells, Journal of theBoston Society for Civil Engineers. Vol. XIII, No. 2, pp. 100—33, April 1955.

32. E. REJSSNER, On Transverse Vibrations of Thin Shallow Elastic Shells, Quarter!.;' ofApplied Mathematics, Vol. 13, No. 2, pp. 169—76, July 1955.

33. R. R. HELDENFELS and L F. Vosru.N, Approximate Analysis of Effects of LargeDeflections and Initial Twist on Torsional Stiffness of a Cantilerer Plate Subjected toThermal Stresses, NACA TN 4067, 1959.

PLATES 1 1

34. E. L. REISS, H. J. GREENBERG and H. B. KELLER, Nonlinear Deflections of ShallowSpherical Shells, Journal of the Aeronautical Sciences, Vol. 24, No. 7, pp. 533-43,July

35. E. L. REISS, Axially Symmetric Buckling of Shallow Spherical Shells under ExternalPressure, Journal of Applied Vol. 25, No. 4, pp. 556—60, December 1958.

36. H. B. KELLER and E. L. REISS, Spherical Cap Snapping, Journal of the Aero/SpaceSciences, Vol. 26, No. 10, pp. 643-52, October 1959.

37. F. B. HILDEBRAND, E. REISSNER and G. B. THOMAS, Notes on the Foundations of theTheory of Sinai! Displacements of Orthotropic Shells, NACA TN 1833, 1949.

38. Y. Y. Yu, Generalized Hamilton's Principle and Variational Equation of Motionin Nonlinear Elasticity Theory, with Application to Plate Theory, Journal of theAcoustical Society of America, Vol. 36, No. 1, pp. 111—19, January 1964.

39. R. WEINSTOCK, Calculus of Variations with Applications to Physics and Engineering,McGraw-Hill, 1952.

40. M. V. BARTON, Vibration of Rectangular and Skew Cantilever Plates, Journal ofApplied Mechanics, Vol. 18, No. 2, pp. 129—34, June 1951.

41. H. J. PLASS JR., J. H. and C. ID. NEWSOM, Application of Reissncrs VariationalPrinciple to Cantilever Plate Deflection and Vibration Problems, Journal of AppliedMechanks, Vol. 29, No. 1, pp. 127—35, March 1962.

42. 1. KOGA, Radio-Frequency Vibrations of Rectangular AT-Cut Quartz Plates, Journalof Applied Physics, Vol. 34, No. 8, pp. 2357—65, August 1963.

43. R. L. BISPLINOHOEF and H. ASHLEY, Principles of Aeroelasticily, John Wiley, 1962.44. V. V. BOLOTIN, Nonconservative Problems of the Theory of Elastic Stability, Translated

by T. K. Lusher and edited by G. Herrmann, Pergarnon Press, 1963.45. S. G. MIKHLIN, Variational Methods in Mathematical Physics, Pergamon Press, 1964.46. K. WASHIZU, Variational Methods Applied to Free Lateral Vibrations of a Plate with

Initial Stresses, Transactions of Japan Society for Aeronautical and Space Sciences,Vol. 6, No.9, pp. 36-42, 1963.

47. L. S. ID. Moiuav, Skew Plates and Slructure.c, Pergamon Press, 1963.

CHAPTER 9

SHELLS

9.1. Geometry before Deformation

We sLall consider theories of thin shells in the present chapter. Let themiddle surface of the shell, denoted by Sm, be taken as a reference curvedsurface which is defined by two curvilinear coordinates tx and fi in such away that the position vector of an arbitrary point in is representedby

rr = fi), (9.1)

as shown in Fig. 9.1. The coordinates and are chosen so as to coincidewith the lines of curvature of the middle surface, and the unit vectors in the

0

FIG. 9.1. Geometry of the shell before and after deformation.(a) before deformation. (b) after deformation.

182

z

U

p

x

SHELLS

directions of and fi are denoted by and respectively:

— — 1

9 2a (.)where

(0) (0)

A — —i- B2 9 3acx'

The length of a line element between two neighboring points in Sm, thecoordinates of which are (x, and + dLx, fi + dfl), is given by

= = A2(dcx)2 + (9.4)t

The unit vector perpendicular to 5,,, is denoted by which is chosen sothat and form a right-handed orthogonal system:

fl(O) x (9.5)

The radii of curvature in the directions of and will be denoted by R,,

and and are tal$en positive when the centers of curvature lie in the posi-tive direction of The geometry of the middle surface gives rise to thefollowing matrix relations:

184 AB8fl

0 0 (9.6a)

o o

— !. o . (9.6b)R1,

nED) 0

Using the above relations and the following identities:821*(O)

= op ' op — op op op

we have,OIB\ I 0$ 1

(9.7)

Specializations to several kinds of shells are given n Appendix E.Refs. 1 a!ld 2. See also Psoblem 5 of Chapter 4 in Appendix H.


and

0 /1 OB\ 0 11 OA\ 48+ +

= 0. (9.8)

These relations are known as the conditions of. Codazzi and of Gauss,respectively.

Next, we shall consider an arbitrary point outside the middle surfaceof the shell. We represent its position vector by:

= fi) + if), (9.9)

where is the distance of the point from the middle surface. The relation(9.9) shows that an arbitrary point in the shell can be specified by the set

which cao be employed as a set of orthogonal curvilinear coordi-nates. Consequently, the forniulatioOs derived in Chapter 4 are applicableand the notation = fi, = will-be used whenever convenient.From Eq. (9.9), we have the local base vectors as

= = 4—

g2 =x =—

g3 — = (9.10)

The position vector connecting two neighboring points fi, and+dc,fi + dfl, C+dC)is -

= A(1 __k-)bOdp+'nwdC,

and its length, denoted by is given by -

3

= gfr(O) =Au-I

where

/ / C'2——-.--,), ——,)

g33 = 1, g23 g31 = g12 = 0. (9.13)

The volume of an infinitesimal parallelepiped, enclosed by the six surfaces:= const, j9 const, = const, + const, + = const,const, is given by

dV=AB(l (9.14)

For later convenience, we shall locate a system of local rectangular Cartesiancoordinates (y'. y2, y3) at the point F"", where the directions of the coordi-

SHELLS, 185

(9.16)

nate axes are taken in coincidence with the unit vectors b'°' and fl(s)at the respectively, as shown in Fig. 9.2. Then, from Eqs. (4.57)and (9.10), we obtain the following geometrical relations:

A(1 — a'y2 = B(l — dy3 = dC. (9.15)

a

Fto. 9.2. A shell element.

Let us now consider the side surface of the shell. It is assumed that themiddle surface is simply connected and that the side surface, denoted by S,is generated by the envelope of normals drawn perpendicular to the middlesurface S,. Let the intersection Curve betWeen Sm and S be denoted by C,and Jet the unit vector drawn outwards on C and perpendicular to S bedenoted by v, as shown in Fig. 9.3. Then, we have the area of an infinitesimalrectangle on S as

dS 1/[m(i +


where / = a' and m a', and s is measured along the curve C. Wehave the following relations on the boundary C:

S

Bd13=±Id.s, (9.17)

as shown in Fig. 9.4. If the direction of increasing s is chosen in such a waythat the local vectors a', s and form a right-hand system, as shown inFig. 9.3, we have

a a a i a a a

where s is the unit tangent vector in the direction of increasing s.

FIG. 9.4. Geometrical relations on the boundary C.

(9.18)

Fio. 9.3. Directions of a', s and

SHELLS 187

9.2. Analysis of Strain

The shell is now assumed to be subject to deformation. The point is

displaced to a new position F, whose position vector is given by

r = + u, (9.19)

where the displacement vector is is a function of(oi. and its components;

denoted by u, v and w, are defined in the directions of and

U = -F + (9.20)

By the use of Eqs. (4.36), (9.9), (9.19) and (9.20), we can calculate the

strains4, defined with respect to the fi, coordinate system in terms of

u, y and w. Then, the strains defined with respect to the (y' y2' .v3)coordinate system may be obtained using the transformation law (4.61)

and the geometrical relations (9.15):

—

_________

— A2(1 — — B2(l —

—, 921— AB(l — —

— .fac — 10.

— A(1 — — B(1 —

it is obvious that the linearized strains (es, c8, ...) defined with respect to

the (y1, y2, y3) system are obtainable from Eqs. (9.21) by ineariz-ing the strains with respect to the displacements.

•We shall now introduce two assumptions for thin shell theories whith willbe developed in the present chapter. First, we assume that transverse normalstress ac is small comparçd to other stress components and may be set

= 0 (9.22)

in the stress—strain relations, thus obtaining

£ E(I — v2)

(ti. + vet,,),(I — v2)

(i.t,, +

= = (9.23)

for linearized theories and

E E-

(J — ,2) + vepp),(I — v2)

+ epp},

= = 2Ge1c, risc = (9.24)

for nonlinear theories. where the stresses as well as the strains are definedwith respect to the local rectangular Cartesian coordinates. Second, we assumethat the displacement vector is is approximated by

u = u0 + (9.25)


where u0 and a1 are functions of (tz, fi) only, their components being definedby

u0 = + ÷ (9.26)a1 = + + (9.27)

and where a geometrical constraint is introduced by requiring that(9.28)

It is seen that Eq. (9.25) under the constraint condition (9.28) provides oneof the simplest expressions for displacements including the effect of trans-verse shear deformation (see Section 8.8 for a similar development).

• We may obtain strain—displacement relations for finite displacementtheory by substituting Eq. (9.25) into Eq. (9.19) and following familiarprocedures. However, we shall be satisfied with deriving those for smalldisplacement theory in the remain&r of this section. To begin with, wefind that due to the assumption of small displacement, Eq. (9.28) is linearizedwith respect to the displacements to yield

w1 = 0, (9.29)

and consequently, Eqs. (9.25) reduce tou=u+Cu1, v=o+Cv1, w=w. (9.30)

Combining Eqs. (9.19), (9.20) and (9.30), we obtainr = + (u + + (v + ÷ . (9.31)

It is seen that Eqs. (9.30) are natural extensions of Eqs. (8.99) of the thinplate to the thin shell.

Next, we shall obtain strain—displacement laTtions. FromE4. (9.31), wehave the blowing relations:

= —+ + m1 + ('21 + m21C)

+ (/3J + m3 1ç')13r Fl

= + +—

+ + m22Cj b

+ (132 +

= + + (9.32)

where it is defined thatlau v ô4 w - lau v8B

1 u I u 931!3

li3w u v

SHELLS 189

and1 0u1 v1 3A I v1

I U1 I u1 t3Bm21 = - m22 = -ff +

-h--,

Ut V1m3 m32

=

By the use of Eqs. (9.9) and (9.31), and with the aid of the abovethe strains4, can be calculated. The strains thus obtained are then linearizedwith respect to the displacements and substituted into Eqs. (9.21) to obtainfollowing strain—displacement relations for a small displacement theoryincluding the effect of transverse shear deformation:

— — —

— 1 — ' — I— — +

9 35(1 — (I — '-

. ( . )

— - —

— I — ' — 1 —

where -

= €8o = = '12 + '21' (9.36)

k,, = —m11, = —m22,

12; 21= —m21 — m12 + -s-- +I%ft

— m12 m21— 0 — 0 '"p

= U1 + '31, = + '32- (9.38)

9.3. AnalySiS of Sfraln under the Kirchhoff-Love Hypothesis

The analysis of strains including the effect of transverse shear deformationhas beeii made in the last section. We shall now proceed to the analysis ofstrain under the Kirchhoff—Love hypothesis that the straight fibres theshell which are perpendicular to middle surface before deformation re-main straight and perpendicular to the deformed middle surface and suffer noextensions.t This is an extension of the Kirchhoff hypothesis for thin platesto thin shells. We observe that a shell theory under the hypothesis is aspecial case of the theory based on Eqs. (9.25) and (9.28).

t The Kirchhoff—Love hypothesis is usually understood to include the assumption= 0 as well. However, only the hypothesis described here will be called the KIrchhoff—

Love in this book.


We consider an arbitrary point of the middle surface having the coordi-nates (x, fi, 0) before deformation, and denote its position vectors before andafteri deformation by and r0, respectively, which arc related to the dis-placement vector u0 introduced in Eq. (9.25) by

r0 = ÷ u0. (9.39)

The.u, it is seen that the hypothesis allows us to express r as

r = r0 + (9A0)

where us a unit normal to the deformed middle surface and is given byOro

(941N

Since

= 40) + + + (9.42)

we may express n in terms of u, v and w as follows:N (0)+ + U

(943)

where

L = 131 + 121132 — '22131,

1*1 = '32 + '12/31 — /11/32, (9.44)

N = 1 + + +111122 — '12'21

From Eqs. (99), (9.19), (9.39) and (9.40), we obtain

-. ii = u0 + — n0). (9.45)

We observe from Eqs. (9.25) and (9.45) that the hypothesis imposes thefollowiu&condition on lii:

= — (9.46)

and reduces the freedom bf'shell deformation to u, v and w only.When a shell problem is confined to a small displacement theory, Eq.

to

U1 = '31, V1 132. (9.47)

Thus, the allows us to eipress the displacement components as

u=u—131C, (9.48)

and the strain-displacement relations as

— — — —

1 — ' — 1 — '

—. — + C2x(1 — — a/RD) '

SHELLS 191

where= —m11, = —m22,

(9.50)

• — — m12 m21— D

— 1 0131 132 I14 - • 1 0131 '32-m11 +- -

— -y-,— 1 0132 OA 1 OR

— m210 — TITI

—— op

+ 71 (9.51)

We note here several formulae which will be useful in subsequent formula-tions:

/12 '21(9.52)

121 + Cth21 '12 + Cm12+ ' 5

= (-k + -k) —(9.54)

where the conditions of Codazzi are used for the proof of Eqs. (9.52).

9'4. A linearized Ildi Shell Theory the Hypothesis

We shall begin this section by prescribing the following thin shell problem.The body forces, together with the forces applied on the upper 'and lowersurfaces of the shell, are defined per unit area of tbe middle surface S11, and'their components are defined by . -

V +' + ' (9.55)

The external force P is applied on the, part S1 ol" the siae boundary S andits components are defined by

= + + (9.56)

The displacement components are prescribed on the remaining part ofthe side boundary S.

We shall a linearized thin shell theory for this problem under theKirchboff-Love hypothesis. The principle of virtual work for this problc!nmaybcwrittenas:

— ff(Fmöu + F 6w) dS = 0, (9.57)1'

t See Eq. (4.84).


where Eqs. (9.48) and as well as the geometrical relations (9.14) and(9.16) have been substituted. Before proceeding to the reduction of Eqs.(9.57), we shall introduce the following notation for the s$ress resultants:

= — —,/ \

(9.58)

= (i—

= [a, (1—

dC,

M, = (i— —i..)

= fTp. (i—

(9.59)

ABC. = ÷ + —

= + + — Mt,, (9.60)

= Ac! + Npam, = Nd + Npm, (9.61)

= M31 + Mq, = + M,m,

= + M,, = — + M,,!, (9.62)

V,, = + . (9.63)

and

= J = fF,HdC. 1', =

f = f (9.64)

= Mj + M,. = — +

Here 11(C) is obtained from Eq. (9.16) as

H(C) = }'[m(l — + 11(1 — C/R,)12. (9.65)

In Eqs; (9.58), an4 (9.64), as well as throughout the present section,the intçgrations with respect to extend from = —h/2 to = h/2, where hdenotes the thickness of the shell. The quantities defined by €qs. (9.58) and(9.59) are stress resultants and moments per unit length of the coordinatecurves and fi of the middle surface as shown in Fig. 9.5. The quantities

and are in-plane stress resultants, while M,, M4 and

are bending, and twisting moments. The quantities and Qp defined by

Eqs. (9.60) are proved equal to shearing and Q,, defined per unitlength of: the curves and fi of the middle swfaçe by considering the equili-brium coöditions with respect to moments of the shell element in the figure.tThe quantities defined by Eqs. (9.64) are prescribed external forces and

t the footnote of Section 7.2 and Section 8.2 for similar developments.

SHELLS 193

moments per unit length along the boundary. It is seen that is the shear-ing force acting in the direction of the normal while and arebending and twisting moments on the boundary. With the aid of theserelations, we have

+ + — —

+ + + +

+

+Cl .C2

Substituting Eq. (9.66) into Eq. (9.57), and using Eqs. (9.18), we obtainfinally,

— If I+ —

+ jôv + + ++

+ ?iIAB] owj dfi

—— (N.

—+

+ [(Va + — (P., + A?,. bw — (Al, — ow 4 di

+ (integrals on C2) = 0. (9.67)

ax

(9.66)

aD

FIG. 9.5. Stress resultants.

- +

194 VARIATIONAL METhODS IN ELASTICITY AND PLASTICiTY

Consequently, the equations of equilibrium are:

+ + - +. =0,

+ =O,(9.68)

The boundary conditions on C1 become:

m.— — , iYp, — —

I'p -

v,, + = v• + M, 2,. (9.69 a, b, c, d)

Equation (9.69c) shows that under the Kirchhoff—Love hypothesis, theactions of the twisting moments M;, and along the boundaryare replaced by that of the shearing forces and respectively. Thisresult is similar to the result we encountered in Section 8.2. Equation (9.67)also suggests that in this approximate theory, the boundary conditions onC2 are specified as follows:

- - 8wu=u, v=v, w=w, (9.70)

we shall obtain relations between the stress resultants and strains.We may combine Eqs. (9.23), (9.49), (9.58) and (9.59) to yield the followingrelations:

NE f [ + — —

(1 — _:211_* +(9.71)

Since the exact evaluation of these integrals lead to very cumbersome stressresultant—strainrelations, Lur'e proposed to expand the integrands of theintegraTs in Eqs. (9.71) in power series of and discard terms higher thanbefore integration, thus

Eh r h2 I I e0 I

= 2 + + (-i- —(j_

—(—3')'.

Eh I h211 l\fEpo= (1— ,2) + +

Gh [VdrPO + (* -k) -(9.72)

SHELLS

I I 1M= —D

= — D(l — v)—

____

= — D(l —— (9.73)

where B = Eh3f12(1 — v2) is the *nding rigidity of the shell. The strainenergy of the shell is calculable with the aid of Eqs. (9.49) and Eq. (3) ofAppendix B, and is expressed in the accuracy of Lur'e's approximation asfollows:

U = ff + +S.,,

— — — 2Mpx,,J AB dfl, (9.74)

where Eqs. (9.72), (9.73) and 0 -

=[Cu 2J[Y40]

1

(9.75)

= Gh [i +— RaRp +

C12 = C21 = D(l—

÷ (9.76)

C22 —v),

have been substituted to express the stress resultants in terms of the dis-placements. We note that the accuracy of Eqs. (9.72) and (9.73) is found anapparent one if we consider the assumptions and hypothesis on which thepresent thin shell theory is based 2) the terms containing ,h2J12 in Eqs.(9.72) and those containing or in Eqs. (9.73) are usually verysmall and may be neglected in comparison with the preceding terms. Conse-quently, these relations have been used very rarely in their original form forpractical purposes.

9.$. Simplified Formulations

Since the thin shell theory derived Section 9.4 leads to rather cum-bersome formulations of the shell problem, we shall be interested in derivingsimpler. formulations in the remainder of this chapter. We shall adopt asimplifying assumption that the shell is so thin that terms of smaller magni-tude may be neglected in geometrical as well as strain—displacement rela-tions. The quantities and h/.Rft may be considered negligible in corn-


parison with unity under this simplifying assumption. To begin with, weadmit that Eqs. (9.14) and (9.16) may reduce tQ

dV= (9.77)

(9.78).

respectvely. Tue factor in Eqs. (9.64) may be taken equal to unity dueto the same reason.

Next, we shall proceed to the redtiction of the strain—displacement rela-tions to simpler form and summarize results of some considerations asfollows:

(a) Linearized thin shell theory including the effect of transverse shear defor-motion. We allow that Eqs. (9.35) reduce to

= — = — Ck,,

= —

= = Ypco' (9.79)

while the displacement components are still given by Eqs. (9.30).(b) Linearized thin shell theory under the Kirchhoff—Love hypothesis. We

allow that Eqs. (9.49) reduce to

= — El, = Epo —

= — (9.80)

while the displacement components are still given by Eqs. (9.48).We shall derive in this connection nonlinear strain-displacement relations

under the Kirchhoff.-Love hypothesis plus the simplifying assumption. Theexact relations may be obtained by employing Eq. (9.45) as an expressionfor the displacements and calculating the strain tensors in a manner similarto the development in Section 9.2. However, we shall be satisfied with ob-taining approximate strain—displacement relations: we shall retain nonlinearterms in the expressions of the Strains of the middle surface, but we Shallretain only linear terms in the formulation of curvatures. Thus, we allowthat the nonlinear strain—displacement relations reduce to

= — = e1,1,0 —

= (9.81)where

= (1 + /)2 + + I,= + (1 + /22)2 + — 1, (9.82)

= (1 + '11)/12 + /21(1 + 122) + 13S32,

while the displacements are still given by Eqs. (9.48). it is obvious that thelinearization of the curvature terms in Eqs. (9.81) and the use of Eqs. (9.48)restrict the field of application of a nonlinear theory founded on theserelations. However, this choice is considered useful in application to shell

SHELLS 197

problems such as buckling or vibrational problems of shells where smalldisplacement motions arc executed about equilibrium configurations withinitial membrane stresses. See Ref. 3 for more detailed considerations onfinite displacement theories of thin shells.

9.6. A Simplified Linear Theory uwder the Kircbhoff-Love Hypothesis

We shall consider again the thin shell problem presented in Section 9.4and derive for it a linear theory by the use of Eqs. (9.48), (9.77), (9.78) and(9.80). The principle of virtual work is employed for the derivation of govern-ing equations. The principle suggests the adoption of the following definitionof the stress resultants:

= f = f dC,

= f = f (9.83)t

M,, f dC, M8 = f= f = f dC, (9.84)t

N—S N—S 985

After sothe calculation, we find that in this simplified linear theory of thethin shell, the equations of equilibrium as well as the mechanical and geo-metrical boundary conditions are derived. in the same• form as those inSection 9.4. However, the stress resultant—strain relations and the expressionof the strain energy of the shell are now given in simpler form as follows:

Eh Eh

= (1 —+ vepoj,

(1 — v2)+

= = .

M = —D(x.., + —D[vx3 +— ii rut i— — F+

= — = Gh[y,,,0 +

= -- MflJR, = +

U= ffj + + 2(1

—

+ D[(,ç, + xp)2 + 2(1 ——

(9.8'9)

t Compare these equations with Eqs. (9.58) and (9.59).Compare these equations with Eqs. (9.72) and (9.73).

198 - VAlUATIONAL METHODS iN ELASTICITY ANt) PLASTICITY

We note here that the second terms in tke right-hand sides of Eqs. (9.88)are frequently neglected in comparison the first terms to obtain thefollowing simpler relations:

=

The use of Eqs. (9.90) for practical purposes may be considered justifiedif we remember the assumptions on which the present thin shell theory isbased. However, Eqs. (9.88) are employed for theoretical presentationsbecause this choice is consistent with the results derived from the principleof virtual work or the principle of minimum potential energy.

9.7. A Nonlinear Thin Shell Theory under the Kfrcbhoff-Love

We shall consider the thin shell problem presented in Section 9.4 andderive for it a nonlinear theory under the Kirchhoff—Love hypothesis bythe use of Eqs. (9.81). The principle of vfrtual work for the present probkmmay be written as

+ +

Sm

—ff(Fou+Fpôv+u,ôw)dS—0.

where Eqs. (9.48), (9.77), (9.78) and (9.81) have been substituted. With theaid of the stress resultants defined by Eqs. (9.83), (9.84) and (9.85), wehave

+ +

+ N3111.+ + + SpJ11 + N/12)o112

+ + NJ21 + ö!21 + (Np + $pJai + Np122) ö122+ + NJ31 + + (a,, + + ô!32) dfl

— f [Me, f313 1 + ô132) dS. . (9.92)cI+c2 .

Comparing Eq. (9.92) with Eq. (9.66), we find that the following replace-ments yield the desired equations of equilibrium and mechanical boundaryconditions for the nonlinear theory:

by + NJ1 + by + +by + NJ21 + by + +by + NJ31 + by + SpJ31 +

(9.93)f Compare these equations with Eqs. (9.72) and (9.73).

See Eq. (4.84).

SHELLS 199

we have for the equations of equilibrium in Sm,

4- + NJ11 + Sd12]) + + +

•''+ + l'1J21 + Sd22] — + SPa!2 +

?aAB=O,

{ALNp + SpJ21 + N,!32]) +4- + NJ21 ÷ Sap122])

+ + SpJ11 + — + Na/u + Sap112]

+ NJ31 + Sd32]) + + + N,!32])

+ SJ1,J + + Np!23]

I

(9.94)

and for the mechanical boundary.conditions on C1,

+ NJ11 + Sap'i 2V+ ÷ SpJ11 + m — = R,, —

[Nap + NJ21 + SapI22]! -I- [Np + SpJ31 + — = N,, —

LQa+ NJ31 + Sap!32J1 + + SpJ35 + Np!32] pt + = +

(9.95)

The boundary are 'still given by Eqs. (9.70). Thestress resultant—strain relations and'1the expression for the strain energy ofthe nonlinear theory are obtainedkbm Rqs.(9.86) (9.87), (9.88) and (9.89)by replacing taO, EpO and respectivcly.t

9.8. A Llne*rlzed Thia Shel the Effect

We shall consider again the problem prescribed in Section 9.4 andfor it a linearized thin shell theory

in which all thO proscribed forces and boundary conditions are

t As mentioned at the end of the last section, we may replace S,p and in Eqs. (9.94)and (9.95) by Nap and Np,, respectively, and employ the simpler relations Nap NPa= Sap 2Gheapo for practical purposes.

200 VARIATIONAL METHODS IN ELASTICITY ANT) PLASTICITY

now assumed to be time-dependent. The principle of virtual work for thepresent dynamical problem may be written as

f + + ± + ôvxl dV

_offf+e(u2 + + w2)dV

— ff (F5 ôu + ôv + F,, i3wJ dSl dt = 0. (9.96)tSt

We shall adopt the simplifying assumption introduced in Section 9.5 andemploy Eqs. (9.30), (9.77), (9.78) and (9.79) for the formulation. Here, weintroduce the following new definitions in addition to Eqs. (9.83), (9.84)and (9.85):

Q,, = Q, (9.97)

m = I,,, = --Qh3AB. (9.98)

The quantities defined by Eqs. (9.97) are shearing forces per unit length ofthe coordinate curves of the middle surface as shown in Fig. 9.5. The quan-tities defined by Eqs. (9.98) are related to tlu mass and mass moment ofinertia of the shell element shown in the same figure. With the aid of the stressresultants thus defined, we have -

fff[oy3e3 + + + +

= ff [N, 5/)1 + N4 d121 + 5131 +SI"

÷ 5112 + + 6132 + Sv1

+ M3 Sm11 + + + Sm22] (9.99)

Substituting Eq. (9.99) into Eqs. (9.96) and using Eqs. (9.30) for the dis-placement components, we have the equations of motion,

+ + N4 — — ÷ ?3AB = mu,

+ + ±/. — N3 — + =

+ + + ?,AB = mw,

(BM5)+ * (A + M4 — — A BQ,, =

(AMp) + + i—. Mp3 — M3 — ABQp = ImVl,

t See Eq. (5.81).

SHRLL.R 201

and the boundary conditions on C1,

N,, QJ+Qpm P.,(9.101)

while it is suggested that the boundary conditions on C2 are specified approxi-mately by:

u=i, v=i,, u1=ü1, v,=U1 (9.102)

The stress resultant—stEam relations and the expression of the strain energyare obtained from.Eqs. (9.23), (9.79); (9.83), (9.84), (9.85) and (9.97), togetherwith the aid of Eq. (3) in Appendix B, as follows:

Eh EhN,, (1 — ,2) + (1 — ,2) (yr10 +

= = (9.103)= —D(lç + + ICR),

= = — D(l — ,) . (9.l04)h2

= + Gh[Ymøo+ (9.105)

Q,, = = (9.106)

u= ,2) + 2(1 — —

+ + 2(1 — — kjcp)]

+ -1- AR (9.107)

The factor k in Eqs. (9.106) and (9.I01)bas been appended to take accountof the non-uniformity of the shearingatrains and over the cross sec-tion. For isotropic shell, the factor k may as mentionedin Section

It is seen from the above formulatlonthat we have five mechanical bouflä-ary conditions on C1 and the same.number of geometrical boundary con-ditions on C2 and compatible with the assumed degree of freedomof the displacement components, is, w, u1 and v1. We have replacedthe action of M,, and Ak,, by that of and in the thin shell theories underthe Kirchhoff—Love hypothesis. However, such replacements are no longernecessary in the thin shell theory includingthe effect of transverse shear de-formation. This is similar to the result we encountered ii) Section 8.8.

9.9. Some Remarks

Sinoe5the of Love's approximate theory, many books havebeen on theories of thin shells (see Refs. I through 11, for example).Many papers concerning shell problems have been published; an extensive


bibliography is given in Ref. 12, while Ref. 13 presents a survey of progresson 1this topic. Theories of thin shells have been proposed by many authorsin the literature, and some discrepancies have been found among them.A comparison of various theories was made in Ref's. 9, 14, 15 and 16.

The thin shell theories developed in thischapter are based on the assump-tions postulated in Section 9.2. As we remarked in Section 8.10, the simulta-neous use of the first and second assumptions mentioned in Section 9.2may lead to an inconsistency in the stress-stiain relations. For improve-ment of the accuracy of thin shell theories, as well as for the completeremoval of the inconsistency, it is necessaiy to abandon the assumptionsand to assume the displacement components as

u = v = fi) w = fl)C. (9.108)m-O m-O

where the number of tepus must be chosen properly.Hildebrand, Reissner and Thomas have presente& a theory of

in which u, v and w are approximated by quadratic functions with respectto namely m = 2 in Eqs. (9.108). Nagbdi has the approximation

u u0 + v = + w = iv0 + + (9.109)

and has developed a theory by the use of Reissner's variational prin-19) He applied his theory to of wave propagation in

cylindrical and concluded that the form of the displacements inEqs. (9.30) needs no improvements, if a theoqis sought in which onlytransverse shear deformation and rotaryinertia effects are Wenote here two papers which arc Mated to the variational formulation ofthin shell

the Rayleigh-Ritz method have provided powerful thols for SoJ!ipg shell problems approxi-mately (see Refs. 23 through 26, for A theory of thin cylindricalshells was proposed by has used for analyz-ing problems of thin cylindrical shells, The of buckling andpost-buckling behavior of shells have been two of the central problems inshell theory.t28• A snap-through theory was by andTsien br the buckling of cyliildrical and sphcrkal shells.'50' 31. 32) As otherproblems of great engineering we mention thermal stressesand thermal buckling of shells,'33' end shell vibrations.U6. 35. 36. 37)

1. V. V. NovozmLov, The Theory of Thin Shelfr, by B. G. Lowe, P. Noord-.hoff Ltd., Groningen, Netherlandi, 1959.

2. A. L. Theory of Elastic Thin Shells, Translated by G. Hermann,Prea, 1961.

3. V. V. Novozuu.ov, Foundations of the Nonlinear Theory of Elasticity. Graylock, ?953.

SHELLS 203

P. M. N*oaDi, Foundations of Elastic Shell Theory, Progress in Solid Mechanics,echted by 1. N. Sneddon and R. Hill, Vol. IV, Chapter 1, North-Holland, 1963.

5. A. E. H. LovE, Mathematical Theory of Elasticity, Cambridge University Press, 4thedition, 1927.

6. S. Tn1osHENKo and S. Theory of Plates and Shells, McGraw-Hill, 1959.

7. W. FL000S, Statik wtd Dynamik tfrr Schalen, Springer Verlag 1934.8. A. B. and W. Theoreiicaj.Elasticity, Oxford University Press, 1954.9. W. S. Ailgemeine Schalen-Theorie mid ihre Anwendw*gen In der Technik,

Akademie-Verlag,, 1958.10. W. FLUOGE, Stresses in Shells, Springer Verlag, 1960.II. Kit. M. Musruu and K. Z. GMOV, No,s-linèar Theory of Thin Elastic Shells, Trans-

lated by 3. Morgenstern, J. 3. Schorr-Kon and PST Staff, Israel Program for Scienti-fic Translations Ltd., 1962.

12. W. A. Bibliography on Shells and Shell-like Structures, David Taylor ModelBasin Report 863,1954. Bibliography on Shells and Shell-like Structanes (1954-1956).Engineering and Industrial Experimental Station, University of Florida, 1957.

13. P. M. NAGUDI, A Survey of Recad Progress in the Theory of Elastic Shells,Mechmkj Reviews, Vol.9, No.9, pp. 365-8, September 1956. -.

14. W. T. A Consistent First Approximation in the General Theory ciaElastic Shells, Proceedings of the Symposiwn ow the Theory of llthi Elastie Shell.

J.U.T.A.M., Delft, pp. 12-33, North-Holland, Amsterdam, 1960.15. D. S. HovolrroN and D. J. JoHNs,, A Cbmparis.n ci the Characteristic Equatioua in

the Theory of Circular Cylindrical Sheffs, The Aer lQnarterly, Vol. 12,Pmrt 3,pp. 228-36, August 1961.

16. R. L. BISPLINGHOFP and H. ASHLEY, Principles of Ae,oelostlciiy, John Wiley, 1962.17. F. B. HILDEBRAND, E. RvssNut and 0. B. THOMAs, Notes on the Fbuwiatkiu, of the

Theory of SmqII Displacements of Orthoiropic Shells, NACA Th 1833, 1949.18. P. M. NAGHDI, Qu the Theory of Thin Elastic Shells, Quarterly of Applles( Mathema-

tics, Vol. ,14, No. 4, pp. 369—80, January 1957.P. M. NAOHDI, The Effect of Transyersc Shear Deformation on the BendingShells of Revolution, Quarterly of Applied Mathematics, Vol. 15, No. 1, pp. 41—52,April 1957.

20. P. M. NAoNDI and P. M. COOPER, Propagation of Elastic Waves in Cylindrical Shells,including the Effects of Transverse Shear and Rotary Inertia, Journal of Acoualcal,,Society of America, Vol. 28, No. 1, pp. 56-63, January 1956.

21. E. Ableitung der Schalenbiegungsgleichungen mit dem CastiglianoathenPrinzip, Zeitschr,fr für Angewandie Mathemasik aid Meckmlk, Vol. 15, No. 112,pp. 101-8, February 1935.

22. E. Variational Considerations for Elastic Beams and Shells, Jourr4,J of theEngineering Mechanics Divalon, Proceedings of the American Society ofCl.1JF4vJ,,eer:,Vol. 88, No. EM 1, pp. 23—57, February 1962.

23. R. SCHMiDT and G. A. The Nonlinear Conical Spring, Trtzissactlo.u of theAmerican Society for Mechanical Engineers, Series B, Vol. 26, No.4, pp.681-2, De-cember P1959.

24. N. C. DAHL, Toroidal-Sheft Expansion Joints, Journal of Applied Meehanlcz, Vol. 20,No. 4, pp. 497—503, December 1953.

25. C. E. TURNER and H. Foiw, Stress and Deflection Studies of Pipejinc EspansiónBellows, Proceedings of the Institute of Mechanical Engineers, Vol. 171, No. 15,pp. 526-52, 1957.

26. P.O. KAsI* and M. B. Stiffness of Curved Circular Tubes with InternalPressure, Journal of Applied Mechanics, Vol. 23, No. 2, pp. 247-54, June 1956.


27. L. H. DONNEL, A New Theory for the Buckling of Thin Cylinders under Axial Com-pression and Bending, Transactions of American Society for Mechanicoi Engineers,Vol. 56, No. II, pp. 795—806, November 1934.

28. S. TIMOSHENKO, Theory of Elastic Stability, McGraw-Hill, 1936.29. H. L. LANGHAAR, General Theory of Buckling, Applied Mechanics Review, Vol. II,

No. Il, pp. 585-8, November 1958.30. T. VON KARMAN and H. S. TSIEN, The Buckling of Sphe,5cal Shells by External Pres-

sure, Journal of the Aeronautical Sciences, Vol. 7, No. 2, pp. 43—50, December 1939.31. T. VON and H. S. TSIEN, The Buckling of Thin Cylindrical Shells Under

Axial Compression, Journal of the Aeronautical Sciences, Vol. 8, No. 8, pp. 303-12,June 1941.

32. H:S. TSIEN, A Theory for the Buckling of Thin Shells, Journal of the AeronauticalSciences, Vol. 9, No. 10, pp. 373—83, August 1942.

33. N. I. HOFF, Buckling of Thin Cylindrical Shell under Hoop Stresses Varying in AxialDirection, Journal of Applied Mechanics, Vol. 24, No. 3, pp. 405—12, September 1957.

34. D. J. Jom4s, D. S. }joucniToN and J. P. H. WEBBER, Buckling due to Thermal Stressof Cylindrical Shells subjected to Axial Temperature Distributions, College of Aero-nautics, Cranfield, CoA Report No. 147, 1961.

35. R. N. AitNou) and 0. B. WARBURTON, The Flexural Vibrations of Thin Cylinders,Proceedings of the Institute of Mechanical Engineers, Vol. 167, No. 1, pp. 62—74, 1953.

36. J. B. BERRY and E. REISSNER, The Effect of an Internal Compressible Fluid Column onthe Breathing Vibrations of a Thin Cylindrical Shell, Journal of theAeronautical Sciences, Vol. 25, No. 5, pp. 288—94, May 1958.

37. J. S. MJXSON and R. W. HERa, An Investigation of the Vibration Characteristics ofPressurized Thin-walled Circular Cylinders Partly Filled with Liquid, NASA TR R—145,1962.

CHAPTER 10

STRUCTURES

10.1. Rediuidaacy

Thus far, variathinal formulations have been developed for simply connect-ed, continuous bodies, the torsion of a bar with a hole treated in Section 6.3being the only exceptionJt will be shown in the present chapter that theseformulations are applicable, withstight modifications, to structured: multiplyconnected continuous bodies built up from basic members or components.For the sake of simplicity, we shall restrict the investigation to thedisplacement theory of structure.

We shall assume that a structure under' can be fictitiouslysplit into a number of simply connected members, the deformation charac-teristics of which have been deth,d with the aid of methods of analysis forsimply connected entire structure are thenreduced to the determination internal force existing at the joints of thesemembers and at the points at which the is supported.

A structure is called redundant or statically indeterminate if the equationsof equilibrium are not sufficient for the determination Of all the internalforces: the degree of redundancy is then the difference between the numberof unknown internal forces arid the numbe* of independent equations ofequilibrium for the structure. Ø this terminology, structuresshould be treated, in general, as multiply connected, continuous bodieswith infinite redundancy., of structures would lead 'toformidable calculations. However, experimental evidence and design cx-,perience have shown that we are justified in. simplifying our analysis ofstructures by approximating the deformations of the members by finitedegree of freedom systems. In other words, structures may be treated asbodies with finite redundancy Wider Wccial circumstances.

Both trusses and frames are of structures in which such simplifi-cations are permitted. All the of a truss are assumed to be pin-jointed and capable of transmitting or compressional axial loadsonly, while frame members are be capable of transmitting axial,bending and torsional forces and moments. In order to make such simpliflca-tions valid, each member must be slender and judiciously connected at thejoints, and externat forces must' be applied. Structures such as


trusses or frames are sometimes called lumped parameter circuits by ana-logy with their electric counterparts.

The principle of virtual work and related variational principles have beenfound extremely effective in analyses of such simplified structures. Anapproach using the principle of minimUm potential energy is usually calleda displacement method,t while another using the principle of minimumcomplementary energy is called a force method.t These two methods havebeen guiding principles for analyses of structures. Due to the limited spaceavailable, we shall concentrate mainly on the analysis of trusses and frames,and put emphasis on the variational formulations. For practical details ofnumerical examples and applications to other structures, we shall be satis-fied withlisting related books in the bibliography for the reader's reference(Refs. 1 through 14).

10.2. Deformation Characteristics of a Trues Member and Presentation of aProblem

We shall consider a truss member under end forces F, as shown in Fig. 10.1,and assume that the end force—elongation relation has been obtained:

(10.1)or conversely = 6(r). (10.2)

The elongation 6 may be considered to be the displacement of one end ofthe member in the direction of the end force P while the other end is fixed.The strain and complementary energies stored in the member are given by

and

U=fP(6)d6 (10.3)

V f6(P)dP. (10.4)

respectively. For an elastic member with uniform cross sectional area A0and original length 1, we have

p = 6, (10.5)

= EA0(10.6)

U = (10.7)

2EAQP2. (10.8)

f They arc also called the stiffness method and the flexibility method. respecti%tly.

STRUCTURES 207

It is obvious that we have the following relations from Eqs. (10.3) and(10.4):

= P. = ô. (10.9), (10.10)

When the bar is oriented arbitrarily with respect to a set of Cattesianreference axes, we require a relation between its elongation and its dis-placements. We shall denote the position vectors of the two ends of thebar before and after deformation by and r1, r2 respectively, whichare related to the displacement vectors, of the two ends of the bar, u1 andu2, by

= ÷ u1, r2 = rr + u2. (10.11)

*d d

Fia. 10.1. Truss member loaded by end forces.

Then, the elongation of the bar, ô12, is given in terms of the displacementsas follows:

— 1r2 — ru — —

= (u2 — u1) —— (10.12)

where higher order orms are neglected due to the assumption of smalldisplacement: .1 , •

Let us nin where rectangular Cartesian coordinates

will be used as a reference system. Let the truss joints be denoted by i;i 1, 2, ..., n and let a which two joints i and j be repre-sented by a double suffix Li; :j = 1, 2, ..., m. The direction cosines of theu-th member before deformation are denoted by A,,, i',1, where the direc-tion from the j-th joint to the i-th point is taken positive. Obviously:

= —A11, = —4U11, Vgj = (10.13)

Denoting the elongation of the y-th member by a,, and the displacementcomponent of the i-th and j-th joints by u,, v1, w, and Uj, Vj, Wj respectively,and using Eq. (10.12), we have the elongation—displacement rela-tions:

= — ÷ (v, — v,)4u,, + (w1 — w,) v,1. (10.14)t

We shall specify the boundary conditions for the truss structure as follows.All the external forces acting on the truss structure are applied at k joints

t The symbol ôg, used throughout the present chapter should not be confused withthe Kronecker symbol defined in the preceding chapters.


of the total is:= Il Z,

(I = 1, 2, ..., k), .(10.15)t

as ahOwn in Fig. 10.2, where -

= = Z1 = 2J (10.16)I I I

In E4s. (10.16), is the internal end force of the y-th member, and thesun4mation with respect toj is taken over all the members which are directlyconnected to the i-tb joint. For the sake of we assume that thetruss structure is rigidly fixed at the remaining (is — k) joints:

u1.=O, v,=0, w4=O, (i=k+1,...,n).In order to simplify the problem further, we assume that the geometricalboundary conditions are sufficient to let the external forces be independentof each other.

X1i+Y1j+z1k

Fso. 10.2. I-th joint and LI-th member.

We assume that the internal force—elongation relations have been obtainedfor each member as

(10.18)or conversely

6,, = (10.19)

t It is not necessary that all the three components of the external force or of the dis-placement be prescribed at a joint. Only the complementary relations between Zand ii,, and 1,, and w1 are necessary. However, we have prescribed our problem asabove to simplify the subsequent formulations.

An extension to a truss problem in which the geometrical boundary conditions aregiven by ü,, v, = w1 (I = k + I, ..., n) is straightforward.

PL i-I

U thmember

Pu

STRUCFURES 209

Conibining Eqs. (10.14) and (10.18) with the boundary conditions (10.15)and (10.17), we have the necessary and sufficient number of equations fordetermining (2m + 3k) unknown quantities F,,, u,, v, and w, where

10.3. VariatIonal Formulations of the Truss Problem

We shall consider the principle of virtual work for the truss problem.Denoting the virtual displacements of the i-th joint by ãu,, and andusing Eqs. (10.15), we have

k— + (1's — 71)ôv, + (Z, — Zjôw,] = 0. (10.20)

I_i

Then recalling Eqs. (10.17), we can transform Eq. (10.20) into:in k

4- + Zgôwg) = 0, (10.21)v_i

where Eq. (10.14) has been substituted. This is the principle of virtual workfor the truss problem. . -

The principle (10.21) suggests that the function for the principle ofminimum potential energy of the truss problem is given' by

in k.17 = Ugj(ô,j) — £ + + Z1w,), (10.22)

u_i . s—iS

where'

=1 (10.23)

and where Eq. (10.14) has been substituted. In the above, the quantitiessubject to vanatio'n are u1, and w4 under the subsidiary conditions (10.17).The function (10.22) may be transformed through familiar procedure tothe generalized form:

— k -

= — (Z1u1 + F,u1 + Z:wt)

— Pgj(ôjj — [(ui — u,) + (v1 — + (w, —

— (X,u1 + + (10.24)

where the quantities subject to. variation are ô,,, uj, v, and w,;= 1, 2, ..., ip and i = 1, 2, ..., n, under no subsidiary conditions.The function for the principle of minimum complementary energy may

be derivedfrom Eq. (10.24) in the usual manner as follows:

= E (10.25)

210 VARIATIONAL METHODS iN ELASTICiTY AND PLA$TICITY

where

=01dI's.,. (10.26)

The independent variables subject to variation in Eq. (10.25) areif = 1, 2, ..., m under the subsidiary conditions (10.15).

Thus, we have derived the principle of minimum complementary energyfrom the principle of minimum potential energy. However, it is obviousthat the principle of minimum complementary energy is derivable alter-natively by use of the principle of complementary virtual work

0, (10.27)

where and are so chosen as to satisfy Eqs. (10.14) and (10.17), and

= 0, =0,' ÔZ, = 0,(I = 1, 2, ..., k) (10.28)

respectively. We note that if Eq. (10.27) holds for any combinatidn ofwhich satisfy Eqs. (10.28), the elongation must be derived from Uj,and w1 as given by Eqs. (10.14) and (10.17).

We shall now derive equations for the displacement components of thejoints where the external forces are applied. We assume that the truss prob-lem has been solved and the applied external forces at the joint areincteased by the amounts dIi, d?1 and d21; i J 2, k, while the geo-mctrical boundary conditions are kept uhged. Then, in a mannersimilar to the development in Section 2.6, we have

dP1, = (u, + Vg dY1 w, dZ1). (10.29)

Equation (1T129) is equivalent to what is known as theoremapplied to the truss structure.

10.4. The Force Method Applied to the Truss Problem

We observe that Eqs. (10.15) consist of 3k equations and are generallyinsufficient for the determination of the m unknowns F,,. In other words,the truss structure redundant, and the degree of redundancyis R = m — 3kby definition. We shall obtain the remaining R equations from the principleof minimum complementary energy (10.25). To begin with, we obtain thegeneral solution of Eqs. (10.15):

R kF,., = + (cx,j,I, + + y,j,Z,),

p..' 1-1

(ij= 1,2,...,m). (10.30)

STRUCFURES 211

The first terms on the right-hand side of Eqs. (10.30) constitute the generalsolution of the homogeneous equations,

= 0, = 0, = 0,I I

(i= l,2,...,k), (10.31)

thus defining a self-equilibrating system of the Internal end forces. Eqs.(10.30) can be written in matrix form as follows:

(F) = (a) tx) + (tx] (I), (10.32)

where the notations { ) and I J denote column and rectangular matrices,respectively, and

- a1.21 al2R ru

fP)= [a]= , (x)=

aa_1.A. t A—l.M.R x1Jflu2.t V32.t

I • a. .: .

(I) = (1,,..., Xk, Yl, ..., 2k,..., 2k). (1o.33)t

Introducing Eq. (10.30) pto the principle of minimumenergy (10.25), and taking variations with respect to we obtain:

= 0 (p = 1,2, ..., R) (10.34)ti..1

or in matrix form[a]' (ö) = 0, (10.35)

where [ ]' denotes the transpose of the matrix ( J, and (t5) {o12, ... ,It is obvious that relations equivalent to Eqs. (10.34) are obtainable fromthe principle of complementary virtual work (10.27). Equations (10.34)provide conditions which must exist between the elongations of the membersin order to ensure that none of the connections between the members ofthe truss are broken after deformation. They are geometrical relations and

t A column matrix is denoted by either one of the following symbols:

(x1, x2, , x,), X, *

xl

xR

of which the former is frequently used for the sake of saving space.

212. VARIATIONAL METHODS IN ELASTICITY AND PLASTICITY

must hold irrespective of the internal end force—elongation relationshipsemployed. Namely, they are the compatibility conditions in the large for thetruss structure, and have the same geometrical meaning as Eq. (6.45), whichwas derived for the torsion of a bar with a hole. By the use of Eqs. (10.30),Eqs. (10.34) are reduced to simultaneous algebraic equations with respectto Xp p = 1, 2, ..., R. By solving these equations, we obtain the values ofthey,, which, in turn, determine and Eqs. (10.30) and (10.19).

When all the members behave elastically, the relations between the ither-nal end forces and elongations are given in matrix form by:

{ô} [C] (F). .. (10.36)

In the trlss structure, (C] is a diagonal matrix as suggested by Eq. (10.6).By the use of Eq. (10.32) and (10.36), we find that Eq. (10.35) provides

(x) = — [G]'[H) (1) (10.37)

for the determination of where

(G] = [a]' f C] [a],(10.38)

[H] [a]' [C]

Substituting Eq. into Eq..l(10.32), we obtain

= [a] [G]' [II]] {t). (10.39)

By the use of Eqs. (10.36) and (10.39), we can determine the elongations ofall the members of the truss structure.• Next, we shall derive equati6ns for the displacement components of thejoints where tke external forces arc applied. Introducing Eqs. (10.32) intoEq. (10.29), and remembering that the external forces are assumed indepen-

• dent of each other, we obtain:m'-.7 . 0

• U1 C$jjj Vj = Wg — )'sj: ii,li-i U—'

(1= 1,2,...,k) (10.40)

in matrix form as• {u) = (Ô), (10.41)

where{u) = {u1, ..., v1, ..., v,, w1, ...,

By of Eqs. (10.36), (10.39) and (10.41), we obtain the displacement• components of the joints where the external forces are applied, and matricesof structural influence coefficients can be derived. The above method con-stitutes the main part of the force method.

A note is made here on the displacement method applied to the trussproblem. A well-known procedure is followed by substituting Eqs. (10.14),

and into Eqs. (10.15) and solving these 3k equations to,determine the unknown displacement components andi = I, 2, ... k. Once the displacement components have been obtained, the

STRUCTURES 213

deformation and internal forces of the truss structure can be determined bythe use of (10.14) and (10.18).

10.5. A Simple Example of a Truss Structure

As an application qf the preceding formulations, we consider the planetruss structure shown in Fig. 10.3. The truss consists of six members andfour joints, namely m = 6 and n 4. The,external forces are applied at thejoints and © in the directions of the x- and y-axes, while at the joint

yA

0

the force is appliedat these joints are:

in th4y-direction only. The equations of equiliorium

p14 + p13 = + (1/12) = V.

"23 + = X2, i'12 + = —

+ (1/V2)P,3 = (10.42)

The geometrical boundary conditions are prescribed at the joints and @

0, u4 = 0, r4 = 0. (10.43)

Since we have five equations of equilibrium (10.42) for six unknown internalend forces, the redun4ancy of the truss is 6—5 = 1, and Eqs. (10.42) can be

Yl

/Fio. 10.3. A truss structure.

as

214 VARIATIONAL METHODS IN ELASTICITY AND ('LASTICITY

written in matrix form as:

—lfJ0O 1 0 0 P13P13 1 00 0 0 0P14 I 0 0 0 0 12P23 — 1 0

(10.44)

P24 1 — 0 0 — —12 0 ?2P34 —I/)200 0 0 —1

We observe that the unknown end force P13 in Eq. (10.44) plays the roleof defined in Eqs. (10.30). in the notation defined in Eq. (lO.33), theright-hand side of Eq. (10.44) provides:

- 00 1 0 0

[aJ [x] = i 0 0 0 0(10.45)

1 003V2 0_—1/J/2 00 0 0 -1

Consequently, noticing {OJ = (612, 614, 624, 034, we obtain fromEq. (10.35) the compatibility condition in the large:

12(013 + 624) + + 623 + 634 0. (10.46)

The displacement compOnents at the joints and are obtained from(10.41): —

= 614, v1 = + 023 — 12624,

= 023, v2 = 623 — 12624,

V3 = (10.47)

10.6. Deformation Characteristics of a Frame Member

Next, we shall deal with a frame structure. To begin with, we considerthe deformatjoff characteristics of a frame member. For the sake of sim-plicity, we shall take a beam which, as shown in Fig. 10.4, is rigidly fixedat one end and is subjected to end forces and moment, .N22, Q M12, atthe other end to a concentrated load P acting in the middle of thespan. We denote components of deformation under the application of theseforces and moment by:

= the displacement of the end © in the direction of N12,the displacement of the end © in the direction of Q12,

= the rotational angle of the end c2j in the direction of M12,the displacement of the point of application in the direction of theexternal force P.

STRUCTURES 215

These quantities may be calculated from Castigliano's theorem:

— aV12 —104812

—12 12 — (3M12

, 12—

)

In the above, the quantity V12 is the complementary energy stdred in thebeam If we employ the beam theory for the analysis of

-frame member, it is given by

- V12= + dx, (1

whereN N1.,, (l0.50a)

o4;(10.501,)

and

M12 — (1— x)Q12 +M=1

M12.—(1—x)Q12,

I

Flo. A cantilever beam.

When the beam is of uniform cross section along the span, we obtain theflexibility matrix of the beam © as follows:

EA00 . 0 0AN

"12

01

1 P 1

0P 1 1

24E1_

N12

Q12

M12

p

(10.51)S


Fqr later convenience, we note that Eqs. (10.51) can be inverted into thestiffness matrix form as follows:

0 oEA0

0

where u1, v1, and u2, v2, 02 are the displacement components in thc x-and y-directions and the rotational angle in the clockwise direction of thejoints® and respectively, as shown in Fig. 10.5. Sincethe beam is in staticequilibrium, we have the following relations among the external forces andmoments: -

12

12 12

FIG. 10. 5. A beam element.

(10.53)

EA0

I

12E1 6E1o 10/2

6E1 4E10 /2 /

I 2E1/3

6E!'j2

0/

6E! 2E1 1—0 /2 /

8P1

0EA0

I,0

12E1 6E10 — /3 •/3

6E1. 2E10 /2 /

U'

V1

U2

V2

02

1

12E!0 /3

6E!0 /2

0 0

6E!/2 2

4E1•1

(10.52)

P

M12

befbre deformotion

STRUCTURES 217

When a curved beam is oriented arbitrarily with respect to a set of carte-sian reference axes and is subject to combined actions of axial, shearing,bending and torsional end forces and moments together with external loadsdistributed along the span, the relations between the external forces and theresulting deformation become complicated. The relations for a straightbeam have been obtained in the flexibility matrix and stiffness matrix formsand are widely used in structural analysis.t

10.7. The Force Method Applied to a Frame Problem

With the above preliminaries, we shall now proceed to the analysis ofa frame structure. Rather than attempting to develop general formulationsfor a three-dimensional frame structure, we shall consider only a simple

Fio. 10.6. A frame structure.

Fio. 10.7. Free body diagram of the frame structure.

t For the stiffness matrsx, see ReL 15.


of the plane frame illustrated in Fig. 10.6 and discover that varia-tional pr&eed in a manner similar to that for truss struc-ture.

We shall be interested in the force method applied to the frame problem.To begin with, let the frame structure be cut fictitiously into several membersand the internal end forces and moments be defined properly for the separatemembers. For example, the structure may be cut into three members, ti),

and ti), as shown in Fig. 10.7, and the internal forces and moments:N12, Q12, M12; N23, Q23, M23; N14, M14 may be defined on ondendof each of these three members, while the asterisked quantities

are defined on the opposite end of member ti). From equilibrium con-ditions for member we have the following rel1tions among these quan-tities:

= N12, = —. F, M2 = M12 — Q121 +3?!. (10.54)

We define geometrical quantities of deformation fopthesc members asfollows. With respect to ©, the end on which the asterisked quantities aredefined is assumed to be held fixed, and components of deformation denotedby and defined as given in the preceding section. In asimilar manner, the quantities and ö arc defined with respect to

in the directions of N23, Q23 and M23, respectively, and the quantitiesand with respect to ti) in the directions of N14, and M14,

respectively. Then, denoting the energies of the members(i), and ti) by V12 (N12, F), V23 (N23, Q23; M23) and V14

(N14, M14), respectively, we have

= 8V12 ,

=110.55)

Let us now consider reassembling these three members into a frame struc-ture. To begin with, the equilibrium conditions at the joints (j) and shouldbe satisfied (see Fig. 10.7):

—N14 + 0, + = 0, —M14 + M& = 0, (10.56)

—N12 + Q3 = 0, —Q12 — N3, = 0, —M12 — M23 = 0. (10.57)

Eliminating the asterisked quantities from Eqs. (10.56) by the use of Eqs.

(10.54), we obtain:

• + = 0, Q1 2 — N14 — P = 0,.• M12 — — Q121 + = 0. (10.58)

Equations (10.57) and (10.58) comprise six equations of equilibrium for thenine internal end forces and moments, thus showing that the redundancyis 9 —6= 3.t These equilibrium equations can be written in matrix form as

t pue to the symmetry property of the problem, reduce the redundancy further.this property will not be taken into account here, since our purpose is to show

the procedure of the force method.

STRUCTURES 219

follows:N12 1 0 0 0

Q12 0 1 0

0 001 0 0 00 00

0 0 00—1 1 4!

where N12, Q12 and M12 are chosen as the independent end forces andmoments. Using the notation defined in Section 10.4, we may write

/ 100 0010 0001 00—i 0 0

[a] = 1 0 0 = 0o o—i o0 0

—1 0 0 00—1 1 4!

We must introduce three conditions of compatibility in the large in orderto solve present problem. They are given by the stationary conditions ofTIC defined by

= V12+ V23 ÷ V14, (10.61)

where tht independent quantities subject to variation are N12, ..., and M14under the subsidiary conditions (10.57) and (10:58). A careful considerationshows that the conditions of compatibility in the large are kiven by

[a]' (Ô} = 0, (10.62)where — IAN AQ AM AN AO AM AN

— 23' "23' "14' "14' '-'14

When written in explicit form, Eq. (10.62) becomes:AN AO — A AQ AN , AN AM! — AU12 '-'23 — "14 — '-', "12 — 23 T 14 — 14 —

M AMj AM_A

Tbese are the conditions of compatibility in the large for the frame structure.By combining Eqs. (10.55), (10.59) and (10.63), we have necessary andsufficient equations the determination of the ninE unknown internal endforces and


Next, we shall determine the displacement of the point of applicationin the direction of the external force P. We assume that the frame problemhas been solved and the external force P is increased by an amount dP. Thenwe have

+ + ... + dP dP (10.64)in a-manner similar to the development in Section 2.6. From Eq. (10.64) weobtain

= [ct]' (b) + t5?;= + (10.65)

We note here that, if the subsidiary conditions (10.57) and arcintroduced into the framework via Lagrange multipliers, the expression(10.61) is transformed into defined asII: = V12 + V23 + V14 +(N12 +Q14)u1

+ — N14 — P) v, + (M12'— M14 — Q12! + +Pl)01— (N12 — Q23) u2 — + N23) v2 — (M12 + 02, (10.66)

where the quantities subject to variation are N12, Q12, M12, N14, Qi4,M14, Q23, U1, u2, v2 and 02 with no subsidiary conditions.The physical meanings of u1, and u2, v2, 02 are the displacementcomponents in the x- and y-directions and the rotational angle in the clock-wise direction of the The expression (10.66) may•be considered as an extension of the Heflinger—Reissner principle to theframe structure. When the mechanical quantities N12, Q12, ... and M23 areeliminated from the expression (10.66) by the use of the stationary conditionssuch that

(IV (2 + u1 — u2 = 0,..., (10.67)

we have the function for the pñnciple of minimum potential energy:I J52j3

El

+ P + V2) + — 01)1, (10.68)

where U12(u1,v1,01,u2, U14(u1, v1,01) and U23(u2, V2. 02) are thestrain energies stored in these members, and the quantities subject to varia-tion are u1, v1, 01, u2; v2 and with no subsidiary conditions.t

We observe that the inversion of Eqs. (10.67) provides a stiffness matrixin the same form as given by Eqs. (10.52). We observe also that by the useof the notations defined before, Eqs. (10.67) can be written as

672=u2—u1,= v2, = —u2, = 02, (10.69)— SQV1, 1114 — —U1, '-'14

and the compatibility conditions in the large (10.63) hold among them.

t See Problems 3 and 4 of JO in Appendix H for the explicit expressions ofU12, U14 and U23.

STRUCTURES 221

A mention is made here of the stiffness matrix method applied to theanalysis of the frame structure."4> A formal procedure begins with derivingthe deformation characteristics all the frame members in stiffness matrixforms. Then, a transformation of coordinates relating to member coordin-ates and absolute coordinates is applied to express all the stiffness matricesin absolute coordinates. Next, equations of equilibrium are derived for aUthe joints with respect to forces and moments, and, by the use of the trans-formed stiffness matrices, these equations are expressed in teñns of thedeformation quantities such as displacements and, rotational angles belong-ing to the joints. It is seen that the equations of equilibrium thus derivedare equivalent to those obtained by applying the principle of minimumpotential energy. By solving these equations, we can determine all thedeformation quantities of the joints. Then, by the use of the matri-ces, all the internal forces and moments at the joints, and consequently, thedeformation and stress of the frame structure can be determined.

As is easily seen, one must struggle with a large number of linear simulta-neous algebraic equations with the deformation quantities at the joints asunknowns in the application of the the stiffness matrix method. The amazingadvance in the development of the high-speed digital computer has madesuch computations a routine calculation.US)

10.8. Notes on the Force Method Applied to Semi-monocoque Structures

Semi-monocoque cons$tuction has wide applications in light structuressuch as airplanes, ships and so forth. A semi-monocoque structure uouallyconsists of panels and stringers, where the panels are used as transmittersof in-plane forces, especially as shear members, and the stringers as trans-mitters of axial forces. Variational principles have been formulated exten-sively for analyses of these structures, reducing them to finite degree-of-deformation or finite redundancy systems.

We shall have some considerations on the force method applied to theanalysis of a semi-monocoque structure,t and review briefly its variationalbackground. In the force method, a semi-monocoque structure is usuallysplit fictitiously into an assemblage of a number of elements consisting ofstringers and rectangular panels. For the sake of simplicity, we assume thateach stringer has uniform cross section and each panel uniform thickness.One of the simplest assumptions on the stress distribution of these membersis as follows: A stringer is assumed to be subjected to end forces P and P*together with uniformly distributed load q as shown in Fig. 10.8. Fromstatic equilibrium, we have

P + qI. (10.70)

t For the force method applied to semi-monocoque structures, see the books of Refs. Ithrough 13 and the papers of Refs. 16 through 24.


The complementary energy of the stringer, V,, is

v3__f2E1A

(10.71).

For a stringer with constant EA0, we obtain

V3— 2EA

[P2! + Pq12 + !q213J (10.72)

y

—0'- —0- p

Fto. 10.8. A stringer under axial forces and distributed shear.

I '1

b

1'

•

X

•0• Fia. 10.9. A shear panel.

A rectangular panel is assumed to be in a state of uniform shearing stressunder a shear flow q, uniformly distributed along the four edges as shownin Fig. 10.9, where q = ti, r is the shearing stress and : is the thickness ofthe panel. The complementary energy of the panel, V,,, is

VP = (10.73)

STRUCTURES 223

By summing the complementary energies of all the meipbers, we obtainthe total, complementary energy of the semi-monocoque structure. Then,the force method reqqires that the total complementary assumes an•absolute mini,mum with respect tp the variation of the iqternal edge forcesunder the subsidiary cOnditions that the internal edge forces are continousbetween adjacent members and the equilibrium conditions at the joints ornodes must be satisfied. The above procedure the main part ofthd force method.

We have seen that in trusses and frames, the deformation quantities suchas elongations and rotational angles are associated with internal end forcesand moments through complementary energy of the members as shownby Eqs. (10.10) and (10.48), and that the principle of minimum comple-mentary energy provides the compatibility conditions in the large existingamong them. However, structural members under more complicatedinternal loadings, the geometrical mealungs of deformation quantitiesassociated with generalized internal forces through complementary energyof the members become liss clear, although the general process of formula-tion remains the same. .For example, let us consider a stringer shown inFig. 10.8, of which the complementary energy is gven by ,Eq. (10.71).Denoting the displacement in the direction of the x-axis by u(x), and employing the relation

P+(1—x)q.=E40(du/dx), (10.74)weobtaiti:

(10.75)

(1 -- x) — (10i6)

thus realizing the physical meaning of these derivatives. it can be said thatalthough geometrical meanings of deformation quantities derived from thecomplementary energy may become obscurà,,a sequence of approximatesolutions obtained via the force method must converge to the actual solutionif the degree of redundancy is ihcreased without limit.

It should be npted here, that, although a solution can be obtained byapplying one of the variational principles, it is in general an approximatesolution. For example, by applying a force method, we can obtain compati-bility conditions in the large which are consistent with the degrees of simpli-fication in establishing the total complementary thergy. However, theygenerally approximations to the exact conditions of compatibility. Ldcalcontinuity of displacement between members is generally violated in theapproximate solution. We consider, for example, the force method appliedto a' plane semi-monocoque 'structure consisting of panels and stringers as


shown in Fig. 10.10. We assume the distribution of internal edge forces ofthese members as shown in Figs. 10.8 and 10.9, and determine the valuesof these internal edge forces by the force method. Since a rectangular panelsubjected to a uniform shear flow q is deformed as shown in Fig. 10.1 1,. inwhich y is the shearing strain given by

= q/Gt, (10.77)

FIG. 10.10. A semi-monocoque plane structure consisting of rectangular panels andstnngers.

0-

Cl

Fzo. IOu. Shear strain v.

it would be required that a geometrical relation

(10.78)

must hold for a joint formed by the four panels and ® in Fig. 10.10in order to ensure that the connections between these panels are not brokenafter - However, conditions of this kind are not satisfied ingeneral. Thus, if we were to calculate the displacement components of themembers independently, using the values of the internal forces obtained

STRUCTURES 225

from the force method, we would find discontinuities of displacements onthe boundaries between the members. For an improvement of the accuracyof the solution, we must assume more complicated internaledge forces instead of the uniformly distributed shear flow. It is obviousthat the principle of minimum complementary energy provides an effectivetool for achieving the improvement.

10.9. Not'es on the Stiffness Matrix Method Applied to Semi-monocoqueStructures

Next, we shall have some considerations on the stiffness matrix methodapplied to semi-monocoque structures following the pioneering work byTurner, Clough, Martin and and review briefly its variationalbackground.t In the stiffness matrix method, a sen'u-moñocoque structure

- Fx2

- t4u2.— -

Fio. 10.12. A stringer element.

is usually split fictitiously into an assemblage of a number of elements con-sisting of stringers and triangular panels. For the sake of simplicity, weassume that each stringer has uniform cross section and each panel uniformthickness. One of the simplest assumptions, on the deformation of thesemembers is as follows: A stringer is in a state Qf strain and itsstrain gy, U%, is

lEA0 2U3 = -y - (u2 — u1) , (10.79)

where u1 and u2 are the displacements of both ends in the axial direction,shown in Fig. 10.12. Defining the end forces by

F — F — (1080)— X2 —

we obtain the stiffness matrix of the stringer:

Fxi1 = EA0 —1 1] fUll (10.81)I I —ii

t For the stiffness method applied to semi-monocoque structure, see also thebooks of Ref's. 10 through 13.

226 VARIATIONAL METHODS IN £LAS11CITY AND PLASTICITY

A triangular panel is assumed in a state of uniform strain:

8u 8v

v by + (c — A) x + C, (10.83)

A, B and C being constants of integratio'n which define rigid body transla-don and rotation of the triangle. Denoting the displacement components

of the three 'vertices of the triangle by (u1, vi), (u2, 02) an4 (u3, 03) respec-

tively, we have the six constants a, b, c, A, B and C, and 4x,nsequently, thestress components of the panel

£ 18u 80'

______

E IOu Ov'• E= (1 = (1 _.,2) (ia + b),

GYx, = Gc,

and the strain energy of the panel

u, +e,)2 +

(a + b)2.+ G(c2 —

(10.84)

= a,80= = b,

from which we obtainu= ax + Ay +

(10.82)

Fyi

Fx1

Fio. 10.13. A triangular panel clenxnt.

STRUCFUR,ES 2274.

m terms of u1, v1, u2, v2, u3 and v3, where I is the thickness of the plate.It is seen that the stress components C, and given by Eqs. (10.84)satisfy the equations of equilibrium:

8r &rXY_o X7Lêx ôy'

Defining the node forces at the vertices by

F F _8U,— 8u1_' " — ,

..., Fp3—

, (10.87)

we obtain

F

F,,[K] is a symmetric matrix given by

Elx

x2 x2y3

— 22x32+

x2 x2y3 x2

• —21X3X23 vx32 21x3

+X2 ' •X2 X2y3

+21X32 — 22x3 4

X2 X2)'3 X2 X2y3 X2

1a1x23 21x3A

11x2

Ty3 373

3, 0-

y3 y3(10.89)

with X1j = — Xj, = (1 — v)/2 and 22 = (1 + v)/2.With these preliminaries, the analysis of a semi-monocoque structure by

the use of the stiffness matrii method proceeds as follows: First, a trans-formation from the member axes to the absolute axes is applied to expressall the stiffness matrices in the absolute coordinates. Next, by the use ofthese stiffness matrices, we obtain equilibrium conditions of all the nodesin terms of the displacement components of the nodes. Since continuityof the displacements along the fictitiously cut edges between the elementshas been satisfied by restricting the displacements between the nodes to the

228 VARIATIONAL METHODS IN ELASTICITY AND

linear variation, we find that the equilibrium equations thus derived areequivalent to those obtained by applying the principle of minimum potentialenergy. By solving these equations, we can determine the displacements ofall thà nodes. Then, the stress of the stringers and triangular panels can becomputed. The above procedure constitutes the main part of the stiffnessmethod. It is observed that the stress components o',, are uniformin each triangle and change discontinuously from one triangle to another.Similar discontinuities exist between neighboring triangles and stringers.For method of smoothing the stress discontinuity, the reader is advisedto reed a paper by Turner and

We remember that although a solution can be obtained by an applicationof a displacement method, it is in general an approximate solution. Byapplying the stiffness matrix method, we can obtain equations of equili-brium which are consistent with the degree of simplification employed inestablishing the total potential energy. However, they are generally approxi-mations of the exact equations of equilibrium. Local continuity of theinternal forces along the fictitiously cut edges or surfaces between elementsis generally violated as mentioned above. We shall consider, for example,a part of a panel which is split fictitiously into several sub-elements as shownin Fig. 10.14 and obtain an interpretation of the equations of equilibriumprovided by the stiffness matrix method. For the sake of simplicity, weassume that no external forces are applied at the i-th node.

y

0.

The equatiou of equilibrium of the i-th node in the direction of the x-axisis given by

0, (10.90)('Ut

K

Fio. 10.14. Assembly of triangular panel elements.

STRUCTURES 229

where 17 is the total potential energy and is the displacement componentof the :-th node in the x-direction. It is obvious that a result equivalent toEq. (10.90) is obtainable from the principle of virtual work for the problem

(10.91)

by requiring that the coefficient of öu, must vanish, where the notatiOn

means that the summation is taken with respect to all the triangles, ande, and are derived from continuous functions u and v such that

(10.92)

Since a, and are so chosen as to satisfy Eqs. (10.86), the contributiondue to âu1 in the (IQ.91) reduces, via integrations by parts, to

+ ou2jt is = 0, (10.93)

where and are the components of the internal stressesdistributed on the U-th edge of the triangles (1, j, j — 1) and (i, J' j,.+ 1) in thedirection of the x-axis respectively, and u11 is the displacement componentsof the ij-th edge in the x-dircction. In Eq. (10.93), the summation with re-spect tojis taken over all the edges which are directly connected to the i-thnode. Since is chosen t9 vary linearly along the edge, we have

— (10.94)

the length of the tj-th edge, and s,, is a measured fromthe i-tb to the J-th node. Then, E4. (10.93) reduces to

+—

= 0, (10.95)

which tequires that the weighted mean of the unbalanced internal stressesalong the edges directly connected to the i-th node should vanish. Equation(10.95) provides an interpretation of Eq. (10.90).

Thus, the deformation and stress obtained by the stiffness matrix method isapproximate. For of the accuracy of the approximatesolution, we may have two First, the triangular stiffness matrixmay be used and the desired accuracy may be obtained by using a sufficientnumber of sub-elements, or, second, a more panel stiffness matrixmay used with fewer sub-elements. It is obvious that the principle ofminimum potential energy provides an effective tool for the second approach.


Bibliography

1. A. S. Nuis and J. S. NEWELL, Airplane structures, John Wiley, 1943.2. D. J. PEERY, Aircraft Structures, McGrflw.HiIl, 1949.3. N. J. Hopp, The Analysis of Structures, John Wiley, 1956.4• P. Stresses in Aircraft and Shell Structures. McGraw-Hill, 1956.5. R. L. BISPUNOHOFF, H. and R. L. Aeroelasticiry, Addison-Wesley,

1955:6. W. S. HEMP, Methods for the Theoretical Analysis of Aircraft AGARD

Lecture Course April, 1957.7. J. H. ARGYRIS, On the Analysis of Complex Elastic Structures, Applied Mechanics

Reviews, Vol. 11, No. 7, pp. 331—8, July 1958.8. J. H. ARGYRIS and S. KELSEY, Energy-Theorems and Structural Analysis, Butterworth,

3960.9. E. C. and F. A. LEaUE, Matrix in Elastomechanics, McGraw-Hill,

1963.10. 3. H. ARGYRIS, Recent Advances in Matrix Methods of Structural Analysis, Pergamon

Press 1964.11. R. H. GALLAGHER, A Correlation Study of Methods of Matrix Structural Analysis,

Pergamon Press 1q64.12. F. L)E VEUBEKE, Editor, Matrix Methods of Structural Analysis, Pergamon Press 1964.13. 0. C. ZIENKIEWICZ and (3. S. HousrEle, Editor, Stress Analysis, John Wiiey,J965.14. H. C. MARTIN, Introduction to Matrix Methods of Structural Analysis, McGraw-Hill,

1966.15. IBM 709017094 FRAN Structure Analysis Program, International BUsiness

Mach inc Corporation, August 21, 1964.16. H. ERNER and H. KOu.zn, Uber die Einleitung von Lángskraften in Venteiften

Zylinderschalen, Jahrbuch der .Deutschen Luftfahrtforsclumg, pp. 464—73, 1937.17. E. EBNER and H. KOLLER, Zur &rechnung des Kraftverlaufs in Versteiften Zylinder-

schalen, Luftfahr:forschung, Vol. 14, No. 12, pp. 607—26, December 1937.18. E. EBNER and H. Uber den Kraft'verlauf in LAngs-und-querverstciften

Scheiben, Luftfahrtforschung, Vol. 15, No, 11, pp. 527—42, October 1938.19. S. LEVY, Computations of Influence Coefficients for Aircraft Structures with Discon-

tinuities and Sweepback, Journal of the Aeronautical Sciences, Vol. 14, No. 10, pp.547—60, October 1947.

20. A. L. LANG and R. L. BI3PLIN0HOn', Some Results of Sweptbavk Wing StructuralStudies, Journal of the Aeronautical Sciences, Vol. 18, No. 11, pp. 705—17, November1951.

21. B. LANGEFORS, Structural Analysis of Swept-back Wings by Mairix-Transformations,SAAB Aircraft Company, Linkoping, Sweden, SAAB TN 3, 1951.

22. B. LANGEFORS, Analysis of Elastic Structures by Matrix Transformation with SpecialRegard to Semi-monoco4ue Structures, Journal of the Aeronautical Sciences, Vol. 19,No. 8, pp. 451—8, July 1952.

23. T. RAND, An Approximate Method for the Calculation of Stresses in Sweptbeck Wings,Journal of the Aeronautical Sciences, Vol. 18, No. 1, pp. 61—3, January 1951.

24. L. !3. WEHLE and W. A. LANSING, A Method for Reducing the Analysis of ComplexRedundant Structures to a Routine Proccdure, Journal of the Aeronautical &iences,Vol. 19, No. 10, pp. 677—84, October 1952.

25. M. J. R. W. CLOUGH, H. C. MARTIN and L. J. Topp, Stiffness andAnalysis of Complex Structures, Journal of the Aeronautical Sciences, Vol. 23, No. 9,pp. 805—23, September 1956.

CHAPTER 11

THE DEFORMATION THEORYOF PLASTICITY

11.1. The Defonnatlon Theory of Plasticity

This chapter will discuss variational principles on the deformationtheory of plasticity.t The deformation theory is characterized among theoriesof plasticity as the one in which relations between instantaneous states ofstress and strain are postulated in such a way, that when the strain is given.,the stress is uniquely determined or vice versa. However, this determinationmay or may not unique in both directions. For example, if the stress isgiventih terms of the strain as

= (11.1)

the inverse relations may or may not be unique in determining the strainin terms of the' stress.

In deriving variational principles in this chapter, we shall assume thatthe stress—strain relations do not change during the loading process. Thisassumption restricts the deformation theory problems which we mayformulate to those in which the loading increases monotonically. Conse-quently, the above assumption in effect renders the deformation theory ofplasticity undistinguishable from the ilOnlinear theory of elasticity discussedin Chapter 3, except for materials which obey a yield condition. Further-more, we shall employ the assumption of small displacements and definea problem of the deformation theory of plasticity as follows

(1) Equations of equilibrium= 0, (11.2)

where body forces are assumed absent for the sake of simplicity;

t It is well established that the deformation theory of plasticity is unsuitable for describ-ing completely the plastic behavior of a metal and should be replaced the flow theoryof plasticity, which follows in Chapter 12. However, this brief chapter is devoted to thedeformation theory of plasticity because of historical interest and its frequent use due tomathematical simplicity.

The summation convention is employed throughout Chapters 11 and 12. Thus, arepeated Roman subscript means summation over the values (I, 2, 3).

231

232 VARIATIONAL METHODS IN ELASTICITY AND'PLASTICITY

(2) Strain—displacement relations

= + Ujg; (11.3)(3) Stress—strain relations

= (11.4)or conversely

(11.5)(4) Boundary conditions

= F, on (11.6)

us = u, on S2. (11.7)

Then, by taking the same steps we took in Chapter 1, we obtain the followingexpressions for the principles of virtual work and complementary virtualwork:

fff.'si dV ffF, öu, dS =0, (11.8)V Si

and

fff Cjj — ff ü,dS = 0. (11.9)V 52

We repeat that these two principles hold independently of the stress—strthrelations.

If Eqs. (11.4) are analytic functiops which the existence of the statefunction A defined by

- = cr,,ãc,,, (11.lO)t

the principle (11.8) leads to the principle of stationary potential energy 4

(11.11)where

11 = fffA(u,) dV— ff F,u, c/S.

On the other hand, if Eqs. (11.5) are analytic functions which assure theexistence of the state function B defined by

oR = (11.13)t

the principle (11.9) leads to the principle of stationary complementaryenergy 4

=0, . . (11.14)

f When the stress system under consideration is uniaxial, such as in a bar under'tension,the existence of the state functions A and B is assured for the deformation theory ofplasticity. This suggests that variational procedures will be extremely powerful in analyzingstructures, if the stresses in the structures can be assumed uniaxial and the deformationtheory of plasticity can be 2. 3)

The appellations "potential energy" and "coi'nplementary energy" seem to be nus-leading in the theory of plasticity. However, we shaU employ them because their mathe-matical definitions are the same as in the theory

DEFORMATION THEORY OP PLASTICITY 233

where= fff B(cr,,) dV — dS. (11.15)

If it is assumed further that Eqs. (11.5) are unique inverse relations ofEqs. (11.4), and vice versa, we can transforim the principle of stationarypotential energy into the principle of stationary complementary energy,and vice versa, in a manner similar to the development in Chapter 2.

Thus, the stationary property of the two functionals and (11.15)is assured under the assumptions mentioned above. the maximumor minimum properties of these functionals cannot be guaranteed unlessthe stress—strain relations are specified more in detail. Following Ref. 4,we shall review some of the variational principles related to the deformationtheory of p!asticity.

11.2. Material

A type of deformation theory called the secant modulus theory in whichthe stress—strain relations are given by

(11.16)

will be discussed in this section. and 4 are the stress and straindeviators defined as = a,1 — c

and âu is the Kronecker symbol. The quantity /4 appearing inEq. (11.16) is assumed to be a positive quantity which depends in generalon the s(ate of strain. It follows immediately from Eq. (11.16) that

S=ul', (11.17)where

= 1' = 14e's,, (11.18)

andSdS=c4da,, (11.19)

It is assumed that S is a single-valued continuous function of r, as shown inFig. 11.1, i.e.

S=S(fl, (11.20)and that the relations

S/i' =js>O, dS/dI'>Ohold throughout the regions of I' and S under consideration. By combiningEqs. (11.16) and (11.17), we have

(11.22)and (11.23)

Only five of the relations in Eqs. (11.22) or (11.23) are independent. There-fore, a sixth relation, will be added, i.e.

a = 3Xe. (11.24)

234 VAlUATIONAL METHODS IN ELASTICITY AN]) PLASTICiTY

where K is the bulk modulus of the material. Under the assumption that theplastic deformation causes no volume change, K can be expressed in termsof Young's modulus E and Poisson's ratio v as follows:

3K (11.25)

With the above preliminaries, we obtain the following expressions for Aand B under the secant modulus theory:

A2(1 2v)

e (11.26)

3(1—2w)a2 ÷fr(s)ds. (11.27)

S

0 1

FIG. 11.1. S — r relation for a strain-hardening material.

By substituting Eqs. (11.26). and (11.27) into Eqs. (11.12) and (11.15)respectively, we have the expressions of the functionals 11 and of theproblem for material obeying the secant modulus theory. We then havetwo variational principles which are called Kachanov's principles (4. 5. 6)

and stated as follows:Kachanov principle I. The exact solution of the problem renders the

functional 11 a minimum with respect to admissible displacement varia-tions.t

Kachanov principle 2. The exact solution of the problem renders the func-tional a minimum with respect to admissible stress variation4

Since [3E/(l — 2v)J ()e)2 ± (S/f3) (f2&,6t1 — ()/F2) (dS/dI')x (r, &,)2, (e, 5e,)2 by Schwarz's inequality 0 from Eq. (11.21),we have 0.

Since 2à2B [3(1 — 2v)/EJ(&i)2 + (uS3) [S25c — (a1 5r1)2] + (1/52) (dl'/dS)x S2 ? by Svhwarz's inequality and dF/dS> 0 from Eq.(11.21), we have 0.

DEFORMATION ThEORY OF PLASTICITY 235

11.3. Perfectly Plastic Material

The secant modulus theory will now be specialized to the case of a per-fectly plastic material obeying the Mises yield condition. The S — r relationwill be assumed as shown in Fig. 11.2: the material elastically forS < k and flows for S V2 k, where k is the yield limit in simple shear.The expressions for A and B, and the stress—strain relations for the perfectlyplastic material may be formally obtained as follows. We replace the S 1'curve in Fig. 11.1 with a broken line such that

S=2G1' for 1'<l'o,S = S0 + — 1's) for 1' (11.28)

where S0 = j/2 k = 2Gf'0 and g9 is a positive constant. We calculate theexpressions for A and B for the relation given by Eqs. (11.28), and drive the

relations from

= =- (11.29)'

We then let approach zero. Thus, we obtain the following expressions forA and B for the perfectly plastic material:

A3E

e2 + GI'2 for F < F0,2(1 — 2v)

A = 2,')e2 + Gfl + k(F — F0) for r> (11.30)

B = 3(1 —2v) + (1

The resulting stress-strain relations obtained, by the above liinitingproce-dures are as follows:

= (1 —'2v)+ 2G4 fdr r <

= (1 —2i') + FEll for F � T0, (11.32)

= (1 —2v)aâgj + for S <

= (1 —2t')aôjj + Àø, for S (2k, (11.33)

where 2 is a positive, indeterminate and finite quantity defined by

Jim (S — S0)/2flS 2. (11.34)s-÷so

A material which exhibits the stress—strain relation of Eqs. (11.32), or equi-valently (11.33), is called a Hencky material for which the Haar—Kárrntht


principle This principle may be stated as follows:

Among arbitrary sets of admissible stress components which satisfy theequations of equilibrium, the mechanical boundary condition on S1and the condition � 2k2, the exact solution renders

licfff[3(l—2v)

(11.35)

an absolute minimum.

S

Fio. 11.2. s — r relation for a perfectly plastic material.

the proof of the Haar—Kármán principle given by Greenberg will befollowed Let ,, and represent the stresses, strains and dis-placements of the exact solution and let the stress components of an ad-missible solution be denoted by In addition, the elastic and plasticportions of ihe strain components will be separated by writing

• Lii = 4 + r. (11.36)

Then, observing that the body consists of a plastic region V. and an elasticregion V1. we may write the first variation of as

= fff4oiY11dv — ffôa,jnjü1 dSV S3

= fff (e,j — dV— ff&r1jfljÜj

V 53

=— fff jUj dV +11 ÔAJ,jfljU, dS

I, SI

+ ff — u,) dS— fff dV

S2 VP

= — (11.37)VP

Since we have= )4), 2 > 0 j1138)

DEFORMATION THEORY OF PLASTICITY 237

from Eq. (11.33), we can write

= — (11.39)

The exact solution satisfies = 2k2 in, the plastic region, while theadmissible solution has been chosen so that � 2k2. Since Schwarz'sinequality proves the following relatioiis:

� � 2k2, (11.40)we have

a1 &iU = — a,,,)

= — ap, � 0, (11.41)

where the equality holds only when = a,,,. Consequently, we concludethat

(11.42)

The functional is a quadratic form with respect to the stress componentsand the second variation can be proved to be positive. Consequently, thefunctional is rendered an absolute minimum for the exact solution.

The above proof shows that, when a subsidjary condition is given in theform of an inequality, it is no longer necessary that the first vajiation mustvanish for an absolute maximum or minimum of An exathplenoted in Ref. 4 is that a maximum of the parabola y = x2 under a subsidiarycondition 0 x � 1 is attained at x = I at which point, however, y'(I) 0.A brief mention is made in-Appendix F concerning variational formulationsof a problem with, a subsidiary condition in the form of an inequality.

11.4. A Special Case of Hencky Material

Variational principles will now be treated for a special case of a Henckymaterial, i.e. one which is assumed to be incompressible and everywhereplastic. The S — rrelation is as shown in Fig. 11.3. We observe that for the

S

a

Fio. 11.3. S — I' relation for a special case of Hencky material.

238 VARIATIONAL METHODS IN ELASTICITY AND PtASTICITY

special case, the expressions (11.30) and (11.31) are reduced to

A = f2k1, (11.43)

B=O, (11.44)

and the corresponding stress—strain relations are

= ñ k(11.45)

Vem,,emn

The principle of minimum potential energy then holds as fo)tows:

Among admissible solutions which satisfy the conditions of compatibility,the geometrical boundary conditions on S2 and the incompressibilitycondition, the actual solutionf renders

(11.46)


This principle is analogous to Markov's principle for the Saint-Venant—Levy—Mises material in the flow theory of plasticity. On the other hand, the prin-ciple of minimum complementary energy is expressed as follows:

Among admissible solutions which satisfy the equations of equilibrium,the yield condition = 2k2 and the mechanical boundary condi-tions on the actual solutiont renders

He = — ff dS (11.47)


This principlç is equivalent to Sadowsky's principle of maximum plasticwork which states that among admissible solutions, the actual solutionrenders

ff dS

an absolute Sadowsky's principle is analogous to Hill's'principle for the Saint-Venant—Levy—Mises material in the flow theory ofplasticity. Proofs of the above two principles are found in Ref. 4 (see alsoSection 12.5 of this book).

Bibliography

I. N. 3. HoFF, The Analysts of Structures, John Wiley, 1956.2. 1. H. ARGYRIS and S. Energy Theorems and Structural Analysis. Butterworth,

1960.3. H. LANGHAAR, Energy Methods in Applied Mechanics, John Wiley, 1962.

1 Except for possible indeterminate uniform hydrostatic stress.

DEFORMATION THEORY OF PLASTICITY 239

4. H. I. On the Variational Principles of Plasticity, Brown University, ONR,NR-041-032, March 1949.

5. L. M. KAcHANOV, Variational Principles for Elastic-Plastic Solids,Mathematika i Mekhanlka, Vol.6, pp. 187—96, 1942. (Translatiqn prepared at BrownUniversity for the Taylor Model Basin in 1946.)

6. A. A. ILvusmN, Some Problems in the Theory of Plastic Deformations, FriklaS',aiaMa:erna: i/ca i Mekhanika, Vol. 7, pp. 245-72, 1943. (Translation prepared at BrownUniversity for the Taylor Model Basin in 1946).

7. A. and Th. v. KAIaAN, Zur Theorie der Spannungszustande in Plastischenund Sandartigen Medien, Nach. der Wiss. zu Go:tingen, pp. 204—18, 1909.

8. M. A. A Principle of Maximum Plastic Resistance, Journal of AppliedMechanics, Vol. 10, No. 2, pp. 65—8, June 1943.

CHAPTER

THE FLOW THEORY OF PLASTICITY

12.1. The Flow Theory of PlasticityIt is well established that unique relations do not exist in generar

stress and strain components in the plastic region; the strain depends notonly the final state of stress, but also on the loading history. Therefore,the relations which have been familiar to us in the theory ofelasticity must be replaced by relations between increments of stress andstrain in developing theories of plasticity. This avenue of the theory ofplasticity is called the incremental strain theory or flow theory of pfasticity.tThe deformation theory of plasticity treated in the last chapter is only aspecial case of the flow theory apd has been found unspitable for af'com-plete description..of the plastic of a metal.

We begin by observing that the flow theory of plasticity employs theEulerian descriptive technique. Namely, a set of values of the rectangularCartesian coordinates which an arbitrary point of a body under considerationoccupies at the generic time is employed for specifying the point duringsubsequent incremental deformations. The stress components 1,, at thegeneric time are defined with respect to these coordinates in a mannersimilar to the definition of initial stresses in Section 5.1.

• Following Prager, let us define a problem in the flow theory of plasticity.At a given instant of time t, a body is assumed to 6e in a state of staticequilibrium, and the state of stress and its loading history are assumedto be known throughout the body. Now, external force increments dF,,i = 1, 2, 3 are prescribed on S1 and displacement increments dü,,i = 1, 2, 3are prescrihcd on S2. Our pro,blem is then to determine increments of thestress and displacement dii, induced in the body under the assumptionsthat the increments are infinitesimal and all the governing equations maybe linearized. Thus, we have:W(1) Equalion.c of equilibrium

— 0,t Refs. I through 6.

Here, d given element of the body (Lagrangian), and differsfrom the change of at a fixed point (Eulerian), denoted by by the amount

— d'a,j Both the original and incremental satisfythe equations of equil1brium: — 0 and da ,j 0. Consequently, we have— C,j.* duk,J = 0, which leads to E4s,'(12.1) un the assumption that increments ofplastic strain are constrained to be of order (l/E) x (the stress-increments), where E isthe Young's modulus of the materlal.U)

240

FLOW THEORY OF PLASTICITY 241

(2) Strain—displacement relations

= + (12.2)

(3) Linear relations between stress increments

and strain increments dek,; (12.3,

(4) Boundary conditions

= dF, on S1, (12.4)

du, = dü, on S2. (12.5)

It is seen that the above problem is defined in a manner similar to an ela?1rcity problem of the small displacement theory, except for the stress—strainrelations. Once the problem inflow theory has been formulated, pro-blems of finite plastic deformation can be analyzed by thtcgrating the result-ing relations along the prescribed loading path.

It is apparent from the above relations that the principles of virtual workand complementary virtual work may be written for the present problem as

• f/f — = 0. (12.6)

andfff de,,ôdo,j dV

— ff = 0, (12.7)

respectively.The above may be considered as quasi-static and in terms

of rate as follows. At a given instant of time, a body is assum4d to be in astate of quasi-static equilibrium. Now, the rates of application of the exter-nal forces F,, I = 1, 2, 3 are prescribed on S1, while the surface velocity13,, 1 1, 2, 3 are prescribed on S2. Our problem is then to the stressrates and velocities v, induced in the body. Here; adót aderivative with respect to time, while denotes a componeflt the velocitywith respect to the rectangular Cartesian coordinates.

The governing equations for the problem expressed in of rate areobtained from Eqs. (12.1) through (12.5) by replacingand with a,,, E,,, F, and t,,, respectively. Two principles cQrrespondingto Eqs. (12.6) and (12.7) may then be written as

and

ffF,ov1dS = 0 (12.8)

f/f ôà,, dV— ff i, âà,jflj dS = 0.


Since the state of stress g1j at the given instant forms a self-equilibratingsystem, we may add the following two principles:

fff,,, dV— ff F, ôv, dS = 0, (12.10)

and

fffi',, ôg,,dV—

&r1,n, dS 0, (12.11)V 32

where F, = agjflj onWith these preliminaries, we shall review some of variational principles

related to the flow theory of

12.2. Sfrala-bardeidng Material

Following Ref. 1, we shall adopt:

= doö,, + + (12.12)

as the incremental stress—strain relations for a strain-hardening material,and call

d4E + 1 . )

,IP_ 11f 14€jj—LiVJfJ

the elastic and plastic strain components, respectively. Here, do and dt4 arethe increment of the average hydrostatic stress and the incremental stressdeviator: do = (1/3) do,, and d4 = do,, — The function h is apositive definite function of The function is called the yield con-dition and thq surface

f=c (12.15)

the yield surface. The parameter c in the above specifies the final state ofstrain-hardening, and its value may vary from point to point throughout thebody. Since df = k,,, we may specify the following loading ter-minology concerning a set ofp4ncremental stresses do,,:

Joading if df>0,/ neutral if df = 0, (12.16)

unloading dfczo.

Refs. I through 6.

FLOW fHEORY OF PLASTICITY

The quantity in Eqs. (12.12) is defined via the above relations in thefollowing manner:

= I where = c and 0,

= 0 where f(au) < c, or where

f(ajj)=c and df<0. (12.17)

The parameter c may be given as a function of the total plastic work:

c = (12.18)

where F is a monotonically increasing positive function and the integral istaken along the loading path. From Eqs. (12.14), (12.15) and (12.18),find that the three functions f, h and F are related by ogj = 1.

By multiplying both sides of Eq. (12.12) by t3f/öa1,, and taking summationswith respect to i andj, we obtain.

1 of '1219-2Gh + .

l'or = 1. This suggests that df> 0 corresponds to (Of/0a1,) 0.With preliminary, we may obtain from Eq. (12.12) the following inVerserelations expressing da11 in terms of

2GE \Oak( / of

= — 2i')

de de The notation is defined•in the following manner:

I here = c and dç, 0.

= 0 < c or where

= c and 0. (12.21)

Equations (12.12) or (12.20) are linear an4 homogeneous in terms of thestrain increments and the stress increments In that sense, they arcsimilar to the stress—strain relations in the linear theory of elasticity, exceptthat they are given in pairs, one set for loading and one set for unloading.

that the coefficients which correspond to the elastic constants are depen-dent on the state of stress and loading history of just before theincremental deformation occurs.

Before proceeding with variational formulations of the problem, weascertain whether or not the incremental stress—strain relations, (12.12)and (12.20), assure the existence of state functions .nf and defined as:

= (12.22)

244 VARIATIONAL METHODS IN ELASTICITY .AND PLASTICITY

and(12.23)

Indeed, we find that they are given by

3E ,

2(1 — 2v)(de)2 + —

, (12.24)

= 3(1± + (12.25)

Consequently, two variational principles may be created for the work-hardening material. The first principle states:

Among admissible solutions which satisfy the conditions of compatibilityand the geometrical boundary conditions, the exact solution rendersthe functional

ii = — dS (12.26)V Si

ah absolute minimum.On the other hand, tue second principle states:

Among admissible solutions which satisfy the equations of equilibriumand the mechanical boundary conditions, the exact solution rendersthe functional

= fff dv— ff dS (12.27)

an absolute minimum.The proof of these principles is given in Ref. 1, together with reference tothe pioneers who contributed to the establishment of the principles.

12.3. Perfectly Plastic Material

By substituting h l/j5 into Eq. (12.12), where fi is assumed to be apositive constant, and letting approach zero in such a way that

Jim = dA> 0 (12.28)

where d% is a positive, indeterminate and finite quantity, we find the follow-ing incremental stress—strain relations for a perfectly plastic material:

= Idcrô,, + (12.29)

Here,= 1 where ft,clu) = c and df = 0,= 0 where f(ou) <c or where

= c and df< 0, (12.30)


and c is a material constant defining the yield conditions of the materialunder consideration.

Simultaneous solution of Eqs. (12.29) yields the following inverse rela-tions:

2G dek:)

= (1 )+ 2Gde, — (12.31)

ôOpQ I

Here, -= I where = c and (afl&rk,) dek, 0.

= 0 wheçe < c, or where

f(ti,,) = c and

d

d= 2(1 — 2v)

(de)2 + Gde,de, — ai , (12.33)

ôcipq)

3(l—2v)(do)2 + (12.34)

for the perfectly plastic materiai.By the use, of the expressions for d and thus derived, we can obtain

two variaiional principles for the perfectly plastic material in a manner

similar to the development in the last section. cxcept that admissible solu-tions in the second principle are now subject to an additional subsidiarycondition in the form of an inequality,t namely, df 0. The proof of theseprinciples is given in Ref. 1, together with reference to the pioneers whocontributed to the establishment of the principles.

12.4. The Prandtl—Reuss Equation

The Prandtl—Reuss equation is a special oase of the stress—strain relationsof Eqs. (12.12) and (12.29) and is based on the assumption that

(12.35)where —

ö = /2

t This is the same situation which we encountered in the Haar—Kármán principle ofthe deformation theory of plasticity.

An overbar does not indicate that the barred quantity is prescribed in the notation &


Introducing the notation = 1'(2/3) Eq. (12.18) is found tobe replaced by

a = (12.37)t

where 1! is a monotonically increasing positive function. Since the relationhH' I is'obtained from Eqs. (12.12), (12,35) and (12.37), we may employthe following equation in place of Eq. (12.12):

= (1 —2,')+ + . (12.38)

Hde= 1 where a = c and da � 0,

= 0 where 0 <c or where

a—c and dO<0. (12.3?)

The inverse form of Eq. (12.38) can be shown to be

(12) + 2Gd4 — (Ii?) (12.40)

Here= 1 where 0 c and a1 de1, 0,

= 0 wherd' 0 <c or where

8 c and < 0. (12.41)

Equations (12.38) and (12.40) a

strain-hardening material. of Eqs. (12.38) and (12.40) are dividedby di, we have the in of rate:

(1 — 2v)+ 2G (H'

+"

(12.43)

/

where the definitions of and are obtained from Eqs. (12.39) and (12.41)

by replacing do and de1, with and s,,, respectively.The above relations may be extended to a perfectly plastic material for

which the yield conditiOn is given by: . /(12.44)

where k is a constant dependent on the material. putting

Jim (12.45)28H

t An overbar does not indicate that the barred quantity is in the notationdv'.


in Eq. (12.38), where dA is a positive, indeterminate and finite quantity, weobtain

= (1 —2v)daö1, + + c'**a'dA (12.46)

Here,= 1 where as', = 2k2 and a1 = 0,= 0 where < 2k2, or where

a' a' = 2k2 and a, da, < 0. (12.47)

The inverse forms of Eqs. (12.46) can be shown to be:

da1,= (1 de tU + 2G d€:, — a,. (12.48)

Here,= 1 where a,', a,', = 2k2 and 0,= 0 where a, < 2Jc2, or where

= 2k2 and <0. (12.49)

Equations (12.46) and (12.48) are called the Prandtl—Reuss equations for aperfectly plastic material. The rate forms of these equations are given by

(—v). a,,

= E+ + (12.50)

(1 —2v)— (12.51)

respectively, where ji 0. The definitions of and in theseequations tare obtained from Eqs. (12.47) and (12.49) by replacingand with and respectively.

Variational prijiciples similar to those given in Sections 12.2 and 12.3have been derived for materials which obey the Prandtl—Reuss equations."

12.5. The Saint-Venant--Levy-Mises Equations

If the elastic strain rates in Eqs. (12.50) are assumed to be negligiblecompared to the plastic strain rates, we have

= isa, where ap,', = 2k2 and a,ã, = 0, (12.52a)

= 0 where eip; < 2k2, or where

= 2k2 and <0. (12.52b)

The inverse relations are

a,', = .(12.53)

l1Ck

248 VARIATIONAL METHODS IN ELASTICITY AND PLASTICiTY

for Eq. (12.52a) only. Materials governed by the above relations are calledrigid-plastic materials. Equations (12.52a, b) are called tJie Saint-Venant—Levy—Mises equations for rigid-plastic materials.

We shall consider. variational principles for a body composed of rigid-plastic material under the assumption that the body is plastic. Theproblem in this section will be defined in a slightly different manner frpmthe previous problems:(1) Equations of equilibrium

auj = 0; (12.54)(2) Yield condition

2k2; (12.55)

(3) Stress—strain rate relalion.c: Eq. (12.53);(4) Rate of strain—velocity relation

= Vg•j + (12.56)

(5) Condition of incompressibility= 0; (12.57)

(6) Boundary conditionsa,jflj = on S1, (12.58)

v1 = on S2. (12.59)

There result two variational principles, the first of which may be statedas follows:

Among admissible solutions which satisfy the conditions of compatibilityand incompressibility, as well as the geometrical boundary conditionson S2, the actual solutiont renders

17 = dv— ffFjv, dS (12.60)


This is called Markov's The proof is as follows. Let the stress,strain rate and velocity of the e,sact solution be denoted by and v1,and the strain rate and velocity of an admissible solution by t and v7'.Then, since

(12.61)

by Schwarz's inequality, and= . (12.62)

by the incompressibility condition, we obtain from Eqs. (12.55), (12.61) and(12.62) the following relation:

cl/s � (1163)

t Except for a possible indeterminate uniform hydrostatic pressure.


On the other hand, Eqs. (12.53) and (12.57) provide:

= (12.64)

Combining Eqs. (12.63) and (12.64), we obtain

V'2 k — — E,,). (12.65)

Integration of Eq. (12.65) through the entire body and integrations byyield: —

fff dV—

F1v1 dS. (12.66)

Since is an arbitrary admissible velocity,Eq. (12.65) proves Markov'sprinciple. The second principle may be stated as follows:

Among admissible solutions which satisfy the equations of equilibrium,yield condition and the mechanical boundary conditions on S2, theactual solutiont'renders

= — dS (12.67)


This is equivalent to Hill's principle of maximum plastic work which statesthat among admissible solutions, the solution renders

fffan absolute rnaximum.(t, 8) The proof is as follows. Let the stress, strainrate and velocity of the actual solution be denoted by and v,, and theStress of an admissible solution by a. Then, from

= 2k2, = 2k2 (12.68)and � = 2k2 (12.69)wehave /

• — a,)a, 0. (12.70)

SubStitution of Eqs. and (12.57) into Eq. (12.70) yields:— � 0. (12.71)

Integration of. Eq. (12.71) throughout the body and integrations by partsprovide:

dS ffan arbitrary admissible stress, Eq. (12.72) proves H in

f Except for a possible indetcrmiaatc uniform hydros*atic


A weak statement can be made on the above two principles that thefunctionals (12.60) and (12.67) arc rendered stationary with respect toadmissible velocity and stress variations, respectively. For example, we canshow

611=0 (12.73)

for the exact solution with respect to admissible displacement variations.The principle can then be generalized into:.

= i'5. k fff )'4,4, dV— ff F1v, dS

— fff ([4, — (J) (Vg,j + v,•1)) — 4, dV

— ff — dS (12.74)S2

where au and a arc Lagrange multipliers which introduce the conditions(12.56), (12.57) an4 (12.59) into the variationalrexpression. The stationarycondition of the functional (12.74) with respect to C12 canbe shown to be

•

— (12.75)

and the expression of the functional (12.67) can be derived through theelimination of and in the usual manner. /

Ihalt Analysis

One of the most successful applications of variational formulations in theflow theory of plastIcity is undoubtedly the theory of limit

a conlinuum or structure, hereafter called a body, which consistsof a material obeying the perfectly plastic Prandtl—Reuss equations (12.50).Surface tractions I = 1, 2, 3 are prescribed on S1, and displacementsare prescribed on such that u1 = 0; 1 = 1, 2, 3. We assume that thesurface tractions are applied in proportional loading, that is, the externaltraction is assumed to be given by 1 = 1,2, 3 where u is a monotonicallyincreasing parameter. When the value of x is sufficiently small, the bodybehaves elastically. As x increases, a point in the body reaches the plasticstate; beyond this value of x the theory of elasticity is no longer applicable.As x increases further, the plastic region of the body spreads gradually,although larger parts of the body may still be in. the elastic state. If the valueof x continues to increase, a state of impending plastic flpw will be reachedin such a way that an increase of plastic strain under co'nstant surfacetions becomes possible for the first time during the loading process. Theset of surface tractions correspond to the impending plastic flow'is


called the collapse load of the body, and the ratio of the collapse load to thedesign load is called the safety factor and is denoted by S. Thus, the safetyfactor is the value of at the collapse load. The problem in limit analysisis to determine the safety factor of the body under the prescribed surfacetractions.

We observe that at the collapse load the elastic strain rates and stressrates are identically zero and the body behaves as Conse-quently, the equations governing the state of impending plastic flow maybe given as follows:(I) Equations of equiibriwn

0; (12.76)(2) Yield condition

op' � 21c2; (12.77)(3) Stress—strain rate relations

= where ap, 2k2, (12.78a)

= 0 where < 2k2; (12.78b)

(4) Rate ofstrain—velocity relation

= + vj,z; (12.79)

(5) Condition of incompressibility— 0; (12.80)

(6) Boundary conditions= on S1, (12.81)

vg =0 on S2. (12.82))

These equations constitute an eigenvalue problem In which the value of Sis determined as an eigenvalue.

The theory of limit analysis puts emphasis on the derivation of upperand lower bound formulae for the safety factor. We shaH restrict theproblemunder consideration to continuous stress and velocity fields for the sake ofbrevityt and introduce the following nomenclature. A set of stress compo-nents will be called statically admissible if it satisfies Eqs. (12.76), (12.77)and

an, = on S1, (12.83)

where m, is a number called a statically admissible multiplier. A set ofvelocity components uj' will be called ad if itEqs. (12.80) and (12.82), and the condition

fFgvtds>0. (12.84).Sj

t An extension to discontinuous velocity fields is found in Ref.

252 VARIATIONAL METHODS iN ELASTICITY PLASTiCITY

A number defined by

= (12.85)

will be called a kinematically admissible multiplier, where .= +We then obtain the following upper and lower bound formulae for thesafety factor:

(12.86)

The proof is given as follows: First, we observe that 'Eq. (12.63) is stillfor the present problem. Integration of Eq. (12.63) through the entire

body and integrations by parts yield:

Sff dS kfff dV (12.87)Si

which, under the assumption (12.84), proves S Second, we observethat Eq. (12.71) is still valid for the present problem. Integration of Eq.(12.71) through the entire body and integrations by parts yield:

(m3 — dS � 0. (12.88)

Since

ffF:vgdS�0Si

for the exact solution, Eq. (12.88) proves m, S.Thus, the upper and lower bound formulae for the safety factor have

been obtained by the simultaneous use of the two variational principles.In that sense, Eqs. (12.86) are analogous to the upper and lower boundformulae for torsional rigidity, which were derived in Section 6.5 from theprinciples of minimum potential and complementary energy, although thedetermination of rigidity is a boundary value problem and not aneigenvalue problem. A variational consideration is.given in Ref. 10 on the.bound formulae, Eq. (12.86).

The theory of limit analysis has been formulated for plane strain problemswhere detailed investigations have been made on the discontinuity of velo.-city and stress fields (see Ref. 2). An excellent example of the plane strainproblem is shown in Ref. 11: a prismatic cylinder having a square sectionand a circular hole at the center is under an uniform pressure,upper and lower bounds for the safety factor are obtained by assuming

velocity as well as stress fields. Limit theory has also provedvery powerful in the analysis of plates, shells and multi-component struc-tures (see Refs. 12 through 16).


12.7. Some Remarks

It has been assumed throughout Chapters 11 and 12 that the displacementcomponents are given in terms of three continuous functions. However,deformation in the plastic region is known to Consist of infinitesimalThis means that the representation of displacements by three continuousfunctions is only an approximation, and suggests that the theory of plasticitymay be improved by the discontinuous character of the displacementinto account. One of the advances in this direction is known as the theoryof dislocation, an qxcellent description of which is given in Ref. 17. A briefmention of variational principles in the theory of creep is made in Appen-dix 0.

Bibliography

1. R. HILL, Mathematical Theory ofPlasticity, Oxford, 1950.2. W. PRAOER and P. G. ix., Theory of Perfectly Plastic Solids, John Wiley,

1951.3. W. PRAGER, An Introduction to Plasticity, Addison-Wesley, 1959.4. P. G. HODGE, ix., The Mathematical Theory of Plasticity, in Elasticity and Plasticity

by J. N. Goodier P. 0. Hodge, Jr., pp. 51—152, John Wiley, 1958.5. D. C. Dxucicmi, Variational Principles in the Mathematical Theory of Plasticity,

Proceedings of Symposia in Applied Mathematics, Vol. 8, pp. 7-22, McGraw-Hill,1958.

6. W. T. KorrER, General Theorems for Elastic-Plastic SOlidS, Progress In Solid Mecha-rdcs, Vol. 1, Chapter IV. pp. 167—221, North-Holland, Amsterdam, Interscience;New York, 1960.

7. A. A. MARKov, On Variational Principles in the Theory of Plasticity, PrikladnaiaMatematika I Mekhanika, Vol. 11, pp. 3 39—50, 1947. (Translation prepared at Brown

8. It. HILL, A Variational Principle of Maximum Plastic Work in Classical Plasticity,Quarterly Journal of Mechanics and Applied Mathematics, Vol. 1, pp. 18—28, 1948.

9. D. C. W. PRAoaa and H. 3. Extended Limit Design Theoremsfor Continuous Media, Quarterly of Applied Mathematics, Vol. 9, No.4, pp. 38 1-9,1952.

10. T.. MuRA and S. L. LEE, Application of Variational Principles to Limit Analysis,Quarterly of Applied Mathemaiiës, Vol. XXI, No. 3, pp. 244-8. October 1963.

11. D. C. DRUCKER, H. J. G and W. PRAGER, The Safety Factor of an Elastic-Plastic Body in Plane Strain, Journal of Applied Mechanics, Vol. 18, No. 4, pp. 371—8,December 1951.

'12. DEN BR0EcK, Theory of Limit Design, John Wiley, 1948.13. B. 0. NEAL, The Plastic Methods of Structural Analysis, John Wiley, 1956.14. L S. Plastic Design of Steel Frames, John Wiley, 1958.15. P. CL Hocos, ix., Plastic_Analysis of Structures, McGraw-Hill, 1959.16. P. CL ix., Limit Analysis of Rotationally Symmetric Plates and Shells, Prentice-

Hail, 1963.17. B. A. BILay,f Continuous Distributions of Dislocations, Progress in Solid Mechanics,

Vol. 1. Chapter VII, pp. 329—98. North-Holland, Amsterdam, Interscience, New York,

1960.

APPENDIX A

EXTREMUM OF A FUNCTION WITHA SUBSIDIARYCONDIT1ON

WE SHALL a problem of finding the extremum of a function undera subsidiary condition., For the sake of simplicity, we shall take a simpleexample to illustrate the procedure.

PROBLEM: Find the extremum of the

z=f(x,y)=x2+y2—2x—4y+6, (1)

under the subsidiary condition:

g(x,y)=2x+y—1=O. (2)

Geometrically speaking, the problem is one of finding the extremal valueof z on the curve of intersection of z fix, y) and g(x, y) = 0. One of theways of solving the problem is to eliminate one of the variables, say y, fromEq. (1) by the use of Eq. (2), thus obtaining

z = f(x, y(x)) = f(x) = 5x2 + 2x + 3, (3)

and then finding the extremum of z by the condition.

df(4)

By solving Eq. (4), we obtainx = —

and find that '4ZCZIT, Y=,. (6)

It is ob$erved that this extremal value proves to be the absolute minimum off*(x)

The method of Lagrange multiplier asserts that the above problem isequivalent to finding the stationary value of the function z1 defined by

z,=x2+y2—2x—4y+6+2(2x+y--1),where the independent variables are x, y and 2 and the stationary conditionsare given by = 2x — 2 + 22 = 0,

(9)

8z1/D2=2x+y—1=0. (10)

254

APPENDIX A 255

By solving these equations, we obtain

y=, A=,and find the stationary value of z, as follows:

_14— T =

If one of the independent variables, say x, is eliminated from z, throughthe use of Eq. (8), the function is transformed into

z,1=y2+2y—22—4y+5, (13)

where there remain only two independent variables, namely A and y. Thestationary conditions are then given by:

(14)

- 8z17/öA—y—22+l=r0, (15)yielding immediately

(16)and consequently

— T — X —

Going further, we shall eliminate x and y from z, through the use of Eqs.(8) and (9). Then, the function is transformed into

z171 = + 3A + 1, (18)

where 2 is the only remaining independent variable. The stationary conditionis then given by

= + 3 0, (19)which gives

(20)and consequently _14_ — 1 _7

— 3 — 2czt, X — )'

It is observed that is the' absolute maximum of the function withrespect to the single variable A. It is obvious that if Eq. (10) is employed asa subsidiary condition, the function z1 to the original function.

Thus, it has been shown that the stationary values are the same for aJithe transformed formulations. The extremal value obtained as the minimumin the function (3) is given as the maximum in the function (18). HoweverZIat and z111 are no longer either maximum or minimum of the functions z,and z,1, respectively.

Bibliography

1. R. and I). IIILBERT, Mathematical Physics, Vol.1, Intersciencc,New York, 1953.

2. C. LANCZOS, The Variational Principles of Mechanics, University of Toronto Press,1949. . -

A THIN PLATE

Fr is a common practice to assume that the transverse normal stress may'be neglected in the stress—strain relations when formulating a thin plateproblem approximately. This assumption reduces Eqs. (1.10) and (1.11) to

F= (1 ' v2)

+ ye,,),

Ea, = (1 2) + e,), = (1)

= 0, Tx, =and

Es, = — Gy,, =Es,, + a,, = (2)

Es, = — + a,), =

respectively. The expressions for the strain and complementary energyfunctions for the above stress—strain relations can be shown to be

E GA

2(1 ,2) + + T + + — (3)

B = + a,)2 + 2(1 + + + — (4)

When Eqs. (3.38) are employed as the stress—strain relations to take accountof nonlinear strain—displacement relations, we obtan relationsing to Eqs. (1), (2) and (3) above by i,,, and with

e,,, e,,, 2e,,, and 2e2,,, respectively.The assumption that the transverse normal stress may be neglected in the

stress—strain relations is frequently employed in thermal stress problems ofa thin plate and reduces Eqs. (5.50) and to

E= (1

(es, + ye,,) — = 2Ge,,,

£ Es° -a, (1 — ,2) (se,. + e,,)

—

0, , = 2Ge,,,,256

APPENDIX B 257

and1 1= — Va,) + e°,

e,, = + a,) + e6, = (6)

= — + a,) + ri,,

respectIvely. The expressions for the strain and complementary energy func-tions for the above stress—strain relations can be shown to be

A = 2(1 _;2) + e,,)2 + + + —

— (1 — v)÷ e,,), (7)

B = [(as + a,)2 + 2(1 + r) + + —

+ + o,). (8)

When linear strain—displacement relations are employed, we obtain relationscorresponding to Eqs. (5), (6) and (7) above by replacing e,,, 2e,2,

and 2er, with e,, Y,z, and y,,, respectively.

APPENDIX C

A BEAM THEORY INCLUDINGTHE EFFECT OF

TRANSVERSE SHEAR DEFORMATION

WE SHALL derive an approximate beam theory, including the effect of trans-verse shear deformation, by employing the generalized principle of minimumcomplementary energy (2.41). Consider a beam of uniform cross sectionwhich is cTamped at x = 0 and in static equilibrium under terminal loads atthe other end x = 1. It is assumed that body forces and surface forces on theside boundary are absent; thus a torsion free bending is in(x, z) plane. The principle (2.41) is written the present asfollows:

wCX Cy C: CX ()y ÔZ

+ (terms on the boundary surfaces),

where u,. r' and w are displacement components and functions of (x, y, z).We assume:

=+ (2)

= Q(x) z), = Q(x) z), (3)

= -tv.. = 0. - (4)

It is seen that Eq. (2) is the same as Eq. (7.29). The two functions andin Eq. (3) are chosen to satisfy

+ — — Z(S

ay 7on the cross section, and

+ = 0 (6)

on the side boundary, where in and n are the direction cosines the normalv drawn outwards on the side boundary, namely m = cos(y, v) andn = cos(z, v). It is required that the functions and so chosen aregood approximations of the stress components and induced in the

258

APPENDIX C - 259

beam. Substituting Eqs. (2), (3) and (4) into Eq. (1) and employing Eq.for the expression of B, we have

fIN2 M2 Q2=J

+ 2E! + 2GkA0

+ N'u0 + (M' — Q) u1 + Q'(v0 + w0)J dx-t- (terms on the both ends), (7)

where it is defined that

kA0 = + dy dz,

anduoAo=ffudydz, u11=ffuzdydz,

v0 = ff dy dz, w0 ff dy dz,

integrations being extended over the cross section of the beam. In the func-tional (7), the quantities subject to variation are N, M, Q, U0, U1, andw0. We have the following stationary conditions:

N = M = E!u. Q = GkA0[(v0 + w0)' + u1J, (10)

N'=O M'—Q=O, Q'=O. (11)

Comparing Eqs. (10) with Eqs. (7.112), (7.113) and (7.114), we observethat, if the quantities u, u1 and w are interpreted as

u=U0, u1=..U1, w=v0+W0,

the two approaches provide equivalent formulations except for the valueof k as far as the present static problem is concerned. The values of -thetransverse shear rigidity are shown below for three cross sections.

(1) Rectangular cross section. Let the bre?dth and height of the section beF' and h, respectively, as shown in Fig. C 1. We then haveU)

(9k, 0, = (1/21) [(/,/2)2 — z2J,

A0=bh,

Circular cross section. Let the radius of the section be a as shown in Fig. C 2.We then 1)

(1 + 2v) I (3 + 2v) 1 1 2 2 1 — 2v2

= 4(1 + v) 7 = 8(1 + v) 7 — 23 + 2v

• A0 = ra2, I = k = 0.851 for v 0.3.

(3) A single-celled, closed, thin-walled circular tube. In a thin-walled tube, theshearing stress on the cross section is assumed to be in the direction of the

260 APPENDIX C

periphery of the For a tube with the radius a and constant thick-ness t shown in Fig. C 3, we have

= (1/,rat)cos2O,

A0=2'wi, k=1. . (15)

y

Fio. CI. A rectangular section.

KI

Fia. C2. A circular section.

F,o. C3. A thin-walled, circular tube.

APPENDIX C 2(1

- Bibliography

1. S. and J. N. Theory of Elasticity, McGraw-Hill, 1951.2. Y. C. FuNG, An Introduction to the Theory of Aeroelasticity, John Wiley, 1955.3. D. J. PEERY, Aircraft Structures, McGraw-Hill, 1949.

APPENDIX D

A THEORY OF PLATE BENDINGINCLUDING THE EFFECT OF

TRANSVERSE SHEAR DEFORMATION

WE SHALL derive an approximate theory of plate bending including theeffect of transverse Shear deformation, employing the generalized principleof minimum complementary energy (2.41). We prescribe that the thin plates in static equilibrium under mechanical boundary conditions

(1)

on S1 and geometrical boundary conditions

U, P. W (2)

on S2, while the external forces to the upper and lower surfaces aregwen by: = o, = 0, - = p, on z = 1,12,

= 0, ir,, = 0, = 0, on z = —h/2. (3)

The body forces are assumed absent. The principle (2.41) can be writtenfor the present problem as follows:

• dxdydzOx Oy Oz / Ox Oy öz /

+ + or,m)P +

+ (integrals on and z = ±h/2), (4)

where I (x, ,') and m = cos (y, v). Following Refs. 1, 2, 3 and 4, wemay choose,

— z M, z — M, z— (h2/6) (h/2)' (h2/6) (h/2)' — (h2/6) (h/2)

=

—

[i— (z)2]

=— I +

(5)

262

D 263

Substituting these equations into Eq. (4) and employing Eq. (2.21) for theexpression of B, we obtain

—iii + if,)2 + 2(1 + — MXM,)SIN

3,

M,)1 +÷ôMx,

5

a

— f [(MJ + M,m) P1 ± + Q,m) W0J dc

+ (integral on C1), (6)

where k = 5/6 and it is defined that'12' r

=Uzdz,

= (4J)fuzaz. pi= = [1 -(7a, b, c,'d,e, 1)

The quantities subject to variation in the functional (6) are M,, M,,,Q,, U1, v1 and w0. The statiopary conditions can be shown to be the

equations of equilibrium,

3Mg, +3X+ a)? — Dx Dy

(8a,b,c)Dx Dy

and the stress resultant—displacement relations,

Dau1 V

= h( Dx Dy / 10(1 v)

a)

M, = D(v Du1 Dy1+ 1+

', ax Dy i 70(1— v) (9b)

Gh3 f 8u1 Dy1

+ Dx)' (9c)

Q. Gkh ( + Q, = Gkh + (9 d, e)

together with the geometrical boundary conditions,

= Ui., v1 I?1, w0 — on C2. (10)

APPENDIX D

The above results suggest that the mechanical boundary conditions can bespecified approximately as follows:

+ = + M,,n =

Q,m on C1. (11)

Unless the surface ioad p is highly concentrated, the last terms in Eqs.(9a) and (9b) may be neglected in comparison to the preceding terms. Then,comparing the above equations with those derived in Section 8.8, we ob-serve that, if the quantities u1, t1 and w are interpreted as

= U1, v1 = v1, w = v,'0. (12)

the two approaches provide equivalent formulations except for the valueof k as far as the present static problem is concerned.

Bibliography

1. E. REISSNER, On the Theory of Bending of Elastic Plates, Journal of Mathematics andPhysics, Vol. 23, No. 4, pp. 184—91, November 1944.

2. E. REISSNER, The Effect of Transverse-Shcar Deformation on the Bending of ElasticPlates, Journal of Applied Mechanics, Vol. 12. No. 2, pp. 69-77, June 1945.

3. E. REIssNER, On Bending of Elastic Plates, Quarterly of Applied Mathematics, Vol. 5,No. 1, pp. 55—68, April 1947.

4. E. KEISSNER, On a Variational Theorem in Elasticity, Journal of Mathematics ,ndPhysics, Vol. XXIX, No. 2, pp. 90-5, July 1950.

APPENDIX E

SPECIALIZATIONS TO SEVERAL KINDSOF SHELLS

EXPLICIT expressions of the geometrical qvantities defined in Chapter 9are shown for several kinds of shells in the following:I. Flat plate (see Fig. El)

= (dx)2 + (dy)2.

(1'22 =

iw ,

= * =(.t C)'

(U (1 c3u= = yx, = — ÷ —,CX C)' (3)' oX

02w

=

A = 1, 8=1,

ill(3$'

112

=

(32

xx = =

AZ

x

Fia.EJ. Flat plate.265

.3uIOx

Ov

Ow'31 =

t3u

— Ox

82w,ex =

3. Spherical shell (see Fig. E3)

x=asinq'cosO, y=asinq'sinO, z=acosq.= +

x=O, A=asinç, B=a, Rp=a.

266 APPENDIX E

2. Cylindrical shell (see Fig. E2)

x=x,= + (a dq)2.

A=I, B=a,IOu

I (Ov122 w).

1 lOw132

I I Ov 1

I 02w Ov\ 11 82w Ot"'C +—).

q' Ox!

EL Cylindrical shell.

111_i k au I 8u

1 &' 1I&'121=1.a

4. Rotationally symmetric shell (see Fig. E4)

= sin, + (R,/I=p, B=R,.

I öu1=— + — '12 R,

'21 = ( sin— u cot 122 = — w).

I 'Ow1 Ow/32

APPENDIX E - 267

1

(sinc,/ Ow

/32

z

Fio. E3. Spherical shell.

268 APPENDIX £

x

y

Fio. E4. Rotationally symmetric shell.

APPENDIX F

A NOTE ON THE HAAR—KARMANPRINCIPLE

EQUATiON (11.42) shows that the Haar—Kármãn principle does not possessa stationary property in the conventional sense. However, if the subsidiarycondition in the form of inequality:

(1)is written as

2k2 — — = 0, (2)where z is a real variable, the functional (11.35) can be generalized as fol-lows:U. 2)

11*

+ fffF_.cr.j.,ui + Q/2) (2k2 — v,fl, — z)J tiz

+ ff — F1) U1 dS, (3)

where u1 and A are Lagrange multipliers which introduce the equations ofequilibrium, mechanical boundary conditions and the yield condition intothe variational expression. It is interesting to observe that the stationaryconditions of the functional (3) with respect to and .z provide:

(l—2v) 1 ,+ =

— E Q&j + + (4)

zA=0. (5)The solutions of Eq. (5) are

z=0, 2=0. (6a,6b)

The first solution corresponds to the plastic state, while the second onecorresponds to the elastic state.

Blbliograpby1. F. A. The Problem of Lagrange with Differential Inequalities as Added

Side Conditions, in, to ;he Calculus of Variations, 1933—1937, Univer-sity of Chicago Press, 1937.

2. A. MIELE, The Calculus of Variations in Applied Aerodynamics and Fliglfl.Mechanics,in, Optimizalion Techniques wish Appllcatwns to Aerospace Systems, edited by G. Leit-mann, Academic Press, 1962.

269

APPENDIX 0

VARIATIONAL. PRINCIPLES INTHEORY OF CREEP

DEFORMATIONS of materials consist not only of elastic and plastic strains,but also of a time-dependent portion, especially at elevated temperature.This portion of the deformation proceeds with the lapse of time, even underconstant external loads, and is known as the phenomenon of creep.t Creepdeformations in structures cause changes in shape, changes in stress distribu-tion and such instabilitIes as creep buckling. Consequently, creep is consi-dered to be one of the decisive factors in the analysis of structures exposedto high temperatures.

Several proposals have been made on the establishment of variationalprinciples in the theory of creep. Wang and Prager have formulated varia-tional principles for a boundary value problem defined (using the notationof Chapter 12) as A body of work-hardening plastic material isassumed to have been deformed, including creep behavior, and at the time 1occupies a region V bounded by a surface S. It is also assumed that thetemperature 8, the stress a,j and the state of are knownthroughout V. We now prescribe an infinitcsjmal temperature changethroughout V. infinitesimal changes of the surface tractions on andinfinitesimal changes dug of the surface displacements on S2. Given relationsbetween incremental components of elastic, plastic, thermal and creep strain,denoted by d4, d4, and d4, respectively, and incremental componentsof stress, temperature and time, the problem is then to find the stress in-crements and displacement incremeflts hi, induced in the body. It isunderstood that the sum of the thermal and creep strains, d4 + d4 can be

• taken as initial strain increments, and the problem is thus reduced to aboundary value problem of a body with initial strain increments in the flowtheory of

Sanders, McConib and Schlechtc have formulated another variationalprinciple for a boundary value problem which may be defined (using thenotation in Section 5.5) as consider that the stresses and thedisplacements w' are known at the time 1. Given the surface force rates P.the surface displacement rates I)', the body forces rates together with

t Refs. I through 5.270

APPENDIX 0 271

relations between stress rates and strain rates, the itress rates andthe displacement rates u' induced in the body. It is understood that the creepstrain rates 4, can be taken as initial strain and the principles derivedin Section 5.5 may be employed for the establishment of variational prin-ciples.

Blbliograpby

I. F. K 0. ODovIsr, Rçcent Advances in Theories of Creep of Engineering Materials,Applied Mechanics Reviews, pp. 517-19, December 1954.

2. N.J. Hon, Approximate Analysis of Structures in the Presence of Moderately LargeCzeep Deformations, Quarterly of Applied Mathematics Vol. 12, No.1, pp. 49-55,April 1954.

3. T. H. H. Stress Distribution and Deformation D.w to Creep, Aerodynamic Hcat-ing of Aircraft Structures in High Speed Flight, Note; for a Special Summer Program,Department of Aeronautical Engineering, Massadiusetta Institute of Technology,pp. 15-1 to 15-34, June 25-July 6, 1956.

4. W. PRAOER, Total Creep and Varying Loads, Journal a! the Asro..aaitkal Sciences,Vol. 24, No.2, pp. 153-5, February 1957.

5. N.J. Hon. editor, High Zifects in Aircr4ft Structures, AGARDograph28, Pergamon Press, 1958.

6. A. 3. and W. PlAceR, Thermal and Creep Effects in Work-Hardening Elastic-Plastic Solids, Vol. 21, No. 5,.pp. 343-4, May 1954.

7. K. Wauiuzu, V Principies In Elasticity and Plasticity, Aeroelastic and Struc-tures Research Laborajory, Massachusetts Institute of Technology, Technical Report25—I 8, March 1955.

8. J. L SANDERS, JR. H. 0. McCotm, Ia, and F. R. San.sana, .4 VariatlonsI Theoremfor Creep with Application to flairs and Columns, NACA TN 4003,1957.

9. T. H. H. PlAN, On the Variational Theorem for Creep, Journal of the Aeeanam'lcalSciences, Vol. 24, No. 11, pp. 846-7, November 1957.

APPENDIX H

PROBLEMS

CHAPTER 1

Problems Related to Sectioss 1.1 aid 1.2

1. Show that by use of Eqs. (1.5) and (1.10), we may express Eqs. (1.4)and (1.12) in terms of displacements as follows:

1 8e Izlu + I 2v Ti = 0

1 .9e F 0, (i)

IJw+i_2,Tj

G 2u 2v

+8v' '8u 8w' 1

and8öx+ 1—2v / ôx/ 0x/ I

43v 3u 2v Ir3v 8w' 1

G —+—)!÷(2—+ e[(ax ay, öy 1 — 2v )

Ow Ou' 'Ow Ov' .1 Ow 2v ' 1G —+—Il+(—+——lm+i2—+ elni=Z,,[(Ox OZJ Oz/ Oz I — 2, / j

respectively, wherezl( ) = ( + ( ),,, + ( e = + v,, +( = O( )fOx, ( ),, = O( )/Oy, and ( = 0( )/Oz. Show also thatthe elasticity problem is reduced to solving equations (i) under the boundaryconditions; equation (ii) and Eqs. (1.14).

2. Show that if the body forces are absent, the conditions of compati-bility, Eqs. (1.15), can be transformed, by the use of Eqs. (1.11) and (1.20),into

1 820=0.0, LIT1+v Ox2 1+vOy6z

LI +1 820 1= 0, +

1 + v OzOx= (i)1+v

+1 020 1=0, l+vOxOy0'1+v

where 4( ) = ( + ( ),,, + ( and 0 = + o, + Showalso that if it is further assumed that the boundary conditions are given

272

APPENDIX H 273

entirely in terms of forces, the elasticity problem is reduced to solvingequations (i) and Eqs. (1.20) under the boundary conditions X, =Y,=

3. We consider two sets of rectangular Cartesian coordinate systems(x, y, z) and j, and denote strain and stress components defined withrespect to these systems by e,, ..., yx,; and ...,,;

respectively, where an overbar is used to distinguish be-tween the two coordinate systems. For the sake of brevity, we shall alsoemploy the following notations frequently:

x=x1, y—x2, z—x3, z=x3,

V,.z = €23 = +V2x = €13., +Yx, = €12 €21,

4YX,=E12=€21;= 1, = C22, C2 = c33, = C23, = 032,

TXZ=U13, Tx,=C12,T,x=C2l.

(1) Show that the following relations hold:3 3

= E 2J cos (i,, Xm) cosm-4 n=l

3 3(1)

= cos (i,, x_) cosrn—I il==1

(2) Show that The following quantities are invariant with respect totransformations from one set of rectangular Cartesian coordinates toanother:

LX + Ep + €z,

+ + — + vL +£x€,Ez + + (Y7zV:xYxy —

+ C, + (ii)

+ + a'a, — + +OxC,Cz + — + +

Show also that these quantities may be written as follows:3

.�:; —2 ,=i

13

i-I. elJ* 1*1 Ek:,—'

2

1 LX �XL CJyCkli3.

274 APPENDIX H

respectively, where,

= 0 when any two of i,j, k are equal,+1 when i, j, k are an even permutaion of 1, 2, 3, (iv)

= —l when i,j, k are an odd permutation of 1 2, 3.

(3) Show that there exist only two independent elastic constants for anisotropic elastic body.

Problems Related to the Conditions of Compatibility

4. Show thatQ

u(Q)= u(P) + + — + +p

Q

v(Q) = v(P)+ + + e,d; — (i)

pQ

w(Q) = w(P) + f {(-— w,) dx + (-i- + dy +

P

apidQ

=+ f — dx

___

—

p

öy 2 ôz/Q

= w,(P)+ f .-! dx

+ — dy2 2t0z Ox)

p (ii)

+ —2 Oz 3xJ J.

Q

wt(Q) = w1(P) +ff(4 —1

8y/p

+1

—.3•y, j

where w, and are components of rotation defined by..3w Ot' Ou 8w r3v Ou

= — 20, = — 2n. = —a,

APPENDIX H 275

while P and Q are two arbitrary points in the bcxiy and integrations aretaken along an arbitrary path bettiveen two points P and Q.

Next, by using these relations, show that the conditions of compatibilityare given by Eqs. (1.15) for a simply-connected body.

5. Consider a doubly-connected body as shown in Fig. H 1, and reduceit to a simply-connected body by means of a barrier surface Q. Take anarbitrary closed circuit C which has initial and final points on Q and cannot

be contracted to a point without passing out of the body. Applying equa-tions (1) and (ii) of Problem 4 to the circuit C, show that even if strains ofthe body are continuous and satisfy the conditions of compatibility, Eqs.(1.15), we have

U1 — Uj = 11 + Paz —

V1 — V1 = 12 + 03X — p1z,

W1 W4 13 + PIY P2X,

where '2' 13 and P2. P3 are constant, and the suffixes f and i arereferred to the final and initial values, respectively. Note: see Ref. 1.1,pp. 221—8, and Ref. 1.20, pp. 99—1104

Problems Related to SectIons 1.6 and 1.7

6. Show that the two-dimensional elasticity problem treated in Section 1.7is reduced to solving the following equations:

t Ref. 1.1 denotes Ref. 1 in the of Chapter 1.

Fio. Hi.

276 APPENDIX H

(1) Displacement method: solve the differential equations

1+,' öe 1+vLlu+ —=0, ——=0 in Sl—vayunder the boundary conditions

[(3u v.)1.t. I —v

+ =(ii)

2f (— + ___)i÷ + = y, c,(1—i') 2 Ox Oy ax ay

where 4( ) = ä2( )/ax2 + 32( )/c3y2 and e = +

(2) Force method: solve the differential equation

zIJF=0 in S (iii)

under the boundary conditions

- dfôF\— T on C, (iv)

where zJJ( ) = )/0x4 + 2a4( )/0x2 e3y2 +

7. Show that the principle of complementary virtual work for the two-dimensional problem treated in Section 1.7 may be given by

ff (ExöCx + e,ãa, + —f(uâX, + vôY,)ds = 0,

where a,, ,, X, and Y, have been expressed in terms of F by use ofEqs. (1.25) and (1.57), and ii, v are Lagrange multipliers. Show also that wemay derive from the above principle the fo'lowing equations:

+ E,, — Yxy. = 0in Sand

u(s)=

dx + (+ Vxy — dy] + ay + b,

v(s) + + e,dy] — ax + c

on C', where

= I [G yr,. — dx + x — + Yxy. y) dY],

while a, b, c are arbitrary constants and s is measured along the boundary C.

PPENDIX H 277

Problem Related to Section 1.9

S. We consider, as an example of Eq. (1.77), a two-dimensional problemin which the stress field is continuous, while the displacement field has aline of discontinuity as shown in Fig. H 2. To begin with, it is assumed

that the line of discontinuity, denoted by dividea the two-dimeusiopalbody R into two subregions and R(2). Two unit vectors, and t(12),are defined on C(12) such that V(12) is the unit normal drawn from R(l)to R(2), and t(12) is obtained by rotating V(12) in a counter-clockwisedirection through 900. The stress components a,, r,) are assumedcontinuous throughout the body R and to satisfy Eqs. (1.24) and (1.53).The tangential stress transmitted across the C(13) linó from R(2) to is

denoted by T(12) and taken positive when it is acting in the direction ofThe displacement components (u, v) are assumed continuous in each ofthe subregions. The displacement components on C(12) of the subregionsR(1) and R(2), taken in the directions of P(I2) and are denoted byV,(1), V,(2) and Vg(j), Vft2), respectively, and continuity of the normal com-ponents, i.e. V,(1) V,(2), is assumed. Then show that we have the diver-gence theorem as follows:

ff + a,e, + dx dy

f + Y,v) ds + f T(12) [V1(1) — ds(12),C C(12)

where Eqs. (1.52) are assumed to hold in each of the subregions. Show alsothat the above relation holds even if the line C(12) does not extend betweentwo points on the boundary, but is a line segment contained in the region R.Note: see Ref. 1.21, pp. 209—13.

Fia. U 2.

278 APPENDIX H

CHAPTER 2

Problems Re'ated to Sections-Li and 2.2

1. Prove Kirchhoif's theorem that the solution of the elasticity problempresented in Section 1.1 is unique.

2. Show that the stationary conditions of 17 defined by Eq. (2.9) are co-incident for an isotropic body with equations (i) and (ii) in Problem I ofChapter 1.

3. Show that, for the two-dimensional problem treated in Sect ion 1.7,the functional for the principle ofminimum complementary energy isgiven by

I ö2F 2 32F 2 Ô2F a2Fil'Jr = ff ++ 2(1 + v) - dx dy

and that the stationary condition is coincident with equation (iii) in Pro-blem 6 Of Chapter 1.

Problems Related to Quadratic Functions

4. We consider a quadratic function with n variables x1, x2, ...,

f(x1, x2, ..., = {x}' [Al (x}

whcre is a symmetric matrix, and is a column matrix:

a11 ...

(x}=

and ( )' denotes the transposed matrix of ). Show that the functionfis positive definite if and only if

D1 > 0, D2 > 0, ..., 0 (1)

where D1, D2 are the principal minors of the matrix A defined by

a11 a12 a11,

a1 , a2 ..., = : (ii)

Note: see Ref. 2.42, pp. 304—8. The relations (i) are useful, Cor exampi;in deriving some relations of inequality among the elastic constants from

APPENDIX H 279

the assumption that the strain energy function is a positive definite functionof the strain components.

5. We consider a function with n variables x1, x2, ...,

(i)

where [A] is a positive definite symmetric matrix, {x} is a column matrix,and {b}' is a row matrix:

{b)' [b1, b2, ..., be].

Show that the stationary conditions off are given by

[A1{x) = {b}

and that the minimum value off is given by

— [A] = — j (b)' (xi,)

where (x3,) denotes the solution of equations (ii).

Problems Related to the Concept of Function Space

Here we cohsider the elasticity problem defined in Section 1.1, assuming,however, that body forces are absent for the sake of simplicity.

6. Show that the' principles of minimum complementary energy andminimum potential energy are given by

(S', S') — [S'S']2 S) — S]2 (I)

and(S", S") — [S", (S, S) —

in vectorial notations, respectively, where

S is the exact solution,S' satisfies Eqs. (1.20) and (1.12),S" satisfies Eqs. (1.5) and (1.14).

The brackcts and denote surface integrals on S1 ans S2. respectivelysuch that

[S", S]1 = •fj- (u"1, + + dS,

S']2 = ff (uX' + iY +

The bracket is so defined that it contains the displacement components ofthe first vector and the stress components of the second one. Note: seeRef. 2.20.

APPENDIX H

7. We cOnsider a special case of the elasticity problem where the boundaryconditions are given by

xv = 1v' = ?,,, = on (i)and

u = r = w = 0 on S2. (ii)Let us take

S' = + S" (iii)•

and determine a,, (p 1, 2, ..., m) and bq (q = 1, 2, ..., si) so that theymake

(iv)and

3(S", S") — ES", (v)

minimum, respectively, where

satisfies Eqs. ((20) and equations (i),satisfies Eqs. (1.20) and homogeneous boundary conditions on S1,namely,

xv = Y, = 0,satisfies Eqs. (1.5) and equations (ii).

Then, show that we have the following inequalities:

(S", S") (S, S) (S', S'). (vi)

8. We consider another special case of the elasticity problem where theboundary conditions are given by

S1. (i)and

u=ü, v=i3, w=* on S2.. (ii)Let us take

S' = S" = + (iii)p=1

and determine a,, (p 1, 2, ..., m)and bq (q = 1,2, ..., that they makc

(S', S') — (iv)and

3 (S", S") (")minimum, respectively, where

1, satisfies Eqs. (1.20) and equations (1),satisfies Eqs. (1.5) and equations (ii),satisfies Eqs. (1.5) and homogeneous boundary conditions on 52,namely,

U V = W = 0.

APPENDIX H 281

Then, show that we have the following inequalities:

(S'. S') (S. S) � (S", S').

Note: From equations (vi) of Problem 7 and equations (vi) of Problem 8,bounds formulae for some scalar quantities are obtainable as exemplifiedin Section 6.5. See also Ref. 2.43.

9.' Obtain the following vectorial equations:

(S". S) [S", + ES", S]2, (i)and

(S, S') = [S, S']1 + (ii)

where S. S', S" are defined in the same manner as in Problem 6. Discussrelations between equation (i) and the unit displacement methOd, and alsorelations between equation (ii) and the upit load method. Note: See Ref. 2.14for the unit displacement method and the load method.

10. We choosç a vector S*, having displacement components

U' = a11x + a12y + a13z,

= a21x ÷ a22y + a23z, (i)

w' = a31x + a32y + a33z,

where Cik (i, k 1, 2, 3) are constants. Show that we have

(S,S') =

ff (u'I,, + v'Y, + w'Z,] dS (ii)Si

or

(S. S') = fff + + '+ dV

= ff 4 vY," + dS (iii)S,+S2 I

where S is the exact solution. Note: The above relations show that if theboundary condiiions are given either entirely in terms of forces or entirelyin terms of displacements, we can calculate the average value of stressesor strains of the exact solution.

Problems Related ;o Section 2.6

11. We consider an elastic body which is held fixed on S2. We apply4two systems of body forces plus surface forces on S1:

I, F, 2, X,, F,, 2,; 1', F', Z', F",

282 APPENDIX II

to the elastic body independently, and denote displacement componentsdue to these forces by

u. v, w; v,

respectively. Then show that Maxwell—Betti's theorem

fff + ?v* + Zw*) dV + ff + F,v* + dS

+ ?v + Z*w)dV+ ff(X,u + F,v + Zw)dS

ho'ds between them.

12. Show that for a concentrated moment A? on S1, Castigliano's theoremprovides:

jay(1)

where 0 is the rotational angle of the local surface (where 2 is applied) inthe direction of 2.

13. Examine relations between the unit displacement method andEq. (2.49). Examine also relations between the unit load method andCastigliano's theorem.

Problem Related to Variational Principles of Elasticity

14. We-divide the ejastic body treated in Section 1.1 into two parts V(j)and V(2) fictitiously, and denote their interface by S(12).

(1) Show that the functional for the principle of minimum potentialenergy, Eq. (2.12), can be written with the use of Lagrange multipliers Px,p, and Pz as follows:

17 = fff [A(u(l), V(l), — (Iu(l) + ?V(1) + ZW(l))] dVV(J)

+ fff[A(u(2),V(2)

— ff + Y,V(1) + dSSi /

+ ff — U(2)) + p,(v(j) — V(2)) + — W(2))] dS (I)3(12)

w crc it is without loss of generality that S1 belongs to 1)' Thein ependent subject to variation in the functional (i) are

,V(j), W(j), U(2), V(2), W(2), Px, pg under subsidiary conditions Eqs. (1.14).Derive also the stationary conditions of the functional (1).

APPENDIX H 2S3

(2) Show that, by the use of Lagrange multipliers qx, q, and q,, the func-tional for the principle of minimum complementary energy, Eq. (2.23), canbe written as follows:

17c = fff B(oX(l), C.v(I), ...,V(t)

+ fjf a,(2), •, Tx,,(2)) dV2)

— ff(uX(2) + +

+ ff + X,(2)) + +S(12)

•+ qZ(Zl,(I) + Z,(2))JdS,

where it is assumed without loss of generality that S2 belongs to V(2).In defining X,(1), ... and Z,(2) on the interface, the outward normals areemployed: the unit normal drawn from V(1) to V(2) is used in defining X,(1),Y,(1) and Z,0), while the unit normal drawn from V(2) to V(1) is used indefining X,(2), and Z,(2). The independent quantities subject tovariation in the functional (ii) are Gc(l), ..., •, ;(2) q,and q,, under subsidiary conditions Eqs. (1.4) and (1.12). Derive also thestationary conditions of the functional (ii).

Problems Related to VariatipOal Formulation

15 We consider an eigenvalue problem of a function u(x) defined in

dI du)/ (I)

with boundary conditions

u'(a) — ocu(a) 0, u'(b) + fiu(b) = 0 (ii)

where is a parameter related to the eigenvalues, and and are specifiedconstants. Show that we have a variational expression for the eigenvalueproblem as follows:

b

H = + ru2 — 2u21 dx

a (iii)1 1

+ xp(a) + flp(b) [u(b)]2,

where the function subject to variation is u(x).

284 APPENDIX H

16. We consider a heat conduction problem, the field equation of whichis given by

+ Q,•, + Q,, =

where Q,, Qj is heat flux and is the heat source. The relation betweenheat flux and temperature gradient is assumed to be

Qxl [c11 c12 c13

Q' I = — I c21 •c22 c23 J 0,,

Q11 ic31 c32 c33 [0,.

where 0 is the temperature, and the are constant and symmetric:

c,1 = c,,;. i,j 1, 2, 3.

The boundary conditions are assumed to' be

QJ + Q,rn K 0) on S1and

0=o on• S2,where (I, m, n) arc direction cosines of the normal drawn outwards from theboundary, K is a constant and 0 and 0 are prescribed. Then show that wehave the followingvariational expression for this problem:

11 = fJf + + c33

+ + + 2c12

— ff KØIO — +02)dS, (vi)

where the-functio!f subject to variation is 0(x, y, z) under tile suBsidiarycondition (v).

CHAPTER

Problems Related to Section 4.1

1. Vectors and tensors are systems of numbers or functions whose com-ponents obey a certain transformation law when the coordinate variables inthe space undergo a that:

= 2 = 1, 2, 3. (I)

An overbar is used to distinguish between two coordinate systemsand A system vA is called a Contravariant vector if its components in

APPENDIX H 285

the new variables satisfy the relations:

(ii)

Similarly, we define a covariant vector by

(in)

a contravariant tensor of order two -a4 by

-(iv)

a mixed tensor of order two asp by

(v)

and a covariant tensor of-order two a4 by

- -• (vi)

In general, a system au:: is called a tensor when its componentsin the new variables satisfy the relations:

ae= ... (vii)

(1) Show that the quantities v4 defined by Eq. (4.15) and v2 defined byEq. (4.18) are contravariant and covariant vectors, respectively.

(2) Show that the quantities defined by Eq. (4.6) and defined byEq. (4.7) are covariant and contravariant tensors of order two, respectively.

(3) Show that the quantities defined

(viii)vi

are a contravariant tensor of order three, where e1" is defined by equa-tions (iv) of Problem 3 of Chapter I, and g is given by Eq. (4.28). Note:see Ref. 4.1, pp. 10—12.

(4) A tensor of order two can be given by any one of the following threeforms: aA; and Show that any one of them can be changed intoanother form by use of the principle of raising and lowering an index of acomponent of the tensor such that

286 APPENDIX H

2. We employ the transformation represented by equation (i) 0. Prob-lem 1 again. By use of the relations:

3 / \= =

show that we have

— = (1)

where is the Christoffel 3-index symbol of the second kind in the ia

coordinate system. By use of Eq. (i), show that i/., defined by Eq: (4.17) aswell as VA., defined by Eq. (4.21) are tensors of order two. Show also that

defined by Eq. (4.22) is a tensor.

3. Discuss geometrical relations between g defined by Eqs. (4.5)and (4.8), respectively, and show that

g1 x g1 =

4. We consider a special case of curvilinear coordinate systems:

= g1 1 + 2g12 d& d22 + g22 +

where g11 , g12 and g22 are functions of X1 and only. Obtain the followingrelations for the Christoffel symbols:

Iii 1

1 iil — g32g11•2),

12 1 22 22

lii =(g21g11,1 + 2g g21,1 — g g11,2),

• U2!(g'2 g22,2 + 31)

1221 =+ —

1121 = 12'iJ+

1 22 + 23 )114 — 1211

— g22,1 g

and all the other are zerb."V

APPENDIX 11 287

Show also that if the variables (&, constitute an orthogonal curvi-linear system, namely, g12 = 0, we have

111 — 1 121 1 ÔBlii I A ocx ' 1221 — B 0fl

— A OA B ORii 11 B2 1221 — A2 ocx'

111 11) 1 04 121_ 121 1 OB

112J 1211 A Ofl' 1121 1211 B

where = cx,cx2 A2 and g22 = B2.

S. We consider an orthogonal curvilinear system defined by

= A2(1— C) (dci)2 + B2 (i — C)2

where A, B, R5 are functions of (cx, only. Choosing cx' cx,

= p = show that the Christoffel symbols at 0 are obtainedas follows:

111 — I 04 121 AOA

1 OA 2 12112!o = 1211o = {12}o = 1211o

11 111 1 121 121131° = — 13lJo

0,

(1 — BOB 121 lOBA2&r.'

(11 — ('1 121 (21

_

1231o — 1321o= 0,

—

('1 —o (211331 o — '

= 0,42 B2

— — R0'— J3 —

— 121 — 1311o = 1231o — — 1331o —

where 1 denotes the value of I 1 at C = o.(I"Jo LFi'I

Problems Related to the Conditions of Compatibility and Stress Functions

6. We consider a simply-connected body, assuming that strain compo-

are given as functions of(&, cx2, tx3), and {{j} are expressed

288 APPENDIX H

in terms off4 by use of Eqs. (4.34) and (4.36). Show that Eqs. (4.42) arenecessary and sufficient conditions for the following equations to be inte-grable:

= GA dxa,= IIZJ}

That is, they are necessary and sufficient conditions for, the existence ofsingle-valued vector functions r and Ga. Obtain relations between equations(1) above and equations (1) and (ii) of Problem 4 of Chapter 1.

7. Show that the conditions of compatibility before deformation are givenby

—o )where is defined from i,y replacing

A

j) •.. with

and discuss the meaning of these conditions.

8. We confine our problem to small displacement theory. Show thatEqs. (4.40) and (4.53) reduce to

= 4 V;p +4 (ii;, + vay.;*), (I)

andT;u + 0, (ü)

respectively. Show also that equations (1) can be derived from equations (ii)by use of the principle of complementary virtual work. Note:

a11a1gi3ai'

9. We confine our problem to the small displacement theory. Show thatthe curvature tensor is reduced to

Rjqg.,w = + fLy. jw — Av

If)1AwI —

imp

or

+ 2 + IrII — —(i)

+ — — f,w; (ii)

where, by definition,

= 4 + — (iii)

Show also that if= (iv)

APPENDIX H 289

the conditions of compatibility given by.

S'1" =0 (A,u= 1,2,3), (v)

where Lb" isdeflned by equation (viii) of Problem 1. Notes: (1) asdefined by equation (iii) is not a tensor (see Ref. 4.1, pp. 61—2). (2)= IfJp;v1;.

10. We confine our problem to the small displacement theory. Show thatthe principle of virtual work may be written as follows:

•

— fffv ... =0 (1)

where are Lagrange multipliers. Show also that we obtain(ii)

from the principle, thus demonstrating that a symmetric covariant tensorplays the role of stress functions in the small displacement theory expressedincurvilinear coordinates. Notes: (1) =0. (2) see Section 1.8 for similar

Related to $ Skew Coordinate Systemt

11. We consider a two-dimensiQnal skew coordinate system ti):

as shown in Fig. H 3, where m is a constant. We confine subsequent formu-lations to small displacement theory and choose & = =

t Ref. 4.14.

yTi

y

FiG. 1-13.

290 APPENDIX H

(1) Derive the relations:1, = COStX, Y.E = 0,1, = —COt = 0, COSCC

= 1, g22 = 1, g12 = g21 = = sins,g1' = cosec2 x, g22 = = g2' = —cos cosec2

111 = 122 = + e,sin2.€x + yxj, Sin i%

21,2 2f2, = +1 = + a, cot2 tx — cot r22 = a, cosec2

= = cot + r, cosec(2) Denoting the displacement 'vector by

i=ug1+rg2,where g1 and g2 are the unit vectors along the E- and respectively,show that

= (u + 123= + v),,

21,2 = (u + + (uCosoc + v),,.

(3) Show that the condition of compatibility is given by

+f22.ff — =0.(4) By use of the principle of virtual work, show that the equations of

equilibrium are given by

(5) By use of the principle of virtual work combined with the conditionof compatibility:

ff — ô122et — sin + = 0,

show that the stress components are expressed in terms of the stress func-tion F: — r 22 — 12 — 21 —T T T —T —

(6) Show that. if the stress—strain relations are given in the (x, y)• coordi-nate system as

cx I, 0

a,= (1

E,,1)V 0

- 002 Vx,

or inversely1—v 0

sy I 0

0 0 2(1+v)

APPENDiX H 291

then from Eqs. (4.76) we have stress—strain relations in the (E. coordinatesystem as follows:

1 +

= (1 1. v2)+ I

I + — vSifl2x2

2112

or inversely

Ifii 1 1 1

2

1 I2

I

[..2f,i] 1.. 2(1 +

Problems Related to Orthogonal Curvilinear Coordinates

12. Derive the following relations:

oj,12 J3,

yg33

— —— — 13' —

— 1 0 1 8 }'g22 .

— j7 &x2 = — i3 31'

1 t3V'g33&x2 i3'

8j3 — 1 8j3 1

3/Ii,

i 83/j. 1

— 12,

where j,; i = 1, 2, 3 are defined by Eq. (4.95). Write these felations forcylindrical polar coordinate systems.

13. Show that for the small displacement theory expressed in the àylin-drical coordinates r, 0 and z (x r cos 0, y = rsin 0, z = z), we have

—1 _2g1, — g22 r , g33 —

e,=8, 68

3u6 I Os,, u0Vre + -

Ou; OUr — I+ )':O — +

292 APPENDIX H

ôa, I 1

+ + + — + ?, = 0,

1&i, 2• y+ ?e0,

3r r r

z

FiG. H 4.

14. Show that for the small displacement theory expressed in the polarcoordinates r, 0 and q (x rsinç,cosO, y = rsmqsinO, z rcosçv),we have

x2=O,

g1 1 1 , g22 = (r sin g33 = r2,

oUr

1 Ou, U U,£8 = . — + + —,rsmq ô6 r r

=r 0q

I .3U lOU8 U8

= Tin9'

+ -r -

H

_18u, U,4814r

- 1 8u, U. 8U.

— r SIfl 9' — 7 +

293

&Ip. I ÔT, I (3Tr,+

2a, — a1 — + Cot 9' + =+ r sin (p 80 7 09' r

÷! +3r, + 2t,cot

+ ?, =0,t3+S•O0 r F

+I

+I

+— cot 9' + + —

Or 7 Oq' F

Fio. H 5.

CHAPTER 5

Problems Related to 5.1 and 5.2

1. Show that the principle (5.5) can be expressed innate system as follows:

a curvilinear coordi-

+ d%2 + = 0,

where and are initial and incremental stresses referred to thecurvilinear coordinate system, respectively, and Eq. (4.40) or (4.41) has beensubstituted.

294 APPENDIX H

2. Show that another expression for the functional of the principle ofstationary potential energy is derived from Eq. as

17=' 1ff + — + Ø(ua)] dV

+ fff [_Po)aua +

for the initial stress problem, where A(eA,I; is given by Eq. (5.10),and Eq. (5.6) has been substituted.

3. We have formulated stability problems in Sections 3.10, 3.11 and 5.2.Discuss relations between these fofmulations.

Problems Related to Sections 5.3 and 5.4

4. Show that if Eqs. (5.32) and (5.33) are given by

=and

- =respectively, we may have

dA =dB = +

and consequently,A

B = +

for the initial strain problem treated in Section 5.3. Compare these relationswith Eqs. (5.43) through (5.53).

5. Show that if we confine the initial strain problem to the small displace-ment theory, we can prove that the actual solution is given by the minimumprQperty of the total potential energy as well as total complementaryenergy.

6. We consider a thermal stress problem of an isotropic elastic body inthe small displacement theory. Show that the funciionàl of the principleof minimum potential energy is given by

17=111 v, w) —+ ÷

dx dy

for a body with free boundary surface, where 0 = — Showalso that by the use of Green's theorem, the above equation is transformed

APPENDIX H 295

into

+

-ff(elu+Onw+ønw)ds

which indicates that the problem is equvalcnt to that of an elastic bodyunder the body forces (—80/ax, —8O/0y, —80/432) and hydraulic pressure—O distributed over the whole surface of the body.

Problems Related to Sectioo 5.7

7. We denote the direction cosines between two rectangular Cartesiancoordinate system (x', x2, x3) and (X1, X2, by.

(.11 x2 x'x1 rn1X2 I '2 m2x3J 13 rn3 n3

and define the direction cosine matrix [U by

[11m1,,113rn2n213 M3 fl3

Show that if the (xt, x2, x3) system is rotated around the x!-, x2-, andx3-axes by the angle of 0 andy,_respectively, the direction Cosine matrixof the new (x', x2, x-') is by

(U, tez(O)I (e*(v)] [LI,

respectively, where

1 0 0 • --fcosOo—SinUlT = 0 cos# , 1e2(0)J1"I 0 1

0—sin# cost L.sinoo'cosoi

—sine0 01

Show also that Eq. (5.102) is obtained 'from the following matrix multi-plication:4

k2(0)]

296 APPENDIX H

8. We have chosen the vector and three sciars öO and as in-dependent quantities in order to derive Eqs. (5.113) and (5.114) fromEq. (5.112). Show that the sanie equations may be obtained by resolvingthe vector r either as

X,j11 + Yo1i +or as

treating (xG, Yu. z6, 0, or 0, respectively as a system ofgeneralized coordinates.

CHAPTEIt 6

Rtleted to Sedlos

1. Show that if the

u = —Oyz, V = Oxz, w mOq4x,y) (i)

are the solution of the Saint-Venant torsion problem, then a family of dis-placements

u = —Oz(y — yo), v = Oz(x — x0),

X0, Yo and are arbitrary constants) are also the solution of thetorsion problem, and that as far as the Saint-Venant torsion problem isconcerned, the center of twist remains undetermined.

2. Show thatJ=J—D

and consequently

where J is defined by Eq. (6.20), 1, is the polar moment of inertia, i.e.

= if (x2 + y2) dx dy, and D = if + dx dy.

3. We consider a doubly-symmetric cross-section. The x- and y-axes aretaken to coincide with the principal axes through the centroid of the cross-section, and the z-axis is taken as the axis of rotation. An additive constant ofthe Saint-Venant warping function is so determined that ff dx dy 0

(see equations (ii) of Problem 1). Then, show that the warping function thusdetermined has the following property:

q(x, y) = — q ( — .r, y) = — q(.v, .—y) = ( — .v, —y).

APPENDIX H 297

4. We consider an approximate determination of the Saint-Venant warp-ing function of a thin-walled open section as showfl in Fig. H 6. Themiddle line of the wall is denoted by C. A coordinate .v is taken along Cand is measured from one end of the middle Une. Two unit vectors t and n,are taken to be tangential to and normal to the middle line,, respectively,

5-0

so that the three unit vectors n, t and i3 constitute a right-handed system.Denoting the position vector of an arbitrary point P on C by 4°', and thatof an arbitrary point Q on the normal drawn at P by we may write

= + (i)

where is measured from the middle line. The equation (i) suggests that aset of parameters (s, may be taken as a curvilinear coordinate systemdefining the section. Denoting the shearing stress in the direction of the

y

s—I

Fio. H 6.

APVENDIX H

tangent on the middle line by;, and the shearing stress in the direction ofthe normal byr, and using the relations

TX2 = GO — = Go + (ii)

we have

(iii)

along the middle line from s 0 to P, and

f; = +-*) + ciojCr,

along the nonnal from P to Q, where

= r = . t. (v), (vi)

The geometrical interpretation of r, and r which belong to the point F isshown in Fig. H 6. With these preliminaries, show that since r, and r maybe taken approximately equal to zero in the thin-walled open section, wehave from equations (iii) and (iv) the value of at Q as follows:

q'= —fr,dc—frd?+q,o,

where is an arbitrary cozistant. Equation (vii) determines the Saint-Venant warping function of the section. Consider also the shearing stressdistribution of the thin-walled open section due to the Saint-Venant tor-sion. Note: see Ref. 6.2, pp. and Refs. 6.7, 6.8, 6.19.

Problem Related to Section 6.2

5. Show that Eq. (6.32) can be derived from Eqs. (6.7), by eliminating w,then expressing and y,, in terms of by use of Eqs. (&8) and

(6.27).

Related to SectIon 6.3

6. By use of the relation:

show that we have

APPENDIX H 299

M_—2ffcbdxdy

2ffcbdxdy +

T= 2A01

Gi

for a simply-connected cross-section, and

22Jck Ak

FiG. Hi.

for a multiply-connected cross-section consisting of an exterior boundary C0and interior boundaries C1, C2, ..., C,, where is the value of on theboundary C,,, and A,, is the area enclosed by the curve C,,. The value of 4on the exterior boundary C0 is taken equal to zero.

7. Show that for a thin-walled closed section as shown ip Fit. H 8, theshearing stress r and the torsional rigidity GJ are given by

M(1)

and

(ii)

respectively, A0 is the area enclosed by the curve C (which is the meanof the outer and inner boundaries), s is measured along C, i(s) is the thick-

of the wail, and is the integral along the closed path C. Note: seeRef. 6.2, pp. 298—9. C

8. We consider an approximate determination of the Saint-Venant warp-ing function of the thin-walled closed section as shown in Fig. H 8. Byuse of equations (iii) and (iv) of Problem 4 plus equations (i) and (ii) ofProblem 7, show that we have

£ 5fds.t fds j — —J

r,ds

C

which determines the Saint-Venant warping function of the section, whereq'° is an arbitrary constant.

9. Consider two thin-walled circular cross-sections, of which one isclosed and the other 'is as shown in Fig. H 9. Show that the torsionalrigidities are given by -

300 . APPENDIX H

H8.

(1)

GJ = for closed section

Fia. H 9.

APPENDIX H 301

andGJ = for open section.

Calculate the ratio for alt = 10, and discuss why thetorsional rigidity of the open section is so drastically lower than that ofthe closed section. Note: see Ref. 6.2, pp. 272-5 and pp.298—9.

10. Show that the Saint-Venant torsion problem of a thin-walled sec-tion with an inner wall as shown in Fig. H 10 can be solved by determining

C

the shearing stress T2, r3 and the twist angle 0 from the-following equa-tions:

11T1 — 12T2 — 13T3 = 0,

2A111T1 + 242t3r3 =

T1-S1 + r2s2 = 2Gs9A1,

r3s3 — = 2G0A2,

where the thickness. 12, 13 are assumed constant along ACB, AJ)B,I4EB, respectively. A1 and A2 are -the areas enclosed by the closed cUlvesA CBD and ADBE, respectively, and- s1, 33 are the length of the curvesACB, ADB, AEB respectively. Note: see Ref. 6.2, pp. 301—2. -

-

- Problems Related to Secflon.6.5 -

- 11. Show that for a multiply-connected cross-section consisting of anexterior boundary C0 and interior boundaries C1, C2, ..., the boundsformulae for the torsional rigidity can be formulated in a manner similar tothose developed in Section 6.5, by replacing Eqs. (6.72) and (6.73) with

M=

A

Fio. H 10.

302 APPENDIX H

andon C0

= ct on Cft; k 1, 2, ...,

respectively, where ck is some constant.

12. Consider a hollow square section as shown in Fig. H 11. Remember-ing the symmetric property of and w, we consider only the region ABCD

y

/0

//./ 8 C

—

_______ _______ ________ ________

.— U •— 0 —.

b

FiG. H 11.

and choose= y) + y)41(x,y) = b(x — b)

çb2(x,y) = (x — b)2and

= b1w1(x,y)

w1(x,y) = x3y — xy3.

Then show that we have the following bounds for the torsional rigidity:

2(b4 — J (b4 ——

(1,6—

a?2

Note: see Ref. 6.14.

APPENDIX H 303

Problems Related to Non-uniform Torsion

13. We consider a torsion problem of a bar which is clamped at one endz 0 and is subjected to a twisting moment 2 at the othea end (z = 1)as shown in Fig. H 12. The bar is assumed to have doubly symmetric

cross-section. Following Reissner's papers (Ref. 6.4), and using the prin-ciple of virtual work or the principle of minimum potential energy, derivethe following relations:

(1) Assuming

u = — y, v 8(z) x, w = fP(z) y) (i)

show that the governing equation and boundary conditions for 8(z) aregiven by

GJiV — Ef8". = A?, (ii)and

9(0) = tV(0) = O"(l) 0, (iii)

respectively, and the strain energy stored in the bar is given by

(GJ(1P)2 + Er(8")21 dz, (iv)

where y) is the Saint-Venant warping function of the cross-section,( )'=d( )/dzand

(v)

The function y) and the x- and y-axes are chosen as in Problem 3.Show also that the present formulation would not close to the exact solu-tion around z = 0, since equations (1) and (iii) combined with the stress—strain relations provide = = 0 at z = 0.

2-0

Fzo. H 12.

304 APPENDIX H

(2) Assuming

u = —6(z)y, v = 0(z) x, w = tx(z) rç(x, y),

show that the governing equations and boundary conditions # andare given by

GJtV—GD(%-iV)=M,1

and6(O)'= = = 0, (viii)

respectively, and the strain energy stored in the bar is given by

+ — Efld)2] dz, (ix)

where -

ff t(,)2 + dx dy.

Show also that the present formujation provides an approximate solution

14. We consider a torsional buckling problem of a bar which is clampedat one end 0), and is to a critical axial load at the other end(z 1) as shown in 13. It is assumed that the bar has doublysymmetric cross-section, and the force F1, changes neither its magnitudenor its direction while the buckling occurs.

Fio. H 13.

(I) We assume that displacement components arc given by

v x sin — y(l — cos i)), (1)

w

where u, v, w are measured just prior to the occurrence of the buckling,y) is the Saint-Venant warping function of the cross-section, i9 is a

Pcr

z-o

APPENDIX H 305

function of z only, and ( )' = d( )fdr. The function q'(x, y) and thex-and y-axes are chosen as in Problem 3. By use of Eq. (5.5) and neglectingterms of higher order, show that the governing equation is finally reduced to

— — = 0, (ii)

and the boundary conditions to

at 1=0and

ErO" = 0, GJ8' — EI',Y" — 0 at z 1, (iii)

wherer=ffq,2 dxdy, ff(xZ—

ff dx dy, =z4JAo.Note: The strain—displacement relations to be used in the above formulationare

Vu = — y), — '(p,, + x),

where the term (O"g')2 is neglected in the expression of e,1 due to itsnegligible contribution to the final result. See Refs. 67, 6.8 and 6.19.

(2) Next, we assume that displacement components arC given by

u•= —x(lv = xsinO — y(1 — cosO), (iv)

w =

where and are functions of z only. Show that we have the governingequations and boundary conditions as follows:

— — 0") — — 0,v

andat z=0

at z=l.

CHAPTER 7.

Related to Sectfoo 7.4

1. We consider free lateral vibration of a beam clamped at one end(x = 0) and supported at the other end (x = I) with a of stiffness k.Show that the functional for the principle of stationary potential energy of

306 APPENDIX H

this problem is given byI 1

it = £I(w")2 dx + k[w(1)]2 — mw2 dx

with subsidiary conditions w(0) = = 0, and derive the governingequation and boundary conditions. Show also that the Rayleigh quotientfor this problem is given by

f EI(w")2 dx + k [w(I)]20

0 *

2. lateral vibration of a' beam with n constraint conditions:

dx = 0 (i = 1, 2, 3, ..., ii), (1)

where 4'1(x); j = 1, 2, ..., n are prescribed functions. Show that the func-tional for the principle of stationary potential efiergy of this problem isgiven by -

II = +JEI (w")2dx — +co2fmw2dx

+ (ii)j=I 0

where I = 1, 2, ..., n are Lagrange multipliers, and derive the stationaryconditions of the functional (ii). Show also that if a constraint is given by

w(a)=0, 0<a<1, (iii)we have

17 = dx — dx + 4uw(a), (iv)

where 1ts is a Lagrange multiplier, and derive the stationary conditions ofthe functional (iv).

3. We consider free lateral vibration of a cantilever beamwith constant angular velocity Q as shown in Fig. H 14. Showtional for the principle stationary potential energy of this problemgiven by

I 1_ I

11 = EI(w")2 dx + +1 — i2(02ffl3)4?2 dx,

APPENDiX H 307

with subsidiary conditions = w'(O) = 0, where is the initial stresscaused in the beam by the centrifugal force:

A0 dx

and is the area of the cross-section. Show also that the governing equa-tion and boundary conditions are obtainable from the principle.as follows:

(ElK")" — — mw2w = 0,and

w=w'=O at x=0,Efw" = (Efw")' 0 at x = 1.

z

///

FIG. H 14.

Problems Related to Section 7.S

4. Using Eqs. (3.19), (7.11) and (7.12), show that the strain of a beamin the finite displacement theory based on the Bernoulli—Euler hypothesisis given by

U' + [(u')2 + (w')2] — z(1 + 0' + 4 z2(0')2, (i)

where 1 + e0 = + u')2 + and n = —sin 0i1 + cosOi3.

5. Using the principle of virtual work and equation (i) of Problem 4,show that equilibrium equations of the beam in the finite displacementtheory are given by

+ u') — cos 0 —Slfl 0

+ — MxxO'l'} + X 0,

— MXO' sin0 +cosO (M(1 + — MxxO'i'} + 2 0,

- 1+80

where = ff dy dz, = ff z dy dz, = £1 dy dz, andwhere I and 2 are the external loads per unit length of the undeformed

308 APPENDDC H

centroid locus in the directions of the x- and z-axes, respectively. Next, showthat the same equations are obtainable from a consideration of equilibriumconditions of a beam element. Note: The internal force normal to the cross-section of the beam is per unit undeformed area.

6. Show that if the centroid locus is assumed inextensional, namely,= 0, and the term containing z2 is neglected, equation (i) of Problem 4

reduces toWI,

z.

Show also that using the above equation and the principle of virtual work,we can derive the beam equation known as Euler's elastica. Note: seeRef. 3.1, pp. 347—51 and Ref. 3.21, pp. 183—6.

Relatiid to Secdo. 7.6

7. We consider a beam shown in Fig. 7.6 and find that the total potentialenergy of the system is given by

H = 3f (El Eu' + 3 (w')212 + El(w")2) dx + Pr., u(l)

post buckling configUration, where u and w are measured from theundeformed state. Applying the results of Section 3.10, especially Eq. (3.85),to the present problem, derive the equation and boundary con-ditions for the buckling and compare them 'with those obtained jn Sec-tiOn 7.6. Note: Rt 3.1, pp. 358-60.

8. A cantilever beam is executing a small disturbed motion under a,follower force P as shown in Fig. H 15, where 0 = w'Q) and is a specified

Fio. H 15.

APPENDIX H 309

constant. Show, by use of the principle of virtual work, that the equation ofmotion and boundary conditions are

(EIw")" + Pw" + = 0,and w=O, w'=O at x=O

EIw" = 0, (EIw")' + P(l — w' = 0 at x = 1

respectively, where a dot denotes differentiation with respect to time.Discuss also whether or not variational ,principles can be formulated forthis problem. Note: see Ref. 3.23.

9. Show that if the effect of transverse shear deformation is taken intoaccount, the functional (7.87) is to be replaced by

11 = + GkA0 (w' + uj)2]dx — dx (i)

where Uj and w are defined in Section 7.7. Show also that by use of func-tional (i), we obtain the governing equations and boundary conditions of theproblem treated in Section 7.6 as follows:

— GkA0(w' + u1) 0,[GkA0(w' + u1)]' — Paw" 0

'and -

u1=0, w=O at x=O;w=O at x=l.

Problems Related to Coupling of Bending and Torthn t

10. Following Trefftz (Refs. 7.3 and 7.4), show that the point (y,, ;)defined by i iyz=_-j-jfzpdydz. (i)

coincidts with the center of shear and center of twist of the cross-sectionof a bar, where the y- and z-axes are taken to coincide with theprincipal axes through the centroid, I, = ff z2 dy dx and 4 ff y2 dy dx.The function q(y, z) is the Saint-Venant warping function of the cross-sec-tion with the x-axis as the axis of rotation and is chosen so that ffip dydz 0.Show also that if z) is the Saint-Venant warping function withthe locus of the point (y,, z,) as the axis of rotation and is so chosen that

= 0,we have

q.'/j, z) = q:{y, z) — z,y + y,z,

where r1 z)dy dx and 1'=t Beams are assumed to have uniform cross-section along she in 16

through 15.

310 APPENDIX H

11. We can calculate the point (yr,;) of a thin-walled open section bythe combined use of equation (vii) of Problem 4 of Chapter 6 and equa-tion (1) of Problem 10 of Chapter 7 as follows:

= 71

=— -- f (/rz Ls) yt ds,

where the term— f r,, dC has been neglected due to its small contribution.

0Show that the point (y1, ;) thus obtained is in coincidence with the tenterof shear derived from the shearing stress distribution due to torsion-freebending. Note: see Ref. 7.32, p. 210 for the shearing stress distribution dueto torsion-free bending and the center of shear of a thin-walled open sec-tion.

12. We can calculate the point (J',, of the thin-walled closed sectionshown in Fig. H 8 by the combined use of equation (I) of Problem 8 ofChapter 6 and equation (i) of Problem 10 of Chapter 7 as follows:

'3=—'C

______

$2A0c o /4 Tt

C

where the term — f r, been neglected due to its small contribution.o

Show that the point (y1, z1) thus obtained is in coincidence with the centerof shear derived from the shearing stress distribution due to torsion-freebending. Note: see Ref. 7.7, p. 474 for the shearing stress distribution of athin-walled closed section due to torsion-free bending.

13. We consider an approximate formulation of a bending-torsion pro-blem of a cantilever beam which is fixed at one end (x 0), and at the otherend (x = I) is subjected to terminal loads:

xp=0, (i)

2A0

zi d.c

1

IC

APPENDIX H 311

We assume the displacement components to be given by

U = ii — yr' — :w' + t9'q

I, =r—;i9 (ii)

where u, v, w and D are functions of x only. The y- and z-axes arc taken incoincidence with the principal axes through the centroid of the cross-sec-tion. The function q4y, z) is the Saint-Venent warping function of the cross-.

and is chosen as mentioned in Problem 10. Show the following•relations:

(I) By using Eq. (1.32) and equations (ii), the principle of virtual workcan be written as

(N ÔU' + It'!: ôt1' — M1 p3w" + Hô8" + Mrô€Y) dx

— P — — Mb1)Q) = 0 (iii)where

N = ffi. dv d:... = — fJ dy d:,

= •1f d,' d:, H = ff dy d:,

= jf — 3) + + y)J dy d:,and

M = (Z, — Yr;) dy 1:.

Note: since the term + dy ti: in equations (iv) aboveis finally found to vanish,we have

= —

(2) The governing equations arc obtained from equation (iii) as

N=O.

to;ether with

- ii = t. = = = = II = ,V' 0 at x = 0;

N=0, M=0, .%1= —P,, M.=O,.4'!:p H=O, IIT—H--M at x=I.

312 APPENDIX H

(3) By using Eqs. (7.2) and equations (ii) and (iv), the stress resultant—displacement relations are given as follows:

N = EA0u',

Al.. = E1:(v" —

= — +H = E(l'tV' — _51:V + y%1,w),

Mr =where I,, I.., ; and rare defined in the same manner as in Problem 10.

(4) Consequently. the problem is reduced to solvingthe differential equa-tions:

EI.(v + = 0,

E17(w + yb)" + P = 0, (ix)

— ;1r + GJO' + = 0,

under the boundary conditions:

= = = 0' = 0 at x =EL(i' — 0, E!,(iv + y,O)" = 0, (x)

E(1O. — :51: + y51,uj" 0 at x 1.

(5) Equations (ix) can be transformed into

= I',,I,, —

— E1,ii', = (xi)

+ = M + ;1'j, —YIP:,where. .

= r — w. = is + p10 (xii)

and / is defined in Problem 10. Equations (xi) indicate the physical mean-ing of the point (y,, the choice of the locus of the point (y5, z5) as thereference axis decouples the governing equations into two groups 'andallows us to treat bending and torsion of be&m separately. Note: seeSections 35 through 38 of Ref. 7.33. See also Ref. 7.28.

Next, derive another approximate formulation of the problem by assum-ing that

U = U — — :si" +(xiii)

w=w+ys9

where ii. v, n'. and are functions of-x only.

APPENDIX H 313

14. We consider an approximate formulation of a torsional—flexuralbuckling problem of a beam which is clamped at one end (x = 0) and issubjected to an axial force at the other end (x = I). as shown in Fig. !l 13.The symmetry of the cross-section is no longer assumed. The displacementcomponents measured from the state just prior to the occurrence of thebuckling are assumed th be

u = u — yr' — + 11j

v=v—y(1 —cos!))-—

iv = w + y II — :(l — cos /)).

where u, r, w and 0 are functions of x only. The y- and z-axes are taken tocoincide with the principal axes through the centroid of the cross-section.The function q(y, z) is the Saint-Venant warping function of the cross-section and is chosen as in Problem 10. By use of Eq. (5.5) and neglectingterms of higher order, show that the governing equations of the bucklingproblem are given by

[El: (r' + PcrVi' = 0.

[LI, 01 + + = 0.

V — :.,L:' + (if/i + PcrI2/I" = 0.

and boundary conditionsS

= ," = = = Il = = 0 at v = 0:

— + Pcrr' = 0. L'l.(:'' — = 0.

EJ,(i''" + + ' 0. + =

E(Pi1 — + — Gil)' + = 0.E(I.I — ;j_, -3- rlu)" = (.) V 1.

where I' = 1' d:. fj. (I:. j.J. ± dy (1:,!p/i40. and 'r' ': arc defined in Problem JO. Note: see Ret's.

7.30 through 7.33.Next, derive another approximate formulation of the prohkm by assuni-

ing thatU = ii — i't' — :**' +

= — 1 — COS II) — = 'ii' 11 . (iv)

Iv = ft + F P —. :11 — 11).

where u, r. it', and 1) are fuulctions of .v only.

314 API'ENDIXH

(P1/I,) (F — x) z,(Pct/21,)

[El r" + Pci141 — v) = 0,

El'!)"" — GID" + — x) L" 0

*10

15. We consider the lateral buckling of a cantilever beam with rectangularcross-section which is clamped at one end (x = 0) and is under a concen-trated load Pc, at the other end (x = I) as shown in Fig. H 16. The y- and2-axes are taken to coincide with the principal axes through the centroid

K

of the cross-section. The stresses caused in the beam by P,, are given by

(i)

where Fr dyd: The force Ps,, which is acting at the

middle point of the upper side of the end section, is assumed to varyneither its magnitude nor its direction whik buckling occurs. We assume

V — y(l — cosD) — (ii)ii' = w + s'sin P — (l — cost),

where is, v, is', are functions of x only. The function z) is the Saint-Venant warping function and is chosen as in Problem 10. Using equations (i),(ii) and Eq. (5.5), show that the governing equations for v and are given by

(iii)

Fic. H 16.

and boundary conditions by

APPENDIX H 315

at x=O;— 0, = 0,

— GJIV + = 0, = 0 at x = 1.

(lv)

Note: see Ref. 7.16. Next, derive another approximate formulation of theproblem by assuming that

U — )'V' — 2W' +

v=v—y(l

where a,, a', w, x only.

Problems Related to a Beam with Small Initial Deflection

(v)

16 Confning our problem to torsion-free bending in the (x, z) plane,consider a beam, thç locus of the centroid of which has a small initial

deflectionz (1)

as shown in Fig. H 17. We represent the positloo vector of an arbitrary pointof the undeformed locus by' -

41'— x11 + z(x)I,r,

and that of an arbitrary point of the undeformed beam by45) + +

(ii)

(iii)

where I,, and 13 are unit vectors in the directions of the x-, and z-axesrespectively. In equation (iii), a tmit normal drawn perpendicular

x-OFic. H 17.

316 APPENDIX H

undeformed locus and is calculated by

x (iv)

where ( )' = d( )/dx. Equation (iii) suggests that the beam is specifiedby the coordinates (x, y, C) which form an orthogonal curvilinear coordi-nate system. Consequently, by taking tx1 = x, tx2 y and = we mayapply the formulatiqn developed in Chapter 4. Next, we define the dis-placement vector of the centroid by

(v)

and employ the Bernoulli-Euler hypothesis to obtain

r=r0+y12+Cn (vi)where

(vii)and

n = )c i2/(r.1. (viii)

With these preliminaries, derive the following relations:

(1) It is assumed hereafter that the beam is slender and the initial deflec-tion is so small that

1. (ix)Then we have -

= + 13 (x)

and observe that the (x, y, C) coordinate system can be taken approximatelyto be locally rectangular Cartesian.

(2) The displacements are assumed to be so small that

u'-'(w')241. (xi)Then, we have

n = — (z' + w') i1 + 13 (xii)and

= 4 (r' . r' — .(xiii)

= u' + z'w' + (w')2 — Cw".

Higher order terms have been neglected in deriving equations (xii) and(xiii) as well as equation (x).

(3) By use of Eq. (4.80) and equation (xiii), equations of equilibrium areobtained as follows:

N' ÷ = 0, M" + I(z' + w') NI' 4 0,

APPENDIX H 317

where it is defined that

N—_f fr'1dydC, (xv)

and where and p1 are distributed external loads per unit length of thex•axis in the directions of the x- and z-axes, respectively.

(4) If mechanical boundary conditions are specified at x = I, we have

M=2 at x=I (xvi)

where P,, and P. are concentrated external forces in the cLrections of thex- and z-axes respectively.

(5) Since the (x, y, C) system is taken to be approximately a locally rect-angular Cartesian system, we may take

(xvii)

and obtain stress resultant-displacement relations as follows:

M=-EJw". .

Note: see Section 8.9 for a similar de pnient applied to. thin shaliowshells.

17. We apply the results of Problem 16 to a anap-through probleqi asshown in Fig. H 18. The total potential energy of.the system is gi* by

j•j= EAof

1["+ z'w' + Pw(l/2),

z.

:Fio.H18.

where z = z(x) is the small initial deflection of the beam, I is the span ofthe beam, i2 = lIAo, and P is the external force applied at x 1/2 in thenegative direction of the z-axis.

318 APPENDIX H

(1) Derive the stationary conditions of the functional 17, where theindependent functions subject to variation are u and w under the boundaryconditions

u=w=O at x=O and xis!.(2) Derive an approximate solution by assuming that

• .z w = _fisln—T —f2sin —r

and noticing that

r'l'EAo

where )O foil, Ii — fill, 12 fill, ft — (4IWfo)Show also that we have the following critical load for the snap-throughproblem:

Note: see Ref. 7.34. See also Refs. 3.19 and 3.20.

CHAPTER 8t

Problems Relat$ te SectieS 8.2 aM 84

1. We consider bending of a square plate with all edges built in and sub-jected to uniform pressure p. Looking for an approximate solution, weassume

w=c(l —E2)2(1

where c is an arbitrary constant, = xI(a12),,, yI(a/2) and a is the sidelength of the square. Using the principle of minimum potential energy andapplying the Rayleigh—Ritz method, show that we obtain

c 0.001 329 pa'/Dand

Exact solution= 0.00133pa'/D O.00126pa'/D= .—0.0513pa2

= 0.0276pa2

t Unless otherwise stated, plates are a.umed to have ciomt.o* thickness and densityin Problems 1 through 11.

APPENDIX H 319.

where v 0.3. The accuracy of the approximate solution is shown inFig. }I 19 for reference by introducing a quantity delined by

ö'w 8'w\

Note: for details of exact solution, see Ref. 8.2, pp. 197—202. See alsoRef. 8.45, pp. 413-19.

2. We consider bending of a solid wing of variableunder the distributed pressure p(x, Assuming that

derive governing diffbrentiel equations and boundary conditions for wij)and (y) by use of the functional (LSI).snd ezplsln the physical meaningsof these equations. Note: see Ref. 8.10, pp. 60-6.

320 APPENDIX H

Problems Related to Seetio 8,5 s.d 8.6

3. Show that Eq. (8.71) can be obtained directly prom Eqs. (8.67) throughthe elimination of u and v by use of the identity

(äu\ (e3v\ 82 öv0j,2 \8x) +

8x2 øxOy+

and expressing e,,0 and in terms of F by use of Eqs. (8.46) and(8.66).

4. We consider the problem of buckling of a uniformly compressed cir-cular plate as shown in Fig. H 21. The plate is simply supported ar r = a.

Fia. H

H21.

APPENDDC H 321

Confining our problem to rotationally symmetric buckling modes, show thatthe principle of virtual work is finally reduced to

+ M6w' + rN1, w'dw'J dr =0,

from which we obtain a differential equation

and boundary conditions

+ rø' + — 1) = 0

r-+O

for the• determination of the critical loads, where ( )' = )/dr,= and D = w'. Note: see Problem 17 of this chapter for the stran—

displacement relations expressed in cylindrical coordinates.

5. We consider the problem of buckling of a circular plate with a con-centric circular hole which is subjected to internal and external pressures

IPe

plus shearing forces as shown in Fig. H fl. Show that the governing equa-tion for the buckling is given by

D44w=

+ +

Plo.

322 H

where— o2(

+1 ô( ) 1 )' / — + .

In equation (1), and 40) are the initial stresses caused by the internalpressure external pressure p, plus the shearing forces r, and;, and aregiven by

— a2b2(p, — ! —

— b2 — a2+

b2 — a2,

—. — — I p.a2 —

— b2 — a2+

b2 — a2' (iii)

The suffixes i and e denote that the quantities are referred to the internaland external boundaries respectively.

Problens Related to Section 8.7

6. A circular plate is subjected to a temperature distribution 0(r). Thesurface of the disc is traction free. Derive from the principle of minimumcOmplementpry energy the governing equations for a, and

— a, + = 0, = a0,

and the boundary conditions

- limfr3cr+(l —v)r2a,J=O, o,(a)=O,r-..0

where is the coefficient of thermal expansion, a is the radius of the plateand ( )' = )/dr. Show that if 0(r) is postulated as a polynomial of r,i.e.

we have£'' 1¼

a, .— — —k—I + b

b

k + 1

7. Show that a thermal stress problem for a plate in large deflection can beformulated from the equatiops developed in the small displacement theory(see Section 8.7) by replacing .i, and y, with er, e,, and respec-tively, aRd that Eqs. (8.70) and (8.71) are generalized to include thermal

APPENDIX H 323

effects as follows:

D,Jtlw + JUT = p + + —

iJ4F +/1 = —

Piiàl'aa, Related to Ltefal .1

8. Weinstein's method mentioned in Section 2.8 may be applied to. freelateral vibrations of a clamped plate. An intennediate problem can be definedas follows: choose a sequence of linearly independent functions p1(x, y),p2(x, y) ... and p,(x, y), all of which are taken to be plane harmonic func-tions, and relax the geometrical boundary conditions of the onginal problem:

w = 0, ôwfO,' = 0 on C (1)

so that they are replaced by

w = 0, fp,(aw/ov)ds= 0, 1 = 1,2, ..., n on C. (ii)

Show that the intermediate problem can be fornulated by the followingvariational expression:

= Df + dxdy ehw2Jf w2 dxdy

nif pg (öw/ôv) di +fqw di, (ill)

where ag; i = 1,2, ..., n q(ir) are multipliers. Show also thatthe following relation is obtained as a natural boundary condition of thefunctional (iii):

D 4 w = E a,p, (iv)


9. Show that the functional for the principle of stationary potential energyof a free lateral vibration problem for a flat plate with initial membranestresses and is given by

TI = + w,,,)2 + 2(1 — —

+ + N? (w•,)2 + — dx dy,

where the plate is assumed to be traction-free on the upper and lowersurfaces (z = ± h/2) and on the C1 part of the side boundary, while it is

324 APPENDiX H

geometrically fixed on the remaining part C2 of the side boundary. Theinitial membrane stresses are so chosen that they satisfy

in S,and

+ =0, + Nrm 0 on C1.

10. We consider the free lateral vibration of a circular plate subject to aninitial stress system:

= — p2), N, fl(a2 — 3r2), = 0,

where is a constant, and a is the radius of the plate. The plate is assumedtraction-free. Confining our problem to rotationally symmetric modes of-vibration where the rigid body mode Ia excluded, show that the functionalfor the principle of stationary potential energy of this problem is given by

a- s'2 '

—2(l—v)"

+ (w')2 — phw2w2 r dr,

where the quantity subject to variation is w(r), under a subsidiary condition

Show also that the governing equation and boundary conditions are givenby

d2 ld d2w ldwD + — (Tr + —

— (r WI' +

andD(rw" + vw') = 0,

D(rw" + vw')' — D (vw" + —

r = 0 r =an approximate solution of Problem 10 by choosing

w = c (r2 —

and applying the Rayleigh—Ritz method, where c is an arbitrary constant.Show that we obtain the following approximate'eigenfrequency of the lowestmode:

(1) From the Rayleigh—Ritz method, we obtain

= 96(1 + v) (D/a4) + 8j9.

APPENDVC H 325

(2) From the modified Rayleigh-Ritz method, we obtain

2 480(1+') Dphw

=

Note: (1) The values of the lowest mode when theinitial stresses are absent are u = 8.8896 for , — * and u 9.0760 forv = 3, where co = u PD/elsa4.

(2) The value of the critical which causes buckling of the lowest modeis fi = _(3.135)2 (1)/a') for, 0.3 (Ref. 8.46).

Problems Related to the Conditions of Compatibility ond Strom Functions

12. We assume the displacement components as given by Eqs. (8.99):

U=u+zu1, v=v+zv1, w=w, ' (1)

where u, v, w, u1 and v1 are functions of (x, y) only. Show that we have

= + ZU1.z, V,z = W,7 + V1,El = V,7 + Vu = + U1,

= 0, = u,, + + z(u1,, + (ii)-

Show also that by use of the principle of virtual work

fff (a. + a, ô€, + + + dx dy

I' _ffpowdxdy+... 0 (iii)

and equation (ii), the equations of equilibrium for the problem presentedin section 8.2 are given by

+ Ni,,, = 0, + N,,, = 0,+ Mi,,, — = 0, + M,,, — Q, = 0, (iv)

+ Q,,, + p

where N,, ..., are defined by Eqs. (8.17), while Q and Q, aredefined by

*12 *12

= f dz, Q, = f dz.—hJ2 —1*

Compare equations (iv) with Eqs. (8.22) and (8.30).

13. We consider the same problem as Problem 12, and write

= —. Viz == €,o — Vu == 0, Yx, = VxyO

326 APPENDIX H

in view of equations (ii) of Problem 12, where ..., and are func-tions of (x, y) only. Then, Eqs. (1.16) are written as follows for the presentproblem:

= R, = — 0,

= + — Vx,o.x, — + ,cx.,, — (ii)

Ux = Xx., + + VxzQj'x —

U,, = — Vxzo.,, —

Show that by use of Lagrange multipliers and !Pa' the principleof virtual work can be written as

ff + N, + 67x,,o + ôyxgo + Qy

- - M, ox, — 2Mg, dx dy

— fff t5.Rz + + p2W,) dx dy dz + = 0 (iii)

and that from the requirement that coefficients of Or,0, ... must vanishin equation (iii), we obtain

= F,,,, N, = == !P1,,, M, = = + Y'2,,), (iv)

= — Q, = +where

F(x, y) = fx3 dz, y) = dz, Lv) = ftp2 dz. (v)

Next, by substituting equations (iv) into equations (iv) of Problem 12, showthat the functions F, !I'I and thus introduced play the role of stressfunctions: Discuss also the role played by the function F* in equation (iii),where y) = f x3z dz.

Problems Related to Curvilinear Coordinates

14. We represent the middle surface of the plate by a nonorthogonal

curvilinear coordinate system such that

= d& + (dz)2,

where g1j; i,j = 1, 2 are functions of (&, only. Using the formulationsin Sections 4.1, 4.2 and 8.1, show that we obtain a plate theory

based on Kirchhoff's hypothesis as follows:

t It is noted here that a roman letter is used in place of (1,2) in Problenis 14 and 18.The summation convention is employed. Thus, a twice-appearing roman letter meanssummation .with fespect to (1, 2).

APPENDIX H 327

(I) The displacement vector is given by

By use of the relations

— + v2g2 + WI3,

3r0 I.

_____

2

where g1 = g2 = orr/&x2, and v', v2, w are functions of (x', tx2)only, we obtain

u + (v2 + !2z)g2 + WI3,I' = 1, (vii)

=g22fg,g22 =g11/g,g12 ( ),= ô((2) The strain—displacement relations for the Kármán theory are given by

( — .k 1k— ;1Z,

122 = + +

2f12 + +

+ +g2jk;i)z; (viii)

Iklwhere = +

fv'. The strain—displacement relations for the small

displacement theory are obtainable by neglecting the underlined terms inequations (viii).

(3) A plate theory based on the Kirchhoff hypothesis is obtainable bythe use of these relations and the principle of virtual work, Eq. (4.80).

15. We represent the middle surface of the plate by a skew coordinatesystem such that

= + 2 di7 + (di7)2 + (dz)2, (1)

where is a constant. The displacement vector of the middk surface isdenoted by

U0 = Ug1 + vg2 + WI3, (ii)

where g1 and g2 are unit vectors in the directions of the andrespectively, and u, v, w are functions of (E, only. Using the results ofProblem 11 of Chapter 4 and Problem 14 of Chapter 8, and confining ourproblem to the small displacement theory, show that we obtain the follow-ing relations based on the Kirchhoff hypothesis:

328 APPENDIX H

(1) We have the displacement and strain-displacement relations asfollows:

a [u — —

+ (v — + cosec2zw,,,)z]g2+ 1443, (iii)

hi = (u + vcostx),8 —

122 = (ucoScx +

v (u cos + v)1 — (iv)

(2) By use of Eq. (4.80), the equations of equilibrium are obtained asfollows:

+ N21•, = 0, N'2, + N22., = 0,+ 2M12.h + +p = 0, (v)

where.[N'', N22, N'2, N21) f (z", T22, T12, T21J dz,

and(j14I 1, dLJrla, M211 JET11, Ta2, .12, z dz.

(3) The equations which correspond to Eqs. (8.49) and (8.34) are ob-tained in the (E, ,i) coordinate system as follows:

= 0, (vi)and

= sin' (vii)respectively, where

a2 82 82

4(j,) —4 cos + 2(1 + 2 cos2a)


16. We represent the mlddk of the plate by an orthogonal curvi-linear coordinate system (m, fi) such that

=

A B are functions of(s, only. Using the results of Problem 14,and denoting the unit vectors in the and a-coordinates by and b'°1,respectively, show that we have the following relations based on the Kirch-hoff hypothesis:

APPENDIX H

(1) We denote the displacement vector of the middle surface by

+ + (II)

where u, v, w are functions of (tt, if) only, and obtain

u= — -z)aw+ (v — + (lii)

(2) The strain—displacement relations for Kdrmán's theory are given by

I ôu v 1 / ow 2 0 /1 1 Owl

I Vv u 1 /8w\2 ii 0 /1 Ow\ 1 OB Owl

— I ,3v u OA I Ou v OB 1 Ow Ow

11— zI_ —I——I —

A2BOfl&IC

i 0/1OIV\ 1 OBOwl(lv)

The sttain—displacement relations for the small displacement theory areobtainable by neglecting the underlined terms. Note: it is obvious that.thescrelations coincide with those obtained from the equations of Problem 9 ofChapter 9 by setting I = I (14 = 0.

17. Show that the displacement and strain-displacement relationsexpressed in a cylindrical coordinate system:

x=rcosO, (i)

are given as follows for Kármdn's theory:

ii = (u — z) a° + (v — ! z) +

8u 1 fOw\2 82w

I Ov 1 / I I 1 82w 1 Owl

IOu lOwOw2e,6=—+——--—+———Or. rOO p rOrOO

Of1OW\ i-z

330 APPENDiX H

where $(O) and are unit vectors in the r- and 0-coordinates, respectively.It is obvious that strain—displacement relations for the small displacementtheory are obtainable by neglecting the underlined terms.

18. We represent the middle surface of the plate by a nonorthogonalcurvilinear coordinate system (&, zx2) such that that

= g,j dcc' + (dz)2

and formulate a plate theory including the effect of transverse shear defor-mation by assuming that

u = ÷ z) g1 + + z) g2 + WI3, (ii)

where w, are functions of (ô1, cc2) only, and g1, g2 are definàdas in Problem 14. Confining our problem to the small displacement theory,show that strain-displacement relations are given by

111

122 = g2, ÷ z,

f330,S S

I I2112 (g1, V0;2 + g2svo;1) + + i) z,

2113 =g1, w1,

2123 g25 + W.2.

Show also that a plate theory including the effect of transverse shear defor-mation is obtainable in the nonorthogonal curvilinear coordinate systemby use of these relations and the principle of virtual work, Eq. (4.80).

CHAPTER 9

Problem Related to SectIon 9.1

I. Show that Eqs. and (9.8) may be derived frojn the conditions

which have been introduced in Problem 7 of Chapter 4.

Problems Related to Sectiem 9.2, 9.3 and 9.4

2. We assume the displacement components

VV+CV1, w=w (i)

- APPENDIX Ii 331

as given by Eqs. (9.30). Confining our problem to the small displacementtheory, show the following relations:

(1) The strain—displacement relations are given by

— +

122 = B2(l — (122 +

133 0,

21,2 = ABt(1 — + + (1 — C/R,) (/21 + Cm21)1

2fr3 = A(u1 + /3k), 2123 = B(v, + 132).

(2) By use of equations (ii), we have

= If EN0, + + ó12, + ôi,2

+ M0,ôm11 + + +

+ Q1ô(u1 + + Q,ô(v1 + 132)1 AB (iii)

where N41, N,,, Nm., M0,, M,. are defined by Eqs. (9.58)and (9.59), while and Q, are defined by

= (1—

Q, =(i

—(iv)

(3) By use of equation (iii) and the principle of virtual work, we mayderive the equations of equilibrium for the problem presented in Section 9.4in the following form:

(BN0,) + (AN,0,) ÷ — - + ?0,AB =0,

FAN,) + + — — g/Q, + =0,

+ (AQ,) + AB + + ?1AB =0, (v)

(BM0,) + (AM,0,) + U,, — — ABQQ, =0,

(AMa) + + U13 — M0,\ ABQ, 0.

Compare these equations with Eqs. (9.60) and (9.68).

332 APIENDIX H

3. Show that equations (v) of Problem 2 are obtainable in a differentway from the following

+ + Bdfl)

+ + + A dczl 49

+ + + = 0,and

x .+ + B 49

+ 49 x + + A

+ [(- + B dPi a

-

which are derived by considering the equilibrium conditions of the shellelement shown in Fig. 9.5 wIth to forces and moments.

4. We represent the unit vectoTs in the directions of the andnates, and the unit vector norma' to the middle surface after deformation,by

a, b, arespectively:

axba= , a= a xShow that}inearization of equations (II) leads to

I 1•kI.I I'. 1.I lII.(O)'U 5 '3211.L '133 1

where terms, higher than the second order with respect to the displacementcomponents (u, v, w) have been. neglected, assuming the displacements ofthe shell to be small. Show also that of equations (iii) withrespect to sand ft leadS to

1 A+T - - Ti +

bI 8A 0112 A42 1j3 OA +r B Oft

+ B Oft'A '32 OA 8133 13 04 — (0)-r

APPENDIX H 333

and

! +— B!31 0131 B!21

A 0x Ofi+

— ! + 0/12 '12 OB B!32A Ofi' A

+op

+ 132 13B B 0/32 i3B B!32op

Problems Related to the Conditions of Compatibility and Stress Fiwctlons

5. Using the results of Problem 9 of Chapter 4 and equations (ii) of•Problem 2, write the Riemann—Christoffel curvature tensor

R2323 R3131 R1212

R1231 R2312 R3123

in terms of!11, '12, '21' m11, m22, m12, m21, u1 + 131 and v1 + 133.After having noticed that the Riemann—Christoffel curvature tensor thus

obtained can be expanded into power series with respect to obtain the

expressions of these tensor components at = 0.

6. By use of Lagrange multipliers X3' and the principle ofvirtual work can be written as

ff + ã122 + + Q,a(v1 + /32)) AB dfi

— 5ff E.r oR1212 + oR1231 + tP2 OR2312 + 1/13 OR3123] jlgdix dfi

• (i)

where the expressions of the Riemann—Christoffel curvature tensor obtainedin Problem S have been substituted. We expand the Christoflèl's symbolsand in power series of and introduce the following notations:

F = f W1 = f W2 = f V'2 dC, = 5 V3 (ii)

Show that from the requirement that the coefficients of 0/22, ... mustvanish in equation (1), we obtain Na, Ni,,, ... and in terms of F, !P1,

and Y'3, thus discovering that the latter play the role of stress functionsin the small displacement theory of shells based on equation (i) of Problem 2.Show also that the stiess functions thus obtained are equivtilent to thosederived in pap. 33-6 Ref. 9.2.

334 APPENDIX H

Problems Related to Other Thsedes of Shells

7. An approximate nonlinear theory for a thin shell has been developedin Section 5.2 of Ref. 9.16, which assumes

(i=u—I31t, v=v—132C, w—w,= — = — Cx,,

where x,, and are given by Eqs. (9.82); (9.50); and

• (iii)

The assumptions Eqs. (9.77) and (9.78) are also employed. Derive the equa-tions of equilibrium, mechanical boundary coziditions and stress resultant-displacement relations for the present approximate theory i*ñp.rethem with those derived in Section 9.7.

8. An approximate small displacement theory for a thin shell is .obithiedby assuming that

18w lOw Ww, (1)

= — = —

(ii)

where u, Up and are given by Eqs. (9.36); and

1 8 (1 Ow\ 1 84 Ow

lOfi Ow\ 1 088w

1 0

1 0(lOw\ 1 OAOw

The assumptions Eqs. (9.77) (9.78) are also employed. By use of theseequations and the principle of virtual work, show that the equations ofequilibrium are given by

(BNIJ• • + + N, -. + Yd,AB =0,

[AN,J•, + [BN4d, + — + F,AB =0, (iv)

++

+ lAB = 0,

APPENDIX H 335

mechanical boundary conditions by

= R,,,, N0,

Q41! + + ÷ M, = 2and stress resuftant-displacement relations by

N.(1

+(1 ,2) + e,oJ,

= = (vi)

= — D(l — (vii)

for the present approximate theory, which is equivalent to the theory of(see Ref. 9.1).

9. An approximate nonlinear theory for a thin shell is obtained by assum-ing

law .18w w=w, (i)

= e41p0 (ii)

where eMo, are given by

1814 vøA w 1 /Ow\2

10v uôB w 1/8w\2

1 81, u 8A 1 öu o .3B 1 8w 8w(jil)

and equations (iii) of Problem 8, respectIvely. The assumptions Eqs. (977)and (9.78) are also employed. With the aid of the principle of virtual work,.show that the equations of equilibrium and mechanical boundary coneditions for the present approximate theory are given by equations (Iv)and (v) of Problem 8, if and are replaced by

18w 18w+ +

and18w 18w

APPENDIX H

respectively, and that stress resultant—displacement relations are obtainablefrom equations (vi) and (vii) of Problem 8 by replacing eao, ego, with

respectively. Note: see Ref. 9.29, p. 189.

10. We consider equilibrium conditions of the shell element shown inFig. 9.5 to obtain the following vectorial equations:

+ W,,b + B dj9] dcx

+ + N,b + Q;n) A dcxj dfl

÷ ± + AB dcx dfl = 0, (I)

and

dcx [Naa + + QnlB do

+ dfl x + + A dcx

+ +

+ [(— + A dcx] = 0 (ii)

where a, b and n are defined in Problem 4 (compare these equationswith equations (i) and (ii) of Problem 3). By use of equations (i) and (ii)thus obtaIned, plus equations (iii) and (iv) of Problem 4, derive equations ofequilibrium in scalar forms for an approximate nonlinear theory of a thinshell based on the K.irchhoff hypothesis and compare them with Eqs. (9,94).

Problems Related to Nonoitbogosal Curvilinear Coordthatest 1

11. We represcnt the middle surface gf the shell before deforn tion bya pair of parameters (cx1, cx2) so that

= (tx1, tx2) (i)

• ¶'See 10.2, 10.4, 10.8 and 10.14..is noted here that a Greek letter will be assigned in place of (1,2,3) and a roman

in place of(1, 2) in Problems 11 and 12. The summation coiwention wil beThus, a 'twice-appearing Greek or roman letter means summation with respect to (1, 2, 3)

or (1, 2), respectively.

APPENDIX H 337

and define the two base vectors in the middle surface and the unit vectornormal to the middle surface by

g1 = = g3. (ii)t

By use of these vectors, we define and by

= . = = (iii)t

Next, we represent the position vector of an arbitrary point of- the shellbefore deformation by

= + (iv)

and employ the set of the three parameters as a system of curvi-linear coordinates, writing = tx3 whenever convenient. With these preli-minaries, show the following geometrical relations:

(1) Concerning the derivatives of the vectors we have the well-knownformulae of Gauss and Weingarten in the theory of differential geometry:

Iii + HJftg3, (v)

= (vi)

'where

} 4 + — (vii)

= . = (viii)

'It is obvious that in the orthogonal curvilinear coordinate system intro-duced in Section 9.1, we have

H11=

H22 = H21 = 0. (ix)

(2) The distance between two neighboring points' (&, and

+ dcc, + dx2, x3 + dx3) is given by

= (g1, — + g"4 d& dcci + (x)

t It is noted that RAp and defined here are different from those inChapter 4. For the purpose of consistency, it may be better to write SeAs andinstead of g2, and respectively. However, we prefer simpler notations as far asProblems 11 and 12 are concerned.

338 APPENDIX H

(3) We calculate Christoffel's symbol in the space defined by equa-

tion (x), and denote its value at 0 by f L We have

{ L =gla (gd. + —

fl 1 =1 I1I11• .1'IJ-'J.o I.JJJo

131 131 =H,,,It/Jo (JiJ0111 121 (31 13) 13) .131 131

t3310 1131o 123L = 132jo= 0.

(4) Introducing a convention that = = 0; x = 1, 2, 3, we canwrite

- 1:;'! (xii)

ga.* = {j (xiii)

= g4 + C { (xiv)

(5) We define components of a vector u(&, as follows:

(xv)Theit we have

= (xvi)

where and throughout Problems 11 and 12 it is defined that

övi (A) -.(xvii)

12. We assume the displacement vector u in the power series of C to besuch that

(1)

where isa function of (&,cc2)only. By use of equation(i), an approximatetheory ol a shell of moderate thickness can be formulated by use of the prin-ciple of virtual work. For exaMple, we may assume

ii = (4 + + + + 4g3 (ii)

APPENDIX 41 339

and obtain 2f&, = of:, + +

= f33 = 0 (lii)where

= += +

+

20f13 = + (iv)

and = 0. In deriving equations (iv), only linear terms are retained,confining our problem to the small displacement

the of the small dis-placement theory of a thin shell which includes the effect of transverseshear deformation and is expressed, in the non-orthogonal curvilinearcoordinate system.

CHAPTER 10

Related te SecIloas 10.2, 103 and 10.4

1. Show that combination of Eqs. (10.36), (10.39) and, (10.41) yields:tEAl — fH][QJ-' [HJJ{Y},

provides the deflection influence coefficients, where

2. We have considered in Section 10.2 a truss problem where the geo-metrical boundary conditions are prescribed by Eqs. (10.17), and bbtainedthe conditions of compatibility (10.34) and the relations (10.40). Show thatif the geometrical boundary conditions are prescribed such that

u1=ü1, (i=k+l,...,n)the conditions of compatibility are given byJr

a,,, 81 — E + + a.1.] = 0,((9) 1—k+l J

and Castigliano's theorem provides:Jr— £ (A,p,,,) +

(U) S—k-fl I I

+ = U,,

2J Pu' 8,, — + L's

+PI� (viiPiia)] = vi,

340 APPENDIX H

£ Yus — + £ (PuVu:) + = WI, -

•(U) i=k+1 J j J

1= 1, 2, ..., k.

Problems Related to SectIon 10.7

3. We consider the beam element shown in Fig. 10.5, and assume that theforce P is absent. Show that the strain energy stored in the beam is given by

tA0 6E1i= 21

(u2 — u1)2 + + — v1)2

+ 1(v2 + v1)(02 (i)

4. We consider the frame structure shown in Fig. 10.6, and cut it ficti-tiously into four members (13, ©, and where the section ischosen perpendicular to the centroid locus of the frame at the point ofapplication of the external force P. Show that the expression of IT! for theprinciple of minimum potential energy for the present problem is given by

17= U15 + U52 + U14 + U23 + Fi,5, (i)

where U15(u1,v1,101,u5,v5,05), U52(u5, v5, u2, 02), U14(u1,

and (/23(u2, v2, 02) are the Strain energies stored in the beam elements,respectively, andtlteir expressions are obtainable by the use of equatiQn (i)of Problem 3. The physical meanings of u1, v1, u2, v2, 02; u5, v5,are the, displacement components in the x- and y-directions, and the rota-tional angle in the clockwise direction of the joints (J), and section (s),respectively.

Show also that by the use of the stationary conditions of 17 with respectto u5, v5 and 817 817__0, —O, —0, (ii)

8v3we may eliminate u5, v3 and from 17 to obtain

iF2!3Ei

+ + V2) — 01)1 (iii)

and that the function 11 thus obtained coincides with Eq. (10.6$).5. We consider a frame structure subject to concentrated forces and

moments as shown in Fig. H 23. Show that we• haveV12 + V23+ V14 +(N12 + ++(Q12—N14—P+?1)v1+ (M12 — M,4 — Q12/ + ++ (—N12 + Q23 + A'2)u2

instead of Eqs. (10.66) for the present frame

H

p V2

341

frame — as6. We consider abeam theory,bering that, from

givenstored in the curved

N2

where a is along the centroid of the curved show thatto the present problem ifl a mannerthe force methOd is 5pplicable

to the develoPment of Seètiofl 10.6.

0

'pS

24.

342 APPENDIX H

Problem Ri14 to &edo. 10.8

7. We consider a plane structure consisting of panelsand stringers as shown in Fig. H 25, and internal forces in thesemembers as shown in Figs. 10.8 and 10.9. Show that by use of the prin-ciple of minimum complementary Into which the equijibrium con.ditions between members are intjoduced by use of Lagrange multipliers,we have the following compatibility condition:

- ijc34dY)

where y,1is.the shearing strain of the panel while u14x)u33(x) andv12(y), are the displacements of the foUr strhgers in the directionsof the x- and taxes, respectively. - -

y ® . C

°k

® *- It. I ®

Problems &elated to Lad Method

8. We consider a truss problem and denote the actual solution of theinternal force and elongation of the (I-th membà by respec-tjvely. Show that if p,, denotes the intenial force in the member due

a unit virtual load acting at the joint for which the deflection is requiredload acts in the direction of the deflection), then the unit load theorem

Provides:

Stlf— -r I

0 a)'

APPENDIX H 343

Next, show that if we substitute the relation

öjj — (ii)into equation (i), and write

(iii)

equation (iii) holds irrespectively of the load-elongation relations. Namely,equation (iii) is applicable to problems of plastic as well as elastic trussproblems. Show also that Eqs. (10447) arc by an application ofthe unit load method.

9. We consider a plane frame structure conaliting of straight members,and employ for the analysis of the structure the elementary beam theoryplus the assumption that the deformation due to dxial force is negligiblecompared to that due to bending moment. The actual solutions of the cur-vature and bending momenta of the (f-tb member arc denoted byand M.Ax) respectively, where x is màsured along the centroid locus. Showthat if mjj(x) denotes the bending moment In the iJh member due to aunit virtual load acting at the point for which the deflection ö is required(the load acts in the direction of the then the unit load theoriçmprovides:

0

Show also that if m1,(x) denotes the moment in the (f4h memberdue to unit virtual external moment acting at the point for which the rots-lion 0 is required (the moment acts in the direction of the rots$On), then'the unit load theorem provides:

.

Next, show that if we substitute the relation

(lii)

into equations (1) and (ii), and write

and

(lv)

O——jjfmiidx,

respectively, equations (iv) and (v) hold irrespectively of the moment—curvature relation. Namely, these equations are appliàble to problems ofplastic as well as elastic frames.- Show also that Eq. (10.65) is obtainableby an application of the unit load method.

344 H

10. Show that the unit load method can be applied also to a three-dimensio-nal frame structure with naturally curved elements. The expression of thecomplementary energy

N2 M2 M2 Q2 Q2I(÷ + + 2W + + 2Gk,A0)

may be helpful for the derivation of the unit load method formulae, where Nis the axial force, 1.1,, and M, are bending moments about the two principalaxes, T is the torsional moment, Q and Q, are shearing forces, and s ismeasured along the centroid.locus. Show also that by substitutions such asthose mentioned in Problems 8 and 9, load formulae whichare applicable to problems of platic frame as well. Note: seeR.efs. 10.1, 10.2, 10.3 and 10.9 for numerica* an4cations of the unit load method. .

Related to

lLWeconsideratnssproblernandasamethattheU-thmeinberhasaninitial excess of length of o". Show that in applying the force method tothe problem, the complementary

k240EJ,1is to be replaced by

12. We consider a plane frame problem and assume that the (I.th mem-ber has initial strain where the (x, z) coordinate System 1. taken ina manner Similar to Chapter 7. Show that in applying the force method tothe problem, the complementary energy

N2 M2

is to be replaced by

+

APPENDIX I

VARIATIONAL PRINCIPLES AS ABASIS FOR THE FINITE

ELEMENT METHOD

Sectloa 1.

Mathematical formulation of a problem for a continuous body is usuallymade by the use of differential equations, as exemplified by the finite dis!placement theory of elasticity introduced in Chapter 3, where mechanical orphysical quantities of the continuous body, such as displacement,strain and so forth, are assumed to be continuous functions of thecoordinates I = 1,2, 3 and the continuous body is treated aàan assemblyof fictitious elements of infinlseshnai magnitude as shown in Fig. 3.1.

On the other hand, the continuous body is divided into a number offictitious elements offinite magnitude, ("finite elements"), and is treated asassembly of these elements in the formulation of the finite element method(often abbreviated FEM). The continuous functions for the mechanical orphysical quantities are now replaced by approximate functions which are

each element, but are continuous and piecewise smooth in thebody. These approximate functions are constructed by the use of

unknown parameters such as values of the quantities at the so-called nodalpoints the use of interpolation functions, in such a way thatdistributions of the quantities in each element may be determined uniquelyonce the values of the unknown parameters have been specified. Thus, we arereplacing the original differential equations by a number of algebraic equa-tions which govern the unknown parameters. Consequently, our nextproblem is how to obtaip the governing equations for the unkngwn para-meters.

It has well established that the variational method provides a powerfuland systematic tool for derivation of the governing equations for theseunknowns. We remember that some mention has been made already inChapter 10 of interrelations between variational methods and FEM. There,the generalized Galerkin method based on the principle of virtual work, and$he Rayleigh—Ritz method based on the of minimum potential.energy, of minimum complementary energy are shown to be

to the structural analysis of variouS finite elements345

346 APPENDIX I

such as truss, frame and semi-monocoque structures. It is noted here that theterminology finite element method may include those techniques based on thegeneralized Galerkin method as well as the Rayleigh—Ritz method.t

It now widely iecognized that Courant was one of the pioneering mathe-maticians in the development of FEM. He presented an approximate solutionof the Saint-Venãnt torsion problem formulated by the use of the principle ofminimum complementary energy, assuming a linear distribution of the stressfunction in each of the assemblage of triangular On the otherhand, the paper by Turner, Clough, Martin and and the work byArgyris and have been regarded as the most important and historicalcontributions among pioneering works in FI3M in the field of structure.Since the appearance of these literatures, the variational method has beenused extensively in the mathematical formulation of FEM. Conversely,the remarkable development of FEM has given great stimulus to the advance-ment of the variational methods: new variational principles such as varia-tional principles with relaxed continuity Herrmann'sprinciples for incompressible and incompressibleand also bending of 12) and so forth have been established duringthe last ten years. The objective of thIs.iiew appendix is to present a briefsurvey of recent developments of variational principles which provide abasis for the formulation of FEM in elasticity and plasticity. For practicalapplications of these principles to the formulation of FEM, the reader isdirected to papers such as Refs. 5, 6 and 7.

As the contents of Refs. 2 and 3 show, the primary purpose of theseworks was to develop a numerical method of analysing the rigidity andstress of an elastic airplane. Since the appearance of these pioneering works,numerous papers have been published concernng applications of FEM in abroad field of engineering FEM is now widely used not only fornumerical analysis of stresses and displacements of elasto-plastic structures,but also for a variety f non-structural problems such as hydrodynamics, heattransfer, seepage and so forth. Thus, the FEM technology, aided by amazingadvances of the digital computer, has been mnking a great contributiOn topractical applications and will be much more developed and much more inuse in the future.

The bibliography of this short appendix is not, intended to be complete.The author is satisfied with referring only to a very limited number of paperswhich are directly and closely related to mathematical formulations intro-duced in this new appendix. The reader is directed to Refs. 17 and 18, forexample, for a complete bibliography for FEM.

f For the sake of simplicity, the generalized Galerkin method will be cafled the Galcrkinmethod in this appendix.

APPENDIX 1 347

.Secdos 2. Varlatlomsi Prbselples for the SmallD of

The first topic of this appendix will be a review of conventional variationalprinciples for the displacement theory of elastostatics, the governingequations of which may be given as follows. t

(1) Equations of equilibrium:

= 0.

(2) relations:

— j(u,,, + (1-2.2)

(3) Stress—strain relations:

= a.,,sekg, (1-2.3)

or convenelyC,, (1-2.4)

(4) MechanIcal bouatdary co,iditlons:

= on S0,

where

= o',1n1. (1-2.6)11

(5) Geometrical boundary conditions:

= u, on (1.2.7)11

In the above, Eqs. (1-2.3) and (1-2.4) are equivalent to Eqs. (1.6) and (1.8),respectively. For later convenience, Eqs. (1.6) and (1.8) are expressed in matrixforms as follows:

= (AXe) (1-2.8)

{e} = [flJ{o) (1-2.9)

where

{a)r (a,, a,,, a,, r,,,, r,,,1— ce,, e,,, c, V,s' Yw Vni,

t convention is employed in this appendix uiAless otherwise stated. Seefootnote on page 231 for the convention.

Notations = 1, 2, 3 are used instead of X, F, 2. respectively.§ Notations?1 and I 1,2,3 areusedinstead of X,, Y,,Z, and I,. 7,, Z,,respectively.¶ Notations a1; I 1, 2, 3 are used instead of!, m, a, respectively.II Notations S0 and are used instead of Si and 52, respectively.

348 APPENDIX I

and where [A] and FBI are positive definite symmetric matrices which obey therelation -

[B] = [A]'. (1-2.10)

The strain energy function A and the complementary energy function Bmaybe written as either

= (1-2.11)

(1-2.12)

(I-2.13)t

= (1-2. 14) t

For later convenience, a notation 4(u,) is *ntroduced here. It is obtainable bysubstituting Eq. (1-2.2) into Eq. (1-2.11) and expressing the strain energyfunction in terms of displacement components:

= + + (1-2.15)

With these preliminaries, the conventional variational principles mentioned inChapter 2 may be summarized as follows.

2.1. PriDcIpIe of Virtual Work

The principle of virtual work may be written as

— — ffTiouidS = 0,V V

where the subsidiary conditions are

öe1, = + (1-2.17)

and

= 0 on S1. (1-2.18)

f Compare with Eqs. (2.2) and (2.20), respectively.Compare Eq. (1.32).

APPENDIX I 349

2.2. PrincIple of Mlnlinwn Potential Energy

The functional for the principle of minimum potential energy may bewritten as

lip = — f1u1]dV — (1-2.19)t1' sq

where the subsidiary conditions are

= a1 on (1-2.20)

2.3. Generalized PrincIple

The functional for the generalized principle may be written as

fff{A(e15) — ju* — u11[e1, — + u,1)fldV

—5 f 1'1u4tLS — —- SI.

with no subsidiary conditions, where a1, and are Lagrange multipliers, ofwhich the physical meanings are given by Eqs. (2.28) and (2.33), respectively.

An alternate expression of the generalized principle may be writte.p in thefollowing form:

flQ2 = 55f{A(e41) — — a11[e1, — j(u1•, + u1.1)]}dV

— 55 — ü1)dS,S0 3..

with no subsidiary The generalized principle, expressed byEqs. (1-2.21) and (1-2.22), is sometimes called the Hu—Washizu principle.

princIple

The functional for the principle may be written as

113 = 55f[4a11(u43 + u,1) — — f1u1JdV

—5 5 TiuiLS — ffT1iu1 — u1)dS, (1-2.23)118,

t Compare with Eq. (2.12). The notation is now widely used for expressing thefunctional for the principle of potential energy.

Compare with Eq. (2.26). The notation will be used in this appendix to express thefunctional for the generalized principlç.

COmpare with Eq. (2.34).I C.mpsre )vIth Eq. (2.37). We note that in Eq. (2.37) have been replaced by T1 in

Eq. (f-2.23).

350 APPENDIX I

with no subsidiary conditions. Integrations by parts lead to the followingalternate expression for the principle:

5ff + + /JuI]dV

— ff(T1 — — if (I-2.24)tSq 5.

where no subsidiary conditions are imposed.

2.5. Principle of Complementary Energy

The functional for the principle of maybe written as follows:

•mc = — 55

V Sii

where the subsidiary conditions are -

fJtj.5+J'I=O in (1!2.26)

and

= on S0. (1-2.27)

2.6. Principle of Complementary Virtual Work

This principle may be written as

55 — SJÔT*uIIS 0,V

where the subsidiary conditions are given by

0 in (1-2.29)

and

=0. on Sq. (1-2.30)

These variational principles are represented in the left-hand column . ofFig. I-i in the form of a flow diagram.

t Compare with (2.41).Compare with Eq. (2.23). -

§ Compare with Eq. (1.50).

APPENDIX 1 351

Conventional Modified variationalprinciples principles for relaxed

continuity requirementsS

Principle of virtual workJ

Modified prir.ciple of[virtual work

IPrinciple of minimum

potential energy mp

Compatible model

I IGeneralized principle

I tllel!inger—Reissnerprinciple

I I

Fio.I-I. A flow diagram for the small displacementtheory of elastostatics.

SectIon 3. DerivatIon of Modified Variational Principlesfrom the Principle Of Potential Energy

The purpose of the present section is tb follow in Fig. I-i an avenue whichstarts from the principle of minimum potential energy and leads to themodified principle of potential energy, the modified generalized principle andfinally to the modified principle. We shall treat a solidbody problem.which is the same as 4efined in the preceding section, exceptthat the regiodIV is now subdivides fictitiously into a finite number of ele-ments: V1, V2, V3, ..., VN. For later convenience, we denote two arbitraryadjacent elements by Va and

Vb in Fig. 1-2, Where tetrahedral elements are used forthe purpose of illustration. Two symbols and will be used whenevernecessary to distinguish the interelement boundary Sab belonging toand OVb, respectively.t

f ØJI, denote the entire boundary of Vi,.

IModified principle ofpotential energy

Hybrid displacementmodel I, II

I I

Mixed model I

; tModified Hellinger—Reissner principle

Modified generalizedprinciple

-

Mixed

I IPrinciple of minimumcomplementary energy

Equilibrium model I

Modified principle ofcomplementary energy

Equilibrium model IIHybrid stress model

Principle of complementaryvirtual .work

Modified principle ofcomplementary virtualwork

352 APPENDIX I

3.1. Principle of MhIIn.R1 Energy

We shall denote displacements in each element by

(1) I 1 2Ut, g,..., j, , , , ,

each of which will be called displacement functions. Then, the ofthese displacement functions may be taken as admissible functions for thefunctional of the principle of minimum potential energy, if they satisfyfollowing requirements:

(I) They are continuous and smgle-val&ed in each element.(ii) They are conforming on interelement boundariès:t

on (1-3.1)

(iii) Those belonging to an element containing satisfy Eq. (1-2.7).Conse4uently, the displacement functions are so chosen as to satisfy therequirements (1), (ii) and (iii), the functional for the principle of minimumpotential energy is given by

ii, — — ffT4u,ds,V. Sc

where the notation means summation over all the elements. The independ-ent quantities subject to variation in H, are up>, ..., ut", ...,(abridged as hereafter).

t See Refi. 14 and 19 for the definition of the terminologyCompare with Eq. (1-2.19).

N11-2. V. V, and S.,.

APPENDIX 1 353

3.2. Modified Principles of Fiiergy

Next, we shall formulate a variational principle in which the subsidiaryconditions (1-3.1) are introduced into the framework of the variationalexpression. By the use of Lagrange multipliers A, defined on SGb, we obtain thefunctional for a modified principle as follows:

= Hp — (1-3.3)

where is given by Eq. (1-3.2), and

H4,,1 = — ur)dS. (1-3.4)

In Eq. (1-3.3), the notation in front of H4,,1 means summation over all theinterelement boundaries. The independent quantities subject to variation in

are and A, under the subsidiary conditions, Eq. (1-2.7). The principlefor the functional will be called the firs: modified principk of potentialenergy with relaxed continuity requirements because the requirement (ii) isrelaxed in and the displacement functions in each element may be chosenindependently without any oncern about the conformity requirement. Aftersome manipulation including integrations by parts, the first variation ofon S4b is shown to be

+ if — + [pb)(u(b)) +

— (us> — ur)oajdS ÷ ..., (1-3.5)

and we obtain the following stationary conditions on S4b:

= ,l,, (1-3.6)

(1.3.7)

where and are obtainable from

= = (1-3.8)

by substituting Eqs. (1-2.2) and (1-2.3) to express and in terms ofand respectively. Needless to say, and are the direction cosinesof the outward-drawn normals on Se,, and respectively, and we have

(1-3.9)

The stationary conditions (1-3.6) indicate the physical meaning of the Lag-range multiplier: A, is equal to on S0b. It is noted here that the modi-fied principle is no longer a minimum principle, but keeps its statioflary property

354 APPENDIX I

only. The functional Thmpl was originally proposed by Jones (20) and laterdeveloped further by

The Iunctional flmpi will be modified slightly. We introduce two functionsand which are defined on Sb and respectively, and obey the

following relation:

+ = 0. (1-3.10)

Then, by writing= (1-3.11)

the integrand of Eq. (1-3.4) can be expressed by

+under the subsidiary condition (1-3.10). Consequently, introducing a new

multiplier defined on Sab, we may write Eq. (1-3.4) in anform denoted by Hab2 as follows:

H4b2 = + 2b)14b) — + (1-3.12)8ab

or

Haba = — + — (1-3.13)SbG

By the use of thus defined, Eq. (1-3.3) may be written in another form asfollows:

— (1-3.14)

This principle will be called the second of potential energywith relaxed continuity requirements, where the independent quantitiessubject to variation are and under the subsidiary conditionsEq. (1-2.7). Among these quantities, in Va, and on may be chosenindependently of in Vb, and on respectively, while defined onSab must be common to and After some manipulation includingpartial integrations, the firstvariation of on Sab is shown to be

=— + —

+ — + —

— + + ... (1-3.15)

and we obtain the following stationary conditions on Sab:

= A(b) = (1—3:16)

= = 1u (1-3.17)

APPENDIX I

+ = (1-3.18)

The stationary conditions (1-3.16) and (1.3.17) indicate the physical meaningof the Lagrange multipliers: and are equal to On

on S:a and Uf on Sab, respectively.If we employ the stationary conditions (1-3.16) in order to

and we may write in another form as follows:

= —

— u,)dS, (1-3.19)

and we obtain

fl.,3 = H, — (1-3.20)

This principle wilt be called the third modified principle of potential energywith relaxed continuity requirements, where the independent quantities subjectto variation are and under the subsidiary conditions Eq. (1-2.7).Among these quantities subject to variation, in may be chosen in-dependently of in V3, while should be common W Sb and Thefunctional and are equivalent to those derived originally by

The modified principles with relaxed continuity requirements willbe called modified principles hereafter for the sake of brevity.

3.3. ModIfied Principle

The modified principles of potential energy thus derived may be generalizedin a familiar manner. We shall start from the functional "mp2 to obtain thefunctional for a generalized principle as follows:

= —V.

— + —

— ffTguiiS — — üg)dS, (l3.21)t

where the independent quantities subject to variation areand with no subsidiary conditions. It can be shown that stationary condi-tions of on S1 provide,

= = (1-3.22)

f Compare with Eq. (1-2.21).

356 APPENDIX I

together with Eqs. (1-3.17) and (1-3.18). Consequently, we may write thefunctional ror the generalized principle in another equivalent form as follows:

—

— — + —

— — aJSS, (h3.23)t30 I

where

..f — + (1-3.24)B..

or— — — (1-3.25)

In Eq. (1-3.23), the independent quantities subject to variation areand with no subsidiary conditions.

S.4. Modified Prhadple

Plimination d from the functional by the use of the $ationaryconditions (1.2.4) leads to the modified functional:

flg. EfJJ[—B(a11) + +

— fJ + — + 7(b))Ws

— If — If — (j-3.26)t

where the independent quantities subject to variation are andwith no subsidiary conditions. Through integrations by parts, we may obtainanother expression for the modified HeHinger—Reissner functional:

= + +V.

— — — — (1-3.27) §S.

where= + (1-3.28)

S..and the independent quantities subject to variations are and-h withno subsidiary conditions,

t Compare with Eq. (1-2.22).Compare with Eq. (1-2.23).

§ Compore with Eq. (1-2.24).

APPENDIX I

Section 4. of Modified Variational Principlesfrom the Principle of Minimum Complementary Energy

The purpose of this section is to follow in Fig. 1-1 an avenue which startsfrom the principle of minimum complementary energy, leading to the modi-fied principle of complementary energy and finally to the modified Hellinger—Reissner principle. We shall treat the same problem as defined in the begin-ning of Section 3, and proceed to a formulation of the principle of minimumcomplementary energy for the assembly of the finite elements.

4.1. Principle of Minimum Energy

We shall denote stresses in each element byg(2) i — 1 2 3U' U'"' U' U'"" ' '

each of which will be called a function for stresses. The assembly of thesefunctions for stresses may be taken as admissible functions for the functionalof the principle of minimum energy, if they satisfy followingrequirements:

(i) They are continuous, single-valued and satisfy Eq. (1-21) in eachelement.

(ii) They satisfy equilibrium conditions on inter-element boundaries:

+ 71(b) = 0 on (1-4.1)

where and are defined by Eqs. (1-3.8).

(iii) Those belonging to an element containing satisfy Eq. (1:2.5).

Consequently, if the functions for stresses are so chosen as to satisfy the require-ments (1), (ii) and (iii), the functional for the pri ciple of minimum comple-mentary energy is given by

lic = — .(1-4.2)tV4. SI,

where the independent quantities subject to variation areNext, we shall formulate a variational principle in which the subsidiary

conditions (1-4.1) are introduced into the framework of the variational expres-sion. By the use of Lagrange multipliers defined on Sab, we obtain the func-tional for a modified principle as follows:

= — (1.43)

f Compare with Eq. (1-2.25).

358 APPENDIX I

where is given by Eq. (1-4.2) and where it is defined that

+ (7-4.4)

and the independent quantities subjected to variation are and under thesubsidiary conditions, Eq. (1-2.1) and (1-2.5). The principle for the functional

will be called the modified principle of eomplemenvaçy energy with relaxedcontinuity requiren*ents, because the requirement (ii) is relaxed in and thefunctions for stresses in each element may be chosen independently without anyconcern about the equilibrium requirements on the interelement boundaries.It is noted here that the modified principle is no longer a nunimum principle,but keeps its stationary property only. The functional was originallyformulated by

4.2. Modified Priaciple

Next, we shall introduce the subsidiary conditions, (1-2.1) and (7.2.5), intothe framework of the variational expression by the use of Lagrange mul-tipliers Then, we may have a functional which is the same as that of themodified Hetlinges—Reissner principle given by Eq. (1-3.27). Needlessto say, it is a simple matter to transform thus obtained into 11mRdefined by Eq. (1-3.26) through integration by parts.

Thus far, two avenues in the flow diagram of Fig. I-I have been traced.Arrows in the diagram show conventional avenues leading from one principleto another. The reader is advised to follow these arrows and familiarize him-self with these transformations.

Several typical finite element models are also listed in the flow diagram,together with the variational principles on which the models are based. Adetailed description of interrelations between these variational principles andrelated finite element models are beyond the intended scope of this appendix.Therefore only a brief mention wiH be made of finite element models basedon the principle of virtual work. For details of these interrelations, thereader is directed to Refs. 5 through 8 and 23, for example.

As mentioned in the Introduction of this book, an approximate method ofsolution based on the principle of virtual work is called the Galerkin method,which may be considered as an application of the method of weightedresiduals. As far as the elastostatic problem in the small displacement theoryis concerned, this method provides a finite element formulation which isequivalent to that obtained by the use of the compatible model. However,the principle of virtual work or its equivalent provides a basis which isbroader than variational principles when applied to problems outside of thesmall displacement elasticity problem. Similar observations may be madeconcerning finite element formulations based the principle of comple-

APPENDIX 1 359

mentary virtual work, the modified principle of virtual work, and the modifiedprinciple of complementary virtual work.

PROI,LEM I. Show that the modified principle of virtual work is given asfollows:

—V.

— Eef — ffTø3u4s 0, (1-4.5)Ssi .

where the subsidiary conditions are given by Eqs. (1-2.17) and (1-2.18).

PROBLEM 2. Show that the modified principle of complementary virtualwork is given as follows:

EJJJE

— + — =0, (1-4.6)83$

where the subsidiary conditions are given by Eqs. (1-2.29) and (1-2.30).

PROBLEM 3. Read Refs. 5, 6, 7 and 22, and show that:(a) Displacements are assumed along all the interelement bound-

aries, and the stiffness matrix of each element is to be obtained by the use ofthe principle of minimum potential energy in the hybrid displacement modelII based on or

(b) Displacements are assumed along all the interelement boundar-ies, and the stiffnesa matrix of each element is to be obtained by the use of theprinciple of minimum complementary energy in the hybrid stress model basedOflhlmc.

PROBLEM 4. Show that by the introduction of a new quantity e defined by

3e = + + 8a'the strain energy function A given by Eq. (2.3) can be generalized as follows:

es,, ..., e, H)

Gv 2

= (1 —+ G(e? + +

+ + +— 2vGH[3e — + r,, +

where His a Lagrange multiplier, which is multiplied by 2vG for the sake ofconvenience.

360 APPENDIX I

Next, show that A(e, e.g,, ..., e, H) is transformed into

A(e:, es,, ..., H)= + + + + ++ + + e,) — i'(l — 2v)H9

through elimination of e by the use of thç stationary condition of ç,yr,; e, H) with respect to e, namely,

3e — (I — 2v)H.

Finally, indicate that A(e2, e,, ..., H) is equivalent to the strain energyfunction derived by Herrmann for nearly incpmpressible 10)

SectIon 5. ConventIonal Variational Prlndplesfor the Bending of a Thin Plate

We shall devote the present and next sections to'the derivation of con-ventional and modified variational principles for the problem defined inSection 8.1, namely, the bending of a thin plate based on the Kirchhoffhypotheses, because problems of plate bending .are frequently treated innumerical examples of various finite element models. We shall first reviewsome fundamental relations of the problem. Unless otherwise stated, we shallemploy the same notation as used in Chapter 8.

We remember that in the bending theory plate, the stress—strainrelations are given by Eq. (8-2), and the function A and comple-mentary energy function B are given by -7

and

A2(I ,,2)(8X + e 4: + + — (I-5.l)t

B = -f- + 2(1 ÷ +

respectively. W,Vremembe the KirchhoLhypothesis imposesgeometrical such that/ U = —zw1, V = W = w,

and

e/= = = = = 0,

t Refer to Eqs. (3) and (4) of Appendix B.Refer to Eqs. (814).

§ Refer to Eq. (8.15).

I 361

where w(x, y) is the displacement of the middle surface in the of thez-axis. Two relations are noted here, since they are frequently used in sub-sequent formulations:

( = I( — m( )..,( m( + l( (I-5.5)t

which hold on the boundary C and

5 Mw JdsCe, +

+ +

where V2, M, and are defined by Eqs. (8.24) and (8.25) by the use ofM1, and

5.1. PrInciple of Minimum Potential Energy

The functional for the principle of minimum potential energy for the platebending problem is given as follows:

Jig, ff[A(w) — pw]dxdy5"

+ I1' + +.i,,w.,)ds,Ce,

where

A(w) + + 2(1 — vXw2.,i, —

and where the subsidiary conditions are

w = *, w,. = W on

The functional (1-5.7) can be derived from Eq. (1-2.19) in a manner similar tothe development in Section 8.2, by first substituting Eqs. (1-5.1), (1-5.3) and(1-5.4) into Eq. (1-2.19) and then performing integrations with respect to z,noticing that

dS=dzds. (1-5.10)

A further partial integration was performed in Section 8.2 to obtain the

t Refer to Eq. (8.20).Refer to Eqs. (8.19), (8.20) and (8.21). Notations Ce, and are used instead of C1 and

C3, respectively.

362 APPENDIX I

mechanical boundary conditions in a manner as given by Eq. (8.31). How-ever, it is preferable for later formulations to write the integration on C,, as itis in Eq. (I-S.?).

5.2. Generalized Principle

The functional (1-5.7) may be transformed through familiar procedure toobtain the functional for a generalized principle:

= xi,, + —S.

+ (3, — + — —pwjdxdy

+ f[— V1w + +cc

+ f[_(w — + —C"

+ (W.a (1-5.11)

where

= ((x' + + 2(1 — — (1-5.12)

and where P1, P2 and are Lagrange multipliers on C, defined later byEqs. (1-5.17). The last term on the right-hand side of Eq. (1-5.11) is obtainablefrom the last term on the right-hand side of Eq. (2.26) of Chapter 2, which is

written for the,present problem as follows:

1112

f 5 ((U — + (V — + (W — (1-5.13)

where p,, and Pz are Lagrange multipliers which introduce the geometricalboundary conditions into the variational expression. By the use of Eq.(1-5.3), (1-5.5) and (1-5.9), we may derive following geometrical relations on

the boundary C,:U = —zw., = .-zQw.,. — mw,),V = = +w=w, (1-514)

and17= —z(1W —

= (1-5.15)

Substitution of Eqs. (1-5.14) and (1-5.15) into the integral (1-5.13) and integra-tibns with respect to z transform the integral into

— W)P3 + (w., — + — (1-5.16)C,,

APPENDiX I

whereP1 = ijp1zdz + niJp11zdz,

P2 = + if p1zdz,

P3 = fp2dz, (1-5.17)

and we obtain the last term in Eq. (1-5.1 1). It is noted here that the Lagrangemultipliers P2 and P3 in Eq. (1-5.11) cannot be assumed independently,because w and wV,, are not on Cf

We may obtain another expression of the generalized variational principlein which the Lagrange multipliers F1, F2 and P3 have been eliminated. Forthis purpose, we may require the coefficients of ow and on of tovanish. After some manipulations including integrations by parts and theuse of Eq. (1-5.6), we find that the first variation of1701 on C1, takes the follow-ing form:

f[(v2 — P3)Ow — — — (M,,, — P2)Owjds +Cu

= + + M,J — (P3 +Cl'

— (M, — + ... . . (1-5.18)

Consequently, the requirement that the coefficients of ow and Ow:, on C1must vanish provides:

Va + Mv,a = Ps + P2,, M, = p1 on (1-5.19)

andM, P2 at the ends of C1. (1-5.20)

We find from Eqs. (1-5.19) and that P1, P2 and P3 may be replaced byM,, and respectively, in thèlntegral (1-5.16). Thus, we may transform

1101 into= x1,) + (x2 —

+ — + 2(x1, — — frwjdxdy

+ f[— ÷ R,w,, + 2,1wjàsC,, 4

+ + M,(w, — ÷ M,(w, — (1-5.21)

where the independent quantities subject to variation are x, x1,, w, M,, M1and with subsidiary conditions that w = at the ends of C1,4

t See Section 8.2.The functional with no subsidiary conditions is obtainable by eliminating P2

and P3 by the use of the stationary conditions of fl01 on C, + It is given by adding— — terms to the R.H.S. of Eq. (1-5.21), vnesns summations

over all the C,,s.

364 APPENDIX I

5.3. Ilellinger-Relssner Principle

We mayO eliminate and from the functional 1102 through the use oftb. stationary conditions, Eq. (8.54), to obtain the functional for the Heilinger—Reissner principle:

HR = — —SI'

— M11) — pw]dxdy

+ + Mw, + 14wjdcC',

+ f (— — W) + — + M,a(W.. — (15.22)C"

where

= [(M2 + + 2(1 + (1-5.23)

By the use of Eq. (1-5.6), the functional (1-5.22) may be transformed intoanother expression of the functional for the HeUinger—Reissner principle:

—17: = M1,)SI,

+ + + + p)wJdxdy

+ ft —(Va — + (M, — + —

\ c',

+ 11— + M,W + (1-5.24)C"

It is obvious that the functional for the principle of minimum complementaryenergy may be derived from Eq. (1-5.24). We repeat here that special care mustbe taken for formulating the mechanical boundary conditions for platebending problems under the Kirchhoff hypothesis.

SectIon 6. DerIvation of Modified Variational Principlesfor the Bending of a Thin Plate

We shall continue to treat the problem defined in the preceditig section,except that the region Sm is now divided into a number of finite elements:

52, ..., 5N, and the whole region is treated as an assembly of these ele-ments. For later convenience, we denote two arbitrary adjacent elements

APPENDIX 1 365

by Sa and and the interelement boundary between Sa and Sb by C..1, asshown in Fig. 1-3. Two symbols and will be used whenever necessaryto distinguish the interelement boundary belonging to and 85,,,respectively. Arrows labelled with Sa and n the same figure denote the

V

0

Fio. 1-3. S.., Sb and C,..

directions of measuring s along the boundaries ofMoreover, two arrows labelled with and denoteOfl C:b and respectively.

and ØS1,, respectively.the outward normals

6.1. PrincIple of Potential Energy

We shall denote the deflection w(x, y) in each element by

U) (2) (b) • (N)w ,w ,..., ,w

The assembly of these displacement functions may be taken as admissiblefunctions for the functional of the principle of minimum potential energy, ifthey satisfy following requirements:

(i) They are continuous and single-valued in each element.(ii) They are conforming on interelement boundaries:

= = Ofl Cab.

(iii) Those belonging to an element containing C, satisfy Eq. (1-5.9).

Consequently, if the displacement functions are chosen to satisfy the require-ments (i), (ii) and (iii), the functional for the principle of minimum potentialenergy is given by

C9,,

x

APPENDIX I

lip = — flw]dxdy3'

+f[— + + (1-6.2)jCo

where the notation means summation over the entire elements. The inde-pendent quantities subject to variation in TI,. are under the subsidiaryconditions (ii) and (iii).

6.2. Modified Prlnciples.of Potential Energy

Next, we shall formulate a variational principle in which the subsidiaryconditions (1-6.1) are introduced into the framework of the variationalexpression. By the use of defined by Eq. (1-3.4) and remembering that

=. =y(b) =

= =

and

= m( +'( ).ia'( = —l( ).'b + m( ).sb'

( = —"( I( ).ab'

where I and m are direction cosines of the normal vi,, we may transfonnHabx into:

= — —

+ J[_A3w"' —

A1 = if + mfa3zdz,

• A2 = —mf ).azdz +

= fA3dz (1-6.4)

are Lagrange derived from (A1, A2, As). Consequently, we have thefollowing funciiónal for the modified principle of potential energy:

limpi = lip — (1.6.5)

t Compare with Eq. (1-5.7).

APPENDIX I 367

where and are given by Eqs. (1-6.2) and (1-6.3), respectively. It isnoted here that the Lagrange multipliers A2 and A3 in HObi cannot beassumed independently, because and and also and are notindependent on Cab.

PROBLEM. Show that HObi may be transformed through integrations byparts into the following form:

H4b1 ÷ —

÷fE—(A3 + — — A2(w(a) — (1-6.6)

where the notation means that values at the ends of Cab are taken. Note:See Eq. (827).

6.3. Modified Generalized Principle

We note (omitting the algebra details) that the following functional for themodified generalized principle may be derived from Eq. (1-6.5):

11mG = EJf[A(x1, + —Sc

+ — + — — jiwjdxdy

+ + MyWi, + A?gw,]dcCo

+f[—P3(w — + — W) + p2(w —

(1-6.7)t

6.4. ModIfied Hellinger—Reissner Principle

By the use of Eqs. (8.54), we may eliminate and x.. from Eq. (1-6.7)to obtain a functional for the modified Hellinger—Reissner principle:

= —

— — B(MZ• — pwJdxdy

— + I ( + Mew., + M,wjdsCo

+$[— p3 (w — —

+ P2(w, — (I-6.8fl

t Compare with Eq. (1-5.21).: Compare with Eq. (5.22).

368 APPENDIX I

PROBLEM 1. Show that by the intrnduction of new functions:

Aia), defined onAr>, /t4b) defined on

and•

defined on C4b,

the expression (1-6.3) may be written in an equivalent form denoted byHaba as follows:

— — — p2) — —

+ b)— — + /4,) /4b)(W — !s1).s,Jbb. (1-6.9)

PROBLEM 2. By replacing !Igbl in Eq. (1-6.8) with HUb2 of Eq. (1-6.9), showthat the stationary conditions of Eq. (1-6.8) on C0b with respect to andallow us to set:

= = = Me>,= = My'>, = (1-6.1O)

in the Eq. (1-6.9) and consequently, we may write Hdba in an equivalent formdenoted by "ab4 as follows:

— 5 — — — — —cb

+ 5 — u1) — + is,) — —. (1-6.11)

6.5. Another Derivation of the Modified Hellinger-ReissnerPrlncipk

Thus far, we have formulated the modified Hellinger—Reissner principlefrom the modified principle of potential energy. Now, we shall trace anotheravenue and derive the modified Hellinger—Reissner principle from Eq.(1-3.27), where the term GOD is given as follows:

= + (1-6.12)

Remembering that the function p2 and /45 in Eq. (1-6.12) correspond toU, V and W on SOb, respectively, and the U, V and w are expressible as givenby Eq. (1.5.3), we may write Eq. (1-6.12) as follows:

h12

GOD = f f + + (y,(a) +h12

+ (f-'6.13)

'Eqs. (I—d. 10) don't hold in general if one or both of the ettd points of is on + C., because the de-termination of these Lagrange muttipliers should be made by the use of the stationary conditions of the func-tional or 11.,. in which H,,1 has been replaced by

APPENIMX I

After some manipulation, we obtain:

= — —

+ — (1.6.14)

where we setW fir, W,0 (1-6.15)

In Eq. (1-6.14), and skould be taken as Lagrange multipliers defined onCIb. Thus, we obtain an expression of the modified Hellinger-Reissnerfunctional as follows: -

M11)

+ + + M1, + p)w)dxdy —

+ jI—(V, Paw + (Al, — Mjw, + (M, —Ca

+ J[— V,P + M,W ÷ (1-6.16)C.

where is given by Eq. (1-6.14).Performing partial integrations, we may transform the functional given in

Eq. (1-6.16) in another form:

— —S.

— M,, Mi,) — pw)dxdy

— — Pi) — —

p) — + — —

+ j r— + 11,w, +Ca

— iP) + M,(w, — 1T) + — *,,)Jfr. (I-6.17)

• 6.6. A Special Case of die Modified Variational Principlesfor the Sedliig of a Thin Plate

As the last topic of this section, we shall consider a special case of themodified variational principles when the displacement functions are so

• The functionals (1-6.16) and (1—6.17 are subject to subsidiary conditions for whichsufficient conditions may be given as follows: (i) = for all the nodalon C1,, and (ii) for all the nodal points on C,.

370 APPENDIX I

chosen that they are continuous along the entire interelementboundaries:

on (1-6.18)

Then, Eq. (1-6.3) reduces to

= .—f + (1-6.19)

By the introduction of new functions and Al") defined on and Ce,,respectively, together With a new Lagrange multiplier 4u, Eq. (1-6.19) may bewritten in an equivalent form as follows:

= —f rArw ÷ — — Ar)]ds, (1-6.20)

or

= — + (1-6.21)c:,

By the use of Eq. (1-6.21), the functional for the modified principle of potentialenergy, Eq. (1-6.5), can be written as follows:

11mP2 = — IHGb2, (1.6.22)

where the independwt quantities subject to variation are andunder the subsidiary conditions, Eq. (1-5.9). Taking variations with respect tothese quantities, we find that the stationary conditions of Ofl

provide= (1-6.23)

—. = (1-6.24)

where and are obtained substituting the stress-resultant and displacement relations, Eq. (8.33), into and respec-tively, to express them in terms of the displacements only. Eqs. (1-6.23) and(1-6.24) indicate the physical meaning of the Lagrange multipliers Al",and We also find that by the use of the stationary conditions, Eq. (1-6.23),we may Al" and to reduce the functional to

=11, — (1-6.25)

where

—

+ (1-6.26)

We may obtain the modified generalized pjinciple and the modifiedHellinger-Reissner principle for this special case by substituting Eq. (1-6.20)

APPENDIX I

or Eq. (1-6.2 1) in place of Hebl into Eq. (1-6.7) and (1-6.8), respectively. Wefind that the stationary conditions on of the functionals derived throughthese substitutions provide

Aia) = = (1-6.27)

Consequently, we find that we may replace in Eqs. (1-6.7) and (1-6.8) bydefined by the following equation:

"ob4 = —

— I + Eu)dSb, (1-6.28)C,:

to obtain alternate expressions of the modified generalized principle and themodified Hellinger—Reissner principle the special case specified by Eq.(1-6.18).

We shall specialize our problem further by ássmning that not only w, butalso M, are continuous along the entire intcrelement boundaries:

= — on - (1-6.29)

Then, Eq. (1-6.28) is reduced to

= — (1-6.30)c;,

and we have an expression of the functional for the modifiedReissner principle as follows:

= — —84

— M,, M11,) — pwJdxdy

+ +CL Cl.

+11— Vw + M,w•, + M,w,JdsC.

+ f(— V1(w — ii)) + M,(w, — + Mjw, — (1-6.31)C"

Through integrations by parts, we may transform Eq. (1-6.31) intoform:

= X5j[—a(M1, M1,, + + M,,)

• + + M1,,) — pw)dxdy

- +C:, •c$.

+11— P1w + (M, — ÷ — MJw]dsCo

+ 5 [— V(w — P) — M,W — (1-6.32)C"

• Inc functionals (1-6.31) and (1—6.32) are subject to subsidiary conditions for whichsufllcknt conditionsmay be $vea as foltowa: w wi') w for all the nodal points on C,,. -

372 APPENDIX I

The functional (1-6.32) was originally formulated and applied to a finiteelement analysis of the plate bending problem by Herrmann.°' 12)

SectIon 7. VarIational Prhiclpks for the SmallDisplacement Theory of Flutodynainlcs

Our next topic will be variational principles fo' the small displacementof theory of clastodynamics, for which the governing equations may be givenas follows:

(1) Equailons of motion:

CUJ +j (1-7.1)

(2) Strain-displacement relations:

LU 4(Ut., + U,•,)

(3) Stress—strain relations:

=

or conversely

Lu = (1-7.4)

(4) Mechanical boundary conditions:

= on S, (1-7.5)

(5) Geometrical boundary conditions:

Ut — on (1-7.6)

where the quantities appearing in these equations, namely, eu, Ut, ,?tI 1,2, 3.Fora

complete definition of the clastodynamic problem, the following initialconditions should be added to the above equatiolis:

ut(xi, x2, 0)

ü1(X1, x2, x8, 0) —

where Ut(O) and are prescribed functions of the space coordinates.Hamilton's principle introduced in Section 5.6 is the best established and

most frequently used variational principle among those derived for the

APPENDIX I

elastodynamic problem. Through transformations and generalizations similarto those for the elastostatic problem, we may create a flow diagram for afamily related to Hamilton's principle as shown in Fig. 1.4. Several papersrelated to this diagram are listed in the bibliography of this

Conventlonol voriotionolprinciples

Variational principlesror relaxed

Here we shall trace only an avenue which leads from the principle of virtualwork to the principle of complementary áergy. The reader ii di.rccted toRefs. 27 and 29 for other routes, includin* the modified variational principleswith relaxed continuity requirements.

71. Prhiciple of Virtual Work

Denoting a virtual variation of u4(t) at time : by öu1(:), we havet

.—jff(iu., — pü1)ôu1dV + f5(r1 — ?1)ôu,dS —0,V

(1.7.8)

where the integrations extend over the entire region of Vand S, at the time t.By integrating Eq. (1-7.8) with i'cspect to time between two limits t t1 and

t It is repeated that öu1(t) is a virtual variation of u,(t) at the tünc I. The reader will findthat the function ui(s) + plays a role of an admissible function in Eq. (1-7.14).

Mediti•d principle ofcomplementary energy

Fio. 1-4. A flow diagram for the small displacementtheory at eiaatodynamics.

374 APPENDIX I

t t2, and employing the convention that values of at t = t1 and t =are prescribed such that

0, 0, (I-7.9)t

together with some manipulation including partial integrations respectto time as well as the space coordinates, we obtain the principle of virtualwork for the problem as follows:

7{oT —

+ 0, (1.7.10)Sc

where

T (1-7.11)

is the kinetic energy of the elastic body, and where the subsidiary conditionsare given by

+ (1-7.12)

andon S1, (1-7.13)

together with Eqs. (1-7.9).

7.2. Hamiltosi's Prbsclpk

If the body forces and external forces on S0 are assumed to be pre-scribed in such a way that they are not subjected to variation, we may derivefrom Eq. the principle of stationary potential energy, or Hamilton'sprinciple, as follows:

t2

of(T—a,)d:=o,

where Tand are given by Eqs. (1-7.11) and (1-2.19), respectively, while thesubsidiary conditions are given by Eqs. (1-7.6) and (1-7.9).

t This convention means that the initial conditions, namely Eqs. (1-7.7), are not takeninto serious consideration in the Hamilton's principle family. Ii may be said that the primaryconcern for the family is derivation of the equations of motion and boundary conditions atthe time:; the initial conditions are of secondary corcern.

Refer to Eq. (5.86).

APPENDIX I 375

7.3. Cewrsllzed Principle

Next, we shall introduce new functions defined by

— 0, (1-7.15)

and write the kinetic energy Tin a generalized form as follows;

T0 — — u1) JdV,- (1.7.16)

where is Lagrange multiplier which introduces the subsidiary condition,Eq. (1-7.15), into the framework of the expression of the kinetic energy. Then,we obtain a generalized principle as follows:

oJ — r102)d: = 0, (F7.17)

where and 11G2 are given by Eqs. (1-7.16) and (1-2.22), respectively, whilethe subsidiary conditions are given by Eqs. (1-7.9).

7.4. Prlncitile

Elimination of v, and from Eq. (1.7.17) by the use of the stationary condi-tions with respect v1 and e11; namely

- pv1 = (1-7.18)

and Eq. (1-7.3), leads to the Hellinger—R.eissner principle:

4 tiff — —J

= o,

where HR is givenby Eq. (1-2.23), while the subsidiary conditions are given byEq. (1-7.9).

Through integrations by parts with respect to time as well as the spacecoordinates, we obtain another expression for the Hdllinger—Reissnerprinciple:

of {_.fff + dV - } at (1-7.20)

where is given by Eq. (1-2.24), and the subsidiary conditions are given by

0pt('i) — 0, = 0. (1-7.21)

376 APPENDIX I

7.5. PrInciple of Stationary Complementary Energy

The principle of stationary complementary energy is obtainable by takingas subsidiary conditions the stationary conditions with respect to the dis-placements, namely:

+ = (1-7.22)

and

= on (1-7.23)

and we obtain

— .fff +

— — 0, (1-7.24)

where Eqs. (1-7.21), (1-7.22) and (1-7.23) are taken as subsidiary conditions.

7.6. Another Expression of the Principle of StationaryComplementary Energy

Next, we shall obtain an alternate expression for the principle of stationarycomplementary energy. First, we introduce the following new notations:

l•u = tf

=

V1 = 131 = il'. (1-7.25)t

Assuming = 0 at t = 0, we may replace Eqs. (1.7.22) and (1-7.23) by thefoHowing equations:

+ = p1 (1-7.26)

and= (1-7.27)

We may eliminate Pt from Eq. (1-7.24) by the use of Eq. (1-7.26) and performpartial integrations .with respect to timetoobtain

? These definitions for v1 and 'r are used in Section 7.6 only.

APPENDIX I 377

—+fJdV

+ ff iividS)d: 0, (1-7.28)V 1,,

where the subsidiary conditions are Eq. (1-7.27) together with

=0, =0. (1-7.29)

Eq. (1-7.28) is another expression of the principle of stationary complemen-tary energy which is expressed in terms of impulse and velocity instead oi'forceand

It is noted here that Hamilton's principle and the principle of virtualwork have been used frequently in mathematical formulations o( the finiteelement method applied to dynamic response problems. An elastic body undercOnsideration is divided into a number of finite elements and Hamilton'sprinciple is applied to obtain asystem of linear algebraic equations which maybe written in a matrix form as follows:

+ (C1(4) + = {Q}, (1-7.30)

where [MI, (CJ and (K] are the inertia, daniping and stiffness matrices,respectively,, while {q) the column vector of nodal displacements, and {Q} isthe external load vector. Eq. (1-7.30) may be solved by either the modesuperposition method or a step-by-step integration procedure. The reader isdirected for Refs. 31 and 32, for example, for further details. It is also notedhere that the principle of stationary complethentary energy has been usedrecently in application to the finite element

7,7. Gurtln's

We have seen that the intal conditions, Eq. (1.7.7), are not taken intoserious consideration in the variational family associated with the Hamilton'sprinciple and, in that sense, none of the family is complete in defining theelastodynanuc problem in the form of variational expressions. Gurtinestablished variational principles which, in contrast to those belonging to theRanulton family, fully characterize the solution of the elastodynamic prob-lem. His formulation begins by first defining the convolution of two functions

t) and w(x, r) by

t) = I — I')w(x, t')d', (1-7.31)

and then observing that cv41 and u,1 satisfy the equations of motion, if andonly if

378 APPENDIX I

g.o'(J•, + = pUs, (1-7.32)

where x denotes the space coordinates {x1, x2, x0), and

g(l) = 1, (1-7.33)

t) = + p(x, t)[ftT(x, o) + 0)]. (1-7.34)

By the use of these relations, Gurtin dcrived a family of variational principleswhich have forms similar to those shown in-Fig. 1-i, except for the presence ofg, the use of convolutions, and the appearance of the initial conditions andthe term p. For details, the reader is directed to Gurtin's original papers. It isnoied that variational formulations using convolution integrals havebeen employed recently in the basic theoretical development of the finiteelement method for time dependent

Section & Finite Displacement Theory of Flastoetatice

In 3.5 we defined a problem of the finite displacement theory ofclastostatics which is usually called a geometrically nonlinear problem,because thesolid body still behaves elastically, although the displacements arefinite and no longer small. We formulated the problem by the use of Kirchhoffstress tensor and Green strain tensor CM in the first part of Chapter 3.tIn the subsequent sections of the chapter, we formulated for the problem theprinciple of virtual work, the principle of stationary potential energy, thegeneralized principle, and the Hellinger—Reissner principle, as represented byEqs. (3.49), (3.68), (3.70) and (3.71). respectively. These variational principlescan be modified into those for relaxed continuity requirements and we obtainthe flow diagram illustrating interrelations between these variational principlesas shown in Fig. 1.5.

&1. Some Remarks on the Flow Diagram

Three comments will be made here with regard to the flow Thefirst comment concerns the principle of complementary energy for thenonlinear biastostatic problem. It can be shown that by the use of the equationsof equilibrium, Eq. (3.27), together with the mechanical boupdary conditions,Eq. (3.42), we may reduce the functional of Eq. (3.71) to

= +

31d

t The stress tensor has been named pseudo.strcss or generalized stress in the footnoteof page 57. It is also called the second Piola-Klrchkoff stress tensor in Ref. 39.

In sections 8 and 9 of this appendix, we shall use subscript Roman letters instead ofsuperscript or subscript Greek ktteri employed in Chapter 3. Thus we write e,g,

instead of w', eM, . ., respectively.§ Thç body forces and the external forces on S0 are assumed dead loads.

APPENDIX I 379

However, since the Qoupling of displacements with stress componentscomplicates the expression of as well as the subsidiary conditions, Eqs.(3.27) and (3.42), there seems to be little merit in deriving the expression for

in the form as shown in Eq. (1-8.1). Consequently, the principle ofcomplementary energy is not listed in the flow diagram of Fig. 1-5.t

variationalprincipiss

Variational princiPiss forrsloxsd COntinultyr.gUTiimsnts

The second observation relales to the variational principles with relaxedcontinuity requirements. It is easily observed that the functional for theprinciple of stationary potential energy is given for a finite element formula-tion as follows:

lip + + 55V. S.

(14.2)

while the functional for the modified principle of potential energy with relaxedcontinuity requirements is given by

— Ii, —

where H, is given by Eq. (1-8.2) and

(1-8.3)

f this does not redues the value of the prlndple.ofmentary energy which may be formulated for incremental theories of the

problem.

Modifiod IIsllinqsr—Rslssnsrprincipi.

Fio. 1-5. A flow diagram for the finite displacement Theodes ofdasiostatics and

380 APPENDIX I

— (1-8.4)all.

In Eq. (1-8.4), the newly introduced functions A4; i = 1, 2, 3 are Lagrangemultipliers, while and arc displacement components belonging to twoadjacent elements a and b, respectively.

The functional may be transformed into another equivalent functionalH12 as follows:

— (1-8.5)

where

55 + — + (1-8.6)

or equivalently

If — ujdS + If — ujdS.5b

The modified principle of potential energy may be generalized in theusual manner to obtainthe functional for the generalized principle:

= +V.

— a45[e1, — + u,4 + u,,1uft1)]dV

— + 55W(u1)dS — fjp4(u4 — ü4)dS. (1-8.8)'0 311

Elimination of e1, from then leads to the functional for the modifiedHellinger-Reissner principle:

= + ui.f +V4

— B(cir45) + (ujJdV —

+ f .f — (1-8.9)3,,

PROBLEM. Show. that the stationary conditions on of the functionalfl01 of Eq. (1-8.8) provide

= Fi", = (1-8.10)

where= + t41), (1-8.11)

J1b) + 4k). (1-8.12)

Show also that the stationary conditions on of the functipnal ofEq. (1-8.5) provide:

APPENDIX I 381

a) = = (1-8.13)

where and rb)(,,(b)) can obtained from Eqs. (1-8.11) and (1-8.12)by substituting Eqs. (3.33) and (3.18) to express and entirely interms of the displacement and respectively.

The third observation relates to the problem of the finite displacementtheory of elastodynamics defined in Section 5.6. It is apparent that we mayobtln for the elastodynamic problem a flow diagram .similar to that shownin Fig. 1-5, if the inertia term is taken into account.

Thus far, several remarks have been made on the flow diagram of Fig. 1-5.It is natural to conclude that we may formulate finite element models cor-responding to these variational principles in a manner similar to those for thesmall displacement elastostatic problem. Among finite element models thusformulated, the most frequently used is the compatible model based on theprinciple of stationary potential energy. This model will be discussed brieflythe .next section.

8.2. A Formulation for the Compatible Model and the ModifiedIncremental Stiffness

A formulation for the compatible model begins by approximating ineach element by

{u} = [S'I(q), (1-8.14)

with the ad of compatible shape functionà, where {u)T = [u1, u2, u3) and {q}is a column vector of nodal displacements. If the total strain energy U isexpressed in terms of

U = (1-8.15)vc

we may obtain the followülg equations by the use of the principle of station-ary potential energy:

= (1-8.16)

where is a column vector of the generalized forces. Since Eq. (1-8.16) arenonlinear, several iterative solution methods have been proposed.

Here, we shall outline an iterative method called the modified incrementalstiffness method, assuming for the sake of simplicity that the elastic body isfixed on We divide the total strain energy in two parts such that

U=UL+UNL (1-8.17)

where tJL is a linear term containing all the quadratic terms with respect todisplacements, while URL is a nonlinear term containing all the remaining

382 APPENDIX I

higher-order products. The stiffness matrix tKI is then derived from

= [K]{q}. (1-8.18)

We now divide the loading path of the solid body problem into a number ofstates:

Q(O), a(N+1),

where and are the initial and final states of the deformation, respec-tively, while is an arbitrary intermediate state. We shall derive an incre-mental formulation for the determination of the f1(N state assuming that thisstate is incrementally close to the state and that the state is known.Penoting the generalized forces and displacements corresponding to the

and states by {Q(N)}, {q(N)) and + {q(N) +respectively,.and by the use of Eqs. (1-8.17) and (1-8.18), we may write Eq.(1-8.16) for the state as follows:'

+ {Aq))+ + Aq))

= {Q(N)} + {AQ }. (1-8.19)

By the use of a Taylor series expansion

+ IXqk) — Ø2UNL

—+

øqgøqjq, +

in which the higher order terms are neglected, we may have

([K]+

{Aq}

(N) a+ {Q(N)) — (C NL(Q )). (1-8.20)

We obtain by solving Eq. (1-8.20), and the displacements correspondingto the state are given by {q(N) ÷

it a unique characteristic of the modified incremental stiffness methodthat the term

{Q(N)) ——

NL(q (1-8.21)

is retained on the right-hand side of Eq. (1-8.20) for an equilibrium check. It isstated in Ref. 40 that the equilibrium check term plays the essential role ofpreventing an approximate solution based on this incremental formulationfrom drifting away from the exact solution.

A review has been given in Ref. 41 on various formulations for solving thegeometrically nonlinear problem numerically. These includes the incremental

APPENDIX 1 383

stiffness procedure, self-correcting incremental procedure (modified incre-mental stiffness method), Newton—Raphson method, perturbation methodand initial-value formulation. Distinguished features of each formulation arediscussed and recommendations are made as to which procedures are the bestsuited. It is also stated in the reference that the treatment of the nonlinearproblem as an initial-value problem opens the door to a arge number ofsolution procedures: For details of these formulations and their applicationsto FEM, the reader is directed to Refs. 40 through 44.

8.3. A Generalized VarIational Principle by the Use of thePlola Streas Teasor

The last topic of this section will be a derivation of another generalizedprinciple from the principle of stationary potential energy, Eq. (3.69). Tobegin with, we find that the strain e,, is a function u,,,, and may be written

e,1 = + + (1-8.22)

where by definition

= (1-8.23)

By the use of Eq. (1-8.22), we may express the strain energy functionin terms

of brevity. Then, by the introduction of Lagrange multipliersand may derive from Eq. (3.69) the following generalized functional:

ficn fff{4z11) + 0(u1) — — u11)}dV

+ — ffp1(uj — ujdS,

where the independent quantities subject to variation are 81j andwith no subsidiary conditions. Taking variations with respect to thesequantities, we obtain the following stationary conditions:

-=

(IJjJ + P1 = 0, (1-8.26)

— = 0, (1-8.21)

= on S, (1-8.28)

aJ(nf = on (1-8.29)

384 APPENDIX I

= on (1-8.30)

These equations indicate the physical meaning of the Lagrange multipliers.It is seen from Eqs. (1-8.25) and (1-8.26) that is the Piola stress tensor.f 4

If the body forces P4 and the external force on are treated as deadloads, we may use Eqs. (1-8.26), (1-8.28) and (1-8.29) for the elimination of u1to transform Eq. (1-8.24) into

fff(A(ct,,) — ÷ ffa,4njutdS, (1-8.31)V Sal

where the independent quantities subject to variation are and under thesubsidiary conditions of Eqs. (1-8.26) and (1-8.28). Thus, the merit of the useof the Piola stress tensor is that the subsidiary conditions are expresSed interms of only in linear forms.

If it were possible to eliminate Eq. (1-8.31) by theuse of Eq. (1-8.25),wç might obtain a functional expressed entirely in terms of and similar inform to that of the principle of minimum complementary energy in the smalldisplacement theory of elasticity. However, this elimination is difficult ingeneral."8> Consequently, it would seem advantageous, for practical applica-tions to FEM, not to struggle with the elimination to obtain the principle ofstationary complementary energy, but to be satisfied with the functional 1102,taking and as independent quantities subject to variation under -thesubsidiary conditions Eqs. (1-8.26) and (1-8.28).

Settle. 9. Two Theories

in the present section, we shall formulate two incremental theories for anonlinear solid body problem with geometrical and material nonlinearity. Thedeformation of the body is characterized by the Features that not only itsdisplacements are finite, but also its strains are no longer small, and thematerial behavior is elastic—plastic.

The formulation of the incremen%al theories begins by dividing the loadingpath of the solid body problem into a number of equilibrium states

Cl(1) a(N), .:., (l&>,

where and Q(!) are the initial and final states of the deformation, re-spectively, while Lv"> is an arbitrary intermediate state. It is assumed that all

t The Piola tensor is also called the Lagrange stress or the first Plola—Kirchhoff stress It is defined by a1 where a, and I, have the same meaningaa a4 introduced in Section 3.2 Unlike the Kirchboff stress tensor as,, the Piola Stress

tensor is generally unsymmetric. -

By combining a, J,,l, with a, + given by Eqs. (3.17) and (3.23). weob(ain + aj,.). which is equivalent to Eq. Q-8.25).

APPENDIX I 385

the state variables such as stresses, strains and displacements, together withthe loading history, are known up to the fr" state. Our problem is then toformulate an incremental theory for determining all the state variables ip the

+ state, under an assumption that the fl(N + 1) state is incrementally Closeto the fl(ti) state and all the governing equations may be linearized withrespect to the incremental quantities. The step characterizing theprocess from the state to the state will be referred to as the(N + l)-th step.

Let the positions of an arbitrary material point of the body in theand states be denoted by P<°>, p(N) and p(N+ 1), respectively, and

the position vectors to these points by and respectively; as

y3

F7o. flUe) QU(+1)

shown in Fig. 1-6, and let the rectangular Cartesian coordinates of the positionsp(O and p(N+ be represented by and Yf, respectively. Then, we

,have

= (1—9.1)

= = + u = + (1-9.2)

f(N+l).....

= (X1 + = (Xf + + (1-9.3)

i = 1,2,3 are die base vectors of the rectangular Cartesian cOordin-ates,whileaandu + + Au1;i = 1,2, 3arethedisplaccmentvectors and their components of the point in the and states,

respectively.

ü(N)

x3

0

, , V1

386 APPENDIX I

9.1. of

We shall denote the familiar Green strain tensors at the and fl(N± 1)

states by et, and et, + respectively. These are defined by

= r(7) . — r9" .

= Ut., + + 14.i Uki, (1-9.4)

and

2(e1, + = . — .

= (Ut + + (U1 + + (Uk + +(1-9.5)

respectively, where ( = ?( It is readily obtained from Eqs. (1-9.4)and ([-9.5) that

= (ók, + Uk t)Auk 5 +

(1-9.6)

On the other hand, we may have another definition of the strain incre-ments for the (N + I )-th step, taking the state as an initial state, and by theuse of the rectangular Cartesian coordinates (X1, X2, X3). l3enoting the strainincrements by A*e*,, we may have

*Ø1.(N+1) 81.(N)

= ax, — ii,—

+ + (I 97)— ax, ax, -

The transformation laws between and are as follows:

axe,, 01$= (1-9.8)uXt OX1

= (1-9.9)

If and are linearized with respect to we obtain

= (ök, + + + (1-9.10)

1911ax, (-

APPENDIX I 387

We note here some of the geometrical relations which are useful in carryingout later formulations. First, we define the Jacobians as follows:

— 0(11, 13)—,

— 0(x1, xa, x3) —

= Y2,= (1-9.12)

0(x1, x2, x3)

and we obtain

0(Y1, Y2, Y3) — Y2, Y3) / 12, —I 9 3

12, 13) — O(x1, x2, x3) I 0(x1, x3, x3) — D(N)( — .1 )

Second, the following relations are also worth noting:

= += + + (1-9.14)-

andOx1 Ox1 . Ox1

OX1 Ox2 Ox3 011 012 013012 013 813 OX2 OX2 OX2

I 9 153x1 Ox2 Ox3 012 813 (.

Ox1 Ox2 Ox3 ?JX1

where ff1 isa unit matrix. Third, if assumed s*i*ll, we tiiay write

:= 1 + (1-9.16)

where =

9.2. Definidoes of Stresses

First, we define the Kirchhoff tensors by the use of the (x1, x2, x3)coordinates, and denote those defined at the points and by and

+ respectively as shown in Fig. 1-7.f These stress tensors are definedper unit area of the state as introduced in Chapter 3.

Second, we define the Euler stress at the point p(N), and denotethem by the Euler stresses are those acting on six surfaces:

= const., + = const.; I = 1, 2, 3

t The Stress tensors and + defined here are respectively the same as and

+ defined in Section 3.11.

--I,.

xl Yl

FIG. 1-7. Definition of Kirçhhoff stress tensors by the use of the(x1, Xi, x,) coordinate system.

x3

Y3

FIG. 1-8. Definition of Euler stress tensors.

of an infinitesimal rectangular parallelepiped containing the pointshown in Fig. 1-8. It should be noted that the Euler stress tensors are definedper unit area of the state and they are taken in the directions of (herectangular Cartesian coordinate axes, namely in the directions of i1; i = 1,2, 3.Following Ref. 45, we have the transformation law between and asfollows:

1 0X10X1= (1-9.17)

Third, we define the Euler stress tensors at the point and

388 APPENDIX I

x3

0

p(N1)

°i2

N)

E E

0

XI Xi Yi

*2

APPENDIK 1 389

obtain the transformation law between + and + as follows:

+ = D<N+1, 8Xk ax,+ &lki). (1-9.18)

Fourth, we define another set of the Kirchhoff stress tensors at the point+ by the use of the (X1, X2, coordinates. We denote its components

'C

x

x,,X1

x2,X2 Y,

Fi;. 1-9. Definition of the Kirchhoff stress tensors by the use of the(Xi, X2, A'3) coordinate system.

by + as shown in Fig. I-9.t The transformation law between+ and + is written in the following form:

÷ = }'3) : + (1-9.19)

X2, X3)

Combining Eqs. (1-9.18) and (1-9.19), and using the relhtion (1-9.13), we obtain

+ = +

Consequently, from Eqs. (1-9.17) and (1-9.20), we obtain

= (1-9.21)

t It is seen that and defined here are respectively the same as anddefined in Section 5.1. It is also observed that defined by Eq. (1-9.7) are the same asdefined by Eq. (5.6). áa, is sometimes called the Truesdell stress increment

SI(F13

E .cT13

390 APPENDIX I

We note here the following relation which can be derived from Eq. (1-9.19)neglecting the terms of higher order product of the incremental displacements

and the incremental stresses

= — —

— + crfJEiekk, (7-9.22)

where it is defined that1 ôAu1\=

—(1-9.23)

Eq. (1-9.16) together with the relation

= -r (1-9.24)

have been used in the derivation.Finally, we define the Jaumann stress increment tensors. We denote the

Euler stresses4acting on the six surfaces of an infinitesimal rectangular paraflele-piped I)R(+1)S(N+l) by + as 6hown in Fig. 7-10. Thedirection cosines of the three of the parallelepiped relative to the rect-angular Cartesian coordinates (X1, X3, X3) are specified in the followingtablet:

Ia 1s

+ 1 )Q(N + 1)1 —

p(N+1)R(N+1) 1

p(N+1)S(N+1)

where AW(f have been defined by Eq. (1-9.23). The quantities representthe rigid body rotation experienced by the rectangular paral-lelepiped P >Q the (N + I)-th step. The stress incrementsthus defined are called the Jaumann stress increment tensors.t47>

Next, we shall derive the relations among and We denotethe matrices of + and + at the point pN+I) by [aE +and (a' + The transformation law may be written in thefollowing form:

It The table indicates that the direction cosines of the vector P(N+1) Q(N+t) relative to

the (X1, X2, coordinate axes are (1, — and

1"u a231"231

C32 G33J

where

APPENDIX 1 391

+ Ao'] = + (1-9.25)

[L)=1

which may be decomposed into the form

where

[U = [11 + (1-9.27)

1-0

=2

0—

0

X:

FIG. I-tO. Definition of the Jaumann stress increment tensors.

Neglecting terms of higher order product, we obtain from Eq. (1-9.25) thefollowing relation:

or= + + [Aco](a'J, (1-9.29)

= — — (1-9.30)

By combining Eq. (1-9.22) with Eq. (1-9.30), we obtain

= — — + c7fA*Ekk. (1-9.31)

Eqs. (1-9.22), (1-9.30) and (1-931) show the relations among andIt is seen that reduces to if the strains may be assumed

small quantities.

(1-9.26)

(1-9.28)

x3

x3

£ J

392 APPENDIX I

9.3. Relations between Stress and Strain Increments

The next step in the incremental formulation is to assume relations betweenthe stress increments and the strain increments. One of the most naturalassumptions may be to postulate the relations between and in thefollowing form:

= (1-9.32)

or in a linearized form

(I-9.33)t

In these equations, mayinclude the effect of past history as mentionedin the flow theory of plasticity introduced in Chapter 12. It is noted here thatsince may be multi-valued in the flow theory of plasticity, some tech-nique is required to choose proper values of for an element under con-sideration, if this incremental theory is applied to a finite elementWe can derive the relations between and with the aid of Eqs. (1-9.9),(1-9.21) and (1-9.32). The result may be written in the following form:

= ClIk4ek(,

or in a linearized form

= (19.35)

where

C — (1-9 36)— ax;An alternate natural assumption may be to postulate the relations between

and in the following form:

= (1-9.37)

Eqs. (1-9.37) have been used frequently in the theoretical development andanalysis of elastic—plastic problems.

If Eqs. (1-9.37) are postulated, we can derive the relations betweenand by the use of Eqs. (1-9.31) and (1-9.37) and obtain as follows:

— + (1-9.38)

which are to be used for Eq. (1-9.33). We can now determine Cilki with the aid ofEqs. (1-9.36) and (1-9.38). The result is

t The linear relations between da1, and dek, derived for the ft w theory of plasticity inChapter 12 may be interpreted as the relations either between and Aek, of Eq.(1.9.33) or between and of Eq. (1-9.31).

APPENDIX I 393

C— øXg ØX,ØX.

— — + (1-9.39)

which are to be used for Eq. (1-9.35). With these preliminaries, we shall nowproceed to formulating the incremental theories.

9.4. An Incremental Theory by the Lagrangian Approseb

First, we shall formulate an incremental theory by the Lagrangian approach.We begin by defining the stresses, strains, displacements, body forces, externalforces acting on S0 and the displacements prescribed on in the and

states by

os,, et,, u1,

at, + e1, + + Aug,

+ + +respectively. Then, in a manner similar to the development of Section 3.11,the principle of virtual work for the L1(N + state is expressed by

÷ + — +

— 5 + 0, (1.9.40)so -

whereon (1-9.411

and where e1, + is given by Eq. (1-9.5). We repeat hereand the surface forces on are defined per unitvolume and inthe state, and that = and arevolume and elementary surface area in the state. 1k terms ofhigher order product of the incremental displacements, we obtain after somemanipulations

+

— + —

— If + = 0. (1-9.42)

If it is assured that the state is in equilibrium, then the terms

ff5 —

— (1-9.41)

394 APPENDIX I

will vanish in Eq. (1-9.42). However, the state may not be in completeequilibrium in this kind of incremental theory due to neglect of the higherorder terms ançl computational inaccuracies. Consequently, it is essential toretain these terms in Eq. (1-9.42) for an equilibrium check, as mentioned in thepreceding section of this appendix. The principle of virtual work thus estab-lished holds irrespective of the incremental stress—strain relations.

An application of Eq. (1-9.42) to a finite element formulation will bediscussed briefly at this point. To begin with, within each finite element isapproximated by

= (1-9.44)k

where x2, x3) are the shape functions and are incremental nodalpoint displacements. We assume that these shape functions are chosen sothat the given by Eq. (1-9.44) are compatible with those of the adjacentelements. Substituting Eqs. (1-9.10), (1-9.35) and (1-9.44) into Eq. (1-9.42),we find that the terms representing the contribution an arbitrary finiteelement to the left-hand side of Eq. (1-9.42) can be expressed in the followingform:

+ + — —

which may also be expressed in a matrix form as

+ [kU)] + — —

=

5ff +

+=

- — 111 + f5V11

ff1 + +

+ JJ (1-9.45)S.,"

and where and S.,., are respectively the region and ihe portion of S.,belonging to the element under consideration. The matrix is the in-cremental stiffness matrix. The matrices )J and are called the initialdisplacement stiffness matrix and the initial stress stiffness matrix, respect-

The matrix may be called the residual matrix. It is a common

APPENDIX 1 395

practice to assemble the terms representing the contributions from all theelements to obtain a system of linear incremental equilibrium equations for theentire structure, which are subsequently solved to determine the state variablesin the + state such as the stresses + the increments of the nodalpoint displacements ..Xu, and so forth.

PROBLEM 1. Show that if applied to the geometrically nonlinear problemfor which the principle of stationary potential energy holds, the formulationdeveloped here is equivalent to that of the modified incremental stiffnessmethod treated in Section 8 of tilis Appendix.

PROBLEM 2. Compare the present method with the Euler method for thestability problem introduced in Section 3.11 of Chapter 3.

9.5. Another Incremental Theory by Combined Use of theEulerian and Lagranglan Approaches

We shall formulate a second incremental theory by combining the Lulerianand Lagrangian As mentiàned in 9.2, we introduce the(X1, X2, X3) coordinates in the state, and denote the Euler stress tensorsby body forces by and the surface forces on S0 by F, in the state.It is noted here that and are defined per unit area and P1 are defined per

•unit volume of the state. On the other hand, we define the Kirchhoffstress tensors + the body forces P1 + and the surface forcesF1 + on S0 in the 1) state, where it is understood that all these quan-tities are defined per unit area and per unit volume of the state. Then, wemay write the principle of virtual work for the state as follows:

+ — (P1 +

— + = 0, (I-9.46)t

where = Au1 on 'S1, (1-9.47)

and where = dX1dX2dX3 and are respectively elementary volumeand elementary surface area of the state: Neglecting the terms of higherorder product of the incremental d.isptacements, we obtain after somemanipulation - -

ff5 + la'-

9X,

— + [a'Mse —

— + = 0. (1-9.48)so

t Eq. (1.9.46) is equivalent to Eq. (5.5) of Chapter 5.

396 APPENDIX I

By the use of Eq. (1-9.48), we can establish a finite element formulation in amanner similar to the development in 9.4. By approximating in eachelement by

= (1-9.49)

where X2, X3) are compatible shape functions, and using Eqs.(1-9.11) and (1-9.33), we find that the terms representing the contribution froman arbitrary finite element to the left hand side of Eq. (1-9.48) can be expressedin the following form:

+ — —S /

which may also be expressed in a matrix form as

+ — —

where

k — IdVN

kl

= f +

+ If (1-9.50)Sal,'

and where is called the incremental geometric stiffness matrix.By assembling the terms representing the contributions from all the elements

to obtain a system of linear equations ic, the entire structure and solvinp these'equations, we can obtain the state variablie in the 1) state. The stresses

+ thus obtained arc now transformed by the use of Eq. (1-9.19) into+ which provide the initial stress for the (N + 2)th step. It should be

noted here that after each succeeding step, total displacements are computed byadding all incremental contributions to update nodal point coordinates, andthe stiffness matrices [kJ and arc recomputed for each step.

The above is an outline of the incremental theory developed in Ref. 50.It is stated in the reference that if the .stroctural response is highly nonlinear,even the above procedure may lead to computed results which are in error.

APPENDIX I 397k

It is also suggested that for this class of problems, Newton—Raphson iterationprocedures can be employed to reduce the error in the nodal point eqailibriumto any desired degree. The reader is directed to Refs. 48 through 51, forfurther details of the incremental theories and other formulations, togetherwith their practical applications to geometrical and material nonlinearproblems.

PROBLEM I. The two incremental theories formulated in Section 9 havebeen made with reference to the rectangular Cartesian coordinate system.Extend the above theories and develop them in the general curvilinear co-ordinate system which have been introduced in Chapter 4.

PROIILEM 2. Show that if the structural response is highly nonlinear, therelations between stress and strain increments as given by Eq. (1-9.32), orgiven in a more general form by

must be employed, and the principles of virtual work, Eq. (1-9.40) and Eq.(1-9.46), must be used without neglecting the terms of higher order product.

PROBLEM 3. Show that Eq. (1-9.46) is equivalent to Eq. (1-9.40). Note:Relations such as Eqs. (1-9.9), and (1-9.20) and = are usefulfor the proof.

PROBLEM 4. Compare the incremental theory formulated in 9.5 with theflow theory of plasticity introduced in Chapter 12.

Section 10. Some Remarks on Discrete Analysis

The term discrete analysis seems to cover a wide spectrum of numericalanalysis methods wherein a system having an infinite number of degrees offreedom is approximated by a system having a finite number of degrees offreedom. Thus, differential or integral equations established for a continuousbody problem are reduced to a finite number of algebraic equations indiscrete analysis.t As is well known, the met'iod of' weighted residuals(abreviated and the finite difference method (abreviated FDM)are two major discrete analysis methods 3 As the list topic of this appendix,we shall examine the MWR because it provides a broader and more flexiblebasis to the formulation of FEM than the variational

Following Ref. 58, we shall take, as an example, a two dimensional heatconduction problem defined by the following differential equation:

+! + Q = 0, in S, (1-10.1)CX 0)7 Oy

t For discrete analysis applied to integral equations, see Refs. 52 through 55, for example.It is stated in Ref. 58 that FDM, which oñgin.illy appeared to be a different process, has

recently been formulated on variational basis and can be identified in FEM terminology.§ I wish to express my gratitude to Professor 0. C. Zienkiewicz for his permission of my

frequent reference to Ref. 58 in the writing of Section JO.

398 APPENDIX I

together with prescribed bound2ly conditions:

ao= 4 on C1, (1-10.2)

0 = 6 on C2, (1-10.3)

where 0, K and. Q are the temperature, the heat conductivity and the heaPsource intensity while ii is the normal drawn outwards on theboundary, and 4 and 6 are prescribed functions of the space coordinates.

10.1. A Variational Principle

A variational principle will be derived here for this problem for laterreference. In a manner similar to the development for the linear elasticityproblem, we begin by writing the following equation:

— + (4)+ ôOdxdy

(lOds=0, (1-10.4)

where dO is a virtual variation of 0, and Eq. (1-10.3) is taken asa subsidiarycondition. If it is assumed that dO is a continuous function in S. integrationsby parts transform Eq. (1-10.4) into

- QoO]dxd.Y

— 5 4M)ds = 0. (1-10.5)Cl

If it is further assumed that K, and are not subjected to variation, we havefrom Eq. (1-10.5) the following var'.ational principle:

= 0, (1-10.6)

where

H = 55(4K(39)2]

—

— f4Ods. (1-10.7)

APPENDIX I 399

.10.2. Method of Weighted Residuals

Returning to the topic of MWR, let us denote an approximate solution for0 by 0 and express it as follows:

0 y) + y). (1-10.8)

where y); I = 1, 2, ..., N are coordinate functions defined in the domainS, and a1; I = 1, 2, ..., N are parameters to be determined. The function

y) is included in Eq. (1-10.8) to take care of some inhomogeneous termsappearing in Eqs. (1-10.1), (1-10.2) and (1-10.3). Introducing Eq. (1-10.8) intoEqs. (1-10.1), (1-10.2) and (1-10.3), we have the so-called residuals defined asfollows:

= (4) +-- (4) + Q in S, (1-10.9)

on C1, (1-10.10)

on C2. (1-10.11)

Unless is an exact solution by chance, these residuals never vanish. Themethod of weighted residuals proposes to determine values of a1 in such a waythat the residuals are reduced to zero in the sense of weighted mean, namely,

+ W1ds + JRC2W(dS = 0,S Cl Ca

1= 1,2, ...,N, (1-10.12)

where W1; I = 1, 2, ..., N are the so-called weighting functions. They may beany functions and have no continuity requirements. They may be discontin-uous functions including the delta function. There are several ways of choos-ing the weighting functions. A different choice leads to a different formulation.Some special cases of these choices will be shown in the following.

10.3. Point Collocation and Subdomain Collocation

If the weighting functions are chosen to be delta functions in such a waythat

W1=t5(x—x8,y---yj;i=1,2,...,N, (1-10.13)

where a is the delta function, and (x1, is the coordinates of a point in S, oron C1, or on C2, we have a formulation called point collocation.

Next, we divide the region 5, and the boundaries C1 and into a numberof subdomains ... and choose the weighting functions in such a waythat

400 APPENI)IX I

= I in the subdomainW1 = 0 elSewhere,

to obtain a formulation called subdosnain collocation.

10.4. GalerkIn Method

Now, we shall choose of Eq. (1-10.8) in such a way that it satisfiesEq. (1-10.3) and write Eq. (1-10.12) as tollowa:

+R1W,ds -0,

i=l,2...,N. (I-10.14)t

If the continuity of the weighting functions is assumed, integrations by partstransform Eq. (1-10.14) into

+— dxdy

—f4Wm 0.Cl

The Galerkin proposes to employ either Eq.(I-1O.14) or Eq. (1-10.15)and to take

= y); I — 1,2, ..., N, (I-id. i

for the determination of the unknown parameters In other words, theweighting functions are taken in coincidence with the coordinate functions inthe Galerkin method.

105. Rayleigh-RItz Method

Needless to say, the Rayleigh—Ritz method asserts that

817—=0;:=i,2,...,N, (1-10.17)8a4

where

17= -— fqodc, (1-10.18)

Cl

t Compare with Eq. (1-10.4).Compare with Eq. (1-10.5).

APPENDIX I 401

and 8 is given by Eq. (1-10.8), while Eq. (1.10.3) is taken as a subsidiarycondition. As expected, the equations obtained from Eq. (1-10.17) ar- equiva-lent to those obtained by the use of the Galerkin method based on Eq. (1-10.15).

As mentioned before, the weighting functions have no continuity require-ments as far as Eq. (1-10.12) are concerned. However, the continuity on theweighting functions is required in transforming Eq. (1-10.14) into Eq. (1-10.15).It is stated in Ref..58 that integrations of Eq. (1-10.14) by parts into Eq.(1-10.15) reduce the continuity requirements on the coordinate functions, butincrease those on the weighting functions.

Thus far, we have seen that MWR includes several methods such ascollocation Galerkin method and variational method, providing abroad basis for the discrete analysis technique and elucidating the features ofindividual methods. MWR can be formulated for almost any problem inengineering science and consequently has universality in applications topractical problems. For further details, the reader is directed to Rcfs. 56through 59, for example.

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APPENDIX I 403

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Structural Mechanics and Design, edited by R. W. dough, Y. Yamamoto and J. T.Oden, The University of Alabama in Huntsville Press, pp. 301—324, 1972.

52.11 L. BISPUNOHOIP and H. Asmiy, Principles of Aeroelastlclt,y, John Wiley & Sons,New York. 1962.

53. H. Asmiy, S. WIDNALL and M. T. LANDAHL, New Directions in Lifting SurfaceTheory, AIAA Journal, Vol. 3, No. 1, pp. 3—16, January 1965.

54. H. ASHLEY, Some Considerations Relative to the Predictions of Unsteady Airloads inLifting Configuration, Journal of Aircraft, Vol.8, No. 10, pp. 747—756, Octoper 1971.

55. J. L. Hass and A. M. 0. SMITh, Calculation of Potential Flow about Arbitrary Bodies, inIn Aeronautical Sciences Vol. 8, edited by 1). Küchemann Pergamon Press,

1967, pp. 1—138.56. S. H. CRANDALL, E.qLaeering Analysis, McGraw-Hill, 1956.57. B. A. FnnAYS0N, The Method of Weighted Residuals and Variational Principles, Aca-

demic Press, 1972.58.0. C. ZwiKiEwIcz, Note on the Element Method and Its Applications, Industrial

Center of Technology, Japan 1972.59.0. C. ZatinuawicZ, Weighted Residual Processes in Finite Element with Particular

Reference to Some Transient and Coupled Probleme, in Leciwes on Rnlte ElementMethods hi Continuum Mechanics, edited by J. T. Oden and E. 11 do Armies a Oliveira,The University of Alabama in Huntsville Press, pp. 415-458, 1973.

APPENDIX 3

NOTES ON THE PRINCIPLE OFVIRTUAL WORK

WE shall make two short notes here on the principle of virtual workexpressed by Eq. (1.32). The first note is concerned with Eq. (1.28). That is:the term*sucb ii

j&i: \( '.'tax +

(lip — Z,), (...), (...)appearing in Eq. (1.28) are taken from Eqs. (1.26) and (1.27), namely, theequations of equilibriuni in Vand the mechanical boundary conditions on S1àf the solid body before the execution of the virtual displacements ôu, ôt' and ow.In other words, a),, ..., and (ôu, ôv, ow) are independent of each other.

The second note is that Eq. (1.32) does not state the first law of thermo-dynamics, but state merely a kind of divergence theorem which is a specialcase of Eq. (1.76). A physical interpretation of Eq. (1.32) may be given asfollows: We consider an infinitesimal rectangular parallelepiped enclosed bythe following six surfaces:

x = const., x + dx conat.;y=const., y+dy—const.;z const., z +dz = const.,

in the body V before thc execution of the inilnitesimal virtual displacements

Ou=Osd+dej+Owk, (J-1)

and denote stresses acting on these six surfaces by

—(al + +

+ + + (oral + i-flj + .... (J2)

Then, the virtual work done, during the virtual displacements, by thesestresses and the body forces acting on this infinitesimal rectangular parallele-piped is given by

405

406 APPENDIX J

+ + ôudydz + [a) + +

+ + 11iJ + rxzk)dX] . [öu + dx] dydz

+ (lou + ?Ov + ZOw)thcdydz,

I øOu øOv IøOu øOv\1(J.3)

where higher order terms are neglected and Eqs. (1.2) have betn substituted.Now, wC divide the solid body fictitiously into a large number of infinitesi-

mal rectangular write relations such as by(J-3) for all the parallelepipeds. If we sum up—these relations over all theparallelepipeds, we find that the terms representing the contribution from the

virtual work done by the stresses acting on the interfaces between the adjacentparallelepipeds are cance!Ied out. Conse4uently, by the use of the fourrelations which hold on the boundary, namely, the relation

(X,Ou + +

= (a2ôu + + + + ±

+ + i-211t3v + (J-4)

together with Eqs. (1.29), (1.30) and (1.12), we finally find that the sums of theterms appearing in the left hand side of Eq. (J-3) is equal to the virtual workdone, during the virtual displacements, by the entire body forces and theprescribed surface forces on S. Thus we obtain:

+ + ... +

= jff(Xou + ?öv +Zâw)dxdydz

+ + Ov + 2, Ow)dS, - (J-5)Si

where 0e, ... and are given in terms of Ou, Ov and Ow as shown in Eq.(1.33). Eq. (3-5) states: The virtual work done by the internal forces is equal tothe virtual work done by the external forces in arbitrary infinitesimal Uirtualdisplacements satisfying the prescribed geometrical boundary conditions. This isan interpretation of the principle of virtual work expressed by Eq. (1.32).

Piwrn.ai 1. Show that the above interpretation is similar to that of thedivergence theorem of Gauss introduced in the footnote of page 14.

PROBLEM 2. Show that the integrand OA . (ôr).AdV in Eq. (3i47) may be

APPENDIX .1 407

interpreted as the virtual work done, during the infinitesimal virtual displace-ments, by the body forces and the surface forces acting on the deformedinfinitesimal paralletepiped.

Note:

—a' . ôrdx2dx3 + (a' + a',dx'). [or + (Or)1dx']dx2dx3

÷ ... + P. Ordx1dx2dx3

= a2. + (higher order terms).

INDEX

Base vector 52contravariant 77covariant 76, 80

Beambending of 134bending-torsion of 310buckling of 144, 304, 308, 313, 314large deflection of 142lateral vibration of 139, 305

Beam theoryelementary 133finite displacement 307including transverse shear deformation

147, 258, 309naturally curved and twisted 150small initial deflection 315

Bending rigidity 158, 195Bernoulli—Euler hypothesis 133BiaDchi's identity 82Body axis 107Boundary conditions

geometrical 10, 61mechanical' 10,60

Boundsof boundary value problems 39of eigenvalues 47, 48of'safety factor 252of torsional rigidity 125, 302

Bulk modulus 234

Calculus of variations 1

Castigliano's theorem 43, 210, 282, 339Center of shear 132, 309, 310Christoffel three-index symbol of the

second kind 78, 90, 286Codazzi, conditions of 184Collapse load 251Compatible models 351Complementary energy 42

ofbeam 139offrame 215,341,344'of panel 222of plate 160,162of stringer 222of torsion bar 119of truss 206

Complementary energy function 30, 31,69, 95, 99, 101

Conditions of compatibility 11, 22, 74,81, 118, 274, 287, 290, 326, 333

inthelarge 24,121,212,219,220,223,275, 339

Conforming 352, 365Conventional variation principles 347,

351, 360, 373, 379Covariant derivative

of base vector 77, 78of tensor 79of vector 79

Creep 270Curvilinear coordinates' 76

d'Alembert's principle 2Deflection influence coefficient 339Deformation theory of plasticity 231

Discrete analysis 397Displacement method 206Divergence theorem 14, 25, 277Dummy load method 25Durchschlag 62

Elastic stability 63Energy criterion for stability 69Entropy 66Equations of equilibrium 8, 56, 83, 92Equilibrium model I, H 351

Euler method 72Euler stress tensor 387, 388Eulerian angle 109,295Eulerian approach 52, 240

Finite element method 345First variation 29,70Flattening instability 62Flexibility matrix 215Flow theory of plasticity 240Follower force 308Force method 206, 210, 217, 221Friedrichs' transformation 36Functional I

Function space 39, 279

409

410 INDEX

Galerkin method 15, 400generalized 6, 15, 49, 74

Gauss, condition of 184Gauss and formulae of 337Generalized coordinates 105Generalized force 106Geometrical and material nonlinearity 384Geometrical nonlinearity 377Green's function 48Green strain tensor 378, 386Gurtin's principle 377

Haar—Kármán principle 235, 269Hamilton's principle 2, 105Heflinger—Reissner principle 35, 220Helmholtz free energy function 66, 100Hencky material 235Herrmann's principles 360, 372Hill's principle 249Hu-Washizu principle 349Hybrid displacement model I, Ii 351Hybrid stress model 351

incremental theoriesby Lagrangian approach 393by Eulerian and Lagrangian approaches

395.Initial strain 98, 344Initial strças 93Internal energy 66

Jacobign 387Jaumann stress increment tensor 390

Kachanov principles 234Kármán's large deflection theory of plate

163Kinematically admissible multiplier 252Kinetic energy 2, 105, 108Kircbhoff hypothesis 153Kirchhoff stress tensor 378, 387, 388, 389Kirchhoff-Love hypothesis 189Kronecker symbol 53

LagraAge multiplier 19, 32Lagrange's equations bi motion 2, 106Lagrangian approach 52, 93Lagrangian function 3, 106Lateral buckling 314

Lattice vector 54Legendre's transformation 3, 35Limit aiialysis 250Loading 242

Marguerró's theory of thin shallow shell173

Markov's principle 248Maxwell—Betti's theorem 282Method of weighted residuals 6, 397, 399Metric tensor

contravariant 77, 80covariant 77, 80

Mises yield condition 235Mixed model I, II 354

Modified incremental stiffness mcthod 381Modified variational principles for relaxed

continuity requirements 351. 357,364, 369, 373. 379

Modulus of rigidity 10

Neutral 242Non-uniform torsion 303

Orthogonal curvilinear coordinates 90,291

Panel 221Perfectly plastic material 235, 244Piola stress tensor 383Plate

buckling of 165, 320-large deflection of 163lateral vibration of 323stretching and bending of 154thermal stress of 168, 322with small initial deflection see Thin

shallow shellPlate theory

including transverse shear deformation170, 262

problem related toin cylindrical coordinates 329in nonorthogonal curvilinear coordi-

nates 326, 330in orthogonal curvilinear coordi-

nates 328in skew coordinates 327

Point collocation 399

INDEX 411

Poisson's ratio 10Positive definite function 27, 40, 67, 278Potential function 2, 28. 67Prandtl-Reuss equation 245Principle of complementary virtual work

3, 17, 23, 24, 210, 232, 241, 276Principle of least work 31

Principle of minimur,s complementaryenergy 29

ofbeam 139of frame 219of plate 160, 162, 169oftorsionbar 118of truss 209

Principle of minimum potential energy 27of beam 138of frame 220, 340of plate 160, 161, 169, 294oftorsionbar 116of 209

Principle of minimum potential energy,generalization of 31

ofbeam 138of plate 160, 162of torsion bar 116of truss 209

Principle of stationary complementaryenergy 45

ofbeam 146of deformation theory of plasticity 232

Principle of stationary free energy 100Principle of stationary potential energy

2, 44, 67, 89, 97, 100of beam 139, 145. 306. 308, 317of deformation theory of plasticity 232of 167, 178

of stationary potential energy,generalization of 44, 68, 90, 95,97, 99, 103

of beam 140, 145of plate 164

Principle of virtual work I, 13, 22. 24, 44,63, 88, 94, 97, 102, l04, 110. 289.405

of beam 135, 143. 144, 148of plasticity 232, 241of plate 155, 163, 166, 171, 176, 326ofshell 191. 198, 200, 333of torsion bar 114of truss 209

Quadratic function 278Qoasi-static 101

Rayleigh,quotient 45, 97, 140, 145, 168method 38, 46, 74, 141, 146,

161, 163, 170, 202, 400modified 47, 142, 146

Rayleigh's principleRedundancy 205Reissner's principle 68Residual matrix 394Riemann—Christoffel curvature tensor

81

Rigid-plastic material 248

Sadowsky's principle of maximum plasticwork 238

Safety factor 251Scalar product of two vectors 52

in function space 40Saint-Venant—Levy—Mises equation 247Saint-Venant principle 5, 121Saint-Venant theory of torsion 113Secant modulus theory 233Second variation 29, 70Semi-monocoque structure 221Shell, geometry of 182

cylindrical 266rotationally symmetric 267spherical 266

Shell theory'In orthogonal curvilinear coordinates

linearized 191, 197nonlinear 198 -

incluring transverse shear deformation199

problem related to, in nonorthogonalcurvilinear coordinates 336

Small displacement theory. 3

in orthogonal curvilinear coordinates 90in rectangular Cartesian coordinates 8problem related to

": coordinales 291

in polar coordinates 292in two-dimensional skew coordinates

• 289Sflap-through 62 202, 317Statically admissible multiplier 251

Stiffness matrix 216, 225, 227incremental 394initial displacement 394initial stress 394incremental geometric 396

Strain 9, 55, 81Strain—displacement relations 9, 55,81,91

412 INDEX

Strain energy 41,105of beam 137, 143, 149of frame 340of panel 226of plate 159, P69of shell 195, 197, 201of stringer 225of torsion bar 119, 303, 304of truss 206

Strain energy function 27, 31, 64, 95,98, 101

Strain-hardening material 233, 242Stress 8, 56, 83Stress function

Airy 13, 160in curvilinear coordinates 289in two-dimensional skew coordinates

290Maxwell 13, 23Morera 13,23of plate theory 160, 326of Saint-Venant torsion 117of shell theory 333

Stress—strain relations 9, 59, 87, 100,256, 290

Stringer 221Subdomain collocation 399Summation convention 53

Tempic—Kato theorem 48Tensor 284Thermal stress 99

of plate 168, 322

Thin shallow shell 173Timoshenko beam theory 149Torsional buckling 304Torsional-flexural buckling 313Torsional rigidity 116, 124, 299, 300,

302Torsion-free bending 133Total plastic work 243

.Transformationof strain 85, 273of stress 58, 86, 273of tensor 285of vector 285

Truesdell stress increment tensor 389Two-dimensional skew coordinate system

289

Unit dispLacement method 25, 281, 28ZUnit load method 25, 281, 282, 342Unloading 242

Vector 284Vector product of two vectors 57

Warping function 115, 297, 300Weinstein's method 48, 323

Yield condition 242Yield surface 242Young's modulus 10

81086474 washizu variational methods in elasticity and plasticity

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