Thin PlateBuckling andPostbuckling

Handbook of

CHAPMAN & HALL/CRC

Thin PlateBuckling andPostbuckling

Handbook of

Boca Raton London New York Washington, D.C.

Frederick BloomDepartment of MathematicsNorthern Illinois University

DeKalb, Illinois

Douglas CoffinInstitute of Paper Science and Technology

Atlanta, Georgia

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2001 by Chapman & Hall/CRC

No claim to original U.S. Government worksInternational Standard Book Number 1-58488-222-0

Library of Congress Card Number 00-050441Printed in the United States of America 1 2 3 4 5 6 7 8 9 0

Printed on acid-free paper

Library of Congress Cataloging-in-Publication Data

Bloom, Frederick, 1944-Handbook of thin plate buckling and postbuckling / Frederick Bloom, Douglas Coffin.

p. cm.Includes bibliographical references and index.ISBN 1-58488-222-01. Plates (Engineering). 2. Buckling (Mechanics). I. Coffin, Douglas. II. Title.

TA660.P6 B55 2000624.1

7765dc21 00-050441 CIP

Contents

Preface

1 Introduction: Plate Buckling and the von KarmanEquations1.1 Buckling Phenomena1.2 The von Karman Equations for Linear Elastic Isotropic

and Orthotropic Thin Plates1.2.1 Rectilinear Coordinates1.2.2 Polar Coordinates

1.3 Boundary Conditions1.3.1 Boundary Conditions on the Deection

1.4 The Linear Equations for Initial Buckling1.5 Figures: Plate Buckling and the von Karman Equations

2 Initial and Postbuckling Behavior of (Perfect) ThinRectangular Plates2.1 Plates with Linear Elastic Behavior

2.1.1 Isotropic Symmetry: Initial Buckling2.1.2 Rectilinear Orthotropic Symmetry:

Initial Buckling2.1.3 Postbuckling Behavior of Isotropic and Rectilinear

Orthotropic Linear Elastic Rectangular Plates2.2 Plates with Nonlinear Elastic Behavior: The

Johnson-Urbanik Generalization of thevon Karman Equations

2.3 Plates which Exhibit Elastic-Plastic orViscoelastic Behavior

2.4 Comparisons of Initial and Postbuckling Behaviorof Rectangular Plates2.4.1 Variations with Respect to Boundary Conditions2.4.2 Variations with Respect to Symmetry

2.4.3 Variations with Respect to Constitutive Response2.5 Initial Buckling/Postbuckling Figures, Graphs, and

Tables: Rectangular Plates

3 Initial and Postbuckling Behavior of ThinCircular Plates3.1 Plates with Linear Elastic Behavior

3.1.1 Isotropic Symmetry: Initial Buckling andPostbuckling Behavior

3.1.2 Cylindrically Orthotropic Symmetry: InitialBuckling and Postbuckling Behavior

3.1.3 Rectilinear Orthotropic Symmetry: InitialBuckling and Postbuckling Behavior

3.2 Plates which Exhibit Nonlinear Elastic,Viscoelastic, or Elastic-Plastic Behavior

3.3 Comparisons of Initial and PostbucklingBehavior of Circular Plates

3.4 Initial Buckling/Postbuckling Figures, Graphs, andTables: Circular Plates

4 Initial and Postbuckling Behavior of (Perfect) ThinAnnular Plates4.1 Plates with Linear Elastic Behavior

4.1.1 Isotropic Symmetry: Initial Buckling4.1.2 Cylindrically Orthotropic Symmetry:

Initial Buckling4.1.3 Isotropic and Cylindrically Orthotropic

Symmetry: Postbuckling Behavior4.1.4 Rectilinear Orthotropic Symmetry: Initial

Buckling and Postbuckling Behavior4.2 Plates which Exhibit Nonlinear Elastic,

Viscoelastic, or Elastic-Plastic Behavior4.3 Comparisons of Initial and Postbuckling

Behavior of Annular Plates4.4 Initial Buckling/Postbuckling Figures, Graphs, and

Tables: Annular Plates

5 Postbuckling Behavior of Imperfect Platesand Secondary Plate Buckling5.1 Imperfection Sensitivity

5.1.1 An Example of Imperfection Bifurcation

5.1.2 Rectangular Plates5.1.3 Circular and Annular Plates

5.2 Secondary Buckling for Thin Plates5.2.1 Secondary Buckling for Circular Plates5.2.2 Secondary Buckling for Rectangular Plates

5.3 Imperfection Buckling and Secondary Buckling Figures,Graphs, and Tables

6 Generalized von Karman Equations for ElasticPlates Subject to Hygroexpansive or ThermalStress Distributions6.1 Rectilinear Coordinates6.2 Polar Coordinates6.3 Boundary Conditions6.4 Thermal Bending and Buckling Equations and Boundary

Conditions

7 Thermal Bending, Buckling, and Postbucklingof Rectangular and Circular Plates7.1 Small Deection Theory7.2 Large Deection Theory7.3 Applications of the Bergers Approximation7.4 Thermal Bending, Buckling, and Postbuckling Figures,

Graphs, and Tables I

8 Other Aspects of Hygrothermal and Thermal Buckling8.1 Hygroexpansive/Thermal Buckling in the

Presence of Imperfections8.2 Buckling of Plates of Variable Thickness8.3 Viscoelastic and Plastic Buckling8.4 Thermal Bending, Buckling, and Postbuckling Figures,

Graphs, and Tables II

References

Preface

In the Spring of 1996, the authors initiated a research program at the In-stitute of Paper Science and Technology (Atlanta, Ga.) whose goal wasto model cockle, i.e., the hygroscopic buckling of paper. In the courseof researching the aforementioned problem, an extensive examinationwas made of the expansive literature on the buckling and postbucklingbehavior of thin plates in an eort to understand how such phenomenawere controlled by the wide variety of factors listed below. A compara-tive study of plate buckling was then produced and published as a pair ofIPST reports; this book presents the essential content of these reports.

The essential factors which inuence critical buckling loads, initialmode shapes, and postbuckling behavior for thin plates are studied,in detail, with specic examples, throughout this treatise; among thefactors discussed which aect the occurrence of initial buckling and theinitial mode shapes into which plates buckle are the following:

(i) Aspect Ratios: Included in our discussions will be examplesof buckling for rectangular plates, circular plates, and annular plates;we will look at how, e.g., critical buckling loads for the various types ofplates are aected by changes in the aspect ratio, associated with theplate, given that all other factors (support conditions, load conditions,etc.) are held xed. For a rectangular plate, the aspect ratio is just theratio of its sides; for a circular plate it is just the plate radius while, foran annular region, it is the ratio of the inner to the outer radius.

(ii) Support Conditions: The edge(s) of a plate may be supportedin many dierent ways; mathematically, an edge support condition isreected in a constraint on the out-of-plane deection of that plate alongits edge(s). The basic types of support that will be considered are thosewhich correspond to having the edges(s) of the plate clamped, free, orsimply supported. For a rectangular plate or an annular plate, one edge(or pair of edges) may be supported in one manner while the other edge(or pair) of edges can be supported in a dierent manner.

(iii) Load Conditions: The edge(s) of a thin plate may be subjected

to compressive, tensile, or shear loadings or to some combinations ofsuch loadings along dierent edges or pairs of edges. The eect of usingdierent load conditions on critical buckling loads and mode shapes forrectangular, circular, and annular plates are illustrated throughout thebook.

(iv) Material Symmetry: For thin plates exhibiting a particularconstitutive response, the material symmetry exhibited by the plate mayhave a marked eect on, e.g., the magnitude of the lowest buckling load,when all other factors, such as aspect ratio or edge support conditionsare held constant. The basic types of material symmetry that will beexamined in this work are isotropic symmetry, rectilinearly orthotropicsymmetry, and cylindrically orthotropic symmetry.

(v) Constitutive Behavior: Our examination of the inuence ofconstitutive response on initial buckling of thin plates was guided by thebehavior observed with respect to the deformation of paper, i.e., thatunder various combinations of loading conditions, paper may exhibitlinear elastic, nonlinear elastic, viscoelastic, or elastic-plastic. Examplesof all four kinds of such constitutive response are presented throughoutthis treatise.

(vi) Hygroscopic or Thermal Stress Distributions: Thin platesare often exposed to variations in temperature or moisture or hygroex-pansive strains which result in local compressive stress distributions thatcan cause buckling even in the absence of applied external loading.

Beyond the factors delineated above, which inuence both criticalbuckling loads and mode shapes for thin plates and also inuence thesubsequent postbuckling behavior, two other important factors comeinto play with respect to postbuckling behavior.

(vii) Initial Imperfections: Thin plates often possess some smallinitial transverse deection (e.g., a paper sheet with an initial cocklewhich is wetted, dried, and experiences further local buckling) or maybe subjected to a small normal loading (transverse) to the initially atunbuckled state of the plate; the presence of either or both of these con-ditions amounts to an initial imperfection which may have a profoundeect on the subsequent postbuckling behavior of the plate.

(viii) Secondary Buckling: After a thin plate has buckled into aninitial mode shape, it is possible that, with increasing load, instead ofobtaining a buckled shape which deforms continuously during postbuck-ling, as the magnitude of the loading increases, a critical load larger thanthe initial buckling load is reached at which the buckle pattern changessuddenly and a new mode shape appears; such phenomena have been

exhibited with respect to plate buckling and will be examined with par-ticular emphasis on the unsymmetric wrinkling that has been observedin circular plates.

Throughout this treatise, several results appear which cannot be foundin earlier books on plate buckling, e.g., the discussion of the Johnson-Urbanik generalization of the von Karman Equations for plates exhibit-ing nonlinear elastic behavior; other results appear which have not pre-viously been published elsewhere, most notably the analysis of initialbuckling for annular rectilinearly orthotropic thin plates, which was pre-sented in the Ph.D. Thesis of the second author.

The authors wish to express their gratitude to the Institute of PaperScience and Technology and its member companies for the support whichmade possible the production of the IPST reports that form the basisfor the present work.

Frederick BloomDeKalb, IllinoisApril 2000

Douglas ConAtlanta, GeorgiaApril 2000

THIS BOOK IS DEDICATED TOthe memory of Sidney Bloom

(1912-1999)and to the memory of

his grandsonScott Bloom(1969-1997)

Chapter 1

Introduction: Plate Buckling and the vonKarman Equations

1.1 Buckling Phenomena

Problems of initial and postbuckling represent a particular class of bi-furcation phenomena; the long history of buckling theory for structuresbegins with the studies by Euler [1] in 1744 of the stability of exiblecompressed beams, an example which we present in some detail below,to illustrate the main ideas underlying the study of initial and post-buckling behavior. Although von Karman formulated the equations forbuckling of thin, linearly elastic plates which bear his name in 1910 [2],a general theory for the postbuckling of elastic structures was not putforth until Koiter wrote his thesis [3] in 1945 (see, also, Koiter [4], [5]);it is in Koiters thesis that the fact that the presence of imperfectionscould give rise to signicant reductions in the critical load required tobuckle a particular structure rst appears. General theories of bifur-cation and stability originated in the mathematical studies of Poincare[6], Lyapunov [7], and Schmidt [8] and employed, as basic mathemati-cal tools, the inverse and implicit function theorems, which can be usedto provide a rigorous justication of the asymptotic and perturbationtype expansions which dominate studies of buckling and postbucklingof structures. Accounts of the modern mathematical approach to bi-furcation theory, including buckling and postbuckling theory, may befound in many recent texts, most notably those of Keller and Antman,[9], Sattinger [10], Iooss and Joseph [11], Chow and Hale [12], and Gol-ubitsky and Schaeer [13], [14]. Among the noteworthy survey articleswhich deal specically with buckling and postbuckling theory are thoseof Potier-Ferry [15], Budiansky [16], and (in the domain of elastic-plasticresponse) Hutchinson [17]. Some of the more recent work in the gen-eral area of bifurcation theory is quite sophisticated and deep from amathematical standpoint, e.g., the work of Golubitsky and Schaeer,

cited above, as well as [18] and [19], which employ singularity theory formaps, an outgrowth of the catastrophe theory of Thom [20]). Besidesproblems in buckling and postbuckling of structures and, in particular,the specic problems associated with the buckling and postbuckling be-havior of thin plates, general ideas underlying bifurcation and nonlinearstability theory have been used to study problems in uid dynamics re-lated to the instabilities of viscous ows as well as branching problemsin nonlinear heat transfer, superconductivity, chemical reaction theory,and many other areas of mathematical physics. Beyond the referencesalready listed, a study of various fundamental issues in branching theory,with applications to a wide variety of problems in physics and engineer-ing, may be found in references [21]-[58], to many of which we will haveoccasion to refer throughout this work.

To illustrate the phenomena of bifurcation within the specic contextof buckling and postbuckling of structures, we will use the example of thebuckling of a thin rod under compression, which is due to Euler, op. cit,1744, and which is probably the simplest and oldest physical examplewhich illustrates this phenomena. In Fig. 1.1, we show a homogeneousthin rod, both of whose ends are pinned, the left end being xed whilethe right end is free to move along the x-axis. In its unloaded state, therod coincides with that portion of the x-axis between x = 0 and x = 1.

Under a compressive load P , a possible state of equilibrium for therod is that of pure compression; however, experience shows that, forsuciently large values of P , transverse deections can occur. Assumingthat the buckling takes place in the x, y plane, we now investigate theequilibrium of forces on a portion of the rod which includes its left end;the forces and moments are taken to be positive, as indicated in Fig.1.2.

Let X be the original x-coordinate of a material point located in aportion of the rod depicted in Fig. 1.2. This point moves, after buckling,to the point with coordinates (X +u, v). We let be the angle betweenthe tangent to the buckled rod and the x-axis and s the arc length along aportion of the rod (measured from the left end). Although more generalconstitutive laws may be considered, we restrict ourselves here to thecase of an inextensible rod in which the Euler-Bernoulli law relates themoment M acting on a cross-section with the curvature d/ds. Thuss = X while

M = EI dds

(1.1)

with EI the (positive) bending stiness. The constitutive relation (1.1)

is supplemented by the geometric relation

dv

ds= sin (1.2)

and the equilibrium condition

M = Pv (1.3)

Combining the above relations, we obtain the pair of rst order non-linear dierential equations

v = dds

, = P/EIdv

ds= sin

(1.4)

with associated boundary conditions

v(0) = v() = 0 (1.5)

A solution of (1.4), (1.5) is a triple (, v, ) and any solution withv(s) 0 is called a buckled state; we note that = 0 implies that v 0and cannot, therefore, generate a buckled state. When = 0, (1.4),(1.5) is equivalent to the boundary value problem

d2

ds2+ sin = 0, 0 < s < (1.6a)

(0) = () = 0 (1.6b)

The actual lateral deection v(s) can then be calculated from byusing the rst equation in (1.4). We note that the rod has an associatedpotential energy of the form

V =12EI

0

(d

ds

)2ds P

[

0

cosds

](1.7)

and that setting the rst variation V = V () = 0 yields the dier-ential equation (1.6a) with the natural boundary conditions (1.6b).

The linearized version of (1.6a,b) for small deections v, and smallangles , is obtained by substituting for sin (a precise mathematicaljustication for considering the linearized problems so generated will beconsidered below); the linearized problem for v then becomes

d2v

ds2+ v = 0

v(0) = v() = 0(1.8)

which has eigenvalues n =n22

2, n = 1, 2, with corresponding

eigenfunctions vn = c sin(nx

). It is desirable to be able to plot v

versus but, unfortunately, v is itself a function (an innite-dimensionalvector) so we content ourselves instead, in Fig. 1.3, with a graph of themaximum deection vmax versus = P/EI. For loads below the rst

critical load P1 =2EI

2no buckling is possible. At the load P1, buckling

can take place in the mode c sin(x

)but the size of the deection is

undetermined due to the presence of the arbitrary constant c. Now, ifan analysis of the buckled behavior of the rod is based entirely on thelinearized problem, then we see that, as the load slightly exceeds P1,the rod must return to its unbuckled state until the second critical load

P2 =42EI

is reached, when buckling can again occur, but now in

the new mode c sin(

2x

), which is still of undetermined size. Except

at the critical loads Pn =n22EI

2there is no buckling. On physical

grounds the picture provided only by the linearized problem is clearlyunsatisfactory.

The nonlinear problem (1.4), (1.5), gives a more reasonable predictionof the buckling phenomena. Again, no buckling is possible until thecompressive load reaches the rst critical load of the linearized theory(the reason for this being discussed below) and as this value is exceededthe possible buckling deformation is completely determined except forsign. For values of the load between the rst two critical loads there aretherefore three possible solutions as shown in Fig. 1.4. When P exceedsthe second critical load a new, determinate, pair of nontrivial solutionsappears, and so forth. The values of corresponding to these criticalloads are said to be branch points (or bifurcation points) of the trivialsolution because new solutions, initially of small size, appear at thesepoints.

A diagram like Fig. 1.4 is called a branching (or bifurcation diagram)and we may justify the statements, above, about the buckling behav-ior for the problem of the compressed thin rod by exhibiting the closed

form solutions of (1.6a,b). The deection v can then be obtained fromthe rst relation in (1.4). We rst note certain simple properties of theboundary value problem (1.4), (1.5), namely, if any one of the triplets(, v, ), (, v, +2n), (,v,), or (,v, ++2n), is a solu-tion so are the other three; all four solutions yield congruent deections.In fact, the rst two are identical, while the third is a reection of therst about the x-axis and the fourth is a reection of the rst about theorigin; in this last case, a reversal in the sign of the load is accounted forby interchanging the roles of the left-hand and right-hand ends of therod. Similar remarks apply to the system (1.6a,b).

All nontrivial lateral deections v(s) of the rod are therefore generatedby those solutions of the initial value problem

d2

ds2+ sin = 0, > 0

(0) = , (0) = 0, 0 < < (1.9)

which have a vanishing derivative at s = .

The initial value problem (1.9) has one and only one solution andthis solution may be interpreted as representing the motion of a simplependulum, with x being the time and the angle between the pendulumand the (downward) vertical position. In view of the initial conditions,(s) will be periodic. If we multiply the equation in (1.9) by d/ds,then this equation becomes a rst order equation which can be solved(explicitly) by elliptic integrals, i.e.

(s) = 2 arc sin[k sn(

s+K)

](1.10)

with k = sin(

2)

K =

2

0

d1 k2 sin2

(1.11)

and

sn u = u (1 + k2)u3

3!+ (1 + 14k2 + k4)

u5

5!(1.12)

(1 + 135k2 + 135k4 + k6)u7

7!+

The parameter K takes on values between

2and + and there is a one-

to-one correspondence between these values and those of in 0 < < .The period of oscillation for is 4K and the condition () = 0 issatised if and only if

K =(

2n

), n = 1, 2, (1.13)

As K >

2, in order for nontrivial solutions to exist, we must have

> 2/2. Also, given a value of > 2/2 there are as many distinctnontrivial solutions (s), with 0 < (0) < , as there are integers nsuch that (/2n)

> /2. Thus, we have for

0 2/2, only the trivial solution2/2 < 42/2, one nontrivial solutionn22/2 < (n+ 1)22/2, n nontrivial solutions

If we then also take into account other values of (0), and the possibilityof negative , we obtain Fig. 1.5, which depicts the maximum value of(s) versus ; the only (physically) signicant portion of this gure is,of course, the part drawn with solid (unbroken) curves, i.e., for > 0and < max < .

The broken curves for > 0 are those attributable to angles that aredetermined only by modulo 2, while the dotted curves for < 0 corre-spond to having the free end of the rod at x = . The deection v(s) isthen obtained from the rst equation in (1.4) and yields the bifurcationdiagram in Fig. 1.4, where it is known that the various branches do notextend to innity because vmax does not grow monotonically (at largeloads the buckled rod may form a knot so that the maximum deectioncan decrease, although the maximum slope must increase.)

We have presented a complete solution of the buckling problem for acompressed thin rod which is based on classical analysis of an associ-ated initial value problem for the boundary value problem (1.6a,b) rsttreated by Euler in 1744. The equivalent problem for the deection v(s)is (1.4), (1.5) and the results of the analysis, based on a closed formsolution using elliptic integrals, seem to indicate a prominent positionfor the linearized (eigenvalue) problem (1.8), or, equivalently, for thelinearization of (1.6a,b) namely,

d2

ds2+ = 0, 0 < s <

(0) = () = 0(1.14)

Indeed, this system governs the initial buckling of the rod for reasonsthat are explained below; our explanation will be phrased in just enoughgenerality so as to make our remarks directly applicable to the study ofbuckling and postbuckling behavior for thin plates. For the case of linearelastic response, the von Karman system of nonlinear partial dieren-tial equations, with associated boundary conditions, which govern platebuckling, are derived in 1.2.

As we will specify below, the nonlinear boundary value problem (1.6a,b), which governs the behavior of the compressed rod, can be put in theform

G(, u) = 0 (1.15)

where is a real number, u is an element of a real Banach space Bwith norm , and G is a nonlinear mapping from R B into B,Rbeing the real numbers. The restriction to real and a real Banachspace B is based on the needs in applications where only real branchingis of interest. Strictly speaking, a solution of (1.15) is an ordered pair(, u) but we often refer to u itself as the solution (either for xed , ordepending parametrically on ); to study branching (or bifurcation) wemust have a simple, explicitly known solution u() of (1.15). We maymake the assumption that

G(, 0) = 0, for all , (1.16)

so that u() = 0 is a solution of (1.15) for all and this solution is thenknown as the basic solution (for the problem considered above, this solu-tion corresponds to the compressed, unbuckled rod). The main problemis to study branching from this basic solution (i.e., the unbuckled state)although within the context of plate buckling we will also discuss, in 5,branching from nontrivial solutions of (1.15), i.e., secondary bucklingof thin plates. Thus, the goal is to nd solutions of (1.15) which are ofsmall norm (small size in the relevant Banach space B); this motivatesthe following:

Denition. We say that = o

is a branch point of (1.15) (equivalentlya bifurcation point or, for the case in which (1.15) represents the equi-librium problem for a structure, such as the compressed rod considered

above, a buckling load) if every neighborhood of (o

, 0), in R B,contains a solution (, u) of (1.15) with u = 0.

We note that the above denition is restricted only to small neighbor-hoods of (

o

, 0); also, the denition is equivalent to the existence of asequence of solutions (n, un) of (1.15), with un = 0 for each n, suchthat (n, un) (o, 0), as n . Before placing the compressed rodproblem within the context of the general formulation (1.15), and thenexplaining the reason for the apparent principal position of the associ-ated linearized problem, it is worthwhile to list some of the principalproblems of bifurcation theory (equivalently, buckling and postbucklingtheory for structures); these are covered by the following questions:

1. Where are the branch-points? What is the relation of the branchpoints to the eigenvalues of some appropriately dened linearizedproblem?

2. How many distinct branches emanate from a branch-point? Is thebranching to the left or right?

3. Can we describe the dependence of the branches on , at least ina neighborhood of the branch point?

4. In a physical problem, which branch does the physical systemfollow?

5. How far can branches be extended? If we are dealing with a Banachspace B, consisting of functions belonging to a certain class, can weguarantee that a particular branch represents a positive function.Does secondary branching occur?

Answers to questions 1-3, above, are the only ones which clearly fallwithin the domain of bifurcation theory; problem 4 is related to stabilitytheory, while problem 5 requires, for an answer, techniques of globalanalysis which fall outside the strict domain of branching theory.

Remarks: Quite often (such as will be the case for the buckling of thinplates governed by the von Karman equations, or some modicationthereof) the functional equation G(, ) = 0, in a Banach space B,subject to the condition (1.16), will assume the particular form

Au u = 0, with A0 = 0 (1.17)

with A a nonlinear operator from B into B.

We now want to indicate how the problem governing buckling of acompressed rod, which we treated above, can be cast in the form (1.15).The situation is actually quite simple; we set

G(, ) =d2

ds2+ sin (1.18)

and for the Banach space B takeB = { C2[0, ], with (0) = () = 0} (1.19)

where C2[0, ] is the set of all functions dened and continuous on [0, ]which have continuous derivatives up to order 2 on the open interval(0, ). The mapping G, as given by (1.18), is a nonlinear dierentialoperator dened on B R+ where R+ is the set of all nonnegative realnumbers.

Now, suppose for the sake of simplicity, that G(, u) = 0 were actuallya scalar equation (like (1.6a)) and suppose, also, that there exists anequilibrium solution of the form (0, u0); by hypothesis (1.16), (, 0) issuch an equilibrium solution for any . Thus G(0, u0) = 0. One mayask the question: does there exist, in a small neighborhood of (0, u0),a unique solution of G(, u) = 0 in the form u = u()? An answer tothis question is provided by the well-known implicit-function theoremof mathematical analysis for whose statement we require the concept ofFrechet derivative.

The concept of Frechet derivative for mappings between Banach orHilbert spaces is dened as in numerous texts; an explanation of thisconcept, which belongs to the engineering literature on buckling andpostbuckling behavior of elastic structures, may be found in the surveypaper of Budiansky [16]. To begin with, we have a Banach or Hilbertspace (B or H, respectively) with a norm that is used to measurethe size of elements in this space; for example, for the function space B,given by (1.19), in the problem associated with buckling of a compressedrod, for in B we may take

=

0

2(s)ds (1.20)

Then, if f is a nonlinear mapping on the (function) space in question, wesay that f has a Frechet derivative fu (or f (u)), at an element (function)u in the space, if f (u) is a linear map such that

limv0

f(u+ v) f(u) f (u)vv = 0 (1.21)

In most cases of importance in applications, the concept of Frechetderivative f (u) is actually equivalent to the denition of the so-calledGateaux derivative of f at u, namely, for v any other element (i.e., func-tion) in the space in question, f (u) is that linear mapping which isdened by

f (u)v lim0

f(u+ v) f(u)

(1.22)

or, equivalently,

f (u)v =

$[f(u+ v)]|=0 (1.23)

For our purposes in this treatise, we may think of the Frechet derivativeof a nonlinear mapping G, such as the one in (1.18), (which dependson the parameter ), as being given by (1.22) or (1.23), e.g., at =0, = 0, the Frechet derivative of G(, u), as dened by (1.18), isgiven by

G(0, 0) =

[G(0, 0 + $)]|=0 (1.24)

with B, as dened in (1.19). With the concept of Frechet derivativein hand, we may state the following version of the implicit function the-orem which, in essence, underlies all of bifurcation (or buckling) theory:

Theorem: Let G (parametrized by , as in (1.18)) be a Frechet dif-ferentiable, nonlinear mapping, on some Banach (or Hilbert) space andsuppose that (0, u0) is an (equilibrium) solution, i.e, G(0, u0) = 0. Ifthe linear map Gu(0, u0) has an inverse which is a bounded (i.e. con-tinuous) mapping then locally, for |0| suciently small, there existsa dierentiable mapping (i.e. function) u() such that G(, u()) = 0.Furthermore, in a suciently small neighborhood of (0, u0), (, u()) isthe only solution of G(, u) = 0

Remarks: Under the hypothesis (1.16), (, 0) is an equilibrium solu-tion for all (note that the mapping G as given by (1.18) satises thiscondition). Thus, if Gu(, 0) is a (linear) map with an inverse, which isbounded, then u() = 0, for all , is the only solution of G(, u()) = 0and one may conclude that no branching (or bifurcation) of solutionscan occur from the trivial solution u() = 0.

As a consequence of the implicit function theorem, one may concludethat branching of solutions can only occur if the linear mapping Gu,evaluated at a specic (equilibrium) solution (c, u0) is singular, so thatGu(c, u0) does not have a bounded inverse; such a point (c, u0) is thena candidate for a branch (or bifurcation) point.

Example (Consider the compressed thin rod, as governed by G(, ) =0, G dened by (1.18), with in the space B given by (1.19). Using thedenition (1.23) of Frechet derivative, we easily nd that for any 0, inthe space given by (1.19), the derivative G(, 0) is the linear mappingL, which acts on a function in B as follows:

L = G(, 0) d2ds2

+ cos0 (1.25)

so that, in particular, for the equilibrium solution (, 0 = 0)

L = G(, 0) d2ds2

+ (1.26)

where, as B, must satisfy (0) = () = 0. But L is invertibleif and only if the only solution of L = 0, for in B, is given by 0.Thus, L is not invertible if there are values = c such that L = 0,for B, has nontrivial solutions, i.e., if there exists = c such that

d2ds2

+ c = 0

(0) = () = 0(1.27)

has at least one nontrivial solution ; such solutions (of 1.27) are, ofcourse, eigenfunctions, corresponding to the eigenvalue = c, whichthen becomes a candidate for being a branch (or bifurcation) point forthe boundary value problem (1.6a,b) associated with the compressed(thin) rod. The eigenvalue problem (1.27) is the linearized problemassociated with (1.6a,b); in the parlance of buckling theory for elasticstructures, = c, such that there exists a nontrivial solution = c(s)of (1.27), is a possible buckling load actually the buckling load di-vided by EI in this case and the eigenfunction c is the associatedbuckling mode. In all cases of interest in classical buckling theory, thelinearization of an equilibrium equation (or set of equations), such as(1.15), about an equilibrium solution (0, u0), leads, in the manner de-scribed above, to an eigenvalue-eigenfunction problem

Gu(c, u0)vc = 0 (1.28)

for the buckling loads c and the associated buckling modes vc; thiswill, of course, be the case for the von Karman equations which we derivein 1.2.

We remark that it is possible to construct equations (and systems ofequations) of the form G(, u) = 0, with G(, 0) = 0, for all , andGu(, 0) not invertible, for which branching does not take place fromeach eigenvalue of the linear operator L = Gu(, 0). However, there aregeneral theorems of bifurcation theory, which apply, e.g., to branchingof solutions of the von Karman equations, and which guarantee thateigenvalues of the linear operator L = Gu(, 0) are not only, by virtueof the implicit function theorem, candidates for branch points, but are,indeed, values of where bifurcation occurs; we state only one suchtheorem: Let G(, u) = Auu, with A a nonlinear operator such thatA(0) = 0, and suppose that A is completely continuous (or compact).Then if

o = 0 is an eigenvalue of odd multiplicity for the linearizedoperator L = A(0),

o

is a branch-point of the basic solution u() 0of the nonlinear problem.

The theorem stated above is a classical result which is due to Lerayand Schauder (see, e.g., [26]): the proof is not constructive (being basedon topological degree theory) and thus little information is obtainedabout the structure of a bifurcating branch. The denition of completelycontinuous for a nonlinear operator A on a Banach or Hilbert space istechnical and may be found in any text (e.g. [26] ) on functional analysis;suce it to say that the operators which are generated in the study ofbuckling and postbuckling of elastic (and, often, viscoelastic) structuresdo conform, in the proper mathematical setting, to the denition of acompletely continuous operator.

1.2 The von Karman Equations for Linear ElasticIsotropic and Orthotropic Thin Plates

In this section we will derive the classical von Karman equations whichgovern the out-of-plane deections of thin isotropic and orthotropic lin-ear elastic plates as well as the linearized equations which mediate theonset of buckling; the equations will be presented in both rectilinearcoordinates and in polar coordinates. We begin with a derivation, inrectilinear coordinates, of the von Karman equations for linear elastic,isotropic, (and then orthotropic), behavior.

1.2.1 Rectilinear Coordinates

We begin with the case of isotropic symmetry. Our derivation followsthat in the text by Troger and Steindl [66] in which the authors beginwith the derivation of the equilibrium equations for shallow shells, un-dergoing moderately large derivations, and then specialize to the caseof the plate equations (which follow as a limiting case corresponding tozero initial curvature); such an approach is of particular interest to usin as much as we will want, later on, to look at imperfection buckling ofthin plates.

In Fig. 1.6, we depict the undeformed middle surface of a shallowshell; this middle surface is represented in Cartesian coordinates by thefunction w = W (x, y) and the displacements of the middle surface corre-sponding to the x, y, z directions are denoted by u, v, and w respectively.

To derive the nonlinear shell equations we

(i) First obtain a relationship between the displacements and thestrain tensor; this is done by extending the usual assumptions made inthe linear theory of plates and shells, and retaining nonlinear terms inthe membrane strains, while still assuming a linear relationship for thebending strains.

(ii) Relate the strain tensor to the stress tensor by using the consti-tutive law representing linear, isotropic elastic behavior.

(iii) Derive the equilibrium equations which relate the stress tensorto the external loading.

We begin (see Fig. 1.7) by considering an innitesimal volume element

dV = hdxdy of the shell. Let

r be the position vector to a point P , inthe interior of the shell, which is located at a distance from the middlesurface; then with

ei, i = 1, 2, 3, the unit vectors along the coordinate

axes

r= xe1 +y

e2 +(W + )

e3 (1.29)

If we employ a form of the Kirchho hypothesis, i.e., that sectionsx = const., y = const., of the undeformed shell, remain plane afterdeformation, and also maintain their angle with the deformed middlesurface, then in terms of the displacement components u, v, w of the

middle surface of the shell, with w,x =w

x, etc.,

u() = u w,x, v() = v w,y, w() = w (1.30)

are the displacement components of P when it moves, after the displace-ment, (which arises, e.g., because of the application of middle surfaceforces at the boundary, or variations in temperature or moisture content)

to a point P . The position vector

r to the point P is then

r= (x+ u w,x) e1 +(y + v w,y) e2+(W + + w)

e3

(1.31)

The components of the strain tensor for this deformation of the shell arethen obtained by computing the dierence of the squares of the lengths

of the dierential line elements dr and d

r ; a length computation basedon (1.31) and (see Fig. 1.6)

r= x

e1 +y

e2 +W

e3 yields

12

[|d

r |2 |dr |2]

= xxdx2 + 2xydxdy + yydy2 (1.32)

where

$xx = u,x +12

(u2,x + v

2,x + w

2,x

)+W,x w,x

w,xx +

$yy = v,y +12

(u,2y +v,

2y +w,

2y

)+W,y w,y

w,yy + xy 2$xy = u,y +v,x +u,x v,y +w,x w,y

+W,x w,y +W,y w,x2w,xy +

(1.33)

and terms of at least third order have been neglected. If we now takeinto consideration the fact that, for stability problems connected withthin-walled structures, the displacement w, which is orthogonal to themiddle surface, is much larger than the displacements u, v in the middlesurface, then the terms quadratic in u and v, in (1.33), may be neglectedin comparison with those in w; we thus obtain the (approximate) kine-matical relations

$xx = u,x +12w,2x +W,x w,xw,xx

$yy = v,y +12w,2y +W,y w,y w,yy

xy = 2$xy = u,y +v,x +w,x w,y +W,x w,y

+W,y w,x2w,xy

(1.34)

For the particular case in which the shallow shell is a plate whose middlesurface, prior to buckling, occupies a region in the x, y plane (so thatW 0) (1.34) reduces further to

$xx = u,x +12w,2xw,xx

$yy = v,y +12w,2y w,yy

xy 2$xy = u,y +v,x +w,x w,y 2w,xy

(1.35)

We now use the generalized Hookes law to describe the material be-havior of the isotropic, linear elastic, shallow shell and we make theusual assumption for thin-walled structures that zz 0. The stresscomponents, xz and yz, which will appear in the equilibrium equa-tions, do not contribute to the constitutive relationship as the corre-sponding strains are zero (due to the Kirchho hypothesis). Thus, aplane strain problem over the thickness h of the shell is obtained forwhich the relevant constitutive equations may be written in the form

hE(u,x +12w,2x +W,x w,x ) = Nx Ny

hE(v,y +12w,2y +W,y w,y ) = Ny Nx

Gh(u,y +v,x +w,x w,y +

W,x w,y +W,y w,x ) = Nxy

(1.36)

where E is Youngs modulus, is the Poisson number, G = E/2(1 + )is the shear modulus, and

Nx = h/2h/2

xxdz

Ny = h/2h/2

yydz

Nxy = h/2h/2

xydz

(1.37)

are the averaged stresses over the shell thickness h, which is assumed tobe small. The bending moment Mx is dened to be

Mx = h/2h/2

xxd (1.38)

and by using (1.34) and (1.36) this may be computed to be

Mx = h/2h/2

E

1 2 ($xx + $yy)d

= K(w,xx +w,yy )(1.39)

with the plate stiness K given by

K =Eh3

12(1 2) (1.40)

With analogous denitions (and computations) for the bending momentsMy and Mxy we nd that{

My = K(w,yy +w,xx )Mxy = (1 )Kw,xy

(1.41)

Although, in a nonlinear theory, the equations of equilibrium for thestresses and loads must be calculated in the deformed geometry, in thenonlinear theory of shallow shells one makes the approximation that theundeformed geometry can still be used for this purpose; this assumptionthen restricts the validity of the equations we obtain to moderately largedeformations. We will, therefore, use the equilibrium equations of linearshell and plate theory, namely, the three force equilibrium equations{

Nx,x +Nxy,y = 0Nxy,x +Ny,y = 0

(1.42)

and

Qxz,x +Qyz,y +Nx(W + w),xx+Ny(W + w),yy +2Nxy(W + w),xy + t = 0

(1.43)

where t(x, y) is a distributed normal loading, and the two moment equi-librium equations

Mxy,x +My,y Qyz = 0(Mxy = Myx)

Myx,y +Mx,x Qxz = 0(1.44)

where the indicated forces and moments are depicted in gures 1.8 and1.9 and the moment equilibrium equation about the z-axis is satisedidentically.

We now proceed by dierentiating the rst equation in (1.44) with re-spect to y, and the second equation with respect to x, and then insertingthe resulting expressions for Qxz,x and Qyz,y in (1.43) so as to obtainthe following equation from which the shear forces have been eliminated:

Mx,xx + 2Mxy,xy +My,yy+Nx(W + w),xx +2Nxy(W + w),xy

+Ny(W + w),yy +t = 0

(1.45)

The next step consists of introducing the Airy stress function (x, y),which is dened so as to satisfy

Nx = ,yy , Ny = ,xx , Nxy = ,xy (1.46)With the above denitions, both equilibrium equations in (1.42) aresatised identically. Finally, if we substitute (1.39), (1.41) and (1.46)into the remaining equilibrium equation, (1.45), we obtain the nonlinearpartial dierential equation

K2w = ,yy (W + w),xx +,xx (W + w),yy (1.47)2,xy (W + w),wy +t(x, y)

for the two unknowns, the (extra) deection w(x, y) and the Airy stressfunction (x, y), where 2 denotes the biharmonic operator in rectilin-ear Cartesian coordinates, i.e.,

2w =4w

x4+ 2

4w

x2y2+4w

y4(1.48)

To obtain a second partial dierential equation for w(x, y) and (x, y),we make use of the identity

(u,x ),yy +(v,y ),xx(u,y +v,x ),xy = 0 (1.49)into which we substitute, from the constitutive relations (1.36), for thedisplacement derivatives u,x , v,y , and u,y +v,x. One then makes useof the denition of the Airy function to replace the stress resultantsNx, Ny, and Nxy; there results the following equation

2 = Eh[(w2,xy w,xx w,yy )

+2W,xy w,xy W,xx w,yy W,yy w,xx] (1.50)

If we introduce the nonlinear (bracket) dierential operator by

[f, g] f,yy g,xx2f,xy g,xy +f,xx g,yy (1.51a)

so that

[f, f ] = 2(f,xx f,yy f,2xy ) (1.51b)

then the system consisting of (1.47) and (1.50) can be written in themore compact form

K2w = [,W + w] + t (1.52a)

1Eh

2 = 12[w,w] [W,w] (1.52b)

In particular, for the deection of a thin plate, which in its undeectedconguration occupies a domain in the x, y plane, so thatW 0, (1.52a),(1.52b) reduce to the von Karman plate equations for an isotropic linearelastic material, namely,

K2w = [, w] + t (1.53a)1Eh

2 = 12[w,w] (1.53b)

An important alternative form for the bracket [, w], which reects theeect of middle surface forces on the deection, is

[, w] = Nx2w

x2+ 2Nxy

2w

xy+Ny

2w

y2(1.54)

Remarks: The curvatures of the plate in planes parallel to the (x, z) and(y, z) planes, are usually denoted by x and y, respectively, while thetwisting curvature is denoted by xy. Strictly speaking, the curvaturex, e.g., is given by

x =

2w

x2{1 +

(w

x

)2} 32 (1.55)

where the minus sign is introduced so that an increase in the bendingmoment Mx results in an increase in x. As w,2x is assumed to be small,

even within the context of the nonlinear theory of shallow shells, thecurvatures are usually approximated by, e.g. x = w,xx, in which casefor linear isotropic response

x = (Mx My)/{(1 2)K}

y = (My Mx)/{(1 2)K}

xy = (1 + )Mxy/ {(1 )K}(1.56)

Remarks: For a plate which is initially stress-free, but possesses aninitial imperfection in the form of a built-in deection, whose mid-surfaceis given by the equation z = w0(x, y), the appropriate modication ofthe von Karman equations (1.53a,b) for a plate exhibiting linear elasticisotropic response may be obtained from the shallow shell equations(1.52a,b) by replacing W (x, y) by w0(x, y). Alternatively, if we set

w(x, y) = w(x, y) + w0(w, y) (1.57)

then we obtain from (1.53a,b) the following imperfection modicationof the usual von Karman equations:

K2(w w0) = [, w] + t1Eh

2 = 12[w w0, w w0][w0, w w0]

(1.58)

In (1.58), w(x, y) represents the net deection, and (1.58) reduces to(1.53a,b) if w0 0. Other modications of (1.53a,b) are needed if theplate is subject to thermal or hygroexpansive strains, or if the stinessK, or the thickness h of the plate are not constant.

Remarks: For a normally loaded (linear elastic, isotropic) plate witha rigid boundary, the in-plane boundary conditions are specied by thevanishing of the displacements u, v, rather than by specication of themiddle surface forces; in such cases, there may be advantages to ex-pressing the large deexion equations for an initially at plate, of con-stant thickness, in terms of the displacements u, v, w. The displacementequations, with some straightforward work, can be shown to have thefollowing form

h2

12

(4w t

K

)= w,xx

{u,x +v,y +

12w,2x +

12w,2y

}+ w,yy

{v,y +u,x +

12w,2y +

12w,2x

}+ (1 )w,xy (u,y +v,x +w,x w,y )

(1.59a)

x

[u,x +v,y +

12

{w,2x +w,

2y

}]+

(1 1 +

)(2u+ w,y2w) = 0

(1.59b)

and

y

[u,x +v,y +

12

{w,2x +w,

2y

}](1.59c)

+(

1 1 +

)(2v + w,y2w) = 0

where 2 is the usual Laplacian operator.We now proceed (still in rectilinear coordinates) to derive the appro-

priate form of the von Karman plate equations for the case of a thin platewhich exhibits linear elastic behavior and has (rectilinear) orthotropicsymmetry. Thus, consider an orthotropic thin plate for which the xand y axes coincide with the principal directions of elasticity; then theconstitutive equations have the form

xxyyxy

= c11 c12 0c21 c22 0

0 0 c66

$xx 1H$yy 2H

xy

(1.60)where we have included in the strain components possible hygroexpan-sive strains iH, i = 1, 2 where the i are the hygroexpansive coef-cients and H represents a humidity change; alternatively, we couldreplace the iH by thermal strains iT with the i thermal expan-sion coecients and T a change in temperature. In (1.60)

c11 = E1/(1 1221)c12 = E221/(1 1221)c21 = E112/(1 1221)c22 = E2/(1 1221)c66 = G12

(1.61)

with E112 = E221 so that c12 = c21. In (1.61), E1, E2, 12, 21, andG12 are, respectively, the Youngs moduli, Poissons ratios, and shearmodulus associated with the principal directions. The constants

Dij = cijh3

12(1.62)

are the associated rigidities (or stiness ratios) of the orthotropic plate,specically,

D11 =E1h

3

12(1 1221) , D22 =E2h

3

12(1 1221) (1.63)

are the bending rigidities about the x and y axes, respectively, while

D66 =G12h

3

12(1.64)

is the twisting rigidity. The ratios D12/D22, D12/D11 are often calledreduced Poissons ratios. For the thin orthotropic plate under consid-eration, the strains $xx, $yy, and xy, the averaged stresses (or stressresultants) Nx, Ny, and Nxy, and the bending moments Mx,My, andMxy are still given by (1.35), (1.37), (1.38), and the relevant expressionsfor My and Mxy, which are analogous to (1.38). Thus, with

xx = c11($xx 1H) + c12($yy 2H)yy = c21($xx 1H) + c22($yy 2H)xy = c66xy

we have

xx = c11(u,x +12w,2xw,xx ) (1.65a)

+c12(v,y +12w,2y w,yy )

(c111H + c122H)

yy = c21(u,x +12w,2xw,xx ) (1.65b)

+c22(v,y +12w,2y w,yy )

(c211H + c222H)

and

xy = c66

(12

(u,y +v,x ) + w,x w,y 2w,xy)

(1.65c)

Using the expressions in (1.65 a,b,c), we now compute the bending

moments Mx,My, and Mxy to be

Mx = c11 h/2h/2

[

(u,x +

12w,2x

) 2w,xx

]d

+c12 h/2h/2

[

(v,y +

12w,2y

) 2w,yy

]d

h/2h/2

(c121H + c222H) d

(1.66a)

My = c21 h/2h/2

[

(u,x +

12w,2x

) 2w,xx

]d

+c22 h/2h/2

[

(v,y +

12w,2y

) 2w,yy

]d

h/2h/2

(c111H + c222H) d

(1.66b)

and Mxy =

c66

h/2h/2

[

(12

[u,y +v,x ] + w,x w,y

) 22w,xy

]d

(1.66c)

At this stage of the calculation, sucient exibility exists to handleany dependence of the (possible) hygroexpansive strains iH, i = 1, 2,on the variable z; one could, e.g., assume that the iH, i = 1, 2 areindependent of z, depend either linearly or quadratically on z, or areeven represented by convergent power series of the form

1H = 0 +

m=1

mzm

2H = 0 +

m=1

mzm

If, however, iH = fi(z), i = 1, 2, with the fi(z) even functions ofz, i.e., fi(z) = fi(z), i = 1, 2, then iH will be an odd functionof ,h/2 < < h/2, and the integrals in both (1.66a) and (1.66b),which involve the hygroexpansive strains, will vanish. For the sake ofsimplicity, we will proceed here by assuming that the hygroexpansivestrains are constant through the thickness of the plate; this assumption

will be relaxed in the discussion of hygroexpansive/thermal buckling inChapter 6. Then

Mx = c11w,xx 3

3

]h/2h/2

c12w,yy 3

3

]h2

h/2

= (c11w,xx +c12w,yy ) h3

12or

Mx = (D11w,xx +D12w,yy ) (1.67a)while, in an analogous fashion,

My = (D21w,xx +D22w,yy ) (1.67b)and

Mxy = 2c66 w,xy 3

3

]h2

h2

= 2D66w,xy(1.67c)

We now set W 0, t 0 in (1.45), to reect the fact that we are dealingwith an initially at plate which is not acted on by a distributed normalload, introduce the Airy stress function through (1.46), and employ theresults in (1.67a,b,c), so as to deduce that

(D11w,xx +D12w,yy ),xx4D66w,xxyy(D21w,xx +D22w,yy ),yy +,yy w,xx

2,xy w,xy +,xx w,yy = 0or

D11w,xxxx +[D12 + 4D66 +D21]w,xxyy (1.68)+D22w,yyyy [, w] = 0

The corresponding modication of the rst von Karman equation for thedeection of a thin, linearly elastic, orthotropic plate, when there existsan initial deection z = w0(x, y), is easily obtained from (1.45) and(1.67a,b,c) by setting W = w0 and dening, as in (1.57), w = w + w0.To obtain the appropriate modication of the second von Karman equa-tion (1.53b) for the case of a linearly elastic, thin, orthotropic plate, we

once again begin with the identity (1.49). We then use the constitutiverelations (1.65 a,b,c) to compute the average stresses (or stress resul-tants) Nx, Ny, and Nxy, introduce the Airy function through (1.46),and solve the resulting equations for u,x , v,y , and u,y +v,x; these, inturn, are substituted into (1.49) and there results the partial dierentialequation

1E1h

,yyyy +1h

(1G12

212E2

),xxyy (1.69)

+1

E2h,xxxx = 12 [w,w]

if one makes use of the fact that E221 = E112.The complete set of von Karman equations in rectilinear Cartesian

coordinates thus consists of (1.68) and (1.69). In an isotropic plateE1 = E2 = E, G12 = G = E/2(1 + ), with 12 = 21 = , andthe system of equations (1.68), (1.69) reduces to the system (1.53a,b),with t 0, where K is the common value of the principal rigiditiesD1 = D11, D2 = D22, and D3 = D1212 + 2D66.

Remarks: The system of equations (1.68), (1.69), as well as their spe-cializations to (1.53 a,b), for the case of linear, isotropic, elastic response,may be obtained from a variational (minimum energy) principle basedon the potential energy

U =12

A

h/2h/2

{c11($xx 1H)2 (1.70)

+2c12($xx 1H)($yy 2H)+c22($yy 2H)2 + c662xy } dzdA

where the outer integral is computed over the area A occupied by theplate. In (1.70), or its equivalent for the case where the plate ex-hibits isotropic response, we must rst substitute from (1.35) in orderto express the integrand as a polynomial expression in the displacementderivatives.

Remarks: In the case of very thin plates, which may have deec-tions many times their thickness, the resistance of the plate to bendingcan, often, be neglected; this amounts, in the case of a plate exhibitingisotropic response, to taking the stiness K = 0, in which case the prob-lem reduces to one of nding the deection of a exible membrane. Theequations which apply in this case were obtained by A. Foppl [69] and

turn out, of course, to be just the von Karman equations (1.53a,b) withK set equal to zero.

1.2.2 Polar Coordinates

In this section we will present the appropriate versions of the vonKarman equations in polar coordinates (actually, cylindrical coordinatesmust be used for the equilibrium equations) for a thin linearly elasticplate. We will present these equations for the following cases: a plateexhibiting isotropic symmetry, both for the general situation as wellas for the special situation in which the deformations are assumed tobe radially symmetric, a plate which exhibits cylindrically orthotropicbehavior (to be dened below), and a plate which exhibits the rectilinearorthotropic behavior that was specied in the last subsection. We beginwith the simplest case, that of a linearly elastic isotropic plate.

It is well known that the operators present in the von Karman equa-tions (1.53a,b) are invariant with respect to changes in the coordinatesystem; in particular, the von Karman equations in polar coordinates fora thin, linearly elastic, isotropic plate may be obtained from (1.53a,b) bytransforming the bracket and biharmonic operators into their equivalentexpressions in the new coordinate system. If ur, u denote, respectively,the displacement components in the middle surface of the plate, whilew = w(r, ) denotes the out of plane displacement, then the strain com-ponents err, er, and e are given by (r = 2er) :

err =urr

+12(w

r)2

2w

r2

e =urr

+1r

u

+1

2r2(w

)2 (1

r

w

r+

1r2

2w

2)

er =ur

ur

+1r(ur

) +1r(w

r)(w

)

2(1r

2w

r 1r2

w

)

(1.71)

With the components of the stress tensor rr, r, , zz, rz andz as shown in Fig. 1.10, (r = r), (rz = zr), and Fr, F the com-ponents of the applied body force in the radial and tangential directions,

the equilibrium equations arerrr

+1r

r

+rr

r+rzz

+ Fr = 0

rr

+1r

+2rr +

zz

+ F = 0(1.72a)

and

rr

(1rw,r +w,rr

)+

(1r2w,

)+r

(2rw,r

)+

1rrz +

rrr

w,r

+

(1r2w,

)+rr

(1rw,

)r

(1rw,r

)+rzr

+rzz

w,r

+1r

z

+zz

(1rw,

)+ Fz = 0

(1.72b)

The transformation of the stress components in Cartesian coordinatesto those in polar coordinates is governed by the formulas:

rr = xx cos2 + yy sin2 + 2xy sin cos = xx sin2 + yy cos2 2xy sin cos r = (yy xx) sin cos + xy

(cos2 sin2 ) (1.73)

with analogous transformation formulae for rz and z. If we set, forthe deection w and the Airy stress function , w (r, ) = w (r cos ,r sin ), (r, ) = (r cos , r sin ), and then drop the superimposedbars in the polar coordinate system, it can be shown directly that thestress resultants (or averaged stresses) Nr, N, and Nr are given interms of by

Nr =1r,r +

1r2

,

N = ,rr

Nr =1r2

, 1r,r

(1.74)

while the operators 2 and [ , ] are given by

2w = w,rrrr +2rw,rrr 1

r2w,rr

+2r2w,rr +

1r3w,r 2

r3w,r

+1r4w, +

4r4w,

(1.75a)

and

[w,] = w,rr

(1r,r +

1r2

,

)+

(1r w,r + 1

r2w,

),rr

2(

1rw,r 1

r2w,

) (1r,r 1

r2,

)or, in view of (1.74),

[w,] = Nrw,rr 2Nr(

1r2w, 1

rw,r

)+ N

(1r w,r + 1

r2w,

)(1.75b)

For the special case of a radially symmetric deformation, in which ur =ur(r), u = 0, and w = w(r), the expressions in (1.75a) and (1.75b) forthe biharmonic and bracket operators reduce to

2w = w,rrrr +2rw,rrr 1

r2w,rr +

1r3w,r (1.76)

and

[w,] = Nrw,rr +N1rw,r (1.77)

Thus, for the von Karman equations for a thin, linearly elastic, isotropicplate, in polar coordinates, we have (with t 0):

K[w,rrrr +

2rw,rrr 1

r2w,rr +

2r2w,rr

+1r3w,r 2

r3w,r +

1r4w, +

4r4w,

]= w,rr

(1r,r +

1r2

,

)+

(1rw,r +

1r2w,

),rr

2(

1rw,r 1

r2w,

) (1r,r 1

r2,

)(1.78)

and

1Eh

[,rrrr +

2r,rrr 1

r2,rr +

2r2

,rr

+1r3

,r 2r3

,r +1r4

, +4r4

,]

= {w,rr

(1rw,r +

1r2w,

)

(1rw,r 1

r2w,

)2} (1.79)

while, for the special case of radial symmetry, these reduce toK[w,rrrr +

2rw,rrr 1

r2w,rr +

1r3w,r ]

=1rw,rr ,r +

1rw,r ,rr

(1.80)

and

1Eh

[,rrrr +

2r,rrr 1

r2,rr +

1r3

,r

]= 1

rw,r w,rr

(1.81)

Remarks: The second product on the right-hand side of equation(1.75b) may be written in the more compact form

2 r

(1r

w

) r

(1r

)

Remarks: For the case of a radially symmetric deformation, it maybe easily shown that isotropic symmetry for the linearly elastic materialyields the relations

Nr =Eh

1 2(e0rr + e

0

)=

Eh

1 2[durdr

+12

(dw

dr

)2+

urr

](1.82a)

and

N =Eh

1 2(e0 + e

0rr

)=

Eh

1 2[urr

+ durdr

+

2

(dw

dr

)2] (1.82b)

Nr = 2Ghe0r (1.82c)

Remarks: In lieu of (1.78), a useful (equivalent) form for the rst vonKarman equation (especially for our later discussion of the buckling ofannular plates) is

K2w = Nrw,rr 2Nr(

1r2w, 1

rw,r

)(1.83)

+N

(1rw,r +

1r2w,

)If rr, r, and are independent of the variable z, in the plate, thenNr, Nr, and N in (1.83), as well as in all the other expressions priorto (1.83), where these stress resultants appear, may be replaced, respec-tively, by hrr, hr, and h.

Remarks: It is easily seen that, for the case of an axially symmetricdeformation of the plate, the relevant equations, i.e., (1.80), (1.81) maybe rewritten in the form

K

{1r

d

drrd

dr

1r

d

drrdw

dr

}=

1r

ddr

d2w

dr2+

1r

d2dr2

dw

dr

(1.84)

and1Eh

{1r

d

drrd

dr

1r

d

drrdw

dr

}= 1

r

dw

dr

d2w

dr2(1.85)

Remarks: From the structure of the operators 2 and [ , ], in the polarcoordinate system, i.e., (1.75a), (1.75b), it is clear that a troublesomesingularity arises at r = 0; the boundary conditions which must beimposed to deal with this diculty at r = 0 will be discussed in the nextsubsection.

The quantities

r =d2w

dr2, =

1r

dw

dr, (1.86)

for the case of radially symmetric deformations of a plate, are the middle-surface curvatures. If the plate is circular, with radius R, then thestrain-energy of bending for the isotropic, linearly elastic plate is

VB = 12K

A

(2r + 2r + 2)dA

= K R

0

[w,2rr +2w,rr

w,rr

+(

1rw,r

)2]rdr

(1.87a)

while the strain-energy of stretching is

VS = Eh2(1 2)

A

(e2rr + 2erre + e

2

)dA

=Eh

1 2 R

0

[(ur,r +

12w,2r )

2

+2(ur,r +

12w,2r

)urr

+(urr

)2]rdr

(1.87b)

If, e.g., we are considering symmetric deformations of a circular plateof radius R, which is compressed (symmetrically) by a uniformly dis-tributed force P per unit length, around its circumference, so that thenet potential energy of loading is

VL = P R

0

w,2r rdr (1.88)

then the total potential energy of the plate is

W (u,w, P ) VB + VS + VL (1.89)

We now turn to the equations, in polar coordinates, for a linearlyelastic, orthotropic, body with cylindrical anisotropy; in this case, thereare three planes of elastic symmetry, one of which is normal to the axisof anisotropy, the second of which passes through that axis, and thethird of which is orthogonal to the rst two. For a plate, we choose therst plane of elastic symmetry to be parallel to the middle plane of theplate; in this case the constitutive relations assume the form

err =1Er

rr E

e = rEr

rr +1E

r =1Gr

r

(1.90)

with Er, E being the Youngs moduli for tension (or compression) inthe radial and tangential directions, respectively, r and the corre-sponding Poissons (principal) ratios, and Gr the shear modulus whichcharacterizes the change of angle between the directions r and . AsEr = Er, the constitutive equations (1.90) can be recast in theform

err =1Er

(rr r)

e =1E

( rr)

r =1Gr

r

(1.91)

so that

rr =Er

1 r err +rE

1 r e

=Er

1 r err +E

1 r e

r = Grr

(1.92)

One may compute the strains err, e, and r by using (1.71) and, then,by employing (1.92), the stresses rr, , and r. The equations ofequilibrium which apply in this situation are (1.72 a,b) and these thenproduce stress components rz and z. The stresses in the cylindri-cally orthotropic plate then lead to stress resultants Nr, N, and Nrand bending and twisting moments Mr,M, and Mr. By employing

straightforward calculations, we are led to the following results for acylindrically orthotropic plate:

Mr = Dr[w,rr +

(1rw,r +

1r2w,

) ]M = D

[rw,rr +

(1rw,r +

1r2w,

)]Mr = Mr = 2Dr

(wr

),r

(1.93)

and

Nr =Erh

1 r

(urr

+12w2,r

)+

rEh

1 r

(urr

+1r

u

+1

2r2w2,

)N =

Erh

1 r

(urr

+12w2,r

)+

E1 r

(urr

+1r

u

+1

2r2w2,

)Nr = Grh

(urr

urr

+1r

ur

+1rw,rw,

)(1.94)

with Dr and D, respectively, the bending stinesses around axes in ther and directions, passing through a given point in the plate, and Drthe twisting rigidity; these are given by

Dr =Erh

3

12(1 r) , D =Eh

3

12(1 r) (1.95)

and

Dr =Gr h3

12(1.96)

whileDr = Dr + 2Dr (1.97)

Using the expressions for Mr,M,Mr in (1.94), those for Nr, N, andNr in (1.74), and a compatibility equation for the displacements in po-lar coordinates, we nd the following form of the von Karman equationsfor a linearly elastic, thin plate exhibiting cylindrically orthotropic sym-metry, (where we have once again introduced the Airy stress functionthrough the relations (1.74)):

Drw,rrrr +2Dr1r2w,rr +D

1r4w, (1.98)

+2Dr1rw,rrr 2Dr 1

r3w,r D 1

r2w,rr

+2(D +Dr)1r4w, +D

1r3w,r

=(

1r,r +

1r2

,

)w,rr +,rr (

1rw,r +

1r2w, )

+2(

1r2

, 1r,r

) (1rw,r 1

r2w,

)and

1E

,rrrr +(

1Gr

2rEr

)1r2

,rr (1.99)

+1Er

1r4

, +2E

1r,rrr

(

1Gr

2rEr

)1r3

,r 1Er

1r2

,rr

+(

21 rEr

+1Gr

)1r4

, +1Er

1r3

,r

= h[w,rr

(1rw,r +

1r2w,

)

(

1rw,r 1

r2w,

)2 ]Equations (1.98), (1.99) may also be obtained directly from many sourcesin the literature, e.g., the paper by Uthgenannt and Brand [70].

Equations (1.98), (1.99), which govern the general deections of acylindrically orthotropic, linearly elastic, thin plate reduce to those whichgovern the deections, in polar coordinates, of an isotropic plate, i.e.(1.78), (1.79) when

Dr = D = Dr = K Eh3

12(1 2) (1.100a)

and

Er = E = E, r = = (1.100b)

Also, for the case of a cylindrically orthotropic plate, undergoingaxisymmetric deformations, with the assumption of radial symmetrythen implying that all derivatives with respect to in (1.98) and (1.99)vanish, these equations reduce to

2rw =1Dr

F (w,) (1.101)

and2r =

12Eh F (w,w) (1.102)

where

2r =d4

dr4+

2r

d3

dr3 r2

d2

dr2+

r3d

dr

= E/Er ( the orthotropy ratio)

F (w,) =1r

[d

dr

(ddr

dw

dr

)] (1.103)

For an isotropic plate undergoing axisymmetric deformations, Dr =D,E = E, = 1, and (1.101), (1.102) specialize to the form givenin (1.84), (1.85)

Remarks: It is often useful to have available the inverted form of theconstitutive relations (1.91) for a cylindrically orthotropic, linearly elas-tic, thin plate, namely,

(rr

)=

Er1 r

(1

) (erre

)(1.104a)

and

r = Grr (1.104b)

The last case to be considered in this subsection concerns the situationin which the linearly elastic, thin plate exhibits rectilinear orthotropicbehavior; thus, the constitutive relations (1.60) apply, (as they would,

e.g., in the case of a linearly elastic paper sheet) but, because we are in-terested in studying deections of circular or annular regions, it is moreappropriate to formulate the corresponding von Karman equations inpolar coordinates instead of rectilinear coordinates. In this situation,we have a mismatch between the elastic symmetry which is built intothe form of the constitutive relations, and the geometry of the regionundergoing buckling; this greatly complicates the structure of the vonKarman equations. It is worth noting that if we make use of the trans-formation

err = exx cos2 + eyy sin2 + xy cos sin

e = exx sin2 + eyy cos2 xy cos sin r = 2(eyy exx) cos sin + xy(cos2 sin2 )

(1.105)

of the principal strains to the polar coordinate system, in conjunctionwith the analogous result (1.73) for the stress components, and the recti-linear orthotropic constitutive relations (1.60), we may transform theseconstitutive equations directly into polar coordinate form; the polar co-ordinate form of the constitutive relations will, indeed, be indicated be-low. However, it is worth noting that the rst of the von Karman equa-tions for this situation has, essentially, been derived in Con [71]andinvolves, of course, using the polar coordinate equivalent for (1.45) withW 0, t 0, namely,

1r(rMr),rr +

1r2M, 1

rM,r +

1rMr,r +Nrw,rr (1.106)

+N

(1rw,r +

1r2w,

)+ 2Nr

(wr

),r

= 0

where the stress resultants Nr, N, and Nr are, once again, given by(1.74) in terms of the Airy function (r, ); we note that the sum ofthe last three terms in (1.106) is (again) identical with the right-handside of (1.75b), i.e., with [w,]. From the work in [71], we deduce thefollowing expressions for the bending moments (which may, of course,be obtained by directly transforming the expressions in (1.67a), (1.67b)and (1.67c) into polar coordinates):

Mr = D1w,rr D12[

1r2w, +

1rw,r

]2D16

(1rw

),r

(1.107a)

M

= D12w,rr D2[

1r2w, +

1rw,r

]2D26

(1rw

),r

(1.107b)

Mr = D16w,rr D26[

1r2w, +

1rw,r

]2D6

(1rw

),r

(1.107c)

where,

D1 = D1 cos4 +D3 cos2 sin2 +D2 sin4

D2 = D1 sin4 +D3 cos2 sin2 +D2 cos4

D12 = 1D2 + (D1 +D2 2D3) cos2 sin2 D6 = D66 + (D1 +D2 2D3) cos2 sin2 D16 =

[(D2 D3) sin2 (D1 D3) cos2

]cos sin

D26 =[(D2 D3) cos2 (D1 D3) sin2

]cos sin

(1.108)

with D1, D2 and D3 the principal rigidities, D1 = D11, D2 = D22, D3 =D212 + 2D66, as dened by (1.63), (1.64). If we now substitute from(1.107a,b,c) into the equilibrium equation (1.106), we obtain the rstof the von Karman partial dierential equations governing the out-of-plane deection of a rectilinear orthotropic, thin, elastic plate in polarcoordinates:

D1w,rrrr +D3[2rw,rrr +

2r

(1rw,r

),r 1

r

(1rw,r

),r +

4r4w,

]+D2

1r4w, +4D16 1

rw,rrr +4D26

1r4w,r

+12r

(D16 D26

) (1rw

),rr

+{D2 D1 +

(D26 D16

)cot 4

}{

32r

(1rw,r

),r +

3r3w,r 4

r4w,

}+

{(D2 D1) tan 2 + 2(D26 D16)

}{

32r4

w, +1r3w,r

}= [w,]

(1.109)

where [w,] is given by (1.74) and (1.75b), and

D3 = D12 + 2D6 (1.110)

Remarks: Inasmuch as the expressions for the moments Mr,M, andMr in (1.107 a,b,c) may be obtained from the expressions for Mx,My,and Mxy in (1.67 a,b,c), and these latter expressions have been derivedby assuming that any existing hygroexpansive strains iH, i = 1, 2,(equivalently, thermal strains iT ) are constant throughout the thick-ness h of the plate, if the iH vary with z in any manner except as anodd function of z (with respect to the middle plane of the plate) then theexpressions for the moments in (1.67a,b,c), and their polar coordinatecounterparts in (1.107a,b,c) would have to be rederived; the new expres-sions obtained for Mr,M, and Mr must then be substituted back into(1.106) so as to obtain the appropriate modication of (1.109), whichapplies in the presence of hygroexpansive strains.

Remarks: The second of the von Karman equations which apply tothe problem of studying the out-of-plane deections of a linear elastic,rectilinearly orthotropic, thin, plate in polar coordinates does not appearin [71] because the primary focus in that work was on studying the initialbuckling problem; nor does the relevant form of this second of the vonKarman equations in polar coordinates for the case of a linearly elastic,

rectilinearly orthotropic plate appear to have been derived anywhere elsein the literature. The calculation, however, which is required to obtainthe equation which complements (1.109) may be carried out in one oftwo ways: rst of all, by transforming the three fourth order partialderivatives ,xxxx ,,xxyy , and ,yyyy which appear on the left-handside of (1.69); the right-hand side of (1.69), in polar coordinates, will beidentical with the right-hand side of (1.79).

Alternatively, to obtain the form of the second of the von Karmanequations, we may rewrite the constitutive relations for a linearly elastic,rectilinearly orthotropic material, i.e.,

exx =1E1

xx 21E2

yy

eyy = 12E1

xx +1E2

yy

xy =1G12

xy

(1.111)

(where we have, for now, not considered the presence of possible hy-groexpansive or hygrothermal strains) in the polar coordinate form

err = a11rr + a12 + a13r

e = a21rr + a22 + a23r

r = a31rr + a32 + a33r

(1.112)

where the aij = aij(), in contrast to the case of a cylindrically or-thotropic material, i.e. (1.91), in which the constitutive coecientsare -independent. The strain components in (1.112) are given by therelations (1.71) in terms of the displacements ur, u, and w, where

h2< a, and r =

x2 + y2; if a = 0, the annulus degenerates

into a circle of radius b and, because of singularities which can developin solutions of the von Karman equations (1.78), (1.79), which apply inthis case, regularity conditions with respect to the deection (as well asthe Airy function) must be satised at r = 0

(i) i is ClampedWe take for i,

{(x, y)|x2 + y2 = Ri

}, i = 1, 2 where R1 a, R2 b;

then is the union of 1 with 2 and, if R1 a = 0, then is just

the circle of radius R2 = b. If 1 is clamped, then as a consequence of(1.129a)

w(Ri, ) = 0, w,r (Ri, ) = 0, 0 < 2 (1.139)

(ii) i is Simply SupportedIn this case, the rst condition w(Ri, ) = 0, 0 < 2, in (1.139)still holds, but the second condition, according to (1.129b), is replacedby Mr = 0 on i; although we obtained (1.78), (1.79) without calcu-lating Mr directly for the isotropic case, in polar coordinates, we mayeasily obtain Mr for the present situation by specializing the rst re-sult in (1.94), for a cylindrically orthotropic thin plate, to the case ofisotropic symmetry. Thus, the second of the simply supported boundaryconditions for w reads[

w,rr +(

1rw,r +

1r2w,

)]r=Ri

= 0, (1.140)

for 0 < 2.

(iii) i is FreeIf i is free then (1.140) applies, for 0 < 2, because, as in thesimply supported case, we still have Mr = 0 at r = Ri. By (1.129c), theother condition at r = Ri is[

Qr +1rMr,

]r=Ri

=[(w),r +

1 r

(1rw,

),r

]r=Ri

= 0(1.141)

where

w = w,rr +1rw,r +

1r2w, (1.142)

Remarks: As was the case for a rectangular plate, for a thin, linearlyelastic, isotropic, annular plate, one may mix and match the varioussets of boundary conditions delineated above, e.g., if the outer radius atr = b is clamped, while the inner radius at r = a is free, the boundaryconditions would read as follows:

w(b, ) = w,r (b, ) = 0, 0 < 2 (1.143a)

[w,rr +

(1rw,r +

1r2w,

)]r=a

= 0, 0 < 2 (1.143b)

[(w) ,r +

1 r

(1rw,

),r

]r=a

= 0, 0 < 2 (1.143c)

Remarks: Suppose that a = 0, so that the annular plate degeneratesto a circular plate of radius b; If the boundary at r = b is clamped,then (1.139) holds with i = 2 and R2 = b. If the boundary at r = bis simply supported, then w(b, ) = 0, 0 2, and, in addition,(1.140) applies with i = 2 and R2 = b. Finally, if the boundary atr = b is free, then (1.140), with i = 2, R2 = b holds, as well as (1.141),with i = 2, R2 = b. For any of these three situations, for the circularplate of radius b, we have a fourth order equation for w (either (1.78), orits specialization, (1.84), to the case of axially symmetric deformations)and only two boundary conditions (at r = b). The missing boundaryconditions which must be imposed arise because of the singularity whichis inherent in the von Karman system (1.78), (1.79)or its axially sym-metric form (1.84), (1.85)at r = 0; the usual assumptions are eitherthat

w|r=0

(i) is ClampedIn this case, the change from isotropic to orthotropic symmetry is incon-sequential; the general conditions in (1.129a) once again translate into(1.130a,b).

(ii) is Simply SupportedBecause we still have w = 0 on , the conditions delineated in (1.130a)still apply in this case. As a consequence of the second condition in(1.129b), however, we have, in lieu of (1.131), the following statements,which are, by virtue of (1.67a,b), equivalent to Mx = 0, for x = 0, x =a, 0 y b, and My = 0, for y = 0, y = b, 0 x a:

D11w,xx +D12w,yy |x=0 = 0, 0 y bD11w,xx +D12w,yy |x=a = 0, 0 y bD21w,xx +D22w,yy |y=0 = 0, 0 x aD21w,xx +D22w,yy |y=b = 0, 0 x a

(1.146)

However, by (1.61) - (1.63), and the fact that E112 = E221:

D12D11

=C12C11

=E221/(1 1221)E1/(1 1221)

andD21D22

=C21C22

=E112/(1 1221)E2/(1 1221)

orD12D11

= 12 andD21D22

= 21 (1.147)

in which case, (1.146) may be rewritten asw,xx +12w,yy |x=0 = 0, 0 y bw,xx +12w,yy |x=a = 0, 0 y b21w,xx +w,yy |y=0 = 0, 0 x a21w,xx +w,yy |y = b = 0, 0 x a

(1.148)

which, clearly, reduce to (1.131) for the isotropic case when 12 = 21 =.

(iii) is FreeIn this case, because we still have Mx = 0 at x = 0, x = a, for 0 y b,and My = 0, at y = 0, y = b, for 0 x a, the conditions in (1.148)hold along each of the respective edges of the rectangle. The second

(general) condition in (1.129c) again becomes (1.134), which reduces to(1.135); for the case of orthotropic symmetry, however, we must now use,in (1.135), the expressions (1.67a,b,c) for Mx,My, and Mxy, respectively.Thus

My,y + 2Mxy,x = D21w,xxy D22w,yyy 4D66w,xxyso that we have, for 0 x a,

(D21 + 4D66)w,xxy +D22w,yyy |y=0 = 0 (1.149)and

(D21 + 4D66)w,xxy +D22w,yyy |y=b = 0 (1.150)for 0 x a. Also,

Mx,x + 2Myx,y = D11w,xxxD12w,xyy 4D66w,xyyso that, for 0 y b,

D11w,xxx +(D12 + 4D66)w,xyy |x=0 = 0 (1.151)and

D11w,xxx +(D12 + 4D66)w,xyy |x=a = 0 (1.152)

Remarks: To check that the boundary conditions (1.149)(1.152), forfree edges on a rectangular orthotropic plate, reduce to those in (1.137),for an isotropic plate, we may note, e.g., that

D21 + 4D66D22

=E112/(1 1221) + 4G12

E2/(1 1221)

= 21 +4G12(1 1221)

E2

so that with isotropic symmetry

D21 + 4D66D22

= +4G(1 2)

E 2

if we use the fact that G =E

2(1 + ). Thus, with the assumption of

isotropic symmetry, (1.149), (1.150) reduce to the rst two conditionsin (1.137) and a similar reduction applies to (1.151), (1.152).

1.3.1.4 Cylindrical Orthotropic Response: CircularGeometry

In this situation, is again the annulus dened by a r b, a 0, b > a, r =

x2 + y2, with the circular domain of radius b corre-

sponding to a = 0. The appropriate constitutive response is given by(1.90) or, equivalently, (1.91), with bending moments and stress resul-tants given as in (1.94) and (1.95). The bending stinesses and twistingrigidity appear in (1.96), while Dr is dened by (1.97). Finally, the vonKarman equations for a thin linearly elastic plate possessing cylindri-cally orthotropic symmetry are exhibited in (1.98) and (1.99). As in thecase of isotropic response, we set i =

{(x, y)|x2 + y2 = Ri

}, i = 1, 2

with R1 = a, R2 = b so that = 1 2. The relevant boundarydata is as follows:

(i) i is ClampedAs in the case of isotropic response and a circular geometry, the bound-ary conditions with respect to w(r, ) reduce to (1.139). If R1 a = 0,we may impose the regularity conditions (1.144) or (1.145) at r = 0,while (1.139) holds for i = 2, i.e., at r = b.

(ii) i is Simply SupportedIn this case, the condition w(Ri, ) = 0, i = 1, 2, 0 < 2, stillapplies but the second condition in (1.139) must be replaced by Mr = 0,at r = a, r = b, which, according to (1.94), means that

[w,rr +

(1rw,r +

1r2w,

)]r=a

= 0

[w,rr +

(1rw,r +

1r2w,

)]r=b

= 0

(1.153)

Remarks: Once again, combined sets of boundary data are possible,e.g., for a cylindrically orthotropic, thin, annular plate, which is linearlyelastic, and has its edge at r = a simply supported, while the edge atr = b is clamped, we would have

w(a, ) = 0,[w,rr +

(1rw,r +

1r2w,

)]r=a

= 0 (1.154a)

w(b, ) = 0,w(r, )

r| r=b = 0, (1.154b)

for 0 < 2.

(iii) i is FreeWith i free, i = 1, 2, (1.153) will still apply but, in lieu of thevanishing of either w or w,r at r = a, b, we must impose the condition[

Qr +1rMr,

]r=Ri

= 0, i = 1, 2 (1.155)

A study of the literature would seem to indicate that the generalform of the free edge boundary condition for a cylindrically orthotropicplate has not been written down; rather, because of the complicatedform that the von Karman equations (1.98), (1.99) take, in the mostgeneral situation, where the deection can depend on , most (if notall) authors, to date, have been content to deal with the axisymmetricform of these equations (and, thus, with the corresponding form of thefree edge boundary condition). The axisymmetric form of the free edgeboundary condition (i.e., the condition that Qr|r=Ri = 0, with w, = 0)is

w,rrr +1rw,rr

(EEr

)w,rr2

|r=Ri = 0 (1.156)

for i = 1, 2. Clearly, for the isotropic, axisymmetric situation, (1.155)reduces to (1.141) because of (1.142), the fact that E = Er, and theassumption that w is independent of .

1.3.1.5 Rectilinear Orthotropic Response: CircularGeometry

Once again is the annulus a r b, a 0, b > a, r =x2 + y2,

with a = 0 yielding a circular plate of radius b. The relevant constitutiveresponse is dened by (1.60), (with the cij as in (1.61) and the iH(or, equivalently, the iT ) taken to be constant through the thicknessof the plate); these relations must be reformulated in polar coordinates,because of the assumed circular geometry of the plate, through the useof (1.73) and the analogous transformation for the components of thestrain tensor (1.105), e.g., the constitutive relations will be the obviousmodications of (1.123a,b,c). The rst von Karman equation is given by(1.106), with bending moments as dened by (1.107a,b,c), (1.108); thisyields the partial dierential equation (1.109). The second of the vonKarman equations for this case comes about, either by directly trans-forming (1.69) into polar coordinates or by substituting the constitutiverelations, in polar coordinate form, into (1.124). The relevant boundaryconditions in this case are now as follows:

(i) i is Clamped

These conditions are, once again, identical with (1.139), for i = 1, 2.

(ii) i is Simply Supported

We again have w(Ri, ) = 0, 0 < 2, i = 1, 2 and, in addition,must require that Mr = 0 for r = a and r = b, 0 < 2; byvirtue of (1.107a), this is equivalent to the following statements for any, 0 < 2,

[D1w,rr +D12

(1r2w, +

1rw,r

)2D16

(1rw

),r

]r=a

= 0(1.157a)

[D1w,rr +D12

(1r2w, +

1rw,r

)2D16

(1rw

),r

]r=b

= 0(1.157b)

where D1, D12, and D16 are dened as in (1.108)

(iii) i is Free

If the boundaries i are free, then both (1.157a) and (1.157b) musthold and, in addition, we have (1.155) where Mr is given by (1.107c)and (1.108); to the best of the authors knowledge Qr has not beencomputed for this situation, to date, and will need to be calculated inthe course of future work on such problems.

Remarks: In general, for the rectilinearly orthotropic, annular plateone will mix dierent types of boundary conditions along the edges at r =a and r = b, e.g., if the inner boundary of the region is simply supported,while the outer boundary is clamped, then we would have w(a, ) =

0, 0 < 2, together with (1.157a) and w(b, ) = 0, w(r, )r

|r=b =0, for 0 < 2.

1.4 The Linear Equations for Initial Buckling

In section 1.1 we considered a general system G(, u) = 0 of equilib-rium equations, parametrized by the real number , and dened for uin some Banach or Hilbert space. We indicated the connection whichexists between the possibility of branching from an equilibrium solu-tion (0, u0) and the existence of a bounded inverse for the linear mapGu(0, u0). In this section, we will indicate how one forms the linearizedequations which govern the onset of buckling in a thin, linearly elastic,plate, i.e., we will show how to obtain the linearized equations whichcontrol initial buckling of a plate from the various sets of nonlinearvon Karman equations we have presented, in both rectilinear and po-lar coordinates, for isotropic and orthotropic response; in the course ofour discussion we will present several sets of boundary conditions forthe Airy function (equivalently, for the forces specied by the variousderivatives of the Airy function on the edge, or edges, of the plate.)

Although we may proceed with a direct discussion of the applicationof Frechet dierentiation to the von Karman equations, as a means ofgenerating the linearized equations of buckling, we note that these equa-tions may also be generated by observing that in all the cases consideredthus far, in both rectilinear and polar coordinates, the von Karman equa-tions enjoy a special structure. Therefore, suppose that, with referenceonce again to Fig. 1.11, which depicts a thin elastic plate that occupiesa region in the x, y plane, 0(x, y) is the stress function producedin the plate, under the action of applied forces on and/or speciedboundary conditions with respect to w, when the plate is not allowedto deect (i.e., 0(x, y) is associated with a state of generalized platestress in ). Suppose further that it is possible to characterize the classof possible loadings of the plate that we are interested in by a singleparameter , which one may think of as being a measure of the strengthof the applied edge forces; in this case we may set

0(x, y) = 0(x, y) (1.158)

where, in writing down (1.158), we are, clearly, thinking of the general-ized state of plane stress (that is represented by the Airy function) asdepending linearly on . We comment later on the fact that it is notalways possible to express 0(x, y) in the form (1.158).

Example: Consider the rectangular plate of length a and width b whichis depicted in Fig. 1.12. Here a compressive thrust of magnitude hb

is applied normal to the edges at x = 0, x = a, for 0 y b. If, e.g.all four edges are simply supported, and the thin plate is isotropic andlinearly elastic, then, along all four edges of the plate, w = w = 0.Referring to the von Karman equations, (1.53a,b), which apply in thiscase, with t 0, for a state of generalized plane stress (1.53a) is satisedidentically while (1.53b) reduces to

20 = 0; 0 < x < a, 0 < y < b (1.159)

subject to the boundary conditions specied above. The solution of thisplane stress boundary value problem may be taken to be

0(x, y) = hy2

2(1.160)

inasmuch as we do not care about linear and constant terms in 0 (be-cause the expressions involving in (1.53a,b) always appear as secondderivatives in the Airy function.) From (1.160) it is clear that we may

take, in accordance with (1.158), 0(x, y) = h2 y2. From (1.160) and

(1.46) we see that, N0x = h, N0y = 0, N0xy = 0.Returning to the general situation we make the following observations:

(i) In every case considered in section 1.2, with respect to the buck-ling of a thin, linearly elastic plate, either for isotropic or orthotropicresponse, and whether it be for the case of rectilinear or circular geom-etry, the structure of the von Karman equations is as follows:

L1w = [, w] (1.161a)

L2 = 12 [w,w] (1.161b)

where L1 and L2 are (usually), variable coecient, linear dierentialoperators whose precise structure is determined by th