126_stressboundryconditions

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Stress boundary conditions for plate bendingFrederic Y.M. Wan

Department of Mathematics, University of California Irvine, Irvine, CA 92697-3875, USA

Received 10 February 2003

Abstract

The determination of the appropriate boundary conditions for a two-dimensional theory of elastic at plates (andshells) consistent with the expected order of accuracy of the theory is both critical and challenging. The reciprocaltheorem of elasticity will be applied in a novel way to obtain the appropriate stress boundary conditions for platebending accurate to all order (with respect to the usual dimensionless thickness parameter) for plates of general edgegeometry and loading. Kirchhoff s two contracted stress boundary conditions are shown to be consistent with a leadingterm (thin plate) approximation theory, but the more general results obtained herein are needed for higher ordertheories. 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Stress; Boundary conditions; Plate bending

1. Introduction

Even with today s computing power, an accurate numerical solution for three-dimensional elastostaticsof plate and shell structures is often impractical and sometime infeasible. When it is practical, a relativelysimple approximate analytical solution for the same problem is often desirable as it allows us to see moreclearly how the behavior of the structure depends on the relevant design parameters. Plate and shell theorieshave been developed over the years to provide ways to obtain such approximate solutions away from theedges of thin structures. From the results of Gregory and Wan (1984, 1985a) and Lin and Wan (1988), wehave now explicit examples showing that the higher order accuracy offered by the governing differential

equations of a higher order plate or shell theory (with respect to a small dimensionless thickness parametere) may not be attained unless commensurate boundary conditions are developed and used for theseequations. These boundary conditions have now been developed for plate and shell problems with specialedge geometries or restricted loading conditions (see Gregory and Wan, 1984, 1985a,b, 1988, 1993; Gregoryet al., 1998). The present paper obtains the appropriate boundary conditions for the Levy (interior) solutionfor plate bending, and hence a two-dimensional plate theory of any order of accuracy, subject to generaladmissible edge tractions at an edge with a general contour. (The same general technique applies also toother types of admissible edge conditions as well.) The classical Kirchhoff contracted stress boundary

International Journal of Solids and Structures 40 (2003) 41074123www.elsevier.com/locate/ijsolstr

E-mail address: [email protected] (F.Y.M. Wan).

0020-7683/03/$ - see front matter

2003 Elsevier Science Ltd. All rights reserved.doi:10.1016/S0020-7683(03)00220-8
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conditions for thin plates (Love, 1944; Koiter, 1964) are shown to be consistent with the leading termasymptotic approximation of the stress boundary conditions obtained. Modications of the Kirchhoff contracted conditions by the new and more general results herein will generally be necessary for a higherorder plate theory.

2. Interior and boundary layer solution

For the purpose of developing appropriate boundary conditions for the plate bending problem, it sufficesto consider a homogeneous, isotropic, linearly elastic plate bounded by two at faces at z h, and anedge, E , spanning the cylindrical surface f f x; y 0; h 6 z 6 hg (see Fig. 1). The plate is subject to nointerior body loading and no surface tractions at the two faces so that

r zx x; y ; h r zy x; y ; h r zz x; y ; h 0 1

for all x; y inside the simple edge curve C dened by f x; y 0, which does not cross itself. Here n and t are in the direction normal and tangent to the edge curve C, respectively. The edge E itself is subject to aprescribed set of admissible tractions so that

r nn rr nn h; z ; r nt rr nt h; z ; r nz rr nz h; z ; 2

along E , where the barred quantities are the prescribed tractions along the edge E and where h is anedgewise variable along C (such as the angular variable of the polar coordinate system in the case of acircular edge). There are no restriction on the prescribed admissible tractions along E except that they giverise only to plate bending (with the in-plane stress components being odd in z and the transverse shearcomponent being even in z ). The case of plate stretching has already been analyzed by Gregory and Wan(1988).

It has been known since the work of Levy (1877) that there is an exact solution of the equations of three-dimensional elasticity theory that is traction free at the two faces of the plate. The expressions for the stressand displacement elds of the plate bending portion of this solution given in terms of a two-dimensionalbiharmonic (mid-plane transverse displacement) function w x; y can be found in Gregory and Wan(1985a). We will give here in polar coordinates only the six quantities needed in the subsequent analysis:

uIr z

1 mo

o r 1 m h2

2 m6

z 2 r 2 w 3

2h

x

y

z

n

t

z

E

Fig. 1. A simply-connected at plate.

4108 F.Y.M. Wan / International Journal of Solids and Structures 40 (2003) 41074123


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uIh z

1 m1r

o

o h1 m h2

2 m6

z 2 r 2 w 4

uI z 1 m z 2

21 mr2 w 5

r Irr Ez

1 m2o 2

o r 2

1r

o

o r

1r 2

o 2

o h2m h2

2 m6

z 2 r 2 w 6

r Ir h Ez

1 m2o

o r

1r

o

o h 1 m h2 2 m6 z 2 r 2 w 7r Irz

E 21 m2

h2 z 2o

o r r 2w 8

Here E is Young s modulus of elasticity (distinguished by the context from the same symbol used for theplate edge), mis Poisson s ratio, and

r 2w o 2wo r 2

1r

o wo r

1r 2

o 2wo h2

with r 2r 2w 0: 9

The general single-valued solution for wr ; h bounded throughout the plate s mid-plane is given by thefollowing Fourier series in the polar angle h:

wr ; h X1

n1fanr n bnr n 2cosnh cnr n d nr n 2sinnhg f a 0 b0r 2g 10

Evidently, the Levy solution reduces to the Kirchhoff thin plate (approximate) solution if terms involvingr 2w are omitted in the displacement elds and the in-plane stress elds. (These terms are of higher order inh=a compared to other terms in the same expressions, with a being a representative length of the plate spansuch as the radius of the circular contour in the case of a circular plate edge.) Similar to the Kirchhoff solution, the Levy solution s simple dependence on the thickness coordinate z makes it generally incapableof satisfying all the prescribed edge conditions along E ; hence, it is only a particular solution of theequations of elasticity. The asymptotic analysis of Friedrichs and Dressler (1961), Gol denveizer (1962), andGol denveizer and Kolos (1965) showed that (1) the complete outer (asymptotic expansion of the exact )solution for the equations of three-dimensional elasticity that is stress free on the two plate faces telescopesto become Levy s exact solution, and (2) the residual between the exact solution and the Levy solutionconsists of only boundary layer phenomena. Such boundary layer residual solution components decayexponentially away from the edge of the plate and become insignicant at a distance large compared to thethickness of the plate. We dene an elastostatic state of the plate to be a boundary layer (or a decaying ) stateif its displacement and stress elds f u; r g satisfy the condition

f u; r g O M eca=h as h ! 0 11

for some maximum modulus M of the prescribed edge tractions, and a positive constant c. (Though theresults of this paper does not depend on the actual value of c, we know from the various special cases that cis greater than unity.) An elastostatic state of the plate is said to be a regular state if its displacement andstress elds have at worst an algebraic growth (in the parameter e h=a) as h ! 0. More recently, Gregory(1992) proved that the exact solution of the plate problem is in fact the sum of the Levy solution andboundary layer residual solution components of the PapkovichFadle type:

F.Y.M. Wan / International Journal of Solids and Structures 40 (2003) 41074123 4109


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f u; r g fuI ; r Ig f ud ; r d g 12

with the superscript I indicating the fact that the elastostatic elds are signicant throughout the interior of the plate while the superscript d indicating decaying elastostatic states. If truncated asymptotic expansionsin a small thickness parameter e h=a should be used for the various elastic elds, we would indicate this by

f u; r g $ f uo ; r o g f uL ; r L g 13

where the superscripts o (for outer ) and L (for layer ) indicate truncated asymptotic expansions of theexact solution with respect to the (previously dened) parameter e. It is important to note here that f uL ; r L gis generally not smaller than O e compared to f uo ; r o g.

The combination of interior and decaying solution components is known to be capable of tting anyadmissible set of prescribed edge conditions along the plate edge E (Gregory, 1979, 1980a,b, 1992). Henceboundary value problems of the elastostatics of at plates are solved in principle. In fact, examples of suchsolutions for specic boundary value problems can be found in Gregory and Wan (1984), Lin and Wan(1993) and Gregory et al. (1997, 1998, 2001). However, for most plate geometries and edge loadings, so-lutions by eigenfunction expansions of the decaying component are also often not feasible or practical. Weare therefore forced to return to the Kirchhoff type approach of determining only the exact or asymptoticinterior solution without calculating the boundary layer solution components as well. In this context, thefollowing question arises naturally: What portion of the prescribed edge data should be assigned to theinterior solution (with the balance going to the residual decaying solution components)? Equivalently, whatconditions must the residual edge data satisfy in order for it to induce only a decaying solution state? Theanswer to either one of these two questions would enable us to determine the interior solution of the platewithout any reference to the supplementary boundary layer states which are much more difficult to obtain.

For the case of stress data prescribed along the edge E , such an assignment of edge data is usually madeby means of a modied form of Saint Venant s principle requiring that the Levy type plate theory solutionand the prescribed edge stresses have the same transverse shear resultant and (the bending and twisting)

moment resultants (Love, 1944; Timoshenko and Goodier, 1951; Reissner, 1963). We put aside for themoment the issue that the three resultant conditions are one too many for the interior solution governed bythe fourth order biharmonic equation and the appropriateness of the resolution of this predicament byKirchhoff s contraction of the three stress boundary conditions into two conditions. There are still anumber of questions and issues regarding this modied Saint Venant principle approach that need to beaddressed:

First, Saint Venant s principle is, strictly speaking, not applicable to our plate bending problem since thetypical linear span of the loaded area is not small compared to the characteristic dimension of the plate.More specically, the circumference of the loaded edge E is not small compared to a representative span of the plate. Why then should a modied form of this principle, taking resultants across thickness only, beexpected to be appropriate for our plate problem?

Second, Saint Venant s principle itself was proved only for some very restricted geometries and loadingconditions which do not include the plate bending problem (see Horgan and Knowles, 1983; Horgan, 1989).If Saint Venant s conjecture or its modied form is expected to be applicable to the plate problem, can weprove that it is in fact appropriate and deduce also the kind of accuracy (in terms of the small parameterh=a) we can expect from this approach?

Third, if the edge conditions are specied in terms of the displacement components or a mixture of stresses and displacements, Saint Venant s principle, even if it should be applicable, is not useful (since notall edge stresses are known); how do we make the proper assignment of the edge data in these cases?

Finally, we recall that the Kirchhoff contracted stress boundary conditions were originally obtained by aphysical argument. Later, Kelvin and Tait derived them mathematically by the direct method of calculus of variations on the basis of certain assumed displacement distributions across the plate thickness (Thomson



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and Tait, 1867; Love, 1944). More recently, Koiter show that the (exact) two-dimensional differentialequations of equilibrium (for both plates and shells) may be re-written, without any approximation, interms of a set of alternative stress resultant and couple variables such four of these correspond to the fourprescribed stress measures of the four Kirchhoff contracted boundary conditions. It is clear from Koiter sanalysis that, even from a strictly mathematical viewpoint, the solution of Kirchhoff s plate theory providesonly an approximation of a (more) complete two-dimensional plate theory (such as a plate theory withtransverse shear deformability, see Reissner, 1985). In relation to the exact three-dimensional elasticitysolution, do these contracted conditions in fact lead to a degree of accuracy in the approximate solutionconsistent with the accuracy expected of the Kirchhoff plate theory or, more generally, of the particular setof plate equations adopted for the analysis?

Some of these questions have been answered for special classes of problems in references cited above.They will be addressed in the following sections for the general plate bending problem considered herein.

3. Reciprocal theorem of elasticity

A general approach has been developed by Gregory and Wan (1984, 1985a, 1988) to properly assign aportion of the edge data to the interior solution to avoid a simultaneous determination of the boundarylayer solution components for thin elastic bodies. The approach is based on the reciprocal theorem of elasticity. We take the rst state in this theorem to be the exact solution of the plate bending problem anddenote it by a superscript (1) . For the second state, denoted by superscript (2) , we take a solution of theelasticity equations for the same plate domain (possibly leaving out a small neighborhood of a solutionsingularity) with no body loads, no face tractions at z h, and no edge tractions along E . With these twoelastostatic states, we apply the reciprocal theorem to a portion of our plate from the edge E inward to theedge of the a ctitious circular hole of radius q centered at a point (possibly a state 2 singularity) sufficientlyfar away from the edge E , with a minimum distance d ) h (see Fig. 2). In that case, the reciprocal relation

takes the form

Z Z E f rr nn u2n rr nt u2t rr nz u2 z gdS Z Z E q fr rr u2r r r hu2h r rz u2 z r 2rr ur r 2r h uh r 2rz u z gdS 14

Note that (i) there are no volume integrals in this relation because there are no body loads in the plateinterior; (ii) there are no surface integrals over the faces because of the traction free conditions of both

2h

n

t

z

E

x y

z

Fig. 2. A simply-connected at plate with a ctitous circular hole.



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elastostatic states on these faces; (iii) a second surface integral over E on the left vanishes because of thetraction free conditions of state (2) along E ; (iv) the superscript (1) for state (1) has been omitted on theright side to emphasize the fact state (1) is the exact solution of the original problem; and (v) the stresscomponents of state (1) along E on the left must be equal to the prescribed edge stresses and have beenwritten in terms of these prescribed quantities instead.

Except for three rigid body displacements and rotations, the only possible non-trivial state (2) for nointerior loading that is also stress free on the two faces and along the edge E must be singular somewhere inthe interior of the plate. There are four families of such singular solutions with their only singularity at thecenter of the ctitious circular hole, r 0, generated by the following singular biharmonic mid-planetransverse displacement elds by way of Levy s formulas for stresses and displacements:

wkcr ; h W kc0 r k cosk h; wksr ; h W ks0 r

k sink h 15

wk 0cr ; h W k

0c0 r

k 2 cosk h; wk 0 sr ; h W k

0 s0 r

k 2 sink h 16

where the multiplicative constants W mq0

may be chosen as appropriate factors that make the four families of generating function dimensionless. For a circular plate of radius a , we may choose W kc0 W ks0 a k andW k

0c0 W k

0 s0 ak 2. However, since our nal results do not depend on the dimension of the singular

biharmonic functions, we will set W kc0 W ks0 W k 0c

0 W k 0 s

0 1 in all subsequent developments for simplicity.For a xed k P 2, any one of these four singular biharmonic (mid-plane transverse displacement)

functions induces a singular interior solution of the Levy type. The relevant Levy s stress components bythemselves do not satisfy the traction free conditions required of state (2) along E . The singular Levysolution will have to be supplemented by additional Levy solutions of the type induced by the generalbounded transverse displacement wr ; hin the form of Eq. (10) as well as decaying solution components tomeet the free edge requirements. We denote the resulting state (2) by f R S ; U S g and the singular portion of these (2) states by f r ; ug with kc; k 0c; ks or k 0 s where k 0 k 2, depending on our choice of thegenerating singular mid-plane transverse displacement. In that case, the reciprocal relation (14) may be

written as

Z Z E f rr nn U S n rr nt U S t rr nz U S z gdS Z Z E q fr rr U S r r r hU S h r rz U S z ur RS rr uhRS r h u z RS r h gdS 17

This form of the reciprocal relation also applies to k 1 though two families of the singular solutionsin (15) corresponding k 0 are no longer singular for this case. They correspond to rigid body rotations andgive rise to no stresses throughout the plate, qualifying them for (2) states without any adjustment bysupplementary decaying states and bounded interior states. The special case of k 0 will be discussed in thenext section.

One other requirement will have to be imposed on state (2) for our method of solution. All (2) statesshould be a regular elastostatic state as dened in the previous section so that its displacement and stresselds have at worst an algebraic growth as h ! 0. We now summarize the specication of state (2) as follows:

It satises the homogeneous governing equations of three-dimensional theory of elasticity in the sameplate domain as state (1) (taken to be the solution of our original problem).

It is stress free on the two faces. It is stress free along the edge E . It has at most one singularity at a minimum distance d away from the edge E large compared to the plate

thickness. It is a regular elastostatic state with at worst an algebraic growth as h ! 0.



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With the circular hole r q far away from the edge E , i.e., h ( d , the decaying solution components of both state (1) and state (2) are exponentially small for r 6 q . Both states being also regular states, each isequal to its interior solution component (including the singular component) except for exponentially smallterms (EST). Furthermore, for a very small hole, the singular portion dominates the interior component of the (2) state. Altogether, the reciprocal relation (17) simplies to

Z Z E f rr nn U S n rr nt U S t rr nz U S z gdS Z

h

h Z 2p

0fr Irr U

r r

Ir hU

h r

Irz U

z u

Ir R

rr u

IhR

r h u

I z R

rz r q gq dh d z 18

Thus, for any of the singular (2) state corresponding to kc; k 0c; ks or k 0 s with k 1; 2; 3; . . . , the Levyinterior solution is related to prescribed stress data along E by the relation (18) except for exponentiallysmall terms. (As indicated earlier, the k 0 case will be discussed in the next section.) It might appear thatby replacing ( R S ; U S ) by (R ; U ), there would be an additional error of the order of q=d m for some m; but

this is not the case as we will show later that the relation (18) is independent of q to the singular (2) states of interest.

4. Determination of the interior solution

We now make use of the reciprocal relation (18) to determine the Fourier coefficients f ak ; bk ; ck ; d k g in theexpansion for wr ; h in (10), and hence Levy s interior solution for our problem. Let R ; U be the sin-gular interior state generated by wkcr ; h r k cosk hand r I ; uIbe the Levy interior solution componentof the exact solution of our original problem generated by the general (non-singular) biharmonic function(10). Orthogonality among the trigonometric functions eliminates all terms on the right-hand side of (18)

except those with a multiplicative factor cos k h. Since the dependence on z and h are now explicit for bothR ; U and r I ; uI, we can carry out the integration with respect to both z and h on the right-hand side of (18) giving us one relation between a k and bk . A second relation for ak and bk can be obtained by using theother singular solution wk

0cr ; h r k 2 cosk h, i.e., with k 0c. Together they determine ak and bk sincethe R S ; U S states associated with wkc and wk

0c needed on the left-hand side of (18) are known.With the help symbolic manipulation software (e.g., Maple), the rather involved calculations on the

right-hand side of (18) were carried out quickly and error free. It turns out that for kc, corresponding tothe rst generating function in (15), the reciprocal relation (18), which is exact up to EST, is independent of q , the radius of the ctitious circular hole. Furthermore, it is a relation for bk alone:

Z Z E

f rr nn U S n rr nt U S t rr nz U

S z gkc dS 8p Dk k 1bk 19

where D 2 Eh3=3l m2is the bending stiffness factor of the plate (with the factor 2/3 changed to the morefamiliar 1/12 if plate thickness is changed from 2 h to h).

The second relation obtained with k 0c, corresponding to the rst generating function in (16), is alsoindependent of q but now involved both ak and bk :

Z Z E f rr nn U S n rr nt U S t rr nz U S z gk 0c dS 8p Dk k 1 a k bk 44 mk 1h2

51 m20

The contribution of the bk term in the relation above is small of order h=d 2 and may be neglected for a

moderately thick plate theory (with an error of the same order).



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The fact that the right-hand side of the reciprocal relation (18) is independent of q for both kc andk 0c implies that the relations (18)(20) all hold for all q 6 d , in particular for q 0. For this choice of q ,the bounded portion of the Levy solution in R S ; U S is absent and the replacement of the singular state

R S ; US

in the relation (17) by

R ; U

maintains the same accuracy (which is exact up to EST) as that

when only the decaying state components are omitted.The formulas (19) and (20) and the corresponding formulas for ck and d k (obtained by setting ks and

k 0 s) completely determine the Levy solution for the original stress boundary value problem withoutsimultaneously determining the boundary layer solution components. It should be clear from the solutionprocess that the same method for nding the Fourier coefficients of the biharmonic generating function w(10) of the relevant Levy solution applies to other set of admissible edge conditions as well. The onlymodication consists of changing the edge conditions for state (2) at E to the homogeneous counterpart of the actual prescribed edge conditions. If all edge conditions of the actual problem are given in terms of thedisplacement components, then all displacement components of state (2) should vanish along E in order forthe left-hand side of the relevant reciprocal relation corresponding to (14) to be in terms of the prescribeddisplacement data and the known singular state (2). In this paper, we will focus only on the case where allthe edge conditions are prescribed in terms of the stress components.

Except for cases involving a rigid body displacement eld, the construction of a singular (2) state,R S ; US , generally requires the use of the boundary layer solution components in order for it to satisfy thestress free conditions along E . However, such a singular state (2) is a canonical problem to be solved onlyonce and the result applies to all stress boundary value problems for the same plate whatever the actualprescribed edge stress distributions may be. It can be generated numerically if necessary as long as thenumerical solution is accurate to the same order of accuracy as the plate theory, i.e., the truncated outerasymptotic solution, used. As we shall see in the next two sections, it is possible, for a few classes of stressboundary value problems, to circumvent the the inclusion of the boundary layer solution components inR S ; US either completely (as in Section 5) or for a (Kirchhoff type) thin plate approximation (see Section 6).

5. Axisymmetric deformation of a circular plate

For solutions single-valued in h, the two singular biharmonic functions for mid-plane transverse dis-placement for the k 0 case are

w0r lnr ; w00r r 2 lnr 21

while the corresponding generating function for the general non-singular axisymmetric Levy interiorsolution wr is as given by the n 0 portion of Eq. (10):

w0 a0 b0r 2 22

Note that the terms associated with the Fourier coefficient a0 corresponds to a rigid body vertical trans-lation and therefore gives rise to no stresses throughout the plate. A unit vertical translation may be used asa (2) state in the reciprocal theorem as it satises all the conditions specifying a (2) state in Section 3. Forsuch a (2) state, the reciprocal relation (18) for the case of a circular edge of radius a simplies to

Z 2p

0 Z h

hf rr rs 1ga d z dh Z

h

h Z 2p

0f r Irz 1gr q q dh d z 23

or, since r Irz 0 for the Levy solution generated by Eq. (22),

2p a Z h

hrr rz d z 2p aQ r 0 24



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The condition (24) is just the requirement that the prescribed transverse shear stress distribution along theplate edge must have no resultant axial force so that the plate is in overall equilibrium. Otherwise, there isno other applied load to balance the axial force.

Parenthetically, we note that a non-self-equilibrating distribution of rr rz that generates a resultant axialforce must be balanced by an equal and opposite axial force in the interior of the plate (such as that due to adistribution of axisymmetric transverse edge shear at an inner edge or a point force at the center of thecircular plate). For the case of a concentrated load at the origin, the Levy solution for w0 now includes asingular term b00 lnr . The rigid body displacement (2) state in (23) now determines the Fourier coefficientb00:

b00 aQ r 4 D

25

Returning to the original problem with a bounded interior solution generated by (22), another (2) statethat determines b0 by way of (18) is obtained with the rst generating function w0r lnr in (21). For this

singular generating function, the relevant stress and displacement components of the Levy solution are

RS rr E z a

1 a 2

r 2; RS r h R

S rz 0 26

U S r z 1h mr a

1 mar i; U S h 0 27

U S r 12

r 2

a1 m 1 ma ln

ar m

z 2

r 28

With this (2) state, the reciprocal relation (18) for a circular edge of radius a becomes

2 Db0 Z h

h z rr rr

m z 2

2arr rz d z 29

As with the k > 0 cases, we have now two exact R S ; U S states for the axisymmetric bending problem.Both of them do not involve any boundary layer solution components. However, they determine only theFourier coefficient b0 but not the rigid body transverse displacement component a 0. Such a level of non-uniqueness is expected when all edge data are prescribed in terms of stresses. In addition to determining b0,we also recovered the overall equilibrium requirement.

With the following sole non-trivial stress component along the edge r a generated by the axisymmetricmid-plane transverse displacement (22),

r Irr

2 Ez

1 mb0 30

the left-hand side of (29) is just the axisymmetric bending moment resultant (also referred to as the bendingstress couple ) M rr (conventionally dened as z r Irr integrated across the plate thickness) so that the relation(29) may be written as a boundary condition on the bending moment resultant at the plate edge:

M rr r a Z h

h z rr rr

m z 2

2arr rz d z 31

An equivalent form of the boundary condition (31), expressed as a necessary condition for the residualboundary data to induce a boundary layer solution state, was previously obtained in Gregory and Wan(1985a).



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The two stress boundary conditions (24) and (31) for plate bending resulting from our use of the re-ciprocal theorem are accurate up to exponentially small terms (because we have replaced f r rr ; r rz g byf r Irr ; r

Irz g). It is signicant then that the condition (31) on the plate bending moment resultant is not

identical to the conventional condition upon the application of (a modied form of) the Saint Venantprinciple (Love, 1944; Timoshenko and Goodier, 1951; Reissner, 1963). (Note that condition (31) coincideswith that obtained by Reiss (1962) if we take rr rz 0 as Reiss did in his paper. However, rr rz needs not vanishidentically even if it should be self-equilibrating.) At the same time, deviation from the conventionalcondition consists only of the new term m z 2=2arr rz ; its contribution to the boundary condition is gene-rally small of order h=a compared to the rst term. In other words, the conventional stress boundaryconditions for plate bending is consistent with the Kirchhoff thin plate approximation. But if we wish toretain the additional accuracy of any higher order plate theory, the condition (31) should be used.

Counter-examples have also been constructed by Gregory and Wan (1985a) to illustrate the appropri-ateness of condition (31) for higher order plate theories. It was shown there that certain three-dimensionaldecaying elastostatic states for the axisymmetric plate bending problem are consistent with (31) but notwith the conventional bending moment condition (corresponding to (31) without the second term). It wasalso shown that the residual solution induced by the residual data after an assignment by (31) for the Levysolution is a decaying state while a similar solution induced by the residual data after an assignment by theconventional bending moment condition is not.

6. Circular plate with unsymmetric edge stresses

For plate bending without axisymmetry, it is generally not possible to avoid the inclusion of decayingsolution components in the determination of the four relevant (2) states. An important exception is the caseof stress edge conditions in the Kirchhoff thin plate approximation of the Levy interior solution. To elu-cidate, it suffices to consider a circular plate with edge stresses along the circular boundary depending on

the edge variable h in the form of cos k hor sin k h for k P 2 so that they are self-equilibrating. The plateresponse to the edge loads will also have the same dependence on h. For the (2) state generated by the rstsingular mid-plane displacement of (15), the following three stress components are required to vanish atr a :

r kcrr cosk h

Ez

1 m2b sk k 1hk 2 mk 2ir

k 4k k 2 1r k 2 h2 2 m

6z 2

a sk k k 11 mr k 2 k k 11 mr k 2 rr kcrr r ; z 32

r kcr hsin

k h

Ez

1

mb sk k k 1 r

k 4k 1

1

mr k 2 h2

2 m6

z 2

a sk k k 1r k 2 k k 1r k 2 rr kcr hr ; z 33

r kcrz cosk h

2 E

1 m2h2 z 2fb sk k k 1r

k 1g rr kcrz r ; z 34

where rr kcij r ; z is the r ; z -dependent portion of the boundary layer components of r kcrz . The correspondingdisplacement components will be needed for the evaluation of the boundary integrals (19) and (20) for a k and bk . For our purpose, we will write these expressions in terms of the relevant mid-plane transversedisplacement wkc:



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wkc f a sk r k b sk r

k 2 r k g cosk h 35

with

ukc z 1 m z 2

21 mr 2 wkc uukc z cosk h

b sk r k 2

2m z 2

1 mk 1r k a sk r

k r k uukc z r ; z cosk h 36

ukcr z

1 mo

o r 1 m h2

2 m6

z 2 r 2 wkc uukcr cosk h

z b sk k 2r k 1

4k k 11 m

r k 1 h2 2 m

6z 2 a sk kr

k 1 kr k 1 uukcr r ; z cosk h

37

ukch z

1 m1r

o

o h1 m h2 2 m

6z 2 r 2 wkc uukch sink h

z b sk kr k 1

4k k 11 m

r k 1 h2 2 m

6z 2 a sk kr

k 1 kr k 1 uukch r ; z sink h 38

We know from the asymptotic analyses of Friedrichs and Dressler (1961), Gol denveizer (1962), Reiss(1962), Gol denveizer and Kolos (1965) and others that the decaying components of the above state (2)displacement elds, f uukc z cosk h; uukcr cosk h; uukch sink hg, are at least O h=a smaller in magnitude than theleading term of their interior counterpart f wkc; z o wkc=o r ; z =r o wkc=o hg. Roughly, because state (2) isstress free along E with r kcry a ; h; z 0, for y r ; h, and z , the decaying components f rr kcrr ; rr kcr h; rr kcrz g in (32) (34) are of the same order of magnitude as their interior counterpart. Now stresses are obtained by dif-

ferentiating the displacement elds and differentiation does not change the order of magnitude of theinterior solution components but increases the magnitude of the decaying components by a factor pro-portional to a=h. Therefore, we have except for terms of the order of h=a ,

ukc z $ wkc f a sk r

k b sk r k 2 r k g cosk h 39

ukcr $ z o

o r wkc z f a sk kr

k 1 b sk k 2r k 1 kr k 1g cosk h 40

ukch $ z r

o

o r wkc kz f a sk 0r

k 1 b sk 0r k 1 r k 1g sink h 41

Upon substituting these asymptotic expressions into (19), we obtained the following leading term ap-

proximation for the Fourier coefficient bk of the Levy solution of our original stress boundary valueproblem:

8p Dk k 1bk Z 2p

0 Z h

hrr rr U kcr rr r hU

kch rr rz U

kc z r a a d z dh

$ Z 2p

0 M rr

o wkc

o r M r h

1r

o wkc

o h Qr w

kc

r aa dh 42

Keeping in mind that the prescribed stresses are proportional to cos k h or sin k h for k P 2,

f rr rr h; z ; rr rz h; z g frr k rr z ; rrk rz z gcosk h; rr r hh; z rr

k r h z sinh 43



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we can integrate the right-hand side of the expression for bk to get

bk $a

8 Dk k 1M

k rr

dwwkc

dr " V

k r ww

kc

#r a44

where

wkcr ; h wwkcr cosk h; V r Qr 1r

o M r ho h

45

f M k rr ; M

k r hg Z

h

hf rr k rr z ; rr

k r h z g z d z ; Q

k r Z

h

hrr k rz z d z 46

Similarly, we have from the rst generating function in (16)

uk 0c

z $ wk 0c f a sk 0r

k b sk 0r k 2 r k 2g cosk h 47

uk 0c

r $ z o

o r wk

0c z f a sk 0kr k 1 b sk 0k 2r

k 1 k 2r k 1g cosk h 48

uk 0c

h $ z r

o

o hwk

0c kz f a sk 0r k 1 b sk 0r

k 1 r k 1g sink h 49

To a leading term asymptotic approximation, we get from (20)

a k $a

8 Dk k 1V

k r ww

k 0c24 M

k rr

dwwk 0c

dr 35r a50

where we have omitted the bk term on the left-hand side of (20) because it is O h2

=a2

compared to thedominant terms in the same equation. With a k and bk given by (44) and (50), we have now the interiorstresses and displacements in terms of the prescribed edge loads at least to a leading term approximation(equivalent to a Kirchhoff type theory). This was accomplished without doing any calculations that involvethe decaying components of the exact solution.

7. Kirchhoffs contracted stress boundary conditions

The leading term of the Levy solution for our plate problem is now seen from the expression for theFourier coefficients ak and bk , (44) and (50), to depend only on the bending moment resultant M rr andthe effective transverse shear resultant V r (see (45)) just as the solution of Kirchhoff

s thin plate theory. Atthe same time, the contraction of the prescribed stresses of the problem to two resultant quantities asproposed by Kirchhoff occurs naturally in the solution process. However, we cannot yet conclude that thestress boundary conditions for a leading term Levy solution are in fact the same as those in the conven-tional thin plate theory.

Suppose we compute from the Levy solution the corresponding moment resultants and transverse shearresultant which appear in conventional two-dimensional plate theories. For a leading term asymptoticapproximation , we have

M rr Z h

hr Irr z d z $ Df a k k k 11 mr

k 2 bk k 1k 2 mk 2 r k g cosk h 51



13/17

M r h Z h

hr Ir h z d z $ D1 mfak k k 1r

k 2 bk k k 1r k g sink h 52

Q r Z h

hr Irz d z $ 4 Dbk k k 1r k 1 cosk h 53

We now use (44) and (50) to express the Fourier coefficients ak and bk of the interior solution of our originalproblem in terms of the prescribed boundary data to get to a leading term approximation

M rr a ; ha k 1 cosk h=8k

M k rr k 124

mdwwk

0c

dr fk 2 mk 2ga2

dwwkc

dr 35r a V

k r k 1 mww

k 0c fk 2 mk 2ga 2wwkc r a 54

V r a ; ha k 2 cosk h=8 V

k

r k 1 mwwk 0c

4 k k ma2

wwkc

r a

M k rr k 124

mdwwk

0c

dr 4 k k ma 2

dwwkc

dr 35r a55

In general, the value of the bending moment resultant and the transverse shear resultant at the edge r aof the circular plate would depend on both M

k rr and V

k r . In order to have these relations identical to the two

Kirchhoff contracted stress boundary conditions, we need to have

k 1 mwwk 0c fk 2 mk 2ga2wwkc r a 0 56

k 124 mdwwk

0c

dr 4 k k ma 2 dwwkc

dr 35r a 0 57

k 124 m

dwwk 0c

dr fk 2 mk 2ga2

dwwkc

dr 35r a 8ka1k 58

k 1 mwwk 0c 4 k k ma2wwkc r a 8a

2k 59

with wwkc and wwk 0c given in (35) and (47), respectively. To evaluate the left-hand side of these four relations,

we need to know the coefficients f a sk ; b sk ; a sk 0; b sk 0g in wwkc and wwk 0c (see (35) and (47)). Unfortunately, the

determination of these coefficients is generally done simultaneously with the unknown coefficients in theeigenfunction expansions for the decaying solution components by the stress free conditions

r kcrr a ; h; z rkcr ha ; h; z r

kcrz a ; h; z 0 60

r k 0c

rr a ; h; z rk 0cr h a ; h; z r

k 0crz a ; h; z 0 61

Given the completeness of the eigenfunctions associated with the decaying state components, we mayassign certain values to a sk and b

sk for instance and then choose the Fourier coefficients in the decaying

solution components to satisfy (60). However, the assigned values must be such that they do not violate therelative magnitude of the various stress components (found in various asymptotic analyses such as those



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referenced earlier). For example, it might appear reasonable to take a sk and b sk to satisfy r

kcrr a ; h; z

r kcr ha ; h; z 0 to leading order so that

r kcrr a ; h; z cosk h $

Ez 1 m2 f b

s

k b k ak

a s

k k k 1ak 2

k k 1ak 2

g 0 62

r kcr ha ; h; z sink h

$Ez

1 mf b sk k k 1a

k a sk k k 1ak 2 k k 1a k 2g 0 63

where b k k 1k 2 mk 2=1 m, leaving the decaying components to absorb the residuals inr kcrr a ; h; z and r kcr ha ; h; z and to ensure r kcrz a ; h; z 0. But this would make rr kcrr and rr kcr h of order h

3=a3 whilerr kcrz remains O h

2=a2. This relative order of magnitude between the decaying components of both thetransverse shear stress and the in-plane stresses is not consistent with the requirement of the differentialequations of equilibrium and is therefore unacceptable. An acceptable assignment would be to require thetwo Kirchhoff contracted resultants to vanish (with the decaying components absorbing the residuals). The

two conditions M kcrr a ; h V

kcr a ; h 0 require:

aa sk k k 1ak 2 bb sk

k 11 m

k 2 mk 2ak k k 1ak 0 64

aa sk k k 1ak 2 bb sk k 14 k 1 ma

k k k 1ak 0 65

giving

a sk $ aa sk

1 m3 m

k 1a2k ; b sk $ bb sk

1 m3 m

ka2k 2 66

and therewith

wwkc 43 m

a k ; ddr

wwkc 4k 3 m

a k 1: 67

Similarly, we have from M k 0c

rr a ; h V k 0c

r a ; h 0

aa sk 0k k 1ak 2 bb sk 0

k 11 m

k 2 mk 2ak k k 1ak 0 68

aa sk 0k k 1ak 2 bb sk 0k 14 k 1 ma

k k k 1ak 0 69

requiring

a sk 0 $ aa

sk 0

k 21 m2 81 mk 3 m1 m a

2k 2; b

sk 0 $ bb

sk 0

1 m3 mk 1a

2k 70

and therewith

wwk 0c

4k 1 m 21 mk 3 m1 m

ak 2;dwwk

0c

dr

4k 1 m 43 m1 m

ak 1 71

Note that our choice of f a sk ; b sk ; a sk 0; b sk 0g by way of (64)(69) does not alter the fact that RS rr , R

S r h and R

S rz (for S

corresponding to both k and k 0) all vanish at the plate edge as required by the specication of a (2) state; thisis ensured by an appropriate choice of the decaying solution components for these singular states. Hencethere is no new error accrued for this step in our method.



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It is then straightforward to verify that the four desired relations (56)(59) are in fact satised identically,and the conventional Kirchhoff contracted stress boundary conditions in fact apply to the leading termLevy interior solution:

M rr a ; h M k rr cosk h; V r a ; h V k r cosk h 72We see then that the Kirchhoff contracted boundary conditions are in fact consistent with a leading termouter solution of the plate bending problem (known more conventionally as the Kirchhoff thin platetheory), at least for a circular plate with edge stresses proportional to cos k h and sin k h for an integer k (with the special cases of k 0 and k 1 requiring special treatment described earlier). The correspondingresults for a circular plate with general edge loads follow immediately after a Fourier decomposition.However, we learned in Section 5 that the conventionally accepted stress boundary conditions for (theKirchhoff theory of) plate bending may not be adequate for higher order plate theories. In the developmentabove leading to the contracted boundary conditions (72), terms of order h=a have been omitted repeatedly,starting with the replacement of the exact singular (2) states f U kc z ; U kcr ; U kch g by f wkc; o wkc=o r ; r 1o wkc=o hg in(42). Hence, higher order accuracy in h=a offered by higher order plate equations may well be lost if thesame contracted stress boundary conditions should be used for these higher order equations.

8. Concluding remarks

While the validity of the Kirchhoff contracted stress boundary conditions for the leading term (thin)plate theory has been established only for circular plate, much of the analysis above carries over to plateswith an arbitrary (but simple) smooth edge and plates with more than one edge. An asymptotic expressionfor bk similar to (42) but for a general smooth edge can be obtained in a similar development:

8p Dk k 1bk

Z Z E

rr nn U kcn rr nt U kct rr nz U

kc z dS

$ Z C M nn o wkc

o n M nt

o wkc

o swkc Qnw

kc d s 73

where o =o n and o =o s are differentiation in direction normal and tangent to the edge curve C, respectively.Upon integration by parts and assuming all stress and displacement components are single-valued, we get

8p Dk k 1bk $ I C M nn o wkc

o n V nwkc d s 74

where

f M nn ; M nt g

Z h

h

f rr nn s; z ; rr nt s; z g z d z ; Qn

Z h

h

rr nz s; z d z

with the effective transverse shear resultant V n is given by V n Qn o =o s M nt . A corresponding expres-sion for ak is

8p Dk k 1a k $ I C " M nn o wk 0c

o n V nwk

0c#d s 75Even without going beyond the expressions for ak and bk , it should be evident from (75) and (74) that theleading term Levy (or Kirchhoff thin plate theory) solution is effectively determined by the bending momentresultant M nn and the effective transverse shear resultant V n of the three prescribed edge stress componentsf rr nn ; rr nt ; rr nz g, only these two edge resultant measures and no others. At the same time, the Kirchhoff



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contracted stress boundary conditions are not likely to be adequate for higher order plate theories, whilethe conditions (19) and (20) are always applicable. The possible inadequacy of the Kirchhoff contractedconditions for higher order interior (or outer) solutions has been known since the asymptotic results of Friedrichs and Dressler (1961) and Gol denveizer (1962). However, unlike these and similar subsequentasymptotic analyses, the solutions (19) and (20) and more generally the approach initiated in Gregory andWan (1985a) make it possible to avoid solving boundary value problems associated with the boundary layercomponents of the exact solution.

Acknowledgements

The author thanks Arthur Leissa for more than 40 years of continual friendship and generous supportsince they rst met through the Haystack Antenna project. He congratulates Arthur for a distinguish careerin the eld of mechanics and wishes him all the best and many more productive years of scholarly activities.

References

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equations of the theory of elasticity. P.M.M. 26, 668686.Gol denveizer, A.L., Kolos, A.V., 1965. On the derivation of two-dimensional equations in the theory of thin elastic plates. P.M.M. 29,

141155.Gregory, R.D., 1979. Green s functions, bi-linear forms, and completeness of the eigenfunctions for the elastostatic strip and wedge.

J. Elasticity 9, 283309.Gregory, R.D., 1980a. The semi-innite strip x P 0, 1 6 y 6 1, completeness of the PapkovichFadle eigenfunctions when / xx0; y ,

/ yy 0; y are prescribed. J. Elasticity 10, 5780.Gregory, R.D., 1980b. The traction boundary value problem for the elastostatic semi-innite strip, existence of solutions, and

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Structures. Elsevier Science, Amsterdam, pp. 3554.Gregory, R.D., Wan, F.Y.M., 1988. On the interior solution for linear problems of elastic plates. J. Appl. Mech. ASME 55, 551559.Gregory, R.D., Wan, F.Y.M., 1993. Correct asymptotic theories for the axisymmetric deformation of thin and moderately thick

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Appl. Math. 100, 6794.Horgan, C.O., 1989. Recent developments concerning Saint-Venant s principle: an update. Appl. Mech. Rev. ASME 42, 295303.Horgan, C.O., Knowles, J.K., 1983. Recent developments concerning Saint Venant s principle. In: Wu, T.Y., Hutchinson, J.W. (Eds.),

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