generalized displacements and the accuracy of classical plate theory

8/3/2019 Generalized Displacements and the Accuracy of Classical Plate Theory

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hf. 3. Solids Structures Vol. 30, No. I, pp. 129-136, 1993Printed in Great Britain.

002&7683 /93 $X00+ .I00 1992 Per&mm Press Ltd

GENERALIZED DISPLACEMENTS AND THEACCURACY OF CLASSICAL PLATE THEORY

Z. RYCHTER~Department of Applied Mathematics, University of Virginia, Thornton Hall,Charlottesville, VA 22903, U.S.A.

(Received 8 October 1991)Abstract-This paper investigates, by means of statically/kinematically admissible constructionsand the hypersphere theorem, the validity of classical (Kirchhoff) plate theory as an approximationto linear elasticity theory. Two levels of accuracy are shown to lead to two entirely differentinterpretations of admissible generalized displacements of classical plate theory. At lower accuracy,a wealth of admissible interpretations is disclosed, leading to extreme flexibility in the modeling ofgeometric boundary conditions. At higher accuracy, that flexibility is lost and only novel generalizeddisplacements, describing lateral deflection and rotations of the faces of the plate, are demonstratedto be admissible. To this end, a refined Kirchhoff hypothesis is formulated which avoids all of thenotorious inconsistencies of the standard Kirchhoff hypothesis. At both levels of accuracy, the effectof large transverse-shear deformability is exposed, to encompass anisotropic and composite plates.

1. INTRODUCTIONThe classical (Kirchhoff) plate theory (CPT) has been the subject of theoretical and practicalinterest for over a century. Its mathematical structure is well understood. Its derivation bydescent from 3-D elasticity theory is achieved either variationally or asymptotically, in thelimit of vanishing plate thickness. However, in real plates, withfinite thickness, the potentialof CPT as a 2-D theory for furnishing adequate 3-D displacement and stress distributionsis, in our view, not fully explored and understood. We plan to show that, because of itsapproximate character, CPT may be interpreted in a variety of ways, depending on theaccuracy desired. To prove this point, we start from the 2-D equations of CPT and construct3-D statically and kinematically admissible solutions. The error of these solutionswith respect to (unknown) elasticity solutions is then evaluated in an energy norm, on thebasis of the hypersphere theorem of Prager and Synge (1947). This approach was firstapplied to CPT by Nordgren (197 1) who found the error to be 0 (h), 2h being the thicknessof the plate. Then Simmonds (1971) and Nordgren (1972) reduced the error to 0(h2).This work demonstrates that each of the two estimates, U(h), 0(/z), is valid in its ownright : their validity domains overlap but are not identical. These domains are found to bedefined by the range of admissible interpretations of generalized displacements-transversedisplacement and cross-section rotations-of CPT. We show that a decrease in the errorof CPT tightens the admissibility range. In particular, at the 0 (h) level of error, we generalizeNordgrens (1971) 3-D kinematically admissible displacement field, based on the so-calledmodified Kirchhoff hypothesis due to Koiter (1970), to a one-parameter family of fields.Here, an infinite number of interpretations for generalized displacements are found tobe admissible, including all known versions such as midplane or average (through-the-thickness) displacement and rotations. Accordingly, geometric boundary conditions canbe modeled in a great variety of ways in that (low accuracy) version of CPT.

At the 0(h2) level of error, most of the generalized displacements from the previouslevel become inadmissible. Specifically, both midplane-associated (classical) and averagedisplacements and rotations are left out. At this level, we use a refined Kirchhoff hypothesisto get rid of all the notorious contradictions of the conventional (and modified) Kirchhoffhypothesis, so that transverse shear and normal strains are accounted for. The refinedhypothesis selects displacements and rotations of the top and bottom surfaces of the plate

t On leave from the Department of Architecture, Technical University of Bialystok, Krakowska 9, 1S-875Bialystok, Poland.129


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130 Z. RYCHTEKas the only admissible (pointwise) generalized displacements, an apparently physicallysound and novel choice for CPT.

In all our developments we investigate and expose the effect of increased transverse-shear deformability, which is frequent in anisotropic and composite plates. This effect isshown to impair the accuracy of 3-D solutions based on CPT, being stronger in the 0(/r)than the 0(h) region.

3. ELASTICITY PROBLEM STATEMENTWithin linear elasticity, consider a plate of constant thickness, which is in static

equilibrium without body forces.~,/l,[i + oxi. = 0. (J13.x (733.3 = 0, (1)

where a(~, z) is the stress tensor, x denotes arbitrary in-plane coordinates, z zz x3 measuresdistance from the middle plane (-_ = 0), Greek indices range over 1, 2 and are summedwhen repeated, ( ),5L a( )/ax and ( ).3 = 8( )/2x3. The constitutive equations, for ahomogeneous, elastic, anisotropic material having symmetry with respect to planes z = con-stant, read

where u(x, Z) is the displacement vector, E the elasticity tensor. As a standard preparationfor using plane stress approximation. the transverse normal strain u~,~ has been eliminatedfrom eqn (2a), giving rise to two tensors :

The top and bottom surfaces share in equal parts a distributed normal load p(x),without tangential tractions,

cJu3= 0, (Tag = +(1/2)p for: = ih, (4)2h being the thickness of the plate. Part S, of the cylindrical edge surface S is subject to thearbitrary forces f,*(s, z), fT(s, z), odd and even in I,

where s runs along the edge of the midplane and n, is a unit normal to S. The remainder,S,, of S is displaced by u,Ys, z), odd in 3, and uT(s, 2). even.

*21, = 21,. 213 = UT on S,,. (6)The solution to eqns (l)-(6) is split into (oYB,033, u,), which fields are odd in Z, and

(CC&u3), which are even. We will seek two-fold approximate solutions : a statically admis-sible stress field 8(x, z), which fulfills only eqns (1), (4), (5)-which do not involvedisplacements, and a kinematically admissible displacement ti(x, z), which conforms to thegeometric boundary conditions (6) and produces, through eqns (2), the stress field I?(G).The distance of (5, ti) from the (unknown) exact solution (a, u) is bounded by the inequalities

which follow from the familiar hypersphere theorem due to Prager and Synge (1947). Thenorms 116 I and lull= IlO) II are based on the positive-definite energy functional


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Accuracy of classical plate theory 131

(A,wwn,, +2&~33%,#33 +4&3~3~3q3 +~43333~33~33) @dz, (8)

where A is the compliance tensor (inverse of E) and F is the middle plane.To concisely expose the effect of transverse-shear deformability, the material tensorsare scaled as

(Earm, E3333, Eg33, &,,) = E(&q, ~93333, &33,&J, (W(A AI% 33339&33) = (l/E)(~am A33339 &I,,), (9b)

E 03~3 = G&3/139 A.3,73 = (l/G)&,,,, PC)

where E, G are chosen to make the barred tensors dimensionless and G( 1). One may thinkof E and G as a generalized in-plane Youngs modulus and generalized transverse-shearmodulus. In well-designed anisotropic and composite plates, typically E >> G.

3. CLASSICAL PLATE THEORYClassical plate theory consists of the gross equilibrium equations

M a,~ = Qe, Q.,a = -P, (10)the constitutive relations

the static boundary conditionsi&n.ns = M*, GKp.f,d,d~ + eon. = H* on so, (12)

and the kinematic boundary conditions*w=w, w,,n,= r* on s,. (13)

M*(s) and H*(s) are a prescribed bending moment and generalized transverse force on theedge,

sh

H*(s) = _h {@elf%, z)l$n +ff(s, z)} dz, (14)

w*(s) and r*(s) are prescribed generalized or gross measures of transverse displacementand rotation about the tangent to the edge,h

w*(s) = I hW3(z)uf(s, z) dz, r*(s) = s W)n&(.r, z) dz,-h -h (15)where s, and s, denote the intersections of S, and S, with the edge of the middle plane, tlstands for a unit tangent to the edge, W,(z) is an even function, and W(z) is an odd function.Equations (14) and (15) reduce the edge data of the 3-D problem of elasticity to the edgedata of CPT. A similar reduction of 3-D statically admissible stresses 5 and kinematicallyadmissible displacements a to 2-D generalized forces A&,,+Qol nd generalized displacementsw, w,~ s postulated in the interior of the plate :


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1 3 2 Z. RYCHTER

Definitions (14) and (16) of the static quantities of CPT are standard and unique. Bycontrast, eqns (15) and (17) describe the kinematic quantities in as general terms as possible.Owing to the arbitrary weight functions W,(z), W(Z), the generalized displacements of CPTmay be given diverse interpretations which offer, importantly, diverse models of geometricedge constraints. Thus, unlike most works on CPT, we do not preussiyrz any particularinterpretation of the generalized displacements ; rather, admissible interpretations will reslrltin Sections 4 and 5 from error analyses of CPT as an approximation to elasticity theory.

4. FAMILY OF LOWER ACCURACY 3-D SOLUTIONS

Our task is to construct Z[M;(Z?), z], ti[w(x), -r] and investigate the relationship betweenthe error i/5-&(&) /( of the 3-D approximate solutions 15, i and the range of associatedinterpretations of 2-D generalized displacements u, br,, of CPT.

We use a standard [see e.g. Nordgren (1971)], statically admissible stress field. with[ = z/h.

&,i = (31/2h)M,,, 6x1 = (3/4h)(l -i>Qa, 63, = (l/4)(3 ga3 >> 6, ?. Thus, the simplestreasonable displacement ti is one that produces a stress 8,, close to eXB nd larger than tiXj.IcrJ3 the details of d,, and e.33 distributions across the thickness are immaterial. Expectingso little of ti, we may find a one-parameter family of conforming displacements fields

where c is a parameter, and

Substitution of eqns (20) and (21) into eqns (2) and use of eqn (11) gives

Subtracting eqns (22) from eqns (18) we find(22)

(23a, b)(23c)


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Accuracy of classical plate theory 133At this point we restrict the parameter c to 0 < c < 1; otherwise, the right-hand sideof eqn (23~) could become arbitrarily large. Thanks to eqns (IO), (11) and (21) all x1-

dependent fields in eqns (1 S), (22) and (23) can be expressed through the single variablew(x) of CPT. Performing this, introducing the results into eqns (7), using eqns (9), andleaving only lowest-order terms in h, yields

(IIs-a ll/ Iltll, Ile -ulllll~ll) G WG) 2W+W2), (24)where L is a constant, having the dimension of length, that captures (integrally) thedependence of eqn (24) on w; we will refer to L as a global characteristic wavelength ofCPT. An explicit formula for L is complex and unnecessary since the relative errorsestimated parametrically in eqn (24) are directly computable from eqns (7) once w (and so8, S) is known.Introducing eqn (20b) into (17a) gives

s sw,(C)4 = 1, w,([)(p-c)=O, o


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134 Z. RVCHTEK

-h lV(;)[d; = 1. (30)

Equation (30) is satisfied by an infinity of weight functions W(i) ; this leads via eqn (17b)to infinitely many measures II,, of cross-section rotation. Two familiar examples are

where it,, represents local rotation at the midsurface (classical choice) and average rotation,respectively. The local interpretation (31a) is easily extended to a family of local rotationsat two equidistant planes i = + [,,,

In another family,

H.,,(_?) = -(l/2hio)[il,(.v.i,,)-li,(x. -[,,I, 0 < (0 6 1. (33)rotation is measured as the difference between in-plane displacements of two equidistantplanes i = kc,. The special case involving top/bottom surfaces (i,, = + 1) was discussedby Rehfield and Murthy (1982) in the context of refined beam theory.

The findings of this section are novel in several respects. First, we have come out witha one-parameter family of 3-D displacement fields in eqns (20). thus generalizing the so-called modified Kirchhoff hypothesis (corresponding to c = 0) due to Koiter (1970) andNordgren (1971). Consequently, our error estimate (24) generalizes Nordgrens (1971) similarresult to all possible versions of CPT which differ from each other by their generalizeddisplacements. The range of admissible interpretations for these displacements has beenidentified. We have shown that one is offered an extreme flexibility in the choice of gener-alized displacements, with several admissible families of transverse displacements Mandrotations it,,. Representatives of these families can enter each specific pair (it, IV,,) in anycombination, thus providing diverse models of geometric boundary conditions. Note thatwhile M,, s numerically related to vv as its derivative, physically they need not be linked.One can go far beyond the historically first, and most popular to this day, selection of IVand )L,,as midplane displacement and rotation. For example, u may measure top surfacedeflection and M,,midsurface rotation or M?may be taken to be an average displacementand kc,, a pointwise rotation. In a concrete situation, one should select an interpretationthat best fits the geometric edge constraints at hand. This is the most one can expect ofCPT which, as a 2-D theory, cannot in general fulfill 3-D elasticity geometric boundaryconditions (6) at all points across the thickness.

5. HIGHER ACCURACY 3-D SOLUTIONS

Equations (18), (19), (22) and (23) indicate that for better accuracy it is necessary toimprove on GIj, making it close to cc3. This is achieved by taking

il, = -gln~.~+(~"-3~)b,., lij = M.+(p-l)g. (34)where

g = (hZ/2)C,pw .,p, b, = - (hP1g.z -&B~QB. (35)

Introduction of eqns (34) and (35) into (2), use of eqn (11) and of the familiar relation


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Accuracy of classical plate theory~&B~A,~~, = 41,

135

where 6,I is the Kronecker delta, produces1amp (35/2h2)M,,+(53-31)(D,8a,+E,3,,C,BCa,)ba,?,

833 = (i-X)E3333G&,p,1on3 = (VW -C2>Qw

W-4Wb)

(36~)Subtraction of eqns (36) from eqns (18) yields

Q-& = (35-r3)(D,pa~+E3333CorBCa~)ba,q,533-633 = (311-13)~14+E3333C,,b,B),

e 1fla3 -0or3 - 0.

(374(37b)

(37c)The 3-D stresses in eqns (18), (36) and (37) can be expressed in terms of a 2-D solution wof CPT by means of eqns (lo), (11) and (35). Performing this, substituting the results intoeqns (7), using eqns (9), and leaving only lowest-order terms in h gives the improved errorestimate for CPT :

(II&-~ll/ll~ll, Ilfi-ull/ll~ll) d @/W21~:+W3), (38)where L, is a refined global wavelength of CPT ; it has a similar meaningon w differently.

to L but dependsEquations (17) and (34) yield

sI

sI

w,(Odi = 1, w3(C)(i- 1) di = 0,-1 -1

(39)

(40)I I

-h s OCdt = 1, s,J(M3 - 30 di = 0.-1 -1The weight functions W,, W which fit eqns (39) and (40) determine admissible interpret-ations of generalized displacements. Looking for pointwise interpretations, we find

W(X) = (1/2)[ti3(X1, 1)+53(X, -I)],

%(Xa) = - (1/2)[&.3(X, 1) +%,3(X, - I)].

(414

(41b)

Equations (41) describe wand w,, as average top/bottom surface displacement and rotations.Such variables are physically sound because edge constraints are often imposed on thefaces. Apparently, this type of generalized displacements have never been used in associationwith CPT [in a higher-order theory of plates, Jemielita (1975) used the same w as we do,but in association with an average rotation]. All familiar interpretations of w and w,, failto comply with eqns (39) and (40) and are not admissible, to name: middle plane dis-placement and rotations (classical, Kirchhoffs choice), average displacement and rotations,or weighted-average displacement of the Reissner type. In that respect this report completesearlier papers on the accuracy of CPT by Simmonds (1971) and Nordgren (1972)], wherethe issue of admissible generalized displacement was not addressed. We have also identifiedthe influence of increased transverse-shear deformability, E/G >>1, in our error formulae(38), an effect of importance in anisotropic and composite plates. This effect is morepronounced at the higher level of accuracy, eqn (38) than at the lower level, eqn (24).Contrary to what has been generally believed so far, the better estimate in eqn (38) doesnot relegate its predecessor (24) because these estimates are associated with two different


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136 2. RYCHTEKversions of CPT : both versions have the same mathematical structure, eqns (lo)-(13), butphysically different generalized displacements and kinematic boundary conditions. Theadvantage of higher accuracy in one version is offset by the availability of versatile gener-alized displacements in the other.

The displacement out field (34) states what may be called a refined Kirchhoff hypothesis:straight lines normal to the undeformed top/bottom surfaces move to bent lines normal tothe deformed surfaces; the normals are also stretched/contracted but leave the initialdistance between the faces unchanged. This statement is free from all the notorious con-tradictions of the standard (h, = y = 0) and modified (h, = 0) Kirchhoff hypotheses: itdoes not contradict plane stress and adequately predicts the parabolic distribution oftransverse shears. The refined hypothesis, in other words, shifts the standard hypothesisfrom the wrong place----the middle surface, where transverse shears reach a maximum. tothe right place-the faces, where there is no shear deformation by definition (zero prescribedtangential tractions).

Under specific circumstances, CPT is capable of higher accuracy than in eqns (38 1.For example, Duva and Simmonds (1990) work on beams implies that CPT is as accurateas we please in the cylindrical bending of a cantilevered plate strip. However in general, theestimate (38) can only be improved upon by shifting from CPT to higher-order platetheories, with additional degrees of freedom. For example, Berdichevski (1973) and Rychter(1988) have found the error of Reissner theory to be 0(h3).

6. CONCLUSIONSThis work furthers the understanding of the classical or Kirchhoff theory of elastic

plates in several ways. Most importantly, it puts the theory in dual perspectives in which thetheory exhibits complementary advantages of greater flexibility in the choice of generalizeddisplacements (and in the modeling of kinematic boundary conditions) or of increasedaccuracy. At lower accuracy, the paper expands the validity of existing error estimates forclassical theory to an infinity of variants with diverse generalized displacements. At higheraccuracy, our work provides a refined Kirchhoff hypothesis which avoids all contradictionsof previous hypotheses and makes the displacements and rotations of the top and bottomsurfaces of the plate represent the best and unique pointwise choice of generalized dis-placements. At both levels of accuracy, the impact of increased transverse-shear deform-ability, important in anisotropic and composite plates, is exposed.

REFERENCESBerdichevski, V. L. (1973). An energy inequality in the theory of plate bending (in Russian). Appl. Muth. Mech

31,941-944.Duva, J. M. and Simmonds, J. G. (1990). Elementary, static beam theory is as accurate as you please. J. .4ppi.

Mech. 51. 134-137.Jemielita. G. (1975). Technical theory of plates with moderate thickness (in Polish). Rozpr. Inryn. 23,483 -499.Koiter, W. T. (1970). On the foundations of the linear theory of thin elastic shells. Proc. Km. Ned. Ak. Wet. B

13, 169-195.Nordgren. R. P. (1971). A bound on the error in plate theory. Q. Appl. Math. 28, 587-595.Nordgren, R. P. (1972). A bound on the error in Reissners theory of plates. Q. Appl. Math. 29,551-556.Prager, W. and Synge, J. L. (1947). Approximations in elasticity based on the concept of function space. Q. Appi.

Math. 5, 241-269.Rehfield, L. W. and Murthy. P. L. N. (1982). Toward a new engineering theory of bending: fundamentals. .4fAAJl20, 693-699.

Reissner, E. (1944). On the theory of bending of elastic plates. J. Math. Phys. 23, 184-191.Rychter, Z. (1988). An improved error estimate for Reissners plate theory. Znt. J. Solids Structures 24, 537- 544.Simmonds, J. G. (1971). An improved error estimate for the error in the classical, linear theory of plate bending.Q, Appl. Math. 29,439447.

generalized displacements and the accuracy of classical plate theory

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