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Numerical methods and refined plate theories
Computers & Structures Vol. 15. No. 4. pp. 351-358. 1982 Printed in Great Britain.
NM-7949182/040351+03.00/0 @ 1982 Pergamon Press Ltd.
NUMERICAL METHODS AND REFINED PLATE THEORIES P. R. S. SPEARE The City University, Northampton Square, London ECIV OHB, England (Received in revised form 20 March 1981;received for publication 28 January 1982) Abstract-A number of refined plate theories are available which, although similar in their basic assumptions, offer a wide choice of variables in which problems may be formulated. The implications for solution of plate problems by numerical methods are discussed and the significance of the formulation chosen is examined. Finite difference and finite element methods are compared for a variety of support and loading conditions.
D E h L Y,
Mm4
NOMENCLATURE flexural stiffness of plate Young’s modulus thickness of plate length of side of square plate
bending and twisting moments per unit width MmM,, P finite difference mesh length uniformly distributed load per unit area shear forces per unit width a,2 B shear stiffness u strain energy W transverse displacement partial detlexions wb, w, n,
t, 2
Cartesian co-ordinate systems x,
Y,z Y shear strain A <+2 ax a$ lJ Poisson’s ratio normal stress ; average rotation shear stress ; Reissner stress function
INTRODUCTION
Classical plate bending theory, in expressing the transverse displacement in the form w = f(x, y), reduces a three dimensional state of stress and deformation to a two dimensional problem and thereby enables solutions to be obtained to a wide range of practical problems. However, a number of approximations are made in so doing, notably that the effects of transverse shear and transverse direct stress are assumed negligible. A refined plate theory is taken in this Paper to mean a small deflexion theory where the transverse displacement is considered to take the form IV= f(x, y) but deformation due to transverse shear is included. (It is not, therefore, an exact three dimensional theory.) Additionally, in some cases the effects of transverse direct stress are also included. Transverse shear deformation becomes significant in situations where the shear stiffness of the plate is small relative to the flexural stiffness. Thick homogeneous plates, sandwich plates and cellular plate systems are typical examples. In thick plates transverse direct stress may also have a significant effect in certain circumstances, but in sandwich plates it is usual to ignore this CAS Vol. IS. No. 4-A
effect, assuming the core to be incompressible in the transverse direction. A number of such refined plate theories have been developed and a range of series solutions obtained, although this is, by virtue of the complexity of the problem, a very limited range. In most practical plate configurations solutions by numerical methods will be necessary and the purpose of this Paper is to examine some aspects of the application of finite difference and finite element methods to representative theories. In order to keep the central issues as clear as possible all theories are considered in isotropic form and only differences in principle for orthotropic conditions noted. THEORETICAL BACKGROUND
There are two theoretical approaches to the inclusion of shear deformation in plate bending theory-the introduction of the associated strain energy in the energy function and the consideration of the state of deformation and stress as the sum of components due to bending and shear. Reissner’s theory [l-3] may be regarded as typical of the first and, in addition, includes the effects of transverse direct stress. A simplified version of this theory, based on the same initial assumptions but of specified order of accuracy, has also been derived[4]. The second approach has been developed with particular reference to sandwich plates and may involve the superposition of curvatures[S] or the use of partial deflexions [6]. The resulting sets of equations appear at first to be somewhat disparate, but it has been shown171 that, if based upon the same initial assumptions and given a consistent definition of shear stiffness, they are in essence the same in the inclusion of deformation due to transverse shear. The apparent differences arise from the range of variables in which the governing equations and formulae for stress resultants may be expressed. The initial assumptions for homogeneous plates are linear distributions with depth of a,, a,, 7XYand quadratic distributions of 7Xz,ryr, The comparison of theories for homogeneous and sandwich plates is based on a shear stiffness of a homogeneous plate being related to flexural rigidity by s
=
5(1-
m
hZ
*
If the effect of transverse direct stress is included in the theory for homogeneous plates then minor modifications
P.R.S.SPE~E
352
usually have to be employed, since hardly any series solutions of the type availablefor classicalplate theory have been found for refined theories. In spite of the wide usage of the finite difference method for plates and slabs with classical theory it has been used very little with refined theories. The theory of Libove and Batdorf has been solved by this me~od[8] for uniform loading using an orthotropic form of eqns (3). The system of simultaneousequations is in the three unknowns, w, QX,Q, and the finite difference treatment involves some special features. The most important of A+_z_h2(2-v)A4 D 10(1-v) these are the use of high order accuracy finite difference equations for first and second order derivatives and the need for off-centre forms at the boundaries in order to A$-$=O. keep the number of fictitious values of the variables in line with the number of boundary equations. Even so an Simpli~edReissner~eory[4]: governingvariable w excess of fictitious unknowns over avaiiable equations arises at the corners and this was overcome either by A2W+h2(2-Y) Afw=A extrapolationor by the use of backwarddifferences. (2) 10(1-v) D Various finite element approaches for plate bending have been used which include shear deformation. The Libove and Batdorf[S]:governingvariables W,QX,i?, effect of the differences between them is in the number of degrees of freedom at nodes and the compatibility achieved between elements. They are most readily understood if the effect of shear deformation is considered to be a relative rotation between the neutral surface and a plane initiaIly normal to it, the actual distribution of shear strain being averaged through the depth of the plate. Hence the total rotation, 4, of a straight line initially normal to the neutrat surface is the sum of the neutral surface slope after deformation, w’, and the average shear strain f
are madeto the equationsbut only to multipliersof certain of the derivatives.Sincethe transversedeflexion,W,is then a function of depth in the plate it becomes a weighted averagerather thanthe deflexionof the neutralsurface,but this is of minor importance in considering the overall deflexion. The governing equilibrium equations for isotropic plates for some of the availabletheories are: Reissner[ l-3): governingvariables W,r~%
#= w’+Y. Partial deflexion theory IS]: A’rv,=; Aw, = -;dzg. (Althoughthe subscripts b and s suggestthat these two partial deffexions are components due to bending and shear, they may be interpreted as such only in very limited circumstances. Further, for orthotropic plates three independentdeflexionterms are necessaryI6J.) The primary effect of there being a number of theoretical approaches available is to leave several options as to the variables which may be chosen for analysis~eflexion, partial deflexions, deflexion and a stress function, deflexionand shear forces. There are two other major points of difference, although these are of secondary importance as far as the immediatepurpose of this Paper is concerned. First, in the form given here eqns (1) and (2) includethe effect of transversedirect stress whereas eqns (3) and (4) do not, since the latter are formulated primarily for sandwich plates for which the core is usually considered to be incompressible.Secondly, the special form of the partial deflexion theory for isotropic plates given here allows only the two conditionsto be satisfiedat each boundary while the other theories require three. In order to obtain a solution to any plate theory problem using a refined theory a numericalmethod will
(5)
Two contrastingappraoches will serve to illustrate the differences which may arise. The refined quad~lateral element of Cloughand Felippa[9]gives three degrees of freedom at each corner node, transverse displacementw, and two rotations of lines initially normal to the neutral surface, & and (6, as deftned by eqn (5). Hence there is a discontinuity allowed in the neutral surface slope between elements. In contrast to this an element[lO] based, with significant simplification,on Reissner’s theory, allows five degrees of freedom at each node by specifying the usual 12 term displacementfunction for defIexionand a four term function for each of the shear strains j$+$$. The result is to achieve continuity of shear strain F along element interfaces and of neutral surface slope at nodes only. Three theories and two numerical methods are selected for inv~s~ation and compa~sou in the Paper. The theories used are Reissner, simplihed Reissner and the partial deflexionmethod.Numericalsolutionsto all three are obtained by use of finite differences and to the Iast two by a finite element method using a localized Ritz formulation. The boundary conditions and type of loading can effect the accuracy of the solution and hence simply supported, fixed and free edges are considered in conjunction with uniform and concentrated loading. The invest~tion thus covers cases typical of most practical problems. For the sake of simplicitythe plates analysed are symmetric, isotropic, and homogeneousbut there is
Numericalmethodsand refinedplate theories no conceptuaf reason why the conclusions concerning the performance of the chosen theories under numericaf computation should not be more generally applicable.
353
Application to Reissner’s theory Since this theory is in terms of two variables the two equations (1) will be satisfied at each mesh point,
together with the appropriate equationsat the boundary. For uniform loadingAq = 0 and hence the first of eqns (1) becomes the normal biharmonic.In the case of conIn selecting a mesh size for finite difference solutions centrated loading an appropriate distributionis assumed account must be taken of the order of accuracy of the in the region one mesh length in each direction from the theories under consideration.This has been shown[4,7] loaded point. The loading vector at this point and adto be of order (plate thickness) 2 with an error term of jacent points must then include the resulting values of order (plate thickness)4. both q and Aq. Since the normal central differencies are of order of A surplus of fictitiousvalues and w and JI arise in the accuracy (mesh length) 2 with an error term of order vicinity of the corner. These are most conveniently (mesh fength) 4 a reasonable mesh length was thought determined by extrapolation rather than by avoiding to be L/12, correspondingroughly to the plate thickness them by writing special forms of the equations at the at which shear deformation begins to become significant, corner incorporatingbackwarddifferences. This also has the merit of enablingconcentrated toads to It should be noted that althoughproblems with maxibe represented realisticallyas a smal!patch load. mum symmetry are considered, I,&itself is not always a symmetric functjon and this must be taken into account EfJ;ect of bending and shear on coeificients in specifying conditions along axis of deflexion symIt is important to note how the bending and shear metry. An example of this arises on the diagonalsof a effects are introduced into the finite difference equations. symmetricallysupported and loaded square plate, along Two broad categories arise: which the requirement QX= Q, leads to &#ax = _-&&lay. (i) Bending and shear effects defined by separate variables(e.g. partial deflexion method). Application to the simplified Reissner theory In this case no coefficient in the finite difference As only one variable is involved in this theory the operators can contain terms arising from both effects; singfeeqn (2) has to be satisfiedat each mesh point. The each coefficientrefates either to bendingor shear. finite difference form of this equation is shown in Fig. 3. (ii) Bending and shear effects defined by common Inspection of the form of the coefficientsin the comvariabfeswholly (e.g. simplifiedReissner)or in part (e.g. putationalmoleculesfor this theory (Figs.2 and 3) shows Reissner).The value of a number of coefficientswill now that there are certain values of plate thicknesswhichwill be determinedby a combinationof the two effects. It will make various terms zero or very smalf.This can somebe shown that under certain circumstancesthis can give times lead to loss of significancein the computation, rise to computationalproblems. although such an occurrence is easily identifiable.On Consider further this second category. Inspection of these occasions the difficultycan be overcome by using eqns (1) and (2) shows that coeffcients relating to w will finite differences of order of accuracy (mesh length) 4 be independent of h for Reissner’s theory but of the for derivatives of third order and lower, which does not form a ij3h*, where (Yand fi are constants, for the lead to an enlargement of the computationalmolecule. Extrapolationwas againused to deal withthe additional simplified Reissner theory. In equations relating to boundary conditions,however, the coefficientswill be of fictitiousvalues at the corners. this fatter form for both theories. Application to the pa~iai de~ex~oamethod
Boundary conditions
The Reissner and simplifiedReissner theories require three conditions to be satisfiedat each boundary. These will be: simplysupported w = M, = 4, = 0 clamped w = #,, = 4, = 0 free M,=M,,=Q,=O
There are two variables, w6and w,, in this theory and hence two finite difference equations to be specifiedat each mesh point. These are eqns (4). but in the form stated they would require the use of backward differences for the second equation at the boundaries since, as already discussed, only two boundary conditions are permitted and hence only two fictitiousvalues can be associated with each boundary point. It has been shown[fj],however, that the second of eqns (4) can be integrated in the form
where 6=-t+
aw 12(1+Y) 5Eh Qn
d, _/4?_w+12(1+v) t
at
5Eh
”
The finite difference forms of these equations are derived from the operators for stress resultantsset out in Figs. 1 and 2. Since only two boundary conditionsare required in the isotropic form of the partial deflexion theory the & = 0 conditionis dropped in the simplysupportedand clamped cases and for free edges M., and Q, are combinedas totat shear, V”= Qn- M~dat.
w,=-gbw,,+C
(7)
where C is an arbitrary constant. If this form is used then there is no need to use backward differences. FIMTEELENA CRAYON General considerations A localized Ritz formulation was used to obtain So!utions to the simplified Reissner and partial deflexion
theories. The applicationof this method to classicalplate bending theory has been described by Walker[~~l. Localized displacement functions are specified which provide continuity of such derivatives as is necessary to
354
P.
R. s.
SPEARE
h’ A= 5P
Mnt
(a)
r
"
t
Fig. 1. (a) Finite differenceequations for M. and M.,-Reissner Theory
(b) Finite difference equation for Q.-Reissner theory
ensure convergence. These functions are specified using generalized co-ordinates. The total potential energy, in&ding that lost by the load, is then evaluated and minimized with respect to these generalized co-ordinates, whose values are then determined from the resulting set of simultaneous equations.
Application to simplified Reissner theory The strain energy of a plate in bending is U=s
1
{ax2+ UY2 t uzz- 2u(u~y t u,u, t up,) t 2(1 t Y)(& t T:, t &I} dx dy dz.
(8)
Numericalmethodsand refinedplate theories
335
1.6
c.
t JL10%vb'
0.1
a2
0.3
-.
Fii. 4. Central deflexionratio. Simply supportedplate: uniform loading(Y= 0) (IV,, = 0.00406qL'/D);Reissner(series solution) and partial detkxions; ------ Reissner, finite difference solution; -. - simplifiedReissner, localized Ritz solution; simplifiedReissner,finite differencesolution.
20
1.8
’
“k
I
/
16
(a)
r
"
1.4
t
1.2
1.0
Fii 2. (a) Fiite diflerence equations for M. and A&,-simplitied Reissner theory (see eqns l&12. (b) Finite difierence equationfor Q,-simplitied Reissnertheory (see eqns 13and 14).
I
I
0.1 0
r
(
h'L
a3
x
Fii. 5. Central detlexion ratio. Simply supportedplate: central point load (V= 0) ( wr,= 0.0116PL’/D); partial degexions; ---- Reissner, finite diflerence solubon; ----simplified Reissner,localizedRitz solution.
Y
38
0.2
l-128
?+
T-F-?w--.-
symmetric
$+
symnetric
Omitting the term in 0: this can be written in terms of stress resultants as w
:;
’
; (I&* t M,.*- 2&M, %
Fii. 3. Finite difference form of eqn (2)-governing equation, simpliied Reissnertheory.
t 2(1 t v)M:,)
(Mx+ My)] dx dy. (9)
P. R. S. SPEARE
356
The expressionfor stress resultants are
(13) (14)
Sub&ion of these equations in (9) leads to the expression for strain energy in terms of deflexion. The loss of potential energy of the toads is calculated from the deflexionunder concentrated loads or the volume of the deflected surface for uniform loads. The order of the displacementfunctions is determined by the continuity requirements between regions. Since fifth order derivativesare the hiit order present in the full energy function knotty of derivatives up to and includingfourth order is necessary. However, a further approximationcan be justiied by consideringthe relative orders of the terms in eqn (9). If the h2 terms in eqns (13) and (14)are omitted then the highestorder derivatives in the energy function are fourth order and the continuity requirementis reduced to third order. This s~pl~cation amounts to ne~ec~ the co~ection which shear deformation makes to the shear forces themselves and was found in a correspondingseries of analyses of beams to have a minimaleffect. In order to achieve continuity of a~ --
a~
a2w
a2w
w+ ax' ay~~~~~~~
a2w a3w a34
a3w
anda
3
ay
10 degrees of freedom are required, which can be achieved by using seventh order polynomials. The approach is then similar to a finite element solution[l2] of Love’s theory for moderately thick plates[ 131. Satisfying the boundary conditions presents little difficultyexcept that for clampedboundaries it is necessary to specify 4” = 0 rather than awlan= 0. Convergencehas been shown to be rapid in classical problemsas the degree of continuity increases[ 11J and a relatively coarse grid with localized regions of side L/4 was found to be satisfactory. Application to partial de@exion method Since the partial detlexion method omits the effects of
transverse direct stress the expression for strain energy will be simpler than that for the simplified Reissner theory. It is given by U=s
1
(M;1tM;Z-ZvM,M,+2(ltv)M:,)
The stress resultants are expressed in terms of derivatives of w, and w,.The momentsare given by the normal
equationsof thin plate theory with wbreplacingw,and the shears are given by Qx=s% Q, =
s?.
(17)
Hence, substitutingin (15)the strain energy is found to be
+5(*~v)[(~)2t(~~]]dxdy.
(18)
Continuity of wb,awdax, awday, a2w$axay and w, is thus required and this is provided by suitable displacement fictions for wband w,, Thus the effects of bending and shear are again separated in the computation as in the finite diflerence method. Localized regions of side L/6 were found to be suitable. rtEsuLT§
Solutionswere obtained by five diierent combinations of theory and numericalmethod: the applicationof finite differences to all three theories under considerationand of the localized Ritz procedure to the simplitkdReissner and partial detlexion theories. The results from six cases are considered here-uniform and concentrated loading on simply supported, clamped and corner supported plates. All five solutions are presented in the lhst case, (uniformly loaded, simply supported plates) for which a series solution of Reissner’s theory is availablell41.In the other cases selected solutions are given where sign&ant points arise. Poisson’s ratio is zero for all cases. Simpiy supported plate with u~ifonn loading, fig. 4
Fire 4 showsthat the results obtainedby use of both finite difference and localized Ritz methods with the partial deflexion theory are, within the limits of plotting accuracy, identical with the series solution. Small differences will, however, occur if v#O, since then the fact that the effects of transverse direct stress are
Numerical methods and refined plate theories
351
neglected in the partial deflexion method will become apparent. The finite diflerence solution of Reissner is also seen to have minimal diiTerences from the series solution. The two solutions to the simplified Reissner theory show slightly greater deviation from the series solution. They are very similar for values of h/L up to 0.2 and are within 4% up to 0.25. At greater values of h/L they both begin to diverge from the series solution although they remain very close to each other. The virtual coincidence of the two partial deflection solutions was a feature of all solutions and no attempt is made to distinguish between them in the following cases. For consistency the results plotted are all those obtained by the finite difference method. Simply supported plate with central point load, Fig. 5 The first significant feature of this graph is the close similarity of the finite difference solution to Reissner’s theory and the partial deflexion results; they lie within 3-4% of each other throughout the range of h/L considered. This difference doubtless results from the representation of the concentrated load. As discussed earlier, in a region of varying load it is necessary, when using Reissner’s theory, to take account of the term involving Aq. Representation of this by the normal central difference operators is the simplest way of modelling this term in the region of the concentrated load, but is doubtless somewhat crude. The small difference which arises in practice between these two sets of results suggests that it is, nonetheless, an acceptable approximation. In contrast to this, the treatment of concentrated loads by the simplified Reissner method was unsatisfactory. The finite difference method gave results with a wide scatter, and these are not shown on the graph. The localized Ritz results fit to a smooth curve but, as can be seen, seriously underestimate the shear component of deflexion. This arises from the fact that continuity of third order derivatives of w is required with this method but in the region of a concentrated load these derivatives, being related to shear forces, will be changing very rapidly. The continuity requirements and the load representation become incompatible with each other under these conditions.
2.5
I
I
0.1
I
02
I
“IL
Q3
Fig. 7. Central deflexion ratio. Clamped plate: central point load (v = 0) (wO=0.00560 PL*/D!; partial deflexions; Reissner, finite diierence solution;----simpliikdReissner,
localizedRitz solution. Clamped plates, Figs. 6 and I The change of boundary condition from simply supported to clamped can be seen from these graphs to have very little effect on the performance on the various methods, despite the fact that the proportion of the total deflexion due to shear is now much greater. The finite dtierence Reissner results and the partial deflexion results are very close to each other for uniform loading. For concentrated loading the difference between these two methods is of the same order as for simply supported plates, again due to the differences in the load representation. For concentrated loads the localized Ritz solution to the simplified Reissner theory fails to account fully for the shear deformation for values of h/L greater than about 0.15. Comer supported plates, Figs. 8 and 9 In the first of these graphs, for uniform loading, the finite difference Reissner and localized Ritz solution to simplified Reissner are shown to yield virtually identical results. For concentrated loading these are compared with
wi w. 1
m
“/L 0.1
a2
a3
0.1
a2
a3
Fii. 8. Central deflexion ratio. Comer supported plate: uniform Fig. 6. Central deflexion ratio. Clamped Plate: uniform loading loading (Y= 0) (w, = 0.0274 qL’/D); _____Reissner, finite simplified Reissner, localized Ritz (V=0) (w,=0.00126 qL’/D); partial deflexions; _____ diierence solution; ----solution. Reissner, finite difference solution.
P. R. S. SPEARE
358
boundary
(Reissner) or separate throughout (partial
deflexion). This does not have much effect when dealii
I
hk Ql
Q2
cl3
Fig, 9. Central deflexion ratio. Corner supportedplate: central point load (V=0) w,,=O.O441Pt*/D); partial deflexions; ----- Reissner, finite difference solution; ----Simplified Reissner, localized Ritz solution. partial deflexion results and here slightly larger discrepancies are apparent for values of h/L greater than 0.15. The manner in which the terms associated with the
concentrated load are dealt with in the finite difference Reissner solution leads to slightly larger deflexions than the partial deflexion results (3% greater at h/L = 0.25). The simplified Reissner results obtained from the localized Ritz solution gave an underestimate of the shear deflexion as in previous cases (5% less than partial deflexion method at h/L = 0.25). DISCUSSION AND CONCLUSIONS
The results given in this Paper are for homogeneous isotropic plates but the operation of the theories and numerical method in dealing with more general configurations can be expected to be very similar. The range of h/L up to 0.3 considered here is equivalent to a range of SL2/D for isotropic sandwich panels with v = 0 down to 56 and the majority of practical cases will fall in this range. Comparing first the performance of the three theories it is clear that for the cases of uniform loading considered the results are all within 3 or 4% of each other for h/L up to 0.25 with larger differences developing at higher values in one or two instances. The theories are thus equally applicable to the range of boundaries considered. However, if the influence of the type of loading is examined a different picture emerges. Whereas the finite difference Reissner and the partial deflexion results agree within 3 or 4% the simplified Reissner theory has not yielded satisfactory solutions by either numerical method for concentrated loads. The difficulty arises from continuity requirements for derivatives connected with shear force in a region where, theoretically, the shear tends to infinity. The partial deflexion theory allows discontinuity of the first derivatives of w, at the load, so that no problem is encountered with this method. Computationally, as has already been noted, the terms dealing with bending and shear are either combined throughout (simplified Reissner), coupled only at the
with uniform loading but in cases of rapidly varying load it has been shown that separation of the two components of deflexion is necessary to achieve consistent accuracy. The computational effort required varies noticeably between the theories and numerical methods. The two factors to be compared are the programming effort and the number of unknowns involved in a particular problem. For the finite difference method it is extremely simple to assemble the operators required, especially if no attempt is made to derive modified operators in the region of the boundary and additional fictitious points are accepted as a penalty. The assembly of the localized Ritz method is much more complex and considerable computational effort is required in the evaluation and minimization of the potential energy before the set of equations for final solution is obtained. It has been shown that a coarser mesh is acceptable in the localized Ritz solutions and hence fewer unknowns are involved than when the finite difference method is used. The simplified Riessner theory, having only one variable has far fewer unknowns in the finite difference method than the Reissner or partial deflexion theories. It should also be noted that in orthotropic cases the partial deflexion method will require three variables and hence will incur some disadvantages. -Cl
E. Reissner, On the theory of bending of elastic plates. I. Math. Phys. 23, 184(1944). E. Reissner, The effect of transverse shear deformation on the bending of elastic plates. 1. Appl. Mech. 12, A69 (1945). E. Reissner, On bending of elastic plates. Quart.Appl. Math. 5, 5s(1947). P. R. S. Speare and K. 0. Kemp, A simplified Reissner theory for plate bending. ht. I. Solids Stmctures. 13, 1073 (1977). C. Libove and S. B. Batdorf, A general small-deflection theory for sandwich plates. N.A.C.A. Rep. 899 (1948). F. J. Plantema, Sandwich Construction. Wiley, New York (1%6). P. R. S. Speare and K. 0. Kemp, Shear deformation in elastic homogeneousand sandwichplates. Pm. Instn Ciu. Engrs Part 2,61,697 (1976). 8. D. G. Williams and J. C. Chapman, Effect of shear deformation on uniformly loaded rectangular orthotropic plates. Proc. Instn. Ciu. Engrs, Suppl. 303 (1%9). 9. R. W. Clough and C. A. Felippa, A relined quadrilateral element for analysis of plate bending. Pm. 2nd Conf on Matrix Methods in Structural Mechanics. Wright-Patterson Air Force Base, Dayton, Ohio (1968). 10. C. W. Pryor, R. M. Barker and D. Frederick, Finite element bending analysis of Reissner plates. J. Engng Me& J&I. A.S.C.E. 96, EM6, %7 (1970). 11. A. C. Walker, Rayleigh-R& method for plate tlexure. I. Engng hfech. h’u. A.S.C.E. 93, EM6, 139(1%7). 12. I. M. Smith, A finite element analysis for “moderately thick” rectangular plates in bending. Int. 1. Mech. Sci 10, 563 (1%8). 13. A. E. H. Love, The Mathematical Theorv of Elasticitv. Cambridge University Press (1927). _ . 14. V. L. Salerno and M. A. Goldbern. Effect of shear deformations on the bending of recta&&r plates. J. Appl. Mech. 27, 54 (l%O).
NM-7949182/040351+03.00/0 @ 1982 Pergamon Press Ltd.
NUMERICAL METHODS AND REFINED PLATE THEORIES P. R. S. SPEARE The City University, Northampton Square, London ECIV OHB, England (Received in revised form 20 March 1981;received for publication 28 January 1982) Abstract-A number of refined plate theories are available which, although similar in their basic assumptions, offer a wide choice of variables in which problems may be formulated. The implications for solution of plate problems by numerical methods are discussed and the significance of the formulation chosen is examined. Finite difference and finite element methods are compared for a variety of support and loading conditions.
D E h L Y,
Mm4
NOMENCLATURE flexural stiffness of plate Young’s modulus thickness of plate length of side of square plate
bending and twisting moments per unit width MmM,, P finite difference mesh length uniformly distributed load per unit area shear forces per unit width a,2 B shear stiffness u strain energy W transverse displacement partial detlexions wb, w, n,
t, 2
Cartesian co-ordinate systems x,
Y,z Y shear strain A <+2 ax a$ lJ Poisson’s ratio normal stress ; average rotation shear stress ; Reissner stress function
INTRODUCTION
Classical plate bending theory, in expressing the transverse displacement in the form w = f(x, y), reduces a three dimensional state of stress and deformation to a two dimensional problem and thereby enables solutions to be obtained to a wide range of practical problems. However, a number of approximations are made in so doing, notably that the effects of transverse shear and transverse direct stress are assumed negligible. A refined plate theory is taken in this Paper to mean a small deflexion theory where the transverse displacement is considered to take the form IV= f(x, y) but deformation due to transverse shear is included. (It is not, therefore, an exact three dimensional theory.) Additionally, in some cases the effects of transverse direct stress are also included. Transverse shear deformation becomes significant in situations where the shear stiffness of the plate is small relative to the flexural stiffness. Thick homogeneous plates, sandwich plates and cellular plate systems are typical examples. In thick plates transverse direct stress may also have a significant effect in certain circumstances, but in sandwich plates it is usual to ignore this CAS Vol. IS. No. 4-A
effect, assuming the core to be incompressible in the transverse direction. A number of such refined plate theories have been developed and a range of series solutions obtained, although this is, by virtue of the complexity of the problem, a very limited range. In most practical plate configurations solutions by numerical methods will be necessary and the purpose of this Paper is to examine some aspects of the application of finite difference and finite element methods to representative theories. In order to keep the central issues as clear as possible all theories are considered in isotropic form and only differences in principle for orthotropic conditions noted. THEORETICAL BACKGROUND
There are two theoretical approaches to the inclusion of shear deformation in plate bending theory-the introduction of the associated strain energy in the energy function and the consideration of the state of deformation and stress as the sum of components due to bending and shear. Reissner’s theory [l-3] may be regarded as typical of the first and, in addition, includes the effects of transverse direct stress. A simplified version of this theory, based on the same initial assumptions but of specified order of accuracy, has also been derived[4]. The second approach has been developed with particular reference to sandwich plates and may involve the superposition of curvatures[S] or the use of partial deflexions [6]. The resulting sets of equations appear at first to be somewhat disparate, but it has been shown171 that, if based upon the same initial assumptions and given a consistent definition of shear stiffness, they are in essence the same in the inclusion of deformation due to transverse shear. The apparent differences arise from the range of variables in which the governing equations and formulae for stress resultants may be expressed. The initial assumptions for homogeneous plates are linear distributions with depth of a,, a,, 7XYand quadratic distributions of 7Xz,ryr, The comparison of theories for homogeneous and sandwich plates is based on a shear stiffness of a homogeneous plate being related to flexural rigidity by s
=
5(1-
m
hZ
*
If the effect of transverse direct stress is included in the theory for homogeneous plates then minor modifications
P.R.S.SPE~E
352
usually have to be employed, since hardly any series solutions of the type availablefor classicalplate theory have been found for refined theories. In spite of the wide usage of the finite difference method for plates and slabs with classical theory it has been used very little with refined theories. The theory of Libove and Batdorf has been solved by this me~od[8] for uniform loading using an orthotropic form of eqns (3). The system of simultaneousequations is in the three unknowns, w, QX,Q, and the finite difference treatment involves some special features. The most important of A+_z_h2(2-v)A4 D 10(1-v) these are the use of high order accuracy finite difference equations for first and second order derivatives and the need for off-centre forms at the boundaries in order to A$-$=O. keep the number of fictitious values of the variables in line with the number of boundary equations. Even so an Simpli~edReissner~eory[4]: governingvariable w excess of fictitious unknowns over avaiiable equations arises at the corners and this was overcome either by A2W+h2(2-Y) Afw=A extrapolationor by the use of backwarddifferences. (2) 10(1-v) D Various finite element approaches for plate bending have been used which include shear deformation. The Libove and Batdorf[S]:governingvariables W,QX,i?, effect of the differences between them is in the number of degrees of freedom at nodes and the compatibility achieved between elements. They are most readily understood if the effect of shear deformation is considered to be a relative rotation between the neutral surface and a plane initiaIly normal to it, the actual distribution of shear strain being averaged through the depth of the plate. Hence the total rotation, 4, of a straight line initially normal to the neutrat surface is the sum of the neutral surface slope after deformation, w’, and the average shear strain f
are madeto the equationsbut only to multipliersof certain of the derivatives.Sincethe transversedeflexion,W,is then a function of depth in the plate it becomes a weighted averagerather thanthe deflexionof the neutralsurface,but this is of minor importance in considering the overall deflexion. The governing equilibrium equations for isotropic plates for some of the availabletheories are: Reissner[ l-3): governingvariables W,r~%
#= w’+Y. Partial deflexion theory IS]: A’rv,=; Aw, = -;dzg. (Althoughthe subscripts b and s suggestthat these two partial deffexions are components due to bending and shear, they may be interpreted as such only in very limited circumstances. Further, for orthotropic plates three independentdeflexionterms are necessaryI6J.) The primary effect of there being a number of theoretical approaches available is to leave several options as to the variables which may be chosen for analysis~eflexion, partial deflexions, deflexion and a stress function, deflexionand shear forces. There are two other major points of difference, although these are of secondary importance as far as the immediatepurpose of this Paper is concerned. First, in the form given here eqns (1) and (2) includethe effect of transversedirect stress whereas eqns (3) and (4) do not, since the latter are formulated primarily for sandwich plates for which the core is usually considered to be incompressible.Secondly, the special form of the partial deflexion theory for isotropic plates given here allows only the two conditionsto be satisfiedat each boundary while the other theories require three. In order to obtain a solution to any plate theory problem using a refined theory a numericalmethod will
(5)
Two contrastingappraoches will serve to illustrate the differences which may arise. The refined quad~lateral element of Cloughand Felippa[9]gives three degrees of freedom at each corner node, transverse displacementw, and two rotations of lines initially normal to the neutral surface, & and (6, as deftned by eqn (5). Hence there is a discontinuity allowed in the neutral surface slope between elements. In contrast to this an element[lO] based, with significant simplification,on Reissner’s theory, allows five degrees of freedom at each node by specifying the usual 12 term displacementfunction for defIexionand a four term function for each of the shear strains j$+$$. The result is to achieve continuity of shear strain F along element interfaces and of neutral surface slope at nodes only. Three theories and two numerical methods are selected for inv~s~ation and compa~sou in the Paper. The theories used are Reissner, simplihed Reissner and the partial deflexionmethod.Numericalsolutionsto all three are obtained by use of finite differences and to the Iast two by a finite element method using a localized Ritz formulation. The boundary conditions and type of loading can effect the accuracy of the solution and hence simply supported, fixed and free edges are considered in conjunction with uniform and concentrated loading. The invest~tion thus covers cases typical of most practical problems. For the sake of simplicitythe plates analysed are symmetric, isotropic, and homogeneousbut there is
Numericalmethodsand refinedplate theories no conceptuaf reason why the conclusions concerning the performance of the chosen theories under numericaf computation should not be more generally applicable.
353
Application to Reissner’s theory Since this theory is in terms of two variables the two equations (1) will be satisfied at each mesh point,
together with the appropriate equationsat the boundary. For uniform loadingAq = 0 and hence the first of eqns (1) becomes the normal biharmonic.In the case of conIn selecting a mesh size for finite difference solutions centrated loading an appropriate distributionis assumed account must be taken of the order of accuracy of the in the region one mesh length in each direction from the theories under consideration.This has been shown[4,7] loaded point. The loading vector at this point and adto be of order (plate thickness) 2 with an error term of jacent points must then include the resulting values of order (plate thickness)4. both q and Aq. Since the normal central differencies are of order of A surplus of fictitiousvalues and w and JI arise in the accuracy (mesh length) 2 with an error term of order vicinity of the corner. These are most conveniently (mesh fength) 4 a reasonable mesh length was thought determined by extrapolation rather than by avoiding to be L/12, correspondingroughly to the plate thickness them by writing special forms of the equations at the at which shear deformation begins to become significant, corner incorporatingbackwarddifferences. This also has the merit of enablingconcentrated toads to It should be noted that althoughproblems with maxibe represented realisticallyas a smal!patch load. mum symmetry are considered, I,&itself is not always a symmetric functjon and this must be taken into account EfJ;ect of bending and shear on coeificients in specifying conditions along axis of deflexion symIt is important to note how the bending and shear metry. An example of this arises on the diagonalsof a effects are introduced into the finite difference equations. symmetricallysupported and loaded square plate, along Two broad categories arise: which the requirement QX= Q, leads to &#ax = _-&&lay. (i) Bending and shear effects defined by separate variables(e.g. partial deflexion method). Application to the simplified Reissner theory In this case no coefficient in the finite difference As only one variable is involved in this theory the operators can contain terms arising from both effects; singfeeqn (2) has to be satisfiedat each mesh point. The each coefficientrefates either to bendingor shear. finite difference form of this equation is shown in Fig. 3. (ii) Bending and shear effects defined by common Inspection of the form of the coefficientsin the comvariabfeswholly (e.g. simplifiedReissner)or in part (e.g. putationalmoleculesfor this theory (Figs.2 and 3) shows Reissner).The value of a number of coefficientswill now that there are certain values of plate thicknesswhichwill be determinedby a combinationof the two effects. It will make various terms zero or very smalf.This can somebe shown that under certain circumstancesthis can give times lead to loss of significancein the computation, rise to computationalproblems. although such an occurrence is easily identifiable.On Consider further this second category. Inspection of these occasions the difficultycan be overcome by using eqns (1) and (2) shows that coeffcients relating to w will finite differences of order of accuracy (mesh length) 4 be independent of h for Reissner’s theory but of the for derivatives of third order and lower, which does not form a ij3h*, where (Yand fi are constants, for the lead to an enlargement of the computationalmolecule. Extrapolationwas againused to deal withthe additional simplified Reissner theory. In equations relating to boundary conditions,however, the coefficientswill be of fictitiousvalues at the corners. this fatter form for both theories. Application to the pa~iai de~ex~oamethod
Boundary conditions
The Reissner and simplifiedReissner theories require three conditions to be satisfiedat each boundary. These will be: simplysupported w = M, = 4, = 0 clamped w = #,, = 4, = 0 free M,=M,,=Q,=O
There are two variables, w6and w,, in this theory and hence two finite difference equations to be specifiedat each mesh point. These are eqns (4). but in the form stated they would require the use of backward differences for the second equation at the boundaries since, as already discussed, only two boundary conditions are permitted and hence only two fictitiousvalues can be associated with each boundary point. It has been shown[fj],however, that the second of eqns (4) can be integrated in the form
where 6=-t+
aw 12(1+Y) 5Eh Qn
d, _/4?_w+12(1+v) t
at
5Eh
”
The finite difference forms of these equations are derived from the operators for stress resultantsset out in Figs. 1 and 2. Since only two boundary conditionsare required in the isotropic form of the partial deflexion theory the & = 0 conditionis dropped in the simplysupportedand clamped cases and for free edges M., and Q, are combinedas totat shear, V”= Qn- M~dat.
w,=-gbw,,+C
(7)
where C is an arbitrary constant. If this form is used then there is no need to use backward differences. FIMTEELENA CRAYON General considerations A localized Ritz formulation was used to obtain So!utions to the simplified Reissner and partial deflexion
theories. The applicationof this method to classicalplate bending theory has been described by Walker[~~l. Localized displacement functions are specified which provide continuity of such derivatives as is necessary to
354
P.
R. s.
SPEARE
h’ A= 5P
Mnt
(a)
r
"
t
Fig. 1. (a) Finite differenceequations for M. and M.,-Reissner Theory
(b) Finite difference equation for Q.-Reissner theory
ensure convergence. These functions are specified using generalized co-ordinates. The total potential energy, in&ding that lost by the load, is then evaluated and minimized with respect to these generalized co-ordinates, whose values are then determined from the resulting set of simultaneous equations.
Application to simplified Reissner theory The strain energy of a plate in bending is U=s
1
{ax2+ UY2 t uzz- 2u(u~y t u,u, t up,) t 2(1 t Y)(& t T:, t &I} dx dy dz.
(8)
Numericalmethodsand refinedplate theories
335
1.6
c.
t JL10%vb'
0.1
a2
0.3
-.
Fii. 4. Central deflexionratio. Simply supportedplate: uniform loading(Y= 0) (IV,, = 0.00406qL'/D);Reissner(series solution) and partial detkxions; ------ Reissner, finite difference solution; -. - simplifiedReissner, localized Ritz solution; simplifiedReissner,finite differencesolution.
20
1.8
’
“k
I
/
16
(a)
r
"
1.4
t
1.2
1.0
Fii 2. (a) Fiite diflerence equations for M. and A&,-simplitied Reissner theory (see eqns l&12. (b) Finite difierence equationfor Q,-simplitied Reissnertheory (see eqns 13and 14).
I
I
0.1 0
r
(
h'L
a3
x
Fii. 5. Central detlexion ratio. Simply supportedplate: central point load (V= 0) ( wr,= 0.0116PL’/D); partial degexions; ---- Reissner, finite diflerence solubon; ----simplified Reissner,localizedRitz solution.
Y
38
0.2
l-128
?+
T-F-?w--.-
symmetric
$+
symnetric
Omitting the term in 0: this can be written in terms of stress resultants as w
:;
’
; (I&* t M,.*- 2&M, %
Fii. 3. Finite difference form of eqn (2)-governing equation, simpliied Reissnertheory.
t 2(1 t v)M:,)
(Mx+ My)] dx dy. (9)
P. R. S. SPEARE
356
The expressionfor stress resultants are
(13) (14)
Sub&ion of these equations in (9) leads to the expression for strain energy in terms of deflexion. The loss of potential energy of the toads is calculated from the deflexionunder concentrated loads or the volume of the deflected surface for uniform loads. The order of the displacementfunctions is determined by the continuity requirements between regions. Since fifth order derivativesare the hiit order present in the full energy function knotty of derivatives up to and includingfourth order is necessary. However, a further approximationcan be justiied by consideringthe relative orders of the terms in eqn (9). If the h2 terms in eqns (13) and (14)are omitted then the highestorder derivatives in the energy function are fourth order and the continuity requirementis reduced to third order. This s~pl~cation amounts to ne~ec~ the co~ection which shear deformation makes to the shear forces themselves and was found in a correspondingseries of analyses of beams to have a minimaleffect. In order to achieve continuity of a~ --
a~
a2w
a2w
w+ ax' ay~~~~~~~
a2w a3w a34
a3w
anda
3
ay
10 degrees of freedom are required, which can be achieved by using seventh order polynomials. The approach is then similar to a finite element solution[l2] of Love’s theory for moderately thick plates[ 131. Satisfying the boundary conditions presents little difficultyexcept that for clampedboundaries it is necessary to specify 4” = 0 rather than awlan= 0. Convergencehas been shown to be rapid in classical problemsas the degree of continuity increases[ 11J and a relatively coarse grid with localized regions of side L/4 was found to be satisfactory. Application to partial de@exion method Since the partial detlexion method omits the effects of
transverse direct stress the expression for strain energy will be simpler than that for the simplified Reissner theory. It is given by U=s
1
(M;1tM;Z-ZvM,M,+2(ltv)M:,)
The stress resultants are expressed in terms of derivatives of w, and w,.The momentsare given by the normal
equationsof thin plate theory with wbreplacingw,and the shears are given by Qx=s% Q, =
s?.
(17)
Hence, substitutingin (15)the strain energy is found to be
+5(*~v)[(~)2t(~~]]dxdy.
(18)
Continuity of wb,awdax, awday, a2w$axay and w, is thus required and this is provided by suitable displacement fictions for wband w,, Thus the effects of bending and shear are again separated in the computation as in the finite diflerence method. Localized regions of side L/6 were found to be suitable. rtEsuLT§
Solutionswere obtained by five diierent combinations of theory and numericalmethod: the applicationof finite differences to all three theories under considerationand of the localized Ritz procedure to the simplitkdReissner and partial detlexion theories. The results from six cases are considered here-uniform and concentrated loading on simply supported, clamped and corner supported plates. All five solutions are presented in the lhst case, (uniformly loaded, simply supported plates) for which a series solution of Reissner’s theory is availablell41.In the other cases selected solutions are given where sign&ant points arise. Poisson’s ratio is zero for all cases. Simpiy supported plate with u~ifonn loading, fig. 4
Fire 4 showsthat the results obtainedby use of both finite difference and localized Ritz methods with the partial deflexion theory are, within the limits of plotting accuracy, identical with the series solution. Small differences will, however, occur if v#O, since then the fact that the effects of transverse direct stress are
Numerical methods and refined plate theories
351
neglected in the partial deflexion method will become apparent. The finite diflerence solution of Reissner is also seen to have minimal diiTerences from the series solution. The two solutions to the simplified Reissner theory show slightly greater deviation from the series solution. They are very similar for values of h/L up to 0.2 and are within 4% up to 0.25. At greater values of h/L they both begin to diverge from the series solution although they remain very close to each other. The virtual coincidence of the two partial deflection solutions was a feature of all solutions and no attempt is made to distinguish between them in the following cases. For consistency the results plotted are all those obtained by the finite difference method. Simply supported plate with central point load, Fig. 5 The first significant feature of this graph is the close similarity of the finite difference solution to Reissner’s theory and the partial deflexion results; they lie within 3-4% of each other throughout the range of h/L considered. This difference doubtless results from the representation of the concentrated load. As discussed earlier, in a region of varying load it is necessary, when using Reissner’s theory, to take account of the term involving Aq. Representation of this by the normal central difference operators is the simplest way of modelling this term in the region of the concentrated load, but is doubtless somewhat crude. The small difference which arises in practice between these two sets of results suggests that it is, nonetheless, an acceptable approximation. In contrast to this, the treatment of concentrated loads by the simplified Reissner method was unsatisfactory. The finite difference method gave results with a wide scatter, and these are not shown on the graph. The localized Ritz results fit to a smooth curve but, as can be seen, seriously underestimate the shear component of deflexion. This arises from the fact that continuity of third order derivatives of w is required with this method but in the region of a concentrated load these derivatives, being related to shear forces, will be changing very rapidly. The continuity requirements and the load representation become incompatible with each other under these conditions.
2.5
I
I
0.1
I
02
I
“IL
Q3
Fig. 7. Central deflexion ratio. Clamped plate: central point load (v = 0) (wO=0.00560 PL*/D!; partial deflexions; Reissner, finite diierence solution;----simpliikdReissner,
localizedRitz solution. Clamped plates, Figs. 6 and I The change of boundary condition from simply supported to clamped can be seen from these graphs to have very little effect on the performance on the various methods, despite the fact that the proportion of the total deflexion due to shear is now much greater. The finite dtierence Reissner results and the partial deflexion results are very close to each other for uniform loading. For concentrated loading the difference between these two methods is of the same order as for simply supported plates, again due to the differences in the load representation. For concentrated loads the localized Ritz solution to the simplified Reissner theory fails to account fully for the shear deformation for values of h/L greater than about 0.15. Comer supported plates, Figs. 8 and 9 In the first of these graphs, for uniform loading, the finite difference Reissner and localized Ritz solution to simplified Reissner are shown to yield virtually identical results. For concentrated loading these are compared with
wi w. 1
m
“/L 0.1
a2
a3
0.1
a2
a3
Fii. 8. Central deflexion ratio. Comer supported plate: uniform Fig. 6. Central deflexion ratio. Clamped Plate: uniform loading loading (Y= 0) (w, = 0.0274 qL’/D); _____Reissner, finite simplified Reissner, localized Ritz (V=0) (w,=0.00126 qL’/D); partial deflexions; _____ diierence solution; ----solution. Reissner, finite difference solution.
P. R. S. SPEARE
358
boundary
(Reissner) or separate throughout (partial
deflexion). This does not have much effect when dealii
I
hk Ql
Q2
cl3
Fig, 9. Central deflexion ratio. Corner supportedplate: central point load (V=0) w,,=O.O441Pt*/D); partial deflexions; ----- Reissner, finite difference solution; ----Simplified Reissner, localized Ritz solution. partial deflexion results and here slightly larger discrepancies are apparent for values of h/L greater than 0.15. The manner in which the terms associated with the
concentrated load are dealt with in the finite difference Reissner solution leads to slightly larger deflexions than the partial deflexion results (3% greater at h/L = 0.25). The simplified Reissner results obtained from the localized Ritz solution gave an underestimate of the shear deflexion as in previous cases (5% less than partial deflexion method at h/L = 0.25). DISCUSSION AND CONCLUSIONS
The results given in this Paper are for homogeneous isotropic plates but the operation of the theories and numerical method in dealing with more general configurations can be expected to be very similar. The range of h/L up to 0.3 considered here is equivalent to a range of SL2/D for isotropic sandwich panels with v = 0 down to 56 and the majority of practical cases will fall in this range. Comparing first the performance of the three theories it is clear that for the cases of uniform loading considered the results are all within 3 or 4% of each other for h/L up to 0.25 with larger differences developing at higher values in one or two instances. The theories are thus equally applicable to the range of boundaries considered. However, if the influence of the type of loading is examined a different picture emerges. Whereas the finite difference Reissner and the partial deflexion results agree within 3 or 4% the simplified Reissner theory has not yielded satisfactory solutions by either numerical method for concentrated loads. The difficulty arises from continuity requirements for derivatives connected with shear force in a region where, theoretically, the shear tends to infinity. The partial deflexion theory allows discontinuity of the first derivatives of w, at the load, so that no problem is encountered with this method. Computationally, as has already been noted, the terms dealing with bending and shear are either combined throughout (simplified Reissner), coupled only at the
with uniform loading but in cases of rapidly varying load it has been shown that separation of the two components of deflexion is necessary to achieve consistent accuracy. The computational effort required varies noticeably between the theories and numerical methods. The two factors to be compared are the programming effort and the number of unknowns involved in a particular problem. For the finite difference method it is extremely simple to assemble the operators required, especially if no attempt is made to derive modified operators in the region of the boundary and additional fictitious points are accepted as a penalty. The assembly of the localized Ritz method is much more complex and considerable computational effort is required in the evaluation and minimization of the potential energy before the set of equations for final solution is obtained. It has been shown that a coarser mesh is acceptable in the localized Ritz solutions and hence fewer unknowns are involved than when the finite difference method is used. The simplified Riessner theory, having only one variable has far fewer unknowns in the finite difference method than the Reissner or partial deflexion theories. It should also be noted that in orthotropic cases the partial deflexion method will require three variables and hence will incur some disadvantages. -Cl
E. Reissner, On the theory of bending of elastic plates. I. Math. Phys. 23, 184(1944). E. Reissner, The effect of transverse shear deformation on the bending of elastic plates. 1. Appl. Mech. 12, A69 (1945). E. Reissner, On bending of elastic plates. Quart.Appl. Math. 5, 5s(1947). P. R. S. Speare and K. 0. Kemp, A simplified Reissner theory for plate bending. ht. I. Solids Stmctures. 13, 1073 (1977). C. Libove and S. B. Batdorf, A general small-deflection theory for sandwich plates. N.A.C.A. Rep. 899 (1948). F. J. Plantema, Sandwich Construction. Wiley, New York (1%6). P. R. S. Speare and K. 0. Kemp, Shear deformation in elastic homogeneousand sandwichplates. Pm. Instn Ciu. Engrs Part 2,61,697 (1976). 8. D. G. Williams and J. C. Chapman, Effect of shear deformation on uniformly loaded rectangular orthotropic plates. Proc. Instn. Ciu. Engrs, Suppl. 303 (1%9). 9. R. W. Clough and C. A. Felippa, A relined quadrilateral element for analysis of plate bending. Pm. 2nd Conf on Matrix Methods in Structural Mechanics. Wright-Patterson Air Force Base, Dayton, Ohio (1968). 10. C. W. Pryor, R. M. Barker and D. Frederick, Finite element bending analysis of Reissner plates. J. Engng Me& J&I. A.S.C.E. 96, EM6, %7 (1970). 11. A. C. Walker, Rayleigh-R& method for plate tlexure. I. Engng hfech. h’u. A.S.C.E. 93, EM6, 139(1%7). 12. I. M. Smith, A finite element analysis for “moderately thick” rectangular plates in bending. Int. 1. Mech. Sci 10, 563 (1%8). 13. A. E. H. Love, The Mathematical Theorv of Elasticitv. Cambridge University Press (1927). _ . 14. V. L. Salerno and M. A. Goldbern. Effect of shear deformations on the bending of recta&&r plates. J. Appl. Mech. 27, 54 (l%O).