A mixed finite element formulation for plate problems

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING. VOL. 4, 67-84 (1972)

A MIXED FINITE ELEMENT FORMULATION FOR PLATE PROBLEMS

A. CHAlTERJEEt AND A. V. SETLURS

Department of Civil Engineering, Indian lnstitute of Technology, Kanpur, India

SUMMARY

A mixed triangular finite element model has been developed for plate bending problems in which effects of shear deformation are included. Linear distribution for all variables is assumed and the matrix equation is obtained through Reissner’s variational principle. In this model, interelement compatibility is completely satisfied whereas the governing equations within the element are satisfied ‘in the mean’. A detailed error analysis is made and convergence of the scheme is proved. Numerical examples of thin and moderately thick plates are presented.

INTRODUCTION

Finite element methods for the analysis of structural problems are now well established and have gained considerable popularity. These methods can be broadly classified into three approaches :

1. Displacement approach which is by far the most popular of the three and which has, as its unknowns, the displacements at the nodal points. Compatibility has to be established through these nodal displacements and this poses the greatest problem in the formulation. Equilibrium equations are obtained by minimizing the potential energy. A number of authors have contri- buted to the development of this approach, notably the groups associated with Clough? Argyris3p4 and Zienkiewic~.~

assumes an equilibrated state of stress within the element, and establishes compatibility through minimization of complementary energy. The main difficulty here lies in the proper selection of the state of stress. The advantage of this approach is that the stresses, which in most cases are the quantities of immediate interest, are obtained directly.

3. In the mixed formulation generalized forces as well as displacements are chosen as the unknowns. Equilibrium and compatibility are satisfied, ‘in the mean’, through special variational formulations such as the Reissner’s variational prin~iple.~ This formulation has been studied by Herrmann,lo, l1 Dunham and Pister12 and Prato.13

For many engineering problems, a knowledge of the stress distribution is of as equal impor- tance as the displacement field. Hence the finite element model should not only be capable of predicting displacements with reasonable accuracy, but also should be able to give a fairly accurate stress field. In some particular cases, this may be achieved by choosing proper mesh sizes and adopting suitable averaging techniques. However, for many problems, this procedure

2. Equilibrium or force approach, pioneered by Fraeijs de Veubeke and developed by

t Doctoral candidate. 3 Assistant Professor.

Received 16 April 1970 Revised 28 August 1970

67 0 1972 by John Wiley & Sons, Ltd.

68 A. CHATTERJEE AND A. V. SETLUR

may be impractical even with higher-order displacement expansions. A likely procedure to overcome this situation has been developed by Pian and his associate^.'^.'^ Even in this approach, due to discontinuities at the interfaces, the fidelity of the solution becomes doubtful. Perhaps, mixed element methods may be best suited for this purpose.

Mixed element models have been recently studied by Herrmann for plate bending problems and by Prato for general shells including the effects of shear deformation. HerrmannlO has developed a triangular model with linear distributions for transverse displacement and bending

9'

I Before and after deformation (Rota tion)

Z'W '

Load deflection and rotations

Y'

Figure 1. Notation

moments. In another paper,'l he has taken linear displacements and constant moments within each element and has continuity of normal moment at the interfaces. Even with this simple model he has shown that the method yields good results. It has to be mentioned here that the shear forces in his approach have to be calculated numerically and will be, at least, one order less accurate than the moment values even if interpolation is employed.

In this paper, a triangular mixed finite element model has been developed for plate bending problems in which shear effects are included. Linear distribution for all variables (transverse deflection, rotations, bending and twisting moments and transverse shears) is assumed and the

A MIXED FINITE ELEMENT FORMULATION 69

resulting matrix equation is obtained by the use of Reissner's variational principle. A detailed error analysis 'of the approximate equation is made and the convergence of the scheme is proved. This formulation has the advantage that

(a) All quantities of interest are obtained directly. (b) Error analysis is straightforward and hence corrections could be obtained. (c) Extension of this formulation to shells of arbitrary shapes using curvi-linear co-ordinates

is feasible. (d) Complications, such as gradual variations of thickness, temperature effects, initial stresses

and dynamic formulation can be handled. The main disadvantage is that the total number of unknowns at a node point is higher than that of the existing formulations. This is partially compensated by the requirement of smaller number of triangles to achieve the same accuracy.

VARIATIONAL FORMULATION

For linear theory of moderately thick plates, the variational formulation considering transverse shear deformation is fairly well known.B Hence without going into details, only the final form, relevant for subsequent derivations, would be presented. The notations used in the sequel have been illustrated in Figure 1.

The non-dimensionalized variational equation in condensed form is

SJ, = llA{ 8nT(gD + E)} dx dy - lsO8bTPds-lsd 8pTr)ds = 0

where B is an 8 x 8 matrix of differential operators given by

a 0 o o - E

0 0 0 0

0 0 0 0

a ax a a 2- 2- 0 0 ay ax

a 0 - 0 12v

aY

1 0 - 0

0 1 - 0

- 0 0 -12

a ax a

aY

0 a aY

--

0 0

0 12v

-48(1+~) 0

0 - 12

0 0

0 0

1

0

a ax --

0

0

0

- 12(1 +.)IS

0

0

1

a aY

--

0

0

0

0

- 12(1 + V ) / S

the non-dimensional state vector,

bT = <Bl P 2 w MI MlZ Mz Vl Vz>


the non-dimensional load vector,

LT=(MI M2 -4 0 0 0 0 0 )

The relation between actual (primed) and non-dimensional quantities are given by (4)

where h is the thickness, E is Young’s modulus and v is Poisson’s ratio. It may be noted that the effect of normal stress is neglected here and hence the additional terms in L normally seen in the Reissner’s formulation are omitted.

The surface integral in equation (1) extends over the entire plate, the first line integral is over the boundary, S,, where stresses are prescribed and the second line integral is over the boundary, S,, where displacements are prescribed. The respective deformation and stress resultant vectors on S, and S, are given by

fiT = (8, A W> on s u

l5T = ( w - w * p,-p: pt-pf:> on Sd (6) RT = (M,-M: M&-M;: V,- V:) on S,

pT = (V, M, 4) on S,

where M,, V, and 8, are the non-dimensional normal moment, shear and rotation along the edge respectively and the asterisked quantities are the corresponding prescribed non-dimensional values. The Euler equations of equation (1) will furnish the first three equations as differential equations of equilibrium and the next five equations as force-deformation relations. The resulting eight equations can be written in matrix form as,

Bb+L = 0 (7) The independent vanishing of the line integrals will provide the required boundary conditions,

and (8) M, = M:, M,, = iUn: and V, = VE on S,

W = w*, /3,=p: and pt=Pf on Sd

FINITE ELEMENT FORMULATION

Within each element, linear distribution has been adopted for the approximate state variable vector, D. It may be noted that the total number of variables used here is larger than that required for a compatible displacement model. However, it has the advantage that, because of linear distribution, continuity of deflection, bending slopes and stresses will be obtained and due to complete polynomial distribution function, completeness and invariance criteriala will be auto- matically satisfied. However, there will be a limitation for this model, when the actual stress field is discontinuous. An indirect approach has been discussed to overcome this difficulty.

To facilitate the choice of trial function, the plate is represented by R number of triangular elements having a total number of nodes equal to P. A typical node, denoted byp, will be common

71 A MIXED FINITE ELEMENT FORMULATION

to M number of elements, as shown in Figure 2. For an element, the nodes are designated as 1, 2 and 3. Local co-ordinates and dimensions of the element have been shown in Figure 3.

I ) X

Figure 2. Typical element m. Typical node p common to M elements

y k X 2 A

Figure 3. Local co-ordinatei and dimensions of an element

A typical quantity, x, at any point ( X I , xz) inside an element, my may be expressed in terms of its nodal values as,

where +T = (1 xlla x2/a>

and 'a' is a typical dimension of the plate, Following equation (9) the distribution of the state vector in terms of its nodal value may be given as,

where


+ = DiaI+T+T+T+T+T+T+T+T]

T = Dia It t t t t t t t ]

DmT = (DIT DzT DaT)

where Di is D at the node, i. Matrix S2 is a 24 x 24. Boolean transformation matrix to rearrange the elements of the vector

< a 1 A 2 1313 1321 1322 Pz3 ... v13 v21 bz v23> to DmT

In the above, the first subscript denotes co-ordinate direction and the second one corresponds to the node number of the element. It should be noted that b and D are not identical. Vector b represents the exact solution surfaces and is the solution of the differential equation (7) with boundary conditions (8); whereas, D is the approximate solution of the finite element method and is valid only within the mth element. Hence the exact state vector b for element m, may be written as

D = +TS2Dm+ em (18) where, em is the associated error vector for the element m. Substituting b from equation (18) in the surface integral portion of equation (I), we get for the element m, the expression

6DmT{ [Km Dm + Lm] + Em} + SemT[Bb + €1 dxl- dx2 (19)

where

and A, is the area of the mth element. The 24 x 24 matrix, Km, and, for linear distribution of load, the exact expression for Lm may be calculated numerically. The expression (19) can be written in a slightly different form as,

i=1,2,3 ] + /IA, s ~ ~ ~ [ B b + ~ ; ] d x ~ d x ~ (23)

where

and partitioning of Km matrix has been shown in the Appendix.

there are only M elements surrounding a typical pth node, the expression (23) gives, Performing summation over all the elements after appropriate superposition and noting that


where the ith node of the element m is p. The independent vanishing of the coefficient of SD,’ in expression (25) gives,

and vanishing of coefficient of SEE gives equation (7). Consider the line integral in equation (1). Designate a typical quantity by x, the conjugate

quantity (with respect to work) by y and associated error terms by E X : and EY. Integrating from node p to p + 1 along the boundary whose outer normal subtends an angle B with x1 axis, as shown in Figure 1, a typical line integral may be written as,

(27)

where x* is the specified value of ,y along s. Since x and y are assumed to be linear functions, the expression (27) may be written as.

r

where sp is the length of the path of integration from p to p + 1 and the barred quantity is the corresponding exact value on boundary s. Since the distribution of x* is known, integrations for the coefficient of Sy, and 6y,+l may be easily performed. It is to be noted that independent vanishing of the coefficient for S E Y will give the exact boundary condition (8). Using standard transformation relations to convert the normal and tangential components (n - t co-ordinate system) to usual quantities, in (xl , x2 system) the coefficient of 6y, may be appropriately added to equation (26) to impose necessary boundary constraints.

Neglecting the error terms, the reduced matrix equation for the entire assemblage of elements which is of a banded nature, may be written as,

KD+L = 0 (29)

where, K is the over-all constrained ‘stiffness’ matrix,

DT={DlT DZT ... DJ}

LT = (LT L; ... LS)

Solving equation (29) by standard algorithms, such as the Gauss elimination procedure, the approximate vector D may be obtained.

ERROR ANALYSIS

Accuracy and convergence study of a proposed finite element model is generally based on numerical evaluation of some problems and comparing the solution with the available analytical results. Such a procedure may be valuable in providing a qualitative character of different models, but is basically deficient in general mathematical justification. Walz and co-workers17 have investigated convergence properties of several finite element displacement models based on classical order of error analysis.


In the following, convergence and error properties of the proposed model are studied to obtain the approximate discretization error involved in the analysis. The discretization error is due to the omission of the term ET in equation (26), which, in turn, is a consequence of the approximation for the distribution of the state variables.

By Taylor series expansion, the functional value of f5 at any point OL(CY. = 1,2,3) can be expressed in terms of b and its derivatives at point i(i = 1,2, and 3). Subtracting appropriate error terms at the nodes, the approximate state vector D can be written as,

where e = amj-aij and E$ is discretization error at the node i. Concentrating the attention on node p (recalling that p is the ith node of element m), substituting equation (31) in equation (26) and since the equation

2 (K$DY+LT)=O m=1, ... M j=1,2,3

has been already solved for approximate solution, it can be shown that

[ K $ ( E ~ - E ~ ) + E T + C e K T , ( g ) i + j=2,3, Z ...a, ] A j = O (32) m=1, ... M j-1,2,3 a=1,2,3 j=1,2

where A i s are given by,

and other A's can be similarly written. For approximate error analysis, the vector E~ may be approximated in a linear form similar to that of the state variable vector D. Following this approximation, equations(22) and (24) will give,

Substituting equation (34) in equation (32) and differentiating the term (as/axj), for linear distribution, it may be shown that,

ET = K g ~ ; n + K z E T + K ~ EF (34)

Equation (35) can be designated as the unconstrained equation for the error vector at the node p, and the expression in the braces as pseudo load. These can be calculated approximately from the known value of D by any numerical technique. Consider now the part of the expression (27) involving Sy,. Following the procedure adopted for the surface integral, the expression containing Sy, can be written as,

where [ is the exact expression for x* - x on s. Again, for approximate solution of error, taking linear distribution we get,


Since the derivatives of [ can be calculated approximately from the previous solution, this expression [after applying proper transformation to (xl, x2) co-ordinate system] has to be added for appropriate boundary to the equation (35) as constraint. Hence the resulting reduced error equation can be solved.

To show the characteristics of the model more clearly all uiis in equation (35) may be divided by a scale factor h where,

aij = ha;

such that when h tends to zero, all the M number of nodes surrounding a typical node p, tend to shrink to p, keeping the relative dimensions of each triangular element to be the same. It can be easily shown that when A+O, equation (35) is of the form

Iim IJ- [ ( B ~ ) + h { } + h ~ { } + ...I dA = O A-tO A

It is seen from equation (33) that expressions in the braces always remain bounded as h tends to zero and hence equation (38) will yield trivial solution e = 0. (Actually, it will be the solution from the uniqueness consideration of elasticity). Convergence to the true solution is thus assured. Monotonicity will be achieved if the lower-order derivatives in the pseudo loading terms retain same sign in the region bounded by the M elements as the element size ( A ) is reduced.

It can be shown that for regular element configuration, the coefficient multiplied by h in the load term is identically zero. Thus, the solution converges quadratically when elements are regular. Also, if the state variable at the nodes of an element are expressed in terms of variables at the centroid of the element by Taylor's series expansion, then, it is seen that the coefficient of h again vanishes, thereby indicating that the solution at the centroid of the element is better behaved and converges quadratically. This result has been demonstrated by several authors.

Finally, near the boundary region, the solution converges linearly because of the presence of the linear term in A.

Similar conclusions can be drawn for the expression (37) which shows the error committed in discretizing the state vector at the boundary of the plate.

DISCONTINUITY OF STRESS FIELD

Discontinuity of stress field in the elastic range may result from the following situations. 1. When there is an abrupt change in the thickness of the plate. 2. When the plate is made up of different materials. Across the discontinuities, the normal moment and shear are continuous, whereas the tangential

components of the stresses are discontinuous. Independent tangential variables may be assumed on either side of the discontinuity and the corresponding equations may be obtained by the independent vanishing of the expressions multiplying the variations of these tangential values. The result is that the number of unknowns and the corresponding number of equations increase at such discontinuities. Also, for such cases, the bandwidth of the resulting simultaneous equations, equation (29), would increase.

NUMERICAL RESULTS

To show numerical convergence and error properties several problems have been solved. Figure 5 shows deflection, w', moment M i , and shear, V;, for a simply supported square plate

having 2, 3, 4 and 6 divisions in a quarter of a plate as shown in Figure 4 (Example 1). It is

76 A. CHATTERJEE A N D A. V. SETLUR

(b)

Figure 4. Arrangement of elements

seen that, in spite of the large element sizes, the results are well behaved and close to the correct solution. Monotonicity of results is clearly seen.

Figure 6 shows deflection, w', and moment, Mi, for a rectangular plate with opposite edges simply supported, one edge fixed and the other edge free (Example 2). Four subdivisions have been taken for one-half of the plate. This example has been taken from Herrmann.lo* l1 The solutions are compared with the series solution given in References 10 and 11. Except near the fixed edges, the finite element solution shows excellent agreement with the series solution.

Figure 7 shows deflection w', moments M i , M i and shear V ; for a square plate with two opposite edges clamped and the other two simply supported for thickness to span ratio 0.01 and 0.1. Except the moments near the fixed edge, other quantities are very well behaved and quite close to the actual solution. Exact solutions for deflections, moments and shears have been drawn for the thin plate.l* In the finite element solutions, differences of moments and shears between the thick and thin plates have been obtained between 2-3 per cent except, only, in the case of shear V i along y = a/2, where the maximum deviation is about 11 per cent. Exact value of the maximum deflection and moment for a thick simply supported plate has been taken from Reference 19.

Lastly, error corrections have been applied using equations (35) to the finite element solutions and the results have been shown in Table I. Since the error at any node has been assumed to be

Table I. Approximate error correction associated with maximum deflection, w' = c4a4/D

Coefficient for maximum deflection, a Finite element

Finite element solution with Example Exact solution error correction Remarks

la 040406 0.00411 0.00380 (Figure 5). For thin S.S. plate

lb 0.00424 0,00428 0.0041 7 (Figure 5). For S.S. plate with

2 0.0582 0.0609 0.0496 (Figure 6). For thin plate with

3 0.001 92 0.001 97 0.00183 (Figure 7). For thin plate with

with 4 x 4 Div

h/s = 0.1 and 6 x 6 Div

4x4Div

6 x 6 Div

the maximum of the corresponding error anywhere in the plate, the wide deviation of the corrected value seems to be justified. Nevertheless, the corrections furnish a bound which gives an approximate idea for the location of the exact solution.


4-0h

I I -

0-1 0.2 0.3 0 . 4 7 5 00'

Sheor q'olong X = 0 / 2 y/u -

IU 44,' = yqb2 1 I I I I

01 0.2 03 0.4. 0.5 Moment M' olong x = 012 Y/O -

1 .o c w'=aqb4/D I I 1 I I

0.1 0-2 0.3 0.4 0.5 Deflection w' olong x=o/2 y/a -

Figure 5. Deflection, moment and shear of S.S. square plate


Maximum deflection 20.0 -

V Series solution 0.0582 qa4/D X 15.0 Q Error 4.65 '/o

Approximate 0 0609 qu4/D - n

0.2 0.4 0.6 0.8 1.0 x / a -

(a)

N s

Figure 6.

12.0 - -

8-0 -

- 4.0 -

'Fixed .-

a = 0.8 E=30x106 u =03 h = 0.01 (Thin plate)

( b ) Deflection and moment for a rectangular plate (thin). (a) Deflection along y =

(b) Moment along x = a


$‘along y at X = U / Z (exoct) 3.0

1.0

V

X S.S.

4’. yqb2 4-01 M;=yqb2

0 3 0 4 05

h/a=o’l lalong yo+ x=u/2 0 h/u=O.Ol ’ h’u=O~O1 ]along x at y = a / ~ . # h h ~ 0 . 1 W’= a qb4/D

30 t

x / a - Deflection w’

Figure 7. Deflection, moment and shear for a square plate

80 A. CHATTERJEE A N D A. V. SETLUR

CONCLUSIONS

A mixed finite element model for plate problems has been developed. This model takes into account the effects due to shear deformation. Because of the linear distribution of the state variables, interelement compatibility is completely satisfied. Equilibrium and compatibility conditions within an element are satisfied ‘in the mean’ through the Reissner variational principle. Since the state variables contain all quantities of interest to the designer, no further effort or approximation is needed to evaluate these.

Error analysis based on the classical order of accuracy approach is developed. Convergence to the true solution has been proved and is shown by numerical experiments. The order of accuracy developed here is useful to obtain a ‘deferred approach to the limit’. A few numerical results have been presented to show the effectiveness of the proposed model. Solutions have also been obtained for various shapes and boundary conditions of the plate and the results agree closely with those published in the literature.

APPENDIX Notation

aB1, aZ2, agl, c32 = Dimensions of an element w, &, pz = Transverse deflection and rotations

M,, M,, Mlz, V,, V, = Moments and shears A, A , = Domain of integration for the entire plate and for the rnth element

respectively Aj , A,, A, = Pseudo load term arising out of the error terms

B = 8 x 8 differential operator matrix n, D = Exact and approximate state vector

Dm, Km, Lm, t, T, 9, $, i3, P,

= Vectors or matrices defined for finite element derivation = Force and displacement vectors on boundary

E = Young’s modulus Em = Unknown loading vector due to discretization error defined in equation

K; = Partitioned matrices of Km f, = Load vector s = Path of line integration

sp = Length of path from pth to p + 1st node on s

(22)

S,, S , = Boundary where displacements or stresses are specified

y, x, 5 = Typical quantities defined in relevant connections X@ = aaj- aij

E~ = Exact error vector for rnth element E$ = Discretization error in ith node for rnth element

X = A scalar factor v = Poisson’s ratio

$2 = Boolean transformation


LOAD VECTOR dl = A/6; d2 = A112

- '1 mll - d2 m12 - d2 m13

- '1 - d2 m22 - d2 m23

4 41 4 42 4 4 3

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

- 4 m u

- 4 m22

dl 4 2

0

0

0

0

0

- 4 m13

4 43

- d2 m23

0

0

0

0

0

- 4 m11

- 4 m21

4 41

0

0

0

0

0

- 4 m12

- 4 m22 4 42 0

0

0

0

0

- 4 m13

- 4 m23

dl 43

0

0

0

0

0


b3, = - a21)!6a2 C2, = (a22-a,2)/6a2

b13 = -a31/6a2

b,, = aZl/6a2

C3, = a3,/6a2 C12 = -a2,/6a2

ELEMENI di = A/6

d, = A112

s3 = 12(l+v)/5


'NT MATRIX

31

4

4

-. . 3 1

' I 4

4

. . -. . . . . .

31

'1 4

I 4


REFERENCES

1. R. W. Clough and J. I. Tocher, ‘Finite element stiffness matrix for the analysis of plate bending’, Proc. Conf. Matrix Meth. Struct. mech., 515-546 (1965).

2. R. W. Clough and C. A. Felippa, ‘A refined quadrilateral element for analysis of plate bending’, Proc. Conf. Matrix Meth. Struct. mech. Wright-Patterson Air Force Base, Ohio, 399-440 (1968).

3. J. H. Argyris, K. E. Buck, I. Fried, H. M. Hilber, G. Mareczek and D. W. Scharp, ‘Some new elements for the matrix displacement method’, Proc. Con$ Matrix Meth. Struct. mech. Wright-Patterson Air Force Base, Ohio, 333-398 (1968).

4. J. H. Argyris, ‘Continua and discontinua’, Proc. Conf. Matrix Meth. Struct. mech. 11-190 (1965). 5 . 0. C. Zienkiewicz and Y . K. Cheung, The Finite Element Method in Structural and Continuum Mechanics,

6. B. Fraeijs de Veubeke, ‘Displacement and equilibrium models in the finite element method‘, in 0. C.

7. B. Fraeijs de Veubeke and G. Sander, ‘An equilibrium model for plate bending’, Znt. J. Solids Struct. 4,

8. L. S. D. Morley, ‘The triangular equilibrium element in the solution of plate bending problems’, Aeronaut. Q.

9. E. Reissner, ‘On a variational theorem in elasticity’, J. Math. Phys. 29, 90-95 (1950).

McGraw-Hill, London, 1967.

Zienkiewicz and G. S. Holister (Eds.), Stress Analysis, Wiley, London, 1965, Chap. 9.

447-468 (1968).

19, 149-169 (1968).

10. L. R. Herrmann, ‘A bending analysis of plates’, Proc. Conf Matrix Meth. Struct. mech., Wright-Patterson Air Force Base, 577-604 (1965).

1 I . L. R. Herrmann, ‘Finite element bending analysis for plates’. J. Engng Mech. Diu.. Proc. Am. SOC. ciu. _ _ Engrs, 93, 13-26 (1967).

12. R. S. Dunham and K. S. Pister. ‘A finite element amlication of the Hellinper-Reissner variational theorem’. Proc. Conf. Matrix Meth. Struct. mech. Wright-Patterson Air Force Base, Ohio, 471-488 (1968).

13. C. A. Prato, ‘Shell finite element method via Reissner’s principle’, Znt. J. Solids. Struct. 5, 11 19-1 133 (1969). 14. T. H. H. Pian and Pin Tong, ‘Rationalization in deriving element stiffness matrix by assumed stress

approach’, Proc. ConJ Matrix Meth. Struct. mech., Wright-Patterson Air Force Base, Ohio, 441-470 (1968).

15. P. Tong and T. H. H. Pian, ‘A variational principle and the convergence of a finite element method based on assumed stress distribution’, Znt. J. Solids Struct. 5, 463472 (1969).

16. C. A. Felippa, Refined Finite Element Analysis of Linear and Nonlinear Two Dimensional Structures, Ph.D. Thesis, University of California, Berkeley, 1966.

17. J. E. Walz, R. E. Fulton and N. J. Cyrus, ‘Accuracy and convergence of finite element approximations’, Proc. ConJ Matrix Meth. Struct. mech., Wright-Patterson Air Force Base, Ohio, 995-1028 (1968).

18. V. L. Salerno and M. A. Goldberg, ‘Effect of shear deformation on the bending of rectangular plates’, J. appl. Mech. Trans. Am. SOC. Mech. Engrs, 27, 54-58 (1960).

19. S. Timoshenko and S. Woinowsky-Krieger, Theory of Plates and Shells, McGraw-Hill, New York. 20. I. M. Smith, ‘A finite element analysis of moderately thick rectangular plates in bending’, Znt. J. mech. Sci.

10, 563-570 (1968).

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A mixed finite element formulation for plate problems