A MIXED ELEMENT FOR LAMINATED PLATES AND …...Mixed element for laminated plates and shells 599 The mass matrix M is defined as in which the D, matrix elements are defined as 1 I

Compvrers & Struc~wes Vol. 37, No. 4, pp. 597611. 1990 Printed in Great Britain.

ca4.s7949/w s3.00 + 0.00 Pergamon Press plc

A MIXED ELEMENT FOR LAMINATED PLATES AND SHELLS

T. E. WILT, A. F. SALEEB and T. Y. CHANG Department of Civil Engineering, University of Akron, Akron, OH 44325-3905, U.S.A.

(Received 23 October 1989)

Abstract-Formulation and numerical evaluation of a simple shear-flexible four-noded quadrilateral laminated composite plate/shell finite-element is presented. The element developed is based on a generalized mixed variational principle with independently assumed displacement and laminate internal strain fields. For the latter, a layer-number-independent polynomial interpolation is utilized, together with a judicious selection of the in-plane (spanwise) distributions for the strain components. This was facilitated by the use of a set of bubble functions as additional kinematic degrees of freedom, which was shown to be crucial for eliminating the locking phenomenon. For dynamic applications, a simplified lumped mass matrix is employed. Finally, an extensive number of critical test cases are given to access the element’s prefotmance and to illustrate its effectiveness in static as well as vibration problems for anisotropic laminated plates and shells.

1. INTRODUCTION

1.1. General

In classical lamination theory (CLT) the usual Love-Kirchhoff assumptions of plane sections remaining plane are effective, thereby neglecting shear deformations totally. When dealing with composite material applications it is well known that this type of theory is too restrictive except in very thin plate or shell applications. Thus, an alternative theory used is that is attributed to the Reissner-Mindlin type of assumptions. In this case, a constant shear angle through the thickness of the plate is assumed and has been referred to as the constant shear angle theory (CST). Typically, some appropriate shear correction factor, K, is chosen to account for the through- thickness shear deformation [l, 21. These type of Reissner-Mindlin elements comprise one class of elements typically used for composite analysis.

It has also been suggested that even this type. of assumption may not be sufficient for laminated plates and shells because of the varying shear moduli through the thickness of the laminate. Thus, a layerwise constant shear angle theory (LCST) has been used in which, as the name implies, a constant shear deformation angle is used for each ply [7-l 11.

Using the terminology of Bert [3], the majority of composite finite elements can be catagorized into two distinct groups, namely, (1) ‘a discrete layer model’ (DLM) and (2) a ‘smeared laminate model’ (SLM). In addition, the above classifications maybe further subdivided into groups which utilize lower- and higher-order approaches. There have been numerous higher-order theories developed (e.g. Reddy [4], Lo et al. [5], Murakami [6]). The criteria for distinguish- ing between a lower-order versus a higher-order approach depends on the order of thickness-coordi-

nate (z) terms included in the expansion of various displacement components. That is, expansions including only up to linear terms in z are described as lower-order approaches, whereas those including higher-than-linear powers of z are collectively referred to as higher-order approaches. It should be noted that application of either approach can be made on the entire laminate, as in the SLM elements, or to individual layers, as in the DLM elements.

In the DLM elements, each layer retains its own identity and in some cases is treated as a ‘sub-element’ of the total element. In terms of shear deformation, the layerwise constant shear angle theory (LCST) has discrete models since each layer is permitted to have its own shear angle. Typical examples are the elements of Spilker [7,8], Spilker et al. [9], and Chaud- huri and Seide [lo]. These types of elements have been found to be computationally expensive [7]. Thus, there has been movement away from a pure DML approach to approaches utilizing constraints [8], assumed stress field simplifications [9], and substructur- ing techniques [1 l] in an effort to improve efficiency.

In the SLM elements, the laminate is not considered as a series of individual layers but a hetero- geneous, anisotropic medium. In terms of shear deformation, the constant shear angle theory (CST) type of elements are in effect smeared models since they do not distinguish the shear strain angles on an individual layer basis. Typical examples are elements of Panda and Natarajan [12], Chang and Sawamiphakdi [13], and Haas and Lee [14]. Lately, it appears that the use of the SML approach in conjunction with a higher-order theory [4-6] is gaining popularity. Elements of Reddy and Phan [15], Kwon and Akin [ 161, Pandya and Kant [17], Rogers and Knight [19] and Lakshminarayana and Murthy [18] are typical examples of such an approach. This

597

598 T. E. WILT et al.

methodology attempts to strike a balance between complexity and stress prediction accuracy without the inefficiency associated with the DML approach. However, possible drawbacks are the higher-order stress and moment resultants which are difficult to interpret physically [ 151.

1.2. Objective and outline of present approach

The objective here is to report some of the results of our recent research in connection with the develop- ment of simple finite-element models for composite plates and shells using mixed methods. In particular, utilizing the degenerated-shell quadrilateral mixed element developed by Saleeb et al. [20,21] for the isotropic case as a starting point, we present details of its ‘extended’ formulation and applications to the static and vibration analysis of anisotropic-elastic laminated plate/shell strutures. In the underlying variational principle for this latter extension to composites, both the displacement and strain fields are approximated independently. Since the ultimate goal of the research is the application of this element for high-temperature metal-matrix composite structural analysis, i.e. nonlinear material analysis, it appears that utilizing a strain assumption, as opposed to a stress assumption, is most desirable. Specifically, in the case of material nonlinearity it is entirely valid to assume a linear variation of bending strains through-the-thickness of the ‘thin’ laminate, while such an assumption for stresses is certainly not valid. Also, choosing a strain field a priori is much easier to justify than it would be for a particular stress field assumption.

In addition the assumed strain field used here represents the entire laminate, and this element can thus be classified as a smeared laminate model, SML type, utilizing a first-order or CST theory. It is noted however, that the present approach cannot yield an accurate prediction on the detailed distribution of transverse shear stresses over the laminate thickness. Methods of performing so-called ‘stress smooth- ing’ [ 111 are currently being investigated.

Finally, an important consideration in the develop ment of any mixed element for plates and shells concerns the judicious selection of the in-plane (spanwise) distributions for the assumed strain components, such that the resulting model exhibits robust performance, i.e. with no kinematic (zero-energy) modes, and avoiding the locking phenomenon [7-9,20,21]. The latter is particularly critical for both thin, as well as moderately thick, plates and shells in view of the strong coupling that exists in the energy terms associated with stretching, bending, and transverse shear actions in a general nonsymmetric laminate. To this end, we use the same strain-field approximation has suggested in [20,21], whose success was clearly demonstrated in a large variety of isotropic test cases. The crucial point concerns the use of a set of bubble functions [i.e. kinematic degrees of freedom (DOF) associated with an interior fifth node]

in the present four-noded quadrilateral composite shell element. It is also interesting to note here that, since the original suggestions presented in [20,21], this approach of using bubble functions to improve the performance of shear-flexible mixed plate elements has been gaining popularity in recent years

(e.g. [321).

2. ELEMENT FORMULATION

2.1. Element st@ness and mass matrices

The derivation of the stiffness and mass matrices is obtained from the functional, rrR, referring to lamina coordinate systems. Justification for choosing such a reference frame is given in detail in the work of Saleeb et al. [20,21]. Thus, the functional can be expressed as

?tR = s[S

[-feTDe+rrIX+fpiiTU]dI’ , v

where a superscript T signifies transposition, dV=dxdydz=]J]drd.rdt, in which r, s, t are natural (isoparametric) coordinates and J is the Jaco- bian-of-coordinate-transformation matrix. In the above equation, u is the displacement vector, e the independent strains, e the geometric (from displace- ments) strains, D the material moduli (stress e = De), p mass density, T prescribed tractions on portion S, of the surface area, and i the time variable.

For brevity, the details of the derivation of the stiffness equations are omitted here and the reader is referred to Saleeb et al. [2O]. From the functional nR, the final element stiffness equations are obtained as follows:

(G?r-‘Gq + Md - Q) 6qT = 0

or MQ+Kq=Q, (2)

where q is the nodal displacement vector, and the stiffness matrix K is defined as

K = GTH-‘G, (3)

in which the matrices, H, G, and Q are given by

1 I I H=

sss PrDP]J] dr d.r dt (4)

-I -I -1

1 I I G=

sss PTDB’]J] dr d.s dt (5)

-1 -I -I

lV?‘]J]dr d.r dt. (6)

Mixed element for laminated plates and shells 599

The mass matrix M is defined as in which the D, matrix elements are defined as

1 I I M=

SH lVpm]JI dr d.s dt. (7)

-I -I -I

EI, 4, =-,

E22

1 - v12v21

D,,=p 1 - h2V21

In the above equations, the arrays P, B’, and m give the strain-field (polynomial) shape functions, the straindisplacement operator, and displacement shaped functions, respectively [20].

-v,2E22 D,2=D2,=~

1 - VlZV21 (11)

2.2. Through-thickness integration for composites

According to eqns (4) and (5) the component matrices for the present ‘smeared’ element stiffness [see eqn (3)] are obtained by simply integrating through-the-thickness and taking into account the differences in material properties of each layer. That is, since the element now represents a multi-layered composite, the material matrix, D, is no longer constant through-the-thickness. Thus, individual layer material matrices, D,, must be formed, transformed, and then combined during integration to form the smeared laminate stiffness.

D33 = G12 9 kDu=aG23, kDSS=dG3l,

where E,, and E22 are the longitudinal and transverse elastic moduli, respectively, the Gs are the transverse and in-plane shear moduli and v is Poisson’s ratio, with all quantities being defined in material fiber coordinates. Now, the layer matrix D, needs to be transformed from material fiber direction to the direction of the laminate coordinate axes. This is accomplished by using the standard coordinate transformation relation:

Specifically, each layer is assumed to be in a state of plane stress (a,, = 0), i.e. the stresses in local

VRI = [TlT[Dl,rn,~ (12)

where [Irr], is the transformed layer material matrix and [Tl, is an appropriate transformation matrix (see Fig. 1 for the definition of the angle e), i.e.

cos* e sin2 8 sin 6 cos e 0 0 -

sin2 e c~~* e -sine case 0 0

PI, = -2sinecose 2sinecos8 c0s2B -sin26 0 0 .

0 0 0 cos e -sine

0 0 0 sin e c0se ,

(13)

material fiber or symmetry-axes coordinates are or- This latter matrix will generally vary along the dered as lamina surface (i.e. t = 0 surface) from one inte-

gration point to another (with different r, stoordi-

{a>, = {g 11~022,T12,T23,713j/ (8) nates), depending on the composite’s material fiber layout, in a generally-distorted (nonrectangular) and

the corresponding layer constitutive relation is

1 D,, D2, 0 0 0 42 D22 0 0 0 D33 0 0 0 0 kDu 0 0 0 0

or symbolically

(9)

curved shell element. In this connection, we note that at each integration point, a local Cartesian reference frame is defined such that two of its axes ar. tangent to the middle surface through the point (see [20] for details). For generally-distorted (nonrectangular) and curved shell elements, these lamina basis vectors, e: , e:, e:, will change at each integration point, i.e. the angle which the material fiber makes with the local lamina coordinate axes changes at each integration point. Therefore, a convenient and unique method of defining the material fiber angle is required.

A variety of approaches were considered for defining the material fiber direction. For example, one could input the material fiber direction as a unit vector, thus uniquely defining its direction in space. Of course, a different unit vector would be required for each element as the fiber moves along the surface. It was felt that requiring the user to define different unit material fiber direction vectors would be tedious and would be somewhat difficult for the user to

(10) calculate.


Y

I Material fiber

Line // to side

I )X

Fig. 1. 0 measured with respect to side 34.

The method chosen for this element measures the material fiber angle, 0*, with respect to a specified side of the distorted element. Specifically, the angle f3 + is measured with respect to side 34 of the element, i.e. the side defined by the local element nodes 3 and 4. Angles measured counterclockwise with respect to side 3-4 are positive (Fig. 1). The element subroutine then internally calculates the appropriate angle t9 that the material fiber makes with respect to the lamina coordinate axes.

This method allows the user to input one material fiber angle for each element since it will be assumed that the fiber lies in the mid-surface plane and this angle is ‘nearly’ constant for all integration points in the particular element. Admittedly, this method can still be somewhat tedious in that the user, depending on the geometry of the structure being modeled, may be required to input a different material fiber angle for each element. However, such generality is required if more complicated geometries other than flatplates, or cylindrical shapes, are to be modeled. A good example is the complex geometry of turbine blades which can assume the shape of a hyperbolic paraboloid.

Thus, the transformation matrix r], uses the internally calculated angle fI to transform the layer material matrix [D], to the local element material axes, and the transformed layer material matrix [61, is now evaluated at each integration point giving the following form of the stress-strain relation, in element local coordinates for layer I:

[:/;[:! !!)I ;jgj.

(14)

3-4

Finally, in order to facilitate the integrations indicated in the above H and G matrices, eqns (4) and (5), the following simple ‘change-of-variable’ procedure, as outlined in [12], is used:

f=-1,; -h,(l-r,)+2&, [ 1 (15) c j-l

which yields, upon differentiating with respect to t,,

dt = ;dt,. c

(See Fig. 2 for definitions of the various quantities used in the above expressions.) This now leads to

P‘1I,P]J(;dt,dr d.r (17) c

P?l,B’].$dt,dr d.s, (18) E

which are evaluated using standard numerical integration; i.e. two Gauss quadrature points per layer, and the indicated summation is over the total number of layers, n.

2.3 Element mass matrix formulation

As mentioned in Sec. 1.2, a feature of the element presented here is the use of bubble functions [20], i.e. the quadrilateral element actually has five nodes, four of which are defined externally and the fifth node is generated internally. For computational efficiency, the fifth node DOF are condensed out of the element stiffness/mass matrices before assembly into the global arrays. It is this feature which presents some difficulty when dealing with the vibration analysis case.


Location of Loyer Integration Points, tL

Fig. 2. Multi-layer composite.

This point can be made more clear by looking at the eigenvalue problem being solved:

@ - ~M){ql = 0, (19)

where 1 is the eigenvalue and K and M have dimen- sions 25 x 25 in this case. For the case of lumped mass matrix, which is being considered here, the above equation can be ‘partitioned’ into components associated with the comer-nodes DOF, q,, and those for the interior (middle-node) DOF, q,,,, thus leading to

If condensation is performed on the above to eliminate the middle node DOF, q,,,, the above equation will reduce to

(&I - A&h,+ W - @a - W,)-‘Kh) = 0.

(21)

As is evident in the above expression, the equation contains both known, stiffness and mass terms, and unknown, 1, quantities. Particularly the term (K,, - lMzz)-’ presents computational difficulties in that some iterative process would be required to solve the above equation, as discussed by Kidder [23]. Most of the literature dealing with condensation in general [24-261 has been applied to the global mass and stiffness matrices. That is, the focus has been on reducing extra structural DOF through the use of condensation techniques. Thus, iterative methods are acceptable on the global mass and stiffness matrices. However, in the present case, the condensation is required on the element level. Thus, the above

methods would have to be applied to each element which would be computationally expensive.

With the above discussion in mind, a suitable alternative to condensation is desired. One such alternative is to neglect the middle node entirely, thereby eliminating the need for any form of condensation. Specifically, the lumped mass matrix M, as defined in eqn (7), is constructed following an algorithm as presented by Hughes [27]. The element lumped-mass matrix is formed in the standard way, i.e.

M=i 1 1

SSI

I

/=I -1 -I -1 p,N’lJ$dr,dr d.s, (22)

c

where the indicated summation is over the total number of layers, n. Here, the shape function matrix N contains the standard four-node shape functions.

The above equation for N applies to the mass quantities associated with both translational and rotational DOF, i.e. initially mvans = mro’ = m. The above quantities are then scaled (normalized) to ensure the total mass of the element is preserved, i.e.

where

M,,, = i m i= I

(23)

and M, is the total mass of the element given by

M,= f: I I I

Iss /=I -1 -I -I p,lJI;dr,dr ds. (25)

C

Finally, the mass quantities associated with the rotational DOF, m”‘, must be adjusted so that rotational inertia is accounted for consistently, i.e.

h* mrol = mrot . 2

12’ (26)

where h, is the laminate thickness. Thus, a diagonal lumped mass matrix with nonzero rotational inertia terms is found.

It is believed that this approach is justifiable for the following reason. The middle node’s additional displacement is very small relative to an average displacement interpolated from the four corner nodes. Thus by using a sufficiently fine mesh, the contribution of the middle node becomes of less importance. Therefore, while it certainly results in significant reduction in computation time for the element, the current assumption of using a lumped mass and neglecting the middle node’s contribution may not lead to significant errors in frequencies. A series of numerical problems were investigated to determine the validity of the approach outlined above, for the


vibration analysis of both isotropic and anisotropic plates/shells, as given later in Sec. 3.2.

3. NUMERICAL STUDIES

In this section, a series of problems taken from various literature sources are used to determine the capability of the element to adequately predict the static and dynamic behavior of composite laminates. These problems involve varied boundary conditions, aspect ratios, loading conditions and laminate configurations. For several of these problems, com- parisons are made with available ‘exact’ analytical solutions as well as other independent finite element solutions. The quantities of interest are predictions of laminate global response, i.e. deflections, and natural frequencies with corresponding vibration modes.

3.1. Static test problems: deflections and stresses

In the following results, the elements of Lakshmi- narayana and Murthy will be referred to as TRIPLT [ 181 and those of Spilker as MQH3T [9] and V2R [8]. These particular elements are chosen because they appear to provide accurate results for comparison. Note, in all of the following tables, the expression for the nondimensional parameter, W is given as

wE,,h3 G = - x 103.

q0b4

3.1.1. A single -layer clamped rectangular plate under uniform pressure. The problem analyzed here is a single-layer rectangular plate subjected to a uniformly distributed pressure load of magnitude q,,. This problem was taken from Lakshminarayana and Murthy [18], and even though no ‘exact’ solution is available, it is felt that the TRIPLT element has been proven to provide accurate results. This problem is considered a difficult test for the element due to the combination of plate aspect ratio, boundary conditions, and high material anisotropy. The geometry of the plate has a ‘planar’ aspect ratio of 2, and has totally clamped boundary conditions (Fig. 3). The

Y

I

I-

b

o/b = 2.0

h/b = 0.01

Fig. 3. Clamped rectangular plate.

material used has the following properties: E,, = 30 x 106, E,, = 0.75 x 106, G,2 = 0.45 x 106, GZ3 = 0.375 x 106, and vu = 0.25.

Because of the demanding nature of the problem, a mesh convergence study was performed to access the element’s behavior as a function of mesh size. Due to a lack of material symmetry, the entire plate was modeled. The results for the TRIPLT element were obtained using an 8 x 8 mesh. Recall that the TRIPLT element uses three nodes with 15 DOF per node, thus giving a total of 1215 DOF for the 8 x 8 mesh. The present element’s mesh contains 605 DOF for the 10 x 10 mesh. As shown in the convergence results in Table 1, for most fiber angles, convergence is attained at the 8 x 8 mesh with the 10 x 10 mesh giving the best results. Thus, the present element is able to model the problem accurately using less DOF.

Table 2 summarizes the above results for a 10 x 10 mesh using the present element and an 8 x 8 mesh for the TRIPLT element. As shown, the present element compares well with the TRIPLT element and uses less total DOF to achieve comparable results.

3.1.2. Two-layer clamped and simply supported square plates under uniform pressure. In this problem,

Table 1. Mesh convergence study for clamped single-layer rectangular plate

Laminate (4 x 4) Present F.E.

(6 x 6) (8 x 8) (10 x 10) TRIPLT (8 x 8)

PI 10.4598 (- 0.74%)

[I51 9.3339 (- 1.18%)

P51 5.9319 (- 2.53%)

1351 2.9553 ( + 1.96%)

1451 1.6414 (+ 16.09%)

1751 1.1334 ( + 24.09%)

1901 0.9962

10.9851 (+ 4.25%)

9.9972 ( + 5.84%)

6.5407 ( + 7.48%)

2.9991 ( + 3.47%)

1.3695 ( - 3.14%)

0.7911 (- 13.39%)

0.6608

10.9592 ( + 4.0%)

10.0391 (+ 6.28%)

6.5245 (+ 7.21%)

2.9269 ( + 0.98%)

1.3644 ( - 3.50%)

0.9000 ( - 1.46%)

0.8116

10.8879 10.5375 (f 3.33%)

10.0215 9.4455 (+ 6.10%)

6.5132 6.0856 (+ 7.03%)

2.9226 2.8985 ( + 0.83%)

1.3820 1.4189 ( - 2.26%)

0.9133 0.9134 (0%)

0.8073 0.8017 . - ( + 24.26%) (- 17.57%) ( + 1.23%) ( + 0.70%)


Table 2. Clamped rectangular plate with uniform pressure

Normalized center deflection, # Laminate TRIPLT Present F.E.

101 10.537 10.888 t151 9.4455 10.022

;:z; 6.0856 2.8985 6.5132 2.9226 WI 1.4139 1.3820 [751 0.9134 0.9133 1901 0.8017 0.8073

a square two-layer plate (a = 10, h = 0.02) subjected to a uniformly distributed pressure q,, = 100 is considered. The plate was analyzed using both simply supported and clamped boundary conditions to access their effect on element performance. The simply supported boundary conditions have, in addition to the transverse displacement, the in-plane displacement normal to the plate edge restrained. The clamped boundary conditions have all DOF restrained along the plate edge. The material properties for the problem are: E,, = 40 x lo6 psi, E,, = 1 x lo6 psi, G12 = 0.5 x lo6 psi, and v,~ = 0.25.

The results are compared to exact solutions and other finite element solutions using the TRIPLT element and the MQH3T and V2R thin plate elements of Spilker et al. [7,9]. Both of these elements contain eight nodes with five DOF per node. A 6 x 6 mesh of MQH3T elements was used for the clamped plate (665 DOF) and a 4 x 4 mesh of V2R elements was used in the simply supported plate (325 DOF). The TRIPLT element used a 6 x 6 mesh (735 DOF). The present element used a 10 x 10 mesh for both cases (605 DOF). Again, due to a lack of material symmetry, the entire plate was modeled.

As is evident from Tables 3 and 4 for displace- ments, as well as the bending moments A4,, and M,,Y for the simply supported plate (Tables 5 and 6), the present element agrees well in both cases with the exact solutions,

3.1.3. Clamped cylinders under internal pressure. The following two problems are chosen to demon- strate the element’s ability to model curved geometries of laminated composites. These problems were taken from Haas and Lee [14] and were chosen because detailed mesh convergence results are presented. This allows a direct comparison to be made between the convergence characteristics of the present element and the element of Haas and Lee, which will be referred to as CSHEL9.

The two problems to be considered are a clamped cylindrical shell under internal pressure and a

Table 3. Clamped square plate with uniform pressure

Normalized center deflection, $ Laminate Exact Present F.E. MQH3T

1 f 51 0.0946 0.1040 0.1083 I+251 0.2355 0.2602 0.2572 1+351 0.2763 0.2914 0.2844 [zt451 0.2890 0.3013 0.2929

Table 4. Simply supported square plate with uniform pressure

Center deflection, w Laminate Exact Present F.E. TRIPLT V2R

[ f 51 592 597 606 595 [+251 984 1003.7 992 983 [k351 945 967.6 952 944 [ * 451 915 937.6 922 914

clamped 90” cylindrical shell, as shown in Fig. 4. The material properties for both problems are: E,, = 7.5 x lo6 psi, Ez2 = 2.0 x lo6 psi, G,r = 1.25 x lo6 psi, G,, = G,, = 0.625 x lo6 psi, and vrr = 0.25. The magnitude of the pressure load is q,, = (6.41 x x)psi. For the problem of the complete cylindrical shell, one octant of the cylinder is modeled by taking advantage of geometric and material symmetry. The result presented by Haas and Lee [14] is also for one octant of the cylinder. For the 90” cylindrical shell, the entire shell was modeled. This is due to the fact that an unsymmetric laminate is considered in one case, thus material symmetry is no longer present. As mentioned previously, for both problems a mesh convergence study was conducted for comparison with the results of Haas and Lee[l4].

For the clamped cylindrical shell, two aspect ratios, R/h = 20 and R/h = 100, are considered, and with three different laminate configurations, i.e. [O], [45/-45],, and [O/90],. The results, in terms of the maximum radial displacement, are presented in Tables 7 and 8. With regards to the mesh sizes used for the present element and those of the CSHEL9 element, one note needs to be made. The CSHEL9 element is nine-noded element with five DOF per node for a total of 45 DOF per element. Thus the mesh sizes used in the present analysis were chosen so as to be comparable to those of Haas and Lee.

As shown in Table 7, the present element achieves a converged solution with the 8 x 8 mesh, i.e. 405

Table 5. Bending moment M,, for a simply supported two-layer square plate

MXX Laminate Exact Present F.E. TRIPLT V2R

[ f 51 1318.0 1324.8 1327.0 1396.2 [k 151 1142.0 1146.9 1150.0 1176.8 1 k 251 843.6 844.9 851.3 855.6 1 f 351 564.6 563.7 573.1 567.8 1 f 451 368.1 360.8 375.3 371.3

Table 6. Bending moment M, for a simply supported two-layer square plate

Laminate MYY

Exact Present F.E. TRIPLT V2R

[k 51 34.2 33.8 34.47 34.99 [* 151 123.4 124.1 124.4 126.4 [+25] 226.0 227.9 228.4 221.5 L&351 304.1 304.2 308.8 305.1 [*451 368.1 360.8 375.3 371.3


w I

L q

Complete Cylinder

SO0 Clamped Cylinder

Fig. 4. Clamped cylinders under internal pressure.

DOF. This is equivalent to the 4 x 4 mesh of Haas and Lee. The values obtained by the present element are within 2% of those obtained by Haas and Lee [ 141.

Finally, the clamped 90” cylinder is considered. In this problem, Haas and Lee used mesh sizes of 6 x 6, 10 x 10, and 12 x 12 and comparable mesh sizes for the present element would be 12 x 12, 20 x 20, and 24 x 24, respectively. Since these latter meshes are rather large, and since the element appears to behave satisfactorily based upon the results of the previous problem, slightly smaller meshes were used in the

analysis, i.e. 9 x 9 and 18 x 18. Table 8 shows the results obtained using the present element as compared to CSHEL9. As shown, it appears as if the element has converged at the 18 x 18 mesh size, with the possible exception being the [O/90/90/0] case. For this case, a solution using a 32 x 32 mesh does appear to be converging.

3.2. Effect of element distortion

One important test of an element’s performance is how well the element behaves when used in a generally-distorted (nonrectangular) geometry. Such

Table 7. Mesh convergence study for clamped cylindrical she11 R/h = 20

Deflection, w x lo3 Present F.E. CSHEL9

Laminate (4 x 4) (8 x 8) (16 x 16) (6 x 6)

PI 0.3841 0.3176 0.3758 0.3781 [45/ - 451 - 451451 0.2359 0.2337 0.233 1 0.2402

10/90/90/01 0.1812 0.17921 0.1787 0.1783

Table 8. Mesh convergence studv for clamned 90” shell R/h = 100

Laminate

IO1 [45/ - 451

[O/90/90/01 r45/- 451 - 451451

(9 x 9)

1.804 0.8306 0.6677 0.8394

Deflection, w x 10’ Present F.E. (18 x 18) (32 x 32)

1.893 1.884 0.8818 - 0.7355 0.7728 0.8854 -

CSHEL9 (12 x 12)

1.877 0.8936 0.7424 0.8965


‘initial’ distortion may be due to the specific problem being modeled which may require some, or all, elements in the finite-element mesh to be distorted. In the majority of all other composite finite-element solutions that we are aware of, no mention is made of that particular element’s performance when distorted. In one article by Haas and Lee [14], a square plate with an intentionally distorted mesh is analyzed, yet the amount of distortion is not severe. In all other references, uniform meshes, with no element distortion, are used. Rather difficult test cases for an element may be developed if ‘challenging’ effects, such as nonsymmetric laminates which have coupling present, and materials which possess high E,, to E,, ratios, are used in conjunction with element distortion.

Thus, it is the objective of the following problems to ‘challenge’ the element, using the conditions discussed above, in order to determine its strengths and weaknesses with regards to modeling composite laminate behavior.

3.2.1. Clamped rectangular plate with uniform pressure. This is the same problem as presented in Sec. 3.1.1, but now using the distorted mesh as shown in Fig. 5. In addition to the element distortion, the following effects are also present; coupling effects of varying degree due to the unidirectional laminate with varying fiber direction, a high ratio of longitudinal versus transverse moduli (E,,/E,, = 40), and the clamped boundary conditions. An 8 x 8 mesh is used in this analysis, and was chosen based upon the previous results of this problem using the undistorted mesh (refer to Table 1). Basically, it is desired to see if the mesh will produce the same results even with element distortion present.

As shown in Table 9, the distorted mesh yields results which are within 2% of those obtained using the undistorted mesh. Thus, it appears that the element is relatively insensitive to distortion effects.

3.2.2. A clamped circular plate with uniform pressure. This problem is also analyzed by Lakshmi- narayana and Murthy [18] using the TRIPLT element. The material is a unidirectional laminate, with the material fibers at an angle 8 = 0 with respect to the global x-axis. It is stated in [18] that a 6 x 6 mesh of TRIPLT elements is used to model one quarter of the plate. No figure of the mesh used or

a 1 k

b -I

Fig. 5. Clamped rectangular plate with distorted mesh.

Table 9. Clamped rectangular plate with uniform pressure, distorted mesh

Laminate

101 [I51 I301 [451

61 [901

Center deflection, G Present F.E.

TRIPLT Undistorted Distorted

10.5375 10.9592 10.8092 9.4455 10.0391 9.9531 6.0856 6.5246 6.4817 2.8985 2.9270 2.9538 1.4139 1.3644 1.4030 0.9134 0.9000 0.9011 0.8017 0.8116 0.7987

other details are given, thus it is difficult to determine the total number of elements used in their analysis, and, more importantly, whether or not any form of element distortion is present. Also, in [18], only the graphite/epoxy material I is considered. In this study, in addition to material I, materials II and III are also considered so as to be able to judge the effect that moduli ratio has when combined with element distortion.

A mesh convergence study was undertaken using mesh sizes of 12, 48, 192, and 300 elements. A representative example of the 48-element mesh used in this analysis is shown in Fig. 6. Tables 10-12 summarize the results for the different mesh sizes, materials, and plate aspect ratios. Some significant and interesting trends can be seen in the results. Note that the quantities in the following tables are normalized deflections, w*, i.e. w* = wD/qOa4, where

D = 3(& + Q,) + 2(&+ 20,) and D,i, &, D,,, and Dss are bending stiffnesses found by laminate theory.

In Table 10, it is noted that the first case appears to present the most difficulty for the element. Specifi- cally, for the large aspect ratio, a/h = 1000 (a very thin plate), with high degree of anisotropy, a rather

I- a

-l

Fig. 6. Finite element mesh for the circular plate.


Table 10. Circular plate using material I

Normalized center deflection, w * No. of elements Exact

0 12 48 192 300 solution

1000 0.1159 0.1144 0.1137 0.1173 0.1249 100 0.1194 0.1241 0.1254 - 50 0.1242 0.1277 0.1285 - 25 0.1373 0.1400 0.1407 16.67 0.1574 0.1600 0.1608 - 10 0.2209 0.2239 0.2248 -

Table 11. Circular plate using material II

Normalized center deflection, W* No. of elements Exact

0 12 48 192 300 solution

1000 0.1152 0.1230 0.1246 0.1265 0.1255 100 0.1192 0.1243 0.1251 - - 50 0.1218 0.1255 0.1262 - - 25 0.1273 0.1302 0.1309 - - 16.67 0.1350 0.1377 0.1385 - - 10 0.1590 0.1619 0.1627 -

fine mesh must he used in order to converge to the exact solution. Also note that as the mesh is refined, there is no monotonic convergence (although the results for all meshes actually differ only in the last two digits, suggesting that a converged solution might have already been obtained!). This trend disap- pears as soon as an aspect ratio less than 1000 is used. As a matter of fact, for all other aspect ratios, convergence is obtained using the 48-element mesh. (Table 10).

In Tables 11 and 12, the element appears to produce good results. That is, for the thin plate, n/h = 1000, the results match the exact solution of Lekhnitskii for both materials II and III. Again, for all aspect ratios, monotonic convergence is observed, with convergence achieved at 48 elements.

3.2.3. Clamped triangular plate subjected to uniform pressure. The last problem considered in this section is that of a triangular plate, clamped on all sides, and subjected to a uniform pressure load. Again, a unidirectional laminate is used with the fiber oriented at an angle 0 = 0 with respect to the global x-axis. The same group of three different material properties is used as described in the previous problem.

The reference solution is taken from Lekhnit- skii [22], which provides an approximate solution for

the thin-plate case. The deflection w,,,,, at the centroid of the triangle can be found from the equation

Wnl,X = 0.000429 * q0a4

D,+D,+D,' (28)

where qO = 1 is the magnitude of the pressure, a = 40 is the side length of the plate and

D,=D,,, D,=D,,, D,=D,2+2Dss.

The quantities D,, , D,,, D,,, and D, are the bending stiffnesses found by laminate theory.

As shown in Fig. 7, a mesh of 48 elements is used in the analysis and was chosen on the basis of the results found from the previous problem. No other finite element solutions for this problem were found in the literature.

The deflection, 8, for various plate aspect ratios and material types is given in Table 13. As is shown a significant difference of approximately +28.7% is noted between the present finite element solution and the reference solution for the thin plate case, a/h = 1000. This difference is consistently present for all three material systems. A finer mesh containing 192 elements was used to verify that convergence had been achieved and the same results were obtained.

Table 12. Circular plate using material III

aih 12

Normalized center deflection, W* No. of elements 48 192 300

Exact solution

1000 0.1163 0.1231 0.1246 0.1255 0.1250 100 0.1193 0.1242 0.1249 - - 50 0.1211 0.1247 0.1253 - - 25 0.1237 0.1264 0.1270 - -

16.67 0.1266 0.1291 0.1297 - 10 0.1355 0.1378 0.1384 -

Mixed element for laminated plates and shells

a

Fig. 7. Finite element mesh for the triangular plate.

In order to check whether or not the above mentioned discrepancy may be attributed to the anisotropy, the same triangular plate was analyzed (8 x 8 mesh, h = 0.04), but now using isotropic material, i.e. E = 30 x 106, v = 0.3. A deflection of w,,, = 2.6171 was obtained from the finite element analysis, as compared to the reference solution which gives a deflection of w, = 2.0821. Again, there is a discrepancy of +25.7%.

At the moment no firm conclusions may be reached as to why the above discrepancy occurs. There is some doubt as to the accuracy of the reference solution because Lekhnitskii does refer to it as a first approximation [22]. Considering the fact that favorable results were obtained for the first two problems reported in this section, it is believed that the results are accurate. An independent finite element analysis will be required to prove or disprove the results.

Table 13. Triangular plate with uniform pressure

Centroidal deflection, G h Material I Material II Material III

0.04 0.1545 0.3457 0.9452 0.4 0.8120 0.3694 0.9975 0.8 0.2168 0.3996 1.0406 1.6 0.3552 0.5088 1.1641 2.4 0.5711 0.6804 1.3517 4.0 1.1813 1.1625 1.9200

PA 0.1213 0.2709 0.7340 h=0.04

3.3. Dynamic test problems

3.3.1. An isotropic cantilevered plate. This first example of an isotropic cantilevered plate is chosen to provide a method for determining how significant neglecting the middle node contribution to the element lumped mass matrix is over a range of frequencies. This particular problem, found in [28], provides both numerial and test results for the first 12 frequencies of vibration. In the analysis performed here, mesh sizes that range from 4 x 4 to 10 x 10, for the entire plate, are used to study the effect of mesh refinement on the convergence of the first six frequencies. As Table 14 shows, reasonable results are obtained for the first two frequencies for all mesh sizes considered (i.e. the error is within 8% for wr). When considering the higher modes, as shown, additional mesh refinement is necessary in order to produce reasonable results (i.e. error within 7% for w6). Thus it appears that for the first six frequencies, the element gives fairly accurate results.

3.3.2. An isotropic cantilevered shallow shell. The second example is that of a moderately thin (a/h = NO), moderately shallow (R,/b = 2.0) cantilevered shell with an aspect ratio of a/b = 1 (Fig. 8). This problem is taken from Leissa et al. [29], in which numerial results are obtained by a Ritz method with results presented for various displacement trial functions.

Results are presented for the the first eight frequencies of vibration using 8 x 8 and 16 x 16 mesh sizes for the entire shell. Two sizes were used to verify

608 T. E. WILT er al.

Table 14. Mesh convergence study for flat isotropic cantilevered plate

Present F.E. Reference Frequency (4 x 4) (6 x 6) (8 x 81 (IO x 10) [281

1 3.34 3.39 3.41 3.42 3.50 2 13.32 13.82 14.07 14.20 14.50 3 19.46 20.43 20.80 20.98 21.70 4 40.94 43.58 44.91 45.62 48.10 5 51.22 55.32 56.98 57.82 60.50 6 71.62 78.81 83.08 85.38 92.30

convergence and were chosen based upon the knowl- edge that it would be reasonable to use a 4 x 4 mesh if only one quarter of the shell had been modeled. The entire shell was modeled in order that the corresponding mode shapes could be plotted. The results taken from [29] are those of the 6 x 5, i.e. largest, solution (90 x 90 determinant size), while the present finite element results contain 405 and 1445 DOF for the 8 x 8 and 16 x 16 meshes, respectively.

Upon examining the results in Table 15, good agreement is generally found for the first four frequencies, with less accuracy for the remaining four frequencies, i.e. the difference varies between approximately -2% and -5%. Leissa et al. 1291 state that the higher frequencies are usually less accurate than the lower frequencies when using approximate methods, which is the case here. Also, the finite element solution does contain significantly more DOF than does the approximate solution, which is especially important for the higher modes. Thus, the overall good results obtained show that lumping the mass at the corner nodes and neglecting the mass contribution of the middle node does not appear to be of major importance.

3.3.3. Four-layer symmetric and antisymmetric square plates. This example is that taken from Reddy [30]. Two different laminate confi~rations are considered: (1) a [45/ - 45/ - 4.51451 symmetric laminate and (2) a [45/ - 45/45/ - 451 antisymmetric laminate. Both simply supported and clamped boundary conditions are used as described in [30]. Two different materials are considered. Material I: Et, = 40 x lo6 psi, Ez2 = 1 x lo6 psi, Gn = 0.6 x

Fig. 8. Shallow isotropic cantilevered shell.

lo6 psi, Gr3 = 0.5 x IO6 psi, and vn = 0.25. Material II: E,, = 25 x 106psi, E,, = 1 x lo6 psi, Gn E 0.5 x lo6 psi, Gr3 = 0.2 x IO6 psi, and viz = 0.25. p = 1 for both material types.

Figure 9 shows the comparison of the present element with that of Reddy and with closed-form solutions where noted. As is shown, the present element shows good agreement in all cases; it even shows better accuracy than that of Reddy, when compared to the closed-form solution for the simply supported material II case.

3.3.4. A laminated cylindrical shell. Next we consider some test problems to access how well the element performs for curved, nonplanar geometries. In particular, a series of laminated shells with different amounts of curvature are investigated with com- parisons to be made to Reddy [31]. In this reference, exact solutions are presented based upon Sander’s shell theory, into which Reddy has included shear deformation effects.

Specifically, the problem chosen is that of a cross- ply, [O/90], cylindrical shell (Fig. 10). The material properties are as follows: E,, = 25 x 106psi, Ez2= 1 x 106psi, G,, = 0.2 x IO6 psi, G,, = G,, = 0.5 x IO6 psi, and vn = 0.25. The boundary conditions are such that the ends are considered to be freely supported, i.e. tangential and transverse dis- placements are restrained. With regards to the shelf geometry, various R,fa ratios were chosen so as to model deep, moderately deep, and shallow shells. In the analysis, a 16 x 16 mesh was used to model the entire shell, No symmetry conditions were used SO as to allow the mode shapes to be plotted.

Table 16 shows the comparison between the present element and the exact solution of Reddy [31] for the first frequency, w,. For the thin shell, and regardless of the R,/a ratio, the present element solutions are within 1% of the exact solution, and

Table 15. Cantilevered shallow cylindrical she11

Present F.E. Reference Frequency (8 x 8) (16 x 16) [291

1 10.3949 10.5337 10.595 2 16.8703 17.0913 16.990 3 29.2692 30.0254 30.649 4 49.7319 41.8442 42.227 5 44.7232 46.5719 47.684 6 59.1120 62.5323 65.450

: 82.3834 87.3282 89.99 1 83.4865 87.990 90.143

M&d element for laminated plates and shells 609

I..

i-.

I

____-e-e 6

-Q- Roddy (IS801 S.S. Mat.1 (45/-45/-45/45) - Closrd Form

A Reddy it980 1 l Present Study >

C.C. Mat. lI(45/-451451-45 1

we -Cl--- Prerent Study ---a--- Reddy(l980)

1

S.S. Mot.It (45/-45/45/-45) ___)__ Bert 8 Chen

(1978)

4 io I; 20 i5 io ;b 40 45 io j5 -

Aspect Rotlo

Fig. 9. Fundamental frequencies versus plate aspect ratio for four-layered laminates.

for thick shells, the difference is 7% for R,/a = 1.0 and decreases to -0.4% for R,/a = lo3 (a plate). Thus, in general, good agreement is observed between the present finite element and the exact solution.

The results in Table 17 show an interesting trend. Depending on the amount of curvature and thickness of the shell, the lowest frequency may not correspond to the first symmetrical mode, as is intuitively ex- pected. For example, in the thin, deep shell, a/h = 100, and R,/a = 1.0, the first symmetric mode corresponds to the third lowest frequency, whereas

the lowest frequency actually corresponds to the first antisymmetric mode. Figure 11 shows the mode shapes corresponding to the first three frequencies for the above case. This type of behavior is peculiar to shell problems, in which curvature has a pronounced effect in increasing the frequencies of the symmetric modes. An interesting and detailed discussion of this type of behavior can be found in Leissa et al. [29]. Thus, in [31] what Reddy refers to as the fundamental frequency may not be the appropriate terminology. We consider the lowest frequency as the fundamental frequency, and in the case of R,/a = 1 .O and

b

Fig. 10. Laminated cylindrical shell.


Table 16. [O/90] Laminated cylindrical shell, fundamental frequency

&la 0 Reddy Present F.E.

1.0 100 65.474 66.1354 10 9.9986 10.7371

2.0 100 34.914 34.9436 10 9.1476 9.5239

3.0 100 24.516 24.7181 10 8.9832 9.1841

103 100 9.6873 9.6505 10 8.8998 8.8650

a/h = 100 this corresponds to the first antisymmetric mode. The ‘fundamental frequency’ which Reddy lists is actually the third frequency, as shown in Table 17.

Interestingly, for thick shells the lowest frequency does correspond to the first symmetric mode for all R,/a ratios. In Table 17, those frequencies which correspond to the first symmetric mode are under- lined; note the change as the curvature and thickness change. This trend was determined by plotting the mode shapes for all frequencies, which are not included here.

Table 17. [O/90] Laminated cylindrical shell, first three frequencies

&la alh

1.0 100 1.0 10

2.0 100 2.0 10

3.0 100 3.0 10

10’ 100 10” 10

Present F.E. QJI 01 03

42.1013 59.2099 66 1354 10 7371 L 19.4124 22.1003

31.3616 34.9436 50.7771 9.5239 20.7523 22.0257

24 7181 28.8023 44.0565 9.1841 20.9788 21.6111

9.6505 26.6131 26.6131 8.8650 21.2503 21.2512

4. CONCLUSIONS

The demonstrated success of the present finite- element mixed model for the anaylsis of thin/ moderately thick laminated plates and shells is mainly attributed to the careful selection of the polynomial- interpolation parameters in its independently- assumed strain field. A crucial point here concerns the use of a set of bubble functions as additional kinematic DOF to facilitate this selection, as was originally suggested for isotropic problems [20,21].

Fig. Il. Mode shapes for first three frequencies of vibration.


In addition to e~~inating the locking problem, this resulted in good element performance which is fairly insensitive to geometric (mesh) distortions.

l2

However, from the standpoint of numerical im- plementation, the introduction of such additional 13. DOF may lead to certain complications, especially for dynamic applications, i.e. when ‘formally’ condensed on the element Ievel, these DOF lead to a

14.

natural-frequency-dependent constraint equation. An approximate approach is utilized here by 15. simply discarding the contributions of the additional DOF in formulating the lumped mass matrix, and this appears to be significant for the compu- 16. tational efficiency of the element. Its justification is primarily provided by the sufficiently accurate results obtained in the various vibration problems presented

17 ’

here.

Acknowtedgemenz-This work is supported by NASA Lewis 18. under research grant number NAG3-901.

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REFERENCES

20.

21.

J. M. Whitney, Shear correction factors for orthotropic laminates under static load. J. appi. Mech. 40, 302-304 (1973). A. K. Noor and J. M. Peters, A poszeriori estimates for the shear correction factors in multilayered composite cylinders. NASA Langley Research Center (1988). C. W. Bert, A critical evaluation of new plate theories applied to laminated composites. Report OU-AMNE- 83-3 (1983). J. N. Reddy, A simple higher-order theory for laminated composite plates. J. appl. Mech. 51, 745-752 (1984). K. H. Lo, R. M. Christensen and E. M. Wu, A high-order theory of plate deformation, part 2: laminated plates. J. appl. Mech. 44, 669-676 (1977). H. Murakami, Laminated composite plate theory with improved in-plane responses. J. appl. Mech. 53,661-t% (1986). R. L. Spilker, A hybrid-stress tinite element foundation for thick multilayer laminates. Compuz. Strucz. 11, 507-514 (1980). R. L. Spilker, Hybrid-stress eight-node elements for thin and thick multilayer laminated plates. hr. J. Numer, 29. Meth. Enmta 18. 801628 (1982). R. L. Spik<r, D. M. Jakobs and B. E. Engelmann, Efficient hybrid stress isoparamet~c elements for mod- 30. erately thick and thin multilayer plates. In Hy&rid and Mixed Finize Element Mezhods (Edited by R. L. Spilker and K. W. Reed). AMD-~01-73. ASME, New York 31. (1985). R. A. Chaudhuri and P. Seide, Triangular finite element for analysis of thick laminated plates. Inz. J. Numer, 32.

22.

23.

24.

25.

26.

27.

28.

Mezh. E&g 24, 1203-1224 (1987). D. R. J. Owen and 2. H. Li. A refined analvsis of laminated plates by finite element displacement

methods--I. Stress and displacement. Cornput. Struct. 26,907-914 (1987). S. C. Panda and R. Natarajan, Finite element analysis of laminated composite plates. Inz. J. Numer. Meth. Engng 14, 69-79 (1979). T. Y. Chang and K. Sawamiphakdi, Large deformation analysis of laminated shells by finite element method. Gompuz. Struct. 13, 331-340 (1981). D. J. Haas and S. W. Lee, A nine-node ~~~-strain finite element for composite plates and shells. Gomput. Struct. 26, 445-452 (1987). J. N. Reddy and N. D. Phan, Analysis of laminated composite plates using a higher-order shear deformation theory. Int. J. Numer. Meth. Engng 21, 2201-2219 (1985). Y. W. Kwon and J. E. Akin, Analysis of layered composite plates using a high-order shear deformation theory. Cornput. Struct. 27, 619-623 (1987). B. N. Pandya and T. Kant, Flexural analysis of laminated composites using refined higher-order Co plate bending elements. Gompuz. Meth. appl. Mech. Engng 66, 173-198 (1988). - H. V. Lakshminarayana and S. S. Murthy, A shear- flexible triangular finite element model for laminated composites plates. Int. J. Numer. Mezh. Engng 20, 591623 (1984). C. A. Rogers and C. E. Knight, An axisymmetric linear/high-order finite element for filament-wound composites-I. Formulation and algorithm. Gompuz. Struct. 29, 265-271 (1988). A. F. Saleeb, T. Y. Chang and W. Graf, A quadrilateral shell element using a mixed fo~ulation. Gomput. Struct. 26, 787-803 (1987). A. F. Saleeb and T. Y. Chang, An efficient quadrilateral element for plate bending analysis. Inz. J. Numer. Mezh. Engng 24, 1123-1155 (1987). S. G. Lekhnitskii, Anisotropic Plates. Gordon & Breach, New York (1968). R. L. Kidder, Reduction of structural frequency equations. AIAA Jnl 11, 892 (1973). M. Paz, Dynamic condensation. AIAA JnI 22,724-727 (1983). M. Paz, Modified dynamic condensation method. J. szruct. Engng, ASGE 115, 234-238 (1989). R. J. Guyan, Reduction of stiffness and mass matrices. AlAA Jnl3, 380 (1965). T. J. R. Hughes, The Finite Element Method. Prentice- Hall, New York (1987). R. G. Anderson, B. M. Irons and 0. C. Zienkiewicz, Vibration and stability of plates using finite elements. Inz. J. Solids Szrucz. 4, 1031-1055 (1968). A. W. Leissa, J. K. Lee and A. J. Wang, Vibrations of cantilevered shallow cylindrical shells of rectangular planform. J. Sound V&r. 78, 311-328 (1981). J. N. Reddy, A penalty plate bending element for the analysis of laminated anisotropic composites plates. Znt. J. Numer. Meth. Engng 15, 1187-1206 (1980). J. N. Reddy, Exact and finite-element analysis of laminated shells. Department of Engineering Science and Mechanics, VP1 (1983). P. M. Pinsky and R. V. Jasti, A mixed finite element fo~ulation for Reissner-Mindlin plates based on the use of bubble functions. Zni. J. Numer. Meth. Engng Zs, 1677-1702 (1989).

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A MIXED ELEMENT FOR LAMINATED PLATES AND …...Mixed element for laminated plates and shells 599 The mass matrix M is defined as in which the D, matrix elements are defined as 1 I