FR.EE VIBRATIONS AND STABILITY ANALYSIS OF LAMINATED ... · theory, when applied to laminated composite plates and shells, provides fairly accurate results (stresses, displacements,

Pergamon

Computers ci S~rucmres Vol. 63, No. 6, pp. 1149-l 163, 1991 0 1997 Published by Elsevier Science Ltd. All rights reserved

Printed in Great Britain

PII: soo45-7949(96)00384-7 004s7949/97 $17.00 + 0.00

FR.EE VIBRATIONS AND STABILITY ANALYSIS OF LAMINATED COMPOSITE PLATES AND SHELLS WITH

HYBRID/MIXED FORMULATION

A. S. Gendyt, A. F. Saleebt and S. N. Mikhailt t Cairo University, Giza, Egypt

$ The University of Akron, Akron, OH 443253905, U.S.A.

(Received 20 April 1995)

Abstract-Modem theory for applications of laminated plates and shells calls for detailed study of the effect of large spatial rotations on the geometric stiffness for stability analysis as well as inertia operators for vibrations. These two issues are carefully examined here in conjunction with the recently developed mixed finite element formulation for plates and shells with low-order displacement/strain interpolations. An extensive set of stability and vibration problems has been solved to demonstrate the effectiveness and general utilities of the formulation described for laminated plate and shells with arbitrary geometry. 0 1997 Published by Elsevier Science. Ltd.

I. INTRODUCTION

Being developed on the basis of hypotheses and assumptions, shell theories and formulations are inherently approximate in nature. In the classical laminate theory (CLT), for instance, the usual Love-Kirchhoff assumption of plane sections remain plane is effective, thereby the shear deformations are totally neglected. It is well accepted that this classical theory, when applied to laminated composite plates and shells, provides fairly accurate results (stresses, displacements, frelquencies, buckling loads) for very thin structure applications, i.e. ratio of total (laminate) thickness-to-side length (and/or smallest initial radius of cu:rvature) << 1. However, most of the advanced present-day composite materials have a low transverse shear modulus and, therefore, transverse shear deformations play an important role in the kinematic assumptions. In order to account for such deformations, an alternative theory attributed to Reissner-Mindlin type of assumption is suggested. In this theory, a constant shear angle through the plate thickness is assumed and, therefore, this theory has been referred to the constant angle shear deformation theory [l, 21. Varying the shear modulus through the thickness of the laminate makes the constant angle shear assumption not sufficient. Thus, a layerwise constant shear angle theory has been developed in which a constant shear deformation angle is assumed for each ply [35:1. Elements based on this latter assumption were found to be computationally expensive [3].

In addition to the above classifications, kinematic assumptions, as tar as the shear deformations are concerned, can be subdivided into two further categories, i.e. lower- and higher-order shear

deformation theories [e.g. 6,7]. The thickness coordinate terms included in the kinematic assumptions are the distinguishing criteria between the lower-order approach and higher-order one. With regard to the latter approach, for example, kinematic assumptions based on a cubic variation in the normal direction for the in-plane displacements and a constant (function of in-plane coordinates only) for the transverse displacement are proposed in Refs [8,9].

From a numerical standpoint, formulations based on Reissner-Mindlin assumptions in general, do not converge to the correct solutions in the limiting case of thin plates and shells due to the well-known shear locking phenomenon, especially when low-order interpolation functions are introduced to represent the displacement field in the finite element level. In order to alleviate the problem of shear locking and membrane locking for curved shell, several approaches have been proposed. Among many others, the most popular approaches are: (i) reduced integrated schemes demonstrated by Hughes et al. [lO--121; (ii) hybrid/mixed approach based on multi-field variational principles[ 13-l 81. The latter approach is utilized to develop the present model.

Our objective in this paper is to develop a simple, shear flexible, shell element model adaptable to isotropic as well as laminated composite material types for free-vibration and stability analyses. To this end, several features are required for a “robust” model: (i) geometric shape generality can be achieved through a quadrilateral finite element; (ii) material generality can be achieved through a multi-layered composite; (iii) minimum number of nodes can be introduced through low-order interpolation functions; (iv) shear deformations can be included through Reissner-Mindlin assumptions; (v) allevia-

1149

1150 A. S. Gendy et al.

tion of shear as well as membrane locking through utilizing the hybrid/mixed variational principles; (vi) facilitate extending this formulation to handle large deformations in space, as well as the connection with other element types, through: (a) utilizing displacements and rotations (not displacement derivatives) to represent the nodal degrees of freedom, (b) applying a second-order precision for representing the incremental rotation in space which leads to second-order accurate geometric stiffness, (c) devel- oping a consistent mass matrix for translational as well as rotational inertia from consistent kinematic assumptions accounts for the effect of rigid finite rotations in space.

In order to fulfill the aforementioned requirements, the five-noded, quadrilateral, mixed element developed by Saleeb et al. [18], for linear static analysis of isotropic materials has been extended for stability and vibration analyses of laminated composite plate and shell structures. In the underlying formulation, the Hellinger-Reissner variational principle is employed in which both the displacement and strain fields are assumed independently. The assumed strain field represents the entire laminate utilizing the first-order (constant) shear deformation theory. It is noted, however, that the present formulation cannot predict an accurate distribution of the transverse shear stress over the laminate thickness [18]. The present element was applied with great success to a number of free vibration and stability problems for isotropic and composite plates and shells. At this stage, the developed five node hybrid/mixed element which includes the shear deformation effects has performed well without any sign of membrane or shear locking.

This paper is presented in the following subsequent sections: general form of incremental variational principle; finite element formulations for accurate geometric and mass matrices; examples and numerical results; and conclusions.

2. GENERAL FORM OF INCREMENTAL VARIATIONAL PRINCIPLE

As with any incremental step-by-step solution, the static and dynamic variables in an equilibrium coniiguration at time “t” are assumed to be known, with the objective being to determine their values in an unknown neighboring equilibrium configuration at a later time “t + At”. A modified Hellinger-Reiss- ner variational principle [e.g. 19,201 provides a starting point for the present incremental analysis. This takes the following form in the updated Lagrangian (UL) description in which the con@ur- ation at time 9” is taken as a reference:

aAa,.,a = 0 (la)

AM = Ar&Au, be) = rrua(f + At) - x&)( 1 b)

AnHR = s

[AuTpl + A’ii - iAeTCAe + aTA? ”

- AeTC(e - ?)]du - A W (lc)

where the integration is carried out over the current volume v;

u, Au

I + Ari

Ae

AC

initial and increment displacement, respectively; acceleration vector at time “t + At”; vector of independently assumed incremental updated Green strains; AZ (linear) + Arj (nonlinear) vector of incremental updated Green “geometric” strains; Alamansi strain vector at time “t” (accumulated from assumed incremental strains); Alamansi geometric strain vector at time

.,t”; anisotropic material stiffness matrix; incremental second Kirchhoff stress vector (also called Truesdell stress rate or increment); true (Cauchy) stress at time “t”; “lumped” work-equivalent term corresponding to prescribed body force and surface tractions in configuration “t + At “,

e

c^

C Aa = CAe

a AW

with

Ai, = f(Auij + Aaj,,); A& = ~Au~,~Au~,, (2b)

and the differentiation is with respect to the current coordinate .x at time “t”. The last term in the bracket of ‘,n’( lc) typically arises from the so-called compatibility mismatch in the hybrid/mixed formulations [21-241.

In the present formulation, three different types of Cartesian coordinate systems are referenced (see Fig. 1). The first system is called tixed, global, reference frame which is defined by its orthogonal base vectors e, (i = 1,2,3) and is used to define element geometry and its translational displacements. The second one is fiber system which is constructed at each node, the associated base vectors are 4 (i = 1,2, 3), where e$ coincides with the nodal fiber direction. This orthogonal fiber triad is used for defining the rotational degrees of freedom at the nodes. The last coordinate system is a lamina system, or a local reference frame, which is defined at each integration point in an element such that two of its axes are tangent to the lamina surface. The corresponding orthogonal tread, e: (i = 1,2,3), with es taken to be normal to the lamina surface, is selected in such a way that, at the integration point

considered, the in-plane lamina-tangent vectors e: and e$, and the corresponding unit covariant basis vector e, and e, which are locally tangent to the “curvilinear” surface coordinates r and s, respectively, share the same angular bisector [18]. This lamina system is Imost convenient for invoking the plane stress assumption in the e: direction. In addition to the main three Cartesian systems defined above, another local reference frame termed “material fiber system” is defined for laminated (composite) element. Its in-plane lamina-tangent vectors are coincident with the principle material axes, that is, parallel and normal to the material fiber directions. The out-of-plane vector is parallel to the lamina vector e4 This material fiber system is convenient for defining the constitutive relations for the laminated composite material.

Analysis of laminated composite plates and shells 1151

the element (lamina) basis. This implies a global- lamina transformation for all the derivative operators in strain-displacement relationships (eqn 2). On the other hand, the independent (assumed) strain increments are assumed directly in the lamina basis.

In order to represent the composite laminated material, the material stiffness matrix, C in eqn (1), is no longer constant through the element thickness. For an individual layer, the material matrix, C, which has known in the material fiber directions, has to be transformed to element, lamina, directions. Assuming plane stress state is the thickness direction, oj3 = 0, the material stiffness matrix in the material fiber direction is

Equation (lc) above is written with respect to (w.r.t.) a fixed global reference frame. However, in application to the present HMSHS element, shown in Fig. 1, all stress/strain quantities will be defined w.r.t.

Cl, Cl2 . .

CZI c22 . . .

c= . . c,, . . L 1 (3) . . . c44 . . . . . css

Typical node @ 1 Ae+k’

2

!%

i-

Y

!?I !?2

X

(8) Geometry, displacement. and fiber basis

(b) Tjpical hmina coordinates

Fig. 1. A typical shell element.


Line //to side 3-4

layer I f 4h

~. layer I

(a) Malerial tibre axes. (b) Integration points of layer I.

Fig. 2. Material fiber directions and location of integration points in a layer.

in which

c,, =-E!!-_-. 1 - V12VZI ’

c,2 = c2, = fg II 21

,-,, = G,2r c, = ;G23; css = :Gu (4)

where El, and EZZ are the longitudinal and transverse elastic moduli, respectively; G12 the in-plane shear modulus; Gz3 and Gs, the transverse shear moduli; v12 and vzl the Poisson’s ratios. The layer material stiffness matrix needs to be transformed from material fiber direction to the plate lamina coordinate axes. This can be accomplished utilizing the standard coordinate transformation relation

C=TTCT (5)

where C is the material stiffness matrix referred to lamina coordinate; T is the transformation matrix [25]. The latter matrix generally varies along the element surface from one integration point to another. This is because the lamina basis vectors (et, e& e$), in general, are not coincident at all integration points within the element. In order to evaluate the transformation matrix T, the material fi&r angle 4*, is specified with respect to a distorted side 3-4 of the element as shown in Fig. 2(a). The transformation angle 4, can be calculated and then used for evaluating the transformation matrix T to transform the layer material matrix, c, to the local (lamina) element material matrix, C. This material stiffness, C, can be evaluated at each integration point to form the incremental stress-strain relations

in the element local coordinates (lamina coordinate) for layer /

Expression (lc) is a basic equation for general nonlinear (incremental) dynamic analysis for shell- type structure accounting for the effect of prebuckling displacements, instability static and dynamic states, as well as post-buckling response. However, restrict- ing the scope of the present paper to free vibration and linearized buckling applications, a simple form is utilized as

AIIHR= I

T [Au p ‘+ “‘ii - fAeTCAe + aTA6]d V - A W.

(7)

Once the finite element discretizations are introduced for the corresponding fields, i.e. incremental displacements Au and the incremental independent strain, Ae, the stationary conditions, BAIIHR = 0, with respect to displacement and strain parameters will then yield the governing equations of the mixed element. The first term in eqn (7) gives the mass matrix, the second and third terms give the linear stiffness matrix; and the geometric stiffness matrix results from the contribution of the quadratic (nonlinear) strains in the last term in the bracket. Explicit forms of various arrays above for the present shell model are given in the next section.


3. FINITE ELEMENT FORMULATIONS

3.1. Element geometry

For the present isoparametric shell element HMSHS, shown in Fig. 1, the geometry in any configuration (at time t) is defined in terms of natural (isoparametric) coordinates (r, s, t) as follows [25- 271:

1 = i, Nktxk + ; i ‘hkNk’e$k’ (8) k-l k-l

where, ‘x are the cartesian global coordinates of an arbitrary point in the shell element; ‘xk, the position vector of nodal point k on the element reference surface; ‘E$~, the c’omponent of a unit pseudo-normal vector emanating from node k in the fiber direction; ‘hk, the fiber dimension (shell thickness) calculated at node k; E the isoparametric normal coordinate; and Nk(r, s), the two-dimensional shape functions associated with node k, i.e. for HMSHS element [27]

Nk = $(l + rrk)(l -t s&) - ;N,, K = 1, 2, 3, 4 (9a)

Ns = (1 - r2)(1 - s2). (ob)

3.2. Kinematics

Five degrees of freedom (DOF) are defined at each nodal point, that is, three translations (a, u, w) along the Cartesian global axes and two rotations (0,, 0,) about mutually-perpendicular axes 4 and e$. In total HMSHS element thus has 25 DOF. Using eqn (9) at time “Cl”, “t” and “t + At”, we thus have (providing no change in the thickness)

u = i Nk’nk + ; i Nkhk(‘dk’ - “ef”‘) (loa) k-l K-I

Au = i Nt Aa + ; i &hk(’ + “efk) - ‘ef”‘) (lob) k-l k-l

with

u = ‘x .- ox , A~u=‘+~‘x--‘x. (104

For small incremental angles A& and At!12, linearized approximation for eqn (lob) [25] is used,

I+ A@, _ @ = Ad x ‘es = (AWe, + A&e2) x lea

= -A&e2 + At92’e, (11)

where Al& and A& are the rotations of the normal vector about basis vectors ‘e, and h from the configuration at time “t” to the conliguration at time “t + At”. Equation (lob) can now be written as

An = i NkAnk K-l

+ f i Nkhk(--A6:‘efk’ + L#%fk’). (12) k=l

The relation in eqn (lOa) is directly employed to evaluate the total element displacements and their derivatives; but the linearized expression in eqn (12) is utilized to calculate the incremental element displacements. This latter expression can be written symbolically in terms of a matrix of the shape functions as

Au = NAq (13)

where the vector of incremental nodal displacement, Aq, is given by

Aq= [Au,, AY,, Aw,, A@“, A@“, . . . , de!“, A@lT (14)

3.3. Assumed strains

For the independent strain field, be, in HMSHS element, several useful guidelines were described in [18,27] based on the following consideration: (i) suppression of kinematic modes; (ii) free shear/ membrane locking; (iii) invariance requirements; (iv) element geometry distortion. With this in mind, a polynomial approximation for strain field was used in the context of linear analysis of plates and shells [27]. This same interpolation is used here for nonlinear case. This incremental lamina strains are assumed as

Be& = /I7 + /38r + l&s + ii/310 + /Lr + 812s)

AeL = /IIS + BBS + b,r (1%

where /Ii (i = 1 + 19) are the generalized strain parameters. Strain field in eqn (15) still needs further modifications to account for element distortion, which is referred to non-orthogonality of vectors e, and e, . An appropriate strain tensor on this distorted configuration, AeU, can be simply obtained by transformation of Ae; as [ 181

Be = j,Ae’$ (16)


where j., is the Jacobian transformation matrix approximated at element’s centroid which can be calculated as

[

ef$e,b e~$e,), 0

j. = ei+(e,), e4.(e,), 0 1 (17) 0 0 1

where (.)O indicates a quantity calculated r = s = T= 0. Now, with eqns (15) and (17), one can express independently assumed incremental lamina strain vector as

Ae = PA/J (18)

where P is a (5 x 19) strain-interpolation matrix and A/Ii (i = 1 -+ 19) is the generalized strain parameter.

3.4. Element stiffness equations

Substituting eqns (12) and (18) into the variational principal of eqn (7), one can arrive at the following set of stiffness matrix after invoking the stationary condition of functional in eqn (7) with respect to AjJ and Aq, respectively [22-251

where

Mq + (KL + KNr.)Aq = 0 - Q, (19)

M = mass matrix (see Section 3.7)

Kr_ = GW-‘G

with

Q, = BTa dV I

(20)

H= PTCPdV; G= s 1

PTCBdV. (21) 0 ”

In the above, M is the mass matrix; KL and KNL, the linear and nonlinear element stiffness matrices, respectively; Q,, the internal force vector; Q, nodal “external” force vector in configuration “t + At”; B, the lamina linear strain-displacement transformation (i.e. AZ = BAq); BNL, the lamina nonlinear strain displacement transformation matrix (i.e. #q&LAq = j,uuArf, dV) d and a, the matrix and vector, respectively of Cauchy stresses at configuration “t”; 4 the acceleration at time “t + At”; Aq the incremental nodal displacement. Details of specific of B, BNL, and d for degenerated shell elements can be found in Ref. [25].

To evaluate expressions in eqns (20) and (21), standard numerical integration is considered for each layer and then summation over the total number of layers, n, is used to account for the variation of material properties and/or material fiber directions along the shell element thickness. In this, 3 x 3 Gauss integration points are used for the lamina directions, and two integration points per layers in the thickness direction. To calculate the T coordinates of a particular integration point in layer L, based on the t, coordinate measured from the mid-surface of this layer as shown in Fig. 2(b), the following simple “change-of-variable” procedure [28] is used

i=-1+; [ -h,(l-r,)+2ih, . 1 (22) /‘I The matrices, N, B, BNL and P are functions of the natural coordinates r, s and ?: Therefore, the volume integration in expressions in eqns (20) and (21) extends over the natural coordinate volume and the volume differential, d V, need also be written in terms of natural coordinates. Utilizing eqn (22), expressions in eqn (21), can be rewritten as

PTC,PIJl 2 dt, dr ds

PTCcBIJI 2 dtc dr ds (23)

with

d;= $ dt,.

3.5. Linearized buckling and free vibration problem

The global stiffness equations (19) provide the following criterion for linearized buckling,

IIKL+KNLII =O (25)

where 11 11 stands for the determinant and KNL is now calculated for a predefined initial stress distribution corresponding to pre-buckling loading state (load factors).

For free vibration analysis, the equilibrium equations can be obtained from eqn (19) as

Mq + K,_q = 0. (26)

The solution of the above equation can be postulated to be of the form

q = 9 sin w(t - to). (27)


Substituting eqn (27) into eqn (26), one obtains the generalized eigenproblem, from which the eigenvalue, w and the corresponding eigenvector a, can be determined

(KL - o*M)cD = 0. (28)

In the present paper, the subspace iteration method described in Ref. [25] is utilized for the eigenproblem solution.

3.6. On the precisfon of geometric stiffness matrix

Consider using the HMSHS element in a “sufficiently refined” mesh to model a “smooth” shell, and assume the shell normals (thickness fibers) are straight and Lnextensible which is customarily employed for any degenerated shell model [e.g. 25, 26, 291, in accordance with the classical Mindlin/Reissner, shear-flexible, theories. By impli- cation, the fiber direction 4 in the current configuration (omitting, for convenience, the configuration indicator subscript) can be viewed as being coincident wllth that of the “true” normal of the element midsurface, i.e. e<. Further, we assume, without loss of generality, that the other two fiber vectors ef and 4 are chosen to coincide with the midsurface tangent vectors e? and e:, respectively. These “preliminaries” enable us to conveniently transform various components of the incremental motion in eqn (10) to the local (lamina) system. Instead of using the linearized expression, i.e. eqn (1 l), for the term associated with the change in direction of mid!;urface normal, a second-order accurate expression which is referred to as Hughes- Winget formula [30, 311 is used, i.e.

with

‘+*‘,: - ‘4 = (A8 + ;A@)‘4 (29a)

A@ = Ar?, = skew(A8) = cvkAflk VW

where cVk is the alternating (or permutation) tensor; A8 the skew-symmetric matrix whose axial vector is A8 which has the generalized incremental rotations (A&, A&, At&) as its components. Utilizing eqn (29), we can easily show that the incremental displacement expression in eqn (10) yields

A& = A& - 7: 11A811* (30)

where overbars denote local components; AP and Au, are the incremental displacements of an arbitrary

point (r, s, ;) in the shell and its anchor point (r, s, o) on the midsurface, respectively; A8 is the rotation vector (A& = 0) of the midsurface normal. Note that AL, A0 and thickness h are interpolated from nodal values.

Clearly, when similarly transformed, the linear kinematic eqn (12) will yield exactly the same expression as in eqn (30) with the exception of the single nonlinear term shown underlined in eqn (30). However, this additional term is associated with a shell-normal extensibility during the incremental motion, i.e. nonzero lamina strain A&,. Based on the assumption of inextensible shell thickness, this term must, therefore, be dropped a priori in any degenerated shell formulation. Thus, the assumption of linearized kinematics, eqn (12), for the present HMSHS element actually implies a second-order precision for the representation of the incremental rotation matrix. In view of its second-order accurate geometric stiffness, the present development for HMSHS fully complies with the classical instability conditions, i.e. the Trefftz buckling criterion for potential-energy formulations or its generalization for various mixed methods given recently in Refs [32]. Therefore, the HMSHS element can be effectively employed to model shell buckling problems.

3.7. On the precision of mass matrix

A similar form of eqn (29) can be used to obtain the orientation of a fiber vector ’ + A’e,, resulting from a rigid finite rotation 0 in space, whose initial components are denoted by ‘e3, i.e.

‘+&e, = R(B)%, (31)

where the orthogonal rotation matrix [33], R, can be approximated as

R N I + 8 (first-order) (32a)

R N I + fi + i& (second-order) (32b)

where I is a (3 x 3) identity matrix. Taking the time derivative (rate of change) of both sides of eqn (31), and noting that the original vector Oel is constant, results in

‘+A’& = %oep = a RT’+Afe = $+++, (33a)

where

&=%RT (33b)

is the familiar skew-symmetric matrix for spatial representation of the angular velocity in terms of the rotational-tensor rate. In the above, the superposed dot signifies time derivative (e.g. % = dR/dt); a superscript “T” the transposition of a matrix and the use of a “hat” symbol conforms to that :n eqn (29).


Substituting eqn (31) into the total displacement expression at time (t + At) yields

‘+A’u=u,+;h(R-I)“e, (34)

where II., h, R and “eX are interpolated from nodal values. The velocity, L and acceleration, ii, can be obtained, respectively, by taking the first and second derivative of eqn (34) with respect to the time. These can easily be written as (after employing eqn (33a))

Wa)

Wb)

where ci, and Care the angular velocity and angular acceleration skew symmetric matrices, respectively, at t + At.

where (a) indicates a scalar (dot) product of vectors. Knowing the fact that the second and third terms in the above equation are linear functions of the thickness i. These two terms will vanish after integrations provided that all the layers have the same material density. Thus, eqn (37) can be rewritten as

.

Substitution of nodal interpolation for displacements and acceleration in eqn (38), and invoking the stationary condition, i.e. 6AzKE = 0, with respect to incremental displacements yields the governing mass matrix for large deformation formulation of the HMSHS element, i.e.

(39)

with

My = s ”

dV (40)

Employing the expression in (31) to obtain the final where orientation of the base vector in the nodal fiber, e3, at time t + At from its orientation at time t, we can rewrite the incremental displacement in eqn (lob) and

pj = NkpNj h& = hk/,.i (41)

the acceleration in eqn (35b) in terms of the base vector e3 at time t as

‘e3 = h, e32, e331T.

Au = An, + i h(R(A8) - I)‘e, (364

‘+Ali=ii,+$h(dM+&~el (36b)

where R(A0) is the orthogonal incremental rotation matrix.

Substituting eqns (36) into the kinetic energy, i.e. first term in eqn (7), yields the following expression after consistent linearization for the mass matrix of the HMSHS element

Specializing eqn (38) for linearized vibration analysis, several assumptions must be considered to eliminate the effect of large deformations. Provided that the initial configuration is at time “o”, and the final configuration is at time “At” due to small rotation vector, AtI, from “0” to “At”, the following limitations can take place

‘e, + oe3 ‘+A’fi (at “t + At”)+ ii, (at “At”);

&(at “t + At”) + 6: (at “At”)

(42)

where the subscript “s” indicates small magnitude. Employing these assumptions, eqn (38) can be written as


Table 1. Comparison between theoretical and test normalized frequencies for a cantilever plate -

WlJ@Z=? Anderson [36] HMSHS

Barton [34] Plunkett [351 (4 x 8) mesh (4 x 8) mesh Mode Ritz Test Test 64 elements 32 elements

1 3.47 3.42 2 14.93 14.52 3 21.26 20.86 4 48.71 46.90 5 - - 6 - -

3.50 3.44 3.44 (s) 14.50 14.77 14.72 (a) 21.70 21.50 21.73 (s) 48.10 48.19 48.30 (a) 60.50 60.54 61.56 (s) 92.30 91.79 92.78 (s)

(s) denotes symmetrical mode; (a) antisymmetrical mode.

Notice that the dc is the final acceleration due to “small” rotation from initial configuration “0” to the final configuration “At”. Thus, eqn (43) can be rewritten utilizing eqn (lla) for small rotation as

(- A0,“e2 + A&Oe,) I> dl/.

This is a typical equation representing a dynamic effect for small deformation formulation [25,29].

4. EXAMPLES AND NUMERICAL RESULTS

Numerical computations are carried out to determine the capability of HMSHS element to predict the dynamic behavior and stability of isotropic and laminated composite plate and shell structures. The elIect of material anisotropy, transverse shear deformations, the number of laminate layers on the frequencies as well as the critical buckling loads are investigated. In this study, comparisons with :available “exact” solutions as well as other independent finite element solutions are presented.

4.1. Vibration analysis

4.1.1. An isotropic cantilever plate. A rectangular plate with dimensions of (L x L/2) and constant

thickness is considered as a first example. The mesh size considered is of 4 x 8 HMSHS elements for the entire plate. Experimental as well as numerical results for this particular problem are available in literature. Barton in Ref. [34] has examined this plate experimentally and also by using the conventional Ritz method; Plunkett in Ref. [35] has obtained the frequencies for this cantilever plate experimentally; and Anderson et ai. in Ref. [36] have obtained the frequencies for this plate using a triangular element based on the displacement-field assumption. The first six frequencies obtained by HMSHS element are given in Table 1 with those obtained in [34-361.

4.1.2. Skew cantilever plates. An isotropic skew cantilever plate of dimensions (L x L) and skew angle a was idealized using (3 x 3) mesh of HMSHS elements. The skew angles were taken as o”, 15”, 30”, 45” and the first two modes were obtained in each case. The results (Table 2) obtained using HMSHS element were compared with those obtained by Barton [34] using the conventional Ritz method as well as experimental test; Dawe [37] using parallelo- gram element; and Anderson [36] using triangular element. The results obtained by the HMSHS element agreed well with the test results of Barton; unlike the results obtained by the conventional Ritz method whose accuracy appeared to be dependent on the skew angle of the plates.

4.1.3. Variable thickness cantilever plates. A variable thickness rectangular cantilever plate of length L and width L/2 is considered. The thickness of the plate varies linearly along the plate width,

Table 2. Normalized frequencies of isotropic skew cantilever plates

Skew angle a0

Barton [34] Dawe [34] Anderson [36] HMSHS Mode Ritz Test 16 elements 8 elements 9 elements

0 1 3.49 3.43 3.47 3.43 3.47 2 8.55 8.32 8.52 8.61 8.64

15 1 3.60 3.44 3.59 3.57 3.58 2 8.87 8.68 8.71 8.60 8.85

30 1 3.96 3.88 3.95 3.98 3.94 2 10.19 9.33 9.42 9.19 9.59

45 1 4.82 4.33 4.59 4.67 4.58 2 13.75 11.21 11.14 11.01 11.47

1158 A. S. Gendy ef al.

Table 3. Normalized frequencies of a variable thickness cantilever plate

olJ@?&% Plunkett [35] Anderson [36]

Mode Test Variable thickness Constant thickness HMSHS

1 2.41 2.40 2.33 2.40 2 10.6 11.27 10.02 11.16 3 14.5 15.14 14.07 14.99 4 28.1 29.41 21.20 29.64 5 34.4 37.30 26.21 35.41 6 47.4 48.99 21.05 49.71

forming a tapered shape with an angle of 3.7” with a maximum thickness “h”. The plate is divided into (6 x 3) HMSHS elements with variable thickness. The first six frequencies were obtained and compared with the experimental results of Plunkett [35] and numerical results of Anderson [36] in Table 3. The frequencies obtained in [36] using constant, average, thickness elements are also presented in Table 3. The results for the variable thickness HMSHS elements agreed well with the experimental ones.

4.1.4. An isotropic cantilever shallow shell. The fourth example is that of a moderately thin, L/h = 100, moderately shallow, R/b = 2.0, cantilever cylindrical shell with an aspect ratio L/b = 1, where L, h, R and b are the axial length, thickness, radius and projected width of the shell, respectively. The shell is modeled using 8 x 8 mesh sizes. The first eight frequencies are tabulated in Table 4 with those obtained in [38] using Ritz method. Examining the results in Table 4 indicated that good agreement was observed for the first six frequencies, i.e. the maximum difference is less than 2%, while less accuracy was observed for the highest two frequencies. Leissa et al. [38] stated that the higher frequencies were usually less accurate than the lower ones when the approximate methods were used, which is the case here.

4.1.5. Comparison with experimental results for composite cantilever plates. The next numerical example compares the first four frequencies of anisotropic cantilever plate with experimental results reported in Ref. [39] along with numerical results in Ref. [40]. The plate is 0.2286 m length and 0.09144 m width. Symmetric laminations were considered comprising eight layers of unidirectional graphite- epoxy laminae. The thickness of each layer is 0.14866 x 1O-3 m. The material propertif-s are:

Table 4. Nondimensional frequencies, 6 = wL2m, for a cantilever shallow shell

Symmetric modes Antisymmetric modes Mode Leissa [38] HMSHS Leissa [38] HMHSS

1 16.990 17.171 10.595 10.603 2 30.649 30.646 42.227 42.904 3 47.684 47.593 65.450 65.833 4 90.143 92.423 89.991 92.249

El, = 128.932 GPa, Ej3 = 10.825 GPa

Gj2 = G13 = 4.206 GPa, Gls = 4.700 GPa

v,2 = ~13 = 0.33, p = 1550 kgn-‘.

Two lamination schemes, namely (O/ -45/O/45), and (90/45/90/-45),, are adopted. The plate is modeled using (8 x 4) HMSHS elements. The first three frequencies are given in Table 5, with those obtained in Refs [39] and [40]. As shown from Table 5, an excellent agreement between the frequencies obtained by HMSHS element and those obtained experimentally and numerically.

4.1.6. Laminated square plates. The effects of material anisotropy, transverse shear deformations, thickness-to-span ratio, and the number of laminated layers on the fundamental frequency of square plates (L x L) are investigated. In this study, simply supported square laminated plates with symmetric and skew symmetric laminations with respect to the middle surface are considered. Fundamental frequencies for the four cases are presented: (1) four-layer equal thickness cross ply (O”/90”/90”/Oo); which is equivalent to a three-layer, h, = h, = h/4 = h2 = h/2, symmetric cross-ply (O”/90”/Oo); (2) two-layer, equal thickness, antisymmetric cross-ply (O”/!W); (3) two-layer equal thickness, antisymmetric angle ply (45”/ - 45”); and (4) eight-layer equal thickness, antisymmetric angle ply (45”/-45”/45/-45/. . .). Each layer is a unidirectional fiber reinforced composite possessing the following material constants: E,/Ez = 40, Gz,/E2 = 0.5, G,JEz = 0.6, v12 = 0.25. The boundary conditions used for the cross-ply and the angle-ply laminate plates are:

(i) Cross-ply boundary conditions

U(X, u) = U(X, L) = u(0, y) = u(L, y) = 0

w(x, 0) = w(x, L) = w(0, y) = w(L, y) = 0

&(O, y) = &(L, y) = &(x, 0) = &(x, L) = 0.

(ii) Angle-ply boundary conditions

u(0, y) = U(L, y) = v(x, 0) = v(x, L) = 0


Table 5. Frequencies for a laminated composite cantilever plates, Hz

(O/-45/0/45), (90/45/90/ -45)* Mode Test [39] Argyris [40] HMSHS Test [39] Argyris [40] HMSHS

1 29.851 29.415 29.395 11.224 11.879 11.880 2 93.553 91.539 90.659 68.601 69.835 71.336 3 190.688 184.613 185.805 97.480 89.520 92.602

Table 6. .Nondimensional fundamental frequencies, 6 = (wLz/h)fi, of cross-ply square plates

[0~/90”/90”/0~] P/~“l L/h Cl’T FSDPT HSDPT HMSHS CPT FSDPT HSDPT HMSHS

5 18.215 10.820 10.989 10.845 10.584 8.757 9.010 8.823 10 18.652 15.083 15.270 15.128 11.011 10.355 10.449 10.456 20 18.767 17.583 17.668 17.648 11.125 10.941 IO.968 10.958 50 18.799 18.590 17.606 18.066 11.127 11.127 11.132 11.253

100 18.804 18.751 18.755 18.833 11.163 11.155 11.156 11.288

Table 7. Nondimensional fundamental frequencies, 6 = (oL2/h)m, of angle-ply square plates

[45/-451 [45/-451451-451. . . ] S-layer Llh CPT FSDPT HSDPT HMSHS CPT FSDPT HSDPT HMSHS

5 13.885 10.335 10.840 10.185 15.708 12.892 12.972 12.848 10 14.439 13.044 13.263 12.816 25.052 19.289 19.266 19.143 20 14.587 14.179 14.246 13.933 25.212 23.259 23.239 23.008 50 14.605 14.561 14.572 14.370 25.258 24.909 24.905 24.611

100 14.636 14.618 14.621 14.576 26.264 25.176 25.174 24.891

w(x, 0) = W(X) L) = w(0, y) = w(L, y) = 0

&(O, y) = &(L, y) = f&(x, 0) = fqx, L) = 0.

The square plate is modeled using (8 x 8) HMSHS elements. The fund.amental frequencies as a function of plate side to thickness ratio are tabulated in Tables 6 and 7 for cross-ply cases, and angle-ply cases, respectively. The classical plate theory (CPT) solution which includes the rotary inertia terms [41]; solutions obtained by the first-order and high-order, shear deformation plate theories [41], termed (FS- DPT) and (HSDPT), respectively, are also provided in Tables 6 and 7. The solutions obtained by HMSHS elements lay between those obtained by (FSDPT) and (HSDPT) in most cases. The classical plate theory (CPT) overestimate the frequencies especially in the plates when the shear deformations are significant, i.e. L/h < 10.

4.1.7. A laminated cylindrical shell. In Ref. [42], exact solutions for laminated shells with different curvatures were p:resented based on Sander’s shell theory into which shear deformations were accounted for. Specifically, thle problem chosen here is that of a cross-ply, [O/90”], cylindrical shell. The material properties are as follows: El, = 25 x IO6 psi; & = 1 x 106psi; Gu = GII = 0.5 x 106psi; Gz3 = 0.2 x lo6 psi; and vu = 0.25. The shell is simply-supported at its botmdaries, i.e. the tangential and transverse displacements are restrained at the boundaries. With regard to the shell geometry, various R/L ratios were considered so as to present

deep, moderately deep, shallow shells; and two different L/h ratios were investigated; i.e. L/h = 100 for thin shells and L/h = 10 for thick shells in which the shear deformations effect is significant; where L is the axial length, R the radius and h the total thickness. One half of the shell is modeled using 8 x 4 HMSHS elements and applying the appropriate boundary conditions at the plane of symmetry. The first symmetric mode for all cases obtained are tabulated with the exact ones obtained by Reddy [42] in Table 8. For thin shells, the solutions using HMSHS element are within 1.6% of the exact solutions and for thick shells, the maximum difference is about 2%.

4.2. Stability analysis

4.2.1. Isotropic plate with cutout. Stability of an isotropic simply-supported square plate (L x L) with central circular hole has been considered here. The

Table 8. Frequencies of [O/W] laminated cylindrical shells; Hz

W

1.0

2.0

3.0

10’

W

100 10

100 10

100 10

100 10

Reddy [42] “exact” HMSHS % Error

65.684 64.939 0.817 9.999 10.127 1.280

34.914 35.478 1.615 9.148 9.389 2.088

24.516 24.861 1.407 8.983 9.059 0.846 9.687 9.558 1.332 8.899 8.704 2.081


1.25 -

Srivatsa & Murty -

Present *

0.75 -

*

S.S.

0.00 ’ I I 1

0.0 0.2 0.4 0.6

d/L

Fig. 3. Buckling stress ratios of a square plate with cut-out.

plate is subjected to a uniform uniaxial stress. The effect of diameter of the circular hole, d, to plate dimension, L, on the buckling strength has been accounted for. The ratio of the buckling strength of this plate to the corresponding one of the unperforated plate is shown in Fig. 3 for various cut-out diameter sizes (as, in Fig. 3, refers to the buckling strength of a solid plate). As shown in Fig. 3, the buckling strength obtained using the HMSHS element is in good agreement with the available one in Ref. [43].

4.2.2. Buckling of laminated square plates. The effect of shear deformations, material anisotropy, and number of layers on the buckling of simply supported square plates have been investigated. The present finite element model has been validated by comparing its results with those obtained by Putcha et al. [44]

slmt#hl edge boundary conditioar:

30.0 1 = _.--., ~~pppf)& -- ---- 2s,o ,, fixed

20.0

15.0

10.0

5.0

l \ ‘. \ ‘.

0.0 - 0 IO 20 -30 40 so 60 70 80 90

Rber qk (deg=W

Fig. 4. Buckling stresses of laminated composite cylindrical panels, R/h = 10.

using mixed tinite element formulation (MFE) based on a refined higher order theory (nine-node quadratic element with eleven DOF per node); Reddy [6] using both first-order (FSDPT) and higher-order (HSDPT) shear deformation plate theory; Noor [45] using 3-D linear elasticity solutions; and the classical plate theory (CPT).

In all plate cases presented here, the laminated cross-ply laminates with equal thickness of individual layers is considered. The material properties used are:

v,2 = 0.25.

The boundary conditions used for the cross-ply simply-supported plates are:

u(x, u) = u(x, L) = &(x, 0) = f$(x, L) = 0,

o(0, v) = VW, y) = &(O, y) = e&Y y) = 0;

W(X, 0) = w(x, L) = w(0, y) = w(L, y) = 0.

Because of symmetry, one quarter of the plate is modeled using (4 x 4) HMSHS elements and applying the appropriate boundary conditions on the planes of symmetry. The use of a quadrant of the plate provides only the symmetric buckling modes, which is the case investigated here. In order to study the shear deformation effect, thick plates of L/h = 10 ratio are considered.

The critical buckling loads of a symmetrical laminated cross-ply are tabulated in Table 9 as a function of the modulus ratio and the number of layers. As observed from the results, the shear deformation effect is more significant for material with a high degree of anisotropy.

In Table 10, the effect of anisotropy and the number of layers on the critical buckling loads for skew symmetric cross-ply laminates are presented. The results obtained using the HMSHS elements are in good agreement with those obtained by Noor [45]. As expected, the classical plate theory overestimates the buckling loads for large numbers of layers and higher modulus ratios.

4.2.3. Buckling of composite cylindrical panels. The laminated composite cylindrical panel is of great interest in weight-sensitive aerospace structures, such as fuselage and wing skin which are composed of stringer stiffened plates of cylindrical panel shape. The stability of such structures subjected to axial compression stress is investigated for general unsymmetric laminate of [$, -@I. The fiber orientation, @, which is measured from the


Table 9. Non-dimensional buckling load, IV = NI L*/Ezzh’, of symmetric cross-ply laminated plates

No. of El,/Ez Reference lavers 3 10 20 30 40

HMSHS 3 5.2217 9.6898 15.1325 19.3262 22.9797 NOOR 1451 5.3044 9.7621 15.0191 19.3040 22.8807 MFE [4;1] - 5.3950 9.9427 15.3001 19.6752 23.3398 HSDPT 161 5.3933 9.9406 15.2980 19.6740 23.3400 FSDF’T [6 j 5.3991 9.9652 15.3510 19.7560 23.4530 CPT 5.7538 I 1.4920 19.7210 27.9360 36.160

HMSHS NOOR MFE HSDPT

5 5.2529 5.3255 5.4112 5.4096

9.8452 15.4339 20.1358 24.1668 9.9603 15.6527 20.4663 24.5929

10.1524 16.0100 21.0023 25.3086 10.1500 16.0080 20.9990 25.3080

FSDPT 5.4093 10.1360 15.9560 20.9080 25.1850 CPT 5.7538 11.4920 19.7120 27.9360 36.1600

HMSHS 9 5.3168 9.9760 15.8462 20.8931 25.2834 NOOR 5.3352 10.0417 15.9153 20.9614 25.3436 MFE 5.4147 10.1991 16.1745 21.3165 25.7908 HSDF’T 5.4313 10.1970 16.1720 21.3150 25.7900 FSDPT 5.4136 10.1890 16.1460 21.2650 25.7150 CPT 5.7538 11.4920 19.7120 27.9360 36.1600

cylindrical shell axial axis ranges from 0 to 90”. The L/R = 1.0; a/R = 0.5, R/h = 10,100 material constants for the present analysis are:

where L is the panel length; R, the panel radius; a,

E,,/Ex = 40, Gu/Ez = 0.6, the panel circumferential; and h the panel thickness. Boundary conditions at curved edges are clamped,

G2,/1$ = 0.5, vu = 0.25; which can be expressed as

~13 = VIZ; G,3 = G23. u # 0, v = w = 0, = 0, = 0 at top edge

The geometry of the panel is, u = 0, v = w = 0, = 0, = 0 at bottom edge.

Table 10. Non-dimensional buckling loads, N = N, L2/E22h3, of skew-symmetric cross-ply laminated plates

No. of E,,/Ez Reference layers 3 10 20 30 40

HMSHS 2 4.6250 6.0563 7.8139 9.4803 11.0792 NOOR 4.6948 6.1181 7.8196 9.3746 10.8167 MFE 4.7769 6.2756 8.1198 9.8751 11.5690 HSDF’T 4.7749 6.2721 8.1151 9.8695 11.5630 FSDPT 4.7718 6.2465 8.0423 9.7347 11.3530 CFT 5.0338 6.7033 8.8158 10.8910 12.9570

HMSHS 4 5.1098 9.0587 14.0829 18.5347 22.5077 NOOR 5.1738 9.0164 13.7429 17.7829 21.2796 MFE 5.2540 9.2344 14.2579 18.6706 22.5822 HSDF’T 5.2523 9.2315 14.2540 18.6670 22.5790 FSDPT 5.2543 9.2552 14.3320 18.8150 22.8060 CPT 5.5738 10.2950 16.9880 23.6750 30.3590

HMSHS 6 5.1988 9.5927 15.1459 19.996 24.278 1 NOOR 5.2673 9.6051 15.0014 19.6394 23.6689 MFE 5.3437 9.7788 15.3548 20.2038 24.4617 HSDPT 5.3420 9.7762 15.3520 20.2010 24.4600 FSDPT 5.3430 9.7893 15.3940 20.2800 24.5770 CPT 5.6738 10.9600 18.5020 26.0420 33.5820

HMSHS 10 5.2449 9.8648 15.6783 20.7258 25.1481 NOOR 5.3159 9.9134 15.6685 20.6347 24.9636 MFE 5.3899 10.0583 15.9169 20.9887 24.4239 HSDPT 5.3882 10.0560 15.9140 20.9860 25.4220 FSDPT 5.3884 10.0600 15.9270 21.0080 25.4500 CPT 5.7250 11.3000 19.2770 27.2540 35.2320


straight edge boundmy conditions:

500.0 f- -free - - - simply supported ---- fixed

400.0 -1 ,__ ,,..... -

N, . .

’ Qel=

.100.0 . . . . . . . ..’ -0’

,.0- 0 IO 20 30 40 50 60 70 80 90

Fiber angle (degrees)

Fig. 5. Buckling stresses of laminated composite cylindrical panels, R/h = 100.

Three different cases of boundary conditions are specified at the straight edges:

Case I free-edge Case II simply-supported; o = w = 0 Case III fixed support; u = v = w = 0, = 0, = 0;

where u is the displacement in the direction of the axial axis and v and w are the displacements in the plane of the panel cross-section.

The (10 x 10) HMSH.5 mesh is adopted for the geometry of the cylindrical panel. The effects of transverse shear deformations, ratio of panel radius-to-thickness, fiber orientation of angle-ply laminates on the critical buckling loads are investigated. The normalized buckling stresses for different fiber angles are presented in Fig. 4 for R/h = 10 and in Fig. 5 for R/h = 100. The buckling stress is significantly influenced by the change of fiber orientations, straight edge boundary conditions and R/h ratio. The maximum buckling stress is obtained near 4 = 20” for thick panel, R/h = 10 and near q5 = 90” for thin panel, R/h = 100, for clamped boundary conditions at the straight edges. For thick panels with clamped boundary conditions, the buckling stress increases with increasing the orientation angle 4 until it reaches the maximum at 4 = 20”. When the orientation angle exceeds 20”, the buckling stress reduces rapidly until 4 = 75”, then it grows up again. For thin panel with similar boundary conditions, the buckling stress remains near constant for orientation angles greater than 40”. On the other hand, in free edge straight panel cases, the buckling stress gradually reduces as the fiber orientation increases for thick as well as thin panel cases.

5. CONCLUSIONS

An extended formulation of the recently developed hybrid/mixed model for plates and shells to the case of stability and vibration analyses of general

laminated composite structures with arbitrary geometry was considered. In this, careful considerations were given to the crucial effects of large spatial rotations on the accuracy of the ensuing geometric stiffness matrix for stability analysis as well as translational and rotational inertia (mass) operators for vibrations. A large number of problems were solved and compared to several existing analytical as well as numerical solutions to demonstrate the effectiveness and general utilities of the formulation developed.

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FR.EE VIBRATIONS AND STABILITY ANALYSIS OF LAMINATED ... · theory, when applied to laminated composite plates and shells, provides fairly accurate results (stresses, displacements,