Geometrically non-linear dynamic analysis of laminated shells using Bézier functions

Pergamon ,n,. J, Non-L,neor Mechan,~~. Vnl 32, No. 3, pp. 425 442. 1997 , 1997 Elsewer Sconce Ltd

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GEOMETRICALLY NON-LINEAR DYNAMIC ANALYSIS OF LAMINATED SHELLS USING BBZIER FUNCTIONS

Vijaya Kumar and Anand V. Singh

Department of Mechanical Engineering, University of Western Ontario, London, Ontario. Canada N6A 5B9

(Received for publication 15 May 1996)

Abstract-In this paper a geometrically non-linear theory for the analysis of open deep laminated shell panels is presented. Parabolic variation of the transverse shear stresses through the thickness of the shell and the effects of rotary inertia are included in the formulation. Linear and non-linear stiffness matrices are derived using orthogonal curvilinear coordinate system for a general doubly curved deep laminated shell. The solution of the equation of motion is based on the Ritz method. Therefore, the Btzier surface patches are used as the admissible displacement fields to represent the shell’s middle surface displacement and rotation components and the resulting equation of motion is solved to obtain the transient response, using Beta-m time integration and Newton-Raphson iterations. A very good convergence of the responses is observed by using only the fifth order BCzier surface patches. The dynamic responses of cross-ply laminated circular and non-circular cylindrical panels pinned at the straight edges under a central point load are studied. Effect of the eccentricity on the dynamic response of non-circular cylindrical panels having the same plan form as the circular cylindrical panel is also examined. 0 1997 Elsevier Science Ltd. All rights reserved.

Keywords: laminated shell, non-circular cylindrical panel, geometrically non-linear vibration

I. INTRODUCTION

Structures made of thin laminated shell elements are increasingly used in aerospace, missile and auto industries due to their high strength to weight ratio. These applications require

such structures to perform under severe loading conditions which induce deflection of the order of the shell thickness or higher. Furthermore, the shells could also buckle under these loads. As small and moderate deformation theories are inadequate in predicting these responses, the need for large deformation and rotation theories arises.

The majority of works on the large deformation analysis of laminated shells are the

extensions of works on laminated plates, isotropic and orthotropic shells. The presence of bending-stretching coupling due to lamination, the anisotropic material properties and the non-linear stiffnesses complicate the analysis of laminated shells. Recently, geometrically non-linear theories were presented by various researchers. Stein [l] presented a two-

dimensional non-linear theory for plates and shells to predict stresses and deformations. Non-linear equations of equilibrium and buckling equations were derived for flat plates and cylindrical shells. Numerical results were presented for the buckling of a thick simply

supported flat rectangular plate in longitudinal compression. A Lagrangian formulation of a refined geometrically non-linear theory of anisotropic laminated shells was presented by Librescu [2]. The theory incorporated the effects of the material anisotropy. structural lamination, high-order dynamical effects as well as the presence of a steady temperature

field. Hirano and Vinson [3] studied the non-linear vibrations of circular cylindrical shells made of laminated materials. The effect of transverse shear on the amplitude frequency curves was examined for the case of a simply supported shell. Dennis and Palazotto [4] presented a large displacement and rotation formulation, which included parabolic transverse shear, to study the static response of laminated shells. In their work, the linear and

Contributed by P. Hagedorn

425

426 V. Kumar and A. V. Singh

non-linear stiffness matrices were derived using a symbolic manipulator program MAC- SYMATM for a cylindrical shell geometry and a displacement based finite element formulation was used for the solution. The theory was then applied to a problem of a transversely loaded isotropic deep arch. Later, Tsai and Palazotto [S] extended the above problem to study the non-linear vibrations of cylindrical shells. Numerical results were presented for isotropic cylindrical shallow panels under impulse and point loads.

Many geometrically non-linear shell finite elements were developed for the non-linear analysis of laminated shells. Formulations of 3D degenerated shell elements based on the total Lagrangian and updated Lagrangian descriptors were presented by Chang and Sawamiphakdi [6] and Chao and Reddy [7], respectively. A nine-noded Lagrangian element based on shallow shell formulation was developed by Noor and Anderson [S]. Noor and Peters [9] gave the non-linear static response of anisotropic shallow cylindrical panels using the Hu-Washizu mixed shallow shell finite element. Another shell element with reduced integration, in which the generalized displacements were approximated with cubic Lagrangian interpolation functions, was developed by Noor and Peters [lo]. Later, Noor and Whitworth [11] used the elements by Noor and Anderson [S] and Noor and Peters [lo] to obtain the non-linear responses of a laminated cylindrical panel under a uniformly distributed pressure and an elliptic toroidal shell subjected to a uniform normal loading, respectively. Non-linear static and dynamic analyses of laminated shallow shell panels were studied by Reddy and Chandrashekhara [12,13], respectively, using a doubly curved shell element. Numerical results were presented for isotropic and laminated shells of cylindrical and spherical geometry. Dennis and Palazotto [14] developed a 36-DOF curved finite element to study the geometrically non-linear static response of cylindrical composite panels with cutouts. Later Tsai and Palazotto [S] used the same element to study the non-linear vibrations of cylindrical shells.

In this study the geometrically non-linear dynamic behavior of fiber-reinforced laminated circular and non-circular cylindrical deep panels is analyzed. The formulation for an arbitrary shell geometry includes parabolic variation of the transverse shear stresses through the thickness and the effects of rotary inertia. The linear and non-linear stiffness matrices are derived along the line of the work by Tsai and Palazotto [S]. The difference is that the present formulation is valid for a general doubly curved deep laminated shell whereas Tsai and Palazotto [S] derived their equations for circular cylindrical shells only. Further- more, they used a displacement based cylindrical finite element for the solution and in the present analysis, the shell middle surface displacement and rotation components are modeled using BCzier surface patches. A combination of Beta-m time integration and Newton-Raph- son iterations is used to solve the resulting equation of motion.

Comparisons of the static and dynamic responses for linear and non-linear cases are made for an isotropic cylindrical panel simply supported at the curved edges and subjected to a half sinusoidal uniformly distributed impulsive loading with those available in the literature. A very good agreement of the responses is observed by using only the fifth order Bezier surface patches. The dynamic responses of cross-ply laminated circular and non- circular cylindrical panels pinned at the straight edges under a central point load are also studied. The effect of the eccentricity on the dynamic response of non-circular cylindrical panels having the same plan form as the circular cylindrical panel is also examined.

2. FORMULATION

A general formulation of the fundamental equations governing the behavior of thin composite shells under finite deformation is presented. The assumptions made are: the transverse shear stresses vary parabolically through the thickness of the shell vanishing at the top and bottom surfaces; the shell is restricted to small strains and made of linear elastic laminated orthotropic material; exact Green’s strain-displacement relations are considered for the in-plane strains (sir, sZ2 and .slZ) and linear strain-displacement relations are considered for the transverse strains (E 23 and s13); and the transverse displacement is assumed to be constant through the thickness of the shell.

The geometry and the coordinate system of the shell element considered here are shown in Fig. 1. The middle surface of the shell is the reference surface and its lines of curvature are

Geometrically non-linear dynamic analysis 421

Fig. I. Geometry and coordinate system of shell.

chosen as the parametric curves which along with their common normal form an orthogonal curvilinear coordinate system. Therefore, any point within the shell body can be located by means of the two orthogonal curvilinear middle surface coordinates (5 1, t2) and a third coordinate [ normal to the surface. To radii of curvature along t1 and t2 are denoted by RI and R2, respectively. The lengths of the middle surface edges (at [ = 0) along the coordinate axes are given as

dsl = g1 d<, and ds2 = gz dt2 (1)

where gi (for i = 1,2) is the square root of the middle surface metric coefficients. The square of the length ds of an infinitesimal element in this triply orthogonal coordinate system is,

As2 = g:(l - c/R,)2 (d51)2 + g:(l - ilR212 (dt212 + W2 (2)

All the metric properties needed to measure lengths, areas and volumes in a shell can be defined from the above expression. In a coordinate system in which the parametric curves are the lines of curvature, if the metric coefficients g: and g: are to determine a surface uniquely except for its position and orientation in space, the following conditions are to be satisfied.

and

(3)

(4)

These three equations satisfy the compatibility conditions of the theory of surfaces and are known as the Gauss-Codazzi relations. It can be easily shown using equations (3) that

& CslU - i/R,)1 = (1 - i/R21 $ 2

&Cgz(l - i/R,)1 = (1 - WI)$ 1

For thin to moderately thick shells, the following approximations are introduced.

1

(1 - i/Ri) %I+; (i=1,2)

1

1

(1 - ilR,)(l - ilRz) z 1 + i(l/R, + l/R,) (6)


Also introduced here are the terms al, u2 and u3 which define the displacement components along tl, t2 and [ directions, respectively, at an arbitrary point within the body of the shell. By considering the transverse displacement u3 to be constant through the thickness and only the linear strain displacement relations the following is obtained

au, E33 = ag = o

Using equations (2) and (5)-(7) and the assumption of linear transverse shear strains in the 3D straindisplacement relations given by Green’s strain tensor [15] the foliowing set of straindisplacement relations is obtained.

El1 = C~l,ll~l + ~2gl,2/(glg2) - C1~31U + ic1) + {C%l/Sl + ~2gl.2Aglg2) - c1u312

+ cu2,1/91 - wI,2/(gIg2)12 + cu3,1/91 +c1~112~ (1 + XCIY2

E22 = cu2,2/g2 + w2,l/hg2) - C2~3lU + ic2) + {Cu2,2/g2 + w2,1Mg1g2) - c2u312

+ C~L2hI2 - ~2s2,1/hs2)12 + cu3,2/g2 + c2u212> (1 + 2&2)/2

El2 = cu2.1 - w1,2/g21u + iCIYS1 + cu1.2 - ~2S2,I/S11(1 + ic2Yg2

+ (h,I + uzs1,2/92 -~3W11C~1,2 --292,hll

+ cu2,2 + w2,1/g, - ~3C2S2IC~2.1 - w1,2lg21

+ cu3.1 + uIcIgIl cu3.2 + u3c2g21) cl+ ih + c2)l/glg2

623 = u2.3 + c2u2(i + k2) + u3,2(1 + b2)/g2

,513 = u1,3 + ClUlU + cc11 + U3,lU + iClYS1 (8)

where (),i = a( )/agi for i = 1,2 and ( ),3 = a( )/a[. Also, parameters cl and c2 are cl = l/R1 and c2 = 1/R2.

Equation (8) represents the straindisplacement relations at any arbitrary point within the body of the shell. This is a very general set of equations in terms of the displacement components ur , u2 and u3 which can be defined in terms of polynomials of the thickness coordinate [ for the analysis of laminated composite shells. In this analysis the following distribution is considered [4, 161

a1(5i, 5290 = 41 - i/R,) + 111/1 + 134$1 + wa’gi)

u2(51, (290 = 41 - i/R,) + i4+2 + t3W2 + w,2/g2)

u3(51,52,0 = w (9)

Here, u, v and w are the middle surface displacements components; 11/r, ti2 are the rotations of the normal; and k = - 4/(3t2). Furthermore, the variables, u, u, w, RI, R2, $i, I,$~, g1 and g2 are functions of the middle surface coordinates ti and t2 only and independent of [. It can also be noted that the above form of the displacement components results in a parabolic variation of the transverse shear stresses.

Expressions for the displacements ur , u2 and u3 given in equation (9) are now substituted into equation (8) and algebraic operations are carried symbolically using MACSYMATM. The expression for each strain component is simplified such that it takes the form of a polynomial in c as given below

E22 =E! + c i”x2p

p=l

429 Geometrically non-linear dynamic analysis

7

Etz=E:+ c ipQp g=1

7

&23=E:+ 1 ipK4p

p=l

7

El3 = E! + c iPKsp p=l

(10)

In the above equation E:, E;, E!, E: and E! are the middle surface strain components and the parameter Kip (i = 1,2, . . ., 5 and p = 1,2, . . ,7) is similar to the curvature terms within the context of classical thin shell theory. The complete expressions for these strain components are given in the work by Kumar [17] and are not presented here. In equation (lo), terms such as c4, 15, . . . , etc. may appear negligibly small to be retained. But these terms are associated with the constant k ( = - 4/3t*) and its square. The components of tiip, ~~~ and ~~~ (p = 4, 5) are multiplied by the constant k, hence terms like 14Ki4 and c5Kis (i = 1,2,3) are actually of the order 2 and 3, respectively, in f or [. Similarly, the components of tiip, tiZp and ~~~ (p = 6,7) are multiplied by the constant k2, therefore, terms like [‘IQ~ and i’tii, (i = 1,2,3) are also of the order 2 and 3, respectively, in t or [. Equation (10) can also be expressed in matrix form as follows,

where,

{El = Cs “) + [K] {&I (11)

{e}T = {Eli E22 812 &23 E13)

{C>‘={i c2 l3 r”...i’I (12)

and

KII K12 K13 K14 “. K17

K21 K22 K23 K24 “. K27

[Kl = K31 K32 K33 K34 “. K37 (13)

K41 K42 K43 0 . . . 0

K51 K52 K53 0 . . . 0

Components of the strain vector (E} and the curvature matrix [K] are functions of the displacements (u, v, w), rotations (11/i, (1/2) and their derivatives with respect to the shell coordinates. They are independent of the thickness coordinate 5. The strain-displacement relations given here are derived in terms of the general shell coordinate system <i and t2 defined by the surface metrics g: and gi and include terms which are significant in one coordinate system but become negligible or zero when another coordinate system is chosen. In other words, these relations are considerably simplified for a particular type of shell (cylindrical, conical, spherical, etc.), because some of the values of g 1. g2, RI and R2 become constant resulting in zero values of their derivatives.

This formulation deals with the geometrically non-linear behavior of composite shell structures. The material remains elastic and the stress-strain relations for nth lamina transformed into shell coordinates (gi, <, and [) is given as [18],

(14)


For convenience, the above equation is expressed as,

{‘x) = C”Q] {c}, (15)

where, C”Q] is the transformed elastic matrix in shell coordinates for the nth layer.

2.1. Strain energy expression The strain energy of a laminated composite shell is written as,

(16)

where A is the middle surface area, t, is the thickness of the nth layer and N is the total number of layers. Using the stress-strain relations given by equation (15), equation (16) is rewritten as,

u = (l/2) SC J

A “$I r. H;r C”QIT (El.di)dA (17)

Substituting equation (11) in the above equation, the following is obtained

u = (l/2) A j1 J J

({E”}T C”QIT {E”) + 2{&“)T rolT [Kl (6) f

n

+ (6)’ [KIT C”QIT [Kl (5)) 4 dA (18)

Expanding the matrix form and rewriting the above equation in terms of the elements of strain vector {E’} and the matrices CnQ] and [K],

(19)

Here, only the elements of the transformed elastic matrix “Qij vary with the layers. Therefore by performing the integration along the thickness, equation (19) is reduced to,

UC; Ji

A i$ jil EjO EPAij + 2 5 i i ~9 Kil ‘Aij

i=l j=l [=I

+ i i E i “jr Kim’+“‘Aij i=l j=1 I=1 m=l I

dA (20)

where

PAij = j, 1. (6” “Qij) dL p = 0, 1,2, . . . ) 14 (21)

The strain components $ and Kil in equation (20) are made of the metric coefficients, radii of curvature, the constant k and the displacements along with their derivatives. In order to obtain the strain energy as a function of displacements and their derivatives, the displacement terms are separated in the strain components as shown below.

EP = {DiOjT {q} + (l/2) {q}T [Eio] (4) for i = 1, 2 and 3

= {DieIT (4) for i = 4 and 5

Kil = (Dil}’ (9) + (l/2) {q}T [Eil] (9) for i < 4 and 1 = 1,2, . . . ,4

= (I/2) {qJT C&l (Q} for i < 4 and I= 5,6,7

= {Dil>’ {q) for i = 4, 5 and I= 1, 2, 3

= 0 for i = 4, 5 and 1> 4 (22)


where (Die} and (DiI} are column vectors of size 18 x 1, the order of each of [Eio] and [Eir] is 18 x 18. These matrices are made entirely of the metric coefficients, radii of curvature and the constant k and independent of the displacement components. The displacement vector {q} used in equation (22) is shown below.

WT = {u U,l u.2 ZJ O,l v,2 w w.1 w.2 W,ll w.22 w,12 $1 $1.1 $1.2 $2 $2,1 +2,21

(23)

Substituting equation (22) in the strain energy equation (20) and rearranging terms in a convenient form, the following expression is obtained.

u = (l/2) s MT @I + Cm11 + (l/4) CNzl) (4 dA (24) A

where [K], [N,] and [N,] are matrices of size 18 x 18. The elements of these matrices are not given in this paper, but can be referred to Kumar [17]. The form of these stiffness matrices are such that different matrices are obtained upon differentiation of the energy expression. This makes the solution procedure cumbersome and computationally ineffic- ient. Therefore, it is necessary to arrange the matrices [K], [N,] and [m,] in a convenient symmetric form so that the same matrices are repeated upon each differentiation of the energy functionals. In the present analysis, the procedure given by Rajasekaran and Murray [19] along with some generalizations introduced by Dennis and Palazotto [4] are used to obtain the symmetric stiffness matrices given in the following strain energy expression.

u = (l/2) J

{qjT (CR1 + (l/3) [NJ + (l/6) CN21) (9) dA (25) A

The generation of these stiffness matrices involves literally hundreds of matrix multiplica- tions and summations, hence it is not computationally economical to obtain them numerically. In the present analysis, they are evaluated symbolically using MACSYMA 417 which creates an optimized FORTRAN code.

2.2. Kinetic energy expression The kinetic energy of the composite shell is obtained by using the following expression

T = (l/2) s( s A ,,!I

P,C@&)~ + @,I2 + @A21 di f.

(26)

where, (‘) = a( )/& (partial differentiation with respect to time z) and p. is the material mass density of the layer n. The expressions for ul, u2 and u3 from equation (9) are now substituted in the above equation to obtain the following matrix form through the simple procedure of separating variables and by rearranging of terms.

T = (l/2) s

{i}’ [M] {i} dA (27) A

where [M] is the mass matrix of an infinitesimal element which contains the coefficients of the inertial terms and mass densities of the layers after integrating over the thickness and

2.3. Work done by the external and body forces In this analysis, in addition to the external surface forces, only the damping forces are

considered in the development of the equation of motion. Therefore, the potential of the forces which includes the work done by the external forces and energy dissipation due to

432

damping are given as,

V. Kumar and A. V. Singh

W = s {4”}T {fe} - 2 s s&it& + u& + r&)didA A n=l t.

= s

A MT Ve> - (91T t-cl ii> dA (29)

where {fe} is the external force vector, gn is the damping coefficient for the nth layer, [C] is the damping matrix and the vector (a) is given as

MT = {a r w *1 t/Q (30)

In practice it is difficult to determine the damping parameters of a shell element since the damping properties are frequency dependent. For this reason the damping matrix [C] is in general constructed using the mass matrix and stiffness matrix of the complete shell structure together with experimental results on the amount of damping. Though the present formulation in its general form includes the effect of damping, it is not considered in all the numerical analysis performed in this work.

3. SOLUTION METHOD

The energy expressions for a laminated shell segment derived so far are in terms of general curvilinear coordinates (ti and t2). Consider a shell segment bounded by four curved edges as shown in Fig. 2. It is assumed that the coordinates (ti and t2) of the four corner points 1, 3, 5 and 7 as well as the four middle points of the edges (2,4,6 and 8) are known. This type of geometry can be conveniently dealt with in normalized coordinates 5 and q which map the surface area into a 2 x 2 square. This mapping procedure is well established in the finite formulation of an eight-node isoparametric shell element [20] and therefore its details are not included here.

The solution procedure used in this analysis is based on the Ritz method which requires properly defined admissible displacement fields. Therefore, described in the following is the application of Bezier surface patches representing the middle surface displacements and rotations

n+1 m+l

u = C C fj(rl)fi(5)puij (31) j=1 i=l

where m and n correspond to the degrees of the Btzier polynomialsf;(<) andfj(r) used in 5 and p directions, respectively, and Paij denote (m + 1) (n + 1) control points used to represent the displacement u. Similarly, the other displacement and rotation components u, w, $i and rjZ can be expressed by the above summation simply by replacing PUij by Pvij,

Pwij, P$lij and P+zij, respectively. The Bezier polynomialfi(5) of degree m (m > 2) is given by [21]

1;:(t) =“Ci-I[(1 + o/2]‘-’ [(l - {)/2]“+‘-’ (32)

for i = 1,2, . . . , m + 1 and mCi = m(m - 1). . . (m - i + 1)/i! Similarly, the polynomialf;(q) is obtained by substituting j for i, q for 4 and n for m in the above equation.

Using equation (31) the displacement vectors (q}, {ij) and (ijj given in equations (23), (28) and (30) are expressed in terms of the control points as follows

1’,“?, = cskl fp>, pi, = CSml PI and (4) = &-I P’> 18x1 /xl 7x1 (xl .5X1 5x1 /xl

(33)

where / = 5(m + 1) (n + 1) is the total number of control points defining the displacements and rotations. The vector {P} is made of the control points PUij, PUij, etc., representing the displacement fields. The matrices [Sk] and [S,] contain the Btzier functions and their derivatives with respect to 5r and t2. The Jacobian matrix is employed here to obtain the derivatives with respect to general shell coordinate system (<r and t2) in terms of the same with respect to normalized coordinates (r and q). Hence, the derivatives of displacements


+-----VI L q=_1

Fig. 2. Normalized and shell coordinate systems

and rotations kl, u,~, u,~, u,~, w,~, w,~, w,ll, WJ~, W,U, $1.1, ti1.2, $2.1 and $2,~) with respect to t1 and t2 are obtained numerically by working in the normalized coordinate system.

The variation of the total potential energy (II = U - T - W) of the shell is used in this analysis to obtain the matrix equation of motion. Now, equation (33) is substituted into the energy functionals (25,27 and 29) to obtain the total potential energy in terms of the control points. Expressing the above equation in terms of the normalized coordinates t and q, the following equation is obtained

1

JY

+1

l-I= C{P)T(CWT (f WI + d CNI + i+ [NJ) CSd {PI -1 -1

- 3 {PI’ CSnlT CM1 C&J {@ + {P}’ CWT Ccl C&J if’)

- PI’ CS,l’ WI 9192 I J(5, rl) I d4 dv

= {P}'(~[~]+i[rn,]+~[~2]){P} -+{P}T[iiQ {P}

+ WTKI P> - W’ IF;> (34)

Here, IJ(l, r)l is the determinant of the Jacobian matrix [20]. Applying Hamilton’s variational principle to equation (34) and taking variation with respect to each of the control points in vector (P}, the following equation of motion is determined

@I P> + cc1 {P> + m + 3cw + m21, P> = m (35)

In equation (35), matrices [Nr] and [N2] are associated with the stiffness matrix [K] of the laminated shell and depend linearly and quadrically, respectively, on the displacement vector {P).

At this stage, the matrix equation for the laminated open shell has been derived in terms of the control points. The procedures used to implement the kinematic boundary conditions within the context of the present formulation are given in the work by Singh and Kumar [22]. The non-linear equation of motion is solved using a combination of Beta-m time integration and Newton-Raphson iterations [S]. The convergence criterion used in this method is based on the following.

(36)

1 {Pn+l (j)>i"+1 j= 1

where 8 = total number of control points (degrees of freedom), P,+ i(j) = value of the jth control point at time r,+ 1 and i = the iteration number.


4. NUMERICAL EXAMPLES

The computer program for the non-linear analysis is developed from the formulation and solution scheme described so far and run on SPARC series SUN Workstation. The stiffness matrices [g], [fir] and [&] are developed with the aid of MACSYMATM. The solution algorithms for the non-linear equation of motion in terms of control points using Beta-m time integration and Newton-Raphson iteration techniques are also developed. The second order Beta-m method (m = 2) is used in the numerical examples studied with the values of the integration parameters PO = pi = 0.5 for unconditional stability. This is equivalent to the Newmark integration method. The convergence criterion e [equation (36)] used in the Newton-Raphson iteration is taken to be 0.1%. Numerical calculations have been done using up to eighth order Btzier polynomials, but results virtually remained unchanged from those which are obtained using only fifth order polynomials. Hence, in all the numerical examples presented in this paper, the fifth order Btzier surface patches are used to represent the middle surface displacement and rotation components. The fifth order Btzier polynomial has six control points and a corresponding surface patch has 36 control points. Also, since there are in all five displacement and rotation components, the total number of unknowns in the equation of motion is 180. This number of unknowns seemed adequate for the analysis, as only one-quarter of the shell structure is required in the model due to its doubly symmetric geometry.

The computer program is organized in such a manner that the same code can be used to study the linear and non-linear responses corresponding to static and dynamic problems. The linear responses are obtained by neglecting the contributions of the non-linear stiffness matrices [mr] and [NJ in the analysis. Similarly the static responses corresponding to linear and non-linear analyses are computed by neglecting the effects of inertia forces (i.e. the value of the mass matrix is ignored). In order to evaluate the stiffness matrices [ii], [R,] and [m2] and the mass matrix [A] completely, the metric scale factors g1 and g2, the radii of curvature RI and R2, and their derivatives with respect to the coordinate axes are to be predefined. These metric scale factors and other necessary parameters for each type of shell are defined in a subroutine to the main computer program. Therefore, by choosing the type of the shell (given as an input to the program) all the required parameters including the metric scale factors and radii of curvature for that particular shell geometry are defined.

The curvilinear coordinate system (s, x and [) shown in Figs 3 and 8 is chosen here for the non-linear analysis of laminated circular and non-circular cylindrical panels made of N orthotropic layers with each layer having different fiber orientation. The circumferential coordinate s follows the median line of the transverse cross-section, the axial coordinate x is fixed to the middle surface and the coordinate [ is normal to both s and x. The corresponding displacement components in the coordinate directions are also indicated in these figures. The lateral boundaries of the shell are assumed to be at distances [ = f t/2 from the median surface, where t is the total thickness of the shell and considered small in comparison with the radius of curvature. The radii of curvature and the metric scale factors of the middle surface are given as

Rr = R,(s), R2 = R, = co, 91 = 1, 92 = 1 (37)

where R, is the radius of curvature along the longitudinal direction. The expression for the varying radius of curvature, R,(s), in the circumferential direction is taken to be the same as that used in the linear free vibration study of non-circular cylinders [24]. This curvature which describes an oval geometry and approximates an ellipse is given as

R,(s) = R&l + scos(2s/R,)] (38)

where the average radius of curvature R. = L,/~z and the eccentricity parameter E is defined as

E = 3Q - (36/35)Q3 (39)

The parameter L, is the circumference of the middle surface of the shell, Q = (a/b - I)/(@ + 1) for 1 < u/b d 2.06, and a and b are the semi-major and semi-minor axes of the shell.


The numerical analyses are performed for an isotropic shallow circular cylindrical shell under impulse load and cross-ply laminated circular and non-circular cylindrical panels under a point load. The linear and non-linear responses obtained from the analysis of isotropic cylindrical panels are compared with those in the literature. New results are presented for laminated deep elliptic cylindrical panels.

4.1. Isotropic shallow circular cylindrical panel under impulse load Initially, the static and dynamic responses at Point A (Fig. 3a) of an isotropic cylindrical

panel corresponding to the linear and non-linear analyses are obtained and compared with those presented by Clough and Wilson [25] and Tsai and Palazotto [S]. Clough and Wilson [25] used a 20 DOF quadrilateral element composed of four flat plate triangles whereas Tsai and Palazotto [S] used a 36 DOF cylindrical shell element in their geometrically non-linear analysis. In the present analysis the concept of BCzier surface patches is applied in the Ritz-type solution procedure. The geometric and material properties of the isotropic cylindrical panel are taken to be the same as those used by the above authors. In Fig. 3, these properties are given in SI units with the corresponding U.S. customary units in the brackets. The panel is subjected to vertical uniformly distributed half-sinusoidal impulse loading shown in Fig. 3(b). The peak intensity of the load is taken to be 4.3 1 kN/m’ (90 psf). The time increment used in the Beta-m integration is 0.02 s, which is approximately 3% of the first natural period of the panel. Due to the doubly symmetric nature of this panel only a quarter of it is analyzed by enforcing proper restraints along the edges at the axes of symmetry. Numerical calculations have been carried out using up to eight order Btzier surface patches for each of the five displacement and rotation components. This results in a total number of 5 x 9 x 9 = 405 unknown control points. But virtually no difference in the deflection has been found when only the fifth order surface patches are used. The fifth order patches create only 180 unknown control points and considerably reduce the computation time.

Four different types of analyses are performed on the cylindrical panel. The first two analyses are done by neglecting the inertia force for both geometric linear and non-linear analysis. Due to the absence of the inertia force, these are considered as static analyses. The transverse displacement w obtained at point A (Fig. 3a) of the panel are compared with those presented by Tsai and Palazotto [S] for both cases. Figure 4 shows the comparison of

0.5

Tie (Set)

(b)

PeakL.oadIntensity=4.31kN/m2(90psf)

R, = R,, = 15.24 m (SO fit-) Open angle = 40” L= 15.24m(SOft.) E = 207 GPa (3 x lo6 psi) i = 0.0762 m (3”) v=o & = 0.0 p= 5.7676kN-sz/m4(11.191 lb-s2/ft4)

Fig. 3. Geometric and material properties and the loading pattern of an isotropic shallow circular cylindrical panel simply supported at the curved edges.


0.16

0.14

h 0.12

5

3

E 0.10

‘4 P jj 0.08 P

8 $ 0.06 L 6 c 0.04

0.02

- Present Analysis

l Tsoi and Palazotto (1991)

L=&= 15.24m t = 0.0762 m

Open angle = 40’ E = 207 GPa

p= 5.7676 kN-s21m4

0.0 0.2 0.4 0.6 0.8 1 .o

Time (Set)

Fig. 4. Comparison of linear static downward deflection at point A of an isotropic circular cylindrical panel (simply supported at curved edges) under impulsive load.

0.14

0.12

~ Present Analysis

l Tsai and Polazotto (1991)

. . - 5 0.10

2

.z

”

0.08

F D

t 0.06

k 2;

j 0.04

0.02

0.00

L =& = 15.24 m t = 0.0762 m &=o.o Open angle = 40” E = 207 GPa

v=o p= 5.7676 kN-s2im4

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 .o

Time (Set)

Fig. 5. Comparison of non-linear static downward deflection at point A of an isotropic circular cylindrical panel (simply supported at curved edges) under impulsive load.

the transverse displacement component for a linear static analysis in which a good agreement of the results is observed. The corresponding non-linear static displacements are plotted in Fig. 5. Here also a good agreement of the results is observed.

1


___ Present Analysis l Tsai and Palazatta (1991)

---. Clough and Wilson (1971)

L=&=15.24m t = 0.0762 m &=o.o Open angle = 40” E = 207 GPa v=o p= 5.7676 kN-s2 i m4

_I

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.6 2.0

Time (Set)

Fig. 6. Comparison of linear dynamic downward deflection at point A of an isotropic circular cylindrical panel (simply supported at curved edges) under impulsive load.

The next two analyses are the geometrically linear and non-linear dynamic analyses where the effects of the inertia force are included. The comparison of the linear dynamic transverse displacement w at point A is given in Fig. 6. The results obtained by the present analysis are plotted as solid curve and studies by Clough and Wilson [ZS] and Tsai and Palazotto [S] are shown by dashed and dotted curves, respectively. The transverse displacements at point A obtained by the present study and by Tsai and Palazotto [5] using the geometrically non-linear dynamic analysis are plotted in Fig. 7. An excellent agreement of the results is observed in this case. The corresponding results obtained by Clough and Wilson [25] are also in very good agreement with those plotted in Fig. 7 and are not shown in the figure as they overlap with the existing curves.

In both of the static cases studied here, the maximum displacements given by the present analysis are slightly higher (11% and 7% in linear and non-linear static analysis, respectively) than those presented by Tsai and Palazotto [S]. Since similar formulations are used in both the studies, this difference is due to a stiff behavior of the finite element used in the study by Tsai and Palazotto [5]. This behavior is not present in the non-linear dynamic analysis, where both studies give very close values of the transverse displacement at point A.

4.2. Cross-ply laminated deep circular and non-circular cylindrical panels Having obtained good comparisons of the results with those in the literature for an

isotropic shallow cylindrical panel, the dynamic response of composite panels, subjected to a point load at the center, is investigated using the geometrically non-linear formulation. The panel is made of 24 cross-plies of graphite/epoxy and the layer arrangement is antisymmetric with respect to the middle surface. The geometric and material properties of the panel are given in Fig. 8. The panel is pinned along the straight edges and free at the curved edges. The boundary conditions for this case can be represented by

at the straight edges. A point load, the dynamic loading history of which is shown in Fig. 8(b), is applied at the center of the panel. The maximum load (8.897 kN) is taken to be


0.20

0.15 - ~ Present Analysis

l Tsai and Palazatto (1991)

z 0.10 -

J e .$ 0.05 -

P F 0

0 0.00 1) c :! L=Ro=15.24m e

:

t = 0.0762 m

-0.05 - & 0.0 =

Open angle = 40’

-0.10 -

-0.15 0.0

E = 207 GPa V

v=o

p= 5.7676 kN-s2/m4

0.2 0.4 0.6 0.6 1.0 1.2 1.4 1.6 1.6 2.0

Time (SW)

Fig. 7. Comparison of non-linear dynamic downward deflection at point A of an isotropic circular cylindrical panel (simply supported at curved edges) under impulsive load.

Fw WI

k_ 0.002

Time (Set)

(b)

& = 0.3048 m E,, = 141 GPa L = 0.4572 m

t=3,048x 105m E22 = 9.24 GPa Grz = Gr3 = 5.96 GPa

c = 0.2565 m C& = 2.96 GPa p = 134.38 N-?/m’ Point Load F, = 8.897 kN

Fig. 8. Geometric and material properties of a cross-ply (24 plies) deep cylindrical panel pinned at the straight edges.

lower than the static buckling load of 12.456 kN for a circular cylindrical panel. Using the doubly symmetric nature of the panels studied here, only a quarter section is analyzed.


& = 0.3048 m L = 0.4572 m r=3.048x 1U3m

c = 0.2565 m p = 134.38 N-s2/m4

El1 = 141 GPa En = 9.24 GPa Grz = Gr3 = 5.96 GPa GB = 2.96 GPa Load = 8.897 kN

-------- Linear Dynamic Response

0.010 0.015

Time (set)

Fig. 9. Linear and non-linear dynamic response of a 24 layer cross-ply laminated circular cylindrical panel under central point load (E = 0.0 and simply supported at the straight edges).

7 t; s g 0.010 ; L C e !-

0.005

& = 0.3048 m Err = 141 GPa L = 0.4572 m t= 3.048 x 10m3 m

E22 = 9.24 GPa Gr2 = Gr3 = 5.96 GPa

c = 0.2565 m Ga = 2.96 GPa p = 134.38 N-s2/m4 Load = 8.897 kN

0.000 ” 0.000 0.005

Linear Dynamic Response

Nonlinear Dynamic Response

0.010 0.015 0.020 0.025

Time (Set)

Fig. 10. Linear and non-linear dynamic response of a 24 layer cross-ply laminated elliptic cylindrical panel under central point load (E = + 0.2 and simply supported at the straight edges).

The linear and non-linear dynamic responses are obtained for three different panel geometries according to the eccentricity parameters F = - 0.2,O and + 0.2. The eccentricity parameters - 0.2 and + 0.2 approximate elliptical (or oval) geometries whereas zero eccentricity represents a circular configuration. The span length c (Fig. 8a) and other dimensions are kept constant in all cases, therefore with changing eccentricities only the

Tra

nsv

erse

C

entr

al

Def

lect

ion

v

(m)

Tra

nsv

erse

C

entr

al

Def

lect

ion

w

(m

)

0 0

P

P

0

B

b 0

I?

B

in

‘0

G

0 cn

---_

_ --

-__

__--

- _-

e

l =_

_ --

-__

___-

--

_>

=__

--

-__

*__-

--

2

c-__

---

_ r_

_---

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---_

__ --

>

c---

-- /c

----

___

‘, ,_

_---

- w

- -.

_ --

--_

.---

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:2

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____

<,--

--

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2-

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._> _-

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c:::y

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_ --

. _-

- _=

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;=I\


0.025

3 0.020 3

: ._

F Elliptic Cylindrical Panel E = -0.2 g 0.015

P Circular Cylindrical Panel a = 0.0 7 s $ 0.010

Z R. = 0.3048 m En = 141 GPa

e L = 0.4572 m En = 9.24 GPa + t = 3.048 x lo5 m = = 0.005 Giz Ga 5.96 GPa c = 0.2565 m Ga = 2.96 GPa p = 134.38 N-s2/m4 Load = 8.897 kN

0.010 0.015

Time (set)

Fig. 13. Non-linear dynamic response of a 24 layer cross-ply laminated elliptic cylindrical panel having eccentricities I: = - 0.2, 0.0 and + 0.2 under a central point load (simply supported at the

straight edges).

deformation pattern when geometrically non-linear theory is used. A periodic motion oscillating about the displacements of 0.017 m and 0.0216 m is observed, respectively, for linear and non-linear dynamic analysis. Also the amplitude of the non-linear oscillations is higher than that of the linear oscillation. Similarly the central transverse displacements using linear and non-linear formulations are shown in Figs 10 and 11 for panels with eccentricities E = + 0.2 and - 0.2, respectively. In the case of eccentricity E = + 0.2 the non-linear oscillations are not perfectly harmonic. The linear and non-linear oscillations are closer than in the case of panels with eccentricities E = 0 and - 0.2. Also the vibratory response is somewhat slower than the other two cases (E = 0 and - 0.2). The variation of the linear dynamic response at the center of the composite panels with different eccentricities (E = - 0.2,O and + 0.2) are shown together in Fig. 12. The deflection of the pane1 with eccentricity E = + 0.2 is seen to be higher than those of panels with eccentricity E = 0 and - 0.2. This is also true in the case of geometrically non-linear analysis (Fig. 13) but the

difference is seen to be less than that of the linear analysis.

5. CONCLUDING REMARK!;

Geometrically linear and non-linear dynamic analyses of isotropic and laminated cylindrical panels are presented in this paper. The formulation includes the parabolic variation of the transverse shear stresses along the thickness of the shell and the effects of rotary inertia. The linear and non-linear dynamic responses of cylindrical panels having different geometric configurations are investigated. The displacement and rotation components of these panels are modeled using Bezier surface patches. All the numerical results presented in this chapter are evaluated using a single Btzier surface patch of order 5. The numerical values calculated are compared with those obtained by finite element analysis in the literature and a good agreement of the results is observed in all four cases, namely, linear static and dynamic and non-linear static and dynamic analyses of an isotropic cylindrical panel.

Deep laminated cylindrical panels having circular and non-circular configurations are also investigated here. The geometrically linear and non-linear responses of cylindrical


panels having eccentricities E = - 0.2, 0 and + 0.2 are obtained and compared with each other. In all cases the periodic oscillations corresponding to the non-linear analysis occur at higher displacements than those in the linear analysis. Also the linear and non-linear responses of panels with eccentricity E = + 0.2 are higher than those corresponding responses of panels with eccentricity E = 0 and -0.2.

4.

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REFERENCES

M. Stein, Nonlinear theory for plates and shells including the effects of transverse shearing. AIAA J. 24, 1537 (1986). L. Librescu, Refined geometrically nonlinear theories of anisotropic laminated shells. Quart. Appl. Math. 45, 1 (1987). Y. Hirano and J. R. Vinson, Nonlinear vibrations of composite material cylinder shells including transverse shear deformation. Design and Analysis ofComposite Material Vessels, ASME 1987 Pressure Vessel and Piping Conference, San Diego, California, pp. 55-63 (1987). S. T. Dennis and A. N. Palazotto, Large displacement and rotation formulation for laminated shells including parabolic transverse shear. Int. J. Non-Linear Mech. 25, 67 (1990). C. T. Tsai and A. N. Palazotto, On the finite element analysis of non-linear vibration for cylindrical shells with high-order shear deformation theory. Int. J. Non-Linear Mech. 26, 379 (1991). T. Y. Chang and K. Sawamiphakdi, Large deformation analysis of laminated shells by finite element method. Computer Structures 13, 331 (1981). W. C. Chao and J. N. Reddy, Analysis of laminated composite shells using a degenerated 3-D element. Int. J. Numer. Methods Engng 20, 1991 (1984). A. K. Noor and C. M. Anderson, Mixed models and reduced/selective integration displacement models for nonlinear shell analysis. Int. J. Numer. Methods Engng 18, 1429 (1982). A. K. Noor and J. M. Peters, Nonlinear analysis of anisotropic panels. AIAA J. 24, 1545 (1986). A. K. Noor and J. M. Peters, Analyis of laminated anisotropic shells of revolution. .I. Engng Mech. Div. Proc. Am. Sot. Civil Eng. 113, 49 (1987). A. K. Noor and S. L. Whitworth, Non-linear analysis of symmetric structures with unsymmetric boundary conditions. Engng Computat. 4, 161 (1987). J. N. Reddy and K. Chadrashekhara, Nonlinear analysis of laminated shells including transverse shear strains. AIAA J. 23, 440 (1985). J. N. Reddy and K. Chandrashekhara, Geometrically non-linear transient analysis of laminated, doubly curved shells. Int. J. Non-Linear Mech. 20, 79 (1985). S. T. Dennis and A. N. Palazotto, Static response of a cylindrical composite panel with cutouts using a geometrically nonlinear theory. AIAA J. 28, 1082 (1990). A. S. Saada, Elasticity: Theory and Applications. Pergamon Press (1974). J. N. Reddy and C. F. Liu, A higher order shear deformation theory of laminated elastic shells. Int. J. Engng Sci. 23, 319 (1985). V. Kumar, Linear and nonlinear dynamic analysis of fiber reinforced laminated shells. Ph.D. thesis, The University of Western Ontario, London, Ontario, Canada (1994). J. R. Vinson and R. L. Sierakowski, The Behaviour ofStructures Composed ofComposite Materials. Martinus Nijhoff (1986). S. Rajasekaran and D. W. Murray, Incremental finite element matrices. ASCE, J. Struct. Div. 99, 2423 (1973). W. Weaver and P. R. Johnston, Finite Elements for Structural Analysis. Prentice-Hall (1984). P. Btzier. The Mathematical Basis of the UNISURF CAD Svstem. Butterworths. London (1986). A. V. Singh and V. Kumar, On free vibrations of fiber reinforced doubly curved deep panels:Parti (Formula- tion). ASME J. Vibrat. Acoustics (in press). M. G. Katona and 0. C. Zienkiewicz, A unified set of single step algorithms Part 3: The Beta-m method, a generalization of the newmark scheme. Int. J. Numer. Methods Engng 21, 1345 (1985). V. Kumar and A. V. Singh, Vibration analysis of non-circular cylindrical shells using B&zier functions. J. Sound Vibrat. 161, 333 (1993). W. Clough and E. L. Wilson, Dynamic finite element analysis of arbitrary thin shells. Computer Structures 1,33 (1971).

Geometrically non-linear dynamic analysis of laminated shells using Bézier functions

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