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Postbuckling and vibration of shear deformable flat and curved panels on a non-linear elastic foundation
Inl. J. Non-Linear Mechanics,
Vol. 32, No. 2, pp. 211-225, 1997 0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 002&7462/97 $17.00 + 0.00
Pergamon
SOO20-7462(%)00057-l
POSTBUCKLING AND VIBRATION OF SHEAR DEFORMABLE FLAT AND CURVED PANELS ON A NON-LINEAR ELASTIC FOUNDATION Liviu Librescu and Weiqing Lin Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0219, U.S.A.
(Received for publication 11 April 1996) Abstract-This study deals with the postbuckling and vibration behavior of flat and shallow curved panels resting on a Winkler linear/non-linear elastic foundation. The considered plate/shell structural model is based on a higher-order shear deformable theory and encompasses a number of effects such as transverse shear, geometric non-linearities and the initial geometric imperfection. Special emphasis is given to the influence played by Winkler’s foundation moduli as well as by the previously mentioned effects upon the postbuckling and vibrational response in the pre/postbuckling ranges and a number of pertinent conclusions are outlined. 0 1997 Elsevier Science Ltd. All rights reserved. Keywords postbuckling, plate/shell, Winkler foundation, hardening and softening non-linear foundation, shear deformability, initial geometric imperfection, buckling and limit load, load-frequency interaction
1. INTRODUCTION
The theory and behavior of beams, plates and shells on elastic foundation occupies a prominent place in the contemporary structural mechanics. In addition to the applications in many civil engineering structures, the recent developments related with solidpropellant rocket motors, the interest for developing further efficient thermal protection systems for space transportation vehicles as well as many other high technological applications have intensified the need for a better understanding of the behavior of plates and shells continuously supported by elastic media. In spite of its practical importance, the research done in this field was mainly devoted to beam structures on elastic foundations (see Cl], where an account of the work done in this field is provided, as well as [2-41). In addition to the fact that the research work addressing the problems of postbuckling and vibration response of plates and shells on elastic foundations (see, e.g. [S-9] and [lo-131, respectively) appears to be rather scarce, the treatment of the problem was done in rather specialized cases. In spite of this, credit for the early work in this area has to be given to Korbut [14], Sinha [15] and Reissner [16], who have pointed out for the first time, several of the features of the response of plates and shells resting on an elastic foundation. As the available specialized literature reveals, little attention has been given to the clarification of the implications of a number of important effects as transverse shear, initial geomtric imperfection, curvature of the panel and loading system upon the postbuckling and vibration of flat and curved panels resting on a linearJnon-linear elastic foundation. It is the goal of this paper to supply pertinent information in connection with this topic. To this end, a rather comprehensive study of the postbuckling and vibrational behavior of flat and curved panels resting on a Winkler foundation, subjected to a complex mechanical loading system and incorporating a number of non-classical features is presented. The mechanical loading investigated consists of a transverse lateral pressure and a system of uniform compressive edge loads acting in the pre/postbuckling ranges. The structural model is based on a higher-order transverse shear deformation theory of shallow shells that incorporates the effects of geometric non-linearities and initial geometric imperfections. 211
212
L. Librescu and Weiqing Lin
Results obtained using a special-purpose analysis, well-suited for parametric studies, are presented for single and three-layer panels. Simply supported panels are considered in which the tangential motion of the unloaded edges in either unrestrained or fully restrained. The results focusing on the postbuckling and load-frequency interaction cover a wide range of geometrical and physical parameters of the considered panels and the strong influence (qualitative and quantitative) played by the linear and non-linear (cubic) elastic foundation moduli as well as by other non-classical effect is emphasized. 2. GENERAL
CONSIDERATIONS
The case of doubly-curved shallow panels of uniform thickness h, symmetrically laminatedof2m+ l(m= 1,2,. . .)t ransversely-isotropic layers is considered, the surface of isotropy being parallel to the mid-surface of the structure. It is assumed: (i) that the shell is supported on the inner surface by an elastic foundation; and (ii) that perfect bonding between the shell and foundation and between the contiguous layers is implied. The points of the 3D space of the panel are referred to a set of curvilinear system of normal coordinates, xi, where x’, (LY = 1,2) denote the tangential coordinates, while x3 = 0 defines the reference surface (denoted henceforth as 0). The components of the metric tensor of the undeformed reference surface are: aap = am-as;
aa3 = aa-a3 = 0;
a33 =a3.a3
= 1,
a’@ = a’.aa;
aa3 = ab.a3 = 0;
a33 = a3.a3
= 1,
(1)
where ai and ai denote the contravariant and covariant base vectors of cr, respectively. The spatial metric tensor components gij of the undeformed shell-space are connected with their 2D counterparts a,@ by relations given in [17, IS] 1 w 933 = g= = 1, gas = g”3 = 0; (2) gap = CL,CLB a,,; where ,u; = 6; - x3b;, (3) plays the role of shifter in the space of the normal coordinates, 6,, and baadenoting the Kronecker delta and the curvature tensor, respectively. In the forthcoming developments, the concept of shallow shells will be used (see, e.g. [19]). Denoting by Z( =.Z(X~)) the amount of deviation of the shell reference surface from a plane Il (measured normal to the plane), consistent with this concept, 2 is assumed to be small when compared with a maximum length of an edge of the shell or with the minimum radius of curvature of 0. For this case, the assumption maxZ,.
< 1,
(4)
gives rise to the result that the metric tensors associated with the system of coordinates on cr and with its projection on the plane II are the same and, in addition, that the curvature tensor of the reference surface behaves as a constant in the differentiation operation. From this result it may be inferred that if the projected coordinate curves on II constitute a Cartesian orthogonal net, then the original ones on g are also to be, on the basis of (4), a Cartesian orthogonal net. Due to the quivalenceof the two metrics, it may also be concluded that the surface covariant differentiations may be done with respect to the metric associated with the plane II and thus it is possible to change the order of the covariant differentiations (since the Riemann-Christoffel tensor associated with the plane vanishes). Consistent with the shallow shell theory we may appropriately assume that & + S;, implying, consequently, gas + aaB and gas + aaB. (5) From (5) it may also be concluded that in this case p = l&l = (g/a)“2 + 1,
(6)
where g = det (gij) and a E det (aas). In order to reduce the 3D elasticity problem to an equivalent 2D one, the equations connecting the covariant derivatives of space tensors with their surface counterparts have to
Flat and curved panels on a non-linear foundation
213
be used. Such relations restricted to the case of shallow surfaces are:
T,,,, = Ta,,- ba,Ti3;
Ta,13
=
Tn.39 -
TJ,,a
=
T3.a
+
K’T,;
T3/,3 =
(7)
T3.3.
For more details concerning the deduction of these expressions, the reader is referred to [17, 181. Here, partial differentiation is denoted by a comma ( ),i E a( )/axi, while ( ) (Ii and ( )llx stand for the covariant differentiations with respect to the space and surface metrics, respectively, while the shifted components are identified by an upper bar. In the above relationships (as well as in the following developments), the Einsteinian summation convention applies to repeated indices where Latin indices range from 1 to 3 while the Greek indices range from 1 to 2.
3. DISPLACEMENT
REPRESENTATION
AND STRAIN
MEASURES
In order to model the geometrically non-linear theory of shear-deformable curved panels, the shifted displacements are represented as (see [20])
shallow
VJxO, x3, t) = u, + x3$, + (x3)% + (x3)3 i,, V3(xU,x3,
t) =
u3,
6%
b)
where Ui G
Ui(Xmy t);
$a(XW,
A., zz &(xW,
t);
t);
L? =
ra(Xw, t).
Based on the above representation of the displacement field, the exact fulfillment of tangential static conditions on the bounding surfaces x3 = f h/2 expressing the absence of shear tractions yields 1, = 0
and
i, = - $(u,,.
+ $=).
(9)
In the light of (9), it is apparent that fulfillment of above mentioned conditions results in a displacement field containing the same unknown functions as the first-order transverse shear deformation theory (FSDT), i.e. u3, U, and tidl. The underlined terms in (8a) have the character of corrective terms enabling one to fulfil the conditions on x3 = + h/2. It iqassumed also the existence of an initial out-of-plane, stress-free geometrical imperfection V3(xw,x3) = G3(xw). By convention, the transverse deflection is measured from the imperfect surface, in the positive, inward direction. Consistent with this, the straindisplacement relationships considered in the spirit of the von K&man’s partially non-linear theory read: 2eij
=
l/il,j
+
vj,li
+
v311iv3//j
+
f3i/iv3ilj
+
v3ilif3i{j.
(10)
Equation (10) used in conjunction with (6)--(g),yields the non-vanishing components of the strain tensor as: e,8 = aafi+ x3x,@ + &3)3&r, em3 =
~~3
+
(x3)2&3,
(lla,b)
where
22,s = - ,,,(+.
+
u3,a
+
&up).
(12a-e)
denote the 2D strain measures of the geometrically non-linear higher-order shell theory. It is easily seen that: (i) within its FSDT counterparts, in (11) the strain measures cab and Aa
214
L. Librescu and Weiqing Lin
become zero quantities; (ii) for *. + - (%a + @u,),
(13)
the strain measures, (12) reduces to the ones associated with the classical shallow shell theory (based on LoveKirchhoff hypothesis); and (iii) that the initial geometric imperfections occur in the membrane strain components, only. In the previous (and forthcoming) equations the covariant differentiation is performed with respect to the metric at II.
4. CONSTITUTIVE
EQUATIONS
As is well-known, the 3D elasticity theory implying small strains but large displacement gradients may be described by linear constitutive equations connecting second Piola-Kirchhoff stress components with Lagrangian strain measures. As a result, the strain-strain relationship for an elastically linear 3D anisotropic body is given by [18]: s@ = FhsWc wp+ 6*~s”3, sa3 = 2Ea3w3c03
(I4a, b)
where p@‘JP= E&P _
E~Zfl33Ea%O,, E3333
(15)
’
In (14) and (15) E’j”” and &?fl@’denote the tensors of elastic and modified elastic moduli, respectively; aA is a tracer identifying the contribution of s33 in the constitutive equations (and later in the governing equations) while sij denotes the second Piola-Kirchhoff stress tensor. In order to express s33 in terms of the basic unknowns, the third equation of motion of the 3D elasticity theory has to be used. For material layers exhibiting transverse-isotropic elastic properties as is considered in the present paper, the tensors of elastic moduli assume the form Cl83 &lm
=
E
l+v
l(aawa8s + a=ca8~) + _
E3333
E’(l -v)
1
_L_amcaa8,
[2
a@
(16a-c)
Here E,v, G (= E/2( 1 + v)) and E’, v’, and G’ denote Young’s modulus, Poisson’s ratio and shear modulus in the plane of isotropy and in the planes normal to the isotropy plane, respectively. Using (11) and (12) in conjunction with (14) and (15) in the equations expressing the stress-resultants and stress-couples
the 2D form of constitutive equations can be obtained. Their explicit expression in terms of the basic unknowns u,, $rr and u3 is not displayed here. In order to represent the governing equations in terms of the unknown displacement quantities, five macroscopic equations of motion are needed. These are derived by taking appropriately the moments of order zero and one of the equations of motion of the 3D non-linear elasticity theory. Upon retaining the non-linearities associated with the transverse deflection only, the 2D version of the equations of motion assumes the form: L”@),= 0, LaB(u3,B+ &)I,
M”Q - Q”3 = 0,
+ b,, Lap + p3101 + P3 - m0ii3 = 0.
(18a-c)
Flat and curved panels on a non-linear
215
foundation
In (1%~) P3 = p3 -
(IQ43
+
R&,
(184
(=p3(x”)) denoting the distributed pressure acting on the outer face, R, and If3 are the linear and cubic Winkler foundation moduli, respectively, while m. is the mass term associated with the unit area of the shell mid-surface. Substitution of stress-resultants and stress-couples expressed in terms of the displacement quantities in (18) yields one possible form of the governing equations of geometrically non-linear shear deformable anisotropic composite panels. This form of the governing equations will not be displayed here.
p3
5. GOVERNING
EQUATIONS
Following the development in [lg, 21-231, the governing dynamic equations of geometrically non-linear theory of curved panels symmetrically composed of transversely-isotropic material layers and incorporating initial geometrical imperfections are reduced to a form which may be viewed as the generalized counterpart of the classical von Karmin-Mushtari-Marguerre large deflection shell theory. Under this form they are: N$
- cawcflP{b,&J, + (%,,B + fi3,a&F,op
and f+l:=o.
(19c)
In (19a, b), F (= F(P, t)) denotes the Airy’s potential function while in (19c) 4 (E&P’, t)) is a potential function associated with the transverse shear rotations ICI=. In addition, D, B, C, M, and S stand for the rigidity quantities of the composite structure whose expression can be found in [23], ( -)I: and ( .)I:$ denote the 2D Laplace and biharmonic operators, respectively, while H denotes the average curvature of the mid-surface. The governing equations (19a, c) have been obtained by using (18b, c) whereas (19b) is based on the compatibility equation caacB%S,al + f%,ar$+* + 3&,l%,orS + tU+&g
+ b&,ni)
= 0,
(20)
referred to as the von Karman type compatibility equation. This equation replaces (Ha) identically fulfilled by expressing I,@ in terms of the Airy’s function F(P, t). The governing equations (19) include the effects of: (i) transverse shear deformation and transverse normal stress; (ii) large deflection (in the sense of the von K&man large deflection theory); (iii) initial geometric imperfections, as well as (iv) the presence of a non-linear elastic foundation. In addition, they fulfill the static conditions on the bounding surfaces of the panel. Specialization of governing equations for ~5,= 0, and of rigidity quantities for c&,= 0 and replacement in the rigidities S and A4 of transverse shear moduli Gikj by K’G;,. (where K2 denotes a transverse shear correction factor), the first-order transverse shear deformation (FSDT) counterpart of the present theory is obtained. The classical von K&r-man-Mushtari-Marguerre equations of large deflection shell theory could easily be obtained from the FSDT variant of the governing equations by considering therein Gikj--+cc. For the case of a single layered shell, the rigidity quantities have to be specialized for h,,, 1j + h/2 and XI=1 (*)-+O.
216
L. Librescu and Weiqing Lin
It should be mentioned that the governing equation (19~) defines the boundary layer effect. Its solution is characterized by a rapid decay when proceeding from the edges towards the interior of the shell. Although uncoupled in the governing equations, the unknown function 4 remains coupled with the other two functions, F and us, in the equations expressing the boundary conditions (in number of five at each edge). As was shown previously [l&21-23], for simply supported boundaries, the function I$ can be rendered decoupled in the boundary condition, and as a result, the boundary layer equation (19~) considered in conjunction with the associated boundary conditions results in the trivial solution 4 G 0.
6. SOLUTION
OF NON-LINEAR WITH
EQUATIONS
RECTANGULAR
OF SHALLOW
PANELS
PLANFORM
The postbuckling and vibration behavior of simply-supported composite doubly curved panels with rectangular planform, (Ii x 12) on II, resting on a non-linear foundation and subjected to a system of compressive edge loading system in the pre/postbuckling regimes and to a lateral pressure field will be analyzed. We will refer the points of CJto a Cartesian orthogonal system of coordinates assumed to be parallel to the panel edges. We consider the panel subjected to a system of uniform in-plane biaxial compressive edges loads Nil and N,, and to a pressure load distributed over the convex outer surface of the panel. Depending upon the tangential behavior at the edges, two cases of simply supported edge conditions labelled as Case (a) and Case (b) will be considered. Case (a): The edges are simply supported and freely movable in the direction normal to the edges in the plane tangent to the surface at the edges, implying that the tangential motion of the unloaded and loaded edges in the normal direction is unconstrained. For this situation the shell edges are referred to herein as movable edges. Case (b): The edges are simply supported. Uniaxial edges loads are acting in the direction of the xi-coordinate. The edges x1 = 0, I1 are considered freely movable (in the tangential direction normal to the edge), the remaining two edges being unloaded and immovable. For this case, the components of the tangential motion normal and parallel to the immovable edges are restrained and unrestrained, respectively. This is equivalent to consider the shear stress resultant at the edge as zero-valued and the normal displacement in the tangential plane as zero-valued in an average sense. The boundary conditions associated with these cases are presented in [23]. By paralleling the developments in the paper it may be shown that the out-of-plane boundary conditions could be expressed in terms of u3 and 4 in an uncoupled form. Moreover, (19~) in conjunction with the associated boundary conditions admits the trivial solution 4~0. In the forthcoming developments we will adopt the point of view documented in the specialized literature (see, e.g. [2]) according to which the imperfection resulting in the most critical conditions is of the same shape as the linear buckling mode. It may readily be shown that the representations for u3 and G3
(21) I
where I, = m7c/11,
pu,= nd2,
(m,n = 1,2,. . .)
fulfill exactly the out-of-plane boundary conditions. The tangential boundary conditions are satisfied on an average. To this end, the potential function F is represented as: F(xx,, 0 = Fr(x,, r) - +((#Nrl
+ (xi)2N~~)-
(22)
Here F,( E F1 (x,, t)) is a particular solution of (19b) [determined in conjunction with (21)] while Nll and N,, denote the normal edge loads (considered positive in compression). Similarly p3(x,) = pmnsin &x1 sin pnx2. (23) In the case of the panel loaded in the directions of the xl-coordinate only, the remaining edges being unloaded and immovable [that is of the Case (b)], the condition for the
Flat and curved
panels on a non-linear
foundation
217
immovable edges x, = 0, lz may be expressed in an average sense as (see [18]). 11 12 uz,2dx1dx2 = 0. ss0 0 This equation considered in conjunction with (12a) provides the fictitious edge load I?~, for which the edges x2 = 0, lz remain immovable. Following the procedure developed in [ 18,21-231, the displacement expansions as given by (21) are substituted into (19b) and the Airy’s stress function is obtained by solving the resulting linear non-homogeneous partial differential equation. The remaining governing equation, (19a), is converted into a set of non-linear ordinary differential equations via Galerkin’s method. This procedure yields the following set of M x N non-linear ordinary differential equations for each set of wave forms determined by the index pair (m, n)
(25) I,s indicates that there is no summation over the indices, I and s, where f r= 1,2,. . . , Mands=1,2,... N. In (25), PI, P2 and P3 are linear, quadratic, and cubic polynomials of the unknown modal amplitudes w,, respectively. The coefficients B,,, C,,, and R, are constants that depend on the material and geometric properties of the shell, & 1 ( = N1 I 1:/rc4D) and L2 ( = NJ, 1:/rr4D) are normalized forms of tangential edge loads, while K1 ( E If, 1:/n4D) and K, ( = I?, Zf/n4D) are dimensionless moduli of Winkler’s foundation. where the symbol
7. STATIC
EQUILIBRIUM
STATES
AND
SMALL
VIBRATIONS
The main emphasis of the present study is the vibrational behavior of flat and curved panels that are loaded quasistatically in the pre/postbuckling ranges as well as their postbuckling response. To obtain equations governing the static pre- and postbuckling equilibrium states and small vibrations about these equilibrium states, the unknown modal amplitudes are expressed as (see [22])
w,,(t)= es+ +w)
(26)
where G,,(t) represents small vibrations about a mean static equilibrium configuration described by W,,. The variations are considered small compared to W,,and the imperfection amplitude i;,, in the sense that
CGW12 4 %s, ks
(27)
for all values of the indices r and s. The equations for the static prebuckling and postbuckling equilibrium states are obtained by discarding the inertia terms given by A,,&, in (25), in which case the solution to the resulting equation is W,. The equations for small vibrations about a given static equilibrium state are then obtained by substituting (26) into (25) and invoking the smallness condition as expressed by (27). The resulting equations of motion are
A,,&(t) +
G,,G,(t)
=0
(28)
where G,s = C&J@,,$s, $“, %s, pm, L”,I , L”2
2,
KI , K2)
r=1,2,...,M;
s= 1,2,. . . , N.
(29)
The constant coefficients & are functions of the material and geometric properties of the panel. Equation (28) governs the small vibrations about a given equilibrium state and are solved by expressing Z&(t) as %,(t) = & exp (iw&)).
(30)
218
L. Librescu and Weiqing Lin
Replacement of (30) into (28) yields an algebraic eigenvalue problem given by
Gsk = ds&%s
r=l,2,...,M;
s=1,2 )...)
N,
.
(31)
r,s The eigenfrequencies o,, in (31) are the unknown quantities to be found and the corresponding amplitudes & are indeterminate. The solution of (28) starts with the determination of the static equilibrium states of the flat/curved panel over a given range of the loading parameters. The static equilibrium configuration for a given flat/curved panel is obtained by solved the static counterpart of the non-linear algebraic system, (25) expressed symbolically as L?(cr,,) = 0. For given panel geometry and mechanical characteristics, as well as for a given set of mechanical loads, the coefficients of _Y(&) are determined and then used to solve the algebraic system for the modal amplitudes. We note that this algebraic system represents the condition of equilibrium, i.e. the first variation of the potential functional being zero (6V = 0). Each solution W,, = Wr*,represents a possible buckled configuration of the panel under prescribed mechanical loads. The stability (instability) of each buckled configuration obtained in this manner is determined according to whether the second variation of the potential 6* V > 0 (CO). The condition can be re-stated in terms of the Jacobian of 9 with respect to W,,evaluated at the buckled equilibrium (w,, = w,*,)whose stability is sought (i.e. J,,,(9)&_,;]. The stability (instability) is decided based upon whether 5,,&Y)I~,S+;, is positive definite (negative definite). For the case m = n = 1 stability (instability) of the buckled configuration can be stated as d.Y/dwll > 0 (CO). After obtaining the static equilibrium configuration of a panel for given loading conditions, the coefficients A,, and G,, in (28) are computed and the linear algebraic eigenvalue problem defined by (31) is solved. Equation (31) also possesses negative eigenvalues that correspond to pure imaginary fundamental vibration frequencies. For the panels investigated herein, the pure imaginary fundamental frequencies correspond to unstable branches of the postbuckling equilibrium path. The existence of such unstable branches in the frequency-load interaction responses also implies the existence of dynamic jumps on the stable part of the branch. Such a behavior corresponds, in the static case counterpart, to the snap-through buckling. At this point it should be noticed the perfect correspondence between the frequency-load interaction and the load-displacement response. Whenever possible, this correspondence will be emphasized in the following numerical illustrations.
8. NUMERICAL
ILLUSTRATIONS
The results presented next aim to emphasize the influence played, in conjunction with other non-classical effects, by the linear/non-linear Winkler foundation upon the postbuckling and small vibration response of simply supported flat and curved panels. In the numerical illustrations it is considered that the panels feature a square projection (I x I) on the plane I-I and consist either of single or three layered panels. For all the results presented herein, for the single-layer panels the fixed ratio E/E’ = 5 and various transverse shear flexibility ratios E/G’ are considered throughout. In the calculations it was considered v = v’ = 0.2. For the symmetrically three-layer panels considered herein the inner layer is twice as thick as the face layers. Concerning the elastic coefficients of the outer layers, these are given in terms of the ratios E/E’ = 5, E/G’ = 10, whereas for the inner layer it is stipulated that E/E’ = 2, E/G’ = 30. The numerical illustrations assoicated with the static postbuckling, are depicted in the plane (J?, 1, 6 + 6,) or (p*, 6 +&,). Herein 6 =W11(1/2,1/2)/h and ~&,=&~;,,(1/2,1/2)/h denote the dimensionless amplitude of the transverse deflection and of the geometric imperfection, respectively, while p* =p3(1/2, 1/2)14/(Dh)is the dimensionless amplitude of the lateral pressure. In addition to these illustrations, other ones are displayed
Flat and curved panels on a non-linear foundation
219
in the plane (2; 1, A1), where Ai denotes the average end-shortening in the x,-direction de&d as I1 12 A,=-; u1,1 dxl dxz. (32) 12 II0 0 These latter plots provide a measure of the additional load-carrying capacity of the panel beyond buckling as well of its postbuckling stiffness and enable one to correlate the theoretical findings with the experimental ones. For the dynamic case, the results are displayed in the form of interaction curves that relate the magnitude of the average compressive edge loads El1 or the lateral pressure amplitude p* to the square of the fundamental frequency O2 ( E co2m014/(z4D)). In order to help understanding of the correlation between the dynamic and postbuckling behavior, several of the figures depicting the load-frequency interaction contain also an inset representing their static postbuckling counterpart. In all the displayed numerical results uniaxial edge compression is considered and unless otherwise specified, the edges are freely movable.
9. DISCUSSION
OF THE NUMERICAL
RESULTS
Figure 1 depicts in the plane (Lll, 6 + 6,) the static postbuckling behavior of geometrically perfect/imperfect flat three-layer plate resting on a linear foundation (KS = 0) and compressed by the edge loads L”,1, whereas the inset depicts its dynamic counterpart. In Figure 2, the counterpart of Fig. 1 in the plane (tll, A,) is considered. The results reveal that the increase of the linear foundation modulus results in the increase of both the buckling bifurcation and of the load-carrying capacity. These figures reflect also the well known fact that the flat panels feature imperfection-insensitivity. The inset of Fig. 1 depicts the frequency-load interaction. The plot shows that the natural frequencies decrease with the increase of the compressive edge load. Whereas for perfect flat panels buckling occurs for those E,, (indicated by the filled circles on the abscissa) rendering 6 2 + 0, in the case of the geometrically imperfect panel the frequencies do not vanish for any value of the compressive edge load. This reverts to the well known conclusion that an imperfect flat panel does not experience buckling bifurcation. The result related with the increase of the buckling load as a result of the increase of K1 emerging from Fig. 1 appears also in Fig. 2. In addition, the inset of Fig. 1 reveals that in the prebuckling range the increase of the linear foundation modulus K1 yields an increase of fundamental frequency, whereas in the 15.0
12.0
2.0 K,=O K,=2 K,=4 K,=6
6.0
3.0
0.0 1
2
Nondimensional
2
4
defkctiun pills imperfection 6ts
5 e
Fig. 1. Effects of a uniform compressive edge load and of the linear Winkler’s foundation modulus on the static postbuckling of a geometrically perfect/imperfect three-layer flat panel (Ii/h = 30, K3 = 0, ho = 0; 0.1). The inset depicts the frequency-load interaction counterpart.
220
L. Librescu
and Weiqing
Lin
4.0
2.0 l9llurcetion
point
0.0 0
0.006
0.012
0.019
Nondimensional Fig. 2. The static postbuckling
of the panel described
0
I 3
Nondimensional
/ 6
0.03
endshortening,
A,
in Fig. 1, displayed
in the plane (I?, r, AI)_
K.=CI
-3
0.024
K.=2
K-=4
1 9
compressive
load
, e
KS6
I
,
12
15
11
Fig. 3. Effects of a uniform compressive edge load and of the linear Winkler’s foundation modulus on the fundamental frequency of a geometrically perfect/imperfect single-layer circular cylindrical panel (II/h = 30, E/G'= 10, II/R1 = 0,12/Rz = 0.3, I& = 0, 6s = 0; 0.05). The inset depicts the static postbuckling counterpart.
postbuckling range the opposite trend holds valid. This reverse in trend is due to the fact that at lower values of Kr, following the buckling bifurcation, the participation of the membrane stiffness yielding the increase of the frequencies, starts at an earlier stage than in the case of the foundation featuring larger values of the linear parameter K1 . Figures 3 and 4 display the effect played by the linear foundation modulus on static postbuckling and frequency-load interaction of a geometrically perfect/imperfect cylindrical panel loaded on the curved edges. The results emerging from these figures restate the
Flat and curved panels on a non-linear foundation
221
15.0
12.0 (I -z i
p “# 9.0 Ir
II 6.0
3.0
.
Blfureofion I..
0.0
0
pold I
.
0.008
*
.
0.016
Nondimensional
.
I
.
.
endshortening,
*I
0.032
0.024 A,
Fig. 4. The static postbuckling of the panel described in Fig. 3, displayed in the plane (Lrt, A,).
3
II
9.0
73 6.0
Nondimensional
compreselve
load,
L,,
Fig. 5. Effects of a uniform compressive load at the curved edges and of the linear Winkler’s foundation modulus on the fundamental frequency of a geometrically imperfect three-layer circular cylindrical panel (II/h = 30, II/R1 = 0.6, Ks = 0,6, = 0.2). The inset depicts the static postbuckling counterpart.
conclusions already obtained in the case of flat panels and related to the effects of the linear foundation modulus K1. In addition, these figures reflect the well known unstable postbuckling behavior featured by the curved panels. However, Figs 3 and 4 reveal that a very small initial geometric imperfection can result in an attenuation of the intensity of the snapping phenomenon and, with the increase of the foundation parameter K1, even in its elimination. Figure 5 diplays the frequency edge-load interaction of a geometrically imperfect circular cylindrical panel featuring a larger geometric imperfection and resting on a linear Winkler foundation, while the inset displays its static postbuckling counterpart represented in the plane (L”,,,6 + 6,). The results reveal that with the increase of K1 larger limit loads are obtained. In addition, it becomes apparent that the increase of K1 is accompanied by an
222
L. Librescu and Weiqing Lin
attenuation of the intensity of the snapping phenomenon, attenuation which imply both the static and the dynamic response behaviors. The effect of the non-linear foundation modulus K3 upon the frequency-load interaction and its static postbuckling counterpart of a geometrically perfect circular cylindrical panel loaded along the curved edges is depicted in Fig. 6, in its inset and in Fig. 7. The results reveal that the increase of the hardening (KS > 0) non-linear foundation modulus the prebuckling behavior remains unaltered, whereas strong attenuations of the jump on the stable parts of the postbuckling paths are experienced. Figures 8 and 9 display the frequency-load interaction and the static counterpart of geometrically perfect and imperfect circular cylindrical panels resting on a non-linear
6
ir ‘3 ----KS>0 -KS=0 _._____K3
< 0
lBifurcation -1
0
point
I
I
I
I
2
4
6
6
Nondimenslonal compressive load
, ‘i
11
Fig. 6. Effects of a compressive edge load and of the cubic Winkler’s foundation modulus on the fundamental frequency of a geometrically perfect single layer circular cylindrical panel (II/h = 20, E/G’ = 10, II/R1 = 0, lZ/R2 = 0.2, Kr = 0). The inset depicts the static postbuckling counterpart.
6.0,,
.
.
..,..,
..,
,
.,..,...v
,
,......I _
_
6.0
V.....‘.....‘.....,.....,.....I 0.012 a 0.606
6.016
0.024
6.03
Nondimensional endrhortenlng, A, Fig. 7. The static postbuckling of the panel described in Fig. 6, displayed in the plane\ (z,
r,
Al).
Flat and curved panels on a non-linear foundation
223
6.0
4.0
2.0
-
lumltlomd
0.0
-2.0
I 2
0
.
.
.
, 4
.
.
.
, 6
I 8
Nondimensional compressive load, L
11
Fig. 8. Effects of a uniform compressive edge load and of the cubic Winkler foundation modulus on the fundamental frequency of a geometrically imperfect three-layer circular cylindrical panel (II/h = 30, II/R1 = 0,Zz/R2 = 0.3, Kr = 0, &, = 0.1).
6.0
j 13
2
4
6
8
Nondimensional external pressure, p’ Fig. 9. Effects of a lateral pressure and of the cubic Winkler’s foundation modulus on the fundamental frequency of a three-layer geometrically perfect circular cylindrical panel (1,/h = 20, II/R1 = 0,/JR2= 0.4). The inset depicts the static postbuckling counterpart.
foundation. The results reveal again that K3 has an influence solely on the postbuckling behavior. In contrast to a softening type foundation (KS < 0), in the case of its hardening counterpart (KS > 0), the intensity of the snap-through buckling is much attenuated and even eliminated. Figure 9 displays the interaction frequency-lateral interaction of a geometrically perfect circular cylindrical panel which is subjected to a pre-load & 1 (zOS(L”, l)Cr) and whose curved edges x1 = 0,l are immovable. The results reveal a similar trend as in the previously displayed case. In addition, the highly detrimental effect upon the static and dynamic post-limit response of the panel supported by a softening non-linear foundation becomes apparent.
224
L. Librescu and Weiqing Lin 12.0
10.0
8.0
6.0
4.0
2.0
0.0 -0.5
0
0.5
1
1.5
2
Nondimensional dehcllon, Fig. 10. Effects of cubic Winkler’s foundation a geometrically perfect circular cylindrical panel edges are immovable while the curved (It/h = 10, E/G’ = 10, It/R1 =
2.5
6
modulus on the static postbuckling behavior of exposed to a lateral pressure rise. The panel straight edges are compressed by the pre-load tlI, 0, &/R, = 0.3, Kr = 0.5, t,, = 3).
12.0
8.0
6.0
Noydimensional dehctlon.
6
Fig. 11. Effects of transverse shear flexibility (measured in terms of the ratio E/G’) upon the postbuckling of a geometrically perfect circular cyli_ndrical panel exposed to a lateral pressure rise, compressed by the uniaxial compressing pre-load Lrl and resting on a Winkler’s foundation. The classical predictions are also included in the graph: f&/h = 20, II/RI = 0, I,/Rz = 0.3, K1 = 0.5, KS=-l,Lrt=3).
Figure 10 presents the dependence pressure amplitude-dimensionless deflection amplitude of a circular cylindrical panel whose straight edges are immovable. It is supposed that the panel is compressed by axial edge loads and that it rests on a Winkler foundation of Ki = 0.5 and varying &. The results of the figure reveal the strong beneficial influence upon the post-limit behavior of the hardening type foundation. Finally, Fig. 11 presents the effect of the transverse shear flexibility upon the static postbuckling of a circular cylindrical panel supported by a Winkler foundation. The results reveal that the classical theory grossly overestimates the limit-load and underestimates the intensity of the snapping phenomenon.
Flat and curved panels on a non-linear foundation
225
10. CONCLUSIONS
A study of the effects played by a linear/non-linear Winkler foundation upon the post buckling response and vibration behavior of flat and curved panels subjected to compressive edge loads and a lateral pressure field was presented. The influence played by the transverse shear flexibility and the initial geometric imperfections was also examined. Although in the present study only uniaxial compression loads have been considered, the theory can also accomodate the case of bi-axial edge loads. The results emphasize the great role played by Winkler’s foundation moduli. In particular, the results show that the increase of K1 and of the hardening foundation modulus results in the increase of the buckling load and of the load-carrying capacity, in the attenuation of the intensity of the snapping phenomenon and even in its removal. REFERENCES 1. D. Hui, Postbuckling behavior of infinite beams on elastic foundations using Koiter’s improved theory. Int. .I. Non-Linear Me&. 23, 113 (1988). 2. J. C. Amazigo, B. Budiansky and G. F. Carrier, Asymptotic analyses of the buckling of imperfect columns on nonlinear elastic foundations. lnt. J. Solids Structures 6, 1341 (1970). 3. G. A. Kardomateas, Effect of an elastic foundation on the buckling and postbuckling of delaminated composites under compressive loads. J. Appt. Mech., Trans. ASME 55, 238 (1988). 4. I. Sheinman and M. Adan, Imperfection sensitivitv of a beam on a nonlinear elastic foundation Int. J. Mech. sci. 33, 753 (1991). 5. C. Massalas, K. Soldatos and G. Tzivanidis, Vibration and stability of a thin elastic plate resting on a non-linear elastic foundation when the deformation is large. J. Sound Vibr. 67. 284 (1979). 6. Y. Nath, Large amplitude response of circular plates on elastic foundations. Int. j. Non~Lin&r Mech. 17,285 (1982). I. M. Stein and P. A. Stein, A solution procedure for behavior of thick plates on a nonlinear foundation and postbuckling behavior of long plates. NASA TP 2174, Sept. (1983). 8. H.-S. Shen, Postbuckling of orthotropic plates on two-parameter elastic foundation. ASCE J. Engng Mech. 121, 50 (1995). 9. H.-S. Shen and F. W. Williams, Postbuckling analysis of imperfect composite laminated plates on non-linear elastic foundation. Int. J. Non-Linear Mech. 30, 651 (1995). 10. J. Ramachandran and P. A. K. Murthy, Nonlinear vibration of a shallow cylindrical panel on an elastic foundation. J. Sound Vibr. 47, 495 (1976). 11. C. Massalas and N. Kafousias, Non-linear vibrations of a shallow cylindrical panel on a non-linear elastic foundation. J. Sound Vibr. 66, 513 (1979). 12. Y. Nath, 0. Mahrenholtz and K. K. Varma, Non-linear dynamic response of a doubly curved shallow shell on an elastic foundation. J. Sound Vibr. 112, 53 (1987). 13. C. Y. Chia, Nonlinear analysis of doubly curved symmetrically laminated shallow shells with rectangular planform. Ingenieur-Archiu 58, 252 (1988). 14. B. A. Korbut, On the stability of shallow cylindrical shells supported on the inner surface by an elastic foundation. In Theory ofShells and Plates, Proceedings of the 4th All-Union Conference on Shells and Plates, 2631 Oct., PO 522 (edited by S. M. Durgarian), Translated by NASA, U.S.A., NSF, Washington, DC and by the Israel Program for Scientific Translations (1962). 15. S. N. Sinha, Large deflections of plates on elastic foundations. J. Ewng _ _ Mech. Div., Proc. of the ASCE EMl, 1 Feb. (1963). _ 16. E. Reissner, On postbuckling behavior and imperfection sensitivity of thin elastic plates on a non-linear elastic foundation. Stud. Appl. Math. XLIX, 45 (1970). 17. P. M. Naghdi, Foundations of elastic shell theory. In Prog. Solid Mech. (edited by I. N. Sneddon and R. Hill), Vol. 4, p. 1. North-Holland, Amsterdam (1963). 18. L. Librescu, Elastostatics and Kinetics of Anisotropic and Heterogeneous Shell-Type Structures. Noordhoff International, Leyden, The Netherlands (1975). 19. A. E. Green and W. Zerna, Theoretical Elasticity. Clarendon Press, Oxford (1968). 20. J. N. Reddy and C. F. Liu, A higher-order theory for geometrically nonlinear analysis of composite laminates. NASA CR-4056, March (1987). 21. L. Librescu, N. K. Chandiramani, M. P. Nemeth and J. H. Starnes Jr, “Postbuckling of laminated flat and curved panels under combined thermal and mechanical loadings. AIAA Paper-93-1563 CP, La Jolla, CA, April (1993). 22. L. Librescu, W. Lin, M. P. Nemeth and J. H. Starnes Jr, “Frequency-Load interaction of geometrically imperfect curved panels subjected to heating. AIAA J. 34, 166 (1996). 23. L. Librescu and M. Stein, A geometrically nonlinear theory of transversely-isotropic laminated composite plates and its use in the postbuckling analysis. Thin-Walled Structures 11, 177 (1991).
Vol. 32, No. 2, pp. 211-225, 1997 0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 002&7462/97 $17.00 + 0.00
Pergamon
SOO20-7462(%)00057-l
POSTBUCKLING AND VIBRATION OF SHEAR DEFORMABLE FLAT AND CURVED PANELS ON A NON-LINEAR ELASTIC FOUNDATION Liviu Librescu and Weiqing Lin Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0219, U.S.A.
(Received for publication 11 April 1996) Abstract-This study deals with the postbuckling and vibration behavior of flat and shallow curved panels resting on a Winkler linear/non-linear elastic foundation. The considered plate/shell structural model is based on a higher-order shear deformable theory and encompasses a number of effects such as transverse shear, geometric non-linearities and the initial geometric imperfection. Special emphasis is given to the influence played by Winkler’s foundation moduli as well as by the previously mentioned effects upon the postbuckling and vibrational response in the pre/postbuckling ranges and a number of pertinent conclusions are outlined. 0 1997 Elsevier Science Ltd. All rights reserved. Keywords postbuckling, plate/shell, Winkler foundation, hardening and softening non-linear foundation, shear deformability, initial geometric imperfection, buckling and limit load, load-frequency interaction
1. INTRODUCTION
The theory and behavior of beams, plates and shells on elastic foundation occupies a prominent place in the contemporary structural mechanics. In addition to the applications in many civil engineering structures, the recent developments related with solidpropellant rocket motors, the interest for developing further efficient thermal protection systems for space transportation vehicles as well as many other high technological applications have intensified the need for a better understanding of the behavior of plates and shells continuously supported by elastic media. In spite of its practical importance, the research done in this field was mainly devoted to beam structures on elastic foundations (see Cl], where an account of the work done in this field is provided, as well as [2-41). In addition to the fact that the research work addressing the problems of postbuckling and vibration response of plates and shells on elastic foundations (see, e.g. [S-9] and [lo-131, respectively) appears to be rather scarce, the treatment of the problem was done in rather specialized cases. In spite of this, credit for the early work in this area has to be given to Korbut [14], Sinha [15] and Reissner [16], who have pointed out for the first time, several of the features of the response of plates and shells resting on an elastic foundation. As the available specialized literature reveals, little attention has been given to the clarification of the implications of a number of important effects as transverse shear, initial geomtric imperfection, curvature of the panel and loading system upon the postbuckling and vibration of flat and curved panels resting on a linearJnon-linear elastic foundation. It is the goal of this paper to supply pertinent information in connection with this topic. To this end, a rather comprehensive study of the postbuckling and vibrational behavior of flat and curved panels resting on a Winkler foundation, subjected to a complex mechanical loading system and incorporating a number of non-classical features is presented. The mechanical loading investigated consists of a transverse lateral pressure and a system of uniform compressive edge loads acting in the pre/postbuckling ranges. The structural model is based on a higher-order transverse shear deformation theory of shallow shells that incorporates the effects of geometric non-linearities and initial geometric imperfections. 211
212
L. Librescu and Weiqing Lin
Results obtained using a special-purpose analysis, well-suited for parametric studies, are presented for single and three-layer panels. Simply supported panels are considered in which the tangential motion of the unloaded edges in either unrestrained or fully restrained. The results focusing on the postbuckling and load-frequency interaction cover a wide range of geometrical and physical parameters of the considered panels and the strong influence (qualitative and quantitative) played by the linear and non-linear (cubic) elastic foundation moduli as well as by other non-classical effect is emphasized. 2. GENERAL
CONSIDERATIONS
The case of doubly-curved shallow panels of uniform thickness h, symmetrically laminatedof2m+ l(m= 1,2,. . .)t ransversely-isotropic layers is considered, the surface of isotropy being parallel to the mid-surface of the structure. It is assumed: (i) that the shell is supported on the inner surface by an elastic foundation; and (ii) that perfect bonding between the shell and foundation and between the contiguous layers is implied. The points of the 3D space of the panel are referred to a set of curvilinear system of normal coordinates, xi, where x’, (LY = 1,2) denote the tangential coordinates, while x3 = 0 defines the reference surface (denoted henceforth as 0). The components of the metric tensor of the undeformed reference surface are: aap = am-as;
aa3 = aa-a3 = 0;
a33 =a3.a3
= 1,
a’@ = a’.aa;
aa3 = ab.a3 = 0;
a33 = a3.a3
= 1,
(1)
where ai and ai denote the contravariant and covariant base vectors of cr, respectively. The spatial metric tensor components gij of the undeformed shell-space are connected with their 2D counterparts a,@ by relations given in [17, IS] 1 w 933 = g= = 1, gas = g”3 = 0; (2) gap = CL,CLB a,,; where ,u; = 6; - x3b;, (3) plays the role of shifter in the space of the normal coordinates, 6,, and baadenoting the Kronecker delta and the curvature tensor, respectively. In the forthcoming developments, the concept of shallow shells will be used (see, e.g. [19]). Denoting by Z( =.Z(X~)) the amount of deviation of the shell reference surface from a plane Il (measured normal to the plane), consistent with this concept, 2 is assumed to be small when compared with a maximum length of an edge of the shell or with the minimum radius of curvature of 0. For this case, the assumption maxZ,.
< 1,
(4)
gives rise to the result that the metric tensors associated with the system of coordinates on cr and with its projection on the plane II are the same and, in addition, that the curvature tensor of the reference surface behaves as a constant in the differentiation operation. From this result it may be inferred that if the projected coordinate curves on II constitute a Cartesian orthogonal net, then the original ones on g are also to be, on the basis of (4), a Cartesian orthogonal net. Due to the quivalenceof the two metrics, it may also be concluded that the surface covariant differentiations may be done with respect to the metric associated with the plane II and thus it is possible to change the order of the covariant differentiations (since the Riemann-Christoffel tensor associated with the plane vanishes). Consistent with the shallow shell theory we may appropriately assume that & + S;, implying, consequently, gas + aaB and gas + aaB. (5) From (5) it may also be concluded that in this case p = l&l = (g/a)“2 + 1,
(6)
where g = det (gij) and a E det (aas). In order to reduce the 3D elasticity problem to an equivalent 2D one, the equations connecting the covariant derivatives of space tensors with their surface counterparts have to
Flat and curved panels on a non-linear foundation
213
be used. Such relations restricted to the case of shallow surfaces are:
T,,,, = Ta,,- ba,Ti3;
Ta,13
=
Tn.39 -
TJ,,a
=
T3.a
+
K’T,;
T3/,3 =
(7)
T3.3.
For more details concerning the deduction of these expressions, the reader is referred to [17, 181. Here, partial differentiation is denoted by a comma ( ),i E a( )/axi, while ( ) (Ii and ( )llx stand for the covariant differentiations with respect to the space and surface metrics, respectively, while the shifted components are identified by an upper bar. In the above relationships (as well as in the following developments), the Einsteinian summation convention applies to repeated indices where Latin indices range from 1 to 3 while the Greek indices range from 1 to 2.
3. DISPLACEMENT
REPRESENTATION
AND STRAIN
MEASURES
In order to model the geometrically non-linear theory of shear-deformable curved panels, the shifted displacements are represented as (see [20])
shallow
VJxO, x3, t) = u, + x3$, + (x3)% + (x3)3 i,, V3(xU,x3,
t) =
u3,
6%
b)
where Ui G
Ui(Xmy t);
$a(XW,
A., zz &(xW,
t);
t);
L? =
ra(Xw, t).
Based on the above representation of the displacement field, the exact fulfillment of tangential static conditions on the bounding surfaces x3 = f h/2 expressing the absence of shear tractions yields 1, = 0
and
i, = - $(u,,.
+ $=).
(9)
In the light of (9), it is apparent that fulfillment of above mentioned conditions results in a displacement field containing the same unknown functions as the first-order transverse shear deformation theory (FSDT), i.e. u3, U, and tidl. The underlined terms in (8a) have the character of corrective terms enabling one to fulfil the conditions on x3 = + h/2. It iqassumed also the existence of an initial out-of-plane, stress-free geometrical imperfection V3(xw,x3) = G3(xw). By convention, the transverse deflection is measured from the imperfect surface, in the positive, inward direction. Consistent with this, the straindisplacement relationships considered in the spirit of the von K&man’s partially non-linear theory read: 2eij
=
l/il,j
+
vj,li
+
v311iv3//j
+
f3i/iv3ilj
+
v3ilif3i{j.
(10)
Equation (10) used in conjunction with (6)--(g),yields the non-vanishing components of the strain tensor as: e,8 = aafi+ x3x,@ + &3)3&r, em3 =
~~3
+
(x3)2&3,
(lla,b)
where
22,s = - ,,,(+.
+
u3,a
+
&up).
(12a-e)
denote the 2D strain measures of the geometrically non-linear higher-order shell theory. It is easily seen that: (i) within its FSDT counterparts, in (11) the strain measures cab and Aa
214
L. Librescu and Weiqing Lin
become zero quantities; (ii) for *. + - (%a + @u,),
(13)
the strain measures, (12) reduces to the ones associated with the classical shallow shell theory (based on LoveKirchhoff hypothesis); and (iii) that the initial geometric imperfections occur in the membrane strain components, only. In the previous (and forthcoming) equations the covariant differentiation is performed with respect to the metric at II.
4. CONSTITUTIVE
EQUATIONS
As is well-known, the 3D elasticity theory implying small strains but large displacement gradients may be described by linear constitutive equations connecting second Piola-Kirchhoff stress components with Lagrangian strain measures. As a result, the strain-strain relationship for an elastically linear 3D anisotropic body is given by [18]: s@ = FhsWc wp+ 6*~s”3, sa3 = 2Ea3w3c03
(I4a, b)
where p@‘JP= E&P _
E~Zfl33Ea%O,, E3333
(15)
’
In (14) and (15) E’j”” and &?fl@’denote the tensors of elastic and modified elastic moduli, respectively; aA is a tracer identifying the contribution of s33 in the constitutive equations (and later in the governing equations) while sij denotes the second Piola-Kirchhoff stress tensor. In order to express s33 in terms of the basic unknowns, the third equation of motion of the 3D elasticity theory has to be used. For material layers exhibiting transverse-isotropic elastic properties as is considered in the present paper, the tensors of elastic moduli assume the form Cl83 &lm
=
E
l+v
l(aawa8s + a=ca8~) + _
E3333
E’(l -v)
1
_L_amcaa8,
[2
a@
(16a-c)
Here E,v, G (= E/2( 1 + v)) and E’, v’, and G’ denote Young’s modulus, Poisson’s ratio and shear modulus in the plane of isotropy and in the planes normal to the isotropy plane, respectively. Using (11) and (12) in conjunction with (14) and (15) in the equations expressing the stress-resultants and stress-couples
the 2D form of constitutive equations can be obtained. Their explicit expression in terms of the basic unknowns u,, $rr and u3 is not displayed here. In order to represent the governing equations in terms of the unknown displacement quantities, five macroscopic equations of motion are needed. These are derived by taking appropriately the moments of order zero and one of the equations of motion of the 3D non-linear elasticity theory. Upon retaining the non-linearities associated with the transverse deflection only, the 2D version of the equations of motion assumes the form: L”@),= 0, LaB(u3,B+ &)I,
M”Q - Q”3 = 0,
+ b,, Lap + p3101 + P3 - m0ii3 = 0.
(18a-c)
Flat and curved panels on a non-linear
215
foundation
In (1%~) P3 = p3 -
(IQ43
+
R&,
(184
(=p3(x”)) denoting the distributed pressure acting on the outer face, R, and If3 are the linear and cubic Winkler foundation moduli, respectively, while m. is the mass term associated with the unit area of the shell mid-surface. Substitution of stress-resultants and stress-couples expressed in terms of the displacement quantities in (18) yields one possible form of the governing equations of geometrically non-linear shear deformable anisotropic composite panels. This form of the governing equations will not be displayed here.
p3
5. GOVERNING
EQUATIONS
Following the development in [lg, 21-231, the governing dynamic equations of geometrically non-linear theory of curved panels symmetrically composed of transversely-isotropic material layers and incorporating initial geometrical imperfections are reduced to a form which may be viewed as the generalized counterpart of the classical von Karmin-Mushtari-Marguerre large deflection shell theory. Under this form they are: N$
- cawcflP{b,&J, + (%,,B + fi3,a&F,op
and f+l:=o.
(19c)
In (19a, b), F (= F(P, t)) denotes the Airy’s potential function while in (19c) 4 (E&P’, t)) is a potential function associated with the transverse shear rotations ICI=. In addition, D, B, C, M, and S stand for the rigidity quantities of the composite structure whose expression can be found in [23], ( -)I: and ( .)I:$ denote the 2D Laplace and biharmonic operators, respectively, while H denotes the average curvature of the mid-surface. The governing equations (19a, c) have been obtained by using (18b, c) whereas (19b) is based on the compatibility equation caacB%S,al + f%,ar$+* + 3&,l%,orS + tU+&g
+ b&,ni)
= 0,
(20)
referred to as the von Karman type compatibility equation. This equation replaces (Ha) identically fulfilled by expressing I,@ in terms of the Airy’s function F(P, t). The governing equations (19) include the effects of: (i) transverse shear deformation and transverse normal stress; (ii) large deflection (in the sense of the von K&man large deflection theory); (iii) initial geometric imperfections, as well as (iv) the presence of a non-linear elastic foundation. In addition, they fulfill the static conditions on the bounding surfaces of the panel. Specialization of governing equations for ~5,= 0, and of rigidity quantities for c&,= 0 and replacement in the rigidities S and A4 of transverse shear moduli Gikj by K’G;,. (where K2 denotes a transverse shear correction factor), the first-order transverse shear deformation (FSDT) counterpart of the present theory is obtained. The classical von K&r-man-Mushtari-Marguerre equations of large deflection shell theory could easily be obtained from the FSDT variant of the governing equations by considering therein Gikj--+cc. For the case of a single layered shell, the rigidity quantities have to be specialized for h,,, 1j + h/2 and XI=1 (*)-+O.
216
L. Librescu and Weiqing Lin
It should be mentioned that the governing equation (19~) defines the boundary layer effect. Its solution is characterized by a rapid decay when proceeding from the edges towards the interior of the shell. Although uncoupled in the governing equations, the unknown function 4 remains coupled with the other two functions, F and us, in the equations expressing the boundary conditions (in number of five at each edge). As was shown previously [l&21-23], for simply supported boundaries, the function I$ can be rendered decoupled in the boundary condition, and as a result, the boundary layer equation (19~) considered in conjunction with the associated boundary conditions results in the trivial solution 4 G 0.
6. SOLUTION
OF NON-LINEAR WITH
EQUATIONS
RECTANGULAR
OF SHALLOW
PANELS
PLANFORM
The postbuckling and vibration behavior of simply-supported composite doubly curved panels with rectangular planform, (Ii x 12) on II, resting on a non-linear foundation and subjected to a system of compressive edge loading system in the pre/postbuckling regimes and to a lateral pressure field will be analyzed. We will refer the points of CJto a Cartesian orthogonal system of coordinates assumed to be parallel to the panel edges. We consider the panel subjected to a system of uniform in-plane biaxial compressive edges loads Nil and N,, and to a pressure load distributed over the convex outer surface of the panel. Depending upon the tangential behavior at the edges, two cases of simply supported edge conditions labelled as Case (a) and Case (b) will be considered. Case (a): The edges are simply supported and freely movable in the direction normal to the edges in the plane tangent to the surface at the edges, implying that the tangential motion of the unloaded and loaded edges in the normal direction is unconstrained. For this situation the shell edges are referred to herein as movable edges. Case (b): The edges are simply supported. Uniaxial edges loads are acting in the direction of the xi-coordinate. The edges x1 = 0, I1 are considered freely movable (in the tangential direction normal to the edge), the remaining two edges being unloaded and immovable. For this case, the components of the tangential motion normal and parallel to the immovable edges are restrained and unrestrained, respectively. This is equivalent to consider the shear stress resultant at the edge as zero-valued and the normal displacement in the tangential plane as zero-valued in an average sense. The boundary conditions associated with these cases are presented in [23]. By paralleling the developments in the paper it may be shown that the out-of-plane boundary conditions could be expressed in terms of u3 and 4 in an uncoupled form. Moreover, (19~) in conjunction with the associated boundary conditions admits the trivial solution 4~0. In the forthcoming developments we will adopt the point of view documented in the specialized literature (see, e.g. [2]) according to which the imperfection resulting in the most critical conditions is of the same shape as the linear buckling mode. It may readily be shown that the representations for u3 and G3
(21) I
where I, = m7c/11,
pu,= nd2,
(m,n = 1,2,. . .)
fulfill exactly the out-of-plane boundary conditions. The tangential boundary conditions are satisfied on an average. To this end, the potential function F is represented as: F(xx,, 0 = Fr(x,, r) - +((#Nrl
+ (xi)2N~~)-
(22)
Here F,( E F1 (x,, t)) is a particular solution of (19b) [determined in conjunction with (21)] while Nll and N,, denote the normal edge loads (considered positive in compression). Similarly p3(x,) = pmnsin &x1 sin pnx2. (23) In the case of the panel loaded in the directions of the xl-coordinate only, the remaining edges being unloaded and immovable [that is of the Case (b)], the condition for the
Flat and curved
panels on a non-linear
foundation
217
immovable edges x, = 0, lz may be expressed in an average sense as (see [18]). 11 12 uz,2dx1dx2 = 0. ss0 0 This equation considered in conjunction with (12a) provides the fictitious edge load I?~, for which the edges x2 = 0, lz remain immovable. Following the procedure developed in [ 18,21-231, the displacement expansions as given by (21) are substituted into (19b) and the Airy’s stress function is obtained by solving the resulting linear non-homogeneous partial differential equation. The remaining governing equation, (19a), is converted into a set of non-linear ordinary differential equations via Galerkin’s method. This procedure yields the following set of M x N non-linear ordinary differential equations for each set of wave forms determined by the index pair (m, n)
(25) I,s indicates that there is no summation over the indices, I and s, where f r= 1,2,. . . , Mands=1,2,... N. In (25), PI, P2 and P3 are linear, quadratic, and cubic polynomials of the unknown modal amplitudes w,, respectively. The coefficients B,,, C,,, and R, are constants that depend on the material and geometric properties of the shell, & 1 ( = N1 I 1:/rc4D) and L2 ( = NJ, 1:/rr4D) are normalized forms of tangential edge loads, while K1 ( E If, 1:/n4D) and K, ( = I?, Zf/n4D) are dimensionless moduli of Winkler’s foundation. where the symbol
7. STATIC
EQUILIBRIUM
STATES
AND
SMALL
VIBRATIONS
The main emphasis of the present study is the vibrational behavior of flat and curved panels that are loaded quasistatically in the pre/postbuckling ranges as well as their postbuckling response. To obtain equations governing the static pre- and postbuckling equilibrium states and small vibrations about these equilibrium states, the unknown modal amplitudes are expressed as (see [22])
w,,(t)= es+ +w)
(26)
where G,,(t) represents small vibrations about a mean static equilibrium configuration described by W,,. The variations are considered small compared to W,,and the imperfection amplitude i;,, in the sense that
CGW12 4 %s, ks
(27)
for all values of the indices r and s. The equations for the static prebuckling and postbuckling equilibrium states are obtained by discarding the inertia terms given by A,,&, in (25), in which case the solution to the resulting equation is W,. The equations for small vibrations about a given static equilibrium state are then obtained by substituting (26) into (25) and invoking the smallness condition as expressed by (27). The resulting equations of motion are
A,,&(t) +
G,,G,(t)
=0
(28)
where G,s = C&J@,,$s, $“, %s, pm, L”,I , L”2
2,
KI , K2)
r=1,2,...,M;
s= 1,2,. . . , N.
(29)
The constant coefficients & are functions of the material and geometric properties of the panel. Equation (28) governs the small vibrations about a given equilibrium state and are solved by expressing Z&(t) as %,(t) = & exp (iw&)).
(30)
218
L. Librescu and Weiqing Lin
Replacement of (30) into (28) yields an algebraic eigenvalue problem given by
Gsk = ds&%s
r=l,2,...,M;
s=1,2 )...)
N,
.
(31)
r,s The eigenfrequencies o,, in (31) are the unknown quantities to be found and the corresponding amplitudes & are indeterminate. The solution of (28) starts with the determination of the static equilibrium states of the flat/curved panel over a given range of the loading parameters. The static equilibrium configuration for a given flat/curved panel is obtained by solved the static counterpart of the non-linear algebraic system, (25) expressed symbolically as L?(cr,,) = 0. For given panel geometry and mechanical characteristics, as well as for a given set of mechanical loads, the coefficients of _Y(&) are determined and then used to solve the algebraic system for the modal amplitudes. We note that this algebraic system represents the condition of equilibrium, i.e. the first variation of the potential functional being zero (6V = 0). Each solution W,, = Wr*,represents a possible buckled configuration of the panel under prescribed mechanical loads. The stability (instability) of each buckled configuration obtained in this manner is determined according to whether the second variation of the potential 6* V > 0 (CO). The condition can be re-stated in terms of the Jacobian of 9 with respect to W,,evaluated at the buckled equilibrium (w,, = w,*,)whose stability is sought (i.e. J,,,(9)&_,;]. The stability (instability) is decided based upon whether 5,,&Y)I~,S+;, is positive definite (negative definite). For the case m = n = 1 stability (instability) of the buckled configuration can be stated as d.Y/dwll > 0 (CO). After obtaining the static equilibrium configuration of a panel for given loading conditions, the coefficients A,, and G,, in (28) are computed and the linear algebraic eigenvalue problem defined by (31) is solved. Equation (31) also possesses negative eigenvalues that correspond to pure imaginary fundamental vibration frequencies. For the panels investigated herein, the pure imaginary fundamental frequencies correspond to unstable branches of the postbuckling equilibrium path. The existence of such unstable branches in the frequency-load interaction responses also implies the existence of dynamic jumps on the stable part of the branch. Such a behavior corresponds, in the static case counterpart, to the snap-through buckling. At this point it should be noticed the perfect correspondence between the frequency-load interaction and the load-displacement response. Whenever possible, this correspondence will be emphasized in the following numerical illustrations.
8. NUMERICAL
ILLUSTRATIONS
The results presented next aim to emphasize the influence played, in conjunction with other non-classical effects, by the linear/non-linear Winkler foundation upon the postbuckling and small vibration response of simply supported flat and curved panels. In the numerical illustrations it is considered that the panels feature a square projection (I x I) on the plane I-I and consist either of single or three layered panels. For all the results presented herein, for the single-layer panels the fixed ratio E/E’ = 5 and various transverse shear flexibility ratios E/G’ are considered throughout. In the calculations it was considered v = v’ = 0.2. For the symmetrically three-layer panels considered herein the inner layer is twice as thick as the face layers. Concerning the elastic coefficients of the outer layers, these are given in terms of the ratios E/E’ = 5, E/G’ = 10, whereas for the inner layer it is stipulated that E/E’ = 2, E/G’ = 30. The numerical illustrations assoicated with the static postbuckling, are depicted in the plane (J?, 1, 6 + 6,) or (p*, 6 +&,). Herein 6 =W11(1/2,1/2)/h and ~&,=&~;,,(1/2,1/2)/h denote the dimensionless amplitude of the transverse deflection and of the geometric imperfection, respectively, while p* =p3(1/2, 1/2)14/(Dh)is the dimensionless amplitude of the lateral pressure. In addition to these illustrations, other ones are displayed
Flat and curved panels on a non-linear foundation
219
in the plane (2; 1, A1), where Ai denotes the average end-shortening in the x,-direction de&d as I1 12 A,=-; u1,1 dxl dxz. (32) 12 II0 0 These latter plots provide a measure of the additional load-carrying capacity of the panel beyond buckling as well of its postbuckling stiffness and enable one to correlate the theoretical findings with the experimental ones. For the dynamic case, the results are displayed in the form of interaction curves that relate the magnitude of the average compressive edge loads El1 or the lateral pressure amplitude p* to the square of the fundamental frequency O2 ( E co2m014/(z4D)). In order to help understanding of the correlation between the dynamic and postbuckling behavior, several of the figures depicting the load-frequency interaction contain also an inset representing their static postbuckling counterpart. In all the displayed numerical results uniaxial edge compression is considered and unless otherwise specified, the edges are freely movable.
9. DISCUSSION
OF THE NUMERICAL
RESULTS
Figure 1 depicts in the plane (Lll, 6 + 6,) the static postbuckling behavior of geometrically perfect/imperfect flat three-layer plate resting on a linear foundation (KS = 0) and compressed by the edge loads L”,1, whereas the inset depicts its dynamic counterpart. In Figure 2, the counterpart of Fig. 1 in the plane (tll, A,) is considered. The results reveal that the increase of the linear foundation modulus results in the increase of both the buckling bifurcation and of the load-carrying capacity. These figures reflect also the well known fact that the flat panels feature imperfection-insensitivity. The inset of Fig. 1 depicts the frequency-load interaction. The plot shows that the natural frequencies decrease with the increase of the compressive edge load. Whereas for perfect flat panels buckling occurs for those E,, (indicated by the filled circles on the abscissa) rendering 6 2 + 0, in the case of the geometrically imperfect panel the frequencies do not vanish for any value of the compressive edge load. This reverts to the well known conclusion that an imperfect flat panel does not experience buckling bifurcation. The result related with the increase of the buckling load as a result of the increase of K1 emerging from Fig. 1 appears also in Fig. 2. In addition, the inset of Fig. 1 reveals that in the prebuckling range the increase of the linear foundation modulus K1 yields an increase of fundamental frequency, whereas in the 15.0
12.0
2.0 K,=O K,=2 K,=4 K,=6
6.0
3.0
0.0 1
2
Nondimensional
2
4
defkctiun pills imperfection 6ts
5 e
Fig. 1. Effects of a uniform compressive edge load and of the linear Winkler’s foundation modulus on the static postbuckling of a geometrically perfect/imperfect three-layer flat panel (Ii/h = 30, K3 = 0, ho = 0; 0.1). The inset depicts the frequency-load interaction counterpart.
220
L. Librescu
and Weiqing
Lin
4.0
2.0 l9llurcetion
point
0.0 0
0.006
0.012
0.019
Nondimensional Fig. 2. The static postbuckling
of the panel described
0
I 3
Nondimensional
/ 6
0.03
endshortening,
A,
in Fig. 1, displayed
in the plane (I?, r, AI)_
K.=CI
-3
0.024
K.=2
K-=4
1 9
compressive
load
, e
KS6
I
,
12
15
11
Fig. 3. Effects of a uniform compressive edge load and of the linear Winkler’s foundation modulus on the fundamental frequency of a geometrically perfect/imperfect single-layer circular cylindrical panel (II/h = 30, E/G'= 10, II/R1 = 0,12/Rz = 0.3, I& = 0, 6s = 0; 0.05). The inset depicts the static postbuckling counterpart.
postbuckling range the opposite trend holds valid. This reverse in trend is due to the fact that at lower values of Kr, following the buckling bifurcation, the participation of the membrane stiffness yielding the increase of the frequencies, starts at an earlier stage than in the case of the foundation featuring larger values of the linear parameter K1 . Figures 3 and 4 display the effect played by the linear foundation modulus on static postbuckling and frequency-load interaction of a geometrically perfect/imperfect cylindrical panel loaded on the curved edges. The results emerging from these figures restate the
Flat and curved panels on a non-linear foundation
221
15.0
12.0 (I -z i
p “# 9.0 Ir
II 6.0
3.0
.
Blfureofion I..
0.0
0
pold I
.
0.008
*
.
0.016
Nondimensional
.
I
.
.
endshortening,
*I
0.032
0.024 A,
Fig. 4. The static postbuckling of the panel described in Fig. 3, displayed in the plane (Lrt, A,).
3
II
9.0
73 6.0
Nondimensional
compreselve
load,
L,,
Fig. 5. Effects of a uniform compressive load at the curved edges and of the linear Winkler’s foundation modulus on the fundamental frequency of a geometrically imperfect three-layer circular cylindrical panel (II/h = 30, II/R1 = 0.6, Ks = 0,6, = 0.2). The inset depicts the static postbuckling counterpart.
conclusions already obtained in the case of flat panels and related to the effects of the linear foundation modulus K1. In addition, these figures reflect the well known unstable postbuckling behavior featured by the curved panels. However, Figs 3 and 4 reveal that a very small initial geometric imperfection can result in an attenuation of the intensity of the snapping phenomenon and, with the increase of the foundation parameter K1, even in its elimination. Figure 5 diplays the frequency edge-load interaction of a geometrically imperfect circular cylindrical panel featuring a larger geometric imperfection and resting on a linear Winkler foundation, while the inset displays its static postbuckling counterpart represented in the plane (L”,,,6 + 6,). The results reveal that with the increase of K1 larger limit loads are obtained. In addition, it becomes apparent that the increase of K1 is accompanied by an
222
L. Librescu and Weiqing Lin
attenuation of the intensity of the snapping phenomenon, attenuation which imply both the static and the dynamic response behaviors. The effect of the non-linear foundation modulus K3 upon the frequency-load interaction and its static postbuckling counterpart of a geometrically perfect circular cylindrical panel loaded along the curved edges is depicted in Fig. 6, in its inset and in Fig. 7. The results reveal that the increase of the hardening (KS > 0) non-linear foundation modulus the prebuckling behavior remains unaltered, whereas strong attenuations of the jump on the stable parts of the postbuckling paths are experienced. Figures 8 and 9 display the frequency-load interaction and the static counterpart of geometrically perfect and imperfect circular cylindrical panels resting on a non-linear
6
ir ‘3 ----KS>0 -KS=0 _._____K3
< 0
lBifurcation -1
0
point
I
I
I
I
2
4
6
6
Nondimenslonal compressive load
, ‘i
11
Fig. 6. Effects of a compressive edge load and of the cubic Winkler’s foundation modulus on the fundamental frequency of a geometrically perfect single layer circular cylindrical panel (II/h = 20, E/G’ = 10, II/R1 = 0, lZ/R2 = 0.2, Kr = 0). The inset depicts the static postbuckling counterpart.
6.0,,
.
.
..,..,
..,
,
.,..,...v
,
,......I _
_
6.0
V.....‘.....‘.....,.....,.....I 0.012 a 0.606
6.016
0.024
6.03
Nondimensional endrhortenlng, A, Fig. 7. The static postbuckling of the panel described in Fig. 6, displayed in the plane\ (z,
r,
Al).
Flat and curved panels on a non-linear foundation
223
6.0
4.0
2.0
-
lumltlomd
0.0
-2.0
I 2
0
.
.
.
, 4
.
.
.
, 6
I 8
Nondimensional compressive load, L
11
Fig. 8. Effects of a uniform compressive edge load and of the cubic Winkler foundation modulus on the fundamental frequency of a geometrically imperfect three-layer circular cylindrical panel (II/h = 30, II/R1 = 0,Zz/R2 = 0.3, Kr = 0, &, = 0.1).
6.0
j 13
2
4
6
8
Nondimensional external pressure, p’ Fig. 9. Effects of a lateral pressure and of the cubic Winkler’s foundation modulus on the fundamental frequency of a three-layer geometrically perfect circular cylindrical panel (1,/h = 20, II/R1 = 0,/JR2= 0.4). The inset depicts the static postbuckling counterpart.
foundation. The results reveal again that K3 has an influence solely on the postbuckling behavior. In contrast to a softening type foundation (KS < 0), in the case of its hardening counterpart (KS > 0), the intensity of the snap-through buckling is much attenuated and even eliminated. Figure 9 displays the interaction frequency-lateral interaction of a geometrically perfect circular cylindrical panel which is subjected to a pre-load & 1 (zOS(L”, l)Cr) and whose curved edges x1 = 0,l are immovable. The results reveal a similar trend as in the previously displayed case. In addition, the highly detrimental effect upon the static and dynamic post-limit response of the panel supported by a softening non-linear foundation becomes apparent.
224
L. Librescu and Weiqing Lin 12.0
10.0
8.0
6.0
4.0
2.0
0.0 -0.5
0
0.5
1
1.5
2
Nondimensional dehcllon, Fig. 10. Effects of cubic Winkler’s foundation a geometrically perfect circular cylindrical panel edges are immovable while the curved (It/h = 10, E/G’ = 10, It/R1 =
2.5
6
modulus on the static postbuckling behavior of exposed to a lateral pressure rise. The panel straight edges are compressed by the pre-load tlI, 0, &/R, = 0.3, Kr = 0.5, t,, = 3).
12.0
8.0
6.0
Noydimensional dehctlon.
6
Fig. 11. Effects of transverse shear flexibility (measured in terms of the ratio E/G’) upon the postbuckling of a geometrically perfect circular cyli_ndrical panel exposed to a lateral pressure rise, compressed by the uniaxial compressing pre-load Lrl and resting on a Winkler’s foundation. The classical predictions are also included in the graph: f&/h = 20, II/RI = 0, I,/Rz = 0.3, K1 = 0.5, KS=-l,Lrt=3).
Figure 10 presents the dependence pressure amplitude-dimensionless deflection amplitude of a circular cylindrical panel whose straight edges are immovable. It is supposed that the panel is compressed by axial edge loads and that it rests on a Winkler foundation of Ki = 0.5 and varying &. The results of the figure reveal the strong beneficial influence upon the post-limit behavior of the hardening type foundation. Finally, Fig. 11 presents the effect of the transverse shear flexibility upon the static postbuckling of a circular cylindrical panel supported by a Winkler foundation. The results reveal that the classical theory grossly overestimates the limit-load and underestimates the intensity of the snapping phenomenon.
Flat and curved panels on a non-linear foundation
225
10. CONCLUSIONS
A study of the effects played by a linear/non-linear Winkler foundation upon the post buckling response and vibration behavior of flat and curved panels subjected to compressive edge loads and a lateral pressure field was presented. The influence played by the transverse shear flexibility and the initial geometric imperfections was also examined. Although in the present study only uniaxial compression loads have been considered, the theory can also accomodate the case of bi-axial edge loads. The results emphasize the great role played by Winkler’s foundation moduli. In particular, the results show that the increase of K1 and of the hardening foundation modulus results in the increase of the buckling load and of the load-carrying capacity, in the attenuation of the intensity of the snapping phenomenon and even in its removal. REFERENCES 1. D. Hui, Postbuckling behavior of infinite beams on elastic foundations using Koiter’s improved theory. Int. .I. Non-Linear Me&. 23, 113 (1988). 2. J. C. Amazigo, B. Budiansky and G. F. Carrier, Asymptotic analyses of the buckling of imperfect columns on nonlinear elastic foundations. lnt. J. Solids Structures 6, 1341 (1970). 3. G. A. Kardomateas, Effect of an elastic foundation on the buckling and postbuckling of delaminated composites under compressive loads. J. Appt. Mech., Trans. ASME 55, 238 (1988). 4. I. Sheinman and M. Adan, Imperfection sensitivitv of a beam on a nonlinear elastic foundation Int. J. Mech. sci. 33, 753 (1991). 5. C. Massalas, K. Soldatos and G. Tzivanidis, Vibration and stability of a thin elastic plate resting on a non-linear elastic foundation when the deformation is large. J. Sound Vibr. 67. 284 (1979). 6. Y. Nath, Large amplitude response of circular plates on elastic foundations. Int. j. Non~Lin&r Mech. 17,285 (1982). I. M. Stein and P. A. Stein, A solution procedure for behavior of thick plates on a nonlinear foundation and postbuckling behavior of long plates. NASA TP 2174, Sept. (1983). 8. H.-S. Shen, Postbuckling of orthotropic plates on two-parameter elastic foundation. ASCE J. Engng Mech. 121, 50 (1995). 9. H.-S. Shen and F. W. Williams, Postbuckling analysis of imperfect composite laminated plates on non-linear elastic foundation. Int. J. Non-Linear Mech. 30, 651 (1995). 10. J. Ramachandran and P. A. K. Murthy, Nonlinear vibration of a shallow cylindrical panel on an elastic foundation. J. Sound Vibr. 47, 495 (1976). 11. C. Massalas and N. Kafousias, Non-linear vibrations of a shallow cylindrical panel on a non-linear elastic foundation. J. Sound Vibr. 66, 513 (1979). 12. Y. Nath, 0. Mahrenholtz and K. K. Varma, Non-linear dynamic response of a doubly curved shallow shell on an elastic foundation. J. Sound Vibr. 112, 53 (1987). 13. C. Y. Chia, Nonlinear analysis of doubly curved symmetrically laminated shallow shells with rectangular planform. Ingenieur-Archiu 58, 252 (1988). 14. B. A. Korbut, On the stability of shallow cylindrical shells supported on the inner surface by an elastic foundation. In Theory ofShells and Plates, Proceedings of the 4th All-Union Conference on Shells and Plates, 2631 Oct., PO 522 (edited by S. M. Durgarian), Translated by NASA, U.S.A., NSF, Washington, DC and by the Israel Program for Scientific Translations (1962). 15. S. N. Sinha, Large deflections of plates on elastic foundations. J. Ewng _ _ Mech. Div., Proc. of the ASCE EMl, 1 Feb. (1963). _ 16. E. Reissner, On postbuckling behavior and imperfection sensitivity of thin elastic plates on a non-linear elastic foundation. Stud. Appl. Math. XLIX, 45 (1970). 17. P. M. Naghdi, Foundations of elastic shell theory. In Prog. Solid Mech. (edited by I. N. Sneddon and R. Hill), Vol. 4, p. 1. North-Holland, Amsterdam (1963). 18. L. Librescu, Elastostatics and Kinetics of Anisotropic and Heterogeneous Shell-Type Structures. Noordhoff International, Leyden, The Netherlands (1975). 19. A. E. Green and W. Zerna, Theoretical Elasticity. Clarendon Press, Oxford (1968). 20. J. N. Reddy and C. F. Liu, A higher-order theory for geometrically nonlinear analysis of composite laminates. NASA CR-4056, March (1987). 21. L. Librescu, N. K. Chandiramani, M. P. Nemeth and J. H. Starnes Jr, “Postbuckling of laminated flat and curved panels under combined thermal and mechanical loadings. AIAA Paper-93-1563 CP, La Jolla, CA, April (1993). 22. L. Librescu, W. Lin, M. P. Nemeth and J. H. Starnes Jr, “Frequency-Load interaction of geometrically imperfect curved panels subjected to heating. AIAA J. 34, 166 (1996). 23. L. Librescu and M. Stein, A geometrically nonlinear theory of transversely-isotropic laminated composite plates and its use in the postbuckling analysis. Thin-Walled Structures 11, 177 (1991).