Non-linear design and control optimization of composite laminated plates with buckling and postbuckling objectives

International Journal of Non-Linear Mechanics 41 (2006) 807 – 824 www.elsevier.com/locate/nlm

Non-linear design and control optimization of composite laminated plates with buckling and postbuckling objectives M.E. Fares ∗ , Y.G. Youssif, A.E. Elshoraky Department of mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Received 14 July 2004; received in revised form 20 April 2006; accepted 4 May 2006

Abstract Multiobjective design and control optimization of composite laminated plates is presented to minimize the postbuckling dynamic response and maximize the buckling load. The control objective aims at dissipating the postbuckling elastic energy of the laminate with the minimum possible expenditure of control energy using a closed-loop distributed force. The layer thicknesses and fiber orientations are taken as design variables. The objectives of the optimization problem are formulated based on a shear deformation theory including the von-Karman non-linear effect for various cases of boundary conditions. The non-linear control problem is solved iteratively until an appropriate convergence criterion is satisfied based on Liapunov–Bellman theory. Liapunov function is taken as a sum of positive definite functions with different degrees. Comparative examples for three-layer symmetric and four-layer antisymmetric laminates are given for various cases of edges conditions. Graphical study is carried out to assess the accuracy of results obtained due to the successive iterations. The influences of the boundary conditions, orthotropy ratio, shear deformation, aspect ratio on the laminate optimal design are elucidated. 䉷 2006 Elsevier Ltd. All rights reserved. Keywords: Structural design and control; Composite laminates; Buckling and postbuckling responses; Non-linear analysis; Shear deformation theory; Various boundary conditions

1. Introduction Laminated composites are being widely used in many industries, mainly because they exhibit high strength-to-weight and stiffness-to-weight ratios, as well as, the other properties which make them ideally suited for use in weight-sensitive structures. One of the most significant uses of advanced composite materials occurs in the aerospace industry, and particularly, in the construction of large space structures which are built with a high degree of flexibility and mostly with very low natural damping. However, serviceability and safety requirements restrict the allowable limits of the dynamical response to external disturbances to specified values. This problem is commonly known as vibration damping [1].

∗ Corresponding author.

E-mail addresses: [email protected], [email protected] (M.E. Fares). 0020-7462/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2006.05.003

Design optimization of composite structures is concerned with the best use of the tailoring capabilities of fiber-reinforced laminates to maximize (or minimize) a given design objective. The vibration damping involves the damping out of the excessive vibrations by means of active structural control. In many research studies, the two techniques of the design optimization and active control were treated as separate issues [2–6]. In the last two decades, a great deal of interest for the interaction between these two techniques has been manifested in the literature with a view towards integrating the design optimization and active control in a single formulation [7–12]. In many engineering applications, it is necessary to maximize the buckling load subjected to such design constraints as strength, frequency, displacement, etc. The problem can be formulated as a minimum weight design problem subjected to buckling and other constraints. Alternatively, the laminate may be optimized with respect to several objectives using a multicriteria design approach. Structures optimized with respect to buckling strength may exhibit low postbuckling resistance [13].

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Thus, for this multiobjective optimization problem, laminates designed with respect to one criterion will perform quite poorly with respect to the other one [14,15]. Therefore, optimization in the postbuckling range becomes an important design consideration for laminates which may be exposed to temperature load or compressive load or combined thermo-mechanical load higher than the buckling load. Many investigators have used integrated approaches to reconciling the conflicting objectives in the design and vibration control of composite structures [16–19]. Further studies on the design and control optimization of composite laminates in the prebuckling and postbuckling ranges can be found in [20–24]. In general, design sensitivity analysis is important to know accurately the effects of the design variable changes on the performance of composite laminates in various cases of boundary conditions. To evaluate these sensitivities efficiently and accurately, it is important to have appropriate techniques associated with good structural models. However, most research studies related to the design and control optimization of composite laminates were carried out based on the classical theories for special cases of boundary conditions. In addition, the optimization problems of composite laminates in the prebuckling range have been extensively studied, but relatively little attention has been directed to such problems in the postbuckling range which need a non-linear (large deflection) analysis. The current work deals with a non-linear multiobjective optimization problem of composite laminated plates subjected to in-plane compressive forces. The design and control objectives are to maximize the buckling load and to minimize the postbuckling dynamic response with minimum expenditure of control energy. The total elastic energy is taken as a measure of the dynamic response. The fiber orientation angles and layer thicknesses are taken as design variables. The optimization objectives are formulated based on a shear deformation theory including the von-Karmen non-linearity [25]. The optimality condition of Liapunov–Bellman theory [26] is used to obtain the optimal control force and controlled buckled deflection iteratively until an appropriate convergence criterion is satisfied. For this purpose, Liapunov function is taken as a sum of positive definite functions with different degrees. Comparative examples are given for three-layer symmetric and four-layer antisymmetric laminated plates with various cases of boundary condition to show the advantages of the present optimization approach. Also, the influences of boundary conditions, material and geometric parameters on the optimization process are studied.

is loaded by a transverse distributed force q(x, y, t) acting as a control force. The present formulation is based on a first-order shear deformation laminate theory accounting for the following Reissner–Mindlin displacements: u1 (x, y, z, t) = u(x, y, t) + z
(x, y, t), u2 (x, y, z, t) = v(x, y, t) + z(x, y, t), u3 (x, y, z, t) = w(x, y, t),

where (u1 , u2 , u3 ) are the displacements along x, y and z directions, respectively, (u, v, w) are the displacements of a point on the mid-plane, and
and  are the slope changes in the x and y directions (i.e. rotations about the y- and x-axes), respectively, due to bending. The present study deals with the postbuckling response characterized by finite deformation. Therefore, the strains associated with the displacement (1) must include the geometric non-linear effect. The strains according to the von-Karman theory take the form [25]: (0)

(0)

1 = 1 + z
,x , 4 = w,y + ,

2 = 2 + z,y , 5 = w,x +
,

(0)

2 , 1 = u,x + 21 w,x

3 = 0, (0)

(1)

6 = 6 + z6 ,

(0)

2 2 = v,y + 21 w,y ,

(0)

6 = v,x + u,y + w,x w,y , (1)

1 =
,x ,

(1)

2 = ,y ,

(2) (1)

6 = ,x +
,y ,

where, a comma denotes partial differentiation with respect to the subscript. On reducing the three-dimensional elasticity problem to a two-dimensional one, the following laminate constitutive equations are obtained: (Ni , Mi , Qm ) (0) (1) (0) (1) = (Aij j + Bij j − Pi , Bij j + Dij j , Amn n ), (i = 1, 2, 6), (m, n = 4, 5).

(3)

The quantities Ni , Mi and Qmn are the in-plane force resultants, moments resultants and transverse shear resultants, defined by (Ni , Mi , Qm ) =

N 


zk

k=1 zk−1

(i , zi , m ) dz.

(4)

The laminate stiffnesses Aij , Bij and Dij are given by

2. Theoretical formulation and basic equations Consider a fiber-reinforced composite laminated rectangular plate composed of N anisotropic layers bounded together in an arbitrary lamination scheme such that each layer possesses one plane of elastic symmetry parallel to the mid-plane of the plate. The laminate is of length a, width b, and total constant thickness h; occupying the space 0
x
a, 0
y
b and −h/2
z
h/2. The plate is subjected to in-plane compressive forces P1 and P2 , and the upper surface of the plate (z = −h/2)

(1)

(Aij , Bij , Dij , Amn ) =

N 


zk

k=1 zk−1

(k)

(cij (1, z, z2 ), c(k) mn K) dz,

(i, j = 1, 2, 6), (m, n = 4, 5), where zk and zk−1 are the top and bottom z-coordinates of the kth lamina, i are the stresses, K is a shear correction factor and (k) cij are the stiffnesses of the kth lamina refereed to the problem coordinates. The governing equations of the laminate may be obtained using the dynamic version of the virtual displacement

M.E. Fares et al. / International Journal of Non-Linear Mechanics 41 (2006) 807 – 824

principle [25], then ¨ N1,x + N6,y = I1 u¨ + I2
, ¨ M1,x + M6,y = Q5 + I2 u¨ + I3
, ¨ N6,x + N2,y = I1 v¨ + I2 , ¨ M6,x + M2,y = Q4 + I2 v¨ + I3 , q + Q5,x + Q4,y = I1 w¨ + N1 w,xx + 2N6 w,xy + N2 w,yy , (I1 , I2 , I3 ) =

N 


zk

(1, z, z2 )(k) dz,

(5)

809

where i , (i = 1, 2,3) are positive constant weighting factors, J1 is the strain energy and J2 is the kinetic energy of the laminate. The functional J3 is a penalty term involving the closedloop control function q ∈ L2 , where L2 denotes the set of all bounded square integrable functions defined on the region {0
x
a, 0
y
b, 0
t < ∞}. Note that the present formulation keeps the constrained functional J of the control objective quadratic and, hence, differentiable and positive definite. This control problem requires finding a solution of the initial boundary value problem (5)–(7) that simultaneously minimize the functional (8). This problem may be solved using Liapunov–Bellman theory.

k=1 zk−1

where, the superposed dot denotes differentiation with respect to time t and (k) is the mass density of the kth lamina. The present control problem accounts for various cases of boundary conditions at the edges, i.e., when the plate edges are simply supported (S), or clamped (C) or free (F ), or when combination of these boundary conditions is prescribed over the edges. These cases of boundary conditions on the edges perpendicular to x-axis (for example) require the following conditions: S: u = w =  = N6 = M1 = 0,

m,n

(6)

Furthermore, let the laminate be subjected to initial disturbances described by the conditions: w(x, y, 0) = A(x, y),

w(x, ˙ y, 0) = B(x, y).

The solution of the system of non-linear partial differential equations (5) under the conditions (6) and (7) cannot be exactly solved. The Galerkin method [27] may be used to obtain an approximate analytic solution for which the displacements (u, v, w,
, ) and the closed-loop control function q may be represented as
(u, v, w,
, , q) = (Umn XY ,y , Vmn X,x Y, Wmn XY , mn X,x Y, mn XY ,y , qmn XY ),

C: u = w =
=  = N6 = 0, F : N1 = N6 = M1 = M6 = Q1 = 0.

4. Solution procedure

(7)

3. Optimal control problem The control objective of the present study aims at minimizing the postbuckling dynamic response of the laminate as time goes to infinity with the minimum possible expenditure of control energy produced by the control force q(x, y, t). The total elastic energy is taken as a measure (criterion) for the dynamic response. This criterion is a functional of the displacements, their spatial derivatives and velocity. The control force q(x, y, t) is introduced in the control objective by taking a performance index which compresses a weight sum of total laminate energy and a penalty functional involving the control force. Then, the control objective may be written in the form J = 1 J1 + 2 J2 + 3 J3 ,     1 ∞ a b h/2 J1 = (cij i j + cmn m n ) dz dy dx dt 2 0 0 0 −h/2 (i, j = 1, 2, 6) (m, n = 4, 5),     1 ∞ a b h/2 2 J2 = (u˙ 1 + u˙ 21 + u˙ 21 )(k) dz dy dx dt, 2 0 0 0 −h/2    1 ∞ a b 2 J3 = (q (x, y, t) dy dx dt, (8) 2 0 0 0

(9)

where Umn , Vmn , Wmn , mn , mn and qmn are unknown functions of time. X(x) and Y (y) are continuous orthonormed functions which must satisfy at least the geometric boundary conditions in (6) to represent approximate shapes for the deflected surface of the vibrating plate. These functions for different cases of boundary conditions are given in Appendix A. Using Eqs. (2) and (3) to obtain the dynamic equations (5) in terms of the displacements (u, v, w,
, , q). Substituting expressions (9) into the resulting equations, and then multiplying each one of the resulting equations by the corresponding eigenfunction, we get, after the integration over the domain of solution, the following non-linear system of equations: 2 U1mn Umn + V1mn Vmn + 1mn mn + 1mn mn + W21mn Wmn ¨ mn , = d52 I1 U¨ mn + d52  2 U2mn Umn + V2mn Vmn + 2mn mn + 2mn mn + W22mn Wmn ¨ mn , = d42 I1 V¨mn + d42 I2 

3mn mn + 3mn mn + W3mn Wmn + (U23 Umn + V23 Vmn 3 + 23 mn + 23 mn )Wmn + W33mn Wmn + d0 qmn ¨ = d0 I1 Wmn ,

U4mn Umn + V4mn Vmn + 4mn mn + 4mn mn + W4mn Wmn 2 ¨ mn , + W24mn Wmn = I2 d52 U¨ mn + I3 d52  U5mn Umn + V5mn Vmn + 5mn mn + 5mn mn 2 ¨ mn , (10) + W5mn Wmn + W25mn Wmn = d42 I2 V¨mn + d42 I3  where, the coefficients U1nm , V1mn , . . . are given in Appendix A. This system of equations may be written in the

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M.E. Fares et al. / International Journal of Non-Linear Mechanics 41 (2006) 807 – 824

following matrix form;

form

¨ = {f }, [[L] + [H ( )]]{ } + [R]{ }

2 3 + b2 Wmn = qmn /I1 , W¨ mn + 2 Wmn + b1 Wmn

{ }T = {Umn , Vmn , Wmn , mn , mn }, {f } = {0, 0, q, 0, 0},

2 = (W3mn + e14 3mn + e15 3mn )/(I1 d0 ), (11)

where [L], [H ] and [R] denote the linear, non-linear (geometric) stiffness and inertia matrices, respectively, and are given in Appendix A. Also, using expressions (2) and (9) with the functional (8) to obtain it in terms of the time-dependent functions Umn , Vmn , Wmn , mn , mn and qmn ,
 ∞ J= Jmn (Umn , Vmn , Wmn , mn , mn , qmn ) dt. (12) m,n

0

Now the optimal control problem is to find a minimum control function q mn from the non-linear differential equations (10) that simultaneously minimize the functional (12) i.e., J (q)
J (q) for all q ∈ L2 × [0, ∞), that is min J = min q

q


m,n

Jmn =




b1 = (e24 3mn + e25 3mn + e11 U23 + e12 V23 + e14 23 + e15 23 )/(I1 d0 ), b2 = (e21 U23 + e22 V23 + e24 23 + e25 23 + W33 )/(I1 d0 ),

where the coefficient eij are given in Appendix B. Eq. (15) may be written in the following iteration form: [W¨ mn + 2 Wmn − qmn /I1 ](r+1) 2 (r) + W (r+1) [b1 Wmn + b2 Wmn ] = 0.

m,n qmn

Since the system of equations (10) is separable, then, the functional (12) depends only on the variables found in (m, n)th equations of the system (11). With the aid of this condition, the problem is reduced to a problem of analytical design of controllers, for every m, n = 1, 2, . . . , ∞. Hence, the minimization problem can be carried out independently for every modal equation. For such problem, Liapunov–Bellman theory [26] is considered as an effective approach to solve it. The necessary and sufficient condition for optimality according to this theory is   dLmn min (13) + Jmn = 0, qmn dt where Lmn is a Liapunov function which must be a positive definite and differentiable on the domain of the problem. The solution of this problem depends on the postbuckling (buckled) displacements which may be obtained by solving the non-linear system (11) iteratively until an appropriate convergence criterion is satisfied. Therefore, a Picard-type successive iteration scheme is employed. Then, the system (11) may be approximated by ¨ } = {f }, [[L] + [H ( r )]]{ r+1 } + [R]{ r

0

2 2 2 + s5 W W˙ mn + s6 W 2 W˙ mn + s7 qmn ) dt,

∈L2

(14)

where r is the iteration number. The non-linear stiffness matrix for (r + 1)th iteration is computed using the solution from rth iteration. Moreover, at the start of the iteration procedure, the solution is assumed to be zero so that the linear solution is obtained at the end of the first iteration. This problem may be simplified by omitting the in-plane inertia terms in system (14). In this case, a non-linear equation of the timedependent functions Wmn and qmn may be obtained in the

(16)

Also, using the system of equations (10) with omitting the inplane inertia terms, we can get the functional (12) in the form
 ∞ 2 2 3 4 (s1 W˙ mn + s2 Wmn + s3 Wmn + s4 Wmn J= m,n

min Jmn .

(15)

(17)

where the coefficients si (i = 1 . . . 7) are dependent on the laminate stiffnesses and given in Appendix B. In this case, the Liapunov–Bellman condition (13) takes the form   jLmn ˙ jLmn ¨ min Wmn + Wmn + Jmn = 0, (18) qmn jWmn jW˙ mn here, Jmn is the integrand of functional (17). To apply this condition to the present non-linear control problem (16) and (17), we shall take the Liapunov function Lmn as a sum of positive definite functions with different degrees in Wmn and W˙ mn , then (2) (3) (4) Lmn (Wmn , W˙ mn ) = L(1) mn + Lmn + Lmn + Lmn + · · · ∞
(i) = Lmn , i=1 2 ˙ ˙2 L(1) mn = C11 Wmn + 2C12 Wmn Wmn + C13 Wmn , 3 2 ˙ ˙2 ˙3 L(2) mn = C21 Wmn + C22 Wmn Wmn + C23 Wmn Wmn + C24 Wmn , 4 3 ˙ 2 ˙2 L(3) mn = C31 Wmn + C32 Wmn Wmn + C33 Wmn Wmn 3 4 + C34 Wmn W˙ mn + C35 W˙ mn ,

(19)

and so on, where the coefficient Cij must satisfy Sylveseter’s conditions [28] ensuring the positive definite property for the function Lmn , and these conditions are given in Appendix B. Note that the Liapunov–Bellman condition (18) must be satisfied in every approximation (iteration) step. The first approximation solution is taken as the solution of the linear control problem described by the equation W¨ mn + 2 W˙ mn = qmn /I1 ,

(20)

M.E. Fares et al. / International Journal of Non-Linear Mechanics 41 (2006) 807 – 824

and the functional
 ∞ (1) (1) (1) 2 2 2 J = Jmn dt, Jmn = s1 W˙ mn +s2 Wmn +s7 qmn . (21) m,n

0

(1)

In this case, the Liapunov function is taken as Lmn = Lmn , and the Liapunov–Bellman condition (18) takes the form   (1) (1) jLmn ˙ jLmn ¨ (1) Wmn + Wmn + Jmn = 0. (22) min qmn jWmn jW˙ mn (1)

Using Eqs. (20), (21) and the expression for Lmn in (19) with the condition (22), we can easily obtain the first approximation (1) for the optimal control force q mn in the form −1 q (1) (C12 Wmn + C13 W˙ mn ). mn = s7 I 1

(23)

Again, substituting expressions (20)–(23) into the condition 2 andW W (22), and equating the coefficients of Wm2 , W˙ mn mn ˙ mn to zero, we get a system of equations, 2C11 − 2C13 2 − 2C12 C13 /(I12 s7 ) = 0, 2C12

2

2 + C22 /(I12 s7) − s2

(24)

C11 = C13 ( 2 I12 s7 + C12 )/(I12 s7 ),   C12 = −I1 2 s7 − 4 s72 I12 + s2 s7 , C13 = I1 2C12 s7 + s1 s7 ,

(25)

(30)

substituting Eqs. (30) and (15) into condition (29), we get the following system of equations: 2C12 b1 + C22 2 − s3 + C22 C12 /(I12 s7 ) = 0, C23 − 3C13 C24 /(I12 s7 ) = 0,

2C22 + s6 − 3C24 2 − 2C23 C13 /(I12 s7 ) − 3C24 C12 /(I12 s7 ) = 0,

(31)

the solution of the above algebraic equations under the condi(2) tion that Vmn is positive definite, is

2 2 + 4 2 b1 s7 I12 C13 + 5s7 I12 C12 s3 − 8b1 s7 I12 C12 2 2 2 2 + 2s3 C13 + 2s6 C12 )/(2 2 s7 C13 I1 2 2 2 + 4 s73 I16 + 2 2 s72 I14 C12 + 2C12 C13 + s7 C12 I1 ),

C23 = C13 I12 s7 ( 2 s6 s7 I12 − 4s7 I12 b1 C12 + s7 C12 2 2 + 2s7 I12 s3 )/(2 2 s7 C13 I1 + 4 s73 I16

(26)

2 2 2 + 2 2 s72 I14 C12 + 2C12 C13 + s7 C12 I1 ),

C24 = I12 s7 C23 /(3C13 ), C22 = I12 s7 (s3 − 2C12 b1 )/(C12 + 2 I12 s7 ). (27)

where and  are unknown coefficients which may be obtained from the initial conditions (6) by expanding them in series, then mn mn + 2Amn , 2mn  a b 4 ( mn , Amn ) = (A, B)XY dx dy. ab 2mn 0 0

(1) 2 ˙ ˙ q (2) mn (Wmn , Wmn ) = q mn − (c22 Wmn + 2c23 Wmn Wmn 2 ˙ + 3c24 Wmn )/(2s7 I1 ),

− 6 2 b1 s72 I14 C12 + 4 2 s7 I12 C12 s6

The solution of above equation can obtained as (1) = e−mn t/2 [ mn cos(vmn t) + mn sin(mn t)], Wmn 2mn = 2mn − 41 2mn ,

(2)

C21 = C13 /3(2 4 b1 s73 I16 + 2 4 s72 I14 s6 + 5 2 s72 I14 s3

2 > 0. The first approximation of where C11 > 0, C11 C13 − C12 (1) the deflection function Wmn may be obtained using expressions (25) and (23) with (20) to get (1) (1) (1) W¨ mn + mn W˙ mn + 2mn Wmn = 0, C13 C12 mn = , 2mn = 2mn + . 2 s7 I1 s7 I12

(1)

using expressions of Lmn and Lmn in (19) and Eq. (15), the condition (29) gives the second approximation for the optimal (2) control force q mn in the form

− 2C23 C12 /(I12 s7 ) = 0,

the solution of this system under the condition that the Liapunov (1) function Lmn is positive definite, is given by



(2)

Lmn = Lmn + Lmn , Then the Liapunov–Bellman condition (18) takes the form  jLmn ˙ jLmn ¨ 2 3 2 min Wmn + Wmn + s1 W˙ mn + s2 Wmn + s3 Wmn qmn jWmn jW˙ mn  4 2 2 2 = 0, (29) +s4 Wmn + s5 W W˙ mn + s6 W 2 W˙ mn + s7 qmn

− 2C23 2 + 3C21 − 2C13 b1 − C13 C22 /(I12 s7 )

= 0,

2 /(I12 s7 ) + 2C12 + s1 = 0, −C13

(1)

811

mn =

(28)

To get the second iteration solution, we consider the nonlinear control problem defined by Eqs. (15), (16) with the functional (17). For this stage, the Liapunov function is taken as

(32) (1)

Substituting the first approximation deflection Wmn in Eq. (16), we get a homogeneous differential equation from which the (2) second approximation of the deflection Wmn may be obtained. (1) (2) (3) For the third approximation, we take Lmn = Lmn + Lmn + Lmn and by following analogues steps as in the second iteration process, we obtain the third approximation for the optimal control (3) force qmn in the form ˙ q (3) mn (Wmn , Wmn )

(1) 3 2 ˙ = q (2) mn + q mn − (C31 Wmn + 2C32 Wmn Wmn 2 3 + 3C33 Wmn W˙ mn + 4C34 W˙ mn )/(2s7 I1 ),

(33)

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M.E. Fares et al. / International Journal of Non-Linear Mechanics 41 (2006) 807 – 824

C31 = 1/4(2C33 2 s7 I12 + 2C13 b2 s7 I12 + 2C23 b1 s7 I12

+ 36C13 C23 C24 s72 I12 2 − 6C22 C24 s72 I14 4

taken as the ratio between the thickness of one of the outer layers to the total laminate thickness, 0

21 . The objective of the design procedure is to maximize the critical buckling load c and minimize the postbuckling vibrational energy. This may be done by solving two-step problem to find, firstly, the optimum ply thickness hk and, secondly the optimal fiber orientation angle k . In general, these objectives conflict with each other necessitating a multiobjective formulation including the postbuckling energy and the critical buckling load. With this situation in mind, the performance index Jˆ(q, hk , k ) of the design problem is specified as

−12b1 C24 s73 I16 2 +36C12 C13 C23 C24 +12C12 C32 I14 s72

Jˆ(q, hk , k ) = 4 /c + J (q, hk , k ),

+ C22 C23 + 2C33 C12 + C32 C13 )/(I12 s7 ), 2 + 8C12 b2 s7 I12 + 4C22 b1 s7 I12 C32 = − 1/4(8C12

− 4s4 s7 I12 )/( 2 s7 I12 + C12 ), 2 2 4 4 2 2 4 2 2 s7 I1 −4C12 C23 s7 I1 +36C13 s7 I12 C32 C33 = 1/8(−27C24

− 6C24 s7 I12 C12 C22 − 12C12 s72 I14 C24 b1 2 2 −54C12 C24 s7 I12 2 −36C24 C13 s7 I12 b1 +12C32 s73 I16 2

2 2 2 2 2 − 4C12 C23 s7 I12 − 27C12 C24 − 12C32 C13 2 2 − 18C22 C24 C13 )/(C13 (3C13 + 4C12 I12 s7 + 4 2 I14 s72 ), 2 C34 = − 1/4(−12C32 C24 C23 + 12I14 s72 C32 + 9 2 I12 s7 C24 2 2 − 6I12 s7 C22 C24 + 9C12 C24 − 4I12 s7 C23 2 − 12I14 s72 C24 b1 )/(3C13 + 4 2 I14 s72 + 4C12 I12 s7 ), 2 2 2 2 C13 − 27 2 I14 s72 C24 − 27I12 s7 C12 C24 C35 = 1/16(−27C24 − 12I12 s7 C24 C13 C23 + 12I16 s73 C32 2 − 6I14 s72 C24 C22 − 12I16 s73 C24 b1 − 4I14 s72 C23 ) 2 2 4 2 2 /(C13 (3C13 + 4 I1 s7 + 4C12 I1 s7 ). (2)

Again use the second approximation of deflection Wmn with (3) Eq. (16), we can get the third iteration deflection Wmn . For the iterations with order more than three (r 4), it is not difficult to prove that L(r) mn = 0,

(3) q (r) mn = q mn

for all r 4,

(34)

this is because the number of the iteration steps depends on the degrees of the non-linear terms included in the control functional and the governing equations. Inserting Eq. (33) and the (3) resulting expression for Wmn into the functional (17), we can get the controlled energy of the laminate as a function of hk (lamina thickness) and k (ply orientation angle). These design parameters may be used to perform further minimization for the total elastic energy.

The critical buckling values c , ( = P2 /P1 ) at which the buckling occurs, may be determined from the system (11) by solving the following eigenvalue problem: ˆ − c [P ]} = 0, or {[L]

(35)

ˆ [P ] are given in Appendix B. The lowwhere the matrices [L], est value of the non-dimensional buckling  with respect to the mode numbers (m, n) is the critical value, i.e c = min , m,n

m, n 1.

where 4 is a positive constant weight factor coefficient. The design procedure aims to determine the optimum values of the optimization variable hk (or ) and  from the following condition: Jˆ(q, opt , opt ) = min Jˆ(q, hk , ), q,

0

1/2, 0

90◦

(38)

6. Numerical results and discussion In this section, numerical results for the optimal control force q, buckled deflection w and the design and control objective Jˆ are presented when all layers of the laminate are made of the same orthotropic material. The engineering constants are introduced instead of the elastic constants from the relations: c11 =

E1 , 1 − 12 21

c44 = G23 ,

c12 =

c55 = G13 ,

12 E2 , 1 − 12 21

c22 =

E2 , 1 − 12 21

c66 = G12 , c16 = c26 = 0,

(39)

where Ej are Young’s moduli; ij are Poisson’s ratios and Gij are shear moduli. The Poisson’s ratios and Young’s moduli are related by the reciprocal relations ij Ej = j i Ei , (i, j = 1, 2). The initial conditions (6) are chosen as w(x, y, 0) = 50X(x)Y (y),

w(x, ˙ y, 0) = 0, m = n = 1. (40)

In all calculations, unless otherwise stated, the following parameters are used

5. Buckling load and design procedure

|L| = 0

(37)

(36)

For the design procedure, we consider three-layer (, 0, ) symmetric laminates and four-layer (, −, , −) antisymmetric laminates. An optimization thickness parameter  may be

E1 /E0 = 25, G12 /E2 = 0.2, G23 /E2 = 0.2, G31 /E2 = 0.5, 12 = 0.25, 1 = 2 = 4 = 1, 3 = 0.8, K = 56 .

(41)

The four letters of the boundary conditions (SSCC, SCCF, . . . , etc.) with their order from left to right refer to the kind of fixing at the plate edges x = 0, x = a, y = 0 and y = b, respectively. The numerical results for the buckled deflection w and optimal control force q displayed in the figures are calculated at the laminate mid-point x = a/2, y = b/2. Figs. 1 and 2 contain plots of the design and control objective Jˆ as a function of the orthotropy ratio E1 /E2 for optimally designed three-layer symmetric laminates with SSSS, SSCS, SSCC

M.E. Fares et al. / International Journal of Non-Linear Mechanics 41 (2006) 807 – 824

100

813

600

SSSS, opt = 45°, opt = 5.0

SSCS, opt = 57.3°, opt = 5.0 500

80 NL2

NL2 400

NL1

NL1

60 Jˆ

Jˆ

L

L

300

40 200 20

100 0

0 0

5

10

15

(a)

20

5

0

25

10

(b)

E1/E2

15

20

25

E1/E2

Fig. 1. Optimization objective Jˆ due to three successive iterations plotted against the orthotropy ratio E1 /E2 for three-layer symmetric laminates, a/b = 1, a/ h = 20.

1400

350

SSCC, opt = 90°, opt = 0.5

1NL2 NL2

1000

3NL1

NL1

250

2L

800 Jˆ

L Jˆ

SSCF, opt = 0°, opt = 0.5

1200

300

200

600 150

400

100

200 0

50 5

0

10

15

20

E1/E2

(a)

0

25

5

10

(b)

15 E1/E2

20

25

Fig. 2. Optimization objective Jˆ due to three successive iterations plotted against the orthotropy ratio E1 /E2 for three-layer symmetric laminates, a/b = 1, a/ h = 20.

750

120 SSSS, opt = 45°, opt = 0.29

SSCS, opt = 57.3°, opt = 0.29

100

600

NL2

NL2 80

NL1

NL1

450 L

Jˆ

Jˆ

L 60

300 40 150

20

0

0 0 (a)

5

10

15 E1/E2

20

25

0 (b)

5

10

15

20

25

E1/E2

Fig. 3. Optimization objective Jˆ due to three successive iterations plotted against the orthotropy ratio E1 /E2 for four-layer antisymmetric laminates, a/b = 1, a/ h = 20.

814

M.E. Fares et al. / International Journal of Non-Linear Mechanics 41 (2006) 807 – 824

4000

800 SSCC, opt = 90°, opt = 0.29

700

3000

600

NL2

NL2

500

NL1

NL1

400

J

J

SSCF, opt = 0°, opt = 0.29

L

L

2000

300 1000

200 100 0

0 0

5

10 15 E1 / E2

(a)

20

0

25

5

10

(b)

15

20

25

E1 / E2

Fig. 4. Optimization objective Jˆ due to three successive iterations plotted against the orthotropy ratio E1 /E2 for four-layer antisymmetric laminates, a/b = 1, a/ h = 20.

250 SSSS, opt = 45°, opt = 0.5

40

SSCS, opt = 57.3°, opt = 0.5

200 NL2 30

NL1

150

NL1

20

100

10

L

50

0

0 1 (a)

NL2

J

J

L

1.5

2 a/b

2.5

3

1 (b)

1.5

2 a/b

2.5

3

Fig. 5. Optimization objective Jˆ due to three successive iterations plotted against the aspect ratio a/b for three-layer symmetric laminates, E1 /E2 =25, a/ h=20.

and SSCF boundary conditions. The solid (L), dashed (NL1) and pointed (NL2) curves are obtained, respectively, due to the first (linear), the second and third (non-linear) iterations. Similar curves for four-layer antisymmetric laminates are presented in Figs. 3 and 4. These figures show that the discrepancies between the first and second iteration results are small, particularly, for laminates with orthotropy ratio E1 /E2 > 10, whereas, these discrepancies between the second and third iteration results are very large. This indicates that the non-linear solution due to the second iteration is very weak to describe the behavior of the laminate in the postbuckling range, and the non-linear solution due to the third iteration is very significant for the accurate design of the laminates with all cases of orthotropy ratio, boundary conditions and lamination schemes. Figs. 5–8 show the optimization objective Jˆ as a function of aspect ratio a/b for optimally designed symmetric and antisymmetric laminates with various cases of boundary conditions. These curves confirm the previous discussion on the accuracy of the non-linear solutions due to the second and third iterations with different

ratios of aspect ratio a/b. Figs. 9–12 show the effect of the transverse shear deformation (a/ h) on the objective Jˆ. It is noticed that the differences between the curves due to the three iteration processes increase with increasing the side-to-thickness ratio. This may be explained by the fact that the thin laminates (a/ h > 20) do not resist the buckling and postbuckling deformation causing a very large deflections. Consequently, the geometric non-linearity effect for these laminates become very significant, and they need high-order iteration analysis to describe accurately their behavior. Figs. 13 and 14 contain plots of the optimal control force q as a function of side-to-thickness ratio for symmetric and antisymmetric laminates. It is worst to note from the previous figures that the results due to the third iteration for the functional Jˆ and optimal control force q have high sensitivity to the variations of the martial and geometric parameters E1 /E2 , a/b and a/ h. This is because the omission of the third iteration amounts to constraining the deformation of the laminates, and hence, they behave as a stiffer. As it is known, the solution due to a higher-order iteration is more

M.E. Fares et al. / International Journal of Non-Linear Mechanics 41 (2006) 807 – 824

250

450

SSCC, opt = 90°, opt = 0.5 200

NL2

815

SSCF, opt = 0°, opt = 0.5

350

NL2

NL1 150

NL1 J

J

L

L

250

100 150

50

50

0 1

1.5

(a)

2 a/b

2.5

1

3

1.5

(b)

2 a/b

2.5

3

Fig. 6. Optimization objective Jˆ due to three successive iterations plotted against the aspect ratio a/b for three-layer symmetric laminates, E1 /E2 =25, a/ h=20.

60

350

SSSS, opt = 45°, opt = 0.29

SSCS, opt = 57.3°, opt = 0.29

300

50

NL2

NL2 NL1

40

NL1

250

L

L J

J

200 30

150 20

100

10

50

0

0 1

1.5

(a)

2 a/b

1

3

2.5

1.5

(b)

2 a/b

2.5

3

Fig. 7. Optimization objective Jˆ due to three successive iterations plotted against the aspect ratio a/b for four-layer antisymmetric laminates, E1 /E2 = 25, a/ h = 20.

300

1200

SSCC, opt = 90°, opt = 0.29

250

SSCF, opt = 0°, opt = 0.29

1000

NL2

NL2

NL1

NL1 800

L

150

J

J

200

400

50

200 0 1

(a)

600

100

0

L

1.5

2 a/b

2.5

1

3 (b)

1.5

2 a/b

2.5

3

Fig. 8. Optimization objective Jˆ due to three successive iterations plotted against the aspect ratio a/b for four-layer antisymmetric laminates, E1 /E2 = 25, a/ h = 20.

816

M.E. Fares et al. / International Journal of Non-Linear Mechanics 41 (2006) 807 – 824

2000

SSSS, opt = 45°, opt = 0.5

140 120

1500

NL2

100

SSCS, opt = 57.3°, opt = 0.5

NL2 NL1

J

J

NL1 80

L

1000

L

60 40

500

20 0

0 10

15

20 a/h

(a)

25

30

10

15

20 a/h

(b)

25

30

Fig. 9. Optimization objective Jˆ due to three successive iterations plotted against the thickness ratio a/ h for three-layer symmetric laminates, a/b = 1, E1 /E2 = 25.

1500

2500

SSCC, opt = 90°, opt = 0.5 1200

SSCF, opt = 0°, opt = 0.5

2000 NL 2

NL 2

NL 1

900

L

Jˆ

Jˆ

NL 1

1500

L 600

1000

300

500

0

0 10

15

(a)

20 a/h

25

30

10

15

20 a/h

(b)

25

30

Fig. 10. Optimization objective Jˆ due to three successive iterations plotted against the thickness ratio a/ h for three-layer symmetric laminates, a/b = 1, E1 /E2 = 25.

600

3000 SSSS, opt = 45°, opt = 0.29

SSCS, opt = 57.3°, opt = 0.29

500

2500 NL 2

NL 2

NL 1

400

NL 1

2000

L

300

Jˆ

Jˆ

L

200

1000

100

500

0

0 10

(a)

1500

15

20 a/h

25

10

30 (b)

15

20 a/h

25

30

Fig. 11. Optimization objective Jˆ due to three successive iterations plotted against the thickness ratio a/ h for four-layer antisymmetric laminates, a/b = 1, E1 /E2 = 25.

M.E. Fares et al. / International Journal of Non-Linear Mechanics 41 (2006) 807 – 824

1500

7500

SSCC, opt = 90°, opt = 0.29

817

SSCF, opt = 0°, opt = 0.29

6500

1200

NL 2 NL 1

NL 1

L

4500

L

Jˆ

Jˆ

900

NL 2 5500

3500

600

2500 300 1500 0

500 10

15

20 a/h

(a)

25

30

10

15

(b)

20 a/h

25

30

Fig. 12. Optimization objective Jˆ due to three successive iterations plotted against the thickness ratio a/ h for four-layer antisymmetric laminates, a/b = 1, E1 /E2 = 25.

2600

450 SSCC, opt = 45°, opt = 0.5

SSCF, opt = 0°, opt = 0.5

NL 2

NL 2

2200

NL 1

NL 1

300

L q

q

L 1800

150 1400

1000

0 10

15

(a)

20 a/h

25

30

10

15

(b)

20 a/h

25

30

Fig. 13. Optimal control force q due to three successive iterations plotted against the thickness ratio a/ h for three-layer symmetric laminates, a/b = 1, E1 /E2 = 25. 1600 SSSS, opt = 45°, opt = 0.29

SSCF, opt = 0°, opt = 0.29.

7000

1400

NL2 NL2

1200 1000

L

L

5000

800

q

q

NL1

6000

NL1

600

4000

400 3000 200 2000

0 10

(a)

15

20 a/h

25

10

30

(b)

15

20 a/h

25

30

Fig. 14. Optimal control force q due to three successive iterations plotted against the thickness ratio a/ h for three-layer antisymmetric laminates, a/b = 1, E1 /E2 = 25.

818

M.E. Fares et al. / International Journal of Non-Linear Mechanics 41 (2006) 807 – 824 5

0.65 hopt

4.8 hE

4.6

0.55 c

J

hE 4.4 4.2

0.45 hopt

4 0.35

3.8 0

15

30

45

60

75

90



(a)

0

15

30

45

60

75

90



(b)

Fig. 15. Controlled elastic energy J (a) and critical buckling load c (b) plotted against the orientation angle , for fourth-layer antisymmetric laminates, hE = hk = h/N , a/ h = 20, a/b = 1.

4

8  = 45°

 = 45°

 = 0°

 = 0°

c

3

c

6

2

4 c = 2.565 ............................................

c = 1.3 ............................................

1

2

0

0 0

1

2

(a)

3

4

0

5

1

2

(b)

w/h

3

4

5

w/h

Fig. 16. The variation of the buckling load c with the postbuckling deflection for a square laminate subjected biaxial compressive for in (a) and uniaxil compressive force in (b).

N = 4, hopt

1N = 4, hopt

0.37

2N = 8, hE 3N = 6, hE

0.35

N = 8, hE

0.75

p1/p2 = 3

N = 6, hE N = 4, hE

4N = 4, hE 0.33

c

c

p1/p2 = 1

0.65

0.31

0.29

0.55

0.27 0

(a)

0.2

0.4

0.6 w/h

0.8

0

1

(b)

0.2

0.4

0.6

0.8

1

w/h

Fig. 17. Effect of number of layers N and optimization design on postbuckling deflection w/ h and buckling and load c for square laminates, a/ h = 20,  = 45◦ .

M.E. Fares et al. / International Journal of Non-Linear Mechanics 41 (2006) 807 – 824

819

3 0.95

hopt , opt hopt ,  = 15° hE,  = 15° 2.1 a/h = 20

c

c

0.75

hopt , opt hopt ,  = 15° hE,  = 15°

0.55

1.2

a /h = 30

0.35

a/b = 3 a/b = 1

0.3

0.15 0

0.2

0.4

0.6

0.8

1

w/h

(a)

0

0.2

0.4

0.6

0.8

1

w/h

(b)

Fig. 18. Effect of plate thickness ratio a/ h (a), and the aspect ratio a/b (b) on buckling and postbuckling responses, p1 /p2 = 1.

0.9

70 hopt , opt

P1 = P 2

hopt ,  = 15°

0.75

hopt ,  = 15° 0.6

P1 = 3P2

45

p1/ p2 = 1

q

c

20

0.45

-5

p1/ p2 = 3 0.3

-30 0.15 0

0.2

0.6

0.4

0.8

1

w/h

Fig. 19. Effect of buckling load ratio on buckling and postbuckling responses.

-55 0

0.2

0.4

0.6

t

Fig. 20. Optimal control force q plotted against time t.

accurate to determine the optimal design of the laminate. The advantage of the present non-linear mathematical procedure is that the number of successive iterations increases with increasing the order of the structural model, and it converges rapidly to appropriate criterion. The numerical results presented in the next figures are obtained due to the third iteration for four-layer antisymmetric laminated plates with simply supported edges (SSSS). Fig. 15 shows the variation of the postbuckling elastic energy J (3 =0) and the non-dimensional buckling load c with the orientation angle . The solid curves represent laminates designed optimally over the thickness, whereas the dashed curves represent laminates with equal thickness layers (hE = h/4). These curves indicate that the design optimization over the thickness reduces significantly the postbuckling energy, and increase the buckling load. Moreover, the postbuckling energy and the buckling load exhibit conflict behaviors against the orientation angle , but,

the minimal values of the elastic energy J (3 = 0) and maximal value of the buckling load take place at  = 45◦ which is the optimal orientation angle. Fig. 16 contains buckling load curves plotted against buckled deflection w/ h in a wide postbuckling range for laminates designed optimally over the thickness. Here, the upper curves represent the behavior of the laminate with best performance with respect to the buckling and postbuckling responses. This figure shows that the optimum values of the orientation angle may vary throughout the postbuckling range, for instance, in the biaxial case with c <1.3, and in uniaxial case withc < 2.56, the optimum value of the orientation angle opt is 45◦ , while, in the deep postbuckling range, the optimal angle  = 0◦ . The effect of the number of layers N on the buckling and postbuckling responses is displayed in Fig. 17. As can be seen from this figure that there is a considerable increase

820

M.E. Fares et al. / International Journal of Non-Linear Mechanics 41 (2006) 807 – 824

50

1200

hE,  = 15° hopt,  = 15°

1000

24

hE, opt hopt , opt

-2 w/h

J

800 600

-28

hE,  = 15°

-54

hE, opt

400

hopt,  = 15°

200

hopt , opt -80

0 0 (a)

0.2

0.4

0.6

t

0

0.2

(b)

0.4 t

0.6

Fig. 21. The variation of the optimization objective Jˆ, and buckled deflection w/ h with respect to time t for optimally designed and nondesigned laminates, a/ h = 20, a/b = 1, E1 /E2 = 25.

in the buckling load, and a considerable decrease in the buckled deflections as the number of layer N increases, and this effect becomes insignificant when N 8. This means that laminates with few layers perform quite poorly with respect to the present optimization objectives, but, the optimization over the thickness and fiber orientations raises remarkably the level of the performance of these laminates, so that optimally designed laminates with N = 4 have higher performance than the non-designed ones with N = 8. The effect of aspect ratio a/b and side-to-thickness ratios a/ h on the buckling and postbuckling responses are presented in Fig. 18. It is shown that the square thin laminates have a weak performance for the present design objectives, and the optimal design procedure is more effective with rectangular moderately thick laminates than with the thinner square ones. Fig. 19 shows the effect of compression load ratio P1 /P2 on the buckling and postbuckling response. With increasing the compression forces, the control and design procedure becomes less effective to improving the performance of the laminates which need more control energy to damp its dynamic response. Figs. 20 and 21 show the variations of the controlled elastic energy J and the control force q with time t for four different laminate designs, which are non-optimal design, partially optimal design over the thickness opt , partially optimal design over the orientation angle opt and optimal design over both opt and opt . All previous cases of optimal design reduce considerably the postbuckling energy of the laminates and raise the level of buckling load as compared to the non-designed ones. But, the optimal design over the orientation angle  is more effective than the optimal design over the thickness, while, the optimal design over both  and  is the most efficient. In addition, the present optimization procedures decrease significantly the expenditure of control energy and the time of the controlling process.

plates. The design and control objectives are the minimization of the postbuckling dynamic response and the maximization of the bucking load. The optimal levels of closed-loop control force, layer thicknesses and fiber orientation angles are determined for three-layer symmetric and four-layer antisymmetric laminated plates with various cases of boundary conditions. The problem objectives are optimized in a unified formulation which includes the postbuckling vibrational energy and buckling load as a weighted sum based on a first-order shear deformation theory. The optimality condition of the Liapunov–Bellman theory is used to solve iteratively the present non-linear optimal control problem. It is found that the iteration processes with order more than three do not add any accuracy for the solution, and the first and second iteration solutions are very weak to determine the laminate optimal design, while, the third iteration solution is the more accurate. Also, the present study indicates that the optimal orientation angle may be changed throughout the postbuckling range. The optimization over the thickness and fiber orientation are very effective to improving significantly the performance of the composite laminates and to reducing the expenditure of control energy. Appendix A The functions X(x) and Y (y) included in the solution (9) for different cases of boundary conditions are SS : X(x) = sin m x,

m = m/a.

CC : X(x) = sin m x − sinh m x − m (cos m x − cosh m x), m = (sin m a − sinh m a)/(cos m a − cosh m a), m = (m + 0.5)/a.

7. Conclusions

CS : X(x) = sin m x − sinh m x − m (cos m x − cosh m x),

An integrated approach is presented to solve a non-linear multiobjective optimization problem of composite laminated

m = (sin m a + sinh m a)/(cos m a + cosh m a), m = (m + 0.25)/a.

M.E. Fares et al. / International Journal of Non-Linear Mechanics 41 (2006) 807 – 824

821

CF : X(x) = sin m x− sinh m x−m (cos m x− cosh m x),

V1mn = A12 d11 + A16 d14 + A26 d12 + A66 d11 ,

m = (sin m a+ sinh m a)/(cos m a + cosh m a), 1 = 1.875/a, 2 = 4.694/a,

V2mn = A22 d22 + 2A26 d12 + A66 d24 ,

3 = 7.855/a, 4 = 10.996/a and m = (m − 0.25)/a for m 5.

W2mn = 0,

1mn = B11 d14 + 2B16 d11 + B66 d12 , 2mn = B16 d24 + B126 d21 + B26 d22 , 1mn = B12 d12 + B16 d11 + B26 d15 + B66 d12 ,

The following coefficients are included in Eq. (10)

2mn = B22 d25 + 2B26 d22 + B66 d21 ,

U5mn = B16 d11 + B126 d12 + B26 d41 ,

W21mn = A11 d16 + A12 d17 + A16 d18 + 2A16 d19 + A26 d110 + A66 d17 + A66 d111

U4mn = B11 d21 + B16 d22 ,

V4mn = B12 d21 + B16 d24 ,

W22mn = A16 d26 + A26 d27 + A66 d82 + A66 d29 + A22 d210 + A26 d27 + A26 d111 + A12 d12 .

V5mn = 2B26 d11 + B26 d14 + B22 d12 , W4mn = −A45 d51 − A55 d52 ,

The matrices of Eq. (11) are given by

W5mn = −A44 d42 − A45 d43 ,

W25mn = B16 d16 + 2B26 d16 + B66 d18 + B66 d19 + B12 d19 + B26 d111 + B22 d45 ,

R33 = I1 d0 , Rij = 0 (i, j = 1, 2, 4, 5), ⎡ ⎤ 0 0 W21 Wmn 0 0 0 W22 Wmn 0 0 ⎥ ⎢ 0 ⎢ ⎥ 2 [H ]= ⎢U23 Wmn V23 Wmn W33 Wmn 23 Wmn 24 Wmn⎥, ⎣ ⎦ 0 0 W42 Wmn 0 0 0 0 W52 Wmn 0 0 ⎡ ⎤ U1 V1 1 1 0 0 ⎢ U2 V2 2 2 ⎥ ⎢ ⎥ [L] = ⎢ 0 0 3 3 W3 − N ⎥ . ⎣ ⎦ U4 V4 4 4 W4 U5 V5 5 5 W5

where

Appendix B

4mn = D11 d25 + D16 d21 − A45 d51 5mn = D11 d14 + D126 d11 + D26 d12 − A45 d44 , 4mn = D12 d22 + D16 d21 − A45 d51 , 5mn = 2D26 d12 + D66 d11 + D22 d41 − A44 d44 , W24mn = B11 d26 + B12 d27 + B16 d28 + B16 d29 ,

A126 = A12 + A66 , D126 = D12 + D66 ,

B126 = B12 + B66 ,

and U3mn = 0, V3mn = 0, N = N11 d31 + 2N12 d32 + N22 d33 , W3mn = A44 d33 + 2A45 d32 + A55 d31 , 3mn = A45 d32 + A55 d31 , 3mn = A44 d33 + A45 d32 , U23mn = A11 d34 + A 16 d35 + A12 d36 + A26 d37 + 2A16 d38 + A66 d36 , V23mn = A12 d34 + A 16 d39 + A22 d36 + A26 d35 + 2A26 d34 + A66 d34 , 23mn = B11 d39 + B 16 d34 + B12 d35 + B26 d36 + 2(B26 d34 + B66 d38 ), 23mn = B12 d35 + B 16 d34 + B22 d37 + B26 d36 + 2(B26 d36 + B66 d38 ), W33mn = 1/2(A11 d42 + A12 d25 + 2A16 d53 )d31 + 1/2(A12 d31 + A22 d42 + 2A26 d52 )d33 + (A16 d42 + A26 d52 + A66 d53 )d32 , U1mn = A11 d11 + 2A16 d12 + A66 d22 , U2mn = A16 d21 + A126 d22 + A26 d23 ,

W1mn = 0,

The parameters eij included in Eq. (15) are     0 V1 1 1   W21 V1    V  0 V2 2 2  W e11 e0 =   , e12 e0 =  22 2  W 4 V4  4  4   W24 V4    W5 V5 5 5 W25 V5    U1 0 1 1    0 2 2  U e21 e0 =  2 ,  U4 W4 4 4    U5 W5 5 5     U1 W21 1 1   U1 V1     U2 W22 2 2   U V2 e22 e0 =   , e14 e0 =  2  U4 W24 4 4   U4 V4    U5 W25 5 5 U5 V5    U1 V1 W21 1     U V2 W22 2  e42 e0 =  2 ,  U4 V4 W24 4    U5 V5 W25 5     U1 V1 1 0   U1 V1     U2 V2 2 0   U V2 e15 e0 =   , e52 e0 =  2  U4 V4 4 W4   U4 V4    U5 V5 5 W5 U 5 V5    U1 V1 1 1     U V2 2 2  e0 = −  2 .  U4 V4 4 3    U5 V5 5 4

1 2 4 5

0 0 W4 W5

1 2 4 5

 1   2  , 4   5

 1   2  , 4   5

 W21   W22  , W24   W25

822

M.E. Fares et al. / International Journal of Non-Linear Mechanics 41 (2006) 807 – 824

The coefficient si in Eq. (17) are given by 2 2 s1 = 1 (d42 e21 +d52 e11 +d42 )I1 +2(d41 e11 e41 +d41 e21 e51 )I2 2 2 + (d42 e41 + d52 e51 )I3 2 2 s2 = 2 (d52 A44 + d42 A55 + d62 A66 e11 + d63 A11 e11 + d63 A12 e11 e21 + d63 A12 e11 e21 + 2d63 B12 e11 e41 + 2d63 D66 e51 e41 + 2d63 B26 e21 e41 + 2d63 B26 e21 e51 + 2d52 A45 e51 e41 + 2d42 A55 e41 + 2d36 A44 e51 2 2 + 2d52 A45 e41 + 2d52 A45 e51 + d42 A55 e41 + d52 D66 e41 2 2 + d52 A44 e51 + d36 D66 e51 + 2d65 D12 e41 e51 2 + 2d21 D16 e41 e51 + 2d12 A16 e11 + 2d21 B11 e11 e41 2 + 2d12 B12 e11 e51 + d16 D11 e41 + 2d12 A26 e11 e21 2 2 +2d12 B22 e21 e51 +d42 A66 e21 +d62 D22 e51 +d21 B12 e41 e21 + 2d21 B66 e21 e51 + 2d21 B66 e21 e41 + 2d12 B26 e21 e51 + 2d65 B16 e11 e41 + 2d62 B26 e11 e51 2 + 2d12 B66 e11 e51 + 2d21 B66 e11 e41 + 2d12 D16 e41 2 + 2d12 D26 e51 + 2d12 D26 e41 e51 + 2d65 A66 e21 e11 2 + 2d21 A26 e21

+ d21 B12 e21 e41 + 2d21 A16 e11 e21

2 + 2d36 B16 e11 e51 + 2d61 B16 e21 e41 + 2d36 A22 e21 ),

s3 = 2 (d210 B12 e51 + 2d62 A66 e11 e12 + 2d65 B26 e22 e51 + 2d65 B66 e22 e51 + 2d21 B66 e22 e41 + 2d29 A66 e11 + d21 B12 e21 e42 + 2d21 D12 e42 e51 + 2d61 D16 e42 e51 + 4d21 A26 e21 e22 + 2d12 A26 e21 e21 + 2d12 B22 e21 e52 + 2d12 A26 e22 e11 + 2d63 A11 e11 e12 + 0.5d66 A12 e11 + d29 B16 e41 + 2d62 A55 e41 e42 + 2d62 D66 e41 e42 +d29 e51 +d19 A22 e21 +2d63 D66 e51 e42 +2d63 D66 e52 e41 + 2d63 B26 e21 e42 + 2d41 A45 e51 e42 + 2d41 A45 e52 + 2d41 A45 e52 e41 + 2d63 A22 e21 e22 + 2d63 B26 e21 e52 + 2d11 A16 e11 + 2d63 B16 e11 e52 + 2d63 B16 e11 e42 + 2d111 B66 e41 +2d111 B66 e51 +d63 A12 e11 e22 +d9 B26 e41 + d63 A21 e11 e22 + d61 B11 e41 + 2d21 B11 e12 e41 + 2d12 B12 e12 e51 + 2d65 B16 e11 e42 + 2d62 B26 e11 e52 + 2d12 B66 e11 e52 + 2d12 B66 e11 e42 + 2d65 B16 e12 e41 + d19 B26 e51 + d28 A11 e11 + 2d52 A44 e52 + 2d111 A26 e21 + 0.5d110 B21 e41 + d21 B12 e22 e41 + d55 A26 e11 + d55 B22 e51 + 2d12 B66 e12 e51 + 2d21 B11 e11 e42 + 2d12 B12 e11 e52 + 2d21 A16 e11 e22 + 2d21 D16 e41 e52 + 2d56 B26 e51 + 2d25 A16 e21 + 2d62 D22 e51 e52 + 4d12 D26 e51 e42 + 2d12 D26 e52 e41 + 2d61 A66 e21 e22 + 2d62 B26 e12 e51 + 2d12 B22 e22 e51 + d63 A12 e21 e12 + 0.5d19 A12 e11 + 0.5d28 A12 e21 + 0.5d28 A12 e21 + 2d63 B16 e12 e51 + 2d63 B16 e12 e41 + 2d63 B26 e22 e51 + 2d63 B26 e22 e41 + 2d53 A45 e42 + 2d52 A44 e51 e52 + 2d65 A66 e21 e12 + 2d19 A66 e21 + 2d61 B16 e21 e42 + 2d21 A16 e21 e12 + 0.5d19 B12 e41 + 2d61 D11 e41 e42 + 4d21 D16 e41 e42 + 2d65 D12 e41 e52 + d29 A16 e11

+ d19 A26 e21 + 2d12 B66 e12 e41 + 2d61 B16 e41 e22 + d21 B12 e41 e22 + d21 B12 e41 e22 + 4d12 A16 e11 e12 + 2d18 B16 e41 + 2d56 B26 e21 e52 + 2d65 B66 e21 e52 + 2d65 B66 e21 e42 + 2d65 A66 e22 e11 + 2d63 D66 e51 e52 + 2d42 A55 e42 + d63 A12 e21 e12 + d12 B12 e42 e21 ),

2 2 s4 = 2 (0.25d24 A11 +0.25d25 A22 +2d43 B16 e42 +2d65 A66 e12 + d25 B11 e42 + d10 B12 e52 + 2d18 A66 e22 + 2d61 B16 e22 e42 + 2d65 B26 e22 e52 + 2d21 B66 e52 e22 + 2d21 B66 e22 e42 + 2d21 D16 e42 e52 + 2d29 B26 e52 + 0.5d110 B12 e42 + 2d62 B26 e12 e52 + 2d12 B66 e12 e52 + 2d56 D12 e42 e52 + 2d21 D16 e42 + 2d12 A26 e22 e12 2 2 + 2d21 A16 e12 e22 + d62 A66 e12 + d61 A66 e22 + 2d21 B11 e12 e42 + 2d12 B12 e12 e52 + 2d66 A66 e12 + d28 A11 e12 +d63 A21 e22 e12 +0.5d19 A21 e12 2 2 +2d65 B16 e12 e42 +2d21 A26 e22 +2d12 D26 e52 +d19 B26 e42 + 2d12 A16 e12 + 2d12 B22 e22 e52 + 0.5d18 B21 e42 2 + d55 B22 e52 + 2d111 A16 e12 + d63 D66 e42 + d42 d53 A16 + 0.5d28 A12 e22 + d110 A26 e22 + d55 A26 e12 2 + d62 D11 e42 + d25 A16 e22 + d210 A16 e12 2 + 2d12 B66 e12 e42 + 2d12 D26 e52 e42 + d62 D22 e52 + d21 B21 e22 e42 + 0.25d63 d34 A21 + d63 A12 e12 e22 2 + d63 A11 e12 + d28 B16 e42 + d28 B16 e52 + 0.5d19 A12 e12 + 2d63 B16 e42 + 2d26 B66 e42 + 2d26 A26 e22 + 2d63 D66 e52 e42 + 2d41 A45 e52 e42 + d42 d52 A66 + 2d63 B16 e12 e52 + d63 B26 e22 e52 + d63 B26 e22 e42 2 + d63 A22 e22 + d19 A22 e22 + d42 A55 e42 + d19 B26 e52 2 + d52 d41 A26 + 0.5d28 A21 e22 + d52 A44 e52 + d63 D66 e52 + 0.25d42 d52 A12 ), 2 2 s5 = 8d32 I2 + 8d22 e22 e52 I2 + 4d52 e52 I3 + 4d42 I3 e42 2 2 + 4d52 e12 + 4d42 e22 ,

s6 = (4d52 e11 e12 + 4d42 e21 e22 )I1 + (4d41 e21 e52 + 4d41 e12 e41 + 4d41 e22 e51 + 4d41 e42 e11 )I2 + (4d52 e51 e52 + 4d42 e41 e42 )I3 . Sylvister’s conditions required for the Liapunov-function (19) are   C11 C12 C21 C22 C23      C11 C12 C21 C22  C12 C13 C22 C23 C24    C C    C C C21 C22 C31 C32 C33  >0,  12 13 22 23  >0,   C21 C22 C31 C32  C22 C23 C32 C33 C34    C22 C23 C32 C33   C23 C34 C33 C34 C35      C11 C12 C21  C C12    > 0, C11 > 0.  C12 C13 C22  >0,  11 C12 C13    C21 C22 C21

M.E. Fares et al. / International Journal of Non-Linear Mechanics 41 (2006) 807 – 824

The matrices ⎡ U1 ⎢ U2 ˆ =⎢ [L] ⎢ 0 ⎣ U4 U ⎡ 5 U1 ⎢ U2 ⎢ [P ] = ⎢ 0 ⎣ U4 U5

of Eq. (35) are given by ⎤ V1 1 1 0 V2 2 2 0 ⎥ ⎥ 0 3 3 W3 ⎥ , ⎦ V4 4 4 W4 V5 5 5 W5 ⎤ V1 1 1 0 V2 2 2 0 ⎥ ⎥ 0 3 3 N ⎥ , ⎦ V4 4 4 0 V5 5 5 0

where a0 = X, a01 = Y, a1 = X,x , a2 = Y,y , a11 = X,xx , a22 = Y,yy , a111 = X,xxx , a222 = Y,yyy , (d11 , d12 , d13 , d14 , d15 , d16 , )  a a = (a11 a2 , a1 a22 , a02 a22 ) 0

0

× a11 a2 , a111 a01 , a1 a22 a11 a2 a0 a2 dx dy, (d17 , d18 , d19 , d110 , d111 )  b a 2 = (a0 a222 , a1 a11 a01 , a0 a1 a22 , a11 a01 a0 a2 ) 0

0

× a01 a2 a12 , a0 a2 dx dy, (d51 , d52 , d53 , d54 , d55 , d56 )  b a = (a0 a222 , a0 a2 , a1 a01 , a0 a1 ) 0

0

× a22 a2 a02 , a1 a22 a0 a01 , a0 a2 dx dy, (d27 , d28 , d29 , d210 , d211 )  a b = (a11 a01 a0 a2 , a01 a2 a12 , a22 a2 a02 , a1 a22 a0 a01 ) 0

0

× a1 a01 a12 , a01 a1 dx dy, (d0 , d31 , d32 , d33 , d34 , d35 )  b a = (a0 a01 , a11 a01 , a1 a2 , a0 a22 , a1 a01 a0 a2 ) 0

0

× a1 a2 a11 a01 , a0 a01 dx dy, (d36 , d37 , d38 , d39 )  b a 2 2 2 = (a0 a22 a11 a01 , a1 a2 a0 a22 , a02 a22 , a12 a22 , a11 a01 ) 0

0

× a0 a01 dx dy, 

a

(d41 , d42 , d43 ) =



b

(a0 a2 , a1 a01 , a0 a01 )a01 a1 dx dy, 0

0

(d61 , d62 , d63 , d64 , d65 , d66 )  b a 2 2 2 2 2 = (a11 a01 , a02 a22 , a12 a22 , a22 a01 , a11 a01 a0 a22 ) 0

0

× a1 a22 a0 dx dy,

823

(d21 , d22 , d23 , d24 , d25 , d26 )  b a 2 = (a11 a2 , a1 a22 , a0 a222 , a111 a01 , a1 a11 a01 , a0 a1 a22 ) 0

0

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[21] G.J. Turvey, I.H. Marshall, Buckling and Postbuckling of Composite Plates, Chapman & Hall, London, 1995. [22] J. Song, T.J. Pence, On the design of three-ply nonlinearly elastic composite plates with optimal resistance to buckling, Struct. Optim. 5 (1992) 45–54. [23] C.A. Conceicfl’ao Ant onio, Optimisation of geometrically non-linear composite structures based on load-displacement control, Compos. Struct. 46 (1999) 345–356. [24] Y.G. Youssif, M.E. Fares, M.A. Hafiz, Optimal control of the dynamic response of anisotropic plate with various boundary conditions, Mech. Res. Commun. 28 (5) (2001) 525–534.

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