Multiobjective design and control optimization for minimum thermal

Struct Multidisc Optim (2005) 30: 89–100 DOI 10.1007/s00158-004-0490-0

R E S E A R C H P A P ER

M.E. Fares · Y.G. Youssif · M.A. Hafiz

Multiobjective design and control optimization for minimum thermal postbuckling dynamic response and maximum buckling temperature of composite laminates

Received: 9 September 2003 / Revised manuscript received: 7 May 2004, 4 August 2004 / Published online: 18 March 2005  Springer-Verlag 2005

Abstract Design and control optimization is presented to minimize the thermal postbuckling dynamic response and to maximize the buckling temperature level of composite laminated plates subjected to thermal distribution varying linearly through the thickness and arbitrarily with respect to the in-plane coordinates. The total elastic energy of the laminates is taken as a measure of the dynamic response. The optimization control problem is solved under constraints on the laminate thickness and the control energy produced by a transverse dynamic load distributed over the upper surface of the laminate. The constrained control objective is expressed as the sum of the total elastic energy and penalty term involving the control force, which may be considered as a measure of the control energy. The thickness of layers and the fibers orientation angles are taken as optimization design variables. The design and control objectives are formulated based on shear deformation theory accounting for the von-Karman nonlinearity. The displacements are chosen as the sum of time-independent displacements due to the static thermal load and time-dependent displacements due to the initial disturbances and the applied control force. Liapunov–Bellman theory is used to obtain the optimal control force, buckled deflections and controlled elastic energy. Numerical examples are presented for angle-ply antisymmetric laminates with simply supported edges. Graphical studies are carried out to show the advantages of the present design and control procedures. Keywords Closed-loop control force · Composite laminated plate · Maximization of the buckling temperature level · Minimization of the postbuckling dynamic response · Optimal design · Shear deformation theory · von-Karman nonlinearity

M.E. Fares · Y.G. Youssif · M.A. Hafiz (B) Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, 35516 Egypt E-mail: [email protected], [email protected]

Introduction Laminates subjected to thermal loads may buckle under the action of forces generated by thermal expansion. In many engineering applications, it is necessary to maximize the buckling temperatures subject to such design constraints as strength, frequency, displacement, stiffness, etc. The problem can be formulated as a minimum-weight design problem subject 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 as reported in Frauenthal (1973). Consequently, optimizations in the postbuckling range become an important design consideration for laminates that may be exposed to temperature load, compressive load or combined thermomechanical load higher than the buckling load. Results reported in Obraztsov and Vasil’ev (1989) indicated that optimum ply angles to maximize the ultimate loadcarrying capacity of symmetric angle-ply laminates coincide with the fiber orientations providing maximum strength in the prebuckled state. Effects of designing for buckling load on the postbuckling behavior as well as optimal designs for maximum postbuckling stiffness were studied in Pandey and Sherbourne (1993) for simply supported symmetric laminates. This work shows that buckling load and postbuckling stiffnesses are described by functions of different nature so that the optimization based on the bending stiffnesses only leads to designs with a weak postbuckling performance. Thus, for this multiobjective optimization problem, laminates designed with respect to one criterion will perform quite poorly with respect to the other one. Improved designs with respect to both criteria can be obtained by introducing plies with fiber orientations providing maximum postbuckling stiffness in the core region of the laminate, whereas, the other plies have orientations providing maximum buckling loads. This procedure has slightly lower buckling load, but, substantially increased postbuckling resistance. Further results on optimization of buckling and postbuckling response of laminates are given in Adali and Duffy (1990a,b, 1993), Song and Pence (1992), Adali and

90

Nissen (1987), Chen et al. (2003), Lee et al. (1999), Yang and Shen (2003), Conceicfl’ao Antonio (1999) and Spallino and Thierauf (2000). In large space structures, the suppression of excessive vibrations represents one of the most pressing and difficult problems facing structural designers. An effective means of suppressing these excessive vibrations is by active structural control. This problem has been a main subject of many studies (Bruch et al. 1990; Wang and Huang 2002; Adali 1984; Turteltaub 2002). During the past two decades, a great deal of interest has been devoted to solve this problem using simultaneously optimal design and active control. This topic has attracted the attention of many researchers (Adali et al. 1991; Sloss et al. 1992; Sadek et al. 1993; Fares et al. 2002, 2004; Grandhi 1989; Rao et al. 1988). In Fares et al. (2002), design and active control optimization was employed to minimize the prebuckling vibrational energy, and to maximize the buckling temperature of composite laminated plates subjected to uniform thermal distributions. A shear deformation theory was used to formulate the optimization objectives. The Galerkin method and Liapunov– Bellman theory were used to determine the optimal control force as a closed-loop distributed function. The same technique was employed in Fares et al. (2004) to minimize the prebuckling dynamic response of composite laminated truncated conical shells. Several studies (Fares et al. 2003; Di Sciuva et al. 2003; Reddy and Liu 1985; Turvey and Marshall 1995; Reddy 1984; Chung and Wang 2002) indicated that the transverse shear deformation theory has a significant effect on the prebuckling and postbuckling responses of composite laminates, and its negligence may lead to 40% or more error in the optimum design variables. However, most of the previous studies are formulated based on classical laminate theories. The current work deals with a multiobjective design and control problem of composite laminates subject to thermal load. The design and control objectives are to maximize the thermal buckling and minimize the thermal postbuckling dynamic response of a laminate with minimum expenditure of control energy produced by a control force distributed over the upper laminate surface. The total elastic energy of the laminate is taken as a measure of the dynamic response. The fiber orientation angles and thicknesses of the plies are taken as design variables. For the design procedure, the total elastic energy, the control energy and the thermal buckling load are set in a unified criterion. The present problem is formulated based on a first-order shear deformation laminate theory including the von-Karman nonlinearity. The displacements are taken as the sum of timeindependent large displacements due to a static postbuckling temperature and time-dependent displacements due to the initial disturbances and control force. The Liapunov– Bellman theory (Gabralyan 1975) is used to determine the optimal control force, controlled deflections and energy. To assess the present control optimization, numerical analysis is carried out for antisymmetric angle-ply laminates with simply supported edges. The influences of the material and geometric parameters on the optimization process are illustrated.

M.E. Fares et al.

Theoretical formulation Consider a fiber-reinforced rectangular laminate of constant thickness h occupying the space 0 ≤ x ≤ a; 0 ≤ y ≤ b; −h/2 ≤ z ≤ h/2. The laminate is composed of N anisotropic layers such that each layer possesses one plane of elastic symmetry parallel to its mid-plane. The laminate is subjected to temperature distribution varying linearly through the thickness and varying arbitrarily with respect to the in-plane coordinates x and y z T(x, y, z) = T0 (x, y) + T1 (x, y) . h

(1)

A control force q(x, y, t) is distributed over the upper surface of the laminate (z = −h/2) to damp the dynamic response under the following initial disturbances: w(x, y, 0) = A∗ (x, y) ;

w(x, y, 0) = B ∗ (x, y) ; ˙

(2)

where the dot denotes differentiation with respect to time. The present formulation accounts for a first-order shear deformation laminate theory based on the following Reissner–Mindlin displacements: u 1 = u + zψ ,

u 2 = v + zφ ,

u3 = w ,

(3)

where (u 1 , u 2 , u 3 ) are the displacements along the x, y and z directions, respectively, (u, v, w) are the displacements of a point on the mid-plane, and ψ and φ are the shear rotations due to bending. The von-Karman nonlinearity strains associated with the displacements (3) are given by: (0)

ε1 = ε1 + zψ,x , ε3 = 0 , ε5 = w,y + φ , ε1(0) = u ,x + 12 w2,x ,

(0)

ε2 = ε2 + zφ,y , ε4 = w,x + ψ ,

ε6 = ε6(0) + zε6(1) ,

ε2(0) = v,y + 12 w2,y ,

ε6(0) = v,x + u ,y + w,x w,y , ε1(1) = ψ,x , ε2(1) = φ,y ,

ε6(1) = φ,x + ψ,y ,

(4)

where a comma denotes partial differentiation with respect to the subscript. The governing equations of the laminate are given in the form (the superposed dot denotes differentiation with respect to time): N1,x + N6,y = I1∗ u¨ I2∗ ψ¨ , N6,x + N2,y = I1∗ v¨ + I2∗ φ¨ ,

Q 1,x + Q 2,y + q = I1∗ w ¨ − (N1 w,x + N6 w,y ),x − (N6 w,y + N2 w,y ),y , M1,x + M6,y − Q 1 = I2∗ u¨ + I3∗ ψ¨ , M6,x + M2,y − Q 2 = I2∗ v¨ + I3∗ φ¨ ,

(5)

where the inertias In∗ and the constitutive equations are given by:

Multiobjective design and control optimization for minimum thermal postbuckling dynamic response and maximum buckling temperature 91

In∗

=

N z k


ρ (k) z n−1 dz ,

(n = 1, 2, 3) ,

k=1z k−1

 (0) (1) (0) (Ni , Mi , Q m−3 ) = Aij ε j + Bij ε j − NiT , Bij ε j  T + Dij ε(1) j − Mi , A mm εn , (i = 1, 2, 6) ,

(m, n = 4, 5)

(6)

The following definitions are used in the preceding equations: (Aij , Bij , Dij , Amn ) =

N z k


  1, z, z 2 , 5/6 dz , c¯ (k) ij

k=1z k−1

(i, j = 1, 2, 6) ,

(m, n = 4, 5) , z N  
k (k) (k) T T N i , Mi = c¯ ij α¯ j (1, z)T dz ,

where σi are the stresses and ξ1 is a positive constant weighting factor. The first term in (9) is the elastic and kinetic energies of the laminate, and the second term is a penalty term involving the control function q ∈ L 2 ; L 2 denotes the set of all bounded square integrable functions on the region {0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ t < ∞}. Note that the present penalty function technique maintains the constrained control objective (9) quadratic and, hence, differentiable. This minimization problem may be solved with the aid of the initial and boundary value problem (2)–(8) based on Liapunov– Bellman theory (Gabralyan 1975).

Solution procedure

(7a) For the present problem in which the laminate is subjected to a static postbuckling thermal load and a dynamic control force q, the displacements functions (u, v, w, ψ, φ) may (i, j = 1, 2, 6) , be chosen as the sum of time-independent displacements k=1z k−1 (u T , vT , wT , ψT , φT ), due to the static postbuckling tem(7b) perature and time-dependent displacements (u L , v L , w L , ψ L , φ L ), due to the control force q as follows: where ρ (k) is the material density of kth layer, NiT and MiT u = u L (x, y, t) + u T (x, y) , v = v L (x, y, t) + vT (x, y) , are resultants and moments due to thermal loading, α¯ (k) j are (k) the coefficients of thermal expansion and c¯ ij are the compli- w = w L (x, y, t) + wT (x, y) , ψ = ψ L (x, y, t) + ψT (x, y) , ance elastic constants of the kth layer. Here, Amn represents φ = φ L (x, y, t) + φT (x, y) . (10) a shear correction factor. For a simply supported laminate, the boundary condiHere, if the control force q is not too large then, the nonlintions at the edges may be written in the form: ear terms of (4) and (5) involving the displacement w L can be omitted. Using (4)–(6), we can easily obtain the laminate v = w = φ = N1 = M1 = 0 at x = 0, a , governing equations in the displacements (u, v, w, ψ, φ). u = w = ψ = N2 = M2 = 0 at y = 0, b . (8) Then, substituting expressions (10) into the resulting governing equations, we can obtain two systems of partial differential equations. The first system is time independent and nonlinear in the displacements (u T , vT , wT , ψT , φT ), and Optimal control problem may be written in the following operator matrix form:         The present control objective aims to minimize the vibra¯ ¯ δ¯ = F¯ , (11) tional postbuckling response of the laminate as time goes [L] + H δ to infinity with the minimum possible expenditure of conT trol energy. The dynamic response of the plate is measured where {δ¯ } = {u T , vT , wT , ψT , φT } , the component of the ¯ and the symmetric matrix of by a functional related to the total elastic energy of the generalized force vector { F} differential operators [L] are given in the Appendix. laminate, which is a function of displacement, its spatial The second system of partial differential equations is derivatives and the velocity. The cost function of the controldesign problem is taken as the sum of the laminate energy time dependent and linear in the displacements (u T , vT , wT , and a penalty term including the control force q(x, y, t). ψT , φT ), and may be written in the following operator maThen, the mathematical formulation of the present control trix form: problem aims to determine the optimal control function q [L]{δ} + [R]{δ¨ } = { f } , (12) that minimizes the functional: where {δ} = {u L , v L , w L , ψ L , φ L }T , and { f } is the force J(q, h k ; θk ) vector. h Using expressions (10), we can rewrite the boundary ∞ a b  2    1 conditions (8) for the two systems of partial differential (k) 2 2 2 = εi σi + ρ u˙ 1 + u˙ 2 + u˙ 3 dz dy dx dt equations (11) and (12) as follows: 2 0 0 0 −h 1) For system (11): 2 ∞ b a vT = wT = φT = N1T = M1T = 0 at x = 0, a, + ξ1 q2 (x, y, t) dx dy dt , (i = 1, 2, . . . , 6) , (9) u T = wT = ψT = N2T = M2T = 0 at y = 0, b . (13) 0 0 0

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2) For system (12): v L = w L = φ L = N1L = M1L = 0 at x = 0, a, u L = w L = ψ L = N2L = M2L = 0 at y = 0, b .

(14)

where Ni = NiL + NiT , Mi = MiL + MiT . Thus, the solution of the present problem leads to the solutions of a nonlinear boundary value problem (11) and (13), and linear initial boundary value problem (2), (12) and (14).

The critical buckling temperature λT , at which buckling occurs, may be found from (17) by solving the following eigenvalue problem     Lˆ − λT RT {∆} = {0} (18) where the matrix [RT ] is given in the Appendix.

The optimal control force For the second initial and boundary value problems (2), (12) and (14), an exact solution can be constructed when the disWe firstly deal with the solution of the nonlinear boundary placements functions (u L , v L , w L , ψ L , φ L ) and the closedvalue problem (11) and (13) to obtain the critical tempera- loop control function q are expressed in the form: ture and postbuckling displacements. This problem cannot u    L Umn sin(λm x) cos(µn y)    be solved exactly, so the Galerkin method may be used.             vL  Then, the displacements (u T , vT , wT , ψT , φT ) may be ex-      cos(λ x) sin(µ y) V mn m n             panding in the form:     ∞  w L 
Wmn sin(λm x) sin(µn y)    , (19) =   ψL  U¯ mn sin(λm x) cos(µn y)     Ψmn cos(λm x) sin(µn y)  uT         m,n=1                             φL  V¯mn cos(λm x) sin(µn y)          Φmn sin(λm x) cos(µn y) vT          ∞       
       q ¯ q sin(λ x) sin(µ y) WT = , (15) Wmn sin(λm x) sin(µn y) mn m n       m,n=1          ψT  ¯     Ψ cos(λ x) sin(µ y) mn m n     where Umn , Vmn , Wmn , Ψmn , Φmn and Q mn are unknown           φT functions of time. ¯ mn sin(λm x) cos(µn y) Φ Substituting expressions (19) into (12), we obtain after where λm = mπ/a, µn = nπ/b and (m, n) are mode num- some mathematical manipulations the following time sysbers. Substituting expressions (15) into (11) and multiplying tem of equations:   each equation by the corresponding eigenfunction, then in-   Lˆ {δ0 } + [R] δ¨0 = { f0 } , (20) tegrating over the domain of solution, we obtain after some mathematical manipulations the following matrix equations:    where {δ0 } = {Umn , Vmn , Wmn , Ψmn , Φmn }T , and { f0 } is Lˆ + [H(∆)] {∆} = {F} , (16) a force vector. The optimal control force q0 and controlled deflections may be found from (20). If the in-plane inertia terms in (20) ˆ and [H] denote the linear and nonlinear (geometwhere [ L] are omitted, then an equation of the time-dependent funcric) stiffness matrices, respectively, which are given in the tions W and Q mn may be obtained in the form: mn Appendix, {∆} and {F} denote the columns:   ¨ mn + ω2mn Wmn = Q mn /I1∗ , W ¯ mn , Ψ¯ mn , Φ ¯ mn , {∆}T = U¯ mn , V¯mn , W 1 ω2mn = 2 ∗ (e1U3 + e2 V3 + e3 Ψ3 + e4 Φ3 − e0 W3 ) . (21) {F}T = {F1 , F2 , F3 , F4 , F5 } . e0 I1 I1 Critical temperature and postbuckling displacements

The postbuckling displacements may be obtained by where the quantities I and e are given in the Appendix. 1 i solving (16) iteratively until an appropriate convergence cri- Also, using expressions (19), we get the cost functional (9) terion is satisfied. In the present study, a Picard-type suc- in the form: cessive iteration scheme is employed. Then, (16) may be ∞ approximated by: 
 
2 2 ˙ mn J= δ1 Wmn + δ2 W + δ3 Q 2mn dt = Jmn ,        Lˆ + H ∆r (17) ∆r+1 = {F} , m,n m,n 0 (22) where r is the iteration number. The nonlinear stiffness matrix for the (r + 1)th iteration is computed using the solution where the quantities δ1 , δ2 and δ3 are given in the Appendix. vector from the rth iteration. Moreover, at the start of the it- Since the system of (22) is separable, the functional (22) deeration procedure, the solution is assumed to be zero so that pends only on the variables found in the (m, n)th equations the linear solution is obtained at the end of the first iteration. of the system. With the aid of this condition, the problem

Multiobjective design and control optimization for minimum thermal postbuckling dynamic response and maximum buckling temperature 93

is reduced to a problem of analytical design of controllers (Gabralyan 1975), for every m, n = 1, 2, . . ., ∞. Now, the optimal control problem is to find a control function q0 (t) that satisfies the conditions:

γmn =

J(q ) ≤ J1 (q) , for all q ∈ L × [0, ∞) , o

2

min J = min q

q

Jmn =

m,n




min Jmn .

2 m,n Q mn ∈L

Omn βmn + 2A∗mn , 2¯vmn

  βmn , A∗mn =

that is:


where βmn and γmn are unknown coefficients which may be obtained from the initial conditions (2) by expanding them in double Fourier series, then:

(23)

4 abω2mn

a b 0

(A∗ , B ∗ ) sin(λx) sin(µy) dx dy .

0

(30)

Hence, the minimization problem (23) can be carried out independently for every modal equation. For such problems, Liapunov–Bellman theory (Gabralyan 1975) is considered as an effective solution approach. The necessary and sufficient conditions, according to Liapunov–Bellman theory, for minimizing the functional (22) is:   ∂Vmn ∂Vmn ˙ ¨ ¯ min (24) Wmn + W + Jmn = 0 , ˙ mn mn Q mn ∂Wmn ∂W

By inserting the expressions (29) and (30) into (22) and (26), we can obtain the total elastic energy and the optimal control force.

Optimal design procedure

The design procedure aims to maximize the critical buckling temperature λT and minimize the postbuckling vibrational energy. This may be done by solving a two-step problem to provided that the Liapunov function Vmn , find firstly the optimum ply thicknesses h k and, secondly, the optimal fiber orientation angles θk . In general, these objec2 2 ˙ mn + C W ˙ mn , (25) Vmn = AWmn + 2BWmn W tives conflict with each other, necessitating a multiobjective formulation including the thermal postbuckling energy and is positive definite, i.e A > 0, C > 0 and AC > B 2 , where critical buckling temperature. With this situation in mind, J¯mn is the integrand of (22). Using the condition (24), we the performance index J (h , θ , qo ) of the design problem 1 k k can obtain the optimal control function in the form: is specified as:      −1  ˙ mn , (31) Q omn = BWmn + C W (26) J1 qo , h k , θk = ξ2 /λT + J qo , h k ; θk ∗ δ3 I1 where ξ2 is a positive constant weighting factor. then, substituting (21), (22), (25) and (26) into (24), and For the numerical purpose, we consider general anti2 , W 2 and W W ˙ mn equating the coefficients of Wmn mn ˙ mn to symmetric laminates with a stacking sequence (θ/ − θ/θ/ zero, we get a system of equations. The general solution of this system under the condition that the Liapunov function is positive definite is given by:        δ1   B = −I1∗2 ω2mn δ3 − δ3 ω4mn δ3 + ∗2  , I1 ! C = −I1∗ δ3 (2B + δ2 ) .

(27)

Using expressions (26) and (27) in (21), one obtains: ˙ mn + Ω2mn Wmn = 0 ¨ mn + Omn W W Omn =

C δ32 I1∗4

,

Ω2mn = ω2mn +

B . δ3 I1∗2

(28)

The solution of (28) under the condition 2Ωmn > αmn , is given by:      o Wmn = e−αmn t/2 βmn cos v¯ mn t + γmn sin v¯ mn t , 1 2 2 = Ω2mn − Omn , v¯ mn 4

(29)

Fig. 1 Geometry and cross section of the antisymmetrically laminated angle-ply plate

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M.E. Fares et al.

−θ/ . . . ) and an even number N of layers with thicknesses (h 1 , h 2 , h 3 , . . . / . . . , h 3 , h 2 , h 1 ), h 1 = h 3 = h 5 = . . . , h 2 = h 4 = h 6 = . . . , (see, Fig. 1). In this case, the ply thickness h k may be generally represented as a function of the number of layers N as follows (Fares et al. 2002, 2004):

Young’s moduli are related by the reciprocal relations vij E j = v ji E i , (i, j = 1, 2). The initial conditions (2) are chosen as:

h k = h N−k+1 =    ηh   ,   2       

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

 k = (1, 3, . . . , N/2 − 1) , N/2 even k = (1, 3, . . . , N/2) ,

 h(4 − ηN )/(2N ) , k = (2, 4, . . . , N/2) ,            h[4 − η(2 + N )]/ (2N − 4) , k = (2, 4, . . . , N/2 − 1) ,

N/2 odd N/2 even

w(x, y, 0) = 0.005 sin(λm x) sin(µn y) ,

E 1 /E 0 = 181 , G 23 /E 0 = 3.39 , a/h = 20 ,

E 2 /E 0 = 10.3 ,

ξ1 = 10ξ2 = 1 ,

G 12 /E 0 = 7.17 ,

G 31 /E 0 = 5.98 ,

α1 /α0 = 0.02 , m =n =1,

w(x, y, 0) = 0 ; ˙ (36)

v12 = 0.28 ,

α2 /α0 = 22.5 , ρ/ρ0 = 1600 ,

N = 4, (37)

N/2 odd , which are typical of graphite/epoxy (T300/5208). Here, (32) E 0 = 1 GPa, α0 = 10−6 (Celsius)−1 and ρ0 = 1 kg/m3 . The where η is a thickness ratio. Therefore, the design procedure numerical results for deflections and force functions are aims to determine the optimum values of thickness ratio ηopt given at the laminate midpoint x = a/2, y = b/2. Figure 2 shows the variation of the postbuckling elastic and the orientation angle θopt from the following conditions: energy J2 (the first term of expression (9)) and the nondi    mensional buckling temperature λT (= 1000αo T¯0 ) with the J1 qo , ηopt , θopt = min J1 qo , η, θk , θ,η orientation angle θ for four-layer laminates subjected to constant temperature (T¯1 = 0; a/h = 20, a/b = 1). The solid ◦ 0 ≤ θk ≤ 90 , 0 ≤ η ≤ ηˆ , curves represent laminates designed optimally over thick" ness, whereas the dashed curves represent laminates with 4/N , N/2 even ηˆ = (33) equi-thickness layers (h E = h/N ). These curves indicate 4/(2 + N ) , N/2 odd that the design optimization over the thickness significantly decreases the postbuckling energy, and increases the buckling temperature. Moreover, the postbuckling energy and the buckling temperature exhibit conflicting behaviors with the orientation angle θ, but the minimal values of the elasNumerical results and discussion tic energy J2 and maximal values of the buckling temIn this section, numerical and graphical studies are carried perature occur at θ = 45◦ , which is the optimal orientation out to illustrate the influence of the present design and con- angle. Figure 3 contains buckling temperature curves plottrol procedures on the buckling temperature and postbuck- ted against buckled deflection wT /h in a wide postbuckling response induced by uniform temperature and nonuni- ling range for square laminates designed optimally over the form sinusoidal temperature. Here, we take all layers of the thickness and subjected to uniform temperature (T¯1 = 0) laminate made of the same orthotropic materials, and sub- and nonuniform temperature (T¯ /T¯ = 0.3). Here, the upper 1 0 jected to the following temperature distributions: curves represent the behavior of the laminates with best re  quired performance with respect to the buckling temperature (T0 , T1 ) = T¯0 , T¯1 sin(πx/a) sin(πy/b) . (34) and postbuckling elastic energy. This figure shows that the optimum values of the orientation angle may vary throughwhere T¯0 is the average over the laminate thickness and T¯1 is out the postbuckling range, for instance, in the two cases the difference between the top and bottom surface tempera- of uniform and nonuniform temperatures when λT < 6.5, tures. the optimum value of the orientation angle θopt is 45◦ , The engineering constants are introduced instead of the whereas in the deep postbuckling range λ > 6.5, θopt = 0◦ . elastic constants using the relations: The present results with results obtained in Fares et al. (2002, 2004) show that the laminate optimal design to miniE1 v12 E 2 mize the prebuckling dynamic response is the same optimal c11 = , c12 = , 1 − v12 v21 1 − v12 v21 design for the laminates in the first stage of the postbuckling range. E2 , c66 = G 12 , c16 = c26 = 0 , (35) c22 = The effect of the aspect ratio a/b on the buckling tem1 − v12 v21 perature and postbuckling deflections are presented in Fig. 4 where E i are Young’s moduli, G ij are the shear mod- for laminates with optimal and non-optimal designs. Rectuli and vij are Poisson’s ratios. The Poisson’s ratios and angular laminates with a/b > 2 have high resistance to

Multiobjective design and control optimization for minimum thermal postbuckling dynamic response and maximum buckling temperature 95

Fig. 2 The variation of the elastic energy J2 (a), and the buckling temperature λT (b) with the orientation angle θ, T1 = 0, h E = h k = h/N

Fig. 3 The variation of the buckling temperature λT with the buckling deflection ωT /h for square laminates with h opt , subjected to nonuniform (a) and uniform (b) temperatures

thermal buckling and low postbuckling deflections, whereas the square laminates has low resistance to thermal buckling and large postbuckling deflections. Note that, the

optimal design procedure is more effective with rectangular laminates than with square ones. The effect of the shear deformation (a/h) on the buckling temperature and

Fig. 4 Effect of the aspect ratio a/b on buckling and postbuckling responses, T¯1 /T¯0 = 0.8

Fig. 5 Effect of plate thickness ratio a/h on the buckling and postbuckling responses, T¯1 /T¯0 = 0.1

96

M.E. Fares et al.

Fig. 6 Effect of number of layers N and optimization design on buckled deflection wT /h and buckling temperature λT for square laminates, a/h = 20, θ = 45◦

postbuckling response are displayed in Fig. 5 for square laminates with T¯1 /T¯0 = 0.1, a/h = 20 or a/h = 30. These curves show that the buckling temperature decreases and the buckled deflections considerably increase with the increase in a/h ratio. This is because the thin laminates with large ratio a/h exhibit low resistance to bending and buckling. Therefore, the optimization procedures significantly raise the level of performance for moderately thick laminates, and slightly raise the performance of thinner ones. Figure 6 shows the effect of the number of layers N on the buckling temperature and postbuckling response for optimal and non-optimal designed laminates with a/h = 20 and a/b = 1. As can be seen from Fig. 6, there is a considerable increase in the buckling temperature and decrease in the buckled deflections as the number of layers N increase from 2 to 8, and this effect becomes insignificant for lami-

nates with N ≥ 8. This means that laminates with few layers perform quite poorly with respect to the present design objectives but, the optimization over the thickness and fiber orientation raises the level of the performance of these laminates remarkably, so that optimally designed four-layer laminates have higher performance than non-designed laminates with N = 8. This may be explained by the fact that, as the number of layers N increases, the bending-stretching coupling coefficients Bij decrease, and the bending-stretching effect makes the laminate more flexible and severely reduces the buckling temperature. The optimization over the thickness and fiber orientation for the present problem is to keep the coupling stiffnesses Bij of the laminates with few layers small as compared with the non-designed cases, particularly, in the case of varying temperature T¯1 /T¯0  = 0. Figure 7 contains plots of buckled deflection-buckling temperature curves for optimal and non-optimal four-layer

Fig. 7 Effect of thermal load ratio on the buckling and postbuckling responses, a/b = 1, a/h = 20

Fig. 8 Effect of orthotropy ratio on the buckling and postbuckling responses, T¯1 /T¯0 = 0.8

Multiobjective design and control optimization for minimum thermal postbuckling dynamic response and maximum buckling temperature 97

Fig. 9 The variation of the elastic energy J2 (a), and the buckling deflection w/h (b) with the time t ∗

laminates with a/h = 20, subjected to different temperature distributions T¯1 /T¯0 (= 0.02, 1). It is noted that the presence of temperature distribution varying through the laminate thickness lowers the buckling temperature. The optimization process increases the resistance to thermal buckling so that it reduces significantly the effect of this temperature on buckling and postbuckling responses, particularly when there are big differences between the temperatures of the upper and lower laminate surfaces (T¯1  T¯0 ). The effect of the orthotropy ratio E 1 /E 2 on the buckling temperatures and buckled deflection can be seen from the results presented in Fig. 8. With increasing orthotropy ratio, the laminate resistance to the thermal buckling increases because laminates with low orthotropy ratio are less stiff. The optimization over the thickness only slightly improves the performance of the laminates, whereas the optimizations over both the thickness and fiber orientation considerably improve the performance of the laminates. Figures 9–11 show the variation of the controlled vibrational energy J¯2 (the integrand of J2 with ξ1 = 0), deflection w/h and the control force q0 with time t ∗ for four different

Fig. 10 Effect of the design optimization on the optimal control force

laminate designs, which are non-optimal designs, a partially optimal design over h opt , a partially optimal design over the fiber orientation θ and an optimal design over both h opt and

Fig. 11 Effect of orthotropy ratio on the optimal control force

Fig. 12 The variation of the control energy J3 (the second term in (9)) with the time t ∗

98

θopt . All previous cases of optimal design considerably reduce the postbuckling total elastic energy of the laminate and raise the level of buckling temperature as compared to the uncontrolled ones. But, the optimal design over θ is more effective than the optimal design over the thickness h k , while, the optimal design over both θ and h k is the most efficient. In addition, the figures indicate that the optimal design procedure not only plays an important role in improving the performance of the laminate, but also contributes significantly to decreasing the expenditure of the control force (see Fig. 12).

Conclusion A multiobjective control problem of composite laminated plates is presented for maximum thermal buckling and minimum postbuckling dynamic response under constraints on the laminate thickness and the control energy. The problem formulation is based on a shear deformation theory including the von-Karman nonlinearity. The solution of the present problem leads to the solutions of a nonlinear boundary value problem and linear initial boundary value problem. The nonlinear boundary value problem is solved iteratively to obtain the buckled displacements and buckling temperature. The linear initial and boundary value problem is solved using Liapunov–Bellman theory to obtain the optimal control force. The optimal layer thickness, fiber orientation angles and closed-loop control force are determined for angleply antisymmetric laminates with simply supported edges subjected to uniform and nonuniform sinusoidal thermal distributions. The present study indicates that the optimum values of the fiber orientation angles may change throughout the postbuckling range. The optimization over the orientation angles is more effective than the optimization over the layer thickness, and the optimization over both the layer thickness and the orientation angles is the most efficient, significantly improving the performance of the angle-ply antisymmetric laminates with respect to buckling temperature and postbuckling response, particularly for thin square laminates with few layers.

Appendix

M.E. Fares et al.

L 33 − A44 d11 − 2A45 d12 − A55 d22 , L 34 = −A44 d1 − A45 d2 , L 35 = −A45 d1 − A55 d2 , L 44 = D11 d11 + 2D16 d12 + D66 d22 − A44 , L 45 = (D12 + D66 )d12 + D16 d11 + D26 d22 − A45 , L 55 = 2D26 d12 + D22 d22 + D66 d11 , and dij denote the differential operators: dij =

∂2 , ∂xi ∂x g

di = di0 =

T T F¯1 = N1,x + N6,y , T T F¯4 = M1,x + M6,y ,

∂ , ∂xi

(i, j = 0, 1, 2)

T T F¯2 = N6,x + N2,y ,

F¯3 = 0 ,

T T F¯5 = M6,x + M2,y ,

H¯ 11 = H¯ 12 = 0 ,

H¯ 13 = 2w,x L 11 + 2w,y L 12 ,

H¯ 14 = H¯ 15 = 0 ,

H¯ 21 = H¯ 22 = 0 ,

H¯ 23 = 2w,y L 22 + 2w,x L 12 ,

H¯ 24 = H¯ 25 = 0 ,

H¯ 33 = 2(A11 u ,x + A16 u ,y + A16 v,x + A12 v,y + B11 ψ,x + B16 ψ,y + B16 φ,x + B12 φ,y )d11 + 2(A12 u ,x + A26 u ,y + A26 v,x + A22 v,y + B12 ψ,x + B26 ψ,y + B26 φ,x + B22 φ,y )d22 + 4(A16 u ,x + A66 u ,y + A26 v,x + A66 v,y + B16 ψ,x + B66 ψ,y + B66 φ,x + B26 φ,y )d12 + 2(w,z )2 × (A11 d11 + A12 d12 + 2A16 d12 ) + 2(w,y )2 × (A11 d11 + A22 d22 + 2A26 d12 ) + 4w,x w,y (A16 d11 + A26 d22 + 2A66 d12 ) , H¯ 41 = H¯ 42 = 0 ,

H¯ 43 = 2w,x L 14 + 2w,y L 24 ,

H¯ 44 = H¯ 45 = 0 ,

H¯ 51 = H¯ 52 = 0 ,

L 13 = 0 ,

H¯ 53 = 2w,x L 15 + 2w,y L 25 , H¯ 54 = H¯ 55 = 0 , # 3 H33 = − µ4n A22 + 2λ2m µ2n A12 + λ4m A11 32 $ 4 2 ¯ mn − λ2m µ2n A66 W , otherwise Hij = 0 . 3

L 14 = B11 d11 + 2B12 d12 + B66 d22 ,

Because of antisymmetry (Jones 1975):

L 15 = (B12 + B66 )d12 + B16 d11 + B26 d22 = L 24 ,

A16 = A26 = A45 = B11 = B12 = B22 = B66

L 22 = 2A26 d12 + A22 d22 + A66 d11 ,

= D16 = D26 = D45 = 0 .

L 11 = A11 d11 + 2A12 d12 + A66 d22 , L 12 = (A12 + A66 )d12 + A16 d11 + A26 d22 ,

L 23 = 0 , L 25 = 2B26 d12 + B22 d22 + B66 d11 ,

The nonzero element of matrix [RT ] is R¯ 33 = −µ2n n 2T − λ2m n 1T , and the elements of force vector {F} are:

Multiobjective design and control optimization for minimum thermal postbuckling dynamic response and maximum buckling temperature 99

T F1 = −n 12 ,

T F2 = −n 12 ,

F4 = −m 1T ,

F5 = m 2T ,

F3 = 0 ,

W2 = 0 ,

Ψ2 = −µ2n B26 − λ2m B16 ,

U3 = 0 ,

V3 = 0 ,

W3 = −(λ2m A55 + µ2n A44 ) ,

where

Ψ3 = −λ2m A55 ,

$ N z k #  
z T T −1 T¯0 + T¯1 n i , m i = (1000T0 α0 ) h

V4 = −µ2n B26 − λ2m B16 ,

(k) (k)

× c¯ ij α¯ j (1, z) dz

(i = 1, 2, 6)

The coefficients ei and δi are: δ2 =

π 2 I1∗ , 8λm µn

δ1 =

 π2 λ2 µ2 D66 (e3 + e4 )2 + A44 µ2n (e0 + e4 )2 8e2o λm µn m n

δ3 = I1 ξ1

ˆ is The linear stiffness matrix [ L]   U1 V1 W1 Ψ1 Φ1   U V2 W2 Ψ2 Φ2      2   Lˆ = U3 V3 W3 Ψ3 Φ3  ,   U V W Ψ Φ  4 4 4 4 4   U5 V5 W5 Ψ5 Φ5

% %U1 % % %U2 e2 = %% %U4 % %U5 % %U1 % % %U2 e4 = %% %U4 % %U5

V1

Ψ1

V2

Ψ2

V4

Ψ4

V5

Ψ5

W1

Ψ1

W2

Ψ2

W4

Ψ4

W5

Ψ5

V1

Ψ1

V2

Ψ2

V4

Ψ4

V5

Ψ5

% Φ1 %% % Φ2 % %, Φ4 %% % Φ5 %

% % W1 % % % W2 e1 = %% % W4 % % W5

% Φ1 %% % Φ2 % %, Φ4 %% % Φ5 %

% %U1 % % %U2 e3 = %% %U4 % %U5

W1 = 0 ,

V1

Ψ1

V2

Ψ2

V4

Ψ4

V5

Ψ5

V1

W1

V2

W2

V4

W4

V5

W5

% Φ1 %% % Φ2 % %, Φ4 %% % Φ5 % % Φ1 %% % Φ2 % %, Φ4 %% % Φ5 %

% W1 %% % W2 % %, W4 %% % W5 %

U1 = −λ2m A11 − µ2n A66 ,

V1 = −λ2m (A12 + A66 ) ,

Ψ1 = −2λ2m B16 ,

U2 = −µ2n (A12 + A66 ) ,

U4 = −2µ2m B16 ,

W4 = −A55 , Φ4 = −µ2n (D12 + D66 ) ,

U5 = −µ2n B26 − λ2m B16 ,

V5 = −2λ2m B26 ,

W5 = −A44 ,

Ψ5 = −λ2m (D12 + D66 ) ,

Φ5 = −µ2n D22 − λ2m D66 − A44 ,

References

+ A55 λ2m (e3 + e0 )2   + e24 λ4m D22 − 2e3 e4 D12 µ2n λ2m + e23 D11 µ4n ,

% %U1 % % %U2 e0 = %% %U4 % %U5

Φ3 = −µ2n A44 ,

Ψ4 = −λ2m D11 − µ2n D66 − A55 ,

k=1z k−1

Φ2 = −2µ2n B26 ,

Φ1 = −µ2n B26 − λ2m B16 ,

V2 = −λ2m A66 − µ2n A22 ,

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