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47) (i ) are with µi denoting the Lagrangian multiplier. 49) Fi(m, n) = 4[D11(m/r)4 + 2(D12 + 2D66)(mn/r)2 + D22n4] – Piπ2(m2/r2 + λin2) with Dij denoting the nondimensional flexural stiffnesses, r = a/b, and λi = Ny/Nx for the i-th loading. 50) The minimum thickness hmin and the optimal ply angle θopt are determined using an iterative solution procedure. Starting from the estimates of h and θ, a minimum thickness satisfying the deflection and buckling constraints is calculated. Next, the least favorable deflection is minimized by computing an optimal ply angle for the value of h computed in the previous step.

Miki et al. (1989, 1990a, 1990b, 1993, 1997), Yang and Ma (1989), Shao et al. (1991, 1992, 1993), Murotsu et al. (1994), Boyer et al. (1997), and Nozomu et al. (1998) applied reliability methods for the optimum design of laminated composites subject to load and material uncertainties. Nonprobabilistic techniques were also employed to study the behavior of structures subject to uncertain load conditions when there is no information available on statistical distributions and probability functions of relevant quantities.

Moreover, the laminate is optimized with respect to ply angle θ subject to a buckling constraint. In terms of the nondimensional quantities, the weight W = h, and consequently the laminate thickness h, is used as the design objective. Let wi(x, y; εi) denote the least favorable deflection at (x, y) subject to the uncertainty level εi. fm Page 42 Monday, June 21, 2004 10:51 PM where Pi is the in-plane load for the i-th loading case and Pcr is the buckling load. The initial deflection under i-th loading is expanded in terms of a (i ) i) Fourier series with the coefficients B (mn = Bmn + B˜ mn , where Bmn refers to the (i ) ˜ deterministic initial deflection and Bmn to the uncertain initial deflection.

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