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In all cases, a two-stage optimization problem needs to be solved. , anti-optimization. In general, these two problems lead to a nested optimization problem and can be solved iteratively (Lombardi and Haftka, 1998; Venter and Haftka, 2000; Zingales and Elishakoff, 2001). A study of the buckling problem under design uncertainties is given in Elishakoff (2000). In the present example a symmetric laminate of thickness H under in-plane loads Nx and Ny is optimized for maximum strength, taking the fiber orientations as design variables (optimization problem) subject to uncertain inplane loads that can produce the least favorable loading conditions for any given lay-up (anti-optimization problem) (Adali and Verijenko, 2000).

There have been various approaches and techniques to deal with the uncertainties in the optimization of composites. 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.

The buckling load remains close to its maximum value while P* increases with increasing a/b. Recent studies on the buckling optimization of composite laminates include those by Joshi and Biggers (1996), Walker et al. (1996), Moh and Hwu (1997), Walker (1998, 1999), Hu and Chen (1999), and Khong (1999). 93 Source: S. , 35: 117–130, 1996, with permission from Elsevier Science. 4 Optimization under Material, Load, and Geometric Uncertainties Optimal design in structures, and in particular composite structures, under deterministic conditions has the drawback that the design becomes unreliable if these conditions change even slightly under service conditions.

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