Optimal design and relaxation for reinforced plates subject to random transverse loads

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We consider a Kirchhoff plate subject to a random transverse load. We reinforce the plate with stiffeners. For a prescribed area fraction of stiffeners we seek their optimal layout so that on average, the plate is as stiff as possible with respect to the random load. We provide a mathematical formulation of this problem. To ensure the existence of a solution we relax the problem to include generalized designs. The relaxation procedure rests upon the derivation of new optimal lower bounds on the compliance energy for the effective elasticity of composites made from stiffeners on multiple scales. Our bounding method follows the program of Hashin and Shtrikman. However our method is novel as no fictitious comparison material is used in the derivation. This relaxation is new and is the extension to the two-dimensional setting of the relaxation given by Cheng and Olhoff (Int. J. Solids Struct., 17 (1981) 305 23, 795-810) for one-dimensional plate problems, when the plate thickness is allowed to take two values. The relaxed formulation of the problem can be solved numerically. © 1994.

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Probabilistic Engineering Mechanics

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