On the Solid Plate Paradox in Structural Optimization∗

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Abstract
This paper discusses the minimum weight design of solid plastic plates subject to constraints on the highest and lowest allowable values of the plate thickness. It is shown that the maximum thickness constraint alone does not ensure a smooth global minimum weight solution because the least weight is furnished, in the limit, by a grillage-like continuum consisting of a dense system of ribs of infinitesimal spacing and uniform depth. The optimal layout of such continua has already been determined for most loading and boundary conditions by the first author and Prager and is found to be the same for (a) plastic limit design, (b) elastic stress design, (c) design for given compliance, and (d) design for given fundamental frequency. Two refinements of the above layout theory are also considered in the current paper. One formulation takes into consideration the weight savings due to rib intersections in high density grillages. The other development deals with minimum as well as maximum thickness constraints and establishes criteria for the occurrence of a combination of a solid plate and one-way ribs in some regions of the optimal solution. The above theory is illustrated with an example.

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