The Dual of Foulkes and Prager‐Shield Criteria

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Abstract
After reviewing existing duality principles for structures with continuously variable cross-section (i.e. structures without segmentation), optimality criteria and duality principles are presented for structures with segment-wise constant cross-section. It is shown that the optimal solution for the latter is associated with a displacement field in which the average strain value for each segment is proportional to the subgradient of the specific cost function with respect to the maximum generalized stress value over that segment. Moreover, the dual problem consists of maximizing the difference of two terms. The first of these is the integral of the product of loads and displacements and the second is the sum of the products of segment sizes (length or area) and the mean ``complementary cost.'' The above principles are illustrated with examples and the optimal solutions are verified by independent methods. It is shown that for special cases the proposed optimality criteria reduce to those by Prager-Shield, Masur and Foulkes.

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