On Complementary Variational Principles for the Linear Theory of Plates

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Abstract
A set of complementary variational principles is developed for the linear theory of plates which yields as special cases the minimum potential energy, minimum complementary energy, and Hellinger-Reissner and Hu-Washizu-type variational principles for the linear theory of plates. The governing (biharmonic) equation is decomposed into a set of lower order (differential) equations involving the deflection, slopes, curvatures, moments, and shear forces. A general variational principle is constructed using Vainberg's theory for the set so that all the dependent variables can be varied independently. When one or more equations of the set are satisfied identically, the lower bounds on the functionals are also established. The theory developed herein is generalized to operator equations of the form Lu = fwith the assumption that the linear operator L is decomposable into lower order linear differential operators. It is believed that the variational principles developed herein can be used for the approximate analysis of plate problems (with relaxed continuity requirements for the trial functions).

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