A Duality Theorem for Plastic Plates

Limit analysis studies the asymptotic behavior of elastic-plastic materials and structures. The asymptotic material properties exist for a class of ductile metals and are designed into optimal structural members such as I-beams and composite plates. The analysis automatically ignores the relatively small elastic deformations. Classical lower and upper bound theorems in the form of inequalities are mathematically incomplete. A duality theorem equates the greatest lower bound and the least upper bound. Although some general statement has been made on the duality relation of limit analysis, each yield criterion will lead to a specific duality theorem. The duality theorem for a class of plastic plates is established in this paper. The family of ~-norms is used to represent the yield functions. Exact solutions for circular plates under a uniform load are obtained for clamped and simply supported boundaries as examples of the specific duality relations. Two classical solutions associated with Tresca and Johansen yield functions are also presented in the spirit of their own duality relations, providing interesting comparison to the new solutions. A class of approximate solutions by a finite element method is presented to show the rapid mesh convergence property of the dual formulation. Complete and general forms of the primal and dual limit analysis problems for the ~-family plates are stated in terms of the components of the moment and curvature matrices.