Remarks on some properties of conic yield restrictions in limit analysis

A major difficulty when applying the kinematic theorem in limit analysis is the derivation of expressions of the dissipation functions and the set of plastically admissible strains. At present, no standard methodology exists. Here, it is shown that they can be readily obtained, provided that the yield restriction can be rewritten as an intersection of cones, and that the expression defining the dual cones is available. This is always possible for the case of self-dual cones and some other classes, and covers many of the well-known criteria. Therefore, a difficult obstacle with respect to the use of the kinematic theorem in conjunction with any numerical method can be overcome. The methodology is illustrated by giving the expressions of the dissipation functions for various conic yield restrictions. A special emphasis is given on upper bound finite element limit analysis. Taking advantage of duality in conic programming, we can obtain the dual problem, where knowledge of the dual cone is not necessary. Therefore, this formulation is feasible for any cone. Finally, it is interesting that the form of the dual problem, for varying yield strength within the finite element, differs from that presented in other papers.