On kinematic method in shakedown theory: I. Duality of extremum problems

A kinematic method for determining the safety factor in shakedown problems is developed. An upper bound kinematic functional is defined on a set of kinematically admissible time-independent velocity fields. Every value of the functional is an upper bound for the safety factor. Using convex analysis methods, conditions are established under which the infimum of the kinematic upper bounds equals the safety factor, in particular, conditions under which it is sufficient to consider only smooth velocity fields for the safety factor calculation. The method generalizes that recently proposed for the case of spherical yield surfaces by Kamenjarzh and Weichert. The extension covers a wide class of yield surfaces and inhomogeneous bodies. A shakedown problem for a beam subjected to a concentrated load is considered as an example.