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The paper discusses a methodology for the evaluation of shakedown and ratchet limits for an elastic perfectly plastic solid subjected to mechanical and thermal cycles of loading. The steady cyclic state is characterised by a minimum theorem that contains the classical shakedown theorems as a special case. For a prescribed class of kinematically admissible strain rate histories, the minimum of the functional is found by a programming method, the Linear Matching Method, which converges to the least upper bound. Three examples are given for a finite element implementation, rolling contact on a half-space, the behaviour of a complex heat exchanger and the behaviour of a regular particulate metal matrix composite subjected to variable temperature.