Mechanical engineering structures and structural components are often subjected to cyclic thermomechanical load-ing which stresses their material beyond its elastic limits well inside the inelastic regime.Depending on the level of loading inelastic strains may lead either to failure,due to low cycle fatigue or ratcheting,or to safety,through elastic shakedown.Thus,it is important to estimate the asymptotic stress state of such structures.This state may be deter-mined by cumbersome incremental time-stepping calculations.Direct methods,alternatively,have big computa-tional advantages as they focus on the characteristics of these states and try to establish them,in a direct way,right from the beginning of the calculations.Among the very few such general-purpose direct methods,a powerful direct method which has been called RSDM has appeared in the literature.The method may directly predict any asymptotic state when the exact time history of the loading is known.The advantage of the method is due to the fact that it addresses the physics of the asymptotic cycle and exploits the cyclic nature of its expected residual stress distribution.Based on RSDM a method for the shakedown analysis of structures,called RSDM-S has also been developed.Despite most direct methods for shakedown,RSDM-S does not need an optimization algorithm for its implementation.Both RSDM and RSDM-S may be implemented in any Finite Element Code.A thorough review of both these methods,together with examples of implementation are presented herein.