The prediction of damage and wear of materials under repeated moving contacts requires first the determination of the limit response in terms of stress and strain. The simulation necessary to calculate the limit response could be performed with classical finite elements and an incremental treatment of the loading history, resulting in lenghtly are repeated calculations. For steady-state moving contacts, the stationary methods allows a quick evaluation of the stabilized state. These procedures cannot be used in the case of an arbitrary cyclic loading resulting, for example, from small oscillatory contacts. In this paper, a new finite element approach is developed for structures subjected to an arbitrary cyclic loading. It is based on an original intergration scheme which allows direct determination of the stabilized state of stresses, strains and internal parameters. This asymptotic state could be either an elastic shakedown, a plastic shakedown or ratchetting for an elastic-plastic material. Elastic and plastic shakedown states are easily obtained with this method. Applications to fretting fatigue and wear in relation to material fatigue properties are presented.