In cold pilgering of tubes, a material element undergoes a series of small incremental deformations (≈100 strokes), alternatively under tensile and compressive stresses. This complex history sometimes results in surface damage, seemingly by low-cycle fatigue. Prior to studying the resistance of diverse potential materials to this kind of complex, non-proportional multi-axial, and non-periodic cycling, a thorough mechanical analysis of the stress states is necessary: the finite element method (FEM) software Forge3(®) has been used, with updated Lagrangian formulation due to the transient character of strains and stresses. The process is periodical, except for the ends of a given preform, which are cut off afterwards. One stroke only should thus be sufficient to analyse the whole process, provided the correct initialisations are done in terms of shape, strains and stresses, but these are parts of the unknown of the problem. This point will be particularly addressed in the following, where it is shown that in the non-work hardening case at least, simulating three strokes leads to an invariant geometry and state of stress, starting from a reasonable estimate of the geometry. Strains and stresses thus obtained will be discussed in detail, together with their probable consequences on the damage and fatigue of the material, to be later correlated with defects. © 2002 Elsevier Science B.V. All rights reserved.