In this paper, a methodology for finite-element elastoplastic analyses of structures subjected to cyclic loading, based on a new form of endochronic (internal time) plasticity theory, is presented. In the present version of the endochronic theory, the rate forms of the stress/strain relations (stated here explicitly for plane stress, plane strain, and three-dimensional cases) are entirely analogous to those of the classical (isotropic or kinematic hardening) plasticity theory and, thus, are no more difficult or expensive to implement in computational algorithms. The problem of a double-edge-cracked panel, subjected to monotonic loading as well as to zero-to-tension or tension-compression cyclic external loading, is analyzed. The elastoplastic stress/strain fields at the root of the notch, for each of these loading conditions, are studied in detail. These results are compared and contrasted with those presented recently by Valanis and Fan, and the noted discrepancies are discussed. This paper is limited to considerations of small strains and small displacements only.