This paper presents plastic mechanism analyses of circular tubular members under cyclic loading. In particular, it provides new methods of analyses for circular hollow sections subjected to a constant amplitude cyclic pure bending (CCPB) and a large axial compression-tension cycle (ACTC). The procedure described herein for CCPB considers both prebuckling deformation and local buckling. The prebuckling deformation was simulated using a progressively deforming elliptical cross-section which was observed during the test. The local buckling analysis was performed using a rigid plastic mechanism analysis. Good agreement was found between the predicted and measured hysteresis. However, the present model over-estimates the strength and absorbed energy. For the ACTC, good agreement was found between the predicted and measured collapse curves within the compression half-cycle. More work is needed to model the effect of tension developed during the remaining of the loading cycle. The present models consider the actual local buckling deformation of the cross-section and also avoid the complexity involved in the numerical analyses since they provide the most commonly used closed-form solutions.