When several concentrated or distributed loads are applied on a circular or elliptic ring and are increased proportionally, the ring first yields at some locations and, at the final stage, collapses as a mechanism. The collapse of a ring cannot take place until the number of yield joints developed in the ring exceeds its number of redundancies. In this paper we shall study the configuration of the collapse mechanism, the corresponding collapse load of rings, and the deformations of a circular ring immediately before plastic collapse takes place. For rings in which the heights of the cross section are small compared with their overall dimensions, the stresses are essentially due to flexure of the ring segments when the ring is under the action of concentrated forces. The stresses due to axial and shear forces are relatively small and may be neglected. Furthermore, at the instant of plastic collapse, the deformation of the ring as a mechanism is unlimited if we assume no strain-hardening of the material in the plastic state. In parts of the ring where the sections are still elastic, or partly elastic and partly plastic, the deformation is contained and may be neglected. Thus the change in geometry of the ring is disregarded; the ring is considered as a mechanism made of perfectly rigid members connected by yield joints that offer constant resisting moments. The positions of these yield joints and the corresponding collapse load are found for several typical loading conditions.