In this paper, a methodology to determine the energy-dissipative characteristics of circular dampers under cyclic in-plane loading is proposed based on the inelastic stress analysis of curved beams with bending and shear coupling. Taking advantages of symmetry, a quarter of the circular damper in form of a curved cantilever beam with shear and bending applied simultaneously to the free end is considered in the analysis. The yield load, the corresponding yield displacement, and consequently the initial stiffness of the damper can be found analytically using the classical theory of elasticity in polar coordinate. The inelastic force-displacement relationship of the curved beam under monotonically increasing (or decreasing) load is obtained by performing the inelastic stress analysis numerically in an incremental manner. The force-displacement relationship so determined is conceived as the skeleton or backbone curve from which the hysteretic loops of the damper can be reconstructed by adopting Masing's rule. This hysteresis characterizes the mechanical behaviour of the circular damper with information of initial stiffness, yield strength, post-yielding stiffness, ductility and energy-dissipative capacity at a specified deformation. The proposed methodology may serve as an effective tool in the preliminary design stage of in-plane type metallic yielding damper to meet the desired specifications before fabrication for component test and practical use.