Abstract: Based on the fundamental differential equations, and with the application of the differential relationship the internal force and displacement, this paper investigates all-roundly the bending and torsion problems of non -ho nicgeneous elastic circular ring with variable cross-section under vertical loods. The general integral formulae arc obtained for the bending and torsional stiffness expressed in the formulae arbitrary functions, through appropriate mathematical transformations. The general solutions for two special case i.e the cicrular ring with stepped and with con stand cross-section, are derived. Lastly, for illustration of the concrete application of these formulae, an expositive example is given and soloved.