Bellows which is utilized for an expansion joint is being widely used in the exhaust systems of vehicle and the piping system of nuclear reactor plants. A shape optimal design of U and annular-typed bellows under axial loads is presented and the results are compared with the existing ones. A theoretical structural analysis employs existing theory using the energy method for the toroidal section, and symmetrical bending of a circular plate augmented by thick-walled cylinder analysis for the annular plate which connects two toroidal sections. The procedure for shape optimization adopts λ, the ratio of bellows radius to corrugation radius, and the number of corrugation and the thicknesses of the structure as design variables. Different radii are assinged to the inner and outer toroidal sections, and it is permissible to vary the average thickness of the two toroidal sections and the interconnecting annular plate independently. The total displacement and weight are considered simultaneously by a multi-objective function. The constraints are imposed on the stress of the structure using maximum distortion energy theory. The displacement and buckling constraint are also considered to prevent overlap of corrugation and the buckling of the toroidal section. The characteristics of the bellows structure is investigated through the optimal design process.