This paper presents analysis of elastic stress and torsion buckling of U-shaped bellows using ring elements. The expansion joint is considered to be composed of the two toroidal sections and inner-connecting annular plates. The general thin shell theory is employed to derive strain-displacement relations of shells and plates, valid for any loadings. Numerical examples under internal pressure or axial loading are described and compared with the results of existing appropriate analysis. The fundamental aspects of torsional buckling, which have not been studied previously, will also be investigated.