The critical external pressure and buckling modes of a closed circular cylindrical shell joined at an edge with an annular plate are obtained by means of asymptotic methods. Two tipe of buckling, corresponding to narrow and wide rings, are analyzed. In both cases approximate solution of shell equations is represented as a sum of slowly varying functions and edge effect integrals. It is shown that a narrow ring can be considered as circular beam. For a wide ring joined with a cylindrical shell the couple buckling problem in the first approximation is reduced to the eigenvalue problem describing the plate buckling. An approximate solution of the last problem is obtained with the help of the perturbation approach. The asymptotic results are in good agreement with the numerical ones obtained by finite element analysis.