The investigation is concerned with axially stiffened cylindrical shells under axial compression. In these structures, two modes are of particular interest, namely on overall long-wave and a local shortwave buckling mode. Numerical results show that in some cases bending of the stringers in the local mode postbuckling solution plays an important role. Exclusion of this effect may lead to an overestimation of the carrying capacity of the shell. Furthermore, it is found that apparently reasonable approximations to the postbuckling fields associated with both the local and the overall mode, as well as with the overall mode alone, may lead to inexact values of the buckling load.