In the design of a modern lightweight structure, it is of technical importance to assure its safety against the buckling under the applied loading conditions. For this issue, the determination of the critical load in an ideal condition is not sufficient, but it is further required to clarify the post-buckling behavior, that is, the behavior of the structure after passing through the critical load. One of the reasons is to estimate the effect of practically unavoidable imperfections on the critical load and the second is to evaluate the ultimate strength to exploit the load-carrying capacity of the structure. For the buckling problem of circular cylindrical shells under axial compression, a number of experimental and theoretical studies have been made by many researchers. In the case of the very thin shell that exhibits elastic buckling, experimental results show that after the primary buckling, secondary buckling takes place accompanying successive reductions in the number of the circumferential waves in each mode change on one-by-one step. In this paper we traced this successive buckling of circular cylindrical shells using some of the general purpose implicit FEM codes currently available. For geometrically nonlinear static problems including buckling and post-buckling, we carried out our studies with two approaches; one is to use the arc length method (the modified Riks method), and the other is stabilizing with the aid of (artificial) damping especially for the local instability. Our analysis procedure consists of the following 2 steps. Before reaching the point exhibiting the comparatively stable state after the primary buckling, the arc length method is applied. After that point, the artificial damping is applied. The results simulate unstable successive buckling and show good agreement with experiments.

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