Transient and steady state stability of cylindrical shells under harmonic axial loads

The transient and steady-state instability of an axially loaded cylindrical shell is discussed in the present paper. Donnell's shallow shell theory is used and the shell spatial discretization is obtained by the Galerkin method. First, an alternative vision of the buckling problem through the evolution and erosion of safe basins using energy and geometric considerations is presented, using an autonomous conservative low dimensional but qualitatively consistent model. Then, the response of the corresponding dissipative system is studied in terms of transient and steady-state behavior. Based on these results, the behavior of the shell under harmonic axial load is investigated through the evolution of basins of attraction. Both parametric instability and escape from a safe pre-buckling well are considered. It is shown that damping has a beneficial influence on the magnitude of the steady-state basins of attraction but must be considered with care when transient stability is of concern. Basin boundaries of forced dissipative systems usually become fractal leading to a complex topological structure and swift erosion under increasing forcing amplitude. We argue that the analysis of the evolution of safe steady-state and transient basins and the specification of appropriate measures of their robustness is an essential step in the derivation of safe design procedures for multi-well and multi-attractor systems. The methodology presented in this work is particularly suited to structural systems liable to unstable bifurcation.