Dynamic stability of doubly curved orthotropic shallow shells under impact

A methodology for the stability analysis of doubly curved orthotropic shell s with simply supported boundary condition and under impact load from the v iewpoint of nonlinear dynamics is studied. The nonlinear governing differen tial equations are derived based on a Donnell-type shallow shell theory, an d the displacement is projected onto the space spanned by the eigenfunction of the linear operator of the motion equation. To analyze the influence of each single mode pair on the response to impact loading, only one term com posed of two half-waves is used in developing the governing equation, where as the first mode pair, which is close to the membrane state, is used in th e numerical examples. For the first case analyzed, it turns out that two ce nters and one saddle mill occur if damping is ignored, whereas the two cent ers become two focuses and the saddle point remains if damping is considere d. The nonlinear behavior also has been investigated by neglecting the infl uence of inertia and damping, and the results show that two saddle-node bif urcations mill occur under certain conditions.