The dynamic buckling loads of some imperfection-sensitive elastic structures subjected to slowly varying time dependent loading are determined using perturbation procedures. First, we consider an elastically imperfect column resting on a softening nonlinear elastic foundation. The governing differential equation has two small parameters. We determine the dynamic buckling load of this column subjected to the stipulated loading for three different cases. The cases are when the small parameters are not related and when they are related first linearly and next quadratically in some way. This idea is next applied to an elastically imperfect spherical cap and the dynamic buckling load of the cap subjected to a slowly varying time dependent loading is determined. The result shows, among other things, that for the case of the cap, the coupling term has no significant contribution to the initial post-buckling phenomenon. By assuming, in the results, that the slowly varying loading function is numerically unity, we obtain the associated step loading results for both the column and the spherical cap. These latter results confirm existing results for columns under step loading and establish new ones for the spherical cap.
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