Localized elastic buckling: non-linearities versus inhomogeneities

Buckling of an elastic beam-column resting on an inhomogeneous Winkler foundation is revisited from the standpoint of multiscale asymptotic analysis. Despite the complexity posed by the presence of non-linearity and variable coefficients in the governing eigenproblems, our analytical strategy is shown to capture successfully both the linear and the incipient post-buckling regimes. In particular, the present investigation produces a non-linear amplitude equation with variable coefficients that accounts for the spatial inhomogeneity in the normal restoring force of the foundation. Localization is here due to the spectral properties of the corresponding linearized buckling operator, whereas the role of the non-linearities remains confined to that of amplifying such behaviour.