In this paper we consider a thin narrow rectangular isotropic plate subjected to a small surface load and supported laterally by a continuous nonlinear elastic foundation. The both short ends of plate are clamped while the longitudinal sides are completely free, so that their points can move along the boundary, along the normal to the boundary, and in a vertical direction. At initial time the uniformly distributed in-plane compressive stresses are suddenly applied to the short ends in the longitudinal direction. Our goal is to find the asymptotic formulas for values of static and dynamic buckling load in the case of the narrow elastic plate and estimate their values as function of the imperfection parameter. We apply the geometrically nonlinear theory for the thin rectangular isotropic plate laterally supported by the continuous softening or stiffening foundation to formulate an associated nonlinear spectral problem for the load parameter. This problem contains a small natural parameter δ - the ratio of the width of the rectangular plate to its length and can be integrated using the asymptotic method developed in the work by Srubshchik, Stolyar and Tsibulin [1]. Accordingly we approximate the solution of the original problem by the leading term of the finite expansion in δ which is described by the motion equations of an axially compressed elastic column on the nonlinear continuous elastic foundation which has only one spatial dimension and can be investigated more comprehensively. The formulas for asymptotic values of the static and dynamic buckling compressive loads are obtained by means of the perturbation theory and by one-term Fourier's approximation respectively. The specific numerical results for these asymptotic values are presented.
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