Bifurcations in von Kármán problem for rectangular, thin, elastic plate resting on elastic foundation of Winkler type

Abstract

This research is devoted to a study of stability problem of linearly elastic isotropic thin rectangular plate resting on linearly elastic foundation (of Winkler type). The plate is simply supported along all four edges and is subjected to a compressive loading of magnitude λ > 0 evenly distributed along two parallel edges, see Figure 1.1. If the loading parameter λ has a small value, then the plate is not deformed and flat (compared to Euler problem of elastic rod). If the loading parameter λ increases to the critical value λ1 (“Euler critical load,” “buckling load”), the plate bifurcation holds,which means that the plate buckles to the bent form. The main purpose of this paper is to give a precise mathematical description of the plate bifurcation. Let us consider corresponding mathematical model. The Cartesian coordinates system (u, v,w) presented in Figure 1.1 is assumed. Themiddle surface of the not buckled thin plate is presented in Cartesian coordinates (u, v,w) by the rectangle