Three-mode bendings of a compressed beam on a double elastic foundation in the modified Vlasov–Leontyev model

In this paper, we present an algorithm for approximate calculation and analysis of supercritical bendings of a longitudinally compressed, elastic beam on a double elastic foundation in the modified Vlasov–Leontiev model. The procedure is based on the Poincaré–Lyapunov–Schmidt variational method, which allows one to reduce the analysis of supercritical deformations of a beam to the analysis of critical points of the key function on the finite-dimensional space of key variables. The Lyapunov–Schmidt method allows one to calculate supercritical bendings of the beam, to determine the stability of bifurcating states, and to analyze the structure of the caustic (discriminant set) in the space of control parameters. The basic idea is the reduction of the problem on bendings of a beam to the discriminant analysis of branching of critical points of a polynomial in three variables (the principal part of the Lyapunov–Schmidt key function).