Rayleigh–Ritz analysis for localized buckling of a strut on a softening foundation by Hermite functions

This paper proposes a Rayleigh–Ritz procedure for localized buckling of a strut on a non-linear elastic foundation. Firstly, the deflected shape of a strut is expanded into a series of Hermite orthogonal functions, which are proved energy-integrable in an infinite region. Secondly, the errors of the numerical integrations of Hermite functions on the infinite region are investigated and the suitable integral limit is proposed. Through the numerical investigation, it is demonstrated that the first thirty Hermite functions are usually enough to approximate the localized buckling pattern. The proposed method overcomes the disadvantages of the traditional methods, in which the trial functions in either Rayleigh–Ritz or Galerkin analysis are based on the perturbation analyses of the corresponding non-linear differential equation.