Treatment of elastically restrained ends for beam buckling in finite difference, microstructured and nonlocal beam models

This paper presents the treatment of elastically restrained ends for the axially, loaded beam-buckling problem for the central finite difference beam model, the microstructured beam model, and Eringen’s nonlocal continuous beam model. The equivalence between the central finite difference beam model and the microstructured beam model is established herein, and these equivalent systems are regarded as belonging to one class of discrete systems since they become indistinguishable. Also, the continualized form of the discrete system is obtained by adopting the continualization method that is based on an exponential displacement function. Three approaches are then proposed for matching the discrete system with Eringen’s nonlocal continuous system for the beam-buckling problem. The approaches depend on the assumption made on the constancy/or varying Eringen’s small length scale coefficient e₀ as well as which one of the discrete or continuum system is taken as the reference system.