Buckling of Timoshenko beam under two-parameter elastic foundations

This paper exposes buckling solutions of a plane, quasi-static Timoshenko beam with small transformation
subjected to a longitudinal force and surrounded by an elastic wall modeled by two-parameter elastic
foundations. A non-dimensional analysis of associated Haringx and Engesser model is performed and buckling
stress and shape are exposed analytically. Relations for rigidity of the wall and buckling solutions were made
for different regimes and for both models using asymptotic approach. Introducing the yield limit gives a simple
criterion in terms of stiffness foundation and slenderness ratio for which buckling or irreversible transformation
occur.