Limit point buckling of a finite beam on a nonlinear foundation

TL;DR

This paper analyzes the buckling behavior of a finite beam on a nonlinear foundation, revealing how imperfections and foundation stiffness influence limit points and stability, with implications for structural engineering.

Contribution

It provides an analytical framework for understanding limit point buckling in finite beams on nonlinear foundations, highlighting the effects of imperfections and foundation stiffness.

Findings

Limit points depend on imperfection size and foundation stiffness.

Localized buckling does not occur in finite beams.

Restoring force models significantly affect decay/growth rates.

Abstract

In this paper, we consider an imperfect finite beam lying on a nonlinear foundation, whose dimensionless stiffness is reduced from $1$ to $k$ as the beam deflection increases. Periodic equilibrium solutions are found analytically and are in good agreement with a numerical resolution, suggesting that localized buckling does not appear for a finite beam. The equilibrium paths may exhibit a limit point whose existence is related to the imperfection size and the stiffness parameter $k$ through an explicit condition. The limit point decreases with the imperfection size while it increases with the stiffness parameter. We show that the decay/growth rate is sensitive to the restoring force model. The analytical results on the limit load may be of particular interest for engineers in structural mechanics