On stability analyses of three classical buckling problems for the elastic strut

Abstract

It is common practice in analyses of the configurations of an elastica to use Jacobi's necessary condition to establish conditions for stability. Analyses of this type date to Born's seminal work on the elastica in 1906 and continue to the present day. Legendre developed a treatment of the second variation which predates Jacobi's. The purpose of this paper is to explore Legendre's treatment with the aid of three classical buckling problems for elastic struts. Central to this treatment is the issue of existence of solutions to a Riccati differential equation. We present two different variational formulations for the buckling problems, both of which lead to the same Riccati equation, and we demonstrate that the conclusions from Legendre and Jacobi's treatments are equivalent for some sets of boundary conditions. In addition, the failure of both treatments to classify stable configurations of a free-free strut are contrasted. © The Author(s) 2011.