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.

Stability of Heterogeneous Beams with Three Supports—Solutions Using Integral Equations

“…If the stability problem is considered f z = 0. The related eigenvalue problem is, therefore, governed by ODE d 4 w…”

“…Buckling of structures and various structural members has been subject to research for a long time and is still a popular topic [1][2][3][4]. When it comes to the stability of beams or columns, the number of available works is numerous.…”

Stability of twisted rods, helices and buckling solutions in three dimensions

“…These filamentary elastic structures have widespread applications in engineering and biology, examples of which include cables, textile industry, DNA experiments, collagen modelling etc [1,2]. One is typically interested in the equilibrium configurations of these rod-like structures, their stability and dynamic evolution and all three questions have been extensively addressed in the literature, see for example [3,4,5,6] and more recently [7,8,9]. However, it is generally recognized that there are still several open non-trivial questions related to three-dimensional analysis of rod equilibria, inclusion of topological and positional constraints and different kinds of boundary conditions.…”

“…The first problem is centered around the stability of the trivial solution or the unbuckled solution in three dimensions (3D), subject to a terminal load and controlled end-rotation with three different types of boundary conditions. This can be regarded as a generalization of the recent two-dimensional analysis of three classical elastic strut problems in [7]. We work with the Euler angle formulation for the rod geometry and work away from polar singularities; this excludes rods with self-intersection or self-contact but still accounts for a large class of physically relevant configurations in an analytically tractable way [3,10,11].…”

“…In particular, the theory of elastic rods has been very successful in the context of modelling DNA molecules, subject to different orientational, positional and topological constraints [4,5]. The theory typically addresses questions related to the equilibria of elastic rods, their stability and dynamic evolution, see for example [6][7][8][9] and more recently [10][11][12]. However, the 3D analysis of rod equilibria and constrained problems still present new challenges, potentially leading to new applications.…”