Journal of Elasticity

, Volume 105, Issue 1–2, pp 117–136 | Cite as

On Stability Analyses of Three Classical Buckling Problems for the Elastic Strut

Open Access
Article

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.

Keywords

Elastica Variational Methods Elastic Stability Buckling 

Mathematics Subject Classification (2000)

34B15 49B10 74G60 74K10 93C10 

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Copyright information

© The Author(s) 2011

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringUniversity of California at BerkeleyBerkeleyUSA

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