The invariant manifold of beam deformations Part 1: the simple circular rod

Article Received: 05 February 1990 DOI :
10.1007/BF00041769

Cite this article as: Roberts, A.J. J Elasticity (1993) 30: 1. doi:10.1007/BF00041769
Abstract The subcentre invariant manifold of elasticity in a thin rod may be used to give a rigorous and appealing approach to deriving one-dimensional beam theories. Here I investigate the analytically simple case of the deformations of a perfectly uniform circular rod. Many, traditionally separate, conventional approximations are derived from within this one approach. Furthermore, I show that beam theories are convergent, at least for the circular rod, and obtain an accurate estimate of the limit of their validity. The approximate evolution equations derived by this invariant manifold approach are complete with appropriate initial conditions, forcing and, in at least one case, boundary conditions.

Key words elastic beam theory subcentre invariant manifolds dynamical systems

AMS (MOS) Classification 73C02 73C10 35A35 58G40

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Authors and Affiliations 1. Department of Applied Mathematics The University of Adelaide Adelaide South Australia