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The Spatial Complexity of Localized Buckling in Rods with Noncircular Cross Section

G. H. M. van der Heijden, A. R. Champneys and J. M. T. Thompson
SIAM Journal on Applied Mathematics
Vol. 59, No. 1 (Sep. - Oct., 1998), pp. 198-221
Stable URL: http://www.jstor.org/stable/118379
Page Count: 24
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The Spatial Complexity of Localized Buckling in Rods with Noncircular Cross Section
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Abstract

We study the postbuckling behavior of long, thin elastic rods subject to end moment and tension. This problem in statics has the well-known Kirchhoff dynamic analogy in rigid body mechanics consisting of a reversible three-degrees-of-freedom Hamiltonian system. For rods with noncircular cross section, this dynamical system is in general nonintegrable and in dimensionless form depends on two parameters: a unified load parameter and a geometric parameter measuring the anisotropy of the cross section. Previous work has given strong evidence of the existence of a countable infinity of localized buckling modes which in the dynamic analogy correspond to N-pulse homoclinic orbits to the trivial solution representing the straight rod. This paper presents a systematic numerical study of a large sample of these buckling modes. The solutions are found by applying a recently developed shooting method which exploits the reversibility of the system. Subsequent continuation of the homoclinic orbits as parameters are varied then yields load-deflection diagrams for rods with varying load and anisotropy. From these results some structure in the multitude of buckling modes can be found, allowing us to present a coherent picture of localized buckling in twisted rods.

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