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Figure 1: First two axi-symmetric and two asymmetric modes of buckling for a compressed strut with guided-guided end conditions: n is the circum- ferential mode number and m the axial mode number. For slender enough cylinders, the n = 1, m = 1 mode is the first mode of buckling.
From:
Riccardo De Pascalis (1), Michel Destrade (2) and Alain Goriely (3)
(1) CNRS, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005 Paris, France
(2) School of Electric, Electronic and Mechanical Engineering, University College Dublin,
Dublin 4, Ireland.
(3) OCCAM, Institute of Mathematics, University of Oxford, UK
“Nonlinear Correction to the Euler buckling formula for compressed cylinders with guided-guided end conditions”, Journal of Elasticity, Vol. 102, No. 2, pp 191-200, February 2011,
DOI: 10.1007/s10659-010-9265-6
ABSTRACT: Euler’s celebrated buckling formula gives the critical load N for the buckling of a slender cylindrical column with radius B and length L as
N/(π3B2) = (E/4)(B/L)2,
where E is Young’s modulus. Its derivation relies on the assumptions that linear elasticity applies to this problem, and that the slenderness (B/L) is an infinitesimal quantity. Here we ask the following question: What is the first nonlinear correction in the right hand-side of this equation when terms up to (B/L)4 are kept? To answer this question, we specialize the exact solution of incremental non-linear elasticity for the homogeneous compression of a thick compressible cylinder with lubricated ends to the theory of third-order elasticity. In particular, we highlight the way second- and third-order constants —including Poisson’s ratio— all appear in the coefficient of (B/L)4.
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