Nonlinear Correction to the Euler Buckling Formula for Compressed Cylinders with Guided-Guided End Conditions

Journal article

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 \[ N / (\pi^3 B^2) = (E/4)(B/L)^2, \] 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 $(B/L)^4$.

Authors: Pascalis, R; Destrade, M; Goriely, A

• Publication status: Published
• Journal: Journal of Elasticity, 102 (2011) 191-200
• Volume: 102
• Issue: 2
• Pages: 191-200
• Publication date: 2013-02-05
• DOI: 10.1007/s10659-010-9265-6
• EISSN: 1573-2681
• ISSN: 0374-3535
• Language: English
• Keywords: cond-mat.soft