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

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/(pi3B2) = (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