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Nonlinear correction to the Euler buckling formula for compressed cylinders with guided-guided end conditions. (English) Zbl 1273.74104
Summary: 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 $N/(\pi^3B^2)=(E/4)(B/L)^2,$ 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 non-linear 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 $$(B/L)^{4}$$.

##### MSC:
 74G60 Bifurcation and buckling 74K10 Rods (beams, columns, shafts, arches, rings, etc.)
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