Buckling of an elastic ring forced by a periodic array of compressive loads

We use an analytical technique based on nonsmooth coordinate transJbrmations to study discreteness effects in the post-buckling state of a circular ring loaded by a periodic array of compressive point loads. The method relies on eliminating singularities due to the point loads in the governing equations, at the expense of increasing the dimensionality of the problem. As a result, the original nonsmooth governing equations are tran.sformed to a larger set of equations with no singularities, together with a set of “smoothening” boundary conditions. The transformed equations are solved by expressing the variables in regular perturbation expansions, and studying an hierarchy of boundary value problems at successive orders of approximation; these problems can be asymptotically solved using techniques from the theory of smooth nonlinear or parametrically varying dynamical systems. As a result, we model analytically discreteness effects' in the post-buckling states of the ring, and estimate the effect of the discrete load distribution on the critical buckling loads. This effect is found to be of very low order, in agreement with numerical results reported in an earlier work.