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

We rise an analytical technique based on nonsmooth coordinate transformatio ns to study discreteness effects ill the post-buckling state of a circular ring loaded by a periodic array of compressive point lends. The method reli es on eliminating singularities due to the point loads in the governing equ ations, at the expense of increasing the dimensionality of the problem. As a result, the original nonsmooth governing equations are transformed to a l arger set of equations with no singularities, together with a set of "smoot hening" boundary conditions. The transformed equations are solved by expres sing the variables in regular perturbation expansions, and studying ail hie rarchy of boundary value problems at successive orders of approximation; th ese problems can be asymptotically solved using techniques from the theory of smooth nonlinear or parametrically varying dynamical systems. As a resul t, we model analytically discreteness effects in the post-buckling states o f the ring, and estimate the effect of the discrete load distribution on th e critical buckling lends. This effect is found to be of very low order; in agreement with numerical results reported in an earlier work.