An Analytical Solution to the Lateral Buckling of an Elastica with Self-Contact

An exact solution to the lateral buckling of an elastic column with self-contact is obtained. By assuming that a counter force exists to maintain the horizontal coordinate at the contact point at the same value as the highest point of the bent beam, we can prevent the shape of the solution from overlapping itself. As the simplest case, we obtain elliptical integral solutions of a laterally post-buckled shape of a linear elastic inextensible column under a concentrated load. For this case, the slope's angle at the contra-flexure point remains constant regardless of the value of the load parameter beyond the minimum critical value. The counter force and several other parameters are expressed in terms of the critical slope angle at the contra-flexure point. Finally, numerical algorithms are suggested for solving the problem, which enable a solution for a nonlinear material or a combined loading case.