Force–displacement characteristics of circular-shaped massless elastica

Abstract

The deformation of circular-shaped elasticas, subjected to unilateral constraints due to ground contact, is studied under vertical loading. For C-shaped elasticas, we consider the cases where the elastica rolls without slipping due to the presence of friction and rolls while slipping due to lack of friction. The elastica behaves as a hardening spring in the absence of friction; in the presence of friction, it behaves as a softening or hardening spring depending on boundary conditions. For O-shaped elasticas, we consider symmetric and asymmetric loadings. Under symmetric loading, the stiffness of the elastica decreases with increase in the load while point contact is maintained; thereafter, the stiffness remains nearly constant while the elastica undergoes large deformation and makes line contact. Similar behavior is observed for a symmetric mass-elastica system. Asymmetric loading is studied by assuming that the initial point of contact of the elastica with the ground is fixed; vertical loading results in both vertical and horizontal motion of the point of loading. For the different statics and dynamics problems considered, the deformed shapes of the elasticas are obtained iteratively using an algorithm that solves a series of two-point boundary value problems.