The clamped-free rod under inclined end forces and transitions between equilibrium configurations

Abstract

We investigate the problem of the straight, inextensible and unshearable clamped-free elastic rod subjected to an inclined end force. Exact analytic solutions representing all equilibrium configurations of the deformed rod are presented in elliptic integral form. Those exact solutions, for a given angle of inclination of the end force and number of inflection points, are characterised by two quantities; the end force and the elliptic modulus. Critical points are discussed and analytic conditions for determining their location are presented. Certain critical points where transitions between two equilibrium configurations whose numbers of inflection points differs by one are pointed out. Simple formulae for the total number of equilibrium configurations for a given end force are given. Applying arguments based on the elastic strain energy of the rod, we discuss scenarios where highly inflectional equilibrium configurations can transition to equilibrium configurations with fewer inflection points.