Greenhill formula for an exact model of elastic rod

Abstract

Greenhill formula for Kirchhoff elastic rod is extended to that of exact model of the rod. Under the assumption of the plane cross section, the configuration of an extensible and shearable elastic rod is expressed as a history of the cross section with arc coordinate. A special solution which describes equilibrium in straight line state of the rod is obtained from a differential equilibrium equation. A linear perturbation equation is derived and its general solution is obtained in which the integral constants are determined by constrained conditions at two ends of the rod. The condition for a non zero solution of the integral constants to exist leads to the Greenhill formula of exact elastic rod model, which shows that the boundary of stable area of the force screw is a closed curve and of symmetry and the inference of extensible and shearable to stability of the rod is dependent on three factors: the difference in flexibility between shear and extension of a section of the rod, the bending stiffness, and the length of the rod.