Elastica of Simple Variable-Arc-Length Beam Subjected to End Moment

Abstract

This paper presents two methods to find the elastica of a bar or a beam of given span length, but unknown arc length. The beam is subjected to a moment at a hinged end that can slide freely over another support. In the first method the differential equation based on large-deflection theory is formulated and solved by using elliptic integrals. The method yields an exact closed-form solution. The critical or maximum applied moment the beam can resist is also obtained by this formulation. Further, the well-known small displacement solution can be obtained from the degeneration of the exact solution by considering small rotations. The second method is based on a variational formulation, which involves the bending strain energy and work done by the end moment. The finite-element discretization of span length instead of bar length is used to solve the problem. Numerical comparisons are given and results from the finite-element method show good agreement with the elliptic integrals solutions.