Large displacements of slender beams in plane: Analytical solution by means of a new hypergeometric function

The goal of this paper is to present an analytical solution for large planar displacements of cantilever beams, avoiding the integration of elliptic integrals. The proposed solution takes advantage of a new hypergeometric function of two variables by which it is possible to obtain the parametric solution of the beam displacement. The solution concerns a cantilever beam subjected to an inclined force and a moment applied at the free end; nevertheless, it is easy to extend it to the case of multiple loads applied in intermediate positions of the beam. The beams have a constant section and initial curvature, the material is elastic, isotropic and homogeneous. It is shown how to extend the results to spring-hinged cantilever or simply supported beams, to loads attached to the beam axis (following forces), or to a cantilever beam having the unsupported end displaced by a rigid cable. Various technical design curves are also provided; these allow an easy and fast estimate of the endpoint displacements. Particular attention is given to the study of the convergence region and speed of this new hypergeometric function. The configurations examined, the methodologies and the procedures to carry on these solutions are explained in detail within the paper. Some comparisons with numerical results support the solution proposed.