Exact Curvature Elastica of a Thin Cantilever Under Terminal Loads

Abstract

A thin, flexible, one side-built-in rod under a concentrated terminal force is studied in its elastic equilibrium configuration. In order to make the problem more tractable, a proper set of state variables is chosen, facing with a second order, nonlinear, but autonomous boundary value problem, in the rotation $\varphi (s)$ pertaining to each $s$-section. The search of the free end rotation $\varphi _{0}$, following the isoperimetric assumption, leads to a numerical sub-problem inside the main BVP. Furthermore, if $x(s)$ and $y(s)$ mean the elastica coordinates parametrized on the arclength $s$, one obtains $x^{\prime }(s)$ and $y^{\prime }(s)$ as elliptic functions of $s$. Finally, some minor changes have been shown in order to pass from a loading force to a more general free-end load combination, consisting of a force and a couple.