Exact curvature elastica of a thin cantilever under terminal loads

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 textit{autonomous} boundary value problem, in the rotation phi pertaining to each s-section. The search of the free end rotation phi0, 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'(s) and y'(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.