Torsional buckling and writhing of a simply supported rod hanging under gravity

The problem of a finite-length simply supported rod hanging under gravity and subject to a prescribed tangential twist Tw is studied using asymptotic and numerical methods. A three-dimensional formulation of the problem is given in which a small parameter ε measures the relative sizes of bending and gravitational forces. For small values of Tw, the rod shape is found by singular perturbation methods and consists of an outer catenary-like solution and an inner boundary layer solution. Large twist Tw = O(1/ε) of an almost straight rod produces a torque on the order of the Greenhill buckling level and is shown numerically to cause buckling into a modulated helix-like spiral with period of O(ε) superimposed onto a parabolic sag across the spanned distance. Multiple scale methods are used in this parameter regime to obtain an approximate description of the postbuckled solution. This analysis is found to capture all the broad features indicated by the numerics. As Tw is further increased, the deformation may localize and the rod jump into a self-intersecting writhed shape