The 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 epsilon^2 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/epsilon) 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(epsilon) 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 post-buckled solution. This analysis is found to capture all the broad features indicated by the numerics. As Tw is further increased, the deformation may localise and the rod jump into a self-intersecting writhed shape.