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 a nd subject to a prescribed tangential twist Tw is studied using asymptotic and numerical methods. A three-dimensional formulation of the problem is gi ven in which a small parameter epsilon (2) measures the relative sizes of b ending and gravitational forces. For small values of Tw, the rod shape is f ound by singular perturbation methods and consists of an outer catenary-lik e solution and an inner boundary layer solution. Large twist Tw = O(1/epsil on) of an almost straight rod produces a torque on the order of the Greenhi ll buckling level and is shown numerically to cause buckling into a modulat ed helix-like spiral with period of O(epsilon) superimposed onto a paraboli c sag across the spanned distance. Multiple scale methods are used in this parameter regime to obtain an approximate description of the postbuckled so lution. This analysis is found to capture all the broad features indicated by the numerics. As Tw is further increased, the deformation may localise a nd the rod jump into a self-intersecting writhed shape. (C) 2001 Elsevier S cience Ltd. All rights reserved.