## The torsional buckling and writhing of a simply-supported rod hanging under gravity

D.M. Stump, A.R. Champneys & G.H.M. van der Heijden

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 eps^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/eps) 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(eps) 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.
keywords: bent and twisted rods, catenary, matched asymptotic expansions

*Int. J. Solids Struct.* **38**, 795-813 (2001)