In this article we examine the qualitative behaviour of non-planar equilibrium states ofnon-linearly elastic rods subject to terminal loads. In our geometrically exact theory, a rod is endowed with enough geometric structure for it to undergo flexure, torsion, axial extension, and shear. The constitutive equations give appropriate stress resultants and couples as non-linear functions of appropriate strains. These constitutive relations must meet minimal conditions ensuring that they be physically reasonable. It turns out that the equilibrium states of such a rod are governed by a boundary value problem for a quasilinear fifteenth-order system of ordinary differential equations.