We consider equilibrium configurations of inextensible, unshearable, isotropic, uniform and naturally straight and prismatic rods when subject to end loads and clamped boundary conditions. In a first paper [Neukirch & Henderson, 2002], we discussed symmetry properties of the equilibrium configurations of the center line of the rod. Here, we are interested in the set of all parameter values that yield equilibrium configurations that fulfill clamped boundary conditions. We call this set the solution manifold and we compute it using a recently introduced continuation algorithm. We then describe the topology of this manifold and how it comprises different interconnected layers. We show that the border set of the different layers is the well-known solution set of buckled rings.