  A convergence result is proved for the equilibrium configurations of a
three-dimensional thin elastic beam, as the diameter h of the cross-section
goes to zero. More precisely, we show that stationary points of the nonlinear
elastic functional E^h, whose energies (per unit cross-section) are bounded by
Ch^2, converge to stationary points of the Gamma-limit of E^h/h^2. This
corresponds to a nonlinear one-dimensional model for inextensible rods,
describing bending and torsion effects. The proof is based on the rigidity
estimate for low-energy deformations by Friesecke, James, and M\"uller and on a
compensated compactness argument in a singular geometry. In addition, possible
concentration effects of the strain are controlled by a careful truncation
argument.
