The subject of this paper is the study of the asymptotic behaviour of the
equilibrium configurations of a nonlinearly elastic thin rod, as the diameter
of the cross-section tends to zero. Convergence results are established
assuming physical growth conditions for the elastic energy density and suitable
scalings of the applied loads, that correspond at the limit to different rod
models: the constrained linear theory, the analogous of von K\'arm\'an plate
theory for rods, and the linear theory.