We derive the shape equations in terms of Euler angles for a uniform elastic filament with circular cross section but free of spontaneous torsion. We show that in general there are planar curve solutions for a closed rod. We study the boundary conditions (i.e., experimental conditions in a force experiment) to form a helical filament under external force and twisting. We find that to form a helix, the Euler angle must be a constant determined by the spontaneous curvatures. We study the elasticity of a helical filament under different conditions. We find that the extension of a helix under fixed and finite torque may subject to a one-step sharp transition with increasing stretching force. However, we show exactly that there is not jump of extension for a helical filament free of external torque. This behavior is quite different from a uniform elastic rod with circular cross section and spontaneous torsion, and provides another very important reason why one cannot observe the sharp jump of extension for most macroscopic helical springs.