Derivation of the elastic force of springs from the curved rod model

Abstract

In this paper the Hook law for cylindrical springs is derived from the linearized elasticity. The cylindrical spring is an elastic curved rod with helicoidal middle curve. The curved rod model is used to obtain the behaviour of the spring subjected to the longitudinal force. The obtained coefficient in Hook's law, expressed in terms of the elastic and geometrical properties of the spring, matches the known coefficient.