Consistent Derivations of Spring Rates for Helical Springs

The spring rates of a coiled helical spring under an axial force and an axially directed torque are derived by a consistent application of Castigliano’s second theorem, and it is shown that the coupling between the two loads may not always be neglected. The spring rate of an extensional spring is derived for the first time through the use of the displacement based principle of minimum total potential energy. The present results are also compared with available derivations of and expressions for the stiffness of a coiled spring.