Answer
The spring rate $k$ for a multi-turn Crest-to-Crest wave spring is derived from the deformation of curved beams. For a spring with $N$ active turns and $Z$ waves per turn, the rate is defined as $k = \frac{E b t^3 Z^4}{1.5 \pi D_m^3 N} \frac{I_D}{O_D}$, where $E$ is the Modulus of Elasticity, $b$ is the radial wall, $t$ is the material thickness, and $D_m$ is the mean diameter. When shim ends are added, the effective number of turns increases slightly because the shim acts as a rigid boundary, reducing the active deflection length. For high-precision applications, the correction factor for the mean diameter $D_m = (D_{outer} + D_{inner})/2$ must account for the radial expansion during compression, as $D_m$ increases slightly, potentially leading to a non-linear stiffening effect as the spring approaches its solid height $H_s$.
