How is the theoretical spring rate ($k$) calculated for a multi-turn Crest-to-Crest wave spring, and what factors cause non-linearity as it approaches solid height?

The theoretical spring rate $k$ for a multi-turn Crest-to-Crest wave spring with shim ends is derived from the formula: $k = \frac{E \cdot b \cdot t^3 \cdot N^4}{D_m^3 \cdot Z \cdot 583}$, where $E$ is the Modulus of Elasticity, $b$ is the radial wall, $t$ is the material thickness, $N$ is the number of waves per turn, $D_m$ is the mean diameter, and $Z$ is the number of active turns. In practice, the load-deflection curve remains linear between 20% and 80% of the available deflection. As the spring approaches 'solid height', the waves begin to 'nest' or touch, which effectively reduces the active length of the beam and increases the number of waves $N$ acting in parallel. This causes an exponential increase in the spring rate, often expressed as $k_{actual} = k_{theoretical} \cdot (1 - \frac{f}{h})^{-1}$ where $f$ is deflection and $h$ is free height, although empirical testing is required for precise solid-height transition modeling.