How do you calculate the minimum groove depth ($d$) required to prevent 'ring pull-out' for a given axial load $P$?

The minimum groove depth $d$ is determined by the requirement that the groove material must support the axial load without yielding. The formula is $d = \frac{P \cdot K}{\pi \cdot D \cdot \sigma_y}$, where $P$ is the load, $K$ is the safety factor, $D$ is the diameter, and $\sigma_y$ is the yield strength of the housing material. Additionally, $d$ must be deep enough to ensure the ring's radial wall $w$ is substantially submerged. A common engineering rule of thumb is that $d$ should be at least $25\%$ to $50\%$ of the ring thickness $T$ to provide mechanical stability. If the groove is too shallow, the dishing moment will cause the ring to fail by 'slipping' over the groove edge rather than by shearing the ring material.