= Bearing thermal end-stop load
{title2=$F=\max(\delta_{\rm th}-g,0)/(C_b+C_s)$}
A <bearing thermal end-stop load> is the axial reaction that appears when relative <thermal expansion> exhausts the free travel of a support arrangement. In a quasistatic one-direction model, let free thermal mismatch be $\delta_{\rm th}$, initial travel to the stop be $g$, shaft compliance be $C_b$ and restraining support compliance be $C_s$. The engaged reaction magnitude is
$$
F=\frac{\max(\delta_{\rm th}-g,0)}{C_b+C_s}.
$$
The gap first closes; additional mismatch is taken up by elastic deformation of the shaft and support. This explains why the <bearing axial displacement allowance> matters. The locating/non-locating arrangement is intended to accommodate relative motion rather than create an unintended thermal restraint.<Rolling bearings by SKF>[Section B.2, printed p.70, support functions and thermal displacement]{ref}
For hypothetical $C_b=2.5\times10^{-8}\,\mathrm{m/N}$, $C_s=10^{-7}\,\mathrm{m/N}$, $g=0.10\,\mathrm{mm}$ and $\delta_{\rm th}=0.30\,\mathrm{mm}$, the effective stiffness is $1/(C_b+C_s)=8\,\mathrm{MN/m}$ and the load is1.6kN. The stored elastic <energy> in this combined linear compliance is
$$
E=\tfrac12F^2(C_b+C_s)=0.16\,\mathrm J.
$$
This is stored energy; the frictionless elastic model does not itself predict heat loss during a complete reversible cycle.
If the housing is fixed and free mismatch is $\alpha L\Delta T$, the <bearing end-stop engagement temperature> increment is $\Delta T_*=g/(\alpha L)$. With hypothetical $\alpha=12\times10^{-6}\,\mathrm{K^{-1}}$ and $L=0.50\,\mathrm m$, the0.10mm gap closes at about16.67K temperature increase. Further heating raises the load according to the engaged stiffness.
The positive-part relation is a simple instance of <unilateral contact complementarity>: the stop supplies a reaction only when the corresponding clearance is exhausted. Shaft inertia, impact, friction or bearing preload require a more complete model. In particular, this static compatibility calculation does not determine a collision force when a moving assembly first hits the stop.
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