= Bearing thermal-expansion constraint load
{title2=$F=EA\alpha\Delta T$}
A <bearing thermal-expansion constraint load> is an axial reaction generated when supports prevent the relative <thermal expansion> that a shaft/housing system would otherwise undergo. For a uniform elastic shaft of length $L$, area $A$, <Young's modulus> $E$ and free extension $\delta_{\rm th}=\alpha L\Delta T$, completely rigid axial restraint gives compressive force magnitude
$$
F=\frac{EA}{L}\delta_{\rm th}=EA\alpha\Delta T.
$$
A locating/non-locating support arrangement is intended to accommodate the required displacement rather than impose this unwanted extra axial constraint.<Rolling bearings by SKF>[Section B.2, printed p.70, axial location and thermal displacement accommodation]{ref}
Actual supports have finite <mechanical compliance>. Let shaft axial compliance be $C_b=L/(EA)$ and the combined restraining support/housing compliance be $C_s$. Compatibility for a fully engaged restraint gives
$$
\delta_{\rm th}=F(C_b+C_s),\qquad F=\frac{\delta_{\rm th}}{C_b+C_s}.
$$
For hypothetical $L=0.50\,\mathrm m$, $E=200\,\mathrm{GPa}$ and $A=100\,\mathrm{mm^2}$, $C_b=2.5\times10^{-8}\,\mathrm{m/N}$. Free extension0.30mm gives12kN against rigid supports. If $C_s=10^{-7}\,\mathrm{m/N}$, the force falls to2.4kN because part of the mismatch is taken up by support displacement.
A gap changes the onset of restraint. If the system permits positive free travel $g$ before reaching an axial stop, a simple one-direction model gives <bearing thermal end-stop load>
$$
F=\frac{\max(\delta_{\rm th}-g,0)}{C_b+C_s}.
$$
With the preceding compliant supports and $g=0.10\,\mathrm{mm}$, the result is1.6kN. Below the gap-closing displacement, this modeled stop contributes no force. Applied <gear axial force>, bearing preload, temperature gradients and other constraints require their own contributions; the simple expression represents one stated thermal compatibility problem.
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