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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 ${\delta }_{\mathrm{th}}=\alpha L\Delta T$, completely rigid axial restraint gives compressive force magnitudeA locating/non-locating support arrangement is intended to accommodate the required displacement rather than impose this unwanted extra axial constraint.^[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 givesFor hypothetical $L=0.50\mathrm{m}$, $E=200\mathrm{GPa}$ and $A=100\mathrm{m}{\mathrm{m}}^2$, $C_b=2.5\times 10^{-8}m/N$. Free extension0.30mm gives12kN against rigid supports. If $C_s=10^{-7}m/N$, 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 loadWith 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.

A non-locating bearing support carries the required radial load while allowing the shaft's intended axial displacement relative to the housing. It complements a locating bearing, which establishes axial position. The movement may occur within the bearing or through a suitable ring-to-seat arrangement; its purpose is to avoid imposing an unwanted axial constraint when dimensions change.^[ref]

An important cause of relative movement is thermal expansion. For an illustrative shaft with constant linear thermal expansion coefficient $\alpha$, length $L$ and uniform temperature increase $\Delta T$, free extension isUsing hypothetical $\alpha =12\times 10^{-6}{\mathrm{K}}^{-1}$, $L=0.50\mathrm{m}$ and $\Delta T=50\mathrm{K}$ gives0.30mm. If the housing expands differently, the required bearing axial displacement allowance depends on the shaft's extension relative to that housing, as well as assembly tolerances.

For contrast within the same mechanical model, constrain both ends of an elastic bar against extension. Axial strain balance gives a compressive bearing thermal-expansion constraint load with magnitude $F=EA\alpha \Delta T$. With hypothetical Young's modulus $E=200\mathrm{GPa}$ and area $A=100\mathrm{m}{\mathrm{m}}^2$, the preceding temperature change gives12kN. Real supports and housings are compliant, so this rigid-constraint calculation illustrates the mechanism rather than predicting a universal bearing load.

A non-locating bearing is not generally a support with no useful constraint: it can still transmit substantial transverse forces. Its permitted axial movement must remain compatible with the shaft's other functional requirements, including gear positioning. The actual support arrangement determines whether thermal movement is accommodated without losing the required location.