Interference Fit Torque⚠ unverified
Mechanical / Joints · Torque transmissible by an interference fit
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| p | p | Pa | 50000000.0 | Contact pressure |
| d | d | m | 0.05 | Diameter |
| L | L | m | 0.08 | Engagement length |
| mu | μ | — | 0.15 | Friction coefficient |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| T | T | N*m | Torque capacity |
The science & history
Understanding the Parameters
-
Contact pressure $p$ — from the interference and geometry (Press Fit Pressure). Torque scales linearly with it, so the interference tolerance directly controls capacity.
-
Engagement length $L$ — a longer fit means more contact area and proportionally more torque; lengthening the hub is the easiest way to raise capacity without changing the fit.
-
Diameter $d$ — appears as $d^2$: once through the contact area ($\pi d L$) and once through the friction radius ($d/2$). A larger shaft gains capacity quickly.
-
Friction coefficient $\mu$ — dry steel-on-steel $\mu \approx 0.1$–$0.15$; contamination, oil, or plating lowers it. It is the least certain input, so designs carry a slip safety factor and sometimes degrease the mating surfaces deliberately.
Derivation (Approaching a Proof)
The interference pressure $p$ acts normal to the cylindrical contact surface of area $A = \pi d L$ (circumference $\times$ length). The total normal force clamping the two parts is
$$N = p\,A = p\,\pi d L.$$
Friction resists relative rotation with a tangential force up to $\mu N$, distributed around the circumference at radius $r = d/2$. The torque it can resist is the friction force times that radius:
$$T = \mu N\cdot\frac{d}{2} = \mu\,(p\,\pi d L)\,\frac{d}{2} = \frac{\pi\,\mu\,p\,L\,d^2}{2}.$$
At this torque the surfaces are on the verge of slipping (the whole interface reaches limiting friction simultaneously, since $p$ and $r$ are uniform). An axial push-off force follows the same logic without the radius: $F_{\text{axial}} = \mu p \pi d L$.
Dimensional check. $T = \mu\,p\,\pi d L\,\dfrac{d}{2} = (\text{–})\cdot\text{Pa}\cdot\text{m}\cdot\text{m} \cdot\text{m} = (\text{N}/\text{m}^2)\cdot\text{m}^3 = \text{N}\cdot\text{m}$ — a torque, as required.
History and Development
Keyless friction drives via shrink and press fits are as old as railway wheelsets and built-up crankshafts. The torque capacity $T = \tfrac12\pi\mu p L d^2$ pairs Lamé's interference-pressure theory (Press Fit Pressure) with Amontons–Coulomb friction to answer the designer's real question — "will the hub slip?" — without a key or spline. Modern practice adds keyless locking assemblies and shrink discs that exploit the same physics with far higher, more controllable interface pressures.
Related Concepts: Press Fit Pressure, Interference Fit Pressure, Keyway Stress Reduction, Pressure Vessel Design, Principal Stresses
Notes: Friction-only (keyless) capacity; carry a slip safety factor. $T \propto p,\,L,\,d^2$. Axial push-off force $= \mu p \pi d L$ (same friction, no radius). $\mu \approx 0.1$–$0.15$ dry steel; degrease to maximise. Get $p$ from Press Fit Pressure (or full Lamé).