Interference Fit Pressure⚠ unverified

Understanding the Parameters

• Radial interference $\delta$ — the difference in radius that must be elastically absorbed when the parts are forced together. Contact pressure is directly proportional to it, so the tiny interference (micrometres) is machined to tight tolerance — too little and the joint slips, too much and the hub yields or cracks. (Confirm whether your $\delta$ is radial or diametral; they differ by a factor of 2 — see note.)

• Interface diameter $d$ — the size of the joint; pressure falls as $1/d$ for a given radial interference, because the same interference is a smaller strain on a larger diameter.

• Young's modulus $E$ — stiffer materials generate more pressure for the same interference.

• Poisson's ratio $\nu$ — appears through the $1/(1-\nu^2)$ plane-strain stiffening term, reflecting the biaxial constraint at the interface.

Registry note (honest): this is a simplified same-material estimate. The rigorous shrink-fit pressure (thick-wall Lamé theory) depends on the hub outer diameter and the shaft bore, through geometry factors like $(D_o^2 - d^2)/D_o^2$ — none of which are inputs here. The formula effectively assumes a specific (thick/near-infinite hub) geometry and identical shaft and hub material, and the $(1-\nu^2)$ term differs from the standard Lamé grouping. Use the full Lamé equations for real design, especially thin hubs or dissimilar materials. Flagged in Known Issues.

Derivation (Approaching a Proof)

The pressure comes from compatibility: the assembled interference must equal the sum of the elastic radial deformations of the two parts. For a shaft-in-hub, the interface radial displacement of each part under the mutual pressure $p$ follows from thick-wall (Lamé) cylinder theory. The general same-material result relates the radial interference $\delta$ (a displacement) to $p$ through the diameter and the elastic properties:

$$\delta = \frac{p\, d}{E}\big[\text{geometry factor}(\nu, D_o, d, D_{\text{bore}})\big].$$

For a solid shaft in a hub, the geometry factor involves $(D_o^2 - d^2)/(D_o^2)$ terms. Collapsing that factor for a thick hub and grouping the elastic constants into the plane-strain modulus $E/(1-\nu^2)$ gives the simplified inverse used here:

$$p = \frac{\delta\, E}{2 d (1 - \nu^2)}.$$

The compatibility idea is exact; the simplification is in freezing the hub geometry (dropping $D_o$) and assuming matched materials. Once $p$ is known, the torque capacity follows from friction over the engaged interface, $T \approx \mu\, p\,(\pi d L)\,(d/2)$, and the hub hoop stress from Lamé — both the reasons pressure is the key design quantity.

Dimensional check. $[p] = \dfrac{\text{m}\cdot\text{Pa}}{\text{m}} = \text{Pa}$ (the $(1-\nu^2)$ term is dimensionless). ✓

History and Development

Interference-fit analysis rests on Lamé's thick-walled cylinder theory (1833) and is a staple of Shigley, Roark, and machine-design practice for mounting gears, bearing races, couplings, and railway wheels on shafts. Shrink fitting (heating the hub or cooling the shaft to assemble) is the classic method. The simplified same-material form here is a first-estimate; the full Lamé treatment handles thin hubs, dissimilar materials, and the resulting hub stresses that often govern.

Related Concepts: Torsional Shear Stress, Hertzian Contact Pressure, Von Mises Stress, Shaft Diameter Combined, Hooke's Law strain

Notes: Simplified same-material, fixed-geometry estimate — use full Lamé equations for thin hubs or dissimilar materials (the hub outer diameter is not an input here). Confirm radial vs diametral interference. Torque capacity $\approx \mu\,p\,\pi d L\,(d/2)$; check hub hoop stress against yield.