Press Fit Pressure⚠ unverified
Mechanical / Joints · Compute the contact pressure in a press fit
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| delta | δ | m | 1.0 | Radial interference between mating parts |
| d | d | m | 1.0 | Nominal interface diameter |
| E | E | Pa | 1.0 | Young's modulus of the material |
| nu | ν | — | 0.3 | Poisson's ratio of the material (dimensionless). Default is 0.3 |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | p | Pa | Contact pressure at the interface, in pascals (Pa). Returns 0.0 when the diameter ``d`` is not positive |
The science & history
Understanding the Parameters
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Radial interference $\delta$ — the amount the shaft radius exceeds the bore radius before assembly. Note it is radial here; the measurable diametral interference is $2\delta$. Pressure scales linearly with $\delta$, so tolerances on $\delta$ directly control the fit.
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Interface diameter $d$ — a larger diameter spreads the interference over more material, giving lower pressure for the same $\delta$ ($p \propto 1/d$).
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Elastic modulus $E$ — stiffer materials generate more pressure for a given interference. The simplified form assumes shaft and hub share the same $E$ and $\nu$.
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$(1-\nu^2)$ term — the plane-strain/biaxial elastic factor from the thick-cylinder solution; it softens the effective stiffness because radial squeeze is accompanied by circumferential (hoop) strain.
Derivation (Approaching a Proof)
Lamé's thick-cylinder theory relates the radial interference to the pressure it produces. In general, for a shaft (subscript s) in a hub (subscript h) meeting at radius $r = d/2$, the total radial interference is the sum of the hub's outward expansion and the shaft's inward contraction:
$$\delta = \frac{p\,r}{E_h}\!\left(\frac{r_o^2 + r^2}{r_o^2 - r^2} + \nu_h\right) + \frac{p\,r}{E_s}\!\left(\frac{r^2 + r_i^2}{r^2 - r_i^2} - \nu_s\right),$$
with $r_o$ the hub outer radius and $r_i$ the shaft inner radius. For a solid shaft ($r_i = 0$), same material ($E_h = E_s = E$, $\nu_h = \nu_s = \nu$), and the particular hub proportion that reduces the geometry factors to unity, this collapses to the compact registry form
$$\delta \approx \frac{2 p\,r\,(1-\nu^2)}{E} = \frac{p\,d\,(1-\nu^2)}{E} \;\Longrightarrow\; p = \frac{\delta\,E}{2\,d\,(1-\nu^2)}.$$
The dropped geometry factors are exactly why the note flags this as a simplified estimate: a thin hub expands far more (higher $\delta$ per $p$, so lower $p$) than this predicts.
Dimensional check. $p = \dfrac{\delta\,E}{2\,d\,(1-\nu^2)} = \dfrac{\text{m}\cdot\text{Pa}}{\text{m}} = \text{Pa}$ — a pressure, as required ($1-\nu^2$ is dimensionless).
History and Development
Gabriel Lamé solved the thick-walled-cylinder elasticity problem in 1833, and it has underpinned interference-fit and pressure-vessel design ever since. Shrink and press fits — heating the hub or pressing the shaft to seat gears, bearing races, and railway wheels without keys — are among the oldest machine elements, and Lamé's equations give the pressure, the assembly force, and the transmissible torque. The simplified same-material form here is the classroom entry point to that full theory.
Related Concepts: Interference Fit Torque, Interference Fit Pressure, Pressure Vessel Design, Principal Stresses, Keyway Stress Reduction
Notes: Simplified same-material, solid-shaft form — the general Lamé result needs hub OD, shaft ID, and both materials' $E,\nu$. $\delta$ is radial (diametral $= 2\delta$). $p \propto \delta$, $p \propto 1/d$. Use full Lamé for design.