Interference Fits: ISO 286 Tolerances and Lamé Stresses

What an interference fit is

An interference (press or shrink) fit joins a shaft into a hub by making the shaft slightly bigger than the hole. Forced together, the two elastically deform against each other and the resulting contact pressure holds them by friction alone — no key, no adhesive. Designing one is a balance: enough interference to transmit the load without slipping, but not so much that the contact pressure over-stresses the hub. Two standards do the work — ISO 286 sets the interference, and Lamé thick-cylinder theory turns it into stress.

ISO 286: where the interference comes from

You don't machine parts to an exact size — you machine them to a tolerance band. ISO 286 is the system of those bands: a letter giving the position of the band relative to nominal (H for a hole starting at nominal, and shaft letters like p, r, s, u sitting progressively above nominal) and a number, the IT grade, giving the band's width. So an "H7/s6" fit means a nominal-based hole (H7) and an interference shaft (s6) that always ends up larger than the hole — guaranteeing interference across the whole tolerance range.

The actual interference isn't a single number; it's a range. The tightest case stacks the largest shaft against the smallest hole; the loosest stacks the smallest shaft against the largest hole. A real design has to work at both ends: enough grip at the loose end, acceptable stress at the tight end.

Lamé theory: interference → pressure → stress

Given the interference, Lamé's thick-walled-cylinder equations give the contact pressure. The interference is shared between the hub expanding outward and the shaft compressing inward, each according to its stiffness (modulus, diameters, Poisson's ratio) — the standard treatment in Shigley's Mechanical Engineering Design. From the pressure you get the stresses: a solid shaft sits under near-uniform hydrostatic compression (σr = σθ = −p), while a hollow shaft or the hub sees its highest stress at the bore, where the hoop stress peaks — that's the location to check against yield. The same pressure also sets the axial insertion force, F = π·f·p·d·L.

The failure mode people forget: temperature

If the shaft and hub are different materials, they expand at different rates. Heating the joint canlose the fit entirely if the hub grows faster than the shaft (interference goes to zero — the part spins), or over-stress it at the cold extreme if the shaft grows faster. The shift is d·(α_shaft − α_hub)·(T − 20 °C), and a proper check evaluates the fit at assembly, operating, and storage temperatures — not just at 20 °C.