A joint with no fastener in it
A press fit transmits torque with nothing but squeeze. Machine a shaft a few hundredths of a millimetre larger than the hole it goes into, force the two together, and the stretched hub clamps the compressed shaft with enough pressure that friction alone carries the load — no key, no setscrew, no adhesive. Gears, bearings, couplings, and flywheels ride on this trick, and the entire design question reduces to a chain of four numbers: how much interference, how much pressure that interference generates, how much grip that pressure buys, and how much stress the parts pay for it.
This guide walks the chain with one running example: a 40 mm (about 1-9/16 in) solid steel shaft pressed into a steel gear hub with an 80 mm outside diameter and a 40 mm engagement length. The fit math happens in metric because that is how the governing standard, ISO 286, writes its tables — the same arithmetic applies to inch fits, but the catalog of standard fits speaks millimetres and micrometres.
H7 and the alphabet of fits: k6 to u6
Interference is rarely specified as a raw number; it is specified as a fit class. In the hole-basis system the hole is held at H7 — its lower limit sits exactly on the nominal size — and the shaft class does the work of creating overlap. The standard ladder climbs from H7/k6 and H7/n6, the transition fits that locate precisely but may assemble with a whisker of clearance, through H7/p6, the light press fit for modest torque and possible disassembly, to H7/s6, the medium drive fit for permanent joints, and H7/u6, the force fit for maximum holding power. Each class is a pair of limit deviations from the ISO 286-2:2010 tables, and the deviations are diametral — the shaft measures that much larger than the hole across the diameter, which is also how a machinist mics the parts.
For the running example, pick H7/s6 at 40 mm nominal. The standard's 30–50 mm band puts the H7 hole at +25/0 µm and the s6 shaft at +59/+43 µm, so the interference can land anywhere from 43 − 25 = 18 µm, when a small shaft meets a big hole, to 59 − 0 = 59 µm when the extremes stack the other way. That threefold spread is not a nuisance to average away — it is the defining fact of interference-fit design, because the loose end and the tight end of the tolerance must each pass a different test.
Interference makes pressure: the thick-cylinder view
Lamé's thick-cylinder theory converts interference into contact pressure. Squeezing an oversized shaft into a hub compresses the shaft and stretches the hub, and the interface pressure p is the diametral interference ratio δ/d divided by the summed elastic compliances of the two parts — terms built from each part's Young's modulus, Poisson's ratio, and wall geometry. Geometry matters as much as material: a thin-walled hub or a hollow shaft is more compliant, so the same interference generates less pressure. That is why a gear with a slender rim grips its shaft more weakly than the fit class alone would suggest.
For the example joint — steel on steel at 200 GPa and ν = 0.3, hub OD twice the shaft diameter, solid shaft — the compliance sum works out to 2.667/E, so p = (δ/d) × E/2.667. At the minimum 18 µm interference that gives p = (0.018/40) × 75 000 ≈ 34 MPa; at the maximum 59 µm it is about 111 MPa. One fit class, one drawing callout — and a factor of three between the pressure the weakest joint in the batch develops and the pressure the tightest one must survive.
Pressure makes grip: torque at the loose end
Friction converts pressure into holding capability. The contact area is π·d·L, the friction force available is μ times the pressure acting over it, and putting that force at the shaft radius gives the transmissible torque T = μ·p·π·d²·L/2, with axial holding force F = μ·p·π·d·L. The critical discipline is which interference to plug in: capacity is evaluated at the minimum, because a production batch spans the whole tolerance and the joint at the loose end grips least. A transition fit's guaranteed capacity is exactly zero for the same reason — H7/k6 can assemble with clearance, so friction drive cannot be promised at all.
The friction coefficient deserves suspicion. Dry steel-on-steel press fits are commonly designed at μ = 0.10 to 0.15, oiled assembly can halve that, and the pressing-on (dynamic) value runs lower than the holding (static) one. Since capacity is directly proportional to μ, it is the least certain number in the chain — use a conservative value and apply a service factor to the torque the joint must carry. At μ = 0.12, the example joint's guaranteed numbers at 18 µm are T = 0.12 × 34 × π × 40² × 40 / 2 ≈ 410 N·m and about 20 kN of axial grip.
The stress bill: hub hoop stress and the yield check
The tight end of the tolerance presents the bill. At maximum interference the pressure peaks, and the largest stress in the joint is the tangential — hoop — stress at the hub bore: σ_t = p(d_o² + d²)/(d_o² − d²). The check is against the hub material's yield strength, and it is not bureaucratic: the fit only keeps its pressure, and therefore its grip, if the hub stays elastic. Yield the bore and some of the interference is spent permanently deforming metal instead of storing clamp.
For the example at 59 µm: p ≈ 111 MPa, and the hoop stress is 111 × (80² + 40²)/(80² − 40²) ≈ 184 MPa — a comfortable pass against a 350 MPa yield. The press that assembles this joint must also deliver the peak-force case: the axial force at maximum interference, about 67 kN here, sets the arbor-press requirement. Both stress and press force belong to the maximum-interference column, exactly opposite the torque check — a press-fit design is always two calculations wearing one part number.
Heat instead of force: ΔT = δ/(αd)
The elegant alternative to 67 kN of pressing is temperature. Heat the hub and its bore grows; grow it past the interference and the shaft drops in with zero force. The required temperature rise is ΔT = δ/(α·d), taken at the maximum interference so the method works for every joint in the batch. With steel's expansion coefficient around 11.5–12 × 10⁻⁶ per °C, the example needs ΔT = 0.059/(11.5 × 10⁻⁶ × 40) ≈ 128 °C above ambient just to reach zero clearance — and shop practice adds an assembly allowance, commonly the same order as the interference again, so the parts slide freely before heat starts flowing into the shaft.
Two cautions ride the method. Keep the target temperature clear of any tempering range if the hub is hardened — grip is worthless in a part that lost its heat treatment on the hot plate. And the arithmetic explains why material pairings matter: an aluminum hub, at roughly double steel's expansion coefficient, reaches the same bore growth at about half the ΔT. When heating is impractical, the mirror-image move is cooling the shaft with dry ice or liquid nitrogen and letting contraction do the work.
A 40 mm shaft in an H7/s6 hub, gripped and heated
Run the whole chain on the example joint. Fit: H7/s6 at 40 mm — hole +25/0, shaft +59/+43, interference 18 to 59 µm diametral. Pressure: with the 80 mm steel hub and solid steel shaft, p spans 34 MPa at the loose end to 111 MPa at the tight end. Grip, taken at minimum interference with μ = 0.12: about 410 N·m of guaranteed torque and 20 kN of axial holding — the numbers the drive design may rely on. Stress and assembly, taken at maximum: 184 MPa of hub hoop stress against 350 MPa yield, a 67 kN press force if assembled cold, or roughly 128 °C of hub heating plus clearance allowance if assembled hot.
Every one of those numbers came from three relations — Lamé's pressure equation, the friction torque formula, and ΔT = δ/(αd) — applied at the correct end of the tolerance. The press fit calculator automates exactly this two-column bookkeeping: pick the nominal size and fit class (or enter interference directly), set geometry and materials, and read guaranteed capacity at minimum interference beside stresses, press force, and shrink temperature at maximum. It is preliminary elastic sizing, as the underlying theory is: surface-finish effects, form error, and centrifugal loosening at speed sit outside the equations.
The fit in one pass
Recap the 40 mm joint: H7/s6 sets 18–59 µm of diametral interference; Lamé turns that into 34–111 MPa of interface pressure across the tolerance; friction at μ = 0.12 turns the guaranteed 34 MPa into 410 N·m of torque and 20 kN of axial grip; the worst-case 111 MPa costs 184 MPa of hub hoop stress (pass at 350 MPa yield) and either a 67 kN press or a 128 °C-plus-allowance shrink cycle. The discipline to carry away is the two-column habit — capacity at minimum interference, stress and assembly effort at maximum — and the humility about μ, the one input that is a judgment call rather than a table lookup. Get those two things right and the alphabet of fits reads like a torque catalog.