How to Calculate Press-Fit and Interference-Fit Contact Pressure (Lamé / DIN 7190)

Lamé theory treats the hub as a thick-walled ring under internal pressure and the shaft as a solid cylinder (or hollow cylinder) under external pressure. When the shaft is pressed in, elastic deformation forces the hub bore to expand and the shaft surface to compress. At the interface the total radial displacement equals the diametral interference δ (in mm). Dividing the interference by the combined compliance of both parts — scaled by the interface diameter d — gives the contact pressure p.

For a solid shaft and a hub with outer diameter D, both made of isotropic linear-elastic materials, the two Lamé compliance factors are derived from Lamé's biharmonic solution for thick-walled cylinders. The hub compliance C_hub depends on the D/d ratio and Poisson's ratio ν_hub; the shaft compliance C_shaft for a solid shaft depends only on ν_shaft (because a solid shaft has no inner bore to amplify its deformation). Note that δ in the formula below is the diametral interference in µm, converted to mm by dividing by 1000.

The contact pressure p acts as an internal pressure on the hub bore. Lamé's solution for a thick-walled ring gives the hoop (circumferential) stress σ_θ and the radial stress σ_r at any radius. The hoop stress is highest at the bore (r = d/2) and falls with increasing r; the radial stress equals −p at the bore and falls to zero at the outer surface. DIN 7190-1 requires the hub to be checked against yielding at the bore using the GEH (von Mises) equivalent stress, not the hoop component alone, because both σ_θ and σ_r are present simultaneously at that critical point.

DIN 7190-1 recommends a minimum burst safety factor SF_burst ≥ 2.0 at the bore. A value between 1.4 and 2.0 warrants a design review; below 1.4 the hub is considered at risk of bursting. Thinner hubs (small D/d ratio) reach higher hoop stresses at lower contact pressures, so the burst check often governs the maximum permissible interference.

The main service outputs — transmissible torque T and axial slip force F_axial — scale with the contact area (π·d·L) and the friction coefficient µ. Because the ISO tolerance fit's minimum-material interference δ_min governs the worst-case grip (the lowest guaranteed contact pressure p_min), capacity checks always use p_min. The assembly press force, conversely, must overcome the highest likely contact pressure p_max (maximum-material interference), so the press must be sized to p_max. DIN 7190-1 also requires a roughness smoothing correction: about 80% of the combined surface roughness heights (Rz_shaft + Rz_hub) are flattened during pressing, reducing the effective interference that creates grip.

For shrink-fit (thermal) assembly the hub is heated (or the shaft is cooled) until the bore has expanded enough to clear the shaft by an assembly clearance allowance δ_cl. The required temperature rise ΔT follows from the thermal expansion relation. For dissimilar material pairs the interference changes with operating temperature: if the hub expands more than the shaft (e.g. aluminium hub on steel shaft), the interference reduces at elevated temperature and can vanish entirely — a thermal loosening check that is especially important for hot-running assemblies.

In practice the interference is not specified as a single number but as a tolerance fit designation such as H7/p6 or H7/s6, which defines a range of possible interferences depending on which part of the tolerance band each component falls in. ISO 286-1:2010 specifies the fundamental deviations and IT-grade tolerances for each letter/number combination: for a given interface diameter d the standard tables give the shaft's upper (es) and lower (ei) deviations and the hole's EI = 0, ES = IT. The minimum interference δ_min = ei_shaft − ES_hole and the maximum δ_max = es_shaft − EI_hole. Torque capacity is governed by δ_min; hub burst stress is governed by δ_max.

DIN 7190-1 §3.3 requires the roughness smoothing loss G = 0.8·(Rz_shaft + Rz_hub) to be subtracted from δ_min before computing the guaranteed grip pressure. If the smoothing loss is comparable to δ_min — which can happen with an H7/k6 transition fit on a rough bore — the joint may have no guaranteed grip at minimum-material condition. The MechanixCalc calculator flags this with a warning and shows the effective interference δ_eff = δ_min − G.