Interference fit calculator: contact pressure, torque capacity and press force

1. Interference fit formulas

Notation: d contact (nominal) diameter, D hub outside diameter, d_i shaft bore (0 for a solid shaft), L contact length, δ diametral interference (shaft diameter minus bore diameter), E Young's modulus, ν Poisson's ratio, μ coefficient of friction. Subscript h for the hub, s for the shaft.

Units must be consistent: mm and MPa give p in MPa, F in N and T in N·mm (÷ 1,000 for N·m); inches and psi give p in psi, F in lbf and T in lbf·in. The interference δ is diametral: if you start from a radial interference, double it.

2. Metric and inch examples

Ø 50 steel shaft in a steel hub (metric units)

Solid Ø 50 mm shaft, hub outside diameter 100 mm, contact length 50 mm, steel E = 210,000 MPa, ν = 0.3. H7/p6 fit: the bore is 0 / +25 µm and the shaft +26 / +42 µm (see the ISO 286 table), an interference of 1 to 42 µm. Calculation at the maximum interference, δ = 0.042 mm, with μ = 0.1:

At the minimum interference of the same H7/p6 (1 µm), the pressure drops to 1.6 MPa and the guaranteed torque to only 31 N·m: a p6 fit is enough to locate a part, not to transmit a large torque. For that you need a tighter fit or a positive connection (key, splines).

Ø 2 in steel shaft (US units)

Solid Ø 2 in shaft, Ø 4 in hub, 2 in long, steel E = 30,000,000 psi, interference δ = 0.0016 in, μ = 0.1: p = 30,000,000 × 0.0016 / 4 × (16 − 4)/16 = 9,000 psi; σ_t = 9,000 × 20/12 = 15,000 psi; F = 0.1 × 9,000 × π × 2 × 2 = 11,310 lbf; T = 11,310 × 1 = 11,310 lbf·in, i.e. 942 lbf·ft.

3. Interference fit calculator

Enter the geometry, the diametral interference and the materials. The calculation runs in your browser with the formulas from section 1; the yield strength is optional.

Default values: the metric example above. In US units, α is entered in 10⁻⁶/°F (steel: 6.7) and ΔT comes out in °F. Expansion coefficients for other materials are in the thermal expansion table.

4. Minimum and maximum interference: two different checks

An interference fit does not have one interference but a range, set by the tolerances of the bore and the shaft. Each end of the range serves one check:

• Minimum interference → guaranteed torque and axial force. This is the value to compare with the torque to transmit, with a safety factor.
• Maximum interference → hub stresses (the highest is the hoop stress at the bore edge), press-in force and shrink-fit temperature.
• Surface finish: at assembly, roughness peaks are flattened and the effective interference is smaller than the measured one. Standard calculation methods subtract part of the roughness of both surfaces; the smoother the surfaces (low Ra), the smaller the loss.
• Service temperature: if the hub expands more than the shaft, the interference drops by (α_h − α_s) × d × ΔT. An aluminium hub (23 × 10⁻⁶/K) on a Ø 50 steel shaft (12 × 10⁻⁶/K) loses 0.044 mm at +80 K: the whole maximum interference of an H7/p6.

5. Press fit or shrink fit

Press fit: the force needed is about F = μ·p·π·d·L calculated at the maximum interference, with the sliding friction coefficient of pressing. It grows with insertion depth, as the contact length increases. A lead-in chamfer on the shaft or the bore guides the parts and avoids galling.

Shrink fit: the hub is heated until its bore exceeds the shaft by a clearance j: ΔT = (δ + j)/(α·d). The shaft can also be cooled (liquid nitrogen, dry ice): the same formula applies with the shaft's coefficient. Shrink fitting does not wear the surfaces and usually gives an effective interference closer to the measured one than pressing. Check that the heating temperature has no effect on the part's metallurgical state (tempering, surface treatment).

6. Assumptions and limits

• Elasticity: the Lamé equations assume that both parts stay elastic. If the hub's equivalent stress exceeds the yield strength, more interference no longer adds torque capacity.
• Uniform pressure: the calculation assumes a hub and shaft of the same length and a constant pressure. In reality, pressure concentrates at the hub edges, which matters in fatigue (fretting).
• Coefficient of friction: published values for steel on steel range from about 0.1 to over 0.2 depending on surfaces, lubrication and assembly method. Torque is proportional to it: without your own data, a low value gives a safe result.
• Speed: at high speed, centrifugal force expands the hub and reduces pressure; this calculation does not include it.

7. FAQ

How do you calculate the contact pressure of an interference fit?

With the Lamé thick-walled cylinder equations. For a solid shaft and a hub of the same material: p = E·δ/(2d) × (D² − d²)/D², where δ is the diametral interference, d the contact diameter and D the hub outside diameter. For two different materials or a hollow shaft, use the general formula with E and ν of each part.

How much torque can an interference fit transmit?

T = μ·p·π·d²·L/2: the torque equals the friction force on the contact surface (μ·p·π·d·L) times the radius d/2. Calculate it with the fit's minimum interference, the one that gives the lowest pressure.

Should you calculate with the minimum or maximum interference?

Both. The minimum interference sets what the joint transmits at least (torque, axial force); the maximum interference sets the stresses, in particular the hoop stress at the hub bore, and the press-in force or the shrink-fit temperature.

What temperature should a hub be heated to for a shrink fit?

ΔT = (δ + j)/(α·d) above the shaft temperature, where δ is the maximum interference, j the clearance wanted at assembly and α the hub's expansion coefficient. For a Ø 50 steel hub with 0.042 mm interference and 0.05 mm clearance: about 153 K. Check that this temperature is compatible with the part's heat treatment.

What coefficient of friction should you use for a shrink fit?

Published values for steel on steel range from about 0.1 to over 0.2 depending on surface condition, lubrication and assembly method. Without data for your own case, a low value such as 0.1 gives a safe torque; your standard or your company's tests take precedence.