How to calculate interference fit pressure

An interference fit grips because the shaft is slightly larger than the hole, so the two parts squeeze each other at a real contact pressure. This is also the interface pressure formula machinists look for when they search press fit pressure or shrink fit pressure. Everything else — the force to press it together, the torque it can carry, whether the hub yields — falls out of that one pressure, and that pressure comes from Lamé thick-cylinder theory.

The pressure formula

p = (E·δ) / (2·d) · (d₀² − d²) / d₀²

δ is the diametral interference (shaft diameter minus hole diameter), d the interface diameter, d₀ the hub outside diameter and E Young's modulus. The first term is the strain the interference imposes; the second, (d₀²−d²)/d₀², is the geometry factor — how much of that strain becomes pressure rather than just stretching a thin hub. This form is for a solid shaft in a hub of the same material; the press fit / interference calculator runs it and the force, torque and stress below. Get the fit class and the interference range first from the hole and shaft fit calculator.

Worked example — 50 mm shaft, 100 mm hub

Steel (E = 200 GPa), d = 50 mm, d₀ = 100 mm, δ = 0.05 mm, μ = 0.15, engagement L = 60 mm:

p = (200000·0.05)/(2·50) · (100²−50²)/100² = 100 · 0.75 = 75 MPa

F = π·μ·p·d·L = π·0.15·75·50·60 ≈ 106 kN · T = F·d/2 ≈ 2.65 kN·m · σ = p·(d₀²+d²)/(d₀²−d²) = 125 MPa

So 0.05 mm of interference makes 75 MPa of grip, takes ~106 kN to press home, and holds ~2.65 kN·m of torque before it slips — while the bore sees a 125 MPa hoop stress, comfortably under steel yield. To shrink-fit instead of pressing, heat the hub by ΔT = δ/(α·d) ≈ 83 °C to open the bore.

Common mistakes

• Using radial instead of diametral interference. δ in this formula is the diameter difference, not the radius difference — halving it halves every result.
• Trusting μ. Friction is the biggest unknown in F and T; dry steel-on-steel runs ~0.1–0.2, lubricated less. Size the grip with a conservative μ and a safety factor, not the optimistic value.
• Forgetting to check the hub. The bore hoop stress σ = p·(d₀²+d²)/(d₀²−d²) can reach yield on a thin hub long before the joint slips — always check it.
• Ignoring temperature and centrifugal loss. A hot or fast-spinning assembly loses interference (the hub grows); the grip you designed cold may not be there in service.

Frequently asked questions

What is the interference fit pressure formula?

From Lamé thick-cylinder theory, the contact pressure for a solid shaft in a same-material hub is p = (E·δ)/(2·d) · (d₀²−d²)/d₀², where δ is the diametral interference (shaft minus hole), d the interface diameter, d₀ the hub outside diameter and E Young's modulus. For d = 50 mm, d₀ = 100 mm, E = 200 GPa, δ = 0.05 mm: p = (200000·0.05)/(2·50) · (100²−50²)/100² = 100·0.75 = 75 MPa.

How do I get the press-in force and torque capacity from the pressure?

Friction on the pressurised interface carries both. The axial press-in (or pull-out) force is F = π·μ·p·d·L and the transmissible torque is T = F·d/2, where μ is the friction coefficient and L the engagement length. With p = 75 MPa, μ = 0.15, d = 50 mm, L = 60 mm: F = π·0.15·75·50·60 ≈ 106 kN and T = 106 kN·25 mm ≈ 2.65 kN·m.

Why does the hub diameter matter so much?

It sets the geometry factor (d₀²−d²)/d₀². A thick hub (large d₀) approaches 1 and gives the full pressure; a thin hub bleeds pressure away because it just stretches. The same thin hub also sees a higher hoop stress σ = p·(d₀²+d²)/(d₀²−d²) at the bore — so reducing d₀ both lowers the grip and raises the yield risk.

How hot does a shrink fit need to get?

To assemble by heating the hub instead of pressing, you need to grow the bore by at least the interference: ΔT = δ/(α·d), where α is the thermal expansion coefficient. For δ = 0.05 mm, d = 50 mm, steel α ≈ 12×10⁻⁶/°C: ΔT = 0.05/(12e-6·50) ≈ 83 °C above the shaft temperature. Add margin for handling clearance.