Thermal Expansion and Shrink Fit Calculator

How is thermal expansion calculated?

Solids grow when heated and shrink when cooled. The change in length is proportional to the original length, the temperature change and the expansion coefficient of the material: ΔL = α × L × ΔT. The coefficient α tells how much a unit length grows per kelvin. It is about 12 × 10⁻⁶ 1/K for steel and about 23 × 10⁻⁶ 1/K for aluminium.

The same relation holds for a diameter. The bore of a heated hub grows just as the diameter of a heated shaft does, which is what shrink fitting relies on.

Example

A 10 metre steel section warms by 50 degrees. It grows by 12 × 10⁻⁶ × 10 × 50 = 0.006 m, or 6 mm. A handy figure: steel grows about 1.2 mm per metre per 100 degrees. Aluminium grows roughly twice as much.

What if expansion is restrained?

A part fixed at both ends cannot grow, so stress builds up: σ = E × α × ΔT. This stress does not depend on the length. In steel, a 50 degree rise gives 210 000 × 12 × 10⁻⁶ × 50 = 126 MPa of compression. That is a large value, which is why long pipelines, rails, bridges and steel structures are given expansion gaps or expansion joints.

Interference fits (press and shrink fits)

When a shaft is made slightly larger than its bore and forced into the hub, the two parts grip each other and the contact pressure carries torque by friction. The pressure follows from the Lamé equations: p = δ / (d × [(K_g + ν_g) / E_g + (K_m − ν_m) / E_m]). Here δ is the diametral interference, and K_g and K_m are factors that depend on the diameter ratios of hub and shaft. Torque capacity is T = μ × p × π × d × L × d / 2.

For 40 microns of interference on a 50 mm diameter, with a 100 mm steel hub on a solid steel shaft, the pressure is 63 MPa and the hoop stress at the hub bore is 105 MPa. To assemble that fit with 20 microns of clearance, the steel hub must be 100 degrees hotter than ambient: ΔT = (δ + c) / (α × d).

Limits

The coefficient changes with temperature, while the calculator holds it constant. The interference fit tabs are a preliminary calculation: surface smoothing, the tolerance band, centrifugal effects and fatigue must be assessed separately to the standard. The stress result is the worst case of fully restrained expansion. Real supports give a little and the stress is lower. If the result approaches the yield strength, or the part is slender and may buckle, a detailed analysis is needed.