Iron Tire Shrink Fit

What is modelled

An iron tire is forged with an inside diameter a little smaller than the wheel. Heating enlarges its circumference in proportion to its temperature rise, so it can be slipped over the wheel with some clearance. On cooling, it tries to return to its smaller size, is held back by the wood, and so carries a hoop tension and presses inward on the wheel.

Equations

• Thermal expansion ΔD = α D ΔT, so ΔT = (δ + c)/(α D) for interference δ and clearance c.
• Hoop strain ε = δ/D, stress σ = E ε; thin-ring pressure p = σ t / a.
• Thick-ring (Lame) pressure on a rigid wheel: p = E u / (a((b²+a²)/(b²-a²) + ν)), u = δ/2.
• Lumped heating m c dT/dt = P - hA (T - Ta) with time constant m c / hA.

Checks run in the page

• Diameter after stepwise heating against the target.
• Thin-ring against Lame pressure within the expected error.
• Newtonian cooling against its exponential.
• Forge heating with losses against its closed form.

Honest simplifications

A rigid wheel (real wood and the felloes give way a little), a lumped uniform tire temperature, constant material properties, a single convection coefficient and no quench water. Illustrative of the heritage craft, not a design tool.

• Press Too loose and see the contact pressure fall with the small interference.
• Press Slow forge: the losses may never let the tire reach the fitting temperature.
• Thicken the tire and compare the thin-ring and Lame pressures.