Fits at operating temperature: when aluminium and steel part company
The formula is trivial, applying it is not
ΔL = α · L · ΔT
α coefficient of linear expansion [1/K]
L original length or diameter [mm]
ΔT temperature difference from 20 °C [K]
Nothing complicated about that. The mistake happens earlier: in deciding which length and which temperature difference to put in. For a fit, the relevant length is the joint diameter, not the length of the part. And the temperature difference is the one between operation and measurement, not between operation and ambient.
| Material | α in 10⁻⁶/K | Expansion at Ø 60 mm and 60 K |
|---|---|---|
| Invar | 1.5 | 5 µm |
| Titanium | 8.6 | 31 µm |
| Grey cast iron | 10.5 | 38 µm |
| Structural steel | 12 | 43 µm |
| Stainless steel | 16 | 58 µm |
| Brass | 18 | 65 µm |
| Aluminium | 23 | 83 µm |
| POM | 110 | 396 µm |
Values from the material list of the thermal expansion calculator. For plastics these are guide values; they depend heavily on filler and moisture.
The aluminium figure is larger than any manufacturing tolerance you would sensibly demand on such a part. With plastic it becomes absurd: the expansion alone exceeds the tolerance by more than an order of magnitude.
The case that keeps coming up
Steel shaft in an aluminium housing, Ø 60 mm, fit H7/g6. At 20 °C the clearance is between 10 and 59 µm. Now the unit runs and the assembly warms to 80 °C, which is 60 K above measuring temperature.
Δ clearance = (α_bore − α_shaft) · D · ΔT
= (23 − 12) · 10⁻⁶ · 60 mm · 60 K
= 0.0396 mm ≈ 40 µm
Clearance at 80 °C: 50 … 99 µm
The clearance has quadrupled without a single part leaving the drawing. For a plain bearing that is often still acceptable; for a location feature it is not. If you have to hold a runout requirement in service, you have a problem here that no fit table mentions.
The other direction is more interesting. The same unit in winter storage at minus 30 °C, that is 50 K below measuring temperature:
Δ clearance = (23 − 12) · 10⁻⁶ · 60 mm · (−50 K)
= −0.033 mm ≈ −33 µm
Clearance at −30 °C: −23 … +26 µm
In the lower branch the clearance is negative. The aluminium housing shrinks more than the steel shaft and grips it. A drive that will not start at minus 30 in the cold and then suddenly works after half an hour of running usually has exactly this cause.
Shrink fits: calculate the temperature instead of guessing
To assemble an interference fit the outer part is heated until the bore is large enough. Large enough means: the maximum interference plus an assembly clearance that still lets the part slide on by hand.
Example: Ø 30 mm, pairing H7/p6, maximum interference 35 µm. With 30 µm of assembly clearance, 65 µm have to be bridged, and the part is steel.
ΔT = ΔD / (α · D)
= 0.065 mm / (12 · 10⁻⁶ · 30 mm)
= 180 K
Oven temperature: 20 °C + 180 K = 200 °C
Two things matter here. First, the part cools on its way to the press, which is why in practice you add another 30 to 50 K. Second, there is an upper limit: with quenched and tempered parts stay well below the tempering temperature or the part loses hardness. For rolling bearings the customary limit is 120 °C; above that the microstructure of the rings changes.
If temperature will not do it, go the other way and cool the inner part in liquid nitrogen or dry ice. At −196 °C a Ø 30 steel shaft shrinks by a good 70 µm, which is enough for most seats.
Where else expansion shows up
In measurement
The reference temperature for dimensional metrology is 20 °C, laid down in ISO 1. A steel part 500 mm long, measured at 28 °C in a workshop in summer, is 48 µm longer than at reference temperature. With a tolerance of ± 0.05 mm the measurement is worthless. Either you measure in a temperature-controlled room, or you correct for it, or you accept that the argument with the supplier cannot be settled.
In mixed construction
An aluminium plate bolted to a steel frame, 800 mm between the outer bolts, 40 K of warming: the plate grows 0.35 mm more than the frame. If both ends are bolted solid, that difference has to go somewhere. It goes into the bolts, into deformation, or into slotted holes if somebody thought of them.
In plastic and printed parts
POM expands about nine times as much as steel. A POM bush of Ø 30 mm in a steel housing loses around 0.12 mm of clearance against the bore over 40 K of warming. With printed parts, expansion is direction-dependent on top of that and the values scatter with the print settings. If you design a fit here, measure the result rather than only calculating it.
What design can do about it
- Same material in the joint. Steel on steel keeps its fit across the whole temperature range, because both partners behave alike.
- Define a fixed point. On long assemblies one place is bolted solid and all the others get slotted holes or floating bearings. In machine building that is the same principle as in bridge building.
- Size the fit for the operating case. If the assembly spends its life at 80 °C, the fit at 20 °C is only an assembly condition. Then you calculate backwards and accept a tighter clearance at room temperature.
- Stack-up with a temperature link. The difference in expansion is a link like any other and belongs in the calculation, not in a footnote.
And if none of that works, there is the honest option: put the temperature range for which the specifications apply on the drawing. It does not solve the problem, but it makes it visible before the first unit stops working in winter.
Frequently asked questions
For a diameter, do I use the radius or the diameter?
The diameter. Expansion is a relative change in length that acts on every distance in the part equally. A bore gets larger when heated, not smaller, because the material around it expands and the edge of the hole moves with it.
Does the coefficient apply across the whole temperature range?
Only approximately. In metals α rises slightly with temperature; between room temperature and roughly 200 °C the error is small enough to ignore. In plastics the value changes abruptly above the glass transition temperature, and there a single coefficient is no basis for anything.
How far may a bearing ring be heated?
Up to 120 °C is the usual limit. Above that the microstructure of the hardened rings can change and dimensional stability is lost. Induction bearing heaters therefore control to that range. Bearings should never be heated with an open flame.
How does thermal expansion enter the stack-up?
As an additional link whose nominal value is the difference in expansion and whose tolerance follows from the uncertainty in temperature. If the operating temperature lies between 60 and 90 °C, the expansion is itself a toleranced dimension. The stack-up calculator takes such a link like any other.