Thermal Expansion and Clearances: Sizing Gaps That Survive Temperature

The Linear Expansion Formula

The change in length of a free, unrestrained part as its temperature changes is ΔL = α · L₀ · ΔT, where L₀ is the original length, ΔT is the temperature change, and α is the coefficient of linear thermal expansion. Its units are reciprocal temperature, written as 1/°C, /K, or equivalently µm/(m·K); the three are interchangeable for a temperature difference.

Growth is proportional to the original length, so a long run accumulates far more movement than a short one from the same temperature swing. A 30 m steel pipe rising 100 °C grows about 36 mm, enough to destroy a rigidly anchored run, while a 30 mm steel pin under the same swing grows about 0.036 mm, which is usually irrelevant. Note also that α itself varies with temperature, so published figures are averages over a stated range and should be treated as typical, not exact.

For isotropic materials the area coefficient is approximately 2α and the volume coefficient approximately 3α. These first-order approximations are perfectly adequate because the neglected terms involve α squared and cubed, quantities on the order of 10⁻¹⁰ for metals. A practical corollary catches people out regularly: a hole in a heated plate gets bigger, not smaller, because it scales up exactly as if it were filled with the surrounding material.

• Linear: ΔL = α · L₀ · ΔT.
• Area: ΔA ≈ 2α · A₀ · ΔT for isotropic materials.
• Volume: ΔV ≈ 3α · V₀ · ΔT for isotropic materials.

Every hole, bore, and gap in a heated part scales up by the same fractional amount as the solid material around it.

Typical Coefficients and Why They Differ

Expansion coefficients are among the more widely spread properties in common engineering use. Carbon and alloy steels typically fall around 11 to 13 × 10⁻⁶ /°C, with 12 × 10⁻⁶ /°C a reasonable working figure. Austenitic stainless steels such as 304 run higher, typically near 17 × 10⁻⁶ /°C, which is why a 304 part and a carbon steel part that fit at room temperature will not fit at 150 °C. Copper is also near 17, and aluminum alloys higher again at roughly 23 × 10⁻⁶ /°C.

The spread widens outside the metals. Common thermoplastics run an order of magnitude above steel, often 70 to 200 × 10⁻⁶ /°C, which is why plastic panels are mounted in slotted holes rather than clamped rigidly. Concrete sits near 10 × 10⁻⁶ /°C, close enough to steel that reinforced concrete works at all. Some carbon fibre laminates are near zero or slightly negative along the fibre direction and anisotropic besides, so the 2α and 3α shortcuts do not apply.

Treat every published coefficient as typical. Values vary with alloy, temper, temperature range, and measurement method, and references disagree at the percent level routinely. That is rarely the limiting factor in a clearance calculation, but it does mean carrying margin rather than sizing a gap to the last micrometre from a single tabulated number. The materials reference lists a coefficient alongside modulus and strength for every entry, which makes comparing two candidates straightforward before you commit to a fit.

• Carbon and alloy steel: typically about 11 to 13 × 10⁻⁶ /°C.
• 304 stainless and copper: typically about 17 × 10⁻⁶ /°C.
• Aluminum alloys: typically about 23 × 10⁻⁶ /°C, roughly twice steel.
• Common thermoplastics: often 70 to 200 × 10⁻⁶ /°C.

Differential Expansion Sets the Clearance

When two parts of the same material heat up together, their fit barely changes, because both grow by the same fraction. What moves a clearance is the difference in expansion between the mating materials, or a difference in their temperatures, or both. That difference is the number worth computing, and it is often far smaller and more manageable than either part's absolute growth.

Take a steel shaft in an aluminum housing bore, both nominally 50 mm, and raise both by 80 °C. The aluminum bore grows 50 × 23 × 10⁻⁶ × 80 = 0.092 mm on diameter; the steel shaft grows 50 × 12 × 10⁻⁶ × 80 = 0.048 mm. The clearance therefore opens by about 0.044 mm. For a plain bearing that may be fine; for a precision locating fit it may be a complete loss of location that a room-temperature inspection will never reveal.

The same arithmetic run backwards is how interference fits are assembled. To install a steel ring with a 100 mm bore onto a shaft with 0.05 mm of interference, the bore must grow by at least 0.05 mm, which takes a temperature rise of about 0.05 / (12 × 10⁻⁶ × 100) ≈ 42 °C above the shaft. Heating the outer part, chilling the inner part, or both is standard practice, and the required difference falls straight out of ΔL = α·L₀·ΔT with the interference substituted for ΔL.

Clearances are governed by the difference in α between mating parts, not by either coefficient alone. If the outer part has the higher coefficient, heat loosens the joint and cold tightens it; if the inner part does, the reverse holds, so check both extremes.

When Expansion Is Restrained: Thermal Stress

If a part is prevented from expanding, the strain it would have taken freely becomes stress instead. For a bar rigidly held between immovable supports, that stress is σ = E · α · ΔT, which follows from setting the restrained thermal strain α·ΔT equal to an elastic strain σ/E. Note what is absent: the length. A restrained member 30 m long and one 30 mm long develop the same thermal stress from the same temperature change.

The numbers get large quickly. A fully restrained steel member with E = 200 GPa and α = 12 × 10⁻⁶ /°C heated 50 °C develops σ = 200 × 10⁹ × 12 × 10⁻⁶ × 50 = 120 MPa, roughly half the 250 MPa yield typical of common structural steel. Aluminum under the same restraint reaches about 79 MPa, lower despite the higher coefficient, because its much lower modulus more than compensates.

Perfect restraint is rare, so real thermal stresses usually land below that prediction, but it is the right upper bound to check. The engineering answer is nearly always to accommodate movement rather than resist it: expansion loops in piping, slotted holes on panels, sliding supports at one end of a long run. Compression is the dangerous direction, since a heated slender strut can buckle well before it reaches the predicted stress.

σ = E · α · ΔT under full restraint, and it does not depend on length. Accommodating movement is nearly always cheaper than resisting it.

Putting It Into a Design

A workable routine is short. Establish the real temperature range the assembly will see, including installation, storage, and transients such as a wash-down or a cold start, rather than the steady operating point alone. Identify the functionally critical dimensions: a bearing clearance, a sealing gap, an alignment path. For each, compute the growth of both mating parts at both extremes and take the difference.

Then compare that differential against the clearance you have and against the manufacturing tolerance stack, because thermal movement and tolerance consume the same gap. The thermal term is often as large as the machining tolerance, which is why the two must be budgeted together. Where the numbers do not close, the options in rough order of preference are to match the coefficients, shorten the dimension accumulating the growth, design the movement in with a slot or flexure, or accept the stress and verify it.

If you want the coefficients side by side while working through a stack, the material properties reference sheet collects them for every material in the database in one printable layout, which is part of what a Premium subscription pays for; the same figures are free to read on each material's own page.