Coefficient of Thermal Expansion: Calculating Thermal Growth and Restrained Stress

A 20-foot aluminum rail sitting in the Texas sun grows almost a quarter of an inch between a cold morning and a hot afternoon. Pin both ends and that growth turns into thousands of psi of stress. Thermal expansion is one of the easiest effects to calculate and one of the most commonly forgotten — until a part buckles, a gap closes, or a bolted joint loosens. Here is how to compute both the movement and the stress it creates.

What the Coefficient of Thermal Expansion Tells You

The linear coefficient of thermal expansion, α (alpha), is how much a material stretches per degree of temperature change, per unit length. It is quoted in units like µin/in·°F or µm/m·°C (both are just “parts per million per degree”). Aluminum’s α is roughly twice steel’s, which is why mixed-material assemblies fight each other when the temperature swings.

Free Expansion: How Much a Part Grows

If a part is free to move, the length change is a one-line calculation:

where ΔL is the change in length, L₀ is the original length, and ΔT is the temperature change. Nothing about load or stiffness enters — unconstrained thermal growth is purely geometric.

Restrained Expansion: When Growth Becomes Stress

Now clamp both ends so the part cannot grow. The strain it “wanted” is still there, but it shows up as elastic stress instead of movement. Because stress = modulus × strain, and thermal strain is α·ΔT:

Notice the length dropped out — fully restrained thermal stress does not depend on how long the part is, only on the material and the temperature change. That surprises people: a short fully-constrained bar builds the same stress as a long one.

Typical Values to Keep in Your Head

The steel-and-concrete match near the bottom is not a coincidence — reinforced concrete works partly because the two expand together. Mismatch two materials that far apart (aluminum on glass, say) and thermal cycling alone will fatigue the joint.

Area and Volume Grow Too

The single α above is the linear coefficient. Surfaces and volumes expand faster: to a very good approximation the area coefficient is 2α and the volumetric coefficient is 3α. That is why a hole in a heated plate gets bigger, not smaller — the metal around it expands outward with everything else. Shrink-fit and interference-fit design leans directly on this: heat the housing, its bore opens up by α·D·ΔT, you drop the shaft in, and it clamps as it cools.

Sizing for the Movement

Once you know ΔL, design the connection to absorb it rather than fight it. A slotted bolt hole should be longer than the fastener diameter by at least the computed ΔL over the run it restrains. Expansion loops in piping, sliding bearing pads under long beams, and gapped facade panels are all just ΔL made into hardware. The mistake is sizing the gap for the average temperature instead of the full extreme-to-extreme swing the part will actually see — use the widest realistic ΔT, not the comfortable one.

Where This Bites in Practice

• Slip-fit vs. press-fit: a shrink fit relies entirely on differential ΔL — heat the hub, drop it on the shaft, let it clamp as it cools.
• Bolted joints: differential expansion between a bolt and the clamped members changes preload with temperature — a joint torqued cold can loosen hot, or vice versa.
• Long runs: pipe, rail, and facade panels need expansion loops or slotted connections sized to the computed ΔL.

Run the Numbers

Grab α for dozens of metals, plastics, and glasses from the Coefficient of Thermal Expansion chart, then use the Engineering Unit Converter to move between °F/°C and psi/MPa cleanly. Choosing the material itself? Start from the alloy comparison in the Materials & Process toolset.