Stand on a long steel bridge on a hot afternoon and you may hear it: a faint clatter as a vehicle crosses the deck. That sound comes from an expansion joint, a deliberate gap in the structure. The bridge is longer in summer than in winter, by more than a centimetre over a typical span, and the joint is there to absorb that growth. Remove the joint and the bridge would not simply stay put — it would push back against its own supports with enormous force.
This article explains how to predict that thermal movement, how it converts into stress when motion is blocked, and why the two outcomes are really the same physics seen from opposite ends.
Why this calculation matters
Almost every material grows when heated and shrinks when cooled. The effect is small per degree, but it accumulates over length and over temperature swings, and it never switches off. Pipelines, railway track, building frames, engine components, printed circuit boards, and precision instruments all live with it.
The danger is not the expansion itself — it is what happens when something prevents it. A pipe run that cannot expand freely will load its anchors and nozzles. A rail welded into a continuous length on a hot day can buckle sideways. A glass dish moved straight from oven to cold water can crack because its surface cools and contracts faster than its core. Designers handle thermal expansion in one of two ways: provide room for the movement, or accept the stress of restraining it. To choose well, you need to be able to compute both numbers.
The core formula
For a solid heated uniformly, the change in any linear dimension is proportional to the original length and to the temperature change:
dL = alpha * L * dT
Here L is the original length, dT is the temperature change, and alpha is the coefficient of linear thermal expansion — a material property, typically a few parts per million per kelvin. Structural steel sits near 12e-6 per K, aluminium near 23e-6, and Invar, an alloy chosen specifically for dimensional stability, near 1e-6.
That equation describes free expansion: the body is unrestrained and simply changes size. The associated thermal strain is:
epsilon_thermal = alpha * dT
Now suppose the body is fully restrained — held rigidly so it cannot change length at all. The thermal strain still "wants" to occur, but it is cancelled by an equal and opposite mechanical strain, and mechanical strain means stress. For a fully restrained member the thermal stress is:
sigma = E * alpha * dT
where E is the elastic modulus. Notice what dropped out: length. Free expansion depends on L; restrained stress does not. A short restrained bar and a long one reach the same stress for the same temperature swing. Most real structures fall between the two extremes — partially restrained — so the actual stress is some fraction of E*alpha*dT, set by the stiffness of the surrounding supports.
A worked example
Consider a steel bridge girder, length L = 30 m, coefficient of thermal expansion alpha = 12e-6 per K, exposed to a temperature swing of dT = 40 K between a cold winter and a hot summer.
Step 1 — free expansion. If the girder can move freely:
dL = alpha * L * dT
dL = 12e-6 * 30 * 40
dL = 0.0144 m = 14.4 mm
The girder grows by 14.4 mm. That is exactly the kind of movement an expansion joint is sized to absorb — and exactly why such joints exist.
Step 2 — fully restrained stress. Now imagine the girder is instead clamped so it cannot lengthen at all. With steel modulus E = 200 GPa:
sigma = E * alpha * dT
sigma = 200e9 * 12e-6 * 40
sigma = 96e6 Pa = 96 MPa
Restraint converts that harmless 14 mm of movement into 96 MPa of internal compressive stress. For typical structural steel with a yield strength around 250 MPa, 96 MPa is a large fraction of the budget spent on temperature alone — before any traffic, wind, or dead load is added. The lesson is stark: the same 40 K swing is either a 14 mm gap to design around or a 96 MPa stress to carry, and the structure does not get to skip both.
Common mistakes
Mixing up the two coefficients. The coefficient of linear expansion alpha applies to length. For area the effective coefficient is about 2*alpha, and for volume about 3*alpha. Using the linear value where a volumetric one belongs underestimates the change threefold.
Forgetting that restrained stress ignores length. It is tempting to think a longer member builds more thermal stress. It does not. E*alpha*dT has no length term. Length governs free movement; stiffness and temperature govern restrained stress.
Assuming the temperature is uniform. The simple formulas assume the whole body reaches the same temperature. A thick part heated suddenly develops a gradient, and the differential expansion between hot and cool regions produces thermal stress even with no external restraint at all. That is how thermal shock cracks brittle materials.
Using the wrong temperature span. Design for the full swing the structure will actually see — the gap between the coldest winter night and the hottest sun-loaded summer surface — not the comfortable difference between two ordinary days.
Treating alpha as a true constant. It drifts with temperature and varies between alloys and tempers of nominally "the same" material. Over a wide range, use a value representative of the operating band rather than a single room-temperature figure.
Try the interactive NovaSolver calculator
Running alpha*L*dT once is easy; comparing materials and temperature spans quickly is where a tool helps. The Thermal Expansion Calculator on NovaSolver lets you pick a material — steel, aluminium, copper, glass, or Invar — or enter a custom alpha and E, then set the initial dimension and temperature change. It returns the linear, area, and volumetric expansion, the thermal strain, the fully restrained thermal stress, and even the deflection of a bimetallic strip, so you can see free movement and restrained stress side by side.
Related calculators
- Thermal expansion simulator — an animated view of how a part grows and deforms as temperature changes.
- Piping thermal expansion calculator — for the partially restrained case of pipe runs, where loops and anchors share the movement.
- 1D heat conduction calculator — to find the temperature distribution that drives the expansion in the first place.
The full collection lives in the thermal engineering tools hub.
Closing note
Thermal expansion is one of the most predictable effects in engineering and one of the most expensive to ignore. The core idea fits in two short equations: free movement is alpha*L*dT, and the price of blocking that movement completely is E*alpha*dT. Real designs land between those poles. Decide early whether your structure will be given room to move or built to carry the stress of staying still — and run both numbers before you commit, because temperature will collect on whichever one you neglected.
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