How To Calculate Thermal Expansion

Thermal expansion is the tendency of matter to change its dimensions as its temperature changes. For most solids the change is close to proportional to both the original length and the temperature change, which makes it unusually easy to predict.

The coefficient of linear expansion is the proportionality constant. It states how much one unit of length grows per one degree of temperature rise, and it is a property of the material rather than of the object.

Coefficients are small numbers, which is why they are quoted per million. Steel sits near 12, aluminium near 23 and concrete near 10, all in units of one millionth per degree Celsius.

Those differences matter. An aluminium component expands roughly twice as much as a steel one over the same temperature range, which is why mixing metals in an assembly is a design problem rather than a detail.

The formula is linear, and that is an approximation. It holds well over ordinary temperature ranges, but the coefficient itself drifts at very low or very high temperatures, so a figure measured near room temperature is not a licence to extrapolate to cryogenic conditions.

Expansion happens in every direction, not just lengthwise. For an isotropic material the area grows by roughly twice the linear fraction and the volume by roughly three times, which is why the same coefficient appears in area and volume calculations with different multipliers.

The area version is slightly larger than twice, because the cross terms count. Expanding by a factor of one plus a small amount and then squaring gives twice the small amount plus its square, and that square is usually negligible but not always.

Strain is the expansion expressed as a fraction of the original length, and it is just the coefficient times the temperature change. It is dimensionless, which makes it the natural way to compare materials independently of the object's size.

If the expansion is prevented, it turns into stress. A restrained member cannot lengthen, so the strain the temperature wanted to produce is absorbed elastically, and the resulting force can be very large.

That is the engineering reason the topic matters. Bridges have expansion joints, railways have gaps, pipelines have loops, and continuous welded rail is laid at a temperature chosen so the stress stays within limits.

Bimetallic strips make deliberate use of the effect. Two metals with different coefficients bonded together must bend when heated, and that bending is the whole mechanism of the classic thermostat.

For the running example, a 10 metre steel span with a coefficient of 12 per million per degree, warmed by 40 degrees, gives a strain of 0.00048.

Multiplied by the original length of 10 metres, the change is 0.0048 metres, which is 4.8 millimetres. The new length is 10.0048 metres.

Four point eight millimetres on a ten metre span sounds negligible, and in many situations it is. Across a hundred metres of bridge it becomes 48 millimetres, which is not.

Cooling works identically with a negative temperature change. A 100 metre steel span cooled by 25 degrees contracts by 0.03 metres, or 30 millimetres, which is why joints have to accommodate movement in both directions.