Calculator of Alexandria

Thermal expansion calculator

How much steel, aluminum, copper, concrete, glass or PVC grows or shrinks with temperature, using ΔL = αLΔT, with the formula explained.

Change in lengthΔL\Delta L
New length (m)LL
Change in length (in)

Formula

ΔL=αL0ΔT\Delta L = \alpha L_0 \Delta T
How it's derived

The formula

ΔL=α L0 ΔT\Delta L = \alpha\,L_0\,\Delta T

ΔL\Delta L is the change in length, α\alpha the material’s coefficient of linear expansion, L0L_0 the starting length and ΔT\Delta T the change in temperature.

Where it comes from

Atoms in a solid jiggle around their resting positions, and the hotter the solid, the bigger the jiggle. The forces between atoms are lopsided: they resist being pushed together more strongly than being pulled apart. So as the jiggling grows, each atom spends more time a little farther from its neighbors, and the average spacing grows.

Over ordinary temperature ranges, the extra spacing is very nearly proportional to the temperature rise. A rod is just a long chain of these spacings, so each part of it grows by the same fraction. That fraction per degree is α\alpha:

ΔLL0=α ΔT\frac{\Delta L}{L_0} = \alpha\,\Delta T

Multiply by L0L_0 and you have the formula. A rod twice as long grows twice as much, and twice the temperature change doubles the growth too.

Because it’s a change in temperature, °C and kelvin give the same answer. A change of 1 °F is only 5/9 of a kelvin, which the calculator handles for you.

Worked example

A 30 m steel rail laid at 0 °C heats to 40 °C in summer sun. With α=12×10−6\alpha = 12 \times 10^{-6} per °C:

ΔL=12×10−6×30×40=0.0144 m\Delta L = 12 \times 10^{-6} \times 30 \times 40 = 0.0144\ \text{m}

That’s 14.4 mm, which is why rails, bridges and long pipes need expansion gaps or joints.

Watch out for

  • Coefficients vary. The values in the list are typical figures near room temperature. Alloys, concrete mixes and plastics vary, and α\alpha itself changes at very high or low temperatures. Use your material’s datasheet for critical work.
  • Areas and volumes. An area grows by about 2α ΔT2\alpha\,\Delta T, and a volume by about 3α ΔT3\alpha\,\Delta T. Liquids are quoted with a volume coefficient instead.
  • Restrained parts build stress. If the ends can’t move, the part doesn’t grow; it pushes instead. That force can buckle rails and crack concrete.

Sources

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