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Technical guide · method and practical tips

Thermal expansion formula and coefficient table

A temperature change modifies a part’s dimensions. Free expansion is straightforward to calculate, but the coefficient must match the project material grade and temperature range.

Reviewed

L₀ΔLΔTΔL = α · L₀ · ΔT
The heated bar becomes slightly longer. The movement looks small, but can become significant over a long length.

1. Linear expansion formula

ΔL = α · L₀ · ΔT

Length change is proportional to initial length L₀, linear expansion coefficient α and temperature difference ΔT.

The sign of ΔT indicates heating or cooling. Final length is Lf = L₀ + ΔL.

2. Thermal expansion coefficient table

The values below are the same presets used by the calculator. They support preliminary estimates and do not replace the exact grade datasheet.

Indicative linear coefficients near room temperature
Materialα [µm/(m·K)]Caution
Steel12Typical indicative value. Check the grade and project temperature range.
Stainless steel14Indicative value: about 14 × 10⁻⁶ K⁻¹, with a possible variation of roughly ±4 depending on the family.
Aluminium23Typical indicative value. Individual alloys may differ.
Concrete12Indicative value depending on mix design, aggregates, and moisture.
Bronze17.5Indicative value. The exact bronze composition affects the coefficient.
Constantan15.2Indicative value for a constantan-type alloy.
Copper17Typical indicative value. Check metallurgical condition and reference temperature.

For stainless steel, metallurgical family matters: an Outokumpu datasheet gives about 10×10⁻⁶ K⁻¹ for several ferritic grades and 16×10⁻⁶ K⁻¹ for several austenitic grades between 20 and 100 °C. The preset value of 14 is deliberately intermediate.

3. Understanding α units

The coefficient can be written in K⁻¹, °C⁻¹ or µm/(m·K). For example, 12 µm/(m·K) equals 12×10⁻⁶ K⁻¹.

For a temperature difference, one kelvin and one degree Celsius have the same magnitude. A Fahrenheit difference must be converted.

4. Why initial length matters

A 20 m part expands ten times more than a 2 m part made from the same material under the same temperature change.

This is why the effect is especially visible in bridges, pipes, rails and long structures.

5. Free or restrained expansion

The formula gives free expansion. If supports prevent movement, thermal force and stress can develop.

Real behaviour depends on support stiffness, geometry, material and any clearances or expansion joints.

Numerical application

Worked example: 2 m steel bar

A 2 m bar is heated from 20 °C to 60 °C. Use α = 12 µm/(m·K), the indicative table value.

  1. Calculate ΔT = 60 − 20 = 40 K.
  2. Convert α = 12 × 10⁻⁶ K⁻¹.
  3. Apply ΔL = 12 × 10⁻⁶ × 2 × 40.
  4. ΔL = 0.00096 m.
  5. Convert: 0.00096 m = 0.96 mm.

Result: The bar freely expands by about 0.96 mm. If this movement is fully restrained, a thermal-stress calculation is required.

Method and example checked against the OpenStax chapter on linear thermal expansion. Open exact source ↗

For checking and further study

Direct references

Each link points to the exact course, standard or publication page used, rather than a generic homepage.

  1. Thermal ExpansionOpenStax — University Physics Volume 2Linear expansion formula, coefficient, examples and thermal stress.
  2. Core range datasheet — Thermal expansionOutokumpuThermal expansion values by stainless-steel family and grade.
  3. The International System of Units, 9th editionInternational Bureau of Weights and Measures (BIPM)Official reference for coherent units, symbols and conversions.
  4. NIST Special Publication 811 — Guide to the SINational Institute of Standards and TechnologyPractical rules for writing and converting physical quantities correctly.