Thermal expansion calculator
Enter your values. The answer and the full worked solution update as you type.
Result
The α values are typical values near room temperature; the exact value depends on the alloy or grade.
Work out how much a part grows or shrinks when its temperature changes. Choose linear expansion for lengths, such as rails, pipes and bridge decks, or volumetric expansion for volumes, pick a material or enter your own coefficient, and get the change, the final size and, for a bar that cannot move, the thermal stress σ = E·α·ΔT.
Step-by-step solution
Coefficient of linear expansion
Temperature change
Change in length
Thermal stress if the ends cannot move
Key formulas
- Thermal strain
- εT = α·ΔT
- Strain
- ε = ΔL/L
How to calculate thermal expansion
- Find the temperature change ΔT = T₂ − T₁. A change of 1 °C is the same as a change of 1 K.
- Look up the coefficient of linear expansion α of the material, for example about 12 × 10⁻⁶/K for steel and 23 × 10⁻⁶/K for aluminium.
- Change in length: ΔL = α·L₀·ΔT, with L₀ and ΔL in the same unit.
- Change in volume: ΔV = β·V₀·ΔT, with β ≈ 3α for a solid.
- If the ends are held, the bar cannot expand and a stress σ = E·α·ΔT builds up instead: compressive when it is heated, tensile when it is cooled.
Questions and answers
Why is β about 3α?
A cube with side L becomes L(1 + αΔT) in every direction, so its volume becomes V(1 + αΔT)³ ≈ V(1 + 3αΔT), because αΔT is very small. This holds for isotropic solids; liquids and gases have their own coefficient of volume expansion.
Where do the values of α come from?
They are typical values near room temperature from standard tables, all × 10⁻⁶/K: structural steel 12, stainless steel 17.3, aluminium 23, copper 17, brass 19, concrete 12, soda-lime glass 9 and PVC 52. Real values depend on the alloy, grade or mix and change slowly with temperature, so use the value your course or data sheet gives.
In which units do I enter the length?
Enter L₀ in metres; ΔL is given in millimetres because the change is usually small. The formula itself works in any unit as long as L₀ and ΔL use the same one.
Why does the restrained stress not depend on the length?
The stress depends on the strain the bar is prevented from taking, αΔT, which does not depend on the length. A fully restrained steel rail of 1 m or of 100 m reaches the same σ = E·α·ΔT = 84 MPa for a rise of 35 K.
Does a hole get bigger or smaller when the plate is heated?
Bigger, by the same fraction as the material around it, as if the hole were filled with the same material. That is why heating a ring makes it easier to fit over a shaft.
Practise it yourself
Practise Axial stress & strain
A calculator gives the answer. In the exam you need the method: practise it with new problems, hints and worked solutions.