Thermal Stress Calculator

Background.

Heat a steel bar and it grows by about 12 millionths of its length for every kelvin. Let it grow and nothing much happens. Clamp both ends so it cannot, and the expansion it was denied turns into force: the bar pushes on whatever holds it, and the metal inside carries a compressive stress. Cool a restrained bar and the same thing happens in reverse, as tension.

For a fully restrained member the size of that stress is σ = E × α × ΔT. The expansion coefficient α times the temperature change gives the strain the bar would have taken up if it were free, and Young's modulus E converts that blocked strain into stress. For steel, with E around 200 GPa and α around 12 × 10⁻⁶ per kelvin, every kelvin of restrained heating adds 2.4 MPa.

Length appears nowhere in that formula. A one-metre bar and a kilometre of welded rail, each held rigidly at both ends, develop the same stress for the same temperature change. That is the problem railways face with continuous welded rail: the track is fastened down at a chosen stress-free temperature, and on a hot afternoon well above it the rail is in heavy compression and can buckle sideways, while on a cold night it is in tension and a weak weld can pull apart.

The number here is the most a rigid restraint can produce, an upper bound. Real structures rarely hold a member perfectly still; bearings slide, joints open and supports flex, which is exactly what expansion joints are for. The formula also assumes a uniform temperature through the member, elastic behaviour and stress along one axis only. Temperature gradients, yielding, creep and buckling itself all fall outside it.

What is thermal stress calculator?

Thermal stress is the stress that builds up in a material when a temperature change tries to make it expand or contract and something stops it. For a straight, uniform member restrained completely along one axis, its magnitude is σ = EαΔT, the stress needed to cancel the free thermal strain αΔT. Restrained heating produces compression and restrained cooling produces tension; the page reports the magnitude in pascals.

How to use this calculator.

1. 01Enter Young's modulus in pascals. Steel is about 200 GPa, entered as 200000000000 or 2e11, and aluminium alloys about 70 GPa (7e10); use the datasheet value for your grade if you have it.
2. 02Enter the linear expansion coefficient per kelvin, about 0.000012 for carbon steel and about 0.000023 for aluminium. A coefficient quoted per °C is the same number.
3. 03Enter the temperature change as a positive number of kelvin: the difference between the temperature at which the member was fixed without stress and the temperature you want to check.
4. 04Compare the result with the material's yield strength, remembering that it assumes rigid restraint, so a real support that gives a little will see less.

The formula.

|σ| = E × α × |ΔT|. The product α × ΔT is the free thermal strain, the fractional change in length the member would show with nothing in its way; for steel heated by 35 K it is 0.00042, or 0.042%. Full restraint holds the actual strain at zero, so the material must carry a mechanical strain equal and opposite to the thermal one, and Hooke's law turns that into stress by multiplying by E. The three inputs enter as a straight product, so doubling any one doubles the stress, and stiff materials that also expand a lot are the most sensitive. When restraint acts in two directions, as in a plate held along all its edges, the stress rises to EαΔT ÷ (1 − ν), where ν is Poisson's ratio; that case is not covered here.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Boresi and Schmidt, Advanced Mechanics of Materials, 6th ed. (PRINT), thermal stress.
2. [2]OpenStax, University Physics Volume 1, section 12.3, Stress, Strain, and Elastic Modulus.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.