Thermal Expansion Calculator
How much it grows, the gap it needs and the stress if it cannot move — for 26 materials.
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Results are estimates from the formulas shown, not a professional design or certification. Have a qualified engineer verify anything safety-critical (structures, electrical installations, gas or pressure systems).
About the Thermal Expansion Calculator
Find how much a part grows or shrinks with temperature, ΔL = α·L₀·ΔT, together with its new length and the exact change in area, (1 + αΔT)² − 1, and in volume, (1 + αΔT)³ − 1. Choose from metals, concrete, glasses and plastics with editable coefficients, or a liquid, which changes only in volume (ΔV = β·V₀·ΔT).
If the ends are held, the calculator gives the thermal stress σ = E·α·ΔT and the force on the restraint — or, with a gap to the stop, the temperature at which the gap closes and the stress after that. Two more calculations cover the jobs people most often need: the expansion gap to leave in a joint for the lowest and highest temperatures, with a setting table for any installation temperature, and the change of a fit between a shaft and its hole made of different materials, with the temperature to heat a hub (or cool a shaft) for a shrink fit.
Without a pass, the free preview shows your inputs, the checks and the first rows of each table by name; the answer, the other figures, the working, Copy result and the CSV come with a Pro pass.
How to use it
- Choose what to work out: the change in size, an expansion gap or a shaft and hole fit, and SI or US customary units (every box can also be switched on its own; changing a unit converts what you typed).
- Pick a material to fill in its expansion coefficient, or type your own α (β for a liquid) from its data sheet.
- Enter the length and the two temperatures. For the restrained stress, say how the ends are held and give E, and optionally the area and the yield strength.
- For a joint, enter the length that moves into it and the installation, lowest and highest temperatures; for a fit, the diameter, both materials and the fit at the first temperature.
- Read the answer, the other figures and the working. Copy the result or download the CSV with a Pro pass, or after unlocking this result.
Examples
α = 12 × 10⁻⁶/°C, L₀ = 10 m, ΔT = 40 °C
ΔL = 12 × 10⁻⁶ × 10,000 mm × 40 = 4.8 mm; new length 10.0048 m; area +0.096 %, volume +0.144 %
E = 200 GPa
σ = 200,000 MPa × 12 × 10⁻⁶ × 40 = 96 MPa in compression, a force of 96 kN. With a 2 mm gap to the stop: the gap closes after 16.7 °C and the stress reaches 56 MPa
Leave 21 mm at 15 °C; the joint closes to nothing at 50 °C and opens to 42 mm at −20 °C
For 0.02 mm assembly clearance, heat the hub by 43.1 °C (to 63.1 °C from 20 °C), or cool the shaft by 83.3 °C. Warming both parts to 120 °C loosens the fit by 0.056 mm, so the interference is lost
Common uses
- Leaving the right gap in steel beams, rails, decking, cladding or pipe runs
- Checking the stress in a pipe, bar or slab that cannot move
- Working out how hot to heat a bearing, gear or hub to fit it on a shaft
- Physics and mechanics of materials homework, with every step shown
Formulas
- Linear expansion: ΔL = α L₀ ΔT, L = L₀ (1 + α ΔT)
- Area and volume of a solid: ΔA ÷ A₀ = (1 + α ΔT)² − 1 ≈ 2α ΔT; ΔV ÷ V₀ = (1 + α ΔT)³ − 1 ≈ 3α ΔT
- Liquid: ΔV = β V₀ ΔT, with β the volumetric coefficient
- Fully restrained member: σ = E α ΔT (compressive when heated, tensile when cooled), F = σ A
- Stop after a gap g: the gap closes when α L₀ ΔT = g; after that σ = E (α L₀ ΔT − g) ÷ L₀
- Expansion joint: gap to set = g_min + α L (T_max − T_install); widest opening = g_min + α L (T_max − T_min)
- Shaft and hole: the fit changes by (α_hole − α_shaft) D ΔT; to turn an interference i into an assembly clearance c, heat the hub by (i + c) ÷ (α_hole D)
A coefficient per °F is the per-°C value divided by 1.8.
About the coefficients
The pure metals use the coefficients near room temperature from the CRC Handbook of Chemistry and Physics (aluminium 23.1, copper 16.5, iron 11.8, nickel 13.4, titanium 8.6, lead 28.9, zinc 30.2 × 10⁻⁶/°C). Alloys, concrete, glasses and plastics vary with composition, so their presets are typical values: carbon steel about 12, austenitic stainless steel about 17.3, aluminium 6061 23.2, concrete about 12 (it depends on the aggregate), window glass about 9, borosilicate glass 3.3, rigid PVC about 52 and polypropylene about 150 × 10⁻⁶/°C. Every coefficient also changes with temperature; for a wide range, type the mean coefficient for that range from the material’s data.
Sources
- CRC Handbook of Chemistry and Physics: linear expansion coefficients of the elements
- Beer, Johnston, DeWolf & Mazurek, Mechanics of Materials: thermal strain and stress in restrained members
- Material presets: minimum yield strengths of ASTM A36 steel, ASTM A240 type 304 stainless steel and aluminium 6061-T6; the 400,000 psi minimum modulus of rigid PVC compound 12454 (ASTM D1784)
Limitations
- Coefficients are near room temperature; they change with temperature and between grades of the same material.
- The restrained stress is elastic: above the yield strength the member yields, and a slender member in compression may buckle first (check it with the column buckling calculator).
- The joint and fit calculations take the whole part at one uniform temperature. Sun on one face, thick sections that heat slowly and parts at different temperatures need their own analysis.
- Results are estimates for planning and checking, not a professional design — have a qualified engineer check expansion joints, piping and anything safety-critical.
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Frequently asked questions
What do I get without a pass?
Without a pass, Thermal Expansion Calculator shows your inputs, the checks and the names of each table’s first rows, with the change in length, the gap or the fit and every other figure hidden. Until you unlock it, the result can’t be downloaded or copied. A Pro, Premium or Ultimate pass, a one-time payment that never renews, unlocks the full result. The pricing page lists the passes and their prices.
How do I calculate thermal expansion?
Multiply the length by the expansion coefficient and the temperature change: ΔL = α × L × ΔT. A 10 m steel member (α ≈ 12 × 10⁻⁶/°C) warmed by 40 °C grows 12 × 10⁻⁶ × 10,000 mm × 40 = 4.8 mm.
How much does steel expand per metre per degree?
About 0.012 mm per metre for each degree Celsius (12 × 10⁻⁶/°C), which is 1.2 mm per 100 m per degree. Aluminium expands about twice as much (23 × 10⁻⁶/°C) and rigid PVC about four times as much (about 52 × 10⁻⁶/°C).
What is the difference between the linear and the volumetric coefficient?
The linear coefficient α describes the change in length; for a solid that expands the same way in every direction, area grows by about 2α and volume by about 3α per degree. Liquids have only a volumetric coefficient β, and it changes a lot with temperature (water’s is about 207 × 10⁻⁶/°C at 20 °C and zero at 4 °C).
How much stress builds up if a part cannot expand?
For a member held at both ends, σ = E × α × ΔT, whatever its length. Steel heated by 40 °C reaches 200,000 MPa × 12 × 10⁻⁶ × 40 = 96 MPa in compression. Over the yield strength the elastic formula no longer holds, and slender members may buckle before that.
How hot should I heat a hub or bearing to fit it on a shaft?
Heat it until its bore has grown by the interference plus a small assembly clearance: ΔT = (interference + clearance) ÷ (α × D). Heating only the hub is the usual way; the calculator also shows how far the shaft alone would have to be cooled instead. Check the part’s own limit on heating before you start.