Thermal Expansion Calculator
Work out how much a length, an area or a volume grows when it warms up. Solve ΔL = αL₀ΔT for any of its values, size an expansion gap for a bridge, rail or pipe, and find the stress when the material is held so it can’t move.
How to Use
- Pick Linear, Area or Volume for how much something grows, or Gap & stress for an expansion joint and the stress when it is held.
- Choose what to solve for: the change, the coefficient, the original size or the temperature change.
- Choose a material to fill in its typical coefficient (and stiffness for Gap & stress), or type your own α in ×10⁻⁶/K, per K or ×10⁻⁶/°F.
- Enter the size and the temperature change in K, °C or °F. A temperature change in °C is the same number in K; use a negative value for cooling.
- Read the answer in its highlighted field and the first readout, see the change drawn (exaggerated so you can see it), and check Show Work for the steps.
Worked Example
A steel bridge. A 100 m steel deck with α ≈ 12 × 10⁻⁶/K, from a winter low to a summer high 60 °C warmer: ΔL = 12 × 10⁻⁶ × 100 × 60 = 0.072 m = 72 mm. The expansion joint must open and close by at least that much.
A rail that can’t move. Held at both ends through a 30 °C rise, the steel takes a strain of 12 × 10⁻⁶ × 30 = 360 × 10⁻⁶. With E = 200 GPa that is σ = 200 × 10⁹ × 360 × 10⁻⁶ = 72 MPa of compression, whatever the length. Left free, a 25 m length would need a 9 mm gap instead.
The common mistake: a Fahrenheit change with a per-kelvin α. A 60 °C swing is a 108 °F swing. Putting 108 into ΔL = αL₀ΔT with α in per kelvin gives 12 × 10⁻⁶ × 100 × 108 = 129.6 mm for the bridge, not 72 mm. A Celsius change and a kelvin change are the same number; a Fahrenheit change must be multiplied by 5/9 first, or α given per °F.
Show Work
Formulas
Typical coefficients near room temperature, rounded, from handbook data (CRC Handbook of Chemistry and Physics for the pure metals; OpenStax College Physics for glass, concrete and liquids; common engineering values for E). Alloys, grades and temperature change them, so use a datasheet for design work.
| Material | α (×10⁻⁶/K, typical) | E (GPa, typical) |
|---|---|---|
| Carbon steel | 12 | 200 |
| Stainless steel 304 | 17.3 | 193 |
| Aluminium | 23.1 | 69 |
| Copper | 16.5 | 117 |
| Brass | 19 | – |
| Concrete | 10–14 (12 used) | about 30 |
| Soda-lime glass | 9 | 70 |
| Borosilicate glass | 3.3 | 64 |
| Invar (36% nickel iron) | 1.2 | – |
| Liquids, volume β | water 207 (20 °C), mercury 181, petrol 950, ethanol 1,100 | – |
From Thermometers to Invar
Thermal expansion was put to work before it was measured. The liquid-in-glass thermometer depends on it, and Daniel Gabriel Fahrenheit’s mercury thermometers of the early 18th century worked because mercury expands steadily and much more than the glass around it. Through the 18th century clockmakers fought it too: John Harrison’s gridiron pendulum combined brass and steel rods so that their expansions cancelled and the pendulum kept its length.
The most famous answer came in 1896, when the Swiss physicist Charles-Édouard Guillaume found that an iron alloy with about 36% nickel hardly expands at all. He called it Invar, for invariable, and it went into precision clocks, surveying tapes and measuring instruments. Guillaume received the Nobel Prize in Physics in 1920 for the discovery.
Engineers meet the same physics in every long structure: expansion joints in bridges and buildings, loops in steam pipes, and the stress management of continuously welded rail, which is laid and anchored so that it never has room to buckle on a hot day.
About This Tool
This calculator works out thermal expansion in one, two or three dimensions and solves for whichever value you are missing: the change, the coefficient, the original size or the temperature change. The Gap & stress mode sizes an expansion gap and shows the stress and force that would build up if the material were held, using E α ΔT. A material list fills in typical coefficients, liquids included, and every value accepts metric or US units, with temperature changes in K, °C or °F.
Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Heat Conduction Calculator, Specific Heat Calculator, and Ideal Gas Law Calculator.
Frequently Asked Questions
What is the formula for thermal expansion?
For a length, ΔL = α × L₀ × ΔT: the expansion coefficient times the original length times the temperature change. A 100 m steel bridge (α ≈ 12 × 10⁻⁶/K) warming by 60 °C grows 12 × 10⁻⁶ × 100 × 60 = 0.072 m, or 72 mm.
How big an expansion gap does a bridge or rail need?
The gap has to take the full change between the coldest and hottest the structure will see. A 25 m steel rail through a 30 °C swing changes by 12 × 10⁻⁶ × 25 × 30 = 9 mm, so the joint needs at least 9 mm, plus whatever it must keep at the hottest.
What stress builds up if expansion is blocked?
σ = E × α × ΔT, independent of the length. Steel (E ≈ 200 GPa) held rigid through 30 °C carries 200 × 10⁹ × 12 × 10⁻⁶ × 30 = 72 MPa, about 10,440 psi, which is why continuously welded rail is laid under tension and anchored firmly.
How do area and volume expansion relate to α?
An area grows by about 2α and a volume by about 3α per degree, because each direction grows by α. A 1 m² aluminium plate (α ≈ 23.1 × 10⁻⁶/K) heated by 100 °C gains 2 × 23.1 × 10⁻⁶ × 100 = 46.2 cm². Liquids have their own volume coefficient β instead.
Why does borosilicate glass survive hot water better than ordinary glass?
It expands far less. Held against a 100 °C difference, soda-lime glass (α ≈ 9 × 10⁻⁶/K, E ≈ 70 GPa) would carry 63 MPa of stress, while borosilicate (α ≈ 3.3 × 10⁻⁶/K, E ≈ 64 GPa) carries only 21.12 MPa, a third as much.
How do I use the Thermal Expansion Calculator?
Just type your numbers. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Do I need to install or sign up for anything?
Not at all — it runs in the browser with nothing to install and no account. After it loads once, it even works without an internet connection.
Is my information private?
Yes. Everything happens in your browser. Nothing you type is sent to a server or saved anywhere.
Common Use Cases
Bridges and expansion joints
A 300 m steel span through a 30 °C swing changes by 108 mm, which the joints at its ends must absorb.
Railway track
Welded steel rail held at both ends through 30 °C builds 72 MPa; with a 7,600 mm² rail section that is 547.2 kN of push.
Fuel and liquids
50 L of petrol (β ≈ 950 × 10⁻⁶/K) warming by 20 °C swells by 0.95 L, so a brim-full tank can overflow in the sun.
Plumbing
A 10 m copper hot-water pipe heated by 60 °C grows 9.9 mm, so long runs need bends or loops to take it up.
Shrink fits
Heating a steel part by 200 °C opens a 50 mm hole by 0.12 mm, enough to slide a shaft in that is locked tight once it cools.
Identifying a metal
A 1 m rod that grows 1.15 mm when heated by 100 °C has α = 11.5 × 10⁻⁶/K, close to carbon steel.
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