Thermal Expansion Calculator – ΔL = αL₀ΔT for Length, Area, Volume and Stress
Almost everything gets bigger when it gets hotter, and the amount is small enough to ignore right up until it is not. A thermal expansion calculator answers the two questions that follow from that: how much bigger does this get, and what happens if you do not let it? This tool works ΔL = α · L₀ · ΔT in one dimension, ΔA = 2α · A₀ · ΔT in two and ΔV = β · V₀ · ΔT in three, and rearranges any of them for whichever quantity you leave blank.
The linear case, and why expansion joints exist
A 100 m carbon-steel bridge girder taken from −10 °C to 35 °C sees ΔT = 45 K. With α = 12.0e-6 /K it grows 12.0e-6 × 100 × 45 = 54.000 mm. That is a thermal strain of ε = αΔT = 540 ppm— five hundredths of one percent, and also the reason every long span carries an expansion joint. Reporting the result in parts per million alongside millimetres is deliberate: ppm is how materials datasheets state α, and it makes the identity “the coefficient of expansion is just strain per kelvin” obvious.
Where the familiar formula starts to lie
ΔL = αL₀ΔT is the first-order expansion of L = L₀(1 + αΔT), and the truncation compounds with dimension. Take a 0.500 × 0.300 m aluminium plate through 80 K: the linear area formula gives 554.400 mm², while expanding each side exactly and differencing gives 554.912 mm². The 0.512 mm² gap is exactly the corner square A₀(αΔT)² that the linear form throws away, and its relative size is always half the thermal strain. The same thing happens in three dimensions: β = 3α is really β = 3α + 3α²ΔT + α³ΔT², which is 0.12 % off for steel over 100 K but 5.77 % off for HDPE over 500 K.
Thermal stress: what restraint costs
Prevent the expansion and it reappears as stress: σ = E · α · ΔT. Carbon steel with E = 200 GPa restrained through 45 K builds 108 MPa, which is 43.2 % of a 250 MPa mild-steel yield from one seasonal swing. Two things surprise people here. First, length does not appear— a 100 m girder and a 100 mm coupon carry the same stress. Second, aluminium expands 1.925× as much as steel yet builds only 71.7 MPa, because its modulus is far lower. The quantity that ranks materials for restrained service is the product Eα, the thermal stress coefficient: 2.400 MPa/K for steel against 1.594 MPa/K for aluminium.
Liquids, containers and bimetallic strips
A brim-full tank overflows by the difference between what the contents gain and what the vessel gains. Sixty litres of gasoline (β = 950e-6 /K) in a steel tank (β = 36.0e-6 /K) warming 25 K spills 1.371 L, not 1.425 L — and for mercury in glass the vessel term hides 15 % of the true expansion, which is precisely how a thermometer is calibrated. Bond two metals with different coefficients and the strip curves toward the low-α side, which is the mechanism inside every mechanical thermostat and circuit breaker.
Practical uses
Size a rail expansion gap from a service temperature range and get both the total travel and the gap to actually set at the install temperature. Work out the shrink fit: a 50 mm steel ring with 40 µm interference needs only 66.67 Kof heating, a hot-water bath rather than a furnace. Or run it backwards — a 2.000 m sample that grew 1.64 mm over 68 K gives α = 12.06e-6 /K, which identifies carbon steel, and lands on the coincidence that makes reinforced concrete possible: concrete and steel share almost exactly the same coefficient, so rebar and concrete move together instead of tearing each other apart.