Thermal expansion calculator

How do you calculate thermal expansion?

Change in length ΔL = α L ΔT, where α is the material’s coefficient of linear expansion, L the original length and ΔT the temperature change. A coefficient per °C is the same per kelvin; per °F it is 1.8 times smaller. The coefficients here are the representative values from OpenStax University Physics, Table 1.2 — real alloys and mixes vary a little.

Example: the Golden Gate Bridge

OpenStax Example 1.2: the steel main span is 1,275 m long at its coldest, and the temperature ranges from −15 °C to 40 °C. ΔL = 12 × 10⁻⁶ /°C × 1,275 m × 55 °C ≈ 0.84 m. The book notes that this change is spread over many expansion joints, so each joint moves only a little.

How much do steel, concrete and aluminum expand?

From Table 1.2: iron or steel 12 × 10⁻⁶ /°C (about 6.7 × 10⁻⁶ /°F), concrete and brick about the same, aluminum 25 × 10⁻⁶ /°C. So a 100 ft steel member warming by 100 °F grows about 0.8 in, and the same length of aluminum about 1.7 in. OpenStax points out that steel and concrete expand almost equally, which is why reinforcing bars in concrete are steel.

How does volume expansion work?

ΔV = β V ΔT, with β ≈ 3α for solids; liquids and gases have only a volume coefficient. In OpenStax Example 1.3 a full 60.0 L steel gas tank warms from 15 °C to 35 °C: the gasoline grows by 950 × 10⁻⁶ × 60 × 20 = 1.14 L and the tank by 35 × 10⁻⁶ × 60 × 20 = 0.04 L, so about 1.10 L spills. Water is the odd one out — between 0 °C and 4 °C it expands as it cools.

What if the material can’t expand?

It is stressed instead: F/A = Y α ΔT, where Y is Young’s modulus. OpenStax Example 1.4: concrete blocks laid tight at 5 °C and heated to 38 °C see 20 × 10⁹ Pa × 12 × 10⁻⁶ /°C × 33 °C ≈ 7.9 × 10⁶ N/m² (about 1,150 psi). The table shows this stress for materials whose Young’s modulus is listed in OpenStax Table 12.1.

Why do I enter a temperature change and not a temperature?

Expansion depends only on how much the temperature changes. A rise of 10 °C is a rise of 18 °F; the 32-degree offset between the scales cancels out in the difference. Going from 60 °F to 160 °F is a change of 100 °F, which equals 55.6 °C, and it is that number you multiply by a coefficient given per °C.

Does a hole get bigger or smaller when the plate is heated?

Bigger. Every dimension of an object, including the size of a hole in it, grows as if the object were a photographic enlargement. The hole expands by the same fraction as the material around it, which is why a metal ring can be loosened over a shaft by heating it.

What the calculator assumes

The linear formula ΔL = α L ΔT treats the coefficient α as a constant. In reality it changes a little with temperature, so the result is good for moderate ranges, and less reliable for large ones. The material is assumed to be free to expand in every direction and to have uniform temperature throughout. The volume formula ΔV = β V ΔT uses the volume coefficient; for an isotropic solid β ≈ 3α because each of the three dimensions grows by α ΔT. For liquids and gases only β exists, and for gases the pressure matters too, so the table value for air applies at atmospheric pressure only. Values come from OpenStax University Physics Volume 2, Table 1.2, and individual alloys vary.

Worked example: a copper pipe and a concrete slab

A 20 ft (240 in) copper pipe heated from 60 °F to 160 °F warms by 100 °F, which is 55.6 °C. With α = 17 × 10⁻⁶ per °C: ΔL = 17 × 10⁻⁶ × 240 × 55.6 = 0.227 in, a little under a quarter inch. A concrete slab 40 ft (480 in) long that warms by 80 °F (44.4 °C) with α = 12 × 10⁻⁶ per °C grows by 0.256 in. Each is small, but long runs of pipe, rails and pavement add up, which is why they need loops, expansion joints or sliding supports.

Material choice

Coefficients differ by more than a factor of 20 across common materials. A 100 m steel bar warmed by 1 °C grows 1.20 mm, while the same bar of Invar, a nickel-iron alloy, grows 0.090 mm, which is why Invar is used in precision instruments. Pyrex's coefficient (3 × 10⁻⁶ per °C) is a third of ordinary glass (9 × 10⁻⁶), so a sudden temperature change stresses it less. And because steel and concrete have nearly the same α, reinforced concrete does not tear itself apart as temperature changes.

Stress when expansion is blocked

If a rigid constraint prevents the expansion, the material is compressed instead. For a steel rail or bar whose ends cannot move, a 30 °C rise produces a stress of Y α ΔT = 20 × 10¹⁰ Pa × 12 × 10⁻⁶ per °C × 30 °C = 72 MPa (about 10.4 thousand psi) in the formula of OpenStax Example 1.4. Whether that damages the part depends on its strength and on how it may buckle, which the formula does not address. The point is to leave gaps or flexible joints rather than rely on strength.

Common mistakes

Mixing coefficient units is the usual error: a coefficient per °C applied to a change in °F gives an answer 1.8 times too large, so convert the coefficient or the temperature difference. Confusing α and β gives results three times off for solids. Forgetting that the new length is the old length plus ΔL, not ΔL itself, is a third. And expecting water to expand uniformly: between 0 °C and 4 °C water contracts on warming, so the table value is for ordinary room temperatures.

Sources: OpenStax University Physics, Volume 2, chapter 1 (thermal expansion, Table 1.2 of coefficients, Examples 1.2 to 1.4) and Table 12.1 of Young's moduli in Volume 1. Related: Young's modulus calculator, specific heat calculator and temperature converter.