Thermal Expansion Calculator

1. Pick what to calculate: Length for a rod, pipe or rail; Volume for a solid block; or Density to see how a solid gets less dense as it warms.
2. Enter the material's linear expansion coefficient in 1/K (0.00001 means 10 parts per million per kelvin) and the initial and final temperatures in kelvin.
3. Enter the starting length, volume or density, then read the change, the final value and how large the change is relative to the original.

Here is what each result means:

Thermal expansion is the change in length, area or volume of a material as its temperature changes. Most solids grow when heated and shrink when cooled because their atoms vibrate harder at higher temperature and sit, on average, a little farther apart.

The size of the effect is set by the coefficient of linear expansion, written alpha, which is the fractional change in length per kelvin. It helps engineers leaving gaps in rails, bridges and pipe runs, builders fitting glass and cladding, and students checking textbook problems. It is not the same as thermal stress: stress only appears when something stops the material from moving freely.

Water is a well-known exception to the usual rule. Its density is greatest near 4 C, so between 0 C and 4 C it contracts slightly as it warms, which a single fixed coefficient cannot model.

It multiplies the coefficient by the starting size and by the temperature change, which is the standard linear model for small changes.

1. Find dT as the final temperature minus the initial temperature. A 1 K step equals a 1 C step, so the coefficient has the same value per K or per C.
2. For length, multiply alpha x L0 x dT to get the change dL, then add it to L0.
3. For volume, use the volume coefficient beta = 3 x alpha. This is the isotropic-solid approximation: the solid expands equally in all three directions.
4. For density, mass stays the same while volume grows by the factor (1 + 3 x alpha x dT), so the new density is rho0 divided by that factor.

Suppose a 2 m bar has a linear coefficient of 0.00001 per K (10 ppm/K) and warms from 300 K to 350 K.

Calculation: dT = 350 - 300 = 50 K. dL = 0.00001 x 2 x 50 = 0.001 m, which is 1 mm. The final length is 2.001 m, a change of 0.05%.

Switch to Volume and the same coefficient applied to a 1 m3 block gives dV = 3 x 0.00001 x 1 x 50 = 0.0015 m3, or 1.5 L. In Density mode, a solid starting at 1,000 kg/m3 falls to 1,000 / 1.0015 = 998.502 kg/m3.

A 1 mm change on a 2 m part is small, but it adds up along long runs. OpenStax works a steel bridge example in which a 1,275 m span with a 55 C swing changes length by about 0.84 m, spread across several expansion joints.

The model is linear, so the result scales directly with each input. Three things move it most.

The Material Coefficient

Coefficients differ by a factor of more than 50 between common materials, so the material choice dominates. Published values also vary slightly by source and alloy: OpenStax lists aluminium at 25 x 10^-6 per C, while HyperPhysics lists 24 x 10^-6 per C at 20 C.

Values are from OpenStax University Physics Volume 2, Table 1.2.

The Size of the Temperature Swing

Doubling dT doubles the change. Use the full range the part will see in service, from the coldest to the hottest condition, not just the change from room temperature.

The Starting Length

A longer part moves more for the same material and temperature. That is why long pipe runs and rails need allowances while small parts often do not. To convert Celsius readings before entering them, use the kelvin to Celsius converter in reverse.

The three modes describe the same physical change from different angles.

OpenStax notes that tabulated volume coefficients for solids are equal to 3 x alpha except for rounding. Liquids are different: they have only a volume coefficient, so for a liquid enter beta divided by 3 in Volume mode. For mass-volume-density conversions at a fixed temperature, the density calculator is the simpler tool.

Sizing Expansion Gaps

Pipe runs, rails, decking and cladding need room to move. Calculate the change over the full service temperature range to size the gap or loop.

Checking Fits and Clearances

A shaft in a bearing or a pin in a hole can bind if the parts expand at different rates. Run each material separately and compare the changes. For the weight of steel parts in the same design, the steel weight calculator helps.

Solving Physics Problems

Textbook questions on rods, rings and bridges use exactly these formulas. The worked output shows every step, which makes it easy to check homework.

1. Entering the Coefficient in Ppm

Typing 10 for a 10 ppm/K material makes the result a million times too large. Enter 0.00001 instead.

2. Entering Celsius as Kelvin

The change dT is the same in both scales, but the tool blocks negative kelvin values. Add 273.15 to Celsius readings, or enter any two numbers that give the right difference.

3. Using a Fahrenheit Difference Directly

A 1 F step is only 5/9 of a kelvin, per NIST. A 90 F swing is a 50 K swing, not 90 K.

4. Using the Volume Coefficient in the Alpha Box

If a table gives beta, divide it by 3 before entering it here, or the volume change will be three times too large.

5. Ignoring Restraint

If the ends are fixed, the part cannot grow, and the expansion becomes stress instead. This tool gives free expansion only.

The linear model is accurate for most solids over ordinary temperature ranges; OpenStax describes a constant coefficient as quite accurate for small changes and sufficient for most practical purposes.

Not for the change. A temperature difference of 1 K equals a difference of 1 C, so dT and the coefficient are the same in either scale. Only the absolute readings differ by 273.15.

Divide by one million. 10 ppm/K is 0.00001 per K, and 17 ppm/K is 0.000017 per K. Enter the decimal, not the ppm figure.

An isotropic solid grows by the same fraction in length, width and height. For small changes those three fractions add, so the volume grows about three times as fast as any one length.

The change comes out negative, meaning the part shrinks, and the density rises. The same formulas apply to cooling as to heating.

Roughly 12 millionths of its length per kelvin, using the OpenStax value for iron or steel. A 10 m steel member warmed by 20 K grows about 2.4 mm; HyperPhysics lists steel at 13 x 10^-6, which gives 2.6 mm.

Yes, in Volume mode, but enter the liquid volume coefficient divided by 3, because liquids are listed by beta. Water is a special case near 4 C, where its expansion is not linear.

No. It gives free expansion. If the part is held fixed, the stress equals Young's modulus times alpha times dT, a relation OpenStax works through for concrete road slabs.

The mass stays the same while the volume grows, so the mass per unit volume falls. The tool divides the starting density by (1 + 3 x alpha x dT).

Most differences come from a different coefficient for the same material, using beta where alpha is needed, or a Fahrenheit difference entered as kelvin. Check the coefficient source and units first.

No. The calculation runs entirely in your browser. Nothing is sent to our servers, and anything you choose to Save stays in this browser only.

• Thermal expansion, coefficients and thermal stress (OpenStax, University Physics Volume 2, Section 1.3).
• Linear and volume expansion relationships (HyperPhysics, Georgia State University).
• Thermal expansion coefficients at 20 C (HyperPhysics table).
• Temperature and temperature-interval conversion factors (NIST SP 811, Appendix B.9).