Thermal Expansion Calculator
Thermal Expansion Calculator
Find how far a bar, pipe, girder, slab or panel grows or shrinks when the temperature changes, from ΔL = α × L × ΔT with sourced coefficients for 14 materials. Or solve for the temperature change that closes a gap, the coefficient from a measurement, or the original length. It also covers area, volume, liquids, restrained stress and expansion joints.
Change in length against temperature
Change in length ΔL = α × L × ΔT, so the new length is L × (1 + α × ΔT). Area changes by about 2 × α × ΔT and volume by about 3 × α × ΔT. A liquid changes by β × V × ΔT. A bar held rigidly at both ends carries a stress of E × α × ΔT. A change of 1 °F is 5/9 of a change of 1 °C or 1 K, and a coefficient per °F is 5/9 of the same coefficient per K. Lengths convert exactly at 0.3048 m per foot and 0.0254 m per inch, and 5/9 is the NIST factor for a temperature interval. Coefficients are typical values with stated ranges, and this tool gives the unrestrained movement, not a design code joint. The CSV, image and printed report contain the values you entered, nothing preloaded.
How to use this tool
Pick what to find, from the movement of a bar to the gap for an expansion joint. Enter a material (or your own coefficient), a length, and either a start and end temperature or a single change.
The input most often wrong is the temperature unit. A change of 1 °F is 5/9 of a change of 1 °C, so choose the unit that matches your numbers and the tool converts for you. The presets are typical values. The shaded band on the chart shows what their published range does to your answer, so use your own alloy, grade or mix when you know it.
What the calculation is doing
A solid grows in proportion to its length and to the temperature change: ΔL = α × L × ΔT. For carbon steel α = 11.7 × 10⁻⁶ per K, so 100 m warmed by 70 K grows 11.7 × 10⁻⁶ × 100 × 70 = 0.0819 m. The strict form, L × exp(α × ΔT), adds only 0.034 mm to that 81.9 mm (0.041%), and even HDPE over 100 K is only 0.72% off, so the tool uses the first order value.
Area changes by about 2 × α × ΔT and volume by about 3 × α × ΔT. For aluminium over 200 K the 3α rule is 0.47% low, so the tool also shows the exact form. Liquids use ΔV = β × V × ΔT, and a container grows by 3α of its own material, so the spill is the difference. A bar held at both ends turns the strain into stress, σ = E × α × ΔT, compression when heated. A joint moves by α × L × (highest − lowest temperature), and the gap at installation is the smallest allowed gap plus the expansion still to come. A change of 1 °F is exactly 5/9 K (NIST SP 811), and lengths use 0.3048 m per foot and 0.0254 m per inch.
Movement and force are two sides of the same strain
An unrestrained bar shows its strain as movement, a held bar as stress, and the stress is the strain times the modulus. A 100 m steel girder moving 81.9 mm is usually harmless, but locked between abutments it would carry 2.34 MPa per kelvin, or 93.6 MPa for a 40 K rise. PVC moves 4.62 times as far as steel yet, held rigid, carries only 0.15 MPa per kelvin against 2.34. Plastic moduli depend on temperature and load time (the Plastics Pipe Institute example uses 172,250 psi for cold HDPE against the preset’s base figure), so enter your own.
Water is where the linear idea breaks down. It is densest at 3.98 °C and expands as it cools to 0 °C, by 0.013%, which is 132 mL in 1,000 L. Above 4 °C its coefficient climbs from 88 × 10⁻⁶ per K at 10 °C to 207 at 20 °C and 357 at 40 °C. The textbook 210 applied over 20 to 60 °C gives 8.4 L for 1,000 L where the density table gives 15.26 L, 45% short, so the tool uses density for water between 0 and 100 °C. Freezing adds about 9% more room, which no coefficient covers.
Movement and restrained stress by material
The same 100 m over a 70 K swing, and the stress each material would carry if held at both ends through a 40 K rise. Cast iron, timber, granite and the plastics rest on one class figure or data sheet, so treat those rows as starting values.
| Material | α, × 10⁻⁶ per K | 100 m over 70 K | E, GPa | Stress per K, MPa | Held, 40 K rise, MPa |
|---|---|---|---|---|---|
| Timber, along the grain | 3.8 | 26.6 mm | 12 | 0.0456 | 1.82 |
| Granite | 8 | 56 mm | 59 | 0.472 | 18.9 |
| Glass, soda lime | 9 | 63 mm | 70 | 0.63 | 25.2 |
| Cast iron, grey | 10.4 | 72.8 mm | 97 | 1.01 | 40.4 |
| Concrete | 10.8 | 75.6 mm | 25 | 0.27 | 10.8 |
| Carbon steel | 11.7 | 81.9 mm | 200 | 2.34 | 93.6 |
| Stainless steel 304 | 16.6 | 116.2 mm | 200 | 3.32 | 133 |
| Copper | 16.9 | 118.3 mm | 117 | 1.98 | 79.1 |
| Brass | 20 | 140 mm | 110 | 2.2 | 88 |
| Aluminium alloys | 23.6 | 165.2 mm | 69 | 1.63 | 65.1 |
| PVC, rigid | 54 | 378 mm | 2.8 | 0.151 | 6.05 |
| Polycarbonate | 70 | 490 mm | 2.35 | 0.165 | 6.58 |
| Acrylic (PMMA) | 70 | 490 mm | 3.3 | 0.231 | 9.24 |
| HDPE | 144 | 1,008 mm | 0.9 | 0.13 | 5.18 |
A worked example you can check on paper
The default view is 100 m of carbon steel between −35 °C and 35 °C. ΔT = 35 − (−35) = 70 K. ΔL = 11.7 × 10⁻⁶ × 100 × 70 = 0.0819 m, so the steel grows 81.9 mm to 100.0819 m. In imperial units, 100 m is 328.084 ft and the range is −31 to 95 °F, so ΔT = 126 °F and the coefficient is 11.7 × 10⁻⁶ × 5/9 = 6.5 × 10⁻⁶ per °F. Then ΔL = 6.5 × 10⁻⁶ × 328.084 × 126 = 0.2687 ft = 3.224 in, which matches 81.9 mm ÷ 25.4.
For a joint, install at 10 °C with a 10 mm smallest gap at 35 °C. Expansion from 10 to 35 °C is 11.7 × 10⁻⁶ × 100 × 25 = 29.25 mm, so the gap at installation is 39.25 mm. Contraction to −35 °C is 11.7 × 10⁻⁶ × 100 × 45 = 52.65 mm, so the widest gap is 91.9 mm, which is 10 mm plus the full 81.9 mm. Backwards, a 30 m steel girder needs 0.006 ÷ (11.7 × 10⁻⁶ × 30) = 17.094 K of warming to grow 6 mm.
Mistakes that catch people out
Treating a change in °F as a change in °C. The girder that moves 81.9 mm over 70 °C moves 45.5 mm over 70 °F, so mixing them is 80% high one way or 44% low the other. Set the unit and let the tool convert.
Subtracting 32 from a difference. The 32 belongs to a reading. From 68 °F to 104 °F the change is 36 °F, which is 20 K, so convert each end or multiply the change by 5/9.
Taking one coefficient as universal. Aluminium alloys run from 22.6 to 24.5, and concrete from 6.9 to 13 depending on the aggregate (limestone near 7.8, chert near 10.8 in FHWA tests). Each plastic grade differs. Values are means over a stated range, often 20 to 100 °C, so over a 300 K swing α itself drifts, and water has no single β at all.
Forgetting what stops the movement. A bar fixed at both ends carries stress, and a joint that closes completely pushes on its neighbour. Use the code for your structure, which sets the temperature range and factors.
Popular calculations
Each opens the calculator with those figures filled in. Several are the worked examples used in the checks below.
Change in length
Temperature change to close or open a gap
Coefficient and original length
Area and volume of solids
Liquids and containers
Restrained stress
Expansion joints
How this was checked
Nine published examples were reproduced. OpenStax University Physics gives 0.84 m for the Golden Gate Bridge and the tool gives 0.8415 m; its gasoline tank spills 1.10 L against 1.0979 L (the tool takes a steel container as 3α = 35.1 × 10⁻⁶, the book 35), and its restrained concrete carries 7.9 MPa against 7.92. Charlotte Pipe’s 80 ft of PVC expands “up to 1.35 inches” against 1.3536 in. The Plastics Pipe Institute handbook gives −3.84 in, −551 psi and −7,800 psi, and the tool gives −3.84 in, −551.2 psi and −7,800 psi. Tennessee DOT’s 0.00936 and 0.00504 in per ft match exactly, and 55 ft of copper moving 7/8 in on a TRU Open Learning page compares with 0.8799 in (0.56% more, since the page rounds to an eighth).
Two independent routes agree. Integrating a linear α(T) numerically matched the tool to 3 × 10⁻¹⁴ over 3,000 random cases, and round trips through the temperature change, the coefficient and the original length recovered the inputs to 8 × 10⁻¹³ over 5,000. The same case in °F and feet, °C and metres, kelvin and a temperature change agreed to 4 × 10⁻¹². Water density, from Tanaka 2001 and Kell 1975, matches the OpenStax Chemistry table within 0.0033 kg/m³. Coefficients come from state DOT manuals quoting AASHTO, FHWA, the ASM Handbook, the Copper Development Association, and single manufacturer sheets for 304 stainless, brass, polycarbonate and acrylic. The −35 to 35 °C bridge range is an example, not a code value.
Common questions
What is the coefficient of thermal expansion?
It is the fractional change in size per degree of temperature change. The linear coefficient α for carbon steel is 11.7 × 10⁻⁶ per K, so a 100 m member grows 11.7 mm for a 10 K rise. Volume expansion is about 3α. Coefficients vary with alloy, mix or grade, so a table value is a typical figure with a spread.
How much does a steel bridge expand?
About 0.82 mm per metre over a 70 K swing. A 100 m steel girder between −35 and 35 °C moves 81.9 mm, or 3.224 in, and 1,000 m would move 0.82 m. Long bridges share that movement among several joints and bearings.
Why is a change of 1 °F only 5/9 of a change of 1 °C?
The Fahrenheit degree is smaller: 180 of them span freezing to boiling water, against 100 Celsius degrees. A change converts by 5/9 and the offset of 32 belongs only to a reading, so 68 °F to 104 °F is a change of 36 °F, which is 20 K.
How do I size an expansion gap?
Find the movement between the highest and lowest temperatures, then set the gap at installation so it closes to the smallest allowed gap at the highest temperature. For 100 m of steel from −35 to 35 °C, installed at 10 °C with a 10 mm minimum, the tool gives 39.25 mm at installation and 91.9 mm at −35 °C. That is unrestrained movement; design codes add the temperature range, load factors (Tennessee DOT multiplies by 1.20), creep and skew.
Does water expand when it cools?
Between 4 °C and 0 °C, yes. Water is densest at 3.98 °C, and 1,000 L at 4 °C becomes 1,000.13 L at 0 °C, a gain of 0.013%. Freezing adds about 9%, because ice at 917 kg/m³ is less dense than water at 999.8 kg/m³.
Why does plastic pipe move so much more than steel pipe?
Its coefficient is larger. PVC at 54 × 10⁻⁶ per K moves 4.6 times as far as carbon steel and HDPE at 144 × 10⁻⁶ per K moves 12.3 times as far, so 100 ft of HDPE cooled by 40 °F shortens by 3.84 in. HDPE is far softer, so a held 40 K rise gives about 5.2 MPa against 93.6 MPa in steel.
Related Calculators and Reference
- Temperature Converter: convert a reading between °C, °F, K and Rankine before you enter it.
- Length Converter: express a movement in the units your drawings use.
- Volume and Capacity Converter: turn a liquid volume change from litres into gallons.
- Stress and Modulus Unit Converter: move a stress or modulus between psi, ksi, MPa and GPa.
- Force Unit Converter: restate the force in a held bar as lbf, kip or newtons.
- Ideal Gas Law Calculator: how a gas expands with temperature compared with a solid.
- OpenStax University Physics Volume 2, section 1.3: the textbook treatment, with the worked examples used in the checks above.