Thermal Expansion Calculator

ΔL = α · L · ΔT and, for an isotropic solid, ΔV / V = 3 · α · ΔT

α

The coefficient of linear thermal expansion, in parts per million per degree — a material constant measured by watching a bar grow under a known temperature change. It is a small number said out loud: aluminium's 23.1×10⁻⁶ means a metre of aluminium gains 23.1 micrometres per kelvin, which is why nothing in daily life looks like it is expanding. The coefficients span more than a factor of thirty across ordinary materials, from about 1.5×10⁻⁶ for Invar — an alloy engineered to be immune to this effect, and used for the pendulum rods and precision instruments that needed it — up to 55×10⁻⁶ and beyond for PVC and other plastics. Picking the right row is therefore most of the work; the arithmetic after that is one multiplication.

L

The length at the starting temperature, in millimetres, in the unit you choose. It is the original length, not the final one, and not the average — a distinction that only shows up once you try to go the other way from a known final length to a coefficient, which is where a surprising number of hand calculations quietly go wrong. The base unit here is the millimetre because that is the unit that keeps the answer readable: a metre of aluminium heated by 40 K grows 0.924 mm, and printing that as 0.000924 m helps nobody. Zero is a legal input — a bar of no length does not expand — but a negative one is not, which is why the field's lower bound is closed at zero and the check further in refuses anything below it.

ΔT

Temperature change in kelvin (or degrees Celsius — the two are the same size, and here it is a change rather than a temperature, so either one is correct). Negative values are normal and mean cooling: steel that cools 40 K contracts, and the expansion comes out negative, which is exactly what the arithmetic should say. The limits of ±500 K cover the range from liquid nitrogen to a furnace and leave the field wide open for the reader who is working in the tens of degrees, which is where almost all real cases live.

ΔL

The change in length, in millimetres, printed alongside the new length so both the amount and the result of adding it are visible at once. The number is small in absolute terms and that is precisely the trap: 0.9 mm sounds trivial until the bar has nowhere to go. A restrained bar cannot simply not expand — the expansion turns into stress instead, and because the elastic modulus is large the force is not small at all. That is the reason railway rails are laid in tension with gaps between them, and why a long bridge has a finger joint at one end rather than a continuous deck. This page gives the free-expansion length change; the stress that appears when expansion is prevented is a second calculation, and it is where the real damage lives.

3 · α · ΔT

The volumetric part, shown as a strain in percent rather than as a volume, because that is what the same measurement gives you for free: each of the three dimensions grows by α · ΔT, so a solid cube's volume grows by about 3 · α · ΔT. Two caveats come with the factor of three. It holds for isotropic materials — metals, glasses, and plastics treated as uniform — but not for wood, which expands differently along the grain than across it, nor for layered composites, nor for single crystals. And it is a small-strain approximation: the exact factor is (1 + α·ΔT)³ − 1, which agrees with 3αΔT to many decimal places at these magnitudes and would not at a temperature change large enough to matter.

Use it before you fix two ends of something to a rigid frame: laying a rail, hanging a pipe run between anchors, glazing a pane that is a tight fit, bolting an aluminium part to steel, or choosing the gap in a bridge deck. Also for the everyday version of the same question — a metal lid that comes off after a hot rinse, or a jar that refuses to open until it is warmed — where the mechanism is that the lid expands first because its coefficient is larger or its mass smaller. It answers a question about free, unrestrained expansion only. If the part is restrained at both ends, this page tells you the length it wants to become, and the far more useful number — the stress that builds up instead — comes from the material's elastic modulus, not from here.

1. First screen: a metre of aluminium, 40 K warmer

   1. ΔL = α · L · ΔT = 23.1×10⁻⁶ · 1000 mm · 40 K
   2. 23.1×10⁻⁶ · 1000 = 0.0231 mm per kelvin
   3. 0.0231 · 40 = 0.924 mm
   4. New length: 1000 + 0.924 = 1000.924 mm
   5. Volumetric strain: 3 · 23.1×10⁻⁶ · 40 = 0.2772%

   Just under a millimetre on a metre, for a temperature change a sunny afternoon could produce — which is why the page exists and why the result is printed to three decimals. The strain row is the same result seen from the other side: 0.28 % of volume is small, but it is not zero, and a part constrained in all three directions cannot absorb it. Note also that the panel prints the coefficient back to you: that row is the material you selected, and it is there so you can check the dropdown before you trust the rest.

2. The classic case: a 25 m steel rail, 60 K of sunshine

   1. ΔL = 12×10⁻⁶ · 25000 mm · 60 K
   2. 12×10⁻⁶ · 25000 = 0.3 mm per kelvin
   3. 0.3 · 60 = 18 mm
   4. New length: 25000 + 18 = 25018 mm

   Eighteen millimetres on a single rail length, from a winter night to a summer afternoon. This is the number that decides how wide a rail gap has to be, and it is why tracks are laid with the gaps they have rather than as one continuous welded ribbon — and why modern continuously welded track is stretched and anchored under controlled tension instead, so that the same 18 mm becomes a manageable stress spread over the whole rail rather than 18 mm of buckling at one joint. The choice is not about whether steel expands; it always does.

3. Cooling, and a negative answer: stainless steel, 40 K colder

   1. ΔL = 17.3×10⁻⁶ · 500 mm · (−40 K)
   2. 17.3×10⁻⁶ · 500 = 0.00865 mm per kelvin
   3. 0.00865 · (−40) = −0.346 mm
   4. New length: 500 − 0.346 = 499.654 mm

   Nothing changes but the sign of the temperature change, and the calculator reports contraction as a negative expansion and a shorter final length. That is the honest convention and it is the one to keep when the parts get assembled: a stainless sleeve shrunk in liquid nitrogen slides into a shaft that it would not fit on at room temperature, and its new length is the 499.654 mm above rather than the 500 mm it was machined to. Both rows are printed because both matter — the amount of shrinkage for the fit, the final length for the assembly drawing.

4. The extreme end of the table: cast iron, 100 m, +500 K

   1. ΔL = 10.8×10⁻⁶ · 100000 mm · 500 K
   2. 10.8×10⁻⁶ · 100000 = 1.08 mm per kelvin
   3. 1.08 · 500 = 540 mm
   4. New length: 100000 + 540 = 100540 mm
   5. Volumetric strain: 3 · 10.8×10⁻⁶ · 500 = 1.62%

   Half a metre, on a hundred metres, for a change from ambient to a furnace. The low coefficient of cast iron is why it was the material of engine blocks and machine beds — it moves least of the common metals — and the example is here to show the scale at which the effect stops being ignorable: 540 mm is not a tolerance, it is a design feature. It also shows why the field's limits are as wide as they are: the page has to survive a reader who really does have a 100 m run of pipe going from cold to hot, and it would rather answer that than refuse it.

The volumetric row assumes the material expands the same way in every direction, which is true of metals and glasses and not true of wood, single crystals, or laminated composites — for those, each axis needs its own coefficient and the factor of three means nothing. The coefficients themselves are quoted at a reference temperature near 20 °C, and α rises slowly with temperature, so an answer for a 500 K swing is an extrapolation from a room-temperature measurement rather than a measurement at that temperature. Published values also disagree: aluminium is 23×10⁻⁶ in one of the sources below and 25×10⁻⁶ in the other, a difference of several percent, so treat the third decimal of any answer as decoration and the first digit as the real information. This page gives the free-expansion length change and says nothing about what happens when the part is held at both ends — the same expansion, resisted, becomes a stress of order E·α·ΔT, which for steel over 60 K is well over a hundred megapascals, enough to buckle a rail or crack a weld, and none of that appears in the numbers above. It also says nothing about time: a thick casting heats unevenly, so the expansion arrives as a gradient and a warping rather than as a uniform change in length. And it assumes the change happens below the material's softening point — a plastic near its glass transition, or anything near melting, is outside what a single coefficient can describe.

What is the thermal expansion formula?

ΔL = α · L · ΔT: the change in length equals the material's coefficient of linear expansion times the original length times the temperature change. For a solid that expands the same way in all directions, the volume grows by three times as much in fractional terms, so the volumetric strain is 3 · α · ΔT — the same relation with a factor that comes from there being three dimensions, not from any extra physics. Both are shown on this page, and the second is reported as a strain in percent rather than as a volume, because that is the form the coefficient gives you directly.

How much does steel expand per degree?

About 12×10⁻⁶ per kelvin, so a metre of steel grows 0.012 mm for each degree — 12 micrometres, roughly an eighth of the thickness of a sheet of paper — paper runs about 0.1 mm. Over the 60 K between a winter night and a summer afternoon, a 25 m rail grows 18 mm, which is why rail gaps exist and why continuously welded track has to be stressed and anchored rather than merely laid. Aluminium is about twice as responsive at 23×10⁻⁶, and PVC is more than four times steel's at 55×10⁻⁶, which is why plastic pipework needs more support spacing and more generous expansion loops than metal pipework of the same run.

Why is the volume change three times the length change?

Because a solid grows in three directions at once. Each dimension grows by the fraction α·ΔT, so a cube's volume grows by (1 + α·ΔT)³ − 1, which for the small numbers involved is 3·α·ΔT to far more decimal places than anyone needs. The factor of three is a consequence of isotropy, though — it assumes the material expands identically along every axis. Wood does not (it moves much more across the grain than along it), nor do single crystals, nor laminated composites, so for those the volumetric row on this page is not an approximation of the right answer but the answer to a different question.

Is the length I enter the original or the final length?

The original one — the length at the starting temperature, before anything heats up. This matters more than it looks: going backwards from a measured final length to a coefficient means dividing by the original length, and using the final length instead introduces a relative error of about α·ΔT, which is negligible for steel and not negligible for a plastic over a large temperature change. The page prints both the change and the new length so you can see which is which, and the change is always relative to what you typed.

Where do expansion joints come in?

They are what you build when you cannot let the expansion be free. A part held at both ends does not stay the length it wants to be; the expansion is resisted and turns into stress instead, of the order of the material's elastic modulus times α·ΔT — for steel over 60 K that is well over a hundred megapascals, which is enough to buckle a rail, bow a pipe run or crack a weld. Expansion joints, loops, slotted bolt holes and sliding bearings are all ways of giving the material somewhere to go, and they exist because this page's calculation has to be paid for one way or another.

Why do different tables give different coefficients?

Because the coefficient is not perfectly constant and not perfectly agreed on. α varies with temperature, so a value quoted at 20 °C and a value averaged over 0–100 °C are different numbers for the same material, and different handbooks quote different alloys under the same name — aluminium in one of the sources below is 23×10⁻⁶ and 25×10⁻⁶ in the other, and 'stainless steel' covers a family whose coefficients run from about 10 to 18×10⁻⁶. For most purposes the disagreement is a few percent and does not change the decision. When it does matter — precision instruments, long runs, tight fits — it is a reason to measure the actual batch rather than to pick a different table.

• 13.2 Thermal expansion of solids and liquids — ΔL = αLΔT with the definition of the coefficient, the β ≈ 3α relation for volume, and a table of coefficients for common metals — OpenStax (Rice University)
• Linear thermal expansion coefficients of materials — the handbook values this page's table is drawn against, including cast iron, brass, stainless and PVC — The Engineering ToolBox