One formula, and a coefficient that varies more than people expect
Length change equals the original length times the coefficient of thermal expansion times the temperature change. The coefficient is a small number, quoted in millionths per degree, and the reason it feels invisible in daily work is that both the length and the temperature change have to be large before the product becomes something you can see. Ten feet of mild steel through a hundred degree Fahrenheit swing moves 0.078 inches, which is about a sixteenth. The same hundred degrees across a two inch part moves 0.0013 inches, which is a thousandth and a bit, and which is enough to ruin a precision fit and nothing else.
The coefficients themselves spread wider than the materials look like they should. Aluminum moves twice as much as steel. Austenitic stainless moves about half again as much as mild steel, which is the reason a stainless weldment distorts more than a mild steel one of identical geometry. Invar barely moves at all. Martensitic stainless like 410 moves less than mild steel. If you are working across two materials, the difference between their coefficients is the number that actually matters.
| Material | Millionths per °F | Millionths per °C | Movement in 10 ft over 100 °F |
|---|---|---|---|
| Invar 36 | 0.7 | 1.3 | 0.008 in |
| Titanium alloy | 4.8 | 8.6 | 0.058 in |
| Concrete | 5.5 | 9.9 | 0.066 in |
| Stainless 410 | 5.6 | 10.1 | 0.067 in |
| Gray cast iron | 5.9 | 10.6 | 0.071 in |
| Mild steel | 6.5 | 11.7 | 0.078 in |
| Stainless 316 | 8.9 | 16.0 | 0.107 in |
| Copper | 9.3 | 16.7 | 0.112 in |
| Stainless 304 | 9.6 | 17.3 | 0.115 in |
| Brass C360 | 11.2 | 20.2 | 0.134 in |
| Aluminum 6061 | 13.1 | 23.6 | 0.157 in |
Concrete sits in that list deliberately. It is close enough to steel that reinforced concrete works at all, and the fact that the two materials expand at nearly the same rate is one of the quiet reasons the combination has been viable for a century.
Shrink fits are the same equation, solved for temperature
An interference fit puts a shaft into a bore that is deliberately smaller than the shaft. Getting them together means temporarily making the bore bigger, the shaft smaller, or both, which means rearranging the same formula: the temperature change needed equals the diameter change wanted, divided by the coefficient times the diameter.
Two things are worth adding to the interference itself. The first is assembly clearance, because a fit that is exactly zero at temperature will bind before the parts are home. A couple of thousandths of slip room is normal. The second is the practical realisation that heating the outer part and cooling the inner part both contribute, so splitting the difference halves the temperature each part has to reach. Dry ice takes a shaft to about minus 110 Fahrenheit without any special equipment, which for a steel fit is worth around ninety degrees of the requirement on its own.
The constraint that catches people is not the temperature, it is the clock. Once the parts touch, they equalise fast, and a shaft that stops half way is not coming out again without cutting something. Everything about the assembly should be set up, aligned and rehearsed cold before anything is heated.
Restrained movement becomes force
The important consequence of thermal expansion is not usually the dimension, it is what happens when the dimension is prevented from changing. A member fixed rigidly at both ends that wants to grow and cannot develops a compressive stress equal to the coefficient times the temperature change times the elastic modulus, and for steel that works out at roughly 190 psi for every degree Fahrenheit. Fifty degrees of restrained rise puts nearly 10,000 psi into the member without any external load at all, which is a substantial fraction of what a mild steel section is good for, and it goes on to buckle it or to tear out whatever is holding it.
This is why long runs get slotted holes, sliding bearings, expansion loops in pipework and gaps between rails. It is also why a weldment distorts: welding heats a small region that expands against the cold metal around it, yields in compression because it cannot expand freely, and then shrinks below its original size on cooling. The distortion in a welded frame is thermal expansion that was restrained and became permanent.
Measuring, and when it matters
A steel rule and the part are usually at different temperatures, and precision measurement standards are defined at 68 Fahrenheit for exactly that reason. For anything at shop tolerances this is noise. It stops being noise when you are working to a thousandth on a part fresh off a cut, when the shop is cold and the gauge came out of a warm pocket, or when a long part is measured at one end of the day and machined at the other. Let things equalise, and if a measurement disagrees with itself, temperature is the first suspect before the instrument.
For the metal itself, the metal weight calculator covers stock weight, the speeds and feeds calculator covers cutting it, and the bolt torque calculator covers the joints that a heated assembly will try to loosen.
Questions people ask
How much does steel expand per degree?
Mild steel moves about 6.5 millionths of its length for every degree Fahrenheit, or 11.7 millionths per degree Celsius. In figures you can use, that is 0.078 inches over ten feet for a hundred degree Fahrenheit rise, or roughly a thirteenth of an inch. A hundred foot run through the same swing moves three quarters of an inch. Nothing about the material or the section changes the rate; only the length and the temperature change matter, which is why a long thin part and a short thick one of the same length move identically.
How hot do I need to heat a part for a shrink fit?
It depends on the diameter and the interference, not on the length or the mass. The temperature change needed is the diameter change wanted divided by the coefficient times the diameter, so a two inch steel bore that has to open two thousandths needs about 154 degrees Fahrenheit of rise. Doubling the diameter halves the temperature needed for the same absolute interference, which is why big fits are relatively easy and small precise ones are not. Cooling the shaft as well as heating the bore splits the requirement in half.
Why does stainless distort more than mild steel when welded?
Two reasons compound. Austenitic stainless expands about fifty percent more than mild steel for the same temperature change, so the heated zone pushes harder against the cold metal around it. It also conducts heat roughly a third as well, so the heat stays concentrated near the weld instead of spreading, which makes the temperature gradient steeper and the expansion more localised. More expansion in a smaller region against colder surroundings is exactly the recipe for distortion, which is why stainless fabrication leans harder on tacking, sequencing, backing bars and fixtures than mild steel does.
Do I need to allow for expansion in a shop project?
Usually not, and occasionally very much so. The test is length times temperature swing. A two foot bracket in a heated shop moves a fraction of a thousandth and nothing cares. A thirty foot rail outdoors through a seasonal swing moves close to a quarter of an inch, and if it is bolted solid at both ends that movement becomes force in the fixings. Anything long, anything exposed to weather, and anything that gets genuinely hot in service needs the movement worked out and somewhere for it to go. Everything else can be ignored with a clear conscience.
Does this apply to plastics and other non-metals?
The formula does; the coefficients in this list do not cover them, and the numbers are much larger. Common plastics expand five to ten times as much as steel, which is why a plastic part fitted precisely into a metal housing at room temperature can seize or fall out when the assembly warms, and why long vinyl and PVC runs need generous expansion allowance. Concrete is included above because it happens to be close to steel. For a specific polymer, look up its coefficient from the material datasheet rather than assuming anything, because the spread between grades of the same plastic family is wide.