TorqueMath

Thermal Expansion Calculator

Length change and thermal stress for 19 engineering materials, how a fit between two parts changes when they run at different temperatures, and the temperature to heat a hub or cool a shaft for shrink fitting – with real low-temperature contraction data for dry ice and liquid nitrogen.

TorqueMath.com · Thermal Expansion Calculator
Calculate
mm
°C
°C
µm/m·K
GPa
MPa
mm²
0 = skip the force. Stress if fully constrained does not depend on the area.
mm
Fit at 20 °C (ISO 286)
µm
µm
Diametral, measured at 20 °C. Negative = clearance (for example −30 and −10 for a running fit).
µm/m·K
µm/m·K
Examples
°C
°C
Assembly method
Cool the shaft in
°C
Assembly clearance
µm
Gap the heated bore must have over the shaft so the parts slide together before the heat flows across. DIN 7190 practice (Roloff/Matek) is about 0.1 % of d; small parts and bearings fitted quickly get by with less.
The heated outer part is
°C
Method and assumptionsMethod: free thermal strain of each part from 20 °C; mean coefficient 20–100 °C, and above 100 °C for steels, austenitic stainless and aluminium the temperature dependence of the Eurocode thermal-strain formulas (EN 1993-1-2, EN 1993-1-4, EN 1999-1-2) scaled to that coefficient, NIST integral contraction tables (interpolated every 10 K) below 20 °C. Fits use free diameters: the interference that remains is the difference of the two thermally changed diameters, measured on nominal d.Basis: α as mean 20–100 °C values, E and minimum yield (0.2 %) at room temperature. ISO 286-2 limits for the fits offered (same tables as the ISO fits and press fit calculators). Stress assumes a straight bar held rigidly at both ends, elastic, no buckling. Parts are assumed at a uniform temperature.Limits: Copper alloys, titanium, Invar, plastics, glass and concrete use a constant 20–100 °C mean α: at a few hundred °C their real expansion is typically 10–20 % larger than shown (Invar much more above 100 °C, so it is limited to 200 °C). E is taken at room temperature. Bearing rings follow ISO 492, not ISO 286: enter their limits as custom values. Plastics: α depends on moisture, filler and direction of flow and can differ by ±30 %; PTFE has phase changes at 19 °C and 30 °C. Brass, bronze, cast iron, cast aluminium and martensitic stainless use a scaled curve of a similar alloy below 20 °C (marked "estimated"). Contact pressure, stresses and torque of the finished fit are not computed here – use the Press Fit calculator.References: ASM Ready Reference: Thermal Properties of Metals (Cverna, 2002) · NIST Cryogenic Material Properties database · NBS Monograph 29 (Corruccini & Gniewek) · ISO 286-2 · DIN 7190-1 · SKF and NSK mounting guidance · EN 1992-1-1 · EN 572-1 · ISO 3585Formulas · How this calculator was checked · TorqueMath v2.7

Options

How it is calculated

Every material has a thermal strain curve ε(T): the change of length per unit length from the reference temperature 20 °C (ISO 1, the temperature at which fits and drawings are specified). Above 20 °C the strain is the mean coefficient α (20–100 °C) times the temperature difference. Below 20 °C the calculator uses measured low-temperature contraction from NIST where it exists, because α falls as metals get colder: steel cooled in liquid nitrogen shrinks about 0.19 %, not the 0.25 % that α × ΔT predicts.

Strain ε(T) = α · (T − 20 °C) T ≥ 20 °C (mean α 20–100 °C) ε(T) = α·(T−20)·α_EC(T)/α_EC(100) T > 100 °C, steels and aluminium (Eurocode strain curve shape) ε(T) = NIST integral contraction table T < 20 °C (steel, stainless, Al, Cu, Ti, Invar, PTFE, PA) Length change L2 = L · (1 + ε(T2)) / (1 + ε(T1)) ≈ L · α · (T2 − T1) Area, volume A2/A = (1 + Δε)², V2/V = (1 + Δε)³ ≈ 2αΔT and 3αΔT Fully constrained σ = E · Δε F = σ · A (compressive when heated) Fit at temperature c(T) = c20 + d · [ ε_outer(T_outer) − ε_inner(T_inner) ] c = clearance, −c = interference Shrink fit ε_hub(T_hub) − ε_shaft(T_shaft) = (i_max + s) / d s = assembly clearance same material, ΔT = (i_max + s) / (α · d) shaft at room temperature shaft at room

Typical cases

Temperature limits used for shrink fitting

Rolling bearings: 120 °C (NSK: "do not heat bearings above 120 °C"; SKF: never above 125 °C unless the maker says otherwise; SKF finds 80 – 90 °C above the shaft is normally enough). Sealed or greased bearings can have a lower limit – check the maker. Hardened and case-hardened parts: 150 °C, as they are usually tempered at 150 – 200 °C and lose hardness if heated beyond their tempering temperature. Age-hardened aluminium (T6): 150 °C, since artificial ageing runs at about 160 – 180 °C and longer heating above it over-ages the alloy. Plain carbon and quenched-and-tempered steel: 350 °C (scale, temper colours). Oil baths: stay well below the flash point of the oil (mineral oils roughly 200 – 250 °C, see the data sheet); above about 150 °C use an oven or an induction heater, never an open flame on finished parts.

Material data: steels, aluminium, copper alloys and titanium from the ASM Ready Reference: Thermal Properties of Metals and the alloy standards (EN 10025, EN 10083, EN 755, EN 1561, EN 1982); Invar from Aperam/Carpenter data; plastics from manufacturer sheets (typical, not guaranteed); soda-lime glass EN 572-1; borosilicate ISO 3585; concrete EN 1992-1-1 (α 10·10⁻⁶ /K, C30/37 Ecm 33 GPa, fctm 2.9 MPa). Low-temperature contraction: NIST cryogenic material property fits for 304 stainless, 6061-T6, OFHC copper, Ti-6Al-4V, Fe-3.25Ni steel (within 2 % of iron in NBS Monograph 29), Invar, PTFE and nylon.

This calculator works out how much a part grows or shrinks with temperature, how a fit changes when the parts run at different temperatures, and how hot the hub or how cold the shaft must be to assemble a shrink fit. It covers linear, area and volume expansion with the stress in a restrained bar, ISO 286 fits between two materials (for example a bearing in an aluminium housing or a steel shaft in a bronze bush) and shrink fitting by heating, freezer, dry ice or liquid nitrogen. Below room temperature it uses measured contraction data instead of a constant coefficient. Mechanics, fitters, designers and students use it to plan assembly, check fits at operating temperature and avoid overheating bearings or hardened parts.

Worked example: steel hub on a Ø50 mm steel shaft, H7/s6

  1. ISO 286 at Ø50 mm: hole H7 0 / +25 µm, shaft s6 +43 / +59 µm. Interference at 20 °C is 43 − 25 = 18 µm to 59 − 0 = 59 µm.
  2. Assembly clearance by the DIN 7190 / Roloff-Matek rule of thumb: 0.001 × 50 mm = 0.050 mm = 50 µm.
  3. The hub bore must grow by the largest interference plus the clearance: 59 + 50 = 109 µm.
  4. Carbon steel α = 11.7 µm/m·K (mean 20–100 °C): with a constant α, ΔT = 0.109 mm / (11.7·10⁻⁶ /K × 50 mm) = 186.3 K. Steel's mean α rises above 100 °C (EN 1993-1-2 strain curve: about 3 % higher at 200 °C), so the bore reaches the needed 2.18 µm/mm at ΔT ≈ 180.5 K. Heat the hub to about 200 °C – below the ~350 °C usually allowed for plain steel parts. With the constant α the answer would be 206 °C, slightly more heat than needed.
  5. If the hub were a bearing (limit 120 °C), cool the shaft instead. In liquid nitrogen (−196 °C) steel shrinks by 1912 µm/m (NIST), so the Ø50 shaft loses 95.6 µm. The hub then needs only 109 − 95.6 = 13.4 µm, ΔT = 23 K: about 43 °C. With α × ΔT the shaft would seem to shrink 126 µm – 32 % too optimistic.

Heat the hub to about 200 °C, or put the shaft in liquid nitrogen and warm the hub to about 43 °C. The finished fit has 18 – 59 µm interference once both parts are back at room temperature.

FAQ

How do I calculate thermal expansion?

Multiply the length by the coefficient of thermal expansion and the temperature change: ΔL = α × L × ΔT. A 1 m steel bar (α ≈ 11.7 µm/m·K) heated by 50 K grows 1000 × 11.7·10⁻⁶ × 50 = 0.585 mm. Area grows about 2αΔT and volume about 3αΔT; the calculator also gives the exact values.

How hot must I heat a hub for a shrink fit?

The bore must grow by the largest interference plus an assembly clearance so the parts slide together: ΔT = (i_max + clearance) / (α × d). For a Ø50 H7/s6 steel fit with 59 µm interference and 50 µm clearance, ΔT = 0.109 / (11.7·10⁻⁶ × 50) ≈ 186 K, so about 206 °C from 20 °C with a constant α; allowing for steel's α rising above 100 °C, about 200 °C is enough.

What is the maximum temperature for heating a bearing?

Bearing makers limit induction or oven heating to about 120 °C (NSK; SKF states 125 °C), because higher temperatures change the steel's hardness and dimensional stability and damage seals and grease. Sealed or greased bearings may have a lower limit – check the maker. If 120 °C is not enough, cool the shaft as well.

How much does steel shrink in liquid nitrogen?

From 20 °C to −196 °C carbon steel contracts about 0.19 % (1.9 µm per mm), measured by NIST. The constant room-temperature α would predict about 0.25 %, overstating the shrinkage by roughly a third, because α falls as the temperature drops. A Ø50 shaft loses about 0.096 mm.

Why does a bearing get loose in an aluminium housing when warm?

Aluminium expands about twice as much as bearing steel (≈ 23 vs 12 µm/m·K). When the housing warms, its bore grows faster than the outer ring, so an interference fit at room temperature can turn into clearance. The calculator shows the temperature at which the fit goes loose for the fit and temperatures you enter.

What stress does thermal expansion cause if the part cannot move?

If both ends are held rigidly, the bar cannot expand and the stress is σ = E × α × ΔT. For steel (E 200 GPa, α 12 µm/m·K) a 30 K rise gives 72 MPa of compression. Long slender bars may buckle before that, which the calculator does not check.

Tools for this job

What you need to build, measure or check the result.

Induction bearing heaterEven, controlled heating for shrink fitsView on Amazon →Infrared thermometerCheck the hub temperature before you fit itView on Amazon →Temperature indicating sticksMark the part and see when it reaches temperatureView on Amazon →

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Checked: 7 results compared with textbooks, standards and maker data by a separate review, 4 corrections made. See the checks →

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