Find how much a component grows or shrinks with temperature, the thermal strain it experiences, and the large stress that builds up if that expansion is fully restrained. Enter the coefficient of thermal expansion (CTE) as a plain number in ×10⁻⁶/°C — for example 12 for steel.
Almost every material expands when heated and contracts when cooled. If the part is free to move, that shows up as a change in length; if it is held in place, the same tendency turns into very large internal stresses. This calculator does both: it gives the free thermal expansion of a member from its coefficient of thermal expansion, and the stress that develops if that expansion is fully prevented. The pairing explains everything from expansion joints in bridges to thermal fatigue in piping.
For a uniform change in temperature, the change in length is ΔL = α·L₀·ΔT, where α is the linear coefficient of thermal expansion (CTE), L₀ the original length, and ΔT the temperature change.
CTE is a small number, usually quoted in units of ×10⁻⁶ per °C (also written ppm/°C or µε/°C). Enter it here as the plain coefficient — 12 for steel, 23 for aluminium — and the calculator applies the ×10⁻⁶ factor internally. A positive ΔT gives expansion; a negative ΔT gives contraction.
The thermal strain is simply ε = ΔL/L₀ = α·ΔT, independent of length. It is the fractional change in size per unit length, the same kind of strain used in stress–strain analysis.
For steel (α ≈ 12×10⁻⁶/°C) heated 50 °C, ε = 12×10⁻⁶ × 50 = 6.0×10⁻⁴, or 0.06%. Small as a fraction, but over a 10 m member it is 6 mm of movement — far more than enough to buckle a rail or crack a rigid joint.
If the member cannot expand or contract, the thermal strain is converted entirely into mechanical strain of the opposite sign, producing a stress σ = E·ε = E·α·ΔT, where E is the elastic modulus.
Crucially, this stress does not depend on length — a short fully-restrained bar develops exactly the same stress as a long one. For steel (E ≈ 200 GPa, α ≈ 12×10⁻⁶/°C) a 50 °C rise gives σ = 200×10⁹ × 12×10⁻⁶ × 50 = 120 MPa, a substantial compressive stress capable of yielding or buckling the member.
Because restrained expansion creates stresses that scale with ΔT and E, designers rarely fight thermal movement head-on — they accommodate it. Bridges use expansion joints and roller/sliding bearings, railways leave gaps or use continuously welded rail with controlled stress, and piping uses expansion loops, bellows, and slip joints.
The calculator lets you see the trade-off directly: the free-expansion figure tells you how much movement a joint must absorb, while the restrained-stress figure tells you how punishing it would be to block that movement instead.
Enter just the coefficient in units of ×10⁻⁶ per °C. For structural steel type 12, for aluminium type 23, for concrete about 10, and for copper about 17. The calculator multiplies your value by 10⁻⁶ automatically, so you should not enter 0.000012 — just 12.
Free expansion ΔL grows with length, but strain (ΔL/L₀) does not — the length cancels out. Since stress depends on strain (σ = E·ε = E·α·ΔT), it is independent of the member length. A 0.1 m and a 100 m fully-restrained steel bar develop the same thermal stress for the same temperature change.
The linear coefficient α describes change in one dimension (length). The volumetric coefficient β describes change in volume and, for isotropic materials, is approximately three times the linear value (β ≈ 3α). This calculator uses the linear coefficient, which is what governs the length change and the restrained axial stress in beams, rails, and pipes.
Full restraint is the upper bound — real supports usually allow some movement, so actual stresses are lower. But partial restraint, friction, and the difference in CTE between bonded materials (bimetallic effects, concrete-steel, coatings) can still generate large local stresses and thermal fatigue. Use the fully-restrained value as a conservative check and refine with a flexibility or finite-element analysis.
Only approximately. The coefficient itself rises gradually with temperature for most metals, and tabulated values are averages over a stated range (often 20–100 °C). For large temperature swings or cryogenic/high-temperature work, use a CTE appropriate to the temperature range, or integrate the temperature-dependent coefficient for best accuracy.