Thermal Stress Calculator — σ = E·α·ΔT
Find the stress that builds up in a structural member when temperature changes but the member cannot expand freely. Enter Young's modulus, the linear thermal expansion coefficient, and the temperature change to get the induced stress and constraining force.
GPa
1/K
K
m²
σ = E · α · ΔT — stress induced in a fully constrained member
- 1
Young's modulus in Pa
200 × 10⁹ = 200,000,000,000 - 2
Thermal strain ε = α × ΔT
0.000012 × 50 = 0.0006 - 3
Thermal stress σ = E × ε ÷ 10⁶ (MPa)
200,000,000,000 × 0.0006 ÷ 10⁶ = 120Heating a constrained member creates compression; cooling creates tension.
How does this calculator work?
A fully constrained member cannot expand when heated, so thermal stress σ = E·α·ΔT builds up — E is Young's modulus, α is the linear expansion coefficient, and ΔT is the temperature change. Heating creates compression; cooling creates tension. The constraining force is F = σ·A, where A is the cross-sectional area.
Formula
How this is calculated
When a solid member is free to expand, a temperature change ΔT produces a thermal strain ε = α·ΔT with no accompanying stress. If the member is fully constrained (ends fixed so it cannot change length), the thermal expansion is prevented and an internal mechanical stress develops: σ = E·α·ΔT, where E is Young's modulus (Pa) and α is the linear coefficient of thermal expansion (1/K). Heating a constrained member causes compressive stress; cooling causes tensile stress.
This formula derives from Hooke's law: if the free thermal strain ε = α·ΔT is blocked, the constraint imposes an equal and opposite mechanical strain, giving σ = E·ε = E·α·ΔT. The cross-sectional area A is not needed for the stress itself, but the constraining force F = σ·A tells you how much force the supports or joints must carry.
The calculator assumes full constraint (zero net deformation), linear elastic behaviour (Hooke's law valid), uniform temperature change, and homogeneous isotropic material. Partial constraint, plasticity, or non-uniform temperatures require more advanced analysis. Typical yield stresses are 250–500 MPa for structural steel, so even moderate ΔT on a fully constrained steel member (σ ≈ 200×0.000012×ΔT GPa = 2.4 MPa/K) can cause plastic deformation at ΔT around 100 K.
Frequently asked questions
Heating a constrained member causes it to try to expand, so the constraint puts it in compression (σ positive in the compressive sense). Cooling tries to shrink the member, and the constraint puts it in tension. The sign depends on your sign convention — this calculator shows the magnitude; apply the sign based on whether the temperature rose or fell.
For partial constraint, only a fraction of the free thermal strain is prevented. If the member can move by δ but its free expansion would be δ_free = α·L·ΔT, the mechanical strain is (δ_free − δ)/L, and the stress is E × that mechanical strain. Full constraint is the worst-case scenario and what this calculator computes.
Without gaps between rail sections, a summer temperature rise of 40–50 K on fully constrained steel (E = 200 GPa, α = 12×10⁻⁶/K) would generate σ ≈ 96–120 MPa of compressive stress, risking buckling of the track (sun kink). Expansion joints allow controlled free expansion so no stress builds up.
Also known as
TG we-Calculate Editorial Team. (2026). Thermal Stress Calculator — σ = E·α·ΔT [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/thermal-stress-calculator
TG we-Calculate Editorial Team. "Thermal Stress Calculator — σ = E·α·ΔT." TG we-Calculate. 2026. https://we-calculate.com/calculator/thermal-stress-calculator.
TG we-Calculate Editorial Team, "Thermal Stress Calculator — σ = E·α·ΔT," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/thermal-stress-calculator
@misc{wecalculate_thermal_stress_calculator, title = {Thermal Stress Calculator — σ = E·α·ΔT}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/thermal-stress-calculator}}, year = {2026}, note = {TG we-Calculate} }
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