Thermal Stress Calculator
Thermal stress in a restrained bar or pipe from a temperature change (σ = E·α·ΔT): stress, force, free expansion and strain for steel, stainless, aluminum, copper, concrete or PVC, checked against yield.
How it works
Thermal strain ε = α × ΔT · restrained stress σ = E × α × ΔT × restraint · force F = σ × A · free expansion ΔL = α × L × ΔT
When a member is heated it wants to grow by α·L·ΔT. If its ends are fixed, that growth is prevented and the material is compressed by exactly the strain it was denied, so the stress is Young’s modulus times α·ΔT — independent of length and area. Cooling a restrained member produces tension instead. Steel gains roughly 2.4 MPa for every degree Celsius it is fully restrained, which is why pipelines, rails and bridges need expansion loops, joints or anchors; slender members may buckle well before they reach yield.
Worked example
A carbon-steel bar 10 m long with a 20 cm² section is heated by 50 °C between fixed supports. Unrestrained it would grow 12 × 10⁻⁶ × 10,000 mm × 50 = 6.00 mm; restrained, the stress is 200,000 MPa × 0.0006 = 120.00 MPa in compression and the end force is 120 × 2,000 mm² = 240.0 kN — 48.0% of the 250 MPa yield strength.
Sources: Engineering ToolBox — Thermal expansion: stress and force · Wikipedia — Thermal stress
Frequently asked questions
Does thermal stress depend on the length of the bar?
No. For a fully restrained member, σ = E·α·ΔT regardless of length or area. Length only changes how much it would have expanded, and area only changes the force on the supports.
Is thermal stress tension or compression?
Heating a restrained member causes compression; cooling causes tension. Concrete is weak in tension, so cooling (shrinkage) cracks are common without joints or reinforcement.
How can thermal stress be reduced?
Allow movement with expansion joints, loops or sliding supports, choose materials with matching expansion coefficients, and install at a mid-range temperature so the swing either way is smaller.
Why does the restrained bar sometimes fail by buckling?
A long, slender member in compression buckles at the Euler critical load, which can be far below the yield stress. Check the compressive force against the buckling load for long pipes and rails.
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