Thermal Stress Calculator
When a part is heated or cooled but cannot freely expand, stress builds up — enough to buckle rail or crack castings. Pick a scenario and material, set the temperature change, and get the thermal expansion, stress and force, checked against yield with a pass/fail and factor of safety.
How Thermal Stress Builds
Heat a free bar and it simply lengthens; cool it and it shrinks. Clamp both ends so it cannot move, and that would-be expansion has nowhere to go — it turns into internal stress instead. The same temperature swing that harmlessly stretches a loose rod can crush or crack a restrained one. This is thermal stress, and it is why bridges have expansion joints, pipelines have loops, and continuous rail can buckle in a heat wave.
The Thermal Stress Formula
For a fully restrained member, thermal stress equals Young’s modulus E times the expansion coefficient alpha times the temperature change. Length does not appear: a short bar and a long bar, equally restrained, reach the same stress. The thermal force is that stress times the cross-section area, and the factor of safety is the yield strength divided by the stress. Partial restraint scales the stress by the restrained fraction; free expansion gives zero stress.
Thermal Expansion vs Thermal Stress
The two are easy to confuse. Thermal expansion is the free change in length, alpha times length times temperature change — it happens whenever a part is free to move, and it carries no stress. Thermal stress appears only when that expansion is blocked. A part can have large expansion and zero stress (free), or zero net movement and huge stress (fully restrained). Aluminium expands about twice as much as steel, yet builds less stress for the same temperature change, because its lower modulus dominates the product E times alpha.
Restraint Scenarios
The calculator covers the common cases: a fixed-fixed member and equivalent fully restrained parts — constrained pipe, beam, rail or track and bolted assembly — all reach the full E-alpha-deltaT stress. Partial restraint applies a fraction for joints that give a little. Free expansion reports movement with no stress. Differential expansion handles two bonded materials of equal area forced to the same length, where the mismatch in expansion coefficients sets up stress even without external restraint.
Common Material Expansion Rates
| Material | α (x10⁻⁶/°C) | E (GPa) | Stress per 100°C, fully restrained (MPa) |
|---|---|---|---|
| Steel | 12 | 200 | 240 |
| Stainless 304 | 17 | 193 | 328 |
| Aluminum 6061 | 23 | 69 | 159 |
| Brass | 19 | 100 | 190 |
| Copper | 17 | 117 | 199 |
| Titanium | 8.6 | 116 | 100 |
Frequently Asked Questions
How do you calculate thermal stress?
For a fully restrained part, multiply Young’s modulus by the thermal expansion coefficient and the temperature change. Partial restraint scales the result by the fraction restrained; free expansion gives zero stress.
Does thermal stress depend on length?
No. Fully restrained thermal stress is independent of length – a short and a long bar reach the same stress for the same temperature change. Length only affects the free expansion and the stored energy.
Is heating compression or tension?
Heating a restrained part produces compression; cooling produces tension. Cooling-induced tension is a common cause of cracking in welds and castings.
What is differential thermal expansion?
When two bonded materials with different expansion coefficients are forced to the same length, the mismatch sets up stress even without external restraint – the basis of bimetallic strips and a frequent cause of joint failure.
How do I reduce thermal stress?
Let the part move: expansion joints, loops, slip or sliding supports and flexible connections all lower the restraint fraction and the stress.
Related calculators
- Thermal Expansion Calculator — the free length change before restraint.
- Stress Calculator — normal stress from force and area.
- Factor of Safety Calculator — margin against yield.
- Hardness Conversion Calculator — hardness and strength of the alloy.