Thermal Expansion Calculator (Growth and Thermal Stress)
Free thermal growth of a member over a temperature change, and the stress and force it develops when it is restrained. Handles partially restrained members and joints with slack that has to be taken up before any load appears, so an expansion loop or a slotted hole can be shown to eliminate the stress entirely rather than merely reduce it.
How to use this calculator
- Enter the reference length, the coefficient of thermal expansion, and the two temperatures. Use a negative service temperature for a cold duty.
- Enter the Young's modulus at the service temperature, since it falls as the metal gets hot.
- Set the restraint from 0 percent for a member that is free to grow to 100 percent for one that is fully anchored, and enter any slack in the supports that must be taken up first.
- Read the free growth, the thermal stress, the force needed to hold the member, and the temperature at which the slack is closed.
Frequently asked questions
Why does thermal stress not depend on the length or the cross section?
Because fully restrained, the strain is set purely by the temperature change and the expansion coefficient, and stress is modulus times strain. A bar 10 mm long and a bar 10 m long reach exactly the same stress, since both are held at the same strain. This is the single most surprising thing about thermal stress, and it is why the cure for a long hot line is flexibility rather than more metal in the wall.
How do I work out the temperature at which a gap closes?
Divide the slack by the expansion coefficient times the length, which gives the temperature change needed to consume it. A 0.5 mm gap in a 1 m steel member with a coefficient of 12 microstrain per kelvin closes after about 42 K. Until that point the member is free and carries no thermal stress at all, which is the entire principle behind expansion loops and slotted holes.
What happens if the thermal stress exceeds the yield strength?
The member yields on the first heat up and then shakes down to a lower stress on subsequent cycles, so it does not necessarily fail. The first cycle does have to be checked against the low cycle fatigue life, because a few hundred such cycles can start and propagate a crack. Thermal fatigue is a cycle counting problem, not a static strength one.
Do cold cases matter as much as hot ones?
They can matter more. A line installed hot and cold in service shrinks, and the resulting tension can pull a support out of a floor, crack a nozzle or overstress a flange. Use the installation temperature as the reference and the service temperature as the extreme, in whichever direction it goes, and check both ends of the range.