Thermal expansion & restraint

Free growth

ΔL = α · L · ΔT

Twelve metres of carbon steel taken from 20 °C to 250 °C grows about 33 mm. That is not a rounding error. It is more than the clearance in most pipe guides and more than the gap most people leave at a wall penetration.

Restrained stress

σ = −α · ΔT · E

Notice what is absent: length. A fully restrained member develops the same thermal stress whether it is 100 mm or 100 m long. Shortening the run does not help. Only letting it move helps.

The numbers are brutal. Carbon steel at ΔT = 100 K, fully restrained, reaches about 240 MPa — effectively at yield before any pressure or dead load has been applied. This is why expansion loops, bellows and slide bearings exist, and why “just anchor both ends” is almost always the wrong answer.

Partial restraint

Real supports are somewhere between free and fixed. The tool scales linearly between the two, which is a simplification but a useful one: it lets you see immediately that even 50 % restraint on a hot line is a serious load path, and that the restraint itself has to be designed for the force it attracts.

Mixed materials

Austenitic stainless expands about forty per cent more than carbon steel for the same temperature rise. Bolt the two together across a heated joint and the differential is the design case, not the average. This catches people out on cladding, on jacketed vessels and on stainless internals inside carbon shells.