Two equations, and the difference between them explains most of the expansion problems I have been asked to look at.
Free thermal growth:
ΔL = α · L · ΔTFully restrained thermal stress:
σ = −α · ΔT · ELook at what happened to L.
The cancellation
It is worth doing the substitution rather than taking my word for it, because the result is genuinely counter-intuitive the first time.
Strain is the growth divided by the original length:
ε = ΔL / L = α · ΔTLength divides out. Stress in the elastic range is σ = E · ε, so:
σ = α · ΔT · EA fully restrained member develops the same thermal stress whether it is 100 mm long or 100 m long. There is no length term because there was never a length term — strain is dimensionless, and stress follows strain.
What that means in practice
Shortening the run does not reduce the stress. This is the mistake I see most often. Somebody has a hot line that is loading its anchors, and the proposed fix is to reroute it shorter. That reduces the movement, which may help the adjacent equipment, but it does nothing whatever to the restrained stress. If it was at yield before, it is at yield now.
Only letting it move helps. Expansion loops, bellows, slide bearings, guides
that permit axial travel. The entire discipline exists because you cannot
engineer your way out of σ = αΔTE by rearranging geometry.
The numbers are brutal, quickly. Carbon steel: α ≈ 12 × 10⁻⁶ /K, E ≈ 200 GPa. So each kelvin of restrained ΔT is worth about 2.4 MPa.
| ΔT | Restrained stress | Fraction of a 260 MPa yield |
|---|---|---|
| 50 K | 120 MPa | 46 % |
| 100 K | 240 MPa | 92 % |
| 150 K | 360 MPa | past yield |
One hundred kelvin — the difference between a cold morning and warm process water — puts a fully restrained carbon steel member effectively at yield before any pressure, dead load or wind has been applied.
The movement still has to go somewhere
Where the member is not restrained, the growth is real and it is larger than people expect. Twelve metres of carbon steel from 20 °C to 250 °C grows about 33 mm.
Thirty-three millimetres is more than the clearance in most pipe guides. It is more than the gap left at a typical wall penetration. It is enough to load a branch connection that somebody sized for pressure alone, and it is more than enough to shear the anchor bolts on a support that was designed as though the line were static.
So the two failure modes are complementary, and you get to pick which one you are designing for:
- Restrain it, and pay in stress.
- Release it, and pay in movement that has to be accommodated everywhere along the run.
There is no third option where nothing happens.
Two traps worth naming
Mixed materials. Austenitic stainless expands roughly 40 % more than carbon steel for the same ΔT — about 17 × 10⁻⁶ /K against 12. Bolt the two together across a heated joint and the differential is your design case, not the average of the two. This catches people on cladding, on jacketed vessels, and on stainless internals inside carbon steel shells, where the assembly is fine at ambient and fights itself at temperature.
Buckling. A restrained member that is heated goes into compression. If it
is at all slender, it will buckle well before it reaches yield, and the buckling
check is not the same calculation. Nothing in σ = αΔTE tells you about
stability. The
thermal expansion tool on the bench says so
explicitly, because it is the failure mode most likely to be missed by the
person who just did the stress sum and felt reassured.
Partial restraint
Real supports are neither free nor fixed. They are somewhere in between, and the in-between is where the arithmetic gets genuinely hard — friction at guides, stiffness of the supporting steel, flexibility of the connected equipment.
Treating it as a linear blend between the two extremes is a simplification, but it is a useful one, because it makes the important point immediately visible: even 50 % restraint on a hot line is a serious load path, and the restraint itself has to be designed for the force it attracts.
That last part gets forgotten a lot. The line is analysed carefully; the bracket holding it is sized by eye.
The version I keep in my head
- Growth scales with length. Stress does not.
- You cannot make thermal stress go away by making the run shorter.
- Every kelvin of restrained ΔT in carbon steel costs about 2.4 MPa.
- If it is restrained and slender, check buckling before you check yield.
- Whatever movement you allow, something downstream has to absorb it — and that something needs to know.
