Thermal Stress Analysis
Theory: stress comes from restrained expansion
Overview
I assumed any temperature difference causes thermal stress. Isn't that right?
A common misconception. The cause isn't the temperature difference itself but expansion that was prevented. A free bar heated uniformly just lengthens with zero stress; fix both ends and the prevented expansion becomes stress. Even with a temperature difference, if it's linear through a plate's thickness the plate simply bends without stress. Stress comes from two things: external restraint and the deviation of the temperature profile from linear. Let's check with numbers.
External restraint
Heating steel (E = 200 GPa, α = 12×10⁻⁶ /K, ν = 0.3) by 100 K gives 240 MPa compression in a bar fixed at both ends and 343 MPa in a biaxially restrained plate — near typical structural steel yield (about 250 MPa), so even 100 K can cause yielding if restrained.
Internal restraint from the temperature profile
For a large unrestrained plate. The second term is expansion from mean temperature and the third bending from the linear component; neither causes stress. Only the deviation from linear remains, and force and moment balance over the thickness (self-equilibrated stress).
Why glasses crack
Pouring boiling water into a thick glass can crack it: the inside tries to expand suddenly while the still-cold outside holds it back, putting the outside in tension — and glass is weak in tension. In the 1910s Corning, troubled by railway signal lantern glass cracking in rain and snow, developed borosilicate glass with about a third of ordinary glass's expansion coefficient. It became the basis of Pyrex cookware — a famous case of reducing thermal stress through the material.
Worked examples
Stress by temperature profile
A large steel plate (no external restraint), 50 K through-thickness difference:
| Profile | Stress |
|---|---|
| Linear (one face 50 K hotter) | 0 (bends only) |
| Linear, bending restrained | ±85.7 MPa |
| Parabolic (both faces 50 K hotter) | Surface −114 MPa / centre +57 MPa |
The parabolic profile's mean is 33 K below the surface, so the surface is held back into compression and the centre pulled into tension; they cancel over the thickness.
Transient quench analysis
A 20 mm steel plate (k = 45 W/mK) is plunged from 300 °C into a 20 °C coolant; solving transient conduction gives the peak surface tension. The upper bound is EαΔT/(1−ν) = 960 MPa.
| Heat transfer coefficient | Biot number | Peak stress | Time |
|---|---|---|---|
| Near-instant cooling | 222 | 855 MPa (0.89 of bound) | 0.03 s |
| 5000 W/m²K | 1.11 | 214 MPa (0.22) | 1.5 s |
| 1000 W/m²K | 0.22 | 61 MPa (0.06) | 2.7 s |
Same 300 °C difference, yet stress varies 14-fold with cooling rate.
The Biot number compares how fast heat leaves the surface with how fast it moves inside. At low Biot the whole plate cools together, so surface-to-mean differences and stress are small. Manson's approximation 1/(1.5 + 3.25/Bi − 0.5e^(−16/Bi)) gives 0.23 at Bi 1.11 and 0.06 at 0.22, matching the calculation. That's why water quenching cracks parts more readily than oil.
Analysis procedure and cautions
- Solve heat transfer, then pass temperatures to the structural model as loads (one-way coupling).
- For transients, peak stress times differ, so pass temperatures at several instants.
- Always set the reference (stress-free) temperature.
- Don't over-restrain; excessive fixity inflates thermal stress.
- Include temperature-dependent properties (E, α, yield); yield falls at high temperature.
“The over-fixed pipe”
A plant steam-pipe analysis with every support fully fixed showed thermal stress over three times allowable. The real pipe had sliding supports free axially, and bends absorbed expansion. Modelling supports as built dropped stress below half the allowable. Thermal stress is set mainly by restraint, so the choice of boundary conditions can change the answer several-fold.
Common mistakes
Mistakes and fixes
| Mistake | Effect | Fix |
|---|---|---|
| Over-restraint | Stress overestimated | Model actual supports |
| Missing reference temperature | Stress from the start | Set stress-free temperature |
| Only final time checked | Peak missed | Search all times |
| Room-temperature properties | High-temperature error | Use temperature-dependent data |
| Coarse thermal mesh | Surface gradient missed | Refine near surfaces |
I'd like to learn related topics.
Try the thermal stress calculator and thermal expansion tool. Related pages include thermal shock analysis, thermal expansion mismatch, thermal buckling and bimetallic strips.
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