Library / Mechanical / MEC-26
Mechanical
Thermal stress in a restrained bar
Stress produced when a fully restrained bar is heated or cooled.
- Ref
- MEC-26
- Rev
- v1
- Field
- Mechanical
- Result
- σ [MPa]
Assumes full axial restraint of a uniform linearly elastic bar, uniform temperature change and free lateral contraction. Tension is positive, so heating a restrained bar gives compression. αT is shown in 10⁻⁶ K⁻¹.
A bar that cannot expand when it heats up pushes against its supports; the push is the stress given here.
Calculator
- Free thermal strain600 µε
- Free expansion of a 1 m bar0.6 mm
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A steel bar heated by 50 K and fully restrained carries about 120 MPa of compression, roughly half the yield strength of mild steel.
- Allowing expansion joints, sliding supports or gaps removes this stress.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (MPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ΔT Temperature change | −200 to 500 K | −1,200 | 480 | 1,680 | +10% | +10 % |
| αT Expansion coefficient | 1 to 30 10⁻⁶ K⁻¹ | −300 | −10 | 290 | +10% | +10 % |
| E Young’s modulus | 10 to 400 GPa | −240 | −6 | 234 | +10% | +10 % |
Sweep of Temperature change
| Temperature change (K) | Thermal stress (MPa) |
|---|---|
| −200 | 480 |
| −112.5 | 270 |
| −25 | 60 |
| 62.5 | −150 |
| 150 | −360 |
| 237.5 | −570 |
| 325 | −780 |
| 412.5 | −990 |
| 500 | −1,200 |
Sweep one variable
Variables and how each one contributes
Thermal stress
What the equation gives you. Everything else feeds it.
Young’s modulus
Elastic stiffness. Directly proportional.
Expansion coefficient
Linear thermal expansion coefficient (steel 12). Directly proportional.
Temperature change
Uniform change in temperature. Directly proportional; heating gives compression.
Worked example with the values above
- Thermal strain: αTΔT = 1.2×10−5 × 50 = 6×10−4.
- Elastic modulus: E = 200,000 MPa.
- σ = −E αT ΔT = −120 MPa (negative: compression).
Assumptions and limits
- Full restraint of a uniform bar.
- Linear elastic behaviour.
Common mistakes
- Forgetting the minus sign.
- Using the wrong α units.
Statistics and uncertainty
Stress depends on three properties each known to 5–10%, so expect about 15% uncertainty.
Where it is used
- Rail, pipe and structure expansion design.
- Joints between dissimilar materials.
Maths you need
- Young’s modulus: stiffness
- Thermal expansion: ΔL = α L ΔT