Thermal Stress Calculator (Restrained Bar)
Stress and force in a bar that cannot expand or contract when its temperature changes, from the material's modulus, expansion coefficient and the temperature change.
Results
The dashed line marks the value you entered.
What this tool does
A bar held rigidly at both ends cannot grow, so the expansion it is stopped from making turns into stress: σ = E × α × ΔT. For steel (E about 200 GPa, α about 12 × 10⁻⁶ per K) that is roughly 2.4 MPa for every degree, so a 100 °C swing would load the bar with about 240 MPa — the order of the yield strength of ordinary structural steel. This is why rails, pipes and bridge decks get expansion joints or loops. The result is a ceiling: real supports always give a little, and a long compressed bar will buckle well before it crushes. Heating gives compression, cooling gives tension.
Formula
σ = E × α × ΔT (bar fully restrained) · F = σ × A · blocked strain = α × ΔT
Variables
| Symbol | Meaning | Unit |
|---|---|---|
em | Young's modulus | GPa |
al | Coefficient of thermal expansion | 10⁻⁶/K |
dt | Temperature change | °C |
ar | Area | mm² |
sy | Yield strength of the material (0 to skip) | MPa |
SG | Thermal stress in the bar | MPa |
FC | Force on the supports | kN |
SN | Strain that is blocked | % |
SR | Stress as a share of the yield strength | % |
Worked example
- Young's modulus200 GPa
- Coefficient of thermal expansion12 10⁻⁶/K
- Temperature change50 °C
- Area500 mm²
- Yield strength of the material (0 to skip)250 MPa
- Thermal stress in the bar120.0 MPa
- Force on the supports60.00 kN
- Strain that is blocked0.0600 %
- Stress as a share of the yield strength48 %
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- Mixing units is the most common source of error. Convert every input to the units shown next to each field before calculating.