Thermal Stress Calculator: Restrained Bar, Pipe or Rail
Find the stress and force in a bar, pipe or rail that is heated or cooled but held at its ends, with or without a gap to grow into.
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Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.
How this calculator works
A bar that is heated grows by its length times the expansion coefficient times the temperature rise. If a support stops it, the bar is squeezed back to its fixed length and a stress builds up. If there is a gap, the bar grows into the gap first, and only the movement left over is stopped.
Enter the length, the temperature change, the expansion coefficient and the modulus of elasticity, and the gap if there is one. The calculator gives the free growth, the part the supports stop, the stress (compression for a rise, tension for a fall), the force on the supports and the temperature rise that would just close the gap.
Formulas used
- \delta_T = \alpha L \Delta T
- \sigma = E\left(\alpha\Delta T - \dfrac{a}{L}\right)
- F = \sigma A
Worked example
With the default numbers in the calculator above:
- Free thermal movement: 2.16 mm
- Movement the supports stop: 2.16 mm
- Thermal stress: 144 MPa (compression)
- Force on the supports: 144 kN
- Temperature rise that would just close the gap: 0 K
- Factor of safety against yield: 1.7361
Change the numbers in the calculator to see the full working, every step, for your own case.
Common questions
Does the cross-section change the stress?
No. The stress depends only on the material, the temperature change and the gap. The cross-section only changes the force on the supports.
Why does a fully fixed steel bar reach such a high stress?
Steel grows about 12 millionths of its length for each degree and is very stiff. A 100 degree rise held at both ends gives about 240 MPa, which is close to the yield strength of mild steel.
What about buckling?
A long slender bar in compression can buckle at a lower stress than it would yield. Check it as a column as well.
How do I avoid the stress?
Let the bar move: expansion joints, loops in pipework, sliding supports or a gap. The calculator shows how big the gap needs to be to give no stress at all.
Related calculators
Further reading: Standard strength of materials texts (thermal stress in restrained members).
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