Thick-Walled Cylinder Calculator (Lame): Hoop and Radial Stress Under Internal and External Pressure
Stresses in a thick pressure vessel wall.
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How this calculator works
Enter the inside and outside radius.
Enter the internal and external pressure.
The calculator gives the stresses at the surfaces and at the radius you choose.
Formulas used
- \sigma_\theta = A + \frac{B}{r^2}
- \sigma_r = A - \frac{B}{r^2}
- A = \frac{p_ia^2 - p_ob^2}{b^2 - a^2},\ B = \frac{(p_i - p_o)a^2b^2}{b^2 - a^2}
Worked example
With the default numbers in the calculator above:
- Hoop stress at the inside surface: 104 MPa
- Radial stress at the inside surface: -40 MPa
- Hoop stress at the outside surface: 64 MPa
- Hoop stress at the radius you chose: 82 MPa
- Radial stress at the radius you chose: -18 MPa
- Largest shear stress (at the inside surface): 72 MPa
- Hoop stress by the thin-wall formula, for comparison: 100 MPa
Change the numbers in the calculator to see the full working, every step, for your own case.
Common questions
When is a cylinder thick-walled?
When the wall is more than about a tenth of the radius. Below that the thin-wall formulas are good enough.
Where is the stress the greatest?
The hoop stress is the greatest at the inside surface.
Does this include the ends?
The Lame equations assume the stress along the length is the same everywhere. A closed end adds an axial stress.
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Got a question about this? Ask an engineering question and get a plain-language answer with references.
Further reading: Any strength of materials text (thick-walled cylinders, Lame's equations). The pressure vessel standard sets design rules.
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