What this calculator does
A pressurised pipe or vessel is stressed in two directions at once. Hoop stress acts around the circumference trying to split it lengthways, and longitudinal stress acts along the axis trying to pull the ends apart.
Hoop stress is always exactly twice the longitudinal stress in a thin-walled cylinder, which is why a failing pipe splits along its length rather than around its circumference. It is also why a sausage splits the way it does when cooked.
The formula
Hoop stress is pressure times diameter divided by twice the wall thickness. Longitudinal stress is the same expression divided by four, making it exactly half. A sphere carries the hoop value halved, which is why spherical vessels are more efficient.
| Term | Meaning |
|---|---|
| Hoop stress (σh) | Circumferential stress, acting around the cylinder wall. |
| Longitudinal stress (σl) | Axial stress along the cylinder, always half the hoop stress in a thin wall. |
| Thin-walled assumption | The approximation that wall thickness is small compared with diameter, generally under about a twentieth. |
The inputs explained
| Field | What to enter |
|---|---|
| Internal pressure (p) (MPa) | Internal gauge pressure in megapascals. |
| Shell diameter (d) (m) | The shell diameter in metres. |
| Wall thickness (t) (m) | The wall thickness in metres. Keep it below about a twentieth of the diameter for the thin-wall assumption to hold. |
| Joint efficiency (η) | Joint efficiency. Use 1 for a seamless shell, or the relevant figure for a welded joint. |
When to use it
Checking a pressure vessel
Hoop stress is the governing figure and must stay within the material's allowable stress with an appropriate safety factor.
Specifying pipe wall thickness
For a given pressure and diameter, the required thickness follows directly from the allowable stress.
Understanding why pipes split lengthways
Since hoop stress is twice the longitudinal stress, failure occurs along the length rather than around the circumference.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does pressure drive the stress in the wall?
The same vessel at a range of internal pressures.
| Internal pressure | Hoop stress (σh) | Longitudinal stress (σl) |
|---|---|---|
| 1 MPa | 30.00 MPa | 15.00 MPa |
| 2 MPa | 60.00 MPa | 30.00 MPa |
| 4 MPa | 120.00 MPa | 60.00 MPa |
| 8 MPa | 240.00 MPa | 120.00 MPa |
Questions
Why is hoop stress twice the longitudinal stress?
Because of the geometry. The area resisting the axial force is the full circumference of the wall, while the area resisting the circumferential force is only two wall thicknesses along the length. Working the areas through gives the factor of two.
Why do spherical vessels use less material?
Because a sphere has no preferred direction, so the stress is the same all round and equal to half the cylindrical hoop value. For the same pressure and diameter a sphere needs about half the wall thickness.
When does the thin-wall assumption fail?
Once the wall thickness exceeds roughly a twentieth of the diameter. Beyond that the stress varies noticeably through the wall and a thick-walled analysis is required.
Does external pressure work the same way?
No. External pressure raises the risk of buckling rather than tensile failure, and buckling depends on geometry and stiffness in a completely different way. It needs its own analysis.
For stress from a direct force, see the mechanical stress calculator. For the stiffness of the material, see the Young's modulus calculator.