Pressure Vessel Stress Calculator Formula
Understand the math behind the pressure vessel stress calculator. Each variable explained with a worked example.
Formulas Used
Hoop (Circumferential) Stress
hoop_stress = pressure * radius / thicknessAxial (Longitudinal) Stress
axial_stress = pressure * radius / (2 * thickness)D/t Ratio
dt_ratio = diameter / thicknessVariables
| Variable | Description | Default |
|---|---|---|
pressure | Internal Pressure (P)(MPa) | 5 |
diameter | Inner Diameter (D)(mm) | 1000 |
thickness | Wall Thickness (t)(mm) | 10 |
radius | Derived value= diameter / 2 | calculated |
How It Works
Thin-Walled Pressure Vessel Stresses
For a cylindrical vessel where the wall thickness is much less than the diameter (D/t > 10), the stresses can be calculated using simplified thin-wall formulas.
Formulas
Hoop stress: sigma_h = P × r / t (circumferential, the larger stress)
Axial stress: sigma_a = P × r / (2t) (longitudinal, half the hoop stress)
The hoop stress is twice the axial stress, which is why pressure vessels fail along longitudinal seams (circumferential stress governs).
Worked Example
A vessel with 1000 mm diameter, 10 mm wall, at 5 MPa pressure.
- 01r = 1000 / 2 = 500 mm
- 02Hoop stress = 5 × 500 / 10 = 250 MPa
- 03Axial stress = 5 × 500 / (2 × 10) = 125 MPa
- 04D/t = 100 (thin-wall assumption valid)
Frequently Asked Questions
When is the thin-wall assumption valid?
The thin-wall formulas are accurate when D/t > 10 (some sources say > 20). For thicker walls, thick-wall (Lame) equations are needed, which account for stress variation through the wall thickness.
Why is hoop stress twice axial stress?
A cylindrical vessel must resist pressure on a larger projected area in the circumferential direction than the longitudinal. The force balance on a longitudinal slice yields twice the stress of a transverse slice.
How do codes like ASME size vessels?
ASME BPVC Section VIII uses modified versions of these formulas with allowable stress, joint efficiency, and corrosion allowance: t = P*R / (S*E - 0.6*P) + C.A., where S is allowable stress and E is joint efficiency.
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