Pressure Vessel Calculator

Internal pressure in a cylindrical vessel creates a biaxial or triaxial stress state. The principal stresses act in three directions: hoop (circumferential), axial (longitudinal), and radial (through-thickness).

Thin-wall approximation (valid when r/t > 10):

$$\sigma_h = \frac{Pr}{t}, \quad \sigma_a = \frac{Pr}{2t}, \quad \sigma_r \approx 0$$

The hoop stress is always twice the axial stress, which is why cylindrical vessels that fail tend to split along their length (perpendicular to the hoop direction). The radial stress varies from −P at the inner surface to 0 at the outer surface but is negligible compared to the other components for thin walls.

Thick-wall (Lamé) equations:

$$\sigma_h = P\frac{r_i^2 + r_o^2}{r_o^2 - r_i^2}, \quad \sigma_a = \frac{Pr_i^2}{r_o^2 - r_i^2}, \quad \sigma_r = -P \text{ (at inner wall)}$$

For thick walls, radial stress is significant and equals −P at the inner surface (compressive). The hoop stress at the inner wall is the maximum stress in the entire vessel.

The von Mises equivalent stress combines all three principal stresses into a single scalar for yield comparison: $$\sigma_{VM} = \sqrt{\frac{1}{2}\left[(\sigma_h - \sigma_a)^2 + (\sigma_a - \sigma_r)^2 + (\sigma_r - \sigma_h)^2\right]}$$ If $$\sigma_{VM} \geq \sigma_y$$, the material begins to yield.

Design codes typically require a safety factor of 3–4 on burst pressure, meaning the allowable working pressure is one-third to one-quarter of the pressure that would cause failure. The ASME code uses allowable stress values that account for material variability, temperature derating, and weld efficiency factors.

The hoop stress is the dominant stress and the primary design driver — it determines the minimum wall thickness required. The axial stress is exactly half the hoop stress for thin walls. The r/t ratio indicates whether the thin-wall approximation is valid: above 10, the thin-wall formulas are accurate within about 5%. The von Mises stress should be compared to the material's yield strength to assess whether yielding will occur.