Hoop Stress Calculator

Consider a thin-walled cylinder of inner radius r and wall thickness t subjected to internal pressure P. Cutting the cylinder along a diametral plane and applying equilibrium gives the hoop stress:

$$\sigma_h = \frac{Pr}{t}$$

Similarly, cutting perpendicular to the axis and balancing forces gives the axial stress:

$$\sigma_a = \frac{Pr}{2t}$$

These are the two principal in-plane stresses. The third principal stress (radial) is approximately zero for thin walls. The von Mises equivalent stress for this biaxial state simplifies to:

$$\sigma_{VM} = \sqrt{\sigma_h^2 - \sigma_h \sigma_a + \sigma_a^2} = \frac{Pr}{t}\frac{\sqrt{3}}{2}$$

The safety factor is computed as the ratio of yield strength to von Mises stress: $$SF = \frac{\sigma_y}{\sigma_{VM}}$$

For design purposes, the minimum wall thickness to achieve a target safety factor of 3 is calculated by setting $$\sigma_{VM} = \sigma_y / 3$$ and solving for t:

$$t_{min} = \frac{Pr\sqrt{3}}{2(\sigma_y/3)} = \frac{3Pr\sqrt{3}}{2\sigma_y}$$

The thin-wall assumption is valid when r/t > 10. For thicker walls, the stress varies significantly through the thickness, and the Lamé equations must be used. The calculator displays the r/t ratio so you can verify the thin-wall assumption holds.

In practice, additional factors affect the required thickness: corrosion allowance (typically 1-3 mm), weld joint efficiency (0.7-1.0), manufacturing tolerances, and dynamic loading. Design codes like ASME BPVC account for all these factors systematically.

The hoop stress is the maximum stress in the vessel wall and the primary design parameter. The safety factor indicates the margin against yielding — values below 2 suggest the design is inadequate for most applications. The minimum thickness for SF=3 tells you the wall thickness needed for a conservative design. Check the r/t ratio: if below 10, use a thick-wall analysis for more accurate results.