Membrane Stresses in Thin-Walled Cylinders and Spheres

For a thin-walled closed cylindrical shell of radius $r$ and thickness $t$ under uniform internal gauge pressure $p$, sufficiently far from edges and local disturbances:

$$N_\theta=pr,\qquad \sigma_\theta=\frac{pr}{t},$$

$$N_z=\frac{pr}{2},\qquad \sigma_z=\frac{pr}{2t}.$$

The hoop stress $\sigma_\theta$ is twice the longitudinal stress $\sigma_z$.

For a thin-walled closed spherical shell:

$$N=\frac{pr}{2},\qquad \sigma=\frac{pr}{2t}.$$

• $p$ — internal gauge pressure;
• $r$ — shell radius;
• $t$ — wall thickness;
• $N_\theta$, $N_z$ — hoop and longitudinal membrane force resultants per unit length;
• $\sigma_\theta$, $\sigma_z$ — hoop and longitudinal membrane stresses;
• $N$, $\sigma$ — membrane resultant and stress in a spherical shell.

These are membrane approximations for $t\ll r$ and uniform pressure. Additional bending and local stresses may occur near edges, openings, supports, and abrupt geometric changes.

The calculator below uses the hoop-stress relation for a thin-walled cylinder.