Hoop, longitudinal and max shear stress in thin cylinder

For a thin cylinder (internal diameter $d$, length $L$, thickness $t$, internal pressure $p$, $t < d/20$) the radial stress is negligible and the wall is in a state of biaxial tension.

Hoop (circumferential) stress

Cut the cylinder by a plane along its axis. The pressure acts on the projected area $d\times L$; two wall sections of area $t\times L$ resist it.

$$p\,d\,L = 2\,\sigma_c\,t\,L \quad\Rightarrow\quad \sigma_c = \frac{pd}{2t}$$

Longitudinal stress

Cut across the cylinder. The pressure on the end area $\frac{\pi}{4}d^2$ is resisted by the ring area $\pi d\,t$.

$$p\,\frac{\pi}{4}d^2 = \sigma_l\,\pi d\,t \quad\Rightarrow\quad \sigma_l = \frac{pd}{4t}$$

Maximum shear stress

$\sigma_c$ and $\sigma_l$ are principal stresses (no shear on these planes), so

$$\tau_{max} = \frac{\sigma_c-\sigma_l}{2} = \frac{pd}{8t}$$

Summary