Types of stresses in thin walled cylinders

A cylinder is called thin walled when the wall thickness is small compared with the diameter ($t < d/20$). Under internal fluid pressure $p$ the wall is stressed in two principal directions; the radial stress is so small that it is neglected.

        hoop (circumferential)
          ^  ^  ^
   sl <--[=======]--> sl
        longitudinal

1. Circumferential (hoop) stress, $\sigma_c$

• Acts tangentially around the circumference, tending to split the cylinder along a longitudinal section.
• Considering half the cylinder: bursting force $p\,d\,L$ = resisting force $2\sigma_c\,t\,L$.

$$\sigma_c = \frac{pd}{2t}$$

2. Longitudinal (axial) stress, $\sigma_l$

• Acts along the axis of the cylinder, tending to separate it along a circular (transverse) section.
• Force on the ends $p\times\frac{\pi}{4}d^2$ = resisting force $\sigma_l\,\pi d\,t$.

$$\sigma_l = \frac{pd}{4t}$$

3. Radial stress

Varies from $-p$ at the inner surface to zero at the outer surface; it is very small compared with the other two and is ignored in thin cylinders.

4. Maximum shear stress

With $\sigma_c$ and $\sigma_l$ as principal stresses, the maximum in-plane shear stress is

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

Notes

• $\sigma_c = 2\sigma_l$, so the hoop stress decides the thickness; a boiler shell fails along a longitudinal seam first.
• Both stresses are tensile, and both are uniform over the thickness.