Hoop Stress

Hoop stress is a circumferential stress formed in a pipe or cylinder subjected to differential internal/external pressure. Hoop stress is tangential to the surface.

Thin-walled Assumption

σ_{θ} = \frac{P r}{t}

Where:

$P$	is the internal pressure (MPa);
$t$	is the wall thickness (mm);
$r$	is the internal radius (mm);
$σ_{θ}$	is the hoop stress (MPa)

The thin-walled assumption is only valid when: $10 t < r$

Thick Walled Cylinders

The radial and circumferential stresses in a thick walled cylinder are given as follows:

\begin{matrix} σ_{r} & = P \frac{a^{2} (r^{2} - b^{2})}{r^{2} (b^{2} - a^{2})} σ_{θ} & = P \frac{a^{2} (r^{2} + b^{2})}{r^{2} (b^{2} - a^{2})} \end{matrix}

Where:

$a$	is the inside radius (mm);
$b$	is the outside radius (mm);
$r$	is the radial distance to the point at which the stress is calculated (mm)

Note that if the cylinder has closed ends there is also an axial load:

σ_{z} = P \frac{a^{2}}{b^{2} - a^{2}}

These stresses can be combined to check for failure against the Tresca criterion which states that yielding begins when the maximum shear stress at a point equals the maximum shear stress at yield in uniaxial tension. That is:

τ_{m a x} >= \frac{Y}{2}

Where the maximum shear stress is calculated by:

τ_{m a x} = \frac{σ_{m a x} - σ_{m i n}}{2}

Note On Nominal Pipe Sizes

The purpose of having nominal pipe sizes with wall thickness defined by "schedules" is ostensibly to allow use of one schedule of pipe across a range of pipe sizes for the same design pressure. It is worth noting that the rate at which wall thickness increases with pipe diameter does not keep up with the corresponding increase in stress as per the equations above.

This is demonstrated in the figure below with the stress in a Schedule 40 pipe per unit of internal pressure plotted at a range of nominal pipe sizes.