Thin cylinders under pressure: hoop and longitudinal stress
Why pipes and tanks split along their length, and how to find the wall thickness of a pressure vessel.
A cylinder is "thin" when its wall thickness t is less than about one-twentieth of its diameter d. Then the stress across the wall can be taken as uniform.
Hoop (circumferential) stress
Cut the cylinder along its length. Pressure p acting on the projected area d × L is resisted by two walls of area t × L:
σh = p d2 t
Longitudinal stress
Cut the cylinder across. Pressure on the end area π d²/4 is resisted by the ring of area π d t:
σl = p d4 t
Why pipes split lengthwise
Hoop stress is twice longitudinal stress. So a pipe that bursts from overpressure opens up along its length, and the longitudinal joints of a riveted or welded shell need more strength than the circumferential ones.
Worked example
A water main of 1.2 m diameter carries a pressure of 1.5 N/mm². Allowable stress is 100 N/mm². Find t.
t = p d2 σh = 1.5 × 12002 × 100 = 9 mm
Thin spheres
A sphere has the same stress in every direction:
σ = p d4 t
So for the same pressure and stress, a sphere needs half the wall thickness of a cylinder. That is why gas holders and LPG storage vessels are often spherical.
Change in volume
The volumetric strain of a thin cylinder is
δVV = p d4 t E (5 − 4μ)
Quick check
In a thin cylinder under internal pressure, hoop stress compared with longitudinal stress is: