Thin-Walled Cylinder Longitudinal Stress
Thin-Walled Cylinder Longitudinal Stress
A thin-walled cylinder under internal pressure — a pipe, a pressure vessel shell, a gas cylinder — develops two distinct stress components: hoop stress acting circumferentially and longitudinal stress acting along the cylinder's axis. The longitudinal stress comes from the pressure pushing outward on the end caps, which pulls the cylindrical wall in tension along its length; it works out to exactly half the hoop stress for the same geometry, a relationship worth knowing for a quick sanity check.Because longitudinal stress is smaller than hoop stress, it is rarely the governing check for a simple pressurized cylinder — but it becomes important where the cylinder also carries axial mechanical loads (like a pipeline under thermal expansion restraint, or a pressure vessel supporting its own weight), where the combined axial stress can exceed what internal pressure alone would suggest.
The longitudinal stress in a thin-walled cylinder is sigma_l = p_i·d_c/(4·t_c), the internal pressure times the cylinder diameter, divided by four times the wall thickness. where p_i is the internal pressure, d_c is the cylinder diameter, t_c is the wall thickness, and sigma_l is the resulting longitudinal stress.
Inputs
p_i := 2 MPa=2 MPa
d_c := 500 mm=500 mm
t_c := 10 mm=10 mm
This thin-wall formula distributes the total axial force from pressure acting on the end cap over the thin circumferential wall area, giving the longitudinal tensile stress in the cylinder wall.
sigma_l := p_i·d_c/(4·t_c)=25 MPa
Results
A longitudinal stress of 25 MPa here is, as expected, exactly half the hoop stress this same geometry would produce (which would work out to 50 MPa) — a useful cross-check when verifying either calculation by hand. Because this formula assumes a thin wall (diameter-to-thickness ratio typically above about 10), it becomes progressively less accurate for thick-walled vessels, where a more detailed Lamé thick-cylinder analysis is needed instead. In practice, the hoop stress — not the longitudinal stress — nearly always governs wall thickness selection for a simple pressurized cylinder, so this check mainly serves to confirm the axial direction is not a hidden governing case.
Assumptions & limitations
- Thin-wall assumption (diameter-to-thickness ratio large enough that stress is treated as uniform through the wall).
- Cylinder is unrestrained and pressure-loaded only — no additional external axial load or thermal restraint included.
- Ends are assumed closed (pressure acting on end caps generates the longitudinal tension).
- Hoop stress, typically the governing case, is calculated separately.