What is hoop stress in a thin cylinder?

Hoop (circumferential) stress is the tensile stress acting around the circumference of a pressurised cylinder, tending to split it along a longitudinal line. It equals sigma_h = p d/(2t), where p is internal pressure, d the diameter, and t the wall thickness.

Why is hoop stress twice the longitudinal stress?

Hoop stress sigma_h = pd/2t resists bursting along a longitudinal seam, while longitudinal stress sigma_l = pd/4t resists the end caps blowing off. Geometry makes the circumferential force per unit length twice the longitudinal, so a cylinder tends to burst along its length, not around its girth.

When is the thin-cylinder theory valid?

The thin-cylinder assumption (uniform stress across the wall) is valid when the wall thickness is small compared with the diameter - typically the diameter-to-thickness ratio exceeds about 20 (t < d/20). Thicker walls need Lame's thick-cylinder theory.

Thin Cylinders — Hoop Stress and Longitudinal Stress with Worked Examples

Boilers, gas bottles, pipelines and pressure tanks are all thin cylinders under internal pressure. Two simple formulas — for hoop and longitudinal stress — govern their design, and they explain why a pressurised cylinder always bursts along its length. Here is the derivation with worked examples.

The Two Stresses

Internal pressure p in a closed cylinder of internal diameter d and wall thickness t (with t ≪ d) sets up two membrane stresses:

Hoop (circumferential) stress σ_h — acts tangentially around the wall, tending to split the cylinder along a longitudinal line.
Longitudinal (axial) stress σ_l — acts along the axis, tending to pull the end caps off.

Derivation of Hoop Stress

Cut the cylinder along a longitudinal plane through the axis and consider a length L. The pressure acts on the projected area (d × L), giving a bursting force p·d·L. This is resisted by hoop stress on the two cut wall edges, each of area t·L:

p · d · L = σ_h · (2 · t · L)   →   σ_h = p d / (2 t)

Derivation of Longitudinal Stress

Cut the cylinder on a transverse (circumferential) plane. The pressure acts on the end area (π/4)d², resisted by longitudinal stress over the ring area π·d·t:

p · (π/4) d² = σ_l · (π d t)   →   σ_l = p d / (4 t)

Therefore σ_h = 2 σ_l — the hoop stress is always twice the longitudinal stress. That is why welded seams run longitudinally and why cylinders split along their length.

Strains in a Thin Cylinder

With both stresses present (biaxial), and Poisson's ratio ν:

Hoop strain      ε_h = (σ_h − ν σ_l)/E = (p d / 2 t E)(1 − ν/2)
Longitudinal strain ε_l = (σ_l − ν σ_h)/E = (p d / 4 t E)(1 − 2ν)
Volumetric strain  ε_v = ε_l + 2 ε_h

Worked Example 1 — Hoop and Longitudinal Stress

A cylindrical pressure vessel of 1.0 m internal diameter and 10 mm wall thickness carries an internal pressure of 2 MPa. Find the hoop and longitudinal stresses.

σ_h = pd/2t = (2 × 1000)/(2 × 10) = 100 MPa
σ_l = pd/4t = (2 × 1000)/(4 × 10) = 50 MPa
Check t/d = 10/1000 = 0.01 ≪ 1/20 → thin-cylinder theory valid ✔

Worked Example 2 — Designing the Wall Thickness

A pipe of 600 mm internal diameter must carry 3 MPa with a permissible hoop stress of 120 MPa. Find the minimum wall thickness.

σ_h = pd/2t → t = pd/(2σ_h) = (3 × 600)/(2 × 120) = 1800/240 = 7.5 mm (adopt 8 mm)

Worked Example 3 — Maximum Shear Stress

For Example 1, find the maximum in-plane shear stress.

τ_{max(in-plane)} = (σ_h − σ_l)/2 = (100 − 50)/2 = 25 MPa
Considering the radial stress ≈ 0 at the outer surface, the absolute max shear = σ_h/2 = 50 MPa.

Practical Notes

Longitudinal welds carry the higher hoop stress, so they are designed and inspected more stringently than circumferential welds.
A joint efficiency factor (< 1) is applied at welds and riveted seams.
Spheres carry pd/4t in all directions — half the hoop stress of a cylinder — which is why high-pressure storage uses spherical vessels.

Common Mistakes

Swapping the hoop and longitudinal formulas (remember: hoop is the bigger one, pd/2t).
Using outer diameter instead of internal diameter without noting the small difference.
Applying thin-cylinder theory to thick walls (t > d/20), where stress varies across the wall.