Pressure Vessel and Hoop Stress

Thin-walled cylinder and sphere stress formulae, hoop stress, longitudinal stress and pressure unit conversions.

Article 10.01Pressure & Fluid3 min read

pressurehoop stressthin wallcylinderspherelongitudinal stressvessel

Thin-Walled Cylinder (r/t ≥ 10)

Hoop (circumferential) stress:
  σ_hoop  =  p · r / t

Longitudinal stress:
  σ_long  =  p · r / (2 · t)

Maximum shear stress:
  τ_max  =  σ_hoop / 2  =  p · r / (2 · t)

Radial stress (inner surface):
  σ_rad  =  −p   (compressive)
  (often neglected for thin walls)

Thin-Walled Sphere

Membrane stress (equal biaxial):
  σ  =  p · r / (2 · t)

(same in all in-plane directions)

Pressure Unit Conversions

From	To	Multiply by
Pa	MPa	1 × 10⁻⁶
MPa	psi	145.038
psi	MPa	0.006895
bar	MPa	0.1
bar	psi	14.5038
atm	MPa	0.101325
kPa	psi	0.145038

Variables

Symbol	Definition	Units
p	Internal pressure	MPa (or Pa)
r	Mean radius of vessel	mm (or m)
t	Wall thickness	mm
σ_hoop	Hoop (circumferential) stress	MPa
σ_long	Longitudinal stress	MPa
r/t	Radius-to-thickness ratio	—

Notes and Limitations

Thin-wall formulae are valid for r/t ≥ 10 — for thick walls, use Lamé equations
The formulae assume uniform internal pressure and no end-cap effects away from discontinuities
At nozzle connections, openings and junctions, stress concentrations occur — use detailed FEA
External pressure buckling is a separate failure mode — not covered by these stress formulae

Related Knowledge

See the Structural Analysis Knowledge category for shell theory and pressure vessel analysis.