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Home/Engineering/Thin-Walled Pressure Vessel Stress

Thin-Walled Pressure Vessel Stress

Cylinder vs sphere membrane stresses: hoop σθ, longitudinal σz, von Mises stress, r/t thin-wall check, and yield safety factor.

Vessel geometry

600 mm
8 mm

Pressure and material

2.5 MPa
250 MPa

Thin-wall formulas assume r/t is large, membrane stress dominates, and local nozzles, welds, heads, and buckling checks are outside the model.

Measured values

Hoop stress σθ187.5MPa
Longitudinal stress σz93.8MPa
von Mises stress162.4MPa
Safety factor1.54
Thin-wall ratio r/t75.0
Yield pressure estimate3.33MPa

This is a membrane-stress teaching model. Real vessel design also checks code allowables, joint efficiency, corrosion allowance, heads, openings, thermal stress, external pressure buckling, fatigue, and proof testing.

Live graphs

About this model

Thin-walled pressure-vessel theory treats wall stresses as membrane stresses when the radius-to-thickness ratio is large. For a closed cylindrical vessel the hoop stress is σθ = pr/t and the longitudinal stress is σz = pr/(2t); for a sphere both principal membrane stresses are pr/(2t). This simulator compares cylinder and sphere behavior, computes the von Mises equivalent stress, estimates a yield safety factor, and warns when r/t is too small for the thin-wall assumption. It is a teaching model only: real vessel design also considers design codes, joint efficiency, corrosion allowance, heads and nozzles, local discontinuity stresses, thermal gradients, external pressure buckling, fatigue, proof testing, and inspection.

Who it's for: Mechanics of materials, pressure-vessel design introductions, process equipment, HVAC, piping, and mechanical engineering courses.

Key terms

  • Thin-walled pressure vessel
  • Hoop stress
  • Longitudinal stress
  • Membrane stress
  • Safety factor

How it works

Thin-wall pressure vessel stress calculator: compare cylindrical and spherical membrane stresses, hoop stress, longitudinal stress, von Mises stress, and yield safety factor.

Key equations

Cylinder: σθ = pr/t, σz = pr/(2t)
Sphere: σθ = σφ = pr/(2t); σvm = sqrt(σθ² − σθσz + σz²)

Frequently asked questions

Why is hoop stress twice the longitudinal stress in a cylinder?
A longitudinal split is resisted by wall area 2tL, giving σθ = pr/t. An end-cap force is resisted by the circumferential wall area 2πrt, giving σz = pr/(2t).
When do thin-wall formulas stop being accurate?
A common rule of thumb is r/t ≥ 10. Below that, stress varies noticeably through the wall and thick-cylinder Lamé equations are more appropriate.