Hoop Stress Calculator (Thin-Wall Pressure Vessel)
Hoop and longitudinal stress in a thin-walled pipe, tank or sphere under internal pressure, the Lamé thick-wall check, factor of safety and the ASME minimum wall thickness.
How it works
Cylinder: hoop σ_h = P·r ÷ t, longitudinal σ_l = P·r ÷ 2t · sphere: σ = P·r ÷ 2t · Lamé (thick wall) σ_max = P·(r_o² + r_i²) ÷ (r_o² − r_i²) · ASME VIII-1 UG-27: t = P·R ÷ (S·E − 0.6·P) cylinder, t = P·R ÷ (2·S·E − 0.2·P) sphere (r, R = inside radius)
Internal pressure tries to split a cylinder along its length, so the hoop (circumferential) stress is twice the longitudinal stress — which is why overpressured pipes and sausages split lengthwise. The thin-wall formulas assume stress is uniform through the wall, accurate when the radius is at least about ten times the thickness; for thicker walls the Lamé solution gives the higher stress at the inner surface. The ASME Boiler and Pressure Vessel Code (Section VIII, Division 1, UG-27) adds small corrections and a joint efficiency E; add corrosion allowance and use the code allowable stress for your material and temperature.
Worked example
A cylindrical tank 500 mm inside diameter with a 5 mm wall at 10 bar (1 MPa) has a hoop stress of 1 × 250 ÷ 5 = 50.00 MPa and a longitudinal stress of 25.00 MPa. With r/t = 50 the thin-wall assumption holds (Lamé gives 50.50 MPa). Against an allowable stress of 138 MPa the factor of safety is 2.73, and the ASME minimum thickness is 1 × 250 ÷ (138 − 0.6) = 1.82 mm before corrosion allowance.
Sources: Wikipedia — Cylinder stress (hoop, axial, Lamé) · Engineering ToolBox — Stress in thin-walled tubes or cylinders
Frequently asked questions
Why is hoop stress twice the longitudinal stress?
Cut a cylinder lengthwise and the pressure acts on the area D·L, resisted by two walls of area t·L; cut it across and the pressure acts on π·D²/4, resisted by the ring π·D·t. Working through gives σ_h = P·D/2t and σ_l = P·D/4t.
When is a pressure vessel thin-walled?
When the inner radius is at least about 10 times the wall thickness (r/t ≥ 10, or D/t ≥ 20). Below that the stress varies through the wall and the Lamé equations should be used.
Is this the same as Barlow’s formula?
Yes, in essence. Barlow’s formula P = 2·S·t ÷ D is the hoop-stress equation rearranged, usually with the outside diameter for a conservative pipe pressure rating.
Should I use inside or outside diameter?
The textbook thin-wall formulas and ASME UG-27 use the inside radius; Barlow’s pipe formula uses the outside diameter, which gives a slightly higher (conservative) stress.
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Engineering estimate for preliminary sizing and education. Verify final designs against the applicable codes, manufacturer data and a qualified engineer.