Thin-wall pressure vessel

Introduction

Air receivers, hydrophore tanks, boilers, heat-exchanger shells, CO₂ bottles and pipelines are all pressure vessels, and their walls work mostly in membrane tension. The thin-wall formulas give that tension directly from pressure, radius and thickness; the ASME Boiler and Pressure Vessel Code turns the same physics into a required thickness with the material's allowable stress, the weld quality and a corrosion allowance. This calculator does both: the textbook hoop and longitudinal stresses with the Lamé thick-wall value for comparison, and the ASME Section VIII Division 1 required thickness and maximum allowable working pressure for cylinders and spheres, plus the axial load the heads must carry.

It is for engineers checking an existing receiver after a thickness measurement, for designers pre-sizing a shell before the full code calculation, and for students meeting pressure-vessel stresses.

Formulas

Thin-wall membrane stresses (inside radius R, wall t, gauge pressure p):

Cylinder:  σ_hoop = p R / t,     σ_long = p R / (2 t)
Sphere:    σ = p R / (2 t)
Valid for t ≤ R/10 (error ≈ t/2R)

Lamé thick-wall maximum hoop stress at the inner surface (r_i = R, r_o = R + t):

Cylinder:  σ_max = p (r_o² + r_i²) / (r_o² − r_i²)
Sphere:    σ_max = p (r_o³ + 2 r_i³) / (2 (r_o³ − r_i³))

ASME Section VIII Division 1, UG-27 (S allowable stress, E joint efficiency, c corrosion allowance):

Cylinder:  t_req = p R / (S E − 0.6 p) + c,      MAWP = S E (t − c) / (R + 0.6 (t − c))
Sphere:    t_req = p R / (2 S E − 0.2 p) + c,    MAWP = 2 S E (t − c) / (R + 0.2 (t − c))
Limits:    t ≤ R/2;  p ≤ 0.385 S E (cylinder), p ≤ 0.665 S E (sphere)

Axial load on each head:

F = p π R²

Joint efficiencies (UW-12): 1.00 full radiography or seamless, 0.85 spot radiography, 0.70 no radiography.

Assumptions and limitations

• Membrane stresses in the shell away from discontinuities: heads, nozzles, supports and welds concentrate stress and need the code's local rules (UG-37 reinforcement, Appendix 1 heads, Division 2 analysis).
• Internal pressure only; external pressure and vacuum are buckling problems (UG-28).
• The allowable stress S is the code value for the material at the design temperature; the calculator does not contain the ASME II-D tables.
• Static pressure; fatigue, thermal gradients and creep above about 370 °C for carbon steel are not covered.
• Mill undertolerance and forming allowances are not added; the corrosion allowance is the only thickness addition.
• The ASME formulas are reproduced for pre-sizing and checking; the code stamp requires the full design by a qualified engineer.

Worked example

Air receiver, cylindrical, 2.0 m inside diameter, 10 mm wall, design pressure 10 bar (1.0 MPa), SA-516 Grade 70 with S = 138 MPa, full radiography (E = 1.0), no corrosion allowance.

1. Thin-wall: t/R = 0.01, well within the thin-wall range. Hoop stress 1.0 × 1 000 / 10 = 100 MPa; longitudinal 50 MPa.
2. Lamé: 1.0 × (1.01² + 1²) / (1.01² − 1²) = 100.5 MPa — thin-wall theory is 0.5 % low, as expected.
3. ASME required thickness: 1.0 × 1 000 / (138 × 1.0 − 0.6 × 1.0) = 7.28 mm; the 10 mm wall has 2.7 mm in hand. With spot radiography (E = 0.85) the requirement rises to 8.57 mm.
4. MAWP of the 10 mm wall: 138 × 10 / (1 000 + 6) = 1.37 MPa = 13.7 bar.
5. End-cap force: 1.0 × 10⁶ × π × 1² = 3.14 MN — the load the dished heads and any flange bolts must carry.

With a 2 mm corrosion allowance the required thickness becomes 9.28 mm and the MAWP of the corroded 8 mm wall 11.0 bar: still above the 10 bar design pressure.

FAQ

Related calculators

• Bolt torque and preload — flange bolts for the end-cap load.
• Pipe flow and pressure drop — the pressures in the connected piping.
• Thermal expansion and restraint stress — thermal stresses added to the pressure stresses.
• Beam deflection and bending stress — saddle supports and lifting of horizontal vessels.