Thin-Wall Pressure Vessel Stress
Finds the membrane stresses in the wall of a thin cylindrical or spherical vessel under internal pressure, or the wall thickness needed to keep them under an allowable stress. Pick an ASME plate or pipe material to fill in its allowable stress and compare it with the result. Use it for a first sizing of tanks, receivers and pipe, before the code calculation.
Equations
Cylinder: σhoop = pr/t σlong = pr/(2t) Sphere: σ = pr/(2t)
p is the internal (gauge) pressure, r the inner radius and t the wall thickness [1]. The hoop stress acts around the circumference and the longitudinal stress along the axis. In a cylinder the hoop stress is twice the longitudinal stress, so it governs, and the required thickness is
Cylinder: t = pr/σallow Sphere: t = pr/(2σallow)
A sphere needs half the wall of a cylinder of the same radius. The material list gives the maximum allowable stress S from ASME Section II, Part D, Table 1A, for −20 to 100 °F [2], converted from ksi to MPa.
Limits and common mistakes
- The formulas assume a thin wall, with r/t of 10 or more; the page warns below that. A thicker wall needs the thick-wall (Lamé) equations.
- This is not the ASME code thickness. The code formula t = PR/(SE − 0.6P) adds the weld joint efficiency E, and a corrosion allowance goes on top [3]. Expect the code to give a thicker wall.
- The stresses are those far from heads, nozzles and supports, where local bending raises them. External pressure (buckling) is not covered.
- Use the allowable stress at the design temperature. The listed values hold only up to 100 °F.
Worked example
A compressed-air receiver, 900 mm inside diameter, is to be rolled from SA-516 Gr. 70 plate for 1.4 MPa. Find the minimum wall, then check the 6 mm plate in stock.
- Choose Vessel Shape Cylindrical and Solve for Required Thickness (given Allowable Stress).
- Choose Material SA-516 Gr. 70 plate (20.0 ksi). The allowable stress fills in as 137.895 MPa.
- Enter Internal Pressure 1.4 MPa and Inner Radius 450 mm, and press Solve. The Solution shows:
Required Wall Thickness (t): 4.56869 mm
r/t ratio at this thickness: 98.4965 - Choose Solve for Stress (given Thickness), enter Wall Thickness 6 mm and press Solve. The Solution shows:
Hoop Stress (σhoop): 105 MPa
Longitudinal Stress (σlong): 52.5 MPa
r/t ratio: 75
Allowable Stress — SA-516 Gr. 70 — carbon steel plate (20.0 ksi): 137.895 MPa and notes that the governing stress is within the allowable stress.
Check: 1.4 × 450/6 = 105 MPa, and 20 ksi × 6.894757 = 137.895 MPa. The 6 mm plate leaves 1.4 mm over the membrane minimum for the joint efficiency and corrosion allowance of the code calculation.
