A pressure vessel is a type of container which is used to store liquids or gases under a pressure different from the ambient pressure. Examples of pressure vessels can be diving cylinder, autoclave, nitrogen tanks, submarine and storage vessels for liquefied gases such as LPG. Different shapes of pressure vessels exist but most generally cylindrical and spherical shapes are used. Spherical vessels are theoretically 2 times stronger than cylindrical ones but due to the manufacturing difficulties, cylindrical ones are generally preferred in the industry.

A pressure vessel is assumed to be a thin walled pressure vessel when the thickness of the vessel is less than 1/20 of its radius. [Ref-2] The walls of thin-walled pressure vessels have little resistance to bending so it may be assumed that the internal forces exerted on a given portion of the wall are tangent to the surface of the vessel. The resulting stress state on vessel is plane stress situation since all stresses are tangent to surface of vessel.

The calculation tool was developed to analyze two types of vessels, cylindrical and spherical type. According to geometric properties and pressure, principal stresses and maximum shear stress on the surface of the vessel can be calculated. The formulas used for the calculations are given in the List of Equations section.

Stresses In Thin-Walled Pressure Vessels
Vessel type
Parameter Symbol Value Unit
Gage pressure of fluid pg
Vessel wall thickness t
Vessel inside radius


Note: Use dot "." as decimal separator.

Parameter Symbol Value Unit
Hoop stress (Principal stress-1) σ1 --- MPa
Longitudinal stress (Principal stress-2) σ2 ---
Maximum shear stress (in plane) τmax(in plane) ---
Maximum shear stress (out plane) τmax(out plane) ---
Thickness to inner radius ratio t/r --- * ---

Note: * Red color :t/r > 1/20 , Green color : t/r < 1/20


Gauge(Gage) Pressure: The pressure relative to atmospheric pressure. Eq: pg=pa-patm : pa is the absolute pressure of the system and patm is atmospheric pressure.

Hoop Stress: Stress acts in tangential direction.  It's the 1st principal stress.

Longitudinal stress: Stress acts in longitudinal direction. It's the  2nd principal stress.

Principal Stress: Maximum and minimum normal stress possible for a specific point on a structural element. Shear stress is 0 at the orientation where principal stresses occur.

Shear stress: A form of a stress acts parallel to the surface (cross section) which has a cutting nature.


Link Usage
Yield Criteria for Ductile Material After calculation of principal stresses on pressure vessel, yield criteria can be checked for ductile material.
Plane Stress Transformations After calculation of principal stresses  on pressure vessel, plane stresses in different orientation can be checked .

List of Equations:

Parameter Symbol Formula
Cylindrical pressure vessel
Hoop stress
σ1 pgr/t
Longitudinal stress σ2 (pgr)/(2t)
Maximum in-plane shear stress τmax(in plane) (pgr)/(4t)
Maximum out-plane shear stress τmax(out plane) (pgr)/(2t)
Spherical pressure vessel
Hoop stress σ1 (pgr)/(2t)
Longitudinal stress σ2 (pgr)/(2t)
Maximum in-plane shear stress τmax(in plane) 0
Maximum out-plane shear stress τmax(out plane) (pgr)/(4t)


Link Usage
Pressure Vessel An example about the calculation of stresses on a pressure vessel, evaluation of yield criteria of material and stress transformation to find shear and perpendicular stresses on welding of the cylindrical body of the pressure vessel.