Stress in Thick-Walled Cylinders or Tubes
Radial and tangential stress in thick-walled cylinders or tubes with closed ends - with internal and external pressure.
When a thick-walled tube or cylinder is subjected to internal and external pressure a hoop and longitudinal stress are produced in the wall.
Stress in Axial Direction
The stress in axial direction at a point in the tube or cylinder wall can be expressed as:
σa = (pi ri2 - po ro2 )/(ro2 - ri2) (1)
where
σa = stress in axial direction (MPa, psi)
pi = internal pressure in the tube or cylinder (MPa, psi)
po = external pressure in the tube or cylinder (MPa, psi)
ri = internal radius of tube or cylinder (mm, in)
ro = external radius of tube or cylinder (mm, in)
Stress in Circumferential Direction - Hoop Stress
The stress in circumferential direction - hoop stress - at a point in the tube or cylinder wall can be expressed as:
σc = [(pi ri2 - po ro2) / (ro2 - ri2)] - [ri2 ro2 (po - pi) / (r2 (ro2 - ri2))] (2)
where
σc = stress in circumferential direction (MPa, psi)
r = radius to point in tube or cylinder wall (mm, in) (ri < r < ro)
maximum stress when r = ri (inside pipe or cylinder)
Resultant Stress
Combined stress in a single point in the cylinder wall cannot be described by a single vector using vector addition. Instead stress tensors (matrixes) describing the linear connection between two physical vectors quantities can be used.
Stress in Radial Direction
The stress in radial direction at a point in the tube or cylinder wall can be expressed as:
σr = [(pi ri2 - po ro2) / (ro2 - ri2)] + [ri2 ro2 (po - pi) / (r2 (ro2 - ri2))] (3)
maximum stress when r = ro (outside pipe or cylinder)
Example - Stress in Thick walled Cylinder
In a cylinder with inside diameter 200 mm (radius 100 mm) and outside diameter 400 mm (radius 200 mm) there is a pressure 100 MPa relative to the outside pressure.
Stress in axial direction can be calculated as
σa = (((100 MPa) (100 mm)2 - (0 MPa) (200 mm)2) / ((200 mm)2 - (100 mm)2)
= 33.3 MPa
Stress in circumferential direction - hoop stress - at the inside wall (100 mm) can be calculated as
σc = [((100 MPa) (100 mm)2 - (0 MPa) (200 mm)2) / ((200 mm)2 - (100 mm)2)] - [(200 mm)2 (100 mm)2 ((0 MPa)- (100 MPa)) / ((100 mm)2 ((200 mm)2 - (100 mm)2))]
= 167 MPa
Stress in radial direction at the inside wall (100 mm) can be calculated as
σr = [((100 MPa) (100 mm)2 - (0 MPa) (200 mm)2) / ((200 mm)2 - (100 mm)2)] + [(200 mm)2 (100 mm)2 ((0 MPa)- (100 MPa)) / ((100 mm)2 ((200 mm)2 - (100 mm)2))]
= -100 MPa
Note! - that in addition stress caused by pressure - stress can be induced in the pipe or cylinder wall by restricted temperature expansion.
Online Thick Walled Pipe & Cylinder Calculator
The calculator below can be used to calculate the stress in thick walled pipes or cylinders with closed ends.
inside pressure (MPa, psi)
outside pressure (MPa, psi)
inside radius (mm, in)
outside radius (mm, in)
radius to point in the wall (mm, in)