# 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:

σ_{a}= (p_{i}r_{i}^{2}- p_{o }r_{o}^{2})/(r_{o}^{2}- r_{i}^{2}) (1)

where

σ_{a}= stress in axial direction (MPa, psi)

p_{i}= internal pressure in the tube or cylinder (MPa, psi)

p_{o}= external pressure in the tube or cylinder (MPa, psi)

r_{i}= internal radius of tube or cylinder (mm, in)

r_{o}= 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:

σ_{c}= [(p_{i }r_{i}^{2}- p_{o}r_{o}^{2}) / (r_{o}^{2}- r_{i}^{2})] - [r_{i}^{2}r_{o}^{2}(p_{o}- p_{i}) / (r^{2}(r_{o}^{2}- r_{i}^{2}))] (2)

where

σ_{c}= stress in circumferential direction (MPa, psi)

r = radius to point in tube or cylinder wall (mm, in) (r_{i}< r < r_{o})

maximum stress when r = r_{i }(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:

σ_{r}= [(p_{i }r_{i}^{2}- p_{o}r_{o}^{2}) / (r_{o}^{2}- r_{i}^{2})] + [r_{i}^{2}r_{o}^{2}(p_{o}- p_{i}) / (r^{2}(r_{o}^{2}- r_{i}^{2}))] (3)

maximum stress when r = r_{o }(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

*σ _{a} = (((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

*σ _{c} = [((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

*σ _{r} = [((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.

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