# 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

The resultant stress in axial and circumferential stress plane can be calculated with vector addition as

*σ _{resultant} = (σ_{a}^{2} + σ_{c}^{2})^{1/2} (3)*

### 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}))] (4)

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. The resulting stress can be calculated with vector addition.

The resulting axial and hoop stress can be calculated as

*σ _{resultant} = ((33.3 MPa)^{2} + (167 MPa)^{2})^{1/2} *

* = 170 MPa*

### 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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