Barlow Equation for pipe internal pressure

Barlow equation named after Perter Barlow, an English mathematician and physicist, is used to relate the hoop stress in the pipe wall to the internal pressure, pipe diameter, and wall thickness.

Hoop stress and Longitudinal stress

Hoop stress is mainly the stress that occurs along the pipe’s circumference direction when pressure is applied and it acts perpendicular to the axial stress direction. This is the largest of the two stresses. It’s also called the circumferential stress.

Longitudinal stress also known as the axial stress, this is the other stress that a pipeline under pressure has, this axial stress acts in a direction parallel to the pipe axis.

Barlow equation

The Barlow equation is valid only for thin walled cylinder pipes, it can be expressed in terms of hoop stress or circumferential stress as shown below;

\[S_h=\ \frac{P\cdot D}{2\cdot t}\]

Or;

\[P\ =\ \frac{2\cdot t\cdot S_h}{D}\]

Where;

Sh = is the hoop or circumferential stress in pipe material, psi
D = is the pipe outside diameter in, inch
P = is the internal pressure, psi
t = is the pipe wall thickness, inch 

The Barlow equation can also be expressed in terms of the longitudinal stress or axial stress as shown below:

\[S_a=\ \frac{P\cdot D}{4\cdot t}\]

Or;

\[P\ =\ \frac{4\cdot t\cdot S_a}{D}\]

Where;

Sa = is the longitudinal or axial stress in pipe material, psi
D = is the pipe outside diameter in, inch
P = is the internal pressure, psi
t = is the pipe wall thickness, inch 

Most pipes transporting gases and liquids generally have thin walls so Barlow equation can mostly be used. However, some petroleum pipes that can have thick walls, they are subject to other equations.

Modified Barlow Equation

For the design of pipelines the equation has been modified to take in three factors, seam joint factor, design factor and temperature de-ration factor.

The Barlow equation below can be used for designing gas pipelines as well as calculating the allowable internal pressure in a pipeline based on a given diameter, wall thickness and pipe material. This is also commonly used as the maximum operating pressure.

\[P\ =\frac{\ 2\cdot t\cdot S\cdot E\cdot F\cdot T}{D}\]

Where;

P = is the internal pipe design pressure, psig or kPa 
D = is the pipe outside diameter in, inch or mm

S = is SMYS of pipe material, psig or kPa
SMYS is the yield stress used in calculation of pipe wall thickness

t = is the pipe wall thickness, inch or mm
E = is the Seam joint factor, 1.0 for seamless and submerged arc welded pipes, SAW pipes

F = is the design factor, usually 0.72 for liquid pipelines, 0.60 for risers offshore, 0.54 for pipes subjected to cold expansion to meet SMYS and heated, other than by welding or stress relieving as a  part of the welding, to a temperature higher than 900oF (482oC) for any period or over 600oF (316oC) for more than 1h. For gas pipelines the design factor ranges from 0.72 for cross-country pipelines to 0.4 in class 4 locations, the class location depends on the population density where the gas pipeline will be located, see the table below

T = is the temperature de-ration factor, 1 for temperatures below 250oF (121oC)

The Barlow equation can be expressed as shown below

\[P\ =\frac{\ 2\cdot t\cdot S\cdot E\cdot F\cdot T}{D}\]