UNDER CONSTRUCTION
Internal / External Pressure in a Pipe:
Derivation from the Theory of Elasticity applied in polar coordinates.
Using the equations of compatibility ( see Elastic
Solids ) the stress function generated is
The polar stresses are defined as:
..............................(1)
.............................................(2)
..........................(3)
In our case the stresses are symmetric and does not depend on so (1) becomes
and (2) remains unchanged
finally (3) is
Evaluate differentials:
Taking B=0 (because at r = 0, ln0 => )
Then
..........................(4)
.......................(5)
Let a and b be the inner and outer radius and Pi and Po be the internal and external pressures, the the boundary conditions are:
into (4) yields
From which
Substituting back into (4) and (5) yields:
These are the stresses for a combination internal and external pressure in a pipe. By considering internal pressure only, these equations reduce even further. This is considered a thick wall fomulation.
JCS
