
47Structural Distress
Summing forces along the x-axis provides the following:
σ
σπ π
σ
2
2
2
2
0
dA pdA
rt pr
pr
=
()()
(2.91)
Relating the results from Equations 2.90 and 2.91,
(2.92)
In other words, given an internal pressure acting on the walls of a cylindrical vessel, the
hoop stress will be twice as large as the longitudinal stress.
Modes of Failure
As shown in Equations 2.91 and 2.92, increased pressure inside a vessel will increase
the stresses tangent to the surface. These stresses can increase to the point of fail-
ure. Acommon type of failure to thin-walled pipes is caused by freezing water. Water
expands volumetrically as it freeze ...