Page 139 - Shigley's Mechanical Engineering Design

bud29281_ch03_071-146.qxd 11/24/09 3:02PM Page 114 ntt 203:MHDQ196:bud29281:0073529281:bud29281_pagefiles:

114 Mechanical Engineering Design
Figure 3–32 p = 0 p = 0
o
o

Distribution of stresses in a t
thick-walled cylinder subjected
to internal pressure.
r
r o
p i i
r i
p i

r
r o

(a) Tangential stress (b) Radial stress
distribution distribution

The equations of set (3–50) are plotted in Fig. 3–32 to show the distribution of stresses
over the wall thickness. It should be realized that longitudinal stresses exist when the
end reactions to the internal pressure are taken by the pressure vessel itself. This stress
is found to be
p i r i 2
σ l = 2 (3–51)
2
r − r
o i
We further note that Eqs. (3–49), (3–50), and (3–51) apply only to sections taken a sig-
nificant distance from the ends and away from any areas of stress concentration.

Thin-Walled Vessels
When the wall thickness of a cylindrical pressure vessel is about one-tenth, or less, of
its radius, the radial stress that results from pressurizing the vessel is quite small com-
pared with the tangential stress. Under these conditions the tangential stress can be
obtained as follows: Let an internal pressure p be exerted on the wall of a cylinder of
thickness t and inside diameter d i . The force tending to separate two halves of a unit
length of the cylinder is pd i . This force is resisted by the tangential stress, also called
the hoop stress, acting uniformly over the stressed area. We then have pd i = 2tσ t , or

pd i
(σ t ) av = (3–52)
2t
This equation gives the average tangential stress and is valid regardless of the wall thick-
ness. For a thin-walled vessel an approximation to the maximum tangential stress is

p(d i + t)
(σ t ) max = (3–53)
2t
where d i + t is the average diameter.
In a closed cylinder, the longitudinal stress σ l exists because of the pressure upon
the ends of the vessel. If we assume this stress is also distributed uniformly over the
wall thickness, we can easily find it to be

pd i
σ l = (3–54)
4t