COMBINED LOADINGS

THIN-WALLED PRESSURE
VESSELS
Cylindrical or spherical pressure vessels are commonly used in industry
to serve as boilers or storage tanks. The stresses acting in the wall of these
vessels can be analyzed in a simple manner provided it has a thin wall,
that is, the inner-radius-to-wall-thickness ratio is 10 or more (r>t ? 10).
Specifically, when r>t = 10 the results of a thin-wall analysis will predict
a stress that is approximately 4% less than the actual maximum stress in
the vessel. For larger r>t ratios this error will be even smaller.
In the following analysis, we will assume the gas pressure in the vessel
is the gage pressure, that is, it is the pressure above atmospheric pressure,
since atmospheric pressure is assumed to exist both inside and outside
the vessel’s wall before the vessel is pressurized
Cylindrical Vessels. The cylindrical vessel in Fig. 8–1a has a wall
thickness t, inner radius r, and is subjected to an internal gas pressure p.
To find the circumferential or hoop stress, we can section the vessel by
planes a, b, and c. A free-body diagram of the back segment along with
its contained gas is then shown in Fig. 8–1b. Here only the loadings in the
x direction are shown. They are caused by the uniform hoop stress s1,
acting on the vessel’s wall, and the pressure acting on the vertical face of
the gas. For equilibrium in the x direction, we require
Fx = 0; 2[s1(t dy)] - p(2r dy) = 0
s1 =
pr
t
(8–1)
The longitudinal stress can be determined by considering the left portion
of section b, Fig. 8–1a. As shown on its free-body diagram, Fig. 8–1c, s2 acts
uniformly throughout the wall, and p acts on the section of the contained
gas. Since the mean radius is approximately equal to the vessel’s inner
radius, equilibrium in the y direction requires
F
y = 0; s2(2prt) - p(pr2) = 0
s2 =
pr
2t
(8–2)
For these two equations,
s1, s2 =???the normal stress in the hoop and longitudinal directions,
respectively. Each is assumed to be constant throughout the
wall of the cylinder, and each subjects the material to tension.
p =???the internal gage pressure developed by the contained gas
r =???the inner radius of the cylinder
t =???the thickness of the wall (r>t ? 1 0 )