76 APPLIED MECHANICS 
90. Thin Cylindrical Shells.—A thin cylindrical shell or pipe (Fig. 88) 
of internal diameter d, thickness ¢, 
and length / is exposed to internal 
fluid pressure of intensity p. Let 
the shell be divided into two equal 
parts by a plane of section contain- 
ing the axis. The resultant R of 
the pressure on either of these 
parts is evidently independent of 
the shape of the other part. Let 
the other part be replaced by a flat 
plate, as shown in Fig. 89. Then the resultant pressure on the flat 
plate is S=pdi. But S must balance R, therefore R=S = pd. 
If f, is the stress in the material of the shell at the plane of section, 
then R= pdl = 2tif, or pd = 2tf,. 
If the shell has longitudinal riveted joints whose efficiency is e, then 
pd = 2th. 
The assumptions made in determining the last two equations are, 
(1) that the stress f, is uniformly distributed over the section of the shell, 
and this is justified if the shell is thin compared with its diameter ; (2) 
that the shell derives no assistance from the ends, and this is justified 
if the cylinder is not very short compared with its diameter. 
The resultant pressure on the ends of the shell is if , and the 
Fig. 88. Fia. 89. 
resistance of the shell to tearing at a section perpendicular to the axis 
is rdtf,, therefore iP =Tdtf, or pd = 4tf,, which shows that the resist- 
ance to tearing at a circumferential section is twice the resistance to tearing 
at a longitudinal section, the effect of the riveted joints being neglected. 
91. Thin Spherical Shells.—By the method of the preceding Article, 
and using the same notation, the resultant pressure on one half of the 
shell is ie , and the resistance to tearing is 7dft/,, therefore {oP = wtf, 
or dp =4tf;. 
92. Centrifugal Tension in a Revolving Hoop.—Each part of a 
hoop revolving about its axis tends to fly outwards because of centri- 
fugal force, and the effect on the hoop is the same as that of an internal 
fluid pressure acting on it. 
Let a be the area of the cross section of the hoop in ainate inches ; 
w the weight of a cubic inch of the material in 
pounds ; v the linear velocity of the hoop in feet 
per second; and d the diameter of the hoop in 
inches. The hoop is supposed to be thin compared 
with its diameter. R R 
The weight of a portion of the hoop 1 inch 
long is aw lbs., and the centrifugal force g of this 
portion is 24awv? /gd. Each inch of hoop will 
have the same amount of centrifugal force acting 
on it, and the result is a uniformly distributed 
radial force acting on the hoop, as shown by the small arrows in Fig. 90. 
Fia. 90. 
