CHAPTER VII 



SPHERES AND CYLINDERS UNDER UNIFORM PRESSURE 



104. Hoop stress. When a hollow sphere or cylinder is subjected 

 to uniform pressure, as in the case of steam boilers, standpipes, gas, 

 water, and steam pipes, fire tubes, etc., the effect of the radial pres- 

 sure is to produce stress in a circumferential direction, called hoop 

 stress. In the case of a cylinder closed at the ends, the pressure on 

 the ends produces longitudinal stress in the side walls in addition to 

 the hoop stress. 



If the thickness of a cylinder or sphere is small as compared with 

 its diameter, it is called a shea In analyzing the stress in a thin 

 shell subjected to uniform pressure, such as that due to water, steam, 

 or gas, it may be assumed that the hoop stress 

 is distributed uniformly over any cross sect 

 of the shell. This assumption will be made in 

 what follows. 



105. Hoop tension in hollow sphere. Con- 

 sider a spherical shell subjected to uniform 

 internal pressure, and suppose that the shell 

 FlGt 8i is cut into hemispheres by a diametral ] 



(Fig. 89). Then, if w denotes the pressure per unit of area within the 

 shell, the resultant force acting on either hemisphere is P = TT 

 where r is the radius of the sphere. If p denotes the unit tensile stress 

 on the circular cross section of the shell, the total stress on this cross 

 section is 2 jrrhp, approximately, where h is the thickness of the sin -11. 

 Consequently, irr^w = 2 irrhp ; whence 



wr 



which gives the hoop tension in terms of the radial pressure. 



From symmetry, the stress is the same on any diametral cross 

 section. Therefore the equivalent stress at any point of the shell is 



122 



