SECTION X 



SPHERES AND CYLINDERS UNDER UNIFORM PRESSURE 



71. 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 compared with 

 its diameter, it is called a shell. 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 section of the shell. This assumption will 

 be made in what follows. 



72. Hoop tension in hollow sphere. Consider 

 a spherical shell subjected to uniform internal 

 pressure, and suppose that the shell is cut into 

 hemispheres by a diametral plane (Fig. 94). 

 Then, if w denotes the pressure per unit of 

 area within the shell, the resultant force act- 



FlG - 94 ing on either hemisphere is P = , where 



d 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 irdhp, approximately, where h is the thickness of the shell. 

 Consequently, 



whence 

 (165) 



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



118 



