SPHERES AND CYLINDERS 



119 



73. Hoop tension in hollow circular cylinder. In the case of a 

 cylindrical shell, its ends hold the cylindrical part together in such 

 a way as to relieve the hoop tension at either extremity. Suppose, 

 then, that the portion of the cylinder considered is so far removed 



from either end that the influence of the 

 end constraint can be assumed to be zero. 

 Suppose the cylinder cut in two by a 

 plane through its axis, and consider a sec- 

 tion cut out of either half cylinder by two 

 planes perpendicular to the axis, at a dis- 

 tance apart equal to c (Fig. 95). Then the 

 resultant internal pressure P on the strip 

 under consideration is P = cdw, and the 



resultant hoop tension is 2 chp, where the letters have the same mean- 

 ing as in the preceding article. Consequently, cdw = 2 chp; whence 



FIG. 95 



(166) 



dw 



n = 



2h 



This result is applicable to shells under both inner and outer 

 pressure, if p is taken to be the excess of the internal over the 

 external pressure. 



74. Longitudinal stress in hollow 

 circular cylinder. If the ends of a 

 cylinder are fastened to the cylin- 

 drical part, the internal pressure 

 against the ends produces longitudi- 

 nal stresses in the side walls. In 

 this case the cylindrical part is 

 subjected both to hoop tension and to longitudinal tension. 



To find the amount of the longitudinal tension, consider a cross 

 section of the cylinder near its center, where the influence of the 

 end restraints can be assumed to be zero (Fig. 96). Then the re- 

 sultant pressure on either end is P = > and the resultant longi- 

 tudinal stress on the cross section is irdhp. Therefore = irdhp ; 

 whence 



