SPHERES AND CYLINDERS 



127 



stress in the outer tube. If, then, this composite tube is subjected 

 to internal pressure, the first effect of the hoop tension thus pro- 

 duced is to relieve the initial compressive stress in the inner tube 

 and increase that in the outer tube. Thus the resultant stress in 

 the inner tube is equal to the difference between the initial stress 

 and that due to the internal pressure, whereas the resultant stress 

 in the outer tube is equal to the sum of these two. In this way the 

 strain is distributed more equally throughout the cylinder. It is 

 evident that the greater the number of tubes used in building 

 up the cylinder, the more nearly can the strain be equalized. 



The preceding discussion of the stress in thick tubes can also be 

 applied to the calculation of the stress in a rotating disk. For ex- 

 ample, a grindstone is strained in precisely the same way as a thick 

 tube under internal pressure, the load in this case being due to 

 centrifugal force instead of to the pressure of a fluid or gas. 



81. Practical formulas for the collapse of tubes under external 

 pressure. A rigorous analysis of the stress in thin tubes, due to 



external pressure, using Poisson's ratio of transverse to longi- 

 tudinal deformation, gives the formula * 



E 



41- 



-f-Y 



1 YW 



or, in terms of the diameter D = 



w = 



This formula, however, is based on the assumptions that the tube 

 is perfectly symmetrical, of uniform thickness, and of homogeneous 

 material conditions which are never fully realized in commercial 

 tubes. From recent experiments on the collapse of tubes, t how- 

 ever, it is now possible to determine the practical limitations of 

 this formula and to so modify it, by a method similar to that by 



* Love, Mathematical Theory of Elasticity, Vol. II, pp. 308-316. 

 t Carman, " Resist, of Tubes to Collapse," Univ., HI. Bull, Vol. Ill, No. 17 Stewart, 

 "Collap. Press. Lap- Welded Steel Tubes," Trans. A.S.M.E., 1906, pp. 730-820. 



