126 RESISTANCE OF MATERIALS 



IV. Clavarino's formula. In this formula each particle of the 

 tube is assumed to be subjected to radial stress, hoop stress, and 

 longitudinal stress, due to a uniform internal pressure acting jointly 

 on the tube wall and its closed ends. The formula also involves 

 Poisson's ratio of lateral contraction, and is theoretically correct, 

 provided the maximum stress does not exceed the elastic limit of 

 the material. Assuming a value of Poisson's ratio = .3 and using 

 the same notation as above, Clavarino's formula is 



* 



whence 



(186) ^ - ' l P 



V. BirniJs formula. This formula is based upon the same 

 assumptions as Clavarino's, except that the longitudinal stress is 

 assumed to be zero. Using the same notation as before and assum- 

 ing Poisson's ratio for steel to be .3, Birnie's formula is 



10 (Z> 2 - 



whence 



(188) 



> lOp + 7w 



80. Thick cylinders built up of concentric tubes. From equations 

 (174) it is evident that in a thick cylinder subjected to internal 

 pressure the stress is greatest on the inside of the cylinder and 

 decreases toward the outside. In order to equalize the stress 

 throughout the cylinder and thus obtain a more economical use 

 of material, the device used consists in forming the cylinder of 

 several concentric tubes and producing an initial compressive stress 

 on the inner ones. For instance, in constructing the barrel of a 

 cannon or the cylinder of a hydraulic press the cylinder is built 

 up of two or more tubes. The outer tubes in this case are made of 

 somewhat smaller diameter than the inner tubes, and each is 

 heated until it has expanded sufficiently to be slipped over the one 

 next smaller. In cooling, the metal of the outer tube contracts, 

 thus producing a compressive stress in the inner tube and a tensile 



