STRENGTH OF PIPES AND BOILERS. 41 



Now this pressure must be held in equilibrium by the 

 forces of cohesion, R, R, acting tangentially on the cross- 

 sections, AE and BH, of the wall of the pipe. Denoting 

 the components of R, R, parallel to MD, by Q, Q, we have 



2Q = 2R sin = ZelT sm a, (5) 



e being the thickness of the pipe and T the strength of each 

 unit of section. 



Therefore, from (4) and (5) we have, 



2elT sin = 2rlp sin ; 



.'. . = ?, (6) 



which shows that the thickness of the pipe is independent 

 of its length. 



Otherwise thus, by the principle of work. 



The whole surface of the interior of the pipe = 2frrl ; and 

 the whole pressure upon the surface = %-rrrlp. Suppose the 

 pipe to rupture longitudinally,* under this pressure, its 

 radius becoming r + dr; then the path described by the 

 pressure will be dr, and the work done by the pressure 



= Znrlp dr. (7) 



The force R, which resists rupture and acts tangentially, 

 = eTl. While the radius of the interior changes from r to 

 r-\-dr, the circumference changes from 2nr to 2-rr (r + dr) ; 

 then the path described by the resistance = 2n dr, and the 

 work done by the resistance 



= ZneTldr. (8) 



* Longitudinal tension produces transverse rupture, and transverse tension pro- 

 duces longitudinal rupture. The stretching tendency to rupture longitudinally is a 

 transverse stretching, i. e., the pipe tends to bulge out all along its length ; hence, 

 transversely, r becomes r+dr. 



