40 



STRENGTH OF PIPES AND BOILERS. 



pressure of p on each unit of surface. Let 2r be the diam- 

 eter MD of the interior, then will the surface pressed be 

 measured by wr 2 , which is the area of the 

 cross-section of the interior, and the 

 whole pressure upon the surface of the 

 end of the pipe and which produces rup- 

 ture will be measured by 



nr*p. (1) 



Let e = AE = the thickness of the 

 pipe ; then the cross-section of the mate- 

 rial of the pipe 



= TT (r + e) 2 rrr 2 = ne (e + 2r). 



Let T denote the strength of the material of which the 

 pipe is composed, for each unit of cross-section ; then the 

 strength of the entire pipe in the direction of the axis 



= ne (e + 2r) T, (2) 



and since the whole pressure in (1) when rupture is about 

 to take place must be held in equilibrium by the strength 

 in (2), we have 



ire (e + 2r) T = 7rr*p, 



rp _ rp 



e = 



(+)* 



since e is usually very small in comparison with 2r. 



(3) 



(2) When the rupture is longitudinal. 



Let EMH be any portion of the wall whose length is I, 

 and let 2 = the angle ECH. Then, since the projection 

 of EMH at right angles to the line MD passing through 

 the centre is a rectangle whose area = 2rl sin a, the mean 

 pressure of the fluid on the wall, EMH 



= 2rl sin p (Art 19). (4) 



