123 



SPHERES AND CYLINDERS 



If, therefore, the cylinder is subjected to a uniform internal pres- 

 sure of amount w i per unit of area, and also to a uniform external 

 pressure of amount w e per unit of area, then p r = w e when r = a, 

 and p r = w { when r = b. Substituting these simultaneous values 

 in equation (173), 



k C k C 



whence 



V - 



2 



a 2 b 2 



k _ w e a 2 w.b 2 

 2~ a 2 -b 2 ' 



Hence, substituting these values of C and in equations (172) 

 and (173), they become 



(174) 



Pr = 



a 2 - b 2 

 a 2 - b 2 



(a 2 -b 2 )r 2 



a 2 b 2 (iv e w i 



(a 2 - b 2 )r 2 



which give the radial and hoop stresses in a thick cylinder subjected 

 to internal and external pressure. Equations (174) are known as 

 Lame's formulas. 



76. Maximum stress in thick cylinder under uniform internal 

 pressure. Consider a thick circular cylinder which is subjected only 

 to internal pressure. Then w e = 0, and equations (174) become 



wb 2 /a 2 A wb 2 



( 175 ) 



Since p h is negative, the hoop stress in this case is tension. 



Since p r and p h both increase as r decreases, the maximum stress 

 occurs on the inner surface of the cylinder, where r = b and p r = w { . 

 Hence 



cm) j=- tp * ( r' + f ) - 



a 2 b 2 



Clearing the latter of fractions, we have ^ = 



b Ph 

 thickness of the tube, h = a 6, is given by 



whence the 



(177) 



