56 THE EQUILIBRIUM OF ELASTIC SOLIDS. 



Let a rectangular elastic beam, whose length is Zirc, be bent into a circular 

 form, so as to be a section of a hollow cylinder, those parts of the beam which 

 lie towards the centre of the circle will be longitudinally compressed, while the 

 opposite parts will be extended. 



The expression for the tangential compression is therefore 



oV r c 



Comparing this value of - with that of Equation (20), 

 r c 



and by (21), q 



wr 



(1 2 \ 

 + ], the equation 



becomes 



dp Zm + Spp 9w 



-J- ^Z 



UT m + 6/x T 

 a linear differential equation, which gives 



(7, may be found by assuming that when r=a iy p = h lt and q may be found 

 from p by equation (21). 



As the expressions thus found are long and cumbrous, it is better to use 

 the following approximations : 



(48) - 



In these expressions a is half the depth of the beam, and y is the distance 

 of any part of the beam from the neutral surface, which in this case is a cylin- 

 dric surface, whose radius is c. 



These expressions suppose c to be large compared with a, since most sub- 

 stances break when -- exceeds a certain small quantity. 



