HOOKS, LINKS, AND SPRINGS 149 



section below the center of gravity of the original section is 



I z ' dy 



Dividing out the constant p and replacing the element of area zdy 

 by dF, this expression for d' becomes 



ydF 



1: 



which is identical with the value of d given by formula (96) above. 

 Consequently, the neutral axis of the original cross section coincides 

 with the gravity axis of the transformed section. 



Now let the moment of inertia of the transformed section be 

 denoted by /'. Then 



in which y' is measured from the gravity axis of the transformed 

 section, that is, from a line through G parallel to OZ; and dF' denotes 

 an element of area of the transformed section ; whence dF' = z'dy'. 

 Therefore, since 



y' = y + d, z' = z - > and dy' = dy, 

 the expression for /' becomes 



or, if the element of area zdy is denoted by dF, 



C(y + dfdF 



/= "/W- 



Tlii< integral, however, is the one which occurs in formula (97). 

 Consequently, if its value from the above equation is substituted in 

 (97), the expression for the unit stress p simplifies into 



(98) p - 



