28 (SENDING. 



The sum of all such moments taken over the whole section 

 becomes the integral. 



r d 2 



/ '-jf-b-y^dy 



Integrating, this becomes 



2 <l- 



:r c ' s 



This is the total moment made up of the sum of all the 

 stresses, varying from to/ c> acting upon all the areas 

 b . 8 y, on one side of the neutral axis. If we suppose that 

 the elastic properties of the material are the same in 

 tension as in compression, then we shall have an equal 

 moment of resistance on the tension side of the neutral 

 axis. That is to say, 



4/" 6 '-8 



ft being equal to f c f, the total moment of resistance of 

 the metal is the sum of these two, or 



,. f b d i 

 o 



and this is equal to the moment of the external forces 

 acting on the section. Calling this M B , we have 



MB -/^p (XXI.) 



= /Z (XXII.) 



Where Z is what is called the " modulus of the section." 



We can go back a step and write the moment again in 

 the form of an integral 



I ~ 2 



which may be written in the form 



L r^ 



= d /by n -dy 



f+i 



l^/ 6 



The integral^/ 6 y 2 d y is called the "moment of 



