BENDING. 29 



inertia" of the section. It is the summation of the 

 products of all small increments of area into the squares of 

 the distances of these areas from the neutral axis. 



The moment of inertia is generally denoted by the 

 letter I. If we call the distance of that part of the 

 section furthest from the neutral axis, and therefore under 

 the maximum stress Y, then the above equation becomes 



^ = ^ ........ (XXIII.) 



Y in the present instance being equivalent to . This 



z 



last equation is perfectly general and applies equally well 

 to sections of all shapes, the moment of inertia varying 

 according to the form of the section. 



Here -"^ = v an< ^ we ^ ave 



that M= /Z, so that 



z = \ (xxiv.) 



or the modulus of the section is equal to the moment of 

 inertia divided by the distance from the neutral axis to the 

 part of the section furthest away from it. 



In working out beam questions which require the use 

 of the moment of inertia, it depends upon what is the form 

 of the section, as to whether the moment of inertia is found 

 by means of ordinary mathematical methods or the more 

 usual formulae, or whether purely graphical methods must be 

 employed. If the section under consideration is a simple 

 and regular one, then the proper formulae can be made use 

 of, or the moment of inertia can be calculated ab initio. 

 But in cases where the section is irregular or complicated, 

 or both, the use of the following well-known graphic method 

 will save time and trouble. 



The principle involved will be best explained by means 

 of a simple example. 



A B C D is an ordinary rectangular beam section, 

 with a neutral axis X Y. As before, consider a small strip 

 of the area a b taken parallel to the neutral axis. The 

 area of this strip is b 8 y } and the stress upon it f y , where 



Jy ~ ' * 



