ANALYSIS OF STRESS IN BEAMS 



41 



47. Graphical method of finding the moment of inertia. If the 

 boundary of a given cross section is not composed of simple curves 

 such as straight lines and circles, it is often difficult to find the 

 moment of inertia by means of the calculus. When such difficulties 

 arise the following graphical method may be used to advantage. 



To explain the method consider a particular case, such as the rail 

 shape shown in Fig. 26, and suppose that it is required to find the 

 center of gravity of the section, and also its moment of inertia about 

 a gravity axis perpendicular to the web. The first step is to draw two 

 lines, AB and CD, par- 

 allel to the required 

 gravity axis, at any 

 convenient distance 

 apart, say /. 



1 f the section is sym- 

 metrical about any axis, 

 Mi.-li as n Y in the fig- 

 ure, it is sufficient to 

 consider the portion 

 on either side of this 

 axis, say the part on 

 tlif ri^ht of OF in the A 

 present case. 



Now suppose tliat 

 the cross section is di- 

 vided into narrow strips 



parallel to AB and CD; let z denote the length of one of these strips, 

 and dy its width. Then, if for each value of z a length d is found, 

 such that 



Fio. 26 



any jxnnt P on the boundary of the original section, with coordinates 

 * and y, will be transformed into a point P' with coordinates z' and y. 

 Suppose this process is carried out for a sufficient number of points, 

 and that the points P' so obtained are joined by a curve, as shown 

 1>> the dotted line in Fig. 26. Let F denote the area of the original 

 curve and F 1 the area of the transformed curve, both of which can 



