192 



RESISTANCE OF MATERIALS 



2. Graphical method. Before proceeding with the explanation of 

 the graphical method it will be necessary to show how the moment 

 of any number of forces with respect to a given point may be 

 obtained from the equilibrium polygon. 



Let PV P z , ^, P denote any set of forces and B the given point 

 about which their moment is required (Fig. 130). First draw the 

 force polygon for these forces, choose any pole 0, and construct 



the corresponding equilib- 

 rium polygon abode. Now 

 in the force diagram, drop 

 a perpendicular oh from the 

 pole on the resultant R. 

 This is called the pole dis- 

 tance of R and will be denoted by H. 

 Also, in the equilibrium diagram draw 

 through the given point B a line par- 

 allel to R, making the intercept xy 

 on the equilibrium polygon. Then 

 the triangle OAE in the force diagram 

 is similar to the triangle xey in the 

 equilibrium diagram, and hence 



D 



or 



E 

 FIG. 130 



r : xy = Hi AE, 

 Rr=Hx xy. 



But Rr is the moment of the result- 

 ant R about B and is equal to the 

 sum of the moments of all the given forces about this point. The 

 following moment theorem may therefore be stated : 



The moment of any system of forces about a given point is equal to 

 the pole distance of their resultant multiplied by the intercept made 

 by the equilibrium polygon on a line drawn through the given point 

 parallel to the resultant. 



The moment of a part of the given set of forces about any point 

 may also be found by this theorem. For example, let it be required 

 to find the moment of JJ and P z about B. The resultant of P r P v is 

 given in amount by AC and acts through the point/, as shown. 

 Hence draw through B a line parallel to this partial resultant, 



