16 



RESISTANCE OF MATERIALS 



forces as adjacent sides (Fig. 7). This single force, which is equiv- 

 alent to the two given forces, is called their resultant. 



Any number of concurrent 

 forces may be thus combined by 

 finding the resultant of any two, 

 combining this with the third, 

 etc. Or, what amounts to the 

 same thing, the given forces may 

 be placed end to end, forming a 

 polygon, and their final resultant 

 will then be the closing side of 

 this polygon (Fig. 8). 



Now, in Fig. 9, let F l and F 2 be any two concurrent forces, and F 

 their resultant. Also let be any given point, and B^ # 2 , <, the angles 

 between OA and the forces 

 F^ F z , F, respectively. Then, 

 taking moments about 0, 



Moment of F l about 



Moment of F 3 about 

 = F 2 x OA sin 2 . 



The sum of these moments is 



M = F l x OA sin B l + F 2 x OA sin 2 = OA (F^ sin B 1 + F 2 sin 2 ). 



But, since F is the resultant 



D 



of F l and 



F sin <f> =F l sin 

 and consequently 



sn 



FIG. 9 



The right member, however, is 

 the moment of the resultant 

 F with respect to 0. Therefore, 

 since is arbitrary, the sum 

 of the moments of any two concurrent forces with respect to a 

 given point is equal to the moment of their resultant with respect 

 to this point. 



