160 



STRENGTH OF MATERIALS 



respectively, it will be found that their resultant, obtained by the 

 above construction, is equal and parallel to R. 

 Since the opposite sides of a parallelogram 

 are equal and parallel, it is more convenient in 

 finding the resultant of two forces to construct 

 half the parallelogram. Thus, in the preceding 

 example, if P 2 is laid off from the end of P p 

 R is the closing side of the triangle so formed 

 (Fig. 114). Such a figure is called a force triangle. 

 In order to find the resultant of several con- 

 current forces lying in the same plane, it is 

 only necessary to combine two of 

 them into a single resultant, com- 

 bine this resultant with a third force, and so on, taking 

 the forces in order around the point in which they meet 

 Thus, in Fig. 115, R l is the resultant of P l and /'.. : ft t is 

 the resultant of ^ and P 3 ; R 9 is the resultant of R t 

 and P 4 ; and R is the resultant of R 9 and P 5 . R is there- 

 fore the resultant of the entire system P,, P t , P f , P 4 , P $ . 



In carrying out this construction it is unnecessary to draw the 



intermediate resultants 



FIG. 113 



. in 



^ tin linal 



resultant in any case 

 being the closii 



the polygon formed by 

 placing the forces end 

 to end in order. Such 

 a timire is called a force 

 polygon. From the ab >\ e 

 construction it U 

 dent that the nece 

 and sufficient condition 

 that a system of concur- 

 rent forces shall be in 

 equilibrium is that their 



force polygon shall close, since in this case their resultant must 



be zero. 



- R 



FIG. 116 



