

SIMPLE STRUCTURES 181 



corresponding to any given system of forces, the force polygon 

 ABODE (Figs. 118 and 119) is first drawn, then any convenient 

 point is chosen and joined to the vertices A, B, C, D, E, of the 

 force polygon, and finally the equilibrium polygon is constructed 

 by drawing its sides parallel to the rays OA, OB, OC, etc. of the 

 force diagram. Since the position of the pole is entirely arbitrary, 

 there is an infinite number of equilibrium polygons corresponding 

 to any given set of forces. The position and magnitude of "the 

 resultant R, however, is independent of the choice of the pole, 

 and will be the same no matter where is placed. 



For a system of concurrent forces (that is, forces which all pass 

 through the same point) the closing of the force polygon is the 

 necessary and sufficient condition for equilibrium. If, however, the 

 forces are not concurrent, or if they are parallel, this condition is 

 necessary but not sufficient, for in this case the given system of 

 forces may be equivalent to a couple, the effect of which would be 

 to produce rotation. To assure equilibrium against rotation, there- 

 fore, it is also necessary that the equilibrium polygon shall close. 



The graphical and analytical conditions for equilibrium are then 

 as follows : 



CONDITIONS OF EQUILIBRIUM 



110. Application of equilibrium polygon to determining reactions. 

 One of the principal applications of the equilibrium polygon is in de- 

 termining the unknown reactions of a beam or truss. To illustrate 

 its use for this purpose, consider a simple beam placed horizontally 

 and bearing a number of vertical loads P P 2 , etc, (Fig. 120). To 

 determine the reactions R l and J? 2 , the force diagram is first con- 

 structed by laying off the loads 7j>, ZJ, etc. to scale on a line AF, 

 choosing any convenient point as pole and drawing the rays OA, 

 OB, etc. The equilibrium polygon corresponding to this force 

 diagram is then constructed, starting from any point, say A', in R^ 



