ARCHES AND ARCHED RIBS 



165 



the force diagram parallel to the closing side A' G' of the equilibrium 

 polygon. Then OH (or P') may be replaced by its components AH 

 and AO, parallel to R^ and A'B 1 respectively ; and similarly, OH may 

 be replaced by its components FH and OF, parallel to R 2 and F' G r 

 respectively. If A and FH are therefore the required reactions. 



Problem 158. A simple beam 20 ft. long supports concentrated loads of 3, 5, 2, 

 and 9 tons at distances of 5, 7, 14, and 18 ft. respectively from the. left support. 

 Calculate the reactions of the supports graphically. 



Problem 159. Construct an equilibrium polygon for a simple beam bearing a 

 uniform load, and show that the reactions are equal. 



133. Equilibrium polygon through two given points. Let it be 

 required to pass an equilibrium polygon through two given points, 

 say M and N (Fig. 122). 



To solve this problem a trial force diagram is first drawn with any 

 arbitrary point as pole, and the corresponding equilibrium polygon 



A 



D 



FIG. 122 



MA'B'C'D'E' constructed, starting from one of the given points, say 

 M. The reactions are then determined by drawing a line OH parallel 

 to the closing side ME' of the equilibrium polygon, as explained in 

 the preceding article. 



The reactions, however, are independent of the choice of the pole 

 in the force diagram, and consequently they must be of amount AH 

 and HE, no matter where is placed. Moreover, if the equilibrium 

 polygon is to pass through both M and N, its closing side must coin- 

 cide with the line MN, and therefore the pole of the force diagram 

 must lie somewhere on a line through H parallel to MN. Let 0' be 



