SIMPLE STRUCTURES 



185 



From Fig. 123 it is evident that the horizontal component of any 

 ray of the force diagram is equal to the pole distance OH. There- 

 fore, if OC is resolved into its vertical and horizontal components, 

 the moment of the vertical component about S is zero, since it passes 

 through this point ; and hence the moment OC d = OH - z, where 

 z is the vertical intercept from the equilibrium polygon to the 

 center of moments S. Having determined 

 the moment at any given point, the stresses 

 at this point can easily be calculated. 



FIG. 123 



114. Relation of equilibrium polygon to bending moment diagram. 

 In the preceding article it was proved that the moment acting at 

 any point of a structure is equal to the pole distance of the force 

 diagram multiplied by the vertical intercept on the equilibrium 

 polygon from the center of moments. For a system of vertical loads, 

 however, the pole distance is a constant. Consequently, the moment 

 acting on any section is proportional to the vertical intercept on the 

 equilibrium polygon from the center of moments. Therefore, if the 

 equilibrium polygon is drawn to such a scale as to make this factor 

 of proportionality equal to unity, the equilibrium polygon will be 

 identical with the bending moment diagram for the given system 

 of loads. 



115. Structures : external forces. The external forces acting 

 upon any stationary structure must be in equilibrium. Hence they 

 may be found, in general, by applying the conditions of equilibrium 

 given in article 109. The conditions of equilibrium may be applied 

 either analytically or graphically. The former method has the ad- 

 vantage of being available under all circumstances; whereas the 



