CHAP. VH.] SIMPLE GIRDERS. 91 



intersections H and I, we have A H = E E', I B' = F F'. Since 

 the triangle A A' H is similar to O A t ^ and B B' I to O B x D 1? 



and since A' H = B I, we have 



A t D t : B t D! ; : A H : B' I ; : E E' : F F' : : x : lx. 



Since A 1 T) t equals the load P! on A C, and B! D x the load P 2 

 on B C, we have P t : P 2 ! * x : lx. 



The same will hold true approximately for concentrated 

 loads. Hence, in order that the moment at any point may be 

 a maximum, the system of loads must have such a position 

 that the Loads either side of this point are to each other as the 

 portions into which the span is divid-ed. 



In PI. 13, Fig. 44, let C D give the moment at C. If the line 

 A B moves so that the horizontal projections of A C and B C 

 remain equal to x and lx, then as long as the ends A and B 

 move on the same straight lines, the point C will also move in a 

 straight line. The point C describes, therefore, a broken line. 

 The verticals between this line and the polygon correspond to 

 the moments for various positions of the load and a given value 

 of x. Evidently the greatest ordinate will be over an angle of 

 the equilibrium polygon which is not under an angle of the 

 line described by C that is, for M maximum, a load must lie 

 upon the cross-section. 



For any cross-section, then, the moment is a maximum when 

 a load is applied at this cross-section. Which of the loads 

 must be so applied is determined by the preceding rule. 



74. Construction of Maximum Moments. After the 

 equilibrium polygon has been constructed, in order to find M 

 for a point C (PI. 13, Fig. 45), we determine two points F and 

 G upon the polygon which are distant horizontally from the 

 load on the given cross-section corresponding to the angle E by 

 distances A C, B C. Then draw F G, and the vertical K E is 

 equal to M when the pole distance is unity. We make C I = E K. 

 In this way we can construct the moments for different loads 

 of the load system at the given cross-section, and thus determine 

 that position of the load which gives the maximum moment at 

 the cross-section. 



Generally when K E y, and the pole distance is a, we have 

 M = ay. The pole distance a is measured to the scale of 

 force, and then y is given by the scale of length. The unit for 



M,in order that M may be equal to y, is evidently th part of 



a 



