Manchester Memoirs, Vol. lix. (191 5), No. %. 9 



one end equal to B, the expression for the characteristic 

 point which lies on the side of the maximum ordinate of 



2 B 

 the triangle, becomes equal to — , so that the point lies 



on the upper line of the triangle. Similarly the length of 

 the perpendicular for the other characteristic point of the 



span becomes — — , and the point is again upon the upper 



line (Fig. 7b). Since a trapezium can always be divided 

 into a rectangle and a triangle and since the heights of 

 the characteristic points above the horizontal base are 

 additive, it follows that these points will lie upon the 

 upper line of the trapezium if the bending moment 

 diagram be of that shape (Fig. 7 c). If then the free 

 ended bending moment diagram for any span consist of 

 a parabola superposed upon a trapezium the characteristic 

 points will lie at distances above the base line equal to 

 the height to the upper line of the trapezium at that 

 point, added to two-thirds the maximum vertical ordinate 

 of the parabola, i.e., 



2 c Wl 2 



- of , 



3 3 



/ being the length of span under consideration. The 

 characteristic points having been determined it only 

 remains to draw a base line which gives a moment at B 

 equal to the cantilever moment, passes over and under 

 each characteristic point on either side of a support by 

 equal amounts and finally gives a zero value at the fuse- 

 lage. Next check the values of P^P n P^ by equating the 

 bending moment at a point immediately to the left of 

 each support to the moments caused by the uniform 

 loading and the load in the wires. Should any serious 

 discrepancy be apparent, the diagram may be modified 

 in the light of our more complete knowledge, but in 

 practice this is rarely if ever necessary. 



