Manchester Memoirs, Vol. lix, (19 15), No. 2. 7 



cription. Produce P 1 P i P s to cut the neutral axis of the 

 spar in points m, «, respectively. Since FB, CG and 

 DH are in practice always small, the distance mB by 

 which 111 lies to the left of B will also be small. If 

 now we consider the forces P X PJP % to be acting at m, n, 

 and 0, the spar will be under the same loadings and 

 moments as before, except that there will now be excess 

 moments equal to the moments produced by the approxi- 

 mately horizontal components of PJPJP^ acting at B, C } D 

 respectively and varying uniformly from B to m i from C 

 to n, and from D to respectively, being zero at m y n, and 

 0. The areas of the excess moment diagrams are small, 

 and therefore for a first approximation they may be neg- 

 lected. Also, since the slope of the beams will be small 

 for a certain distance at either side of any prop, it follows 

 that the three points m, n, and will lie sensibly in a 

 straight line. The slope of the spar at these points will 

 therefore be given by the distance by which the base line 

 passes over or under characteristic points on either side 

 of the prop, and the heights of these characteristic points 

 will be given by two-thirds the maximum height of the 

 parabolas on the equivalent spans mn, no, oE respectively. 

 The bending moment diagram, Fig. 5, for this modi- 

 fied system may be drawn according to the methods 

 already indicated. It may be noted here that although 

 they have been sketched in the figure for the sake of 

 clearness, it is unnecessary to draw the actual parabolas 

 for the detached spans since it is only required to know 

 the maximum heights in order to determine the charac- 

 teristic points. Let M mi M n , M be the moments at ;//, ;/, 

 and as obtained from this diagram. Take now the 

 portion of the spar lying to the left of n and write down 

 the bending moment at n in terms of the distributed load 

 and P x . This value if equated to the moment at 11 as 



