162 METHODS OF LOADING [CHAP. XL 



positive moments at the right end of first span, abcde. From 

 e draw a line to left support. At the other supports, in like 

 manner, we determine the positive moments, and join the 

 points e e in second and third spans, and e and right support 

 in fourth. 



We have already seen (Art. 103) that within the fixed points 

 the moments are negative wherever a load may be placed. 

 The maximum, therefore, occurs for full load. "VVe have, there- 

 fore, found for 3d and 4th spans the parabola for full load, 

 precisely as in Fig. 63. These parabolas are given in broken 

 lines in the Fig. a. By subtraction of the positive moments 

 outside the fixed points from the positive moments at the same 

 points for total load given by these parabolas, we obtain directly 

 the lower curves as far as the points e in each span. 



The second parabola (partly full, partly broken) is all that is 

 needed to complete our Fig. To obtain this we have simply, 

 in 2d and 3d spans, to make the vertical through centre of 

 line e e equal to its length already laid off for total load, viz. : 



/Z\ 2 



Jj9X 2 l-J, Art. 98 (2), produce ee to intersections with sup- 

 port verticals, and join these intersections with the extremities 

 of the first verticals above. We can then construct the para- 

 bola, which completes our diagram, and gives us Fig. a. 



1O6. Practical simplifications In practice, the construc- 

 tions given in Figs. 63, 71 and 72, admit of many simplifica- 

 tions, which, in order to avoid confusion at first, have been dis- 

 regarded. The whole solution, given for the sake of clearness 

 in three separate Plates, can be performed upon a single sheet, 

 since the Fig. for the second equilibrium polygon in Figs. 63, 

 71 and 72 may be combined in one. Indeed, the lines neces- 

 sary for the construction of the maximum shearing forces can 

 be applied directly to the determination of the maximum 

 moments. It is therefore unnecessary to divide the construc- 

 tion into separate sheets. 



2. The cross-lines in the end spans can be omitted, since 

 all that is required are the distances to be laid off upon the 

 end verticals, and these when found can be laid off at once. 



3. We can apply the second equilibrium polygon -directly 

 in order to find the moments for the dead and moving load. 

 Thus the transferring of ordinates from one Fig. to another is 

 avoided. 



