CHAP. IX.] LOADED AND UNLOADED SPANS. 145 



, This property of the second equilibrium polygon was first 

 made known by Culmann. 



97. Arbitrary Loading. According to the above properties 

 of the second equilibrium polygon, the general course of pro- 

 cedure for any given case of loading is then as follows [Fig. 60, 

 PL 16] : 



1. Construct all the fixed points A I a I 3 --- K t K 2 , etc. (Art. 

 93), and draw verticals through them. 



2. Construct the cross-lines for every span, Art. 88. 



3. *Make A C equal to C^ Q! as given by the cross-lines, and 

 draw a line through C and A x to intersection D 2 , with vertical 

 through I 2 . Then make D 2 C 2 equal to O 2 QJJ, and draw a line 

 through C 2 and A 2 to D 3 , and so on. Precisely the same con- 

 struction holds for the other way from the right end. Thus 

 A 4 E 4 is equal to R 4 P 4 , etc. 



4. In this way we obtain for each of the middle sides of the 

 second equilibrium polygon two points, C and F x , C 2 and F 2 , 

 etc. ; A and E x , D 2 and E 2 , and so on ; so that now we can ac- 

 tually draw these middle Bides. 



The intersections of these lines with the support verticals 

 give, according to Art. 88, the moments at the supports. For 

 spans whose length is \these moments are given directly; for 

 other spans the construction of Art. 74 must be applied. The 

 following simple construction may also be applied. Let 

 I K be the intersections of the verticals through the fixed 

 points, with the line A B joining the supports [Fig. 61, PI. 16]. 



Make I D' = I D , KF' = KF , C' D' = O Q 



E'F' = R P t\ and draw C' F' and E' D'. These lines cut 



the support verticals in M' and N', so that A M' and B N' are 



the moments. 



By the construction errors accumulate from one span to the 

 next, so that the diagram must be made with care. We have 

 also several checks, viz. : 1. The intersection of the middle sides 

 must lie in the vertical through the intersection of the cross- 

 lines. 2. The prolongation of the middle sides must intersect 

 in the limited third vertical. 3. The corresponding intersec 

 tions of the middle sides with the third verticals must lie in a 

 straight line through the support. 



