136 CONTTNTTOtrS GIRDERS. [CHAP. VHI 



If the construction of the moments over the supports is our 

 sole purpose, as is in practice the case, the polygon need not be 

 drawn. We have only to find our fixed points, and note the 

 intersection of the sides with the verticals through the supports, 

 without drawing the sides themselves. In the preceding Arts- 

 we have purposely considered only the particular case of uni- 

 form loading, and have taken only three spans, in order to 

 familiarize the reader with the nature of the problem and the 

 method of its solution. In order to attain a clear understand- 

 ing of the subject as thus far developed, he would do well to 

 take some particular case, as, for instance, that of a girder of 

 two or three or four spans of given length, the end spans being 

 equal, and intermediate spans equal and say one-fourth longer 

 than the ends, and work out by diagram the moments at the 

 supports for a uniform load over the whole length of girder. 

 For two spans the moment at the centre support should be 



^p P, I being the length of span, p the load per unit of length. 



For three spans the moment at the two inner supports is 



1 + n* 

 4/0 I t > \ P V, where n I = the length of end spans. Thus, if 



3 91 



n = -7, we have TTVTT P P- For four spans the moment at the 



4: 1152^ 



1 + 2 n 3 



aecond and fourth supports is 7- pF, and at the middle 



4 (3 4- 4 n)-^ 



1 + 2 n ?$ 

 support A /o , A \~ P t' fy these formulae the graphical 



results may be checked. 



When the reader has thus become thoroughly familiar with 

 the principles of the preceding Arts, and their practical appli- 

 cation, he will be ready to resume at this point the more gen- 

 eral development which follows. 



8. The Second Equilibrium Polygon. We see, there- 

 fore, that the actual form, of the elastic line is not required to 

 be known. Only the outer forces and their moments are sought, 

 and to determine these it is sufficient to know the position of 

 the tangents to the elastic line at the supports. Thus the first 

 line of the equilibrium polygon [Fig. 52, 5] being given in 

 position, by the aid of the middle, third, and limited third 

 verticals and the known point K, all the other sides may be 



