CHAP. EX.] LOADED AND UNLOADED SPANS. 



CHAPTER IX. 



CONTINUOUS GIRDER LOADED AND UNLOADED SPANS. 



91. Unloaded Span. If the span is unloaded, we have to 

 construct the second equilibrium polygon, only the two forces 



- M' I and - M" I. If the position of the end tangents is known, 



<a . 2 



the polygon is completely determined. If we prolong the 

 middle side U V to intersections M and N with the end ver- 

 ticals [PI. 15, Fig. 55], then, by the preceding Art., A M = 



M' ( l \\ B N = M" (-J ; therefore, A M : B N : : M' : M". 



If now we draw A B intersecting U V in I, the moment at this 

 point is zero. That is, the intersection I of the line joining the 

 supports with the middle side of the polygon is the point ofin- 

 Jlection of the elastic line. 



92. Two successive Unloaded Spans. Prolong the two 

 middle sides U V and U t V t [PI. 15, Fig. 56] of the equilibrium 

 polygon for the two spans ^ and l^. The point of intersection 

 W is a point in the resultant of the forces at V and U^ Since 



these forces are - M x 4 and - M x ^, we have W V : W U :: 



'2 2i 



4 : 1 . But the horizontal projection of V U is - (7 tt + k), 



o 



therefore that of V W is - ^ and of U W , - 1 Q ; while that of 



o ' 3 



B W is - (^ ZQ). The vertical through W we have called the 



o 



limited third vertical. Its position is, as we see, easily found, 

 and depends simply upon the length of the spans. 



Let us now consider more closely the intersections I and L of 

 the middle sides with the straight line joining the supports. 

 We have 



LU O :LW O ::U O U I: W O W. 



