128 CONTINUOUS GIRDERS. [CHAP. VIET. 



real size, then in the diagram the ordinates of the elastic line 

 will be given in true size. 



Equation (5) may also be written 



. 



that is, the elastic curve is an equilibrium curve, or catenary, 

 whose horizontal force H is E I or . , and whose corresponding 



variable load per unit of length is M or -j- respectively. 



If we divide, then, the moment area by verticals into a number 

 of smaller areas, and consider these areas as forces acting at 

 their centres of gravity, these forces determine, as we have seen 

 (Art. 43), an equilibrium polygon which is tangent to the elas- 

 tic carve at the verticals which separate the areas. Thus we 

 can construct any number of tangents to the elastic curve ; areas, 

 which are positive or negative, must, of course, be laid off in the 

 force polygon in opposite directions. 



If we divide the moment area by lines which are not vertical 

 [PI. 14, Fig. 51], the directions of the outer polygon sides are 

 the same as for vertical divisions, because the vertical height 

 between the corresponding outer sides in the force polygon ia 

 in any case always equal to the total load. 



The two outer polygon sides for any method of division are, 

 therefore, tangents to the elastic curve at the ends of the same, 

 Here also we can, of course, have negative areas. 



82. Effect of End Moments. A beam or girder continuous 

 over three or more supports differs from a beam simply resting 

 upon its supports, in that, in addition to the outer forces, we 

 have acting at each intermediate support a moment or couple, 



But, as we have seen, Art. 23, the effect of these moments or 

 couples will be simply to shift the closing line of the equilibrium 

 polygon through a certain distance. Thus [PI. 14, Fig. 52 (#)], 

 if the span ^ were uniformly loaded and simply supported at 

 the extremities A and B, the equilibrium curve, or curve of mo- 

 ments, would, as we know (Art. 44), be a parabola A D B. If, 

 however, the beam is continuous, we have at A and B moments 

 or couples acting, and the closing line A B is shifted to some 

 position as A' B'. If now we consider the moment area, we 

 see that by the shifting of the closing line the former moment 

 area, which we shall call the positive area, is diminished, while 

 to the right and left we have negative areas A A' C and B B' C.' 



