CHAP. VIII.] CONTINUOUS GIRDERS. 129 



It is evident that these areas have also a corresponding action 

 upon the elastic line. For a positive moment area this last ia 

 conceive upwards, while for negative areas it is convex upwards. 

 At the points of transition C and C' we have the inflection 

 points. This follows easily if we only hold fast the manner in 

 which the elastic line is constructed, viz., by dividing the mo- 

 ment area into laminae and regarding the area of each as a 

 force. The forces thus obtained must plainly act, some upwards 

 and some downwards, and the corresponding equilibrium poly- 

 gon or elastic line must be in part convex upwards and in part 

 convex downwards, and hence at the points of transition we 

 must \\&VQ points of inflection where the moment is zero. 



83. Division of the moment Area. We shall assume the 

 cross-section of beam constant. Regarding the elastic line 

 simply as an equilibrium polygon, we can apply the principle 

 that the order in which the forces are taken is indifferent (Art. 

 6) when the resultant only is desired. Since in the considera- 

 tion of a single span only the first and last sides are of impor- 

 tance, we can, so long as we consider a single span only, take 

 then the laminae or divisions of the moment area in any order 

 we please. More than this, we can, as we have seen in Art. 81, 

 divide the moment area into laminae not vertical ; for example, 

 we may in any span distinguish three parts, one positive and 

 two negative, and consider each as a force acting at the centre 

 of gravity of the corresponding area. [This holds good only 

 for constant cross-section. For variable cross-section the hori 

 zontal force E I is variable.] Still further, we can divide the 

 moment area for a single span into a positive area, which is pre- 

 cisely the same as for a non-continuous beam, and into a nega- 

 tive area, which will be evidently a trapezoid. 



This is of great importance. To understand it fully we refer 

 to PI. 14, Fig. 52. Here, in the second span, we see that the 

 real moment area consists of a positive part, viz., the parabola 

 C D C', and two negative parts A A' C and B B' C'. Instead of 

 these we may take the entire parabolic area A D B and the tra- 

 pezoid A A' B' B, or, finally, instead of this trapezoid, we may 

 take the two triangles A A' B' and B B' A. The parabolic 

 area is positive, the triangular areas are negative. 



If we assume the load as uniformly distributed, the first area 

 will be always parabolic, and we may, therefore, call it the 

 parabolic area. 



