130 CONTINUOUS GIRDERS. [CHAP. VIIL 



By this division of the moment area we have obtained a great 

 advantage. While the three areas C D C', A A' C and B B' C' 

 are all three dependent upon the moments at the supports A A' 

 and B B', we have by this new division to do with three areas, 

 of which the first is entirely independent of the moments at the 

 supports, the second depends only upon that to the left, and the 

 third only upon that to the right. 



84. Properties of the Equilibrium Polygon. Let us con- 

 sider now the case of a beam over four supports, that is, of 

 three spans l^ l and 4 the first and last being, as is usually the 

 case, equal, and the two first loaded with both live and dead 

 load, the last with dead load only. The parabolas for the ver- 

 tical loads [PI. 14, Fig. 52] may be constructed by means of a 

 force polygon, or the ordinates at the centre calculated, and the 

 parabolas then drawn. The moments at the supports are A A! 

 and B B'. Although these are unknown, it is not necessary to 

 assume them at first. They may be directly constructed. 



Thus, if we conceive the moment areas in each span divided 

 into positive parabolic areas and negative triangles, we have in 

 the first and last span one, in the middle two triangles. If we 

 consider these areas as forces acting at the corresponding centres 

 of gravity, we shall obtain an equilibrium polygon of the form 

 given in Fig. 52 (&). That is, this polygon must have eight 

 sides, and its angles must be somewhere on the verticals through, 

 the centres of gravity of the parabolic and triangular areas. 

 The parabolic areas act downwards, the triangular areas up- 

 wards. The problem is, to make these last so great that this 

 polygon shall pass through all the points of support. 



One of the properties of the polygon we have, therefore, just 

 noticed, viz. : its angles must lie in the verticals through the 

 middle points of the spans and through the points distant from 

 A and B one-third of the spans on each side (i.e., the centres of 

 gravity of the triangles). 



If we prolong the second and fourth sides of the polygon, 

 they intersect in a point M, the point of application of the 

 resultant of the two contiguous triangular area forces (Art. 44). 



The areas of these two triangles are -A.A'l and ^ A A' 4, that 



is, the areas are as the spans ^ and ^. 



Then by the principle of Art. 18 the resultant divides the 

 distance between the forces into two portions, which are to each 



