60 MOMENT OF ROTATION PARALLEL FORCES. [CHAP. V. 



to the ends. The points thus obtained form a curve, and the 

 greatest ordinate between this curve and the polygon gives the 

 greatest moment which can act at the given cross-section. 

 This greatest ordinate will always be found at an angle of the 

 polygon, and hence a weight must always rest upon the cross- 

 Bection. Since the cross-section itself must lie upon this ordi- 

 nate, we have directly the position of the span with reference 

 to the given forces. The closing line for this position being 

 then drawn, a parallel to it in the force polygon gives the reac- 

 tions for this position. 



The reader will do well to make the construction indicated 

 for an assumed span and system of weights, to convenient scales, 

 checking the results by computation.* 



The above method applies more particularly to solid or 

 "plate " girders, beams, or trusses. It may of course be applied 

 to framed structures also, such as those illustrated in chapter 

 first. Thus the moment at any point, divided by the depth of 

 truss at that point, gives the strain in flanges. The more pre- 

 ferable, as perhaps also the simplest method of determining the 

 strains in such cases, however, is to find the reactions due to 

 each individual weight. Each reaction can then be followed 

 through the structure, as explained in that chapter, and the 

 strains in every member for every weight in every position can 

 thus be obtained and tabulated. An inspection of the table 

 will then give at once the strains due to the united action of 

 any desired number of these weights. 



We have thus two methods for the solution of such cases ; 

 first, by the composition and resolution of forces, and, second, 

 by the equilibrium polygon and moments of rupture, and may, 

 if we choose, check the results obtained by one method by the 

 other. In most practical cases involving framed structures, 

 however, the first method is preferable as being simpler, quicker 

 of application, and of superior accuracy. 



For solid-built beams or "plate girders," etc., the second 

 method comes more especially into play. The determination 

 of the strains in a structure of this kind from the known mo- 

 ment of rupture at any point, requires a knowledge of the 

 moment of inertia of the cross-section at that point, and this 

 may also be found by the Graphical method. 



* This construction is given in Art. 15, Fig. VIII., of the Appendix. 



