CHAP. V.] MOMENT OF ROTATION PARALLEL FORCES. 53 



passes then from 2 to V , and beyond these two limits it can 

 never pass. 



Onr construction, then, is simply to lay off the load in oppo- 

 site directions perpendicularly from each end of the axisAo L, 

 and join the end points A E and L E'. The intersections of 

 these lines with the diagram of shear give the points 2 and 

 V required. 



47. Load Systems* Concentrated loads occur in general 

 in practice in a certain succession, as for instance the forces 

 acting at the points of contact of the wheels of a train of cars 

 passing over the beam, and it is necessary then to investigate 

 the influence of different positions of the train. It evidently 

 amounts to the same thing whether we suppose the weights to 

 move over the beam, or suppose the weights stationary and the 

 beam to move. In either case we obtain every possible posi- 

 tion of every weight relatively to the ends of the beam. 



The severest load to which we can subject a railway truss, for 

 example, is when the span is filled with locomotives. If we 

 suppose, for illustration, in round numbers, the distance between 

 the three axles of the locomotive 3 ft. 6 in., between the 

 axles of the Jtender 5 ft. in., between the foremost tender 

 and the back locomotive axle 4 ft., and the entire length of 

 locomotive and tender 34 ft. 6 in., and then suppose the weight 

 upon each locomotive axle 13 tons, and upon each tender axle 

 8 tons, we have a system of weights in fixed order and at fixed 

 distances, and the truss should be investigated for a series of 

 these systems, as many as can be placed upon the span, passing 

 >ver it from one end to the other. 



In PI. 9, Fig. 32 (a), we assume two such locomotives as 

 shown by P^o, and construct the force and equilibrium poly- 

 gons. The forces are symmetrically arranged with respect to 

 a central point, and the pole in the force polygon is therefore 

 taken perpendicular from the middle of the force line. 



Now the system of forces being as represented, suppose the 

 span to shift. Thus suppose the span of a given length repre- 

 sented by S x St in the Fig. Then 6 is the line closing the 

 polygon for this position of the span, and a parallel to 6 in 

 the force polygon, viz., C L gives the reactions at the ends. 

 Let now the span move from S r S x to s s s & ; we have a new po- 



* Element* der Graphischen Statik, Bauschinger. 



