CHAP. VII.] SIMPLE GIRDEE8. 89 



when there is a load of the system at that point, and the maxi- 

 mum is positive or negative, according as the load lies just to 

 the right or left of the point. 



Since for a single load (Art. 70) S is positive or negative, ac- 

 cording as the load is to the right or left, S will be in general 

 a positive or negative maximum when all the loads lie to the 

 right or left, and the heaviest nearest the cross-section. Only 

 in cases where a small load precedes, can S be greatest when 

 the second load lies upon the point in question. 



If P is the resultant of all the loads and /3 its distance from 

 the right support, 



Vi = P and therefore S = P ^ P t . 



l> L 



Therefore S will vary as the first power of x, the distance 

 of the cross-section from the left support, provided that no 

 wheel passes beyond the support. Therefore, between any two 

 cross- sections for which the load on the girder remains the 

 same, the shea/r S is represented by the ordinates to a, straight 

 line. 



72. Construction of the Maximum Shearing Forces. 

 Construct the force polygon with the given loads ; choose a 

 pole O [PL 13, Fig. 42 (a)] and draw the corresponding equi- 

 librium polygon. It is required to determine the shear S at a 

 cross-section distant x from the left support, under the suppo- 

 sition that the first load P x of the system, moving towards the 

 left, acts at this cross-section. 



Determine upon the outer side P! A of the polygon passing 

 through the point P!, a point A distant from P t by the distance 

 , and then find the point B upon the polygon distant from A 

 by Z, the length of span, and draw A B. Parallel to A B draw 

 O L in the force polygon, then A L = V t = S, the shear at P t . 

 Drop a vertical through B intersecting P t A produced, in M ; 

 then the triangles O A L and A M B are similar, and there- 

 fore S = A L = B M , when a is the pole distance. If we 



v 



choose a = I, then S = B M. 



Hence : the maximum shearing forces are proportional to 

 the vertical segments between the equilibrium polygon and the 

 prolongation of the outer side taken at the end of the system, 

 or are equal to these segments if the pole distance is taken equal 

 to the span / provided that the last load is at the cross-section. 



