88 SIMPLE GIRDERS. [CHAP. VH. 



BO that in this case the moment at any point is directly given 

 by the ordinate of the polygon at that point. It is this impor- 

 tant property of the equilibrium polygon which renders it espe- 

 cially serviceable in the graphical solution of this and similar 

 problems. 



70. Concentrated Load Variable Position Shearing 

 Force. If the load lies to the right of any given cross-section, 

 then the shearing force at this cross-section will be S^V^ or, 

 since we regard a force to the left acting up as positive, S is 

 positive. As the load P moves towards the left, V^ or S in- 

 creases. When the load is to the left of the cross-section, the 

 shearing force at the cross-section is S = V t P, and since P 

 is always greater than V\, S is negative. The nearer P ap- 

 proaches the cross-section, the smaller is S algebraically. 



Hence : a concentrated load causes a positive or negative 

 shear, according as it is to the right or left of the cross-section 

 considered, and the shearing force is greater the nearer the load 

 i to the cross-section. 



Moments. If the load lies to the right of the cross-section, 

 the moment is M = V t x, x being the distance of the cross- 

 section from the left support. M is therefore negative and in- 

 creases with V t ; that is, as the load approaches the cross-sec- 

 tion. If the load is on the left of the cross-section, M = V 2 

 (I x), V 2 being the reaction at the right support. Here also 

 M is negative and increases with V 3 ; that is, as the load ap- 

 proaches the cross-section. 



Hence : a concentrated load wherever it lies causes in every 

 cross-section a negative moment, which for any cross-section is 

 a maximum, when the load is applied at that cross-section. 



71. Poition of a given System of Concentrated Loads 

 causing Maximum Shearing Force. If PI is the sum of all 

 the loads to the left of any cross-section, the shear at that cross- 

 section is S = Vj P,. As the system moves to the left with- 

 out any load passing off the girder or any load passing the cross- 

 section, V a and therefore S increases as long as S is positive, or 

 as long as V t > P v If a load passes off the girder, then for the 

 remaining loads S increases anew as the system moves to the 

 left, until a load of the system passes the cross-section in ques- 

 tion. The same holds good for a system moving to the right, 

 where S is negative. 



Hence : the shearing force is a maximum for any point^ 



