38 MOMENT OF ROTATION PARALLEL FORCES. [CHAP. V 



The points of application of the resultants of all the forces 

 right and left of d are then at the intersection of this tangent 

 with a g, or at an infinite distance. 



At d then we have a couple, the resultant of which is as we 

 have seen (Art. 20), an indefinitely small force acting at an 

 indefinitely great distance. That is, with reference to d, the 

 forces acting right and left cannot be replaced by a single 

 force. 



Hence generally : at the point of maximum moment (" cross 

 section of rupture"}, the resultant of the outer forces on either 

 side reduces to an indefinitely small and distant force, the 

 direction of which is reversed at this point, and the point of 

 application of which changes from one side to the other of the 

 equilibrium polygon.* 



The " cross section of rupture " then, is that point where the 

 weight of that portion of the girder between it and the end is 

 equal to the reaction at that end, or where the resultant changes 

 sign. 



The value of the moment at this point, is therefore equal to 

 the product of the reaction at one end into its distance from 

 the point of application of the equal resultant of all the loads 

 between that end and the point. 



Thus for a beam uniformly loaded with w per unit of length, 



the reaction at each end is -- From the above, the cross sec- 



a 



tion of rupture is then at the middle. The point of application 

 of the resultant of the forces acting between one end and the 



7, . wl I wl? 



middle is at -j-j hence the maximum moment is-^- x-; = -K-. 



39. Beam with Two Equal and Opposite Forces beyond 

 the Supports. The ordinates to the equilibrium polygon thus 

 give, as it were, a picture or simultaneous view of the change 

 and relative amount of the moments at any point. The point 

 where the moment is greatest, i.e., where the beam is most 

 strained, is at once determined by simple inspection. 



Let us take as an example a beam with two equal and oppo- 

 site forces beyond the supports. Thus, Fig. 21, PI. 6, suppose 

 the beam has supports at A and B, the forces being taken in 

 the order as represented by P t P 2 . We first construct the force 



* Die Graphische Statik. Culmann, p. 127. 



