BENDING-MOMENT AND SHEAR DIAGRAMS 37 



supply a moment which balances the sum of the moments of the 

 external forces on either side of the point. This resisting moment 

 in the beam is called the stress couple or bending moment, and is 

 evidently equal to the resultant external moment at the point in 

 question. Consequently 



The bending moment at any cross section of a beam is equal to the 

 sum of the moments of the external forces on one side of this point, 

 about the centroid of the section. 



For example, in Fig. 38, consider a cross section mn at an arbitrary 

 distance x from the left support. Then for the portion of the beam 

 on the left of mn the mo- 

 ment of R l about the cen- 

 troid of the section is Rjc, 

 and the moment of F[ about 

 the same point is I( (x d^). 

 Therefore the total bend- 

 ing moment at the section 

 mn is 



FIG. 38 



As another example, con- 

 sider a beam of length I 

 bearing a uniform load of amount w per unit of length. Then the 



total load on the beam is wl, and each reaction is ' Therefore, 



taking a section at a distance x from the left support and consider- 

 ing only the forces on the left of the section, the total bending 

 moment at this point is 



wl x wx -. 



O O O v -/* 



From this relation it is evident that M = when x = or x = I, and 



attains its maximum value when x = - ; that is, the bending moment 



2 



is zero at each end of the beam and a maximum at the center. 



28. Bending-moment and shear diagrams. Since in general the 

 bending moment and shear vary from point to point along a 

 beam, it is desirable to show graphically the moment and shear at 



