36 RESISTANCE OF MATERIALS 



Now, applying the condition ^ vertical forces = 0, we have 



J^ + JR.-P^O, 



and inserting in this equation the value just found for R^ and then 

 solving for /?,, the result is 



26. Vertical shear. By applying the conditions ^F = 0, ^M= 0, 

 as just explained, all the external forces acting on the beam may 

 be found. The beam may then be supposed to be cut in two at any 

 point and these conditions applied to the portion on either side of 

 the section. 



In general, the sum of the external forces on one side of any 

 arbitrary cross section will not be identically zero. If, then, the 

 condition of equilibrium ^F = is satisfied for the portion of the 

 beam on one side of the section, the stress in the material at this 

 point must supply a force equal in amount and opposite in direc- 

 tion to the resultant of the external forces on one side of this point. 

 This resisting force, or resultant of the vertical stresses in the plane 

 of the cross section, which balances the external forces on one side 

 of the section, is called the vertical shear. Therefore 



The vertical shear on any cross section = the algebraic sum of the 

 external vertical forces on either side of the section. 



For instance, suppose that a beam 10 ft. long bears a uniform 

 load of 300 lb./ft., and it is required to find the vertical shear on 

 a section 4 ft. from the left support. In this case the total load 

 on the beam is 3000 lb., and, since the load is uniform, each reac- 

 tion is 1500 lb. The load on the left of the given section is then 

 4 x 300 = 1200 lb., and therefore the shear at the section is 

 1500 - 1200 = 300 lb. 



27. Bending moment. In applying the condition Vlf = to the 

 portion of a beam on either side of any cross section, the center of 

 moments is taken at the centroid of the section. Since the position 

 of the cross section is arbitrary, it is obvious that the sum of the mo- 

 ments of the forces on one side of the section about its centroid will 

 not in general be zero. Therefore, to satisfy the condition V|4f = 0, 

 the normal stresses in the beam at the section considered must 



