48 STEENGTH OF MATERIALS 



Article 37. For a simple beam placed horizontally and supporting a 

 system of vertical loads, the plane of the moment is perpendicular to 

 the plane of the section, and the single force is a vertical shear Ivin- 

 in the plane of the section. Therefore, since the portion of the beam 

 on either side of the section must be in equilibrium, the v -r 

 shear is equal to the algebraic sum of the external forces on either 

 side of the section. Thus, if the portion of the beam on the li- 

 the section is considered, the vertical shear on the section is equal 

 to the reaction of the left support minus the sum of the loads on the 

 left of the section. 



Problem 54. A beam 10 ft. long bears a uniform load of 800 Ib./ft. Find the 

 vertical shear on a section 4 ft. from the left support 



Solution. The total load on the beam is 3000 Ib. Therefore, since the load is 

 uniform, each reaction is equal to 1600 Ib. The load on the left of the secti 

 300 x 4 = 1200 Ib. Therefore the vertical shear on the section is 1500 - 1200 = 8m IK 



Problem 55. Find the vertical shear at the center and ends of the beam in the 

 preceding problem. 



Problem 56. A beam 12 ft. long bears loads of 1, J, and 8 tons at distances of 

 2, 6, and 7 ft. respectively from the left support Kind the vertical shear at -. 



end of the beam, and also at a 

 p f point between each pair of loads. 



51. Maximum bending 

 moment. 



P ' ! 



. 



ing moment at any j 

 abeam is Defined as the sum 

 of the moments, almm 



\ 

 ~* 



M neutral axis of a cross sec- 



tion through the point, of all 

 the external forces on -iiln -r 

 FIG. 34 side of the section. Thus, if 



the portion of the beam on 



the left of the section is considered, the external moment at this j.. 

 is the moment of the reaction of the left support about the m-,. 

 axis of the section, minus the sum of the moments of the loads 

 between the left support and the section, about the same neutral axis. 



For example in Fig. 34 the moment of RI about the neutral axis of the *r. 

 rnn is Bjx, and the moment of P, about the same axia is P, (x - d,). Ther- 

 the total external moment acting on the section mn is 



