50 BROOMALL : 



The relation between horizontal forces and horizontal 

 shear may be expressed by the equation : 



V„ - dS (i) 



where Vh = total horizontal shear on bottom of block, 

 S = algebraic sum of horizontal forces on one 

 side of block. 



L,et V = unit horizontal shear 



b = thickness or width of beam 



^. Vh dS 



Tnen v 



b . dl b . dl 



dS 

 Or b . V = — (2) 



dl ^ 



Equation (2) shows that the absolute value of the hori- 

 zontal unit shear (and hence also the vertical unit shear) at 

 any point is proportional to the rate at which the horizontal 

 forces on the block are changing value. As we pass from 

 the middle toward the support, along any level line, the hori- 

 zontal forces become less and less by regular decrements from 

 the previous value, true subtractions. By the time the end of 

 the beam is reached these decrements have reduced the hori- 

 zontal forces to zero. The values of the horizontal and verti- 

 cal shear also change, becoming greater, but not by regular 

 additions to a previous value. 



The horizontal shear may be regarded as the physical 

 lever arm of the couples formed by the direct stresses in the 

 upper and lower portions of the beam. Without this hori- 

 zontal shear the parts of the beam would not work together 

 as a harmonious whole. 



Let us consider the question of horizontal shear as regards 

 its vertical distribution. In Figure 3 is indicated the manner 

 of production of horizontal shear, and its effect upon an ele" 

 ment of the bod}' at the shearing plane. The lower arrows in 

 A and B and upper arrow of C represent the companion force 

 of the shear couple produced bj^ the resistance of the adjacent 



