RECTANGULAR BEAMS 



41 



axis since the statical moment Q has its maximum value for a 

 point on this axis, and b is constant. 



6. If a beam is of constant cross-section throughout, the 

 maximum values of / and v will occur at the section where M 

 and V respectively have maximum values. 



The case of longitudinal shear is shown in Fig. 13. Let the 

 fiber stresses at section m be represented by f t and those at 

 section n by / 2 , while the longitudinal variations of the fiber 

 stresses between these two sections are indicated at section n by 

 the cross-shaded areas. This increase of horizontal stress from 

 one section to another (which we know to be true since the bend- 



m n 



Neutral plane 



&V59V 



n 



FIG. 13. 



ing moment M increases from the ends toward the center of 

 span and with it the intensity of the horizontal stresses) induces 

 a force at every longitudinal layer tending to slide the upper 

 portion past the lower; and this sliding or shearing force, which 

 increases at every layer, attains its maximum intensity at the 

 neutral plane. 



In addition to the longitudinal or horizontal shear at any 

 point, as explained above, there co-exists a vertical shear and 

 the intensity of this vertical shear is equal to the intensity of 

 the horizontal shear. This may be proved as follows: 



Fig. 14 represents an infinitely small portion of the side of 

 a beam at any given point. The sides of the element, as far as 

 any difference in the result is concerned, may each be represented 

 by h, and the breadth of the beam at this point will be denoted 

 by b. Now there are two sets of shearing forces acting upon it, 

 one vertical and the other horizontal; and these shears form two 

 pairs of couples, acting as indicated by the vertical and horizontal 

 arrows. For an infinitesimal distance (h) the horizontal fiber 



