108 SHEARING STRESSES IN A SOLID BEAM. Art. 75. 



The same result is obtained by differentiating the value of 

 MX above and substituting $ x for its value. 



The importance of this simple relation is apparent when it 

 is remembered that the moment is constant, is a maximum, or 



is a minimum when - = 0. It follows that, 



(J JC . 



1. Where the shear is zero or passes through zero, the mo- 

 ment is constant or is a maximum or a minimum. 



2. Where the shear is uniform, the increase of the moment 

 is uniform. 



3. That the area of the shear diagram to the left of any 

 section is equal to the moment at the section since 



CdM= Csdx = M. 



The shear and moment diagrams of Figs. 94 to 112 should 

 be carefully compared to understand these relations. It should 

 be remembered that the moment is not a maximum where it 

 passes from a positive to a negative value, and that the shear be- 

 comes zero or passes through zero at a support. 



That the moment is dependent upon the shear (a resultant) 

 and the distance through which it acts is shown in Fig. 81, where 



75. Shearing Stresses in a Solid Beam. In addition to the 

 bending stresses (normal) at a section, there are, in general, also 

 shearing stresses (tangential) in order that "the sum of the ver- 

 tical components" shall be zero as was shown in Art. 57. The 

 shear (the sum of the vertical components of the external forces) 

 is equal to the shearing stress (the sum of the vertical compon- 

 ents of the internal forces). 



It is usually assumed that shearing stresses are uniformly 

 distributed over the cross section of a beam. This is a reasonable 

 assumption for an I-shaped section and perhaps also in certain 

 other cases as will be pointed out below. 



As was shown in Art. 74, the shear and the bending moment 

 are dependent upon each other: it follows that the shearing 

 stresses and bending stresses are interdependent and that the dis- 

 tribution of the shearing stress depends upon the distribution of 

 the bending stress, that is, upon the theory of flexure. 



In order to fully investigate the stresses at any point of a 



