42 RESISTANCE OF MATERIALS 



where the summation includes only the loads on the left of the 

 section. For an adjacent section distant Ax from run, that is, at a 

 distance x + bx from the origin, the moment is 



M' = R l (x + As) - ^ P(x + Az - d). 



Let A Jlf denote the difference between these two moments. Then 

 M' - M= Rx - 



.-- 



But, by definition, the shear S at the given section mn is 



S = R^P. 

 Consequently, 



-* 



This relation also holds for a beam uniformly loaded. Thus, if 

 w denotes the uniform load per foot of length, and I is the span in 

 feet, the moment in this case at any section distant x from the left 



support is wl wa? 



M = -x- , 



and at a section distant A# from this it is 



Therefore the change in the moment is now 



= M' - M= A - wx . Az - 



If Ax is assumed to be small, its square may be neglected in com- 

 parison with the other terms. In this case, dropping the last term, 

 we have 



wl 



' 



Evidently the same relation holds for any combination of uniform 

 and concentrated loads. The general fundamental relation between 

 the shear and moment diagrams is therefore 



