BENDING-MOMENT AND SHEAR DIAGRAMS 43 



Since - represents the rate at which the ordinate to the moment 



A# 



diagram is changing, this relation may be expressed in words by 

 saying that 



The rate of change of the moment is equal to the shear. 



From this result important properties of the two diagrams may 

 be deduced, as explained in the next paragraph. 



30. Properties of shear and moment diagrams. Consider the 

 highest point of any given moment diagram for instance, of 

 those shown in Figs. 39-43. 



Since the moment increases up to this point and decreases after 

 it passes it, the change in the moment &.M, corresponding to an 

 increase A# in the abscissa, must be positive on one side of the 

 point and negative on the other. Since Aa; is positive in both cases, 



the ratio - - changes sign in passing the point. But since - = S, 



\.r> L\X 



this means that the shear changes from positive to negative in pass- 

 ing the given point, and therefore must pass through zero at the 

 point in question. 



The same reasoning evidently holds for the lowest point of 

 the moment diagram. Therefore, at the section where the moment 

 is greatest or least the shear is either zero or passes through zero 

 in passing the point. 



I By referring to the diagrams in the preceding article or in 

 Table XIII it will be observed that this is true in each case. 

 If the moment is constant, then AJf == and consequently =0. 

 That is to say, where the moment is constant the shear is zero. 



For a system of concentrated loads the equations for moment 

 and shear, as shown in article 29, are 



The first of these represents an inclined straight line, and the 

 second a horizontal straight line. Therefore, for concentrated loads 

 the moment diagram is a broken line and the shear diagram is a series 

 of horizontal lines or steps. 



