62 KESISTANCE OF MATERIALS 



Now for any small arc, Az, the deflection Adf at any point at a dis- 

 tance x, measured from the tangent to the arc at the initial point 

 (Fig. 62), is 



Hence the total deflection for any finite portion of the arc AB, 

 measured from one end A to the tangent at the other end B, is 



d = Ao! =^x^ = JL ^(.IfAz)*. 



But JfA# denotes the area of a small vertical strip of the moment 

 diagram of altitude M and base A#, and ^\(M&x)x is the sum of 

 the static moments of all these elements of area with respect to the 

 point A. From the results of Section II, however, this is equal to 

 the area of the moment diagram between A and B multiplied by 

 the distance of its centroid from A. That is, if A ab denotes the area 

 of the moment diagram between the points A and 2?, and X Q is the 

 distance of the centroid of A nh from A, then 



Therefore d = A ab X Q , 



or, in general, 



(36) d = (static moment of the moment diagram). 

 El 



The angular deflection <j> between any two points A and B, that 

 is, the angle between the tangents to the elastic curve at these two 

 points, is given by 



Therefore, since ^Mkx denotes the area A^ of the moment diagram 

 between the two points in question, and since for the small deflec- 

 tions which actually occur in practice we may assume </> = tan </> 

 without introducing any appreciable error, 



(37) 



