220 CONTINUOUS QIBDEE. [CHAP. XHL 



144. Concentrated Load In any Span Moments at Sup- 

 ports. It only remains to consider a concentrated load at any 

 point. If the formulae for this case do not prove to be too com- 

 plex or intricate for practice, we may consider the case, so far 

 as equal spans are concerned, as fully solved. 



We have seen that the "theorem of three moments," so far 

 as uniform loads are concerned, enables us to solve the case 

 thoroughly. It is more especially as regards concentrated or 

 partial loads that the opinion widely prevails as to the impossi- 

 bility of obtaining practically useful formulae ; and this, not- 

 withstanding that it has been shown by Sresse, WinMer, Wey- 

 rauch, and many others, that the theorem of three momenta 

 can be extended to include concentrated loads also. 



The Theorem as thus extended is as follows: 



k-i + 2 M m 



where, by our notation (Fig. 89, Art. 130), a' and a are the dis- 

 tances of P m _i, P m , from the nearest left supports."* 



By the aid of this theorem, we are able to deduce the follow- 

 ing formulae : 



For moments left of /*, and including support r, that is 



when m <r + ~L, M m = c^ 8 ~ r+2 8 ~ r+1 . 



For moments right of r + 1, including support r + 1, or 



A c r + A' c r+l 

 when m > r, Mm = <? B _ m + 2 



In these formulae, c represents, as above, the Clapeyronian 

 number, and A A' stand for the following expressions: 



k being the fraction y, or the ratio of the distance of the weight 



9 



P from the left support, to the length of span. 



145. Illustration of Application of above Formulae. 



These formulae are by no means difficult of application. Let 



* For demonstration of this Theorem, Bee Supplement to this chapter. 



