of Continuous Girders. 181 



gent at q, and we have 



6EI/' = M/+2M7-P/ 2 (&-& 3 ). . . . (III.) 



If we consider the origin at q we can also find another expression 

 for /', depending on M', M", and P' ; this is of course analogous 

 to the value of t written above, or 



6EI*'= -2Wl'-W'l ! + T't 2 (2k-M* + k 3 ), . (IIP.) 



a! 

 where k denotes y, and is not necessarily the same in the two 



expressions. Comparing then the two values of t l , we get the 

 Theorem of Three Moments for concentrated loads, or 



m+2W(l+l') + M"l t =¥P{k-k 3 )+V'l' 2 {2k-3k* + }c i ). (IV.) 



If there be many loads, we have only to prefix to the terms in- 

 volving P and P' the sign of summation §, For uniformly dis- 

 tributed loads w and «/per unit of length, we place £P = J wd(kl) 

 and 2P ; = KiddiJeV) and integrate between the required limits. 

 If the loads extend over the whole span, the first integral is taken 

 between M=0 and M—l t the second between 7cl' = Q and kl' = l'. 

 Then 



Ml+2M(l+l , ) + Wl'=iwl 3 + ±w l l' 3 , . . (V.) 



which is the theorem as first deduced by Clapeyron*. 



With the above formulae as a basis, I propose to develop two 

 expressions by which the moment at any support can be directly 

 and easily determined without the application of the theorem of 

 three moments, and which are in so simple a form that their use 

 is far preferable to the tedious solution of the equations arising 

 in the process as ordinarily followed. 



Remembering that all girders to be investigated are subject 

 to the four conditions mentioned above, they may be regarded 

 for the purposes of this article as forming three classes. The 

 first class includes all girders whose two ends rest free upon the 

 abutments, the second where one end is resting and the other 

 walled in or fastened horizontally, and the third where both ends 

 are fixed horizontally. 



Case I. Ends resting free upon abutments. 

 I L l r Is 



AAA A fl-^1 A A A 



1 2 3 r * p r-H s s + i 



Let the girder consist of any number of unequal spans, the 



* Comptes Rendus, 1857. The extension to concentrated loads was 

 made by Bresse and Winkler independently about 1863. Vide La Meca- 

 nique Appliquee and Die Elaslicitats-Lehre. 



