ART. 3.] SUPPLEMENT tO CHAP. XHT. 245 





x= lr, y becomes AT+I. If also we put =- = Jc, or a = Ic l^, and insert also 



^r 



for Sj. its value as given in (2 a), we find for ^ 



7 a f T 



( r 0X3 1 



We see, then, that the equation of the curve is completely determined, 

 when we know M r and M r+1 , the moments at the supports. These, as we 

 shall see in the next Art., are readily found by the remarkable "theorem 

 of three moments," already alluded to in Art. 144. 



3. Theorem of Three Moments. 



"' 



In the Fig. we have represented a portion of a continuous girder, the 

 spans being li li . . . If, etc., and the supports 1,2 ... r, etc. Upon the 

 spans ZT_I and IT are the loads P r _i and Pr, whose distances from the near- 

 est left-hand supports are Jc lj._i and Ic 1 T ; k being any fraction expressing 

 the ratio of the distance to the length of span. 



The equation of the elastic line between P r and the r + 1 th support is 

 given by (4), and the tangent of the angle which the curve makes with the 

 axis of abscissas is given by (3 a). If in (8 a) we substitute for Sr its 

 value from (2 a), and for tr its value from (5), and make at the same time 



cL 'u 

 x = ZT, then - becomes Zr+i, the tangent at r + 1, and we have 



a x 



<r+l = ^ +1 ~ **" + ^ [M r ^ + 2 M r+1 Z, - P r 



Remove now the origin from o to n, and we may derive an expression 

 for tr by simply diminishing each of the indices above by unity ; therefore 



t, = **-**-* + 



TI Xj 



Now, comparing these two equations, we may eliminate the tangents, 

 and thus obtain 



M r _! Jr.! + 2 M r (?,._! + Z r ) 



_ 6 B I 



I? 



which is the most general form of the theorem of three moments for a 

 girder of constant cross-section. 



When the ends of the girder are merely supported, the end moments are, 

 of course, zero. Then, for each of the piers, we may write an equation of 

 the above form, and thus have as many equations as there are unknown 

 moments. 



