FLEXURE OF BEAMS 



81 



and first appeared in the Comptes Rendus for December, 1857. The 

 following is a simplified proof of the theorem for the case of 

 uniform loading. 



Let A y B, C be three consecutive piers of a continuous beam at 

 the same height, and let M a , M b , M c and R^ R b , R c denote the bend- 

 ing moments and reactions at these three points respectively (Fig. 65). 





Also, K't /j and /, denote the lengths of the two spans considered, 

 and w v w t the unit loads on them. Then, taking A as origin, the 

 differential equation of AB is 



(42) 

 Integrating twice, 



and 





. ~ 



At A, x = and y = ; hence C, = 0. At B, x = /, and y = ; 

 hence 



2 6 24 



In equation (42), if x = /,, A'/'-'' = M b . Therefore 



If (-) denotes the slope of the elastic curve AB at B, then, from 

 \dx/ b 



equation (43), 



