SECTION VI 



CONTINUOUS BEAMS 



42. Theorem of three moments for uniform loads. A continuous 

 beam, or girder, is one which is supported at several points of its 

 length. The reactions and moments in this case are statically 

 indeterminate ; that is to say, the ordinary static conditions 

 of equilibrium, V.F=0, Vlf = 0, are insufficient to determine 

 them. To solve the problem it is necessary also to take into 

 account the deflections of the beam. 



FIG. 70 



The simplest method of finding the reactions and moments 

 at the supports for a continuous beam is by applying what is 

 known as the theorem of three moments. This theorem establishes 

 a relation between the moments at three consecutive supports 

 of a continuous beam and the loads on the two included spans, 

 and was first published by Clapeyron in 1857. The following 

 proof of the theorem, however, is very much simpler than any 

 previously given, 



70 



