160 



EQUATIONS FOR CONTINUOUS BEAMS. Art. 97. 



decrease uniformly, so that if it were raised half as much as 

 necessary to make the reactions zero, that is, if it were raised a 





distance - the end reactions would be j 3 6 wl\ if it were lowered 



a distance ' *f the center reaction would 1 e | wt. 



97. General Equations for Continuous Beams. 1 It was 



shown in Art. 95 that the stresses in a span of a continuous beam 

 become statically determinate when the moments at the supports 

 are known; if, therefore, an equation can be dedueed involving 

 the moments at three successive supports, a continuous beam of n 

 spans having w+1 supports and n 1 unknown moments at sup- 

 ports (the moments at the end supports being zero), will furnish 

 n 1 equations from which to determine the n 1 unknowns. 



Such an equation (for uniform loads) was published by 

 Clapeyron in 1857 and, as in the special cases above, is based on 

 the equation of the elastic line. The deduction of this equation 

 for uniform load follows. 



Fig. 110 shows two successive spans of a continuous beam 

 of an indefinite number of spans; J/ q , J/ r , and 3/ s are the mo- 

 ments at the supports ; 8.1 is the shear just to the right of q and 

 $ s is the shear just to the left of s. The moments at the supports 

 may be assumed positive. 

 For span qr 



- 



- Ely = 



When x =- Z r , V -- and (/) becomes 

 = I MJ1 + IS - -frwA + dl r and 



from (a) and (39) 

 + ( C 2 = 0) 



(C) 



(<f) 



(e) 

 (/) 



(g) 



At r, from (g) and (e), when x = l r 



J For a complete trcatniont of tlio rontiimous bojnii with variable 

 moment of inertia, see Howe's The Theory of the Continuous Girder. 

 For a pr:rpbirjil treatment of the continnoius beam see Eddy's Researches 

 in Graphical Statics. 



