80 



whence 



STRENGTH OF MATERIALS 



The equations of both branches of the elastic curve are now determined except 

 that the reaction E s is still unknown. Since B is assumed to be on th- 

 with 0, its ordinate is zero. Therefore, to determine i2 8 , put x = 

 equation (41) ; whence R _ 5 p 



From symmetry BI = E s . Therefore 



Problem 87. Determine the reactions of the supports for a beam simply 

 supported at its center and ends, and bearing a uniform load 

 of length. 



Solution. If the end supports were removed, the beam wot 

 cantilevers, AB and BC (Fig. 64), each of length I and bearing a uniform load. 





Ki... 



From Problem 66, the deflection at the end of such a beam is D = - Hut the 



s /-. / 



reaction E s (or #1) must be of such amount as to counteract thia deflect inn 

 from Problem 76, the deflection at the end of a cantilever bearing a single c<>: 



trated load R 3 is D = 



whence 



3 El 



Therefore 



3EI 



toft 



From symmetry, EI = Eg. Consequently. 



E 2 = 2wl - (Ri + J? 8 ) = f wi. 



Having found the reactions of the supports, the equations of the elastic curves can 



be determined as in the preceding problems. 



71. Theorem of three moments. The theorem of three nnnn 

 is an algebraic relation between the bending moments at three con- 

 secutive piers of a continuous beam. The theorem is due to Clapeyron, 



