94 



STEEXGTH OF MATERIALS 



79. Application of Castigliano's theorem to continuous beams. 



Castigliano's theorem affords still another means of determining tlu* 

 unknown reactions of a continuous beam; for the reactions may be 

 included among the loads on the beam, and since the points of applica- 

 tion of these reactions are assumed to be fixed, their deflections are zero. 

 Therefore, if P h is one of the reactions, D k = 0, and consequently 



A condition equation of this kind can be found for each reaction, and 

 from the system of simultaneous equations so obtained tlu* unknown 

 reactions may be calculated. The following problems illustrate the 



application of tin- ilu-<>- 



rem. 



5 Problem 98. A uniformly 



loaded liram of ItMiirth . 



supported at its center and 

 r) j-n.ls. Kind the reactions of 



tin- supports by means of Cas- 



ti-liano's theorem. 



FIG. 75 



Solution. Let w denote the 

 unit load on the beam (Fig. 75). 

 From symmetry, P! = P 8 . Also, by taking moments about B, 



For a point in the first opening at a distance x from the left support, 



consequently, 



w= \ r*M*dx = I rpp _ p lW j4 tfin 

 2 Jo El 2 Ell 3 ' 4 h ~2o" 



The work of deformation for the other half of the beam is of the same amount. 



Therefore the total work of deformation is 



= Ei[~T ~ J T~ + ~26"J' 



Since Pi is a function of P 2 , the partial derivative of FT with respect to P, is 

 E- J_pPi* 8 cP l wl* , 



aPi~.BiL~T~'api~~T'; 



Since P l = wl-?*,? p -* = -l 

 2 dP 2 2* 



