182 GRAPHIC AND ANALYTIC [CHAP. XII. 



(2 -f n) & - (2 -f 3 n + 3 n 2 ) k 



These reactions, it will be observed, when added together 

 R A + 2 R D + R c are equal to P, as should be the case. 



By the application of these formulae, which are for any par- 

 ticular case by no means intricate, we can find the reactions at 

 A and C as also at D or D ; and then starting, say, from A, can 

 follow the reaction there through the frame by the method of 

 Art. 114. A negative reaction indicates that the support tends 

 to rise, and unless more than counterbalanced by the positive 

 reaction due to uniform load, the end where this negative reac- 

 tion occurs must be latched down. 



121. Support in Pivot pan are not on a level Reac- 

 tions for live load, however, are the same as for level sup- 

 ports. The three supports of a pivot span should not be on a 

 level. It is evident that if this were the case, the first time the 

 draw is opened the two cantilevers deflect and it would be diffi- 

 cult to shut it again. The centre support should therefore be 

 raised until the reactions at the end supports are zero, that is, 

 until thejjust bear. The centre support is then raised by an 

 amount equal to the deflection of the beam when open, due to 

 the dead load. Even when shut, then, there are no reactions at 

 the end supports except when the moving load comes on. Now 

 this being the condition of things, it may seem strange to assert 

 that these reactions are precisely the same as for three level sup- 

 ports, and yet such is the fact. If the beam originally straight 

 were held down at the lower ends by negative reactions, then 

 the reactions would have to be investigated for supports out of 

 level, and a load would diminish these negative reactions, or 

 might even cause them to become positive. But such is not 

 the state of things. The end reactions are in the beginning 

 zero, and any load gives, therefore, at once positive reaction at 

 its end support. This positive reaction is just wliat it looula, 

 be for the same beam over three level supports. 



An analytical discussion of the case would be out of place 

 here, but assuming the expression to which such a discussion 

 would lead us, we may show that this is so. 



Thus, for a beam over three supports A, B, and C, not on a 

 level, GI being the distance of A below B, and <% the distance 

 of C below B, the modulus of elasticity being E and the mo- 



