76 RESISTANCE OF MATERIALS 



adding these equations the general theorem of three moments for 

 any number of concentrated loads is found to be 



(78) 



44. Effect of unequal settlement of supports. In deriving the 

 theorem of three moments the supports were assumed to be at a fixed 

 elevation in the same line. If their relative elevation changes, owing 

 to unequal settlement of the supports or to other causes, the effect 

 in general is to increase the stress in the member. To take account 

 of this effect in applying the theorem, suppose that the supports 

 were originally in line, and denote the settlement of three consecu- 

 tive supports A, B, C from their original level by h^ A 2 , A 3 , respec- 

 tively. Then the difference in elevation between A and B is 7^ A 2 , 

 and between B and C is \ \- Thus, if A settles more than B, 

 ^ A 2 is positive and the deflection at A is increased by this 

 amount. If A settles less than B, ^ h 2 is negative and the 

 deflection at A is decreased by this amount, etc. In general, then, 

 equations (70) and (74) for the deflections at A and C become 



Substituting these values of D A and D c in the relation 



5d= -Es, 



' '. 



the result, after combining terms and changing the signs of 

 ^ M*> is 



(79) M^ + 2 M,(l, + 1 2 ) + M 3 l z 



