Loaded Shaft supported in Three Bearings. 645 



Then 

 6C, M+ i _ 6C, 



6G ?n+ i _ 6G- 

 / 3 ~ Z 



= ^ A*» + 3(/z n , ~ 1W [Y m (2 - s m ) - X^t.-s^ +2^1 



v 6C, r Y 9 Mn 





(>H r _ 6G, _ , f v ± M 

 7 3 / 3 



6K w+ i 6K ?l 



^^ + 8(^-l) 5?l Yi2-6- n ), 



and similar expressions for the second span. 



Also ? 1 = rV; ^1=0; G^=0. 



J l J-l 



Further 2Y+-^+'^=0, 



- 1 + p ~r ~[>T~ — u > 

 and Y/ + Y'Z' - X^ + M x = 0. 



The above equations give C, C, G, G', K, K', H, H', 

 Y, Y' in terms of X x , M t corresponding to the portions of 

 the shaft of. different diameters. Then the displacement and 



slope equations written above give u and -=- at the position 



t in terms of X 1; Mj. Likewise for r and — in the second 

 span, due to Xi, Mi on the first span. ( z , , 



t i-i i i ■ • e du do 



In like manner we obtain expressions tor w, -7-, v, — - r 



as </: ' 



corresponding to other loads X and J\I on the spans. 



The total deflexion due to all the loads acting simultaneously 

 is obtained by adding the separate deflexions for each loud 

 regarded as acting separately, so that we obtain the value 



of u, r, -7^-, -=-, by the addition of the deflexions and slopes 



given by above equations. Thus we obtain equations for the 



