CONTINUOUS BEAMS 



73 



The distance of the load in any span from the adjoining support on 

 the left will be denoted by kl, where I is the length of the span and 

 & is a proper fraction ; that is, kl is some fractional part of the span 

 (Fig. 72). Thus, if the load is at the middle of the span, k = \ r ; if 

 it is at the quarter point, k = ^, etc. 



Now consider a portion of the beam extending over three con- 

 secutive supports A, B, and (7, and let M^ M^ M g denote the 

 moments, and JB 1 , R^ E 8 the reactions, at these supports. Then, to 

 obtain the theorem of three moments, calculate the deflections of 

 A and C measured from the tangent to the elastic curve at B. To 

 calculate the deflection of A, suppose the beam to be cut by a plane 

 just inside the sup- 

 port at A, and call 

 the shear on the 

 section S*. Then, 

 considering the end 

 B as fixed, calculate 

 the deflection of A 

 by treating the part AB as a cantilever subjected to the moment M^ 

 the shear S* regarded as a load, and the concentrated load JJ. Call- 

 ing these three partial deflections d^ d^ c? 3 , respectively, we have 



FIG. 72 



a - 

 *- ~ 



d = - 



+ 



El 2 El/ 



(Eq. (50), Art. 39) 

 (Eq. (38), Art. 37) 

 (Eq. (44), Art. 37) 



In the present notation the quantities a and b in the expression for 



d s are 



a = distance from fixed end = ^ kjL$ 



b = distance from free end = k^. 

 Substituting these values of a and 5, the equation for d B becomes 



