FLEXUKE OF BEAMS 



95 



Therefore 



By Castigliano's theorem, = 0. Therefore 

 5P 2 



EI _ 8 3 

 whence 



P! = |^. 



Substituting in this expression the value of PI in terms of P 2 , 



Problem 99. A uniformly loaded beam extends over three openings of equal 

 span. Find the reactions at 

 the supports. 



Solution. Let I denote the 

 length of each span and w the 

 unit load (Fig. 76). 



From symmetry, PI = P 4 

 and P a = P 8 . Also, by taking 

 moments about B, 



fr > 



FIG. 76 

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



and therefore, as in the preceding problem, 



w - - _ * 



~ 3 4 



PI = P 4 , IT has the same value for the third opening, that is, W 9 = 

 the second opening 



M = P\x + P 2 (x I) 



and therefore 



In 



8 3 ' 20 ' 3 4 12 J 



Hence the total work of deformation for all three openings is 



8 



Tlicrefore 



i _ 



l 



4 p f . a 





12 



