300 PRACTICAL MATHEMATICS 



This gives the value of y for any point distant x feet from the 

 centre, and y is greatest when x = a. 



W 



Wa 3 



24EI 



WZ 3 



since a = - 



~ 192EF ~ 2 



This also gives the maximum value of the deflection. 

 In order to obtain 8, the deflection at any point, 



= Vmax ~ V 



W/z 3 W 



W 



W 



Since the bending moment at any point is (a x) u and 



Wa 

 U= 



WWn 

 . , u/ 



men M. = -(a x) -- 



W 



= (a - 2x) 

 4 



Thus when x = a, M = -Wa ; when x = 0, M = -Wa ; and 



4 4 



when x = -, M = 0. 



B 



Therefore the bending moment increases uniformly from -- Wa 



4 



at the point of fixing to + -Wa at the centre of the beam, and 



4 



at a point half-way it vanishes. 



159. (6) A beam of length I feet fixed at both ends, carrying 

 a uniformly distributed load of w tons per foot run. 



The effect of keeping each end horizontal is the same as applying 

 at each end a couple of magnitude u, which acts in a clockwise 

 direction. Also, since at the ends and at the centre the direction 



Qftl 



of the beam is horizontal, at these points -p = 0. 



The vertical reaction at each point of fixing is wa tons, where a 

 is the length of the half span. 



