298 PRACTICAL MATHEMATICS 



But at O, where x = 0, y = 0. Then Const = 

 and 



2EIV2 12 



w 



24EP 



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

 centre, and y is greatest when x = a. 



24EI 



5re>Z 4 I 



= -=^ } since a = - 



5WZ 3 

 " 384EI 



where W = wl tons, the total load on the beam. 



This also gives the maximum value of the deflection. 

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



" = I/max y 



W 



158. (5) A beam of length Z ft, fixed at both ends, carrying a 

 concentrated load W tons at the middle. 



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 



dll 



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



W 



The vertical reaction at each point of fixing is tons. 



M 



Let a = the half span, and let the centre of the beam be taken 

 as the origin. 



Let A be a section situated at a distance x ft. from O. 



W 



Bending moment at A = (a x} u ft. tons 



2 



Hence El -^ = (a x} u 



CvCC *j 



_,_ dii W / 1 ox 



and El -- = (ax -x 2 } ux + Const 



dx 2 \ 2 



