SUPPLEMENT TO CHAP. VII. [CHAP. ITL 



For the breaking weight, then, from (8) 



18 



or 4 times as much as for a beam of same length loaded uniformly and 

 fixed at one end. 

 For the change of shape, we have 



<Z* y _ p x (lx) 

 ~d~x* ~ 2EI 

 The constants of integration are determined by the conditions that, for 



z = , --=0; * = 0, y = ; and x = I, y = 0. Integrating, then, twice 



& Ci X 



under these conditions, we have 



This is greatest at the centre, or for x = ; hence the maximum deflection is 



8 



A = - ^ , or only isths of a beam of the same length fixed at one 



end and uniformly loaded. 



16. Beam supported at one end and fixed at tbe oilier 

 Constant cros-section Concentrated load. Let the left 

 end be fixed horizontally so that the tangent to the deflected curve at that 



d *u 

 point is always horizontal, and therefore = 0. 



d 3! 



Let the distance of tbe weight P from left be a, and the distance of any 

 point x. 

 Then, for x less than a, we have 



M = V (lx) + P (ax) ; 

 for x greater than a, 



M = V (l-ai), 

 where V is the reaction at the free end, and is so far unknown. 



If we put M = -^ and M' = -^-, and integrate as usual, and remem- 

 d x' 1 a x* 



ber that f or x = 0, ~ = 0, and for x = a, -^ = -^ we have 

 d x d x d * 



J2 = -,-'-, 



Integrating again and determining the constants by the conditions that, 

 for x = 0, y = 0, and for x = a, y = y", we have 



y' = ^ [V * (3 Z-aO-F (3 *-a) a*]. 



