118 SUPPLEMENT TO CHAP. TO. [CHAP. 



For breadth constant, h = hi-r. 

 For similar cross-sections, h = hi 



The first is in plan a parabola ; the second, in elevation a triangle ; the 

 third, a paraboloid of revolution. 



For the change of shape, we have, by proceeding in the same manner as in 

 Art 12, A = 2 Ao, A = 4 Ao, and A = 3 A ia the three cases, where A is the 

 deflection of a similar beam of constant cross-section 61 h\. 



14. Beam supported at both the ends Constant cross* 

 section Concentrated load. Let the weight P be distant from the 

 left end by a distance Zi and from the right end by Z 2 . Let the distance of 

 any point from the left end be x. For the upward reaction at the left end, 



Z a 

 Vi x Z = P Z or V t = P . 



The moment, then, at any point between the left end and F, for * less 

 than Z,, isM - _ . For any point to the right of P, or x greater 



than Z,, M' = ~ + P [a? Zi I. Instead of this, however, we may 



take the reaction at the other end, V, = P -/ ; and then for * greater than Z t , 



i 



The moment is evidently greatest at the point of application of the load, 



P 1 I 

 or for x = l t Hence the maximum moment is v -. 



(a) Breaking weight. 



2 T I F I I 



From (11) M = -_ = ' , or, for the breaking weight, P = 



ll i 



2 T I I 1 Till- I 



ii I For rectangular cross-section, I = TO & A* and P o Z Z 



1 1 



For a load in the 1 middle, Zi = I, = I and Max. M = -r P I, and P = 



a 4 



8TI 

 . , , or 4 times as great as for a beam of same length fixed at one end 



and free at the other. 

 (J) Change of shape. 

 We have, then, from (8), for x less than Zi, 



d">y P I, x , d* y' P Z! (Z-) 



~ and = - 



Integrating, we have 



dy_ PZig* 



" ~ 



For x = Zi, these two values of --sac equal, and hence, since Z, = I Zi, we 



a x 



