CHAP, m.] SUPPLEMENT TO CHAP. VH. 117 



For similar cross-sections, we have 



9 r 6 as 8 I /> ~ 9 36 P P 



If we call the floZwme of the beam of constant cross-section V, then in 



1 3 



the first case the volume Vj = - V ; in the second, V a = V ; in the third, 



& O 



V,=4v ; or 



V : V 9 : V, : V, = 30 : 20 : 18 : 15. 

 The maximum deflections, as we see above, are as 



9 3 



2 Ao, = Ao, Ao, or as 20, 18, and 15. 



5 a 



That is, the deflections at the ends for a beam of uniform strength in the 

 three cases are as the volumes. 



13. Beam as before fixed at one end Uniform load 

 Constant cross-section. If p is the load per unit of length, we have 

 for the moment at any point distant x from the free end, 



x p x* p x 2 d* y 



M=pxx - =: , a nd hence ^^y = J^T> 



pi* 

 This moment is greatest for x = I, and hence Max. M = r . 



For the ~breaking weight, then, from (11) t 



pi* STI 4TI 



__ = __ or pl=^j-, 



or twice as great as for an equal weight at the end. 

 For the change of shape, we integrate twice, precisely as before, the ex- 



d 2 y p x* 

 pression -= $ = _ _, and obtain thus 



(t X a ii * 



The maximum deflection, then, is 



tf) IT 9 



A = 8~EI' 



or only fths as great as for an equal load at the end. 

 2. Constant strength. 



"We have, as before, from (11) M = ^- = ^ , whence T = - ; or 



a n 41 



for rectangular cross-section, I = I h 3 and T = '~^. If Jj Ai are the 



12 o h* 



breadth and heighth of the fixed end section, then, since T must be always 

 constant, 



8 p I* I A* x* 



For height constant, & = 5i | j ) 



