,638 PROCEEDINGS OF SECTIO^- H. 



We can simplify this equation, for solution, graphically, by 

 writing- Ii = ?il, and solving the resulting equation for rt, viz : — 



n 



+ 6n- — Gn +1=0. 



A few values of this expression are calculated for different 

 values of n, and the results plotted in Fig. 18. There, the abscissae 

 represent values of li and ordinates values of iv^ + Qn^ - Qn x I. 

 The value of h, for which this expression vanishes, is about 0'2143Z. 

 For any value of h, greater than this, 11^ and K3 will act upwards. 

 F'or any lesser value, they will act downwards. 



If, then I, >0-2148/, R, = li, = ~ - K, 



wl 

 and if I, <0-2143Z, lij =R.,' = R.,- -, • 



We can, next, readily lind the bending moment at any point of 

 the beam. Calculating a few values for the bending moments, corre- 

 sponding to certain values of Z^, we obtain two of the curves, shown 

 on Fig 19, where abscissae represent values of li, and ordinates 

 represent bending moments. One of the said curves shows the bend- 

 ing moments at mid-span ; the other shows them at the posts. It 

 will be observed that the bending moments at the posts exceed those 

 at mid-span for all values of li, and that the minimum value of the 

 bending moment at a post corresponds to the value 0'357Z for ^i. 



We shall Jiext investigate the bending moments in the end bays, 

 corresponding to different values of li, and, with this information, 

 added to the foregoing, we shall decide as to the most favourable 

 positions for the posts, having regard to all the bending moments 

 endured by the beam. 



Bending Moments in- End Bays. — We have seen that, when the 



posts are further from the ends than 0'2143 /, the forces Rj and R3, 



act upwards. For any given value of /i, then, greater than 0'2143/. 



the bending moment in the end bay will be a maximum at some 



pai-ticular point in that bay. The expression for the bending moment 



in the end bay, at a distance, x, from the end, is — 



ivx- 

 ±t, X — -— = M, say. 



To find for wliat value of x M is a maximum, we write — 



= iv, — wx = O 



dx 



and notice that —- — is negative ; therefore M is a maximum when 

 dx'- 

 T> Ti - ' 



X =. ~ , and the value of M is then — ' 

 w 2 IV 



Calculating a few values of this bending moment for certain 

 values of l-^, and plotting them, we obtain the curve marked "Max. 

 B.Ms, in end bay" in Fig. 19. We observe that this curve crosses 

 both of the curves previously plotted. The most important point on 

 this diagram for our purpose is that at 0'357/, where the curve of 

 maximum bending moments in end bay cuts the curve of bending 

 moments at top of post. We see that, when the posts are placed 

 0'357 I from the ends of the beam, the bending moment over a post 



