14 



Then the equation to be solved, in order to find the 



deflection, is 



d p 



d x 



El 



(l + p 2 ) 2 



where p = 



W / I 

 2 \ 2 



dy 



— x 



d x 



In the usual text-book solutions the denominator on the 

 left-hand side is taken as equivalent to unity, but an exact 

 solution may be obtained as follows : — 



Integrating the above equation as it stands, we have 



E I p W I W x 2 



= x (since x = o when p=o) 



V (l + ffi 2 2 4 



4^7 

 Put a 2 = 



F 



W 

 (lx-x 2 ) 2 



from which we get 



l + p 2 



fl 

 \2 



(H" 



V = 



v' 



I 2 



4 



I 



)T 



— x 



J ) -J 



I 



Now put — 



2 



I 



X = — cos <£ 

 2 



I 



.*. d x = — sin <f> (I <f> 

 2 

 .*. deflection at centre = y 



I 



2 — sin 2 <f> — sin <£ d <jf> 

 4. 9 



J 



( p ■ ' 



V I & 4 sin 4 <f> 



\ 16 



