Buckling of Deep Beams. 203 



Now the value of P which makes b infinite will make t 

 infinite except at the fixed end. That is, the beam is 

 unstable for that value of P which makes the coefficient of b 

 zero in equation (93). The condition for instability is, 

 therefore, 



.gfr/i— 1_ + * 2 i 



K«V 4.5^4.5.8.9 / 



+ { 1 -ri- + Tror 8 -l" '-' (94) 



It should be observed that this equation is independent of 

 p, which shows that the stability is not affected by putting 

 the load a litttle to one side of the centre. The only result 

 of displacing the load laterally is to put a torsion on the 

 beam, thus giving a new equilibrium state of the beam from 

 which instability begins. 



Let us write„ for shortness, 



/w=i-A-» 



4.5 4.5.8.9 



F« = 1-^t+, 



3.4 ' 3.4.7.8 

 Then our equation for the load is 



-g/M + F(*)=0 (95) 



An approximate solution of this is the solution of the 

 equation 



F(«)=0, (96) 



since the other term is small because q is small. 



Let 8 X denote the solution of (96), and let P x be the 

 corresponding value of P. P 2 is, of course, the value of P 

 found in Case 3. In the small term in equation (95) we 

 may use the approximate value P x for P. Then writing 8 



for x?-^> and (s^-j-z) for s, equation (95) becomes 



-Sfl*i + «) + £(* + *)=*<>• 



Since z is small, 



F (s, + z) = F(sj ) -f zF'(si) approximately 



Then taking account of the first powers of 8 and z only 

 we get 



-S/W+*Fl«i)«0. 



