Buckling of Deep Beams. 303 



If we make the substitutions 



s = \m?x 2 

 and then , 



T = ZS*, 



the resulting differential equation is 



dh 

 ds 2 



1 dz /, 1 \ 



+ sl s + ( 1 -ie?y=°' ■ ■ • (») 



which is the equation for Bessel Functions of order £. 

 The solution is 



z = aJ i (s)+bJ_ i (s) (21) 



It is quite easy to solve (19) at once by a series of powers 

 of x. The solution in series form, obtained either by using 

 (21) or directly from (19) is 



f m 4 .v 5 mV 1 



, . f ., m 4 .r 4 m s x s i 



At the free end, where # = 0, there is no twisting couple 

 and therefore 



-j- = when # = 0. 



This makes a = 0. 



Another condition is that t = at the fix:ed end where 

 x=l. This means that m is given by the equation 



1 "U f 3.i,7.8 "*°' a01 ' * ' (23) 



An approximate solution of this equation can be got by 

 dropping all the terms except the first three. The result 

 can then be improved by using the approximate result in 

 the remaining terms. This method is not very laborious. 

 Or we can get the solution from a formula for the zeros 

 of Bessel functions, for this last equation amounts to the 

 same as 



J.l(4m»P) = (24) 



The solution of this equation is approximately 



im 2 Z 2 =2-006, 



whence PZ 2 = 4-012 y/EnUK (25) 



