Buckling of Beep Beams. 215 



Consequently 



or W/ 2 P/ 2 ■ 



I^86 + e9 = ^ E " CK < 156 > 



Now we see that, whether W is small compared with P, 

 or P small compared with W, the result expressed by (140) 

 is nearly true. Then it is sure to be nearly true for all 

 other positive values of the ratio W : P. The result may be 

 expressed roughly in the following form : — A load on the 

 free end of the clamped-free beam has approximately the 

 same effect in buckling as three times that load distributed 

 uniformly along the beam. 



There is one assumption in the working of the last case 

 that should not be passed over without justifying it. It is 

 the assumption, made in the squaring of G, that P is small 

 compared with wx. This, of course, is quite true everywhere 

 except where x is small. But if we consider that the actual 

 term neglected, namely P 2 ^ 2 , is itself small in the region 

 where there is any possibility of error, and that the assump- 

 tion is wrong only over a very small range of values of ar, 

 it is clear that the error made is negligible.. 



There is still another point of view that will show the 

 justification for this assumption. The actual method of 

 solution consists in dropping a term from G 2 , and if we had 

 added a term of the same order we could have got 



Then, by changing the variable to (x-\ ), our differential 



equation would have reduced to the same form as in Case 6 

 where P was zero. It is easy to show by this method that 



the solution of Case 6, with lx+ — J for x and (l+— \ 



for I, applies correctly to the present case provided powers 



WlP 



of beyond the first are neglected. But this is precisely 



the solution we have obtained by taking a different value 

 of G 2 . Then it follows that the term neglected in G 2 does 

 not affect the result to our degree of approximation. 



Case 14. — Beam carrying a small uniformly distributed 

 load W and a much larger concentrated load P at the middle, 



