Buckling oj Deep Beams. 



201 



Without assuming that R is small we see that equation (85; 

 gives a pair of roots with opposite signs. The negative root 

 indicates a tension, and thus we see that the beam could 

 buckle under a tension applied in a line parallel to the 

 unstrained central line but not along it. Since the sum of 

 the roots of the equation is negative, it follows that the 

 tension that would cause buckling is greater than the thrust 

 that must be applied in the same line. Moreover, if p is^ 

 small then the tension at which buckling occurs is very 

 great, and an approximate value is obtained by dropping the 

 term on the right of (So). This approximate value is 



R= 



is 



This is a very remarkable result in that this tension 

 independent of the flexural rigidity of the beam. 



the first paper the loads were all taken on the centre 



In 



of the sections of the beam. Two cases will now be worked 

 out to show the effect of taking a load slightly off the centre 

 of the beam. 



Case 10. — Beam built into a support at one end and free 

 at the other where a load P is applied. 



Plan of Central 



The load P, before the beam is strained, is situated at 

 (o, p, q) referred to three rectangular axes through the free 

 end, as indicated in fig. 13. 



