Buckling of Deep Becu 



197 



perpendicular to the twisted depth at D (see fig. 10). The 

 former component, of magnitude Gtt, bends the central line 

 in a plane perpendicular to the depth of the beam, and this 

 plane of bending is everywhere nearly horizontal. If there 



Fig. 10. 



is an end couple M on the beam acting in a horizontal plane, 

 as in clamping the end, then the total couple at D causing 

 bending in a horizontal plane is Gt+M. 



Then, since the curvature produced by this couple is the 

 cause of the deflexion y, we get 



EC^ = Gt+M. 



doc 2 



(72) 



Equations (71) and (72) are the general differential 

 equations, which, together with the end-conditions of the 

 beam, determine the buckling load when there is no tension 

 or thrust in the beam, and when the load is applied at the 

 centre line of the beam. ™ 



Substituting in (71) the value of -y-^ from (72) w r e get 



dT 



dx 



^(Gr + M). 



(73) 



From the meaning of Kn we have 



KnX (angle of twist per unit length) = torque, 



that is, 



Kn— =1. 

 ax 



Therefore (73) becomes finally 



G(Gt + M). 



(74) 



(75) 



It is clear that a tension in the beam would help to 

 stabilize it, and that a thrust would make it less stable, for 

 the beam could buckle under a thrust alone. If a thrust R 

 is applied at the ends of the beam, then an extra term — R# 



