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XXX. The Buckling of Deep Beams. 

 By J. Prescott, M.A., D.Sc. (Mane.) *. 



IT is a very well-known fact that a loaded beam may 

 buckle sideways if the depth is much greater than the 

 breadth, but so far no one seems to have given the mathe- 

 matical theory of the subject. That theory is supplied in 

 this paper. It is essentially a question of stability of the 

 same type as Euler's problem of the strut but of rather more 

 complexity. A sketch of a buckled beam is shown in fig. 1 : 

 it is drawn to suit Case 2. 



Fur. 1. 



It will be seen that the buckling load depends on the 

 torsional rigidity of the beam as well as on the flexural 

 rigidity for bending in a horizontal plane. 



The first case to be considered, and one which leads to 

 quite simple mathematics, is the case of a beam under a 

 uniform bending moment. To make the problem quite clear, 

 suppose a long strip of steel, such as a steel rule a yard long, 

 is acted on at its ends by a pair of opposite couples the planes 

 of which are parallel to the faces of the strip. If the couples 

 are increased gradually there will be a certain limiting mag- 

 nitude of the couples for which the strip is unstable, and at 

 this stage it will bend sideways ; that is, bend in a plane 

 perpendicular to the one in which the couples are acting. 

 The magnitude of this buckling couple will depend, of 

 course, on the way in which the ends are held. 



The method of attack is to assume that buckling has 

 actually occuired and to find what couples at the end will 

 maintain the buckled state of the beam. 



* Communicated by the Author. 

 Phil. Mag. S. 6. Vol. 36. No. 214. Oct. 1918. X 



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