Buckling of Deep Beams. 195 



carried out in the Manchester College of Technology by 

 Mr. H. Carrington. 



The same notation as in the first paper will be used, and 

 it will be useful to explain the notation again here 



E = Young's modulus, 

 EC = the flexural rigidity of the beam for horizontal 



bending, 

 /i = the modulus of rigidity, 

 Km = the torsional rigidity of the beam, 

 T = the angle of twist of the beam at any point D, 

 < y = the deflexion of the central line of the beam from 



the naturally straight state at the same point, 

 t ?,' = the abscissa of D referred to an t r-axis taken 



along the unstrained central line of the beam. 



The coordinate y is a horizontal one, the assumption being 

 made that the depth of the beam is so much greater than 

 the width that the vertical deflexion is negligible compared 

 with the horizontal deflexion when buckling occurs. More- 

 over, the vertical plane containing the central line in the 

 unstrained state is supposed to be a plane of symmetry of 

 the beam. 



In the first paper the differential equations for each beam 

 were worked out separately. It will be shown here that 

 they all come under one general form. 



Suppose ODB is the plan of the central line of a uniform 

 beam which is bent sideways as shown in fig. 8. At the 



Fisr. 8. 



same time the beam is twisted so that, when r is positive* 

 the lower edge oi the beun is further from the straight 

 line OX than the corresponding upper edge. 



The force on the part of the beam on one side of D may 

 be considered as made up of the following four actions 

 at D:— 



(1) a bending moment in the vertical plane touching 



the central line of the beam at D due to vertical 

 forces ; 



(2) a torque about the central line ; 



(3) a vertical shearing force at D ; 



(4) a possible horizontal couple due to fixing the ends. 



02 



