298 Dr. J. Prescott on the 



Case 1. — A pair of equal and opposite couples G act at the 

 ends A and B in a vertical plane, as shown in fig. 2 a, causing 

 the beam to buckle like a strut, to a greater extent on the thrust 



Fig-. 2 a. 



¥\g. 2 b. 



Plan of Central Line of Section. 



side than on the tension side, as fig. 2 a indicates. The deflexion 

 of the beam in the vertical plane is assumed to be negligible, 

 while the deflexion of the central line of the section in the 

 horizontal plane is comparatively large. This means that 

 the beam is twisted and bent, the curve of the central line 

 being nearly a horizontal curve. Moreover, the amount of 

 twist is so small everywhere that the displacement of a point 

 on the central line may be considered to be perpendicular to 

 the faces of the beam. 



The couples G are represented as vectors in fig 2 b so that 

 they can be resolved in the required directions. 



Let E denote Young's modulus for the beam, C the least 

 moment of inertia of the section, Kn the torsional rigidity, 

 n being the modulus of rigidity. 



r is the angle of twist at any point R the coordinates of 

 which are #, y, with A as origin. 



Since the upper edge of the section through R is bent 

 further out than the lower edge it follows that the couple G 

 at the end A has a component about a line parallel to the 

 twisted depth at R. This component has a magnitude Gt 

 and tends to increase the curvature of the central line. 

 Consequently, by the usual equation for a loaded beam, 



■0g--.Gr (1) 



