Closed and Open Tubes. 537 



the boundary of the strip should be approximately parallel 

 over nearly the whole length of the strip. The formula will 

 fail to apply, for example, to a strip, however thin, made of 

 a number of small circles strung together by thin strips 

 whose widths are infinitesimal compared with the diameters 

 of the circles, such a section, in fact, as would be obtained 

 by cutting through a string of spherical beads, the circles 

 being supposed to be complete. 



If the central line of the strip is straight equation (5G) 

 gives 



Q=4nrl y (57) 



J y being the moment of inertia of the strip about the central 

 line. It should be noticed that our results show that the 

 torque is the same for the same twist whether the central 

 line is straight or not. For example, a circular tube split 

 longitudinally has the same strength under torsion whether 

 it is left in its circular shape or flattened out into a plane 

 sheet. 



We can verify equation (57) by means of some accurate 

 results worked out by St. Venaut's analysis. Thus, for a 

 rectangle of length a and width 6, when a>'3b, the torque 

 is approximately 



Q = hiTbhc(l-0'MO-) 

 =4nrI y (l-0:630^ (58) 



If the width b is so great as one-third of the length, the 

 formula (57) is wrong by a little over 20 per cent. But if 

 b is one-tenth of a then the result is wrong by just over 

 G per cent. 



If the rectangular section were bent into any curved 

 shape, the lines of shear stress would be distributed in the 

 section in almost precisely the same way as when the section 

 was rectangular. Consequently there would he just about 

 the same proportional error in equation (57) for a curved 

 section as for a straight section, always assuming that ihe 

 thickness is small compared with the radius of curvature. 



To take another example. The accurate torque in an 

 elliptic section with semi-axes a and 1> is 



a d b z 



Q = 7rnT— — r „ (59) 



a~ + b~ 



Phil. Mag. Ser. G. Vol. 40. No. 239. Nov. 1920. 2 N 



