T H E 

 LONDON, EDINBURGH, and DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 

 'tlOVU t SIXTH series.] 



NOVEMBER 1920. 



LXIV. Z7i£ Torsion of Closed and Open Tabes. By John 

 Prescott, M.A.* JD.Sc, Lecturer in Mathematics in the 

 Manchester College of Technology * . 



THE object of the present paper is to deduce, from St. 

 Venant's theory of torsion, approximate formulae for 

 the torsion of closed tubes, and of rods whose sections are 

 long thin strips, such, for example, as split tubes. Although 

 the results are only approximate the percentage errors they 

 contain approach zero as the ratio of the length to the 

 breadth of the section approaches infinity. The results have 

 thus the same sort of limitations as the ordinary beam theory, 

 which applies accurately only to rods whose lengths are 

 infinitely greater than their lateral dimensions. 



Axes of x and y are taken in the plane of one normal 

 section of the rod, and the ~-axis along the untwisted axis. 

 The displacements of a particle parallel to the three axes 

 are u, v, iv ; the three shear stresses are Si, S 2 , S 3 ; the first 

 of these acts on x;j planes and zx planes, the second on 

 yz planes and xy planes, the third on zx planes and yz planes. 

 Fig. 1 shows the relation of Si and S 2 to the axes of x and y. 

 The usual assumption in St. Venant's theory is that the 

 section at z is twisted through tz, so that r is the angle 

 of twist per unit length. This gives, as the component 



* Communicated by the Author. 



Phil. Mag. S. 6. Vol. 40. No. 2W. Nov. 1920. 2 M 



