Closed and Open Tubes. 527 



and each one of these curves has the same property as a 

 boundary, namely, the property that there is no component 

 shear stress perpendicular to the curve. At every point of 

 one of these curves then the resultant shear stress is along 

 the curve itself. That is, the curve 



is a curve whose direction at every point is the direction 

 of the resultant shear stress at that point. Each of these 

 curves, obtained by giving different values to k within the 

 range of variation of f inside the section, is called a Shear 

 Line. The shear lines form a system of closed curves inside 

 the section, and, in the case of a tube, each of these closed 

 curves encloses the inner boundary and is enclosed by the 

 outer boundary. In the case of a thin tube whose two 

 boundaries are nearly parallel curves, it is quite clear from 

 continuity that all the shear lines are also nearly parallel 

 curves. 



Suppose now that we are dealing with a thin tube such 

 that the normal to one boundary is approximately normal to 

 the other boundary also. Let t denote the thickness of the 

 tube at any point ; that is, t is the length, intercepted 

 between the boundaries, of the normal to one of the 

 boundaries, say the inner boundary for definiteness. Let ds 

 now represent an element of length perpendicular to t at its 

 mid-point. Then, since the shear stress must be approxi- 

 mately constant across the thickness, the mean shear stress 

 across t will be, by {%€>), 



S=+n S f, ...... (27) 



where &£ is the excess of f at the inner, over that at the 

 outer, boundary, and this shear stress is approximately the 

 resultant shear stress at the middle of t. 



Fkr. 4. 



The Torque. — Let the curve KLM (fig. 4) be the locus of 

 the mid-points of the thickness t, and let OH be the per- 

 pendicular from the axis of twist on the tangent to tins 



