534 Dr. J. Prescott on the lorsion of 



If we imagine the section to contain two rectangular 

 pieces of different widths equation (49) gives S in either 

 portion at a considerable distance from the ends and from 

 the junction of the two parts. If shear lines are drawn for 

 equal increments of f it is clear that there must be some 



Fig-. 7. 



shear lines in the wider portion which do not run in the 

 narrower portion, as is shown in fig. 7. The boundary itself 

 is one shear line and the extra lines in the wider part are 

 found inside the lines coming from the narrower part. What 

 happens in this extreme case must happen wherever the 

 section broadens. That is, new lines come into existence 

 where the section broadens and some of the lines close up 

 again where the section narrows. 



The statement was made above that the distribution of 

 shear stress and shear lines is approximately the same in the 

 portion PP'Q'Q as in a long thin rectangle, and now we 

 have found that the shear stress at distance x from the 

 central line of a long thin rectangle, at points not near the 

 ends, is %vrx. As the result for the torque obtained on this 

 assumption is rather unexpected, we shall make quite sure 

 that the shear stress is the same in a thin curved strip. For 

 this purpose we shall find the shear stress in a thin circular 

 split tube at points not near the open ends. 



Let the mean radius of the thin split tube be a and its 

 thickness t, so that the radii of the boundaries are a + ^t. 

 Now the general solution for -ty is contained in the equation 



w + iyfr =J\x + if/) 



=f(re i9 ), (50) 



r and 6 being polar coordinates, with the pole at the origin 

 of the xy axes. Let this origin be taken at the common 

 centre of the boundary circles. Now, since the shear lines 

 are circles except near the open ends, f must be a function 

 of r alone. But, since 



it follows that yjr is a function of r when f is a function of r. 

 Then we have to find a form of the function f in equation 

 (50) which makes ty a function of r and not of 6. The only 



