CmCULAR CYLINDERS UNDER CERTAIN PRACTICAL SYSTEMS OF LOAD. 231 
§ 32. Discussion of the Results. 
From the above tables we see that the radii in each cross-section do not remain 
straight lines, but assume distorted shapes, which are shown, on a very exaggerated 
scale, on Diagram 16, where, for each of the ten cross-sections, the curve ot 
(y — v)/vq, which indicates the deviation from the straight line in the distorted 
form of the radii, has been plotted. The variation from the straight line increases 
rapidly as we approach the region where the stress is applied, as can be seen from 
curves (l)-(4) on Diagram 16. On the other hand, towards the ends, the distortion 
remains fairly constant. The distorted radii meet the bounding circles at right 
angles when r(f) = 0 at the surface, but they meet it at a finite angle where 
r(f) = T. 
From the values of (f)Z and i f we see that as soon as we get at all away from the 
ends the conditions that 7'(f) = 0, (f)Z = jxTZ, v — tvz, which hold for uniform torsion, 
are very closely satisfied, and that, more generally, except where the abrupt change 
takes place in the shearing stress at the surface, the approximate expressions given 
in § 28 do not differ widely from the true exjDressions, the law that if varies as 
the square of the radius being, near the ends, tolerably well verified. It is to be 
noted also that, where the approximations would give a discontinuity in r<^ inside the 
material (viz. at 2 = ’be), the true values are almost exactly the mean of the two 
discontinuous values obtained from the approximate formulse assumed correct. 
In like manner (jiz is nearly the same as (f)z', except near z = 'Sc, where, as we have 
seen, an infinite stress really occurs, of which the approximations give no hint. 
We note, however, that (f)Z does not strictly vary as r all over the section, Ijeing 
smaller than should be expected inside and larger at the boundary. 
The theoretical result, that the stress is infinite where the transverse applied shear 
is discontinuous, throws much light on the case of a cylinder whose cross-section 
abruptly changes, as in fig. 1, with the difference that now the stress applied to the 
collar is transverse. We see that in such a case we should expect the material to 
give way at the points of sudden change. This conclusion is in accordance with 
practical experience, the tail ends of propeller shafts, for instance, breaking almost 
invariably in this manner. 
§ 33. General Conclusion. 
This example concludes the series of three which it was proposed to treat of. The 
object has been to obtain a clear idea of the effects of certain surface distributions of 
stress which come much nearer to the cases arising in practice than does the uniform 
distribution ordinarily taken. 
