TORSION OF CERTAIN FORMS OF SHAFTING. 31 1 



hyperbolas. Although the analysis is worked out for the case where the two hyper- 

 bolic segments are not symmetrical, I have not given any numerical examples of this 

 case, as the sections obtained by taking two hyperbolas curved the same way, as in 

 fig. 1, do not correspond to any intersecting practical case: the section is too broad 

 at the ends and too narrow at the bend to be any fair representation of the angle 

 iron. 



The section (fig. 2) bounded by an ellipse and the two branches of a confocal 

 hyperbola is, on the other hand, an approximate representation of a well-known 

 section, much used in engineering practice, the rail section. 



This section I have worked out for various values of the eccentricity of the 

 ellipse and of the angle between the asymptotes of the hyperbola. 



The four sections in fig. 2, where this angle is 120, give the best representation 

 of the rail section. 



The numerical results are tabulated so as to show the ratio of the torsional rigidity 

 of this section to that of the circular section of the same area, and also the same ratio 

 for the maximum stress. 



The ratio of these two ratios gives us a kind of measure of the usefulness or 

 " efficiency " of the section. 



In the case of the sections of fig. 2 I have investigated at length the position of 

 the fail- points, or points of maximum strain and stress, the maximum strain, in the 

 case of torsion, being coincident with the maximum stress. It is found that for the 

 two smaller ellipses the maximum stress occurs at the point B where the section 

 is thinnest. For the two larger ellipses the maximum stress occurs at points 

 F, F, F, F, symmetrically distributed round the contour, and lying on the broad 

 sides of the section. The critical section, when these two cases pass into one another, 

 can be calculated and is shown as qq, <jq in fig. 2. In figs. 2-5 the corresponding 

 points belonging to the different sections are distinguished by suffixes. 



The changes in the stresses are shown by the curves in fig. 9, (p. 340) in which the 

 abscissa represents the quantity a whose hyperbolic cosine and sine are proportional 

 to the major and minor axes of the ellipse respectively, and in which the ordinates 

 represent the stresses at A, B, F, divided by the maximum stress of the circular 

 section of equal area. The curves are in certain parts only roughly drawn, but they 

 suffice to show the manner in which the stresses vary. It is seen that the stress at 

 B separates from the maximum stress after the critical value a = T225, and gradually 

 diminishes, compared with the stresses at A and F. 



This result might have been expected from the investigations of DE SAINT- VENANT 

 upon certain sections bounded by curves of the fourth degree. These investigations 

 appear, however, not to have been sufficiently noticed. THOMSON and TAIT, in their 

 ' Natural Philosophy,' and BoussiNESQ, in his researches on torsion (' Journal de 

 Mathenmtiques,' Sdrie II., vol. 16, p. 200), both conclude that the fail-points are at 

 the points of the cross-section nearest to the centre, and BOUSSINESQ even gives an 



