348 



MD. L. X. C. VILOX OX Till; I.'KSISTAXCK TO 



The same remarks which were made concerning the solution in 5 apply here. 

 The critical values are ft = ir/4, ft = 3ir/4. The limits to which the values of ?, M 

 and the stresses, given in (30)-(33), tend, are easily obtained if required. In practice, 

 however, the other solution would probably be used. 



22. Numerical Results. Effects of Kcyways upon Torsional Rigidity. 



I have worked out numerically six cases of this type of section. The sections 

 selected are shown in figs. 6-8. The bounding ellipses are a = ir/G and a. = ir/2, the 

 bounding hyperbolas are ft = ir/2, 37T/4 and IT, giving respectively (a) the half ellipse, 

 (b) the ellipse with a rectangular hyperbolic keyway, (c) the ellipse with a single 

 thin keyway or slit. 



The values of the torsion moment and of its ratio to the torsion moment of the 

 equal circular section, are shown in the tables below. 



TABLE of M/^trc 4 . 



TABLE of M/M . 



y = - 6 



When we look at these results we see that the torsional rigidity of these sections 

 is always less than that of the circular section. The sections consisting of a complete 

 ellipse with one fine slit up to the focus are weaker than the half-ellipse or the 

 sections with a broad keyway. This is more particularly shown in the case of the 

 more elongated ellipse, a = Tr/6. 



With regard to the effects of slits, or thin keyways, it is interesting to compare the 

 values of M/MO for the ellipses a = TT/G, a = ir/2, when (i.) there are no slits, (ii.) 

 there is one slit, (iii.) there are two slits 



