TORSION* OF CERTAIN* FORMS OF SH.U ll.\.. 





We find 



It follows that, in the first case, the cutting of one thin key way lowers the rigidity 

 of the section by 144 per cent., and of two keyways by 2'29 per cent. Hence th- 

 effect of two such keyways is slightly greater, if anything, than twice the effect of a 

 single keyway, in the case of the more elongated ellipse. The dinWencr is, however, 

 practically negligible. 



In the other case the result is different. The reduction of the torsional rigidity in 

 very great : it amounts to 23'08 per cent, for one keyway and to 42*68 per cent, for 

 two keyways. Here we see the effect of two keyways is rather less than twice the 

 effect of one. 



We may infer, however, from these two results that we may in practice, without 

 very large error, if we have a number of keyways cut symmetrically into a section, 

 .mil \\c know the effect on the torsional rigidity of any single keyway, assume that 

 the effect of all the keyways is the sum of their separate effects. 



Another important point which is brought out by the above results is that the 

 effect of such a keyway upon the torsional rigidity is by no meuns simply proportional 

 to the depth of the keyway, but increases according to some much more rapid law. 



Thus, for the ellipse a = Tr/6, the depth of the keyway = '123 (semi-major axis). 

 For the ellipse a. = ir/'2, the depth = '601 (semi-major axis). Thus, when the depth 

 of the keyway is only decreased to one-fifth of what it was before, the reduction of 

 torsional rigidity falls from 23 per cent, to 1 per cent., or nearly in the ratio of the 

 xi/nnrea of the depths of the keyways. 



This result may explain the fact that, when keyways of only moderate deptli are 

 cut into shafts, the decrease of torsional rigidity is by no means so great as would 

 have been inferred from DE SAIXT-VKNANT'S results for a circular section, with a thin 

 keyway or slit extending right up to the centre. 



If we suppose, which appears reasonable, that the effect of such a slit upon an 

 ellipse, which is not very elongated, does not differ much from the effect on a circle, 

 we see that a keyway, whose depth equals about one-eighth of the radius, will decrease 

 the torsional rigidity by only about 1 per cent. 



Now, when we make a = oo , we get the case of the circle with a keyway going 

 right up to the centre. The reduction of torsional rigidity is then 44 per cent, about. 



