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COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS. 345 
VIII. 
On the Stability of a Rotating Shaft, Subjected Simultaneously to 
End Thrust and Twist. 
By R. V. SourTHWELL and BARBARA S. GouGH, of the 
National Physical Laboratory. 
$1. Nearly forty years ago, Sir George Greenhill worked out the criterion of 
stability for a long shaft, ‘simply supported’ (i.e., by journals which impose no con- 
straint upon its direction) at each end, and subjected simultaneously to end thrust 
and twist.' His results showed that the influence of twist is relatively unimportant, 
and may, for most practical purposes, be neglected : the criterion then reduces to 
Kuler’s familiar formula for the critical load of a free-ended shaft. 
When the shaft is not subjected to twist, but revolves, end thrust may combine 
_ with the rotation in bringing about instability of another type, which reveals itself 
& 
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CR i ee 
tion: additional contours may be obtained from these, 
and interpolation, 
this paper, but solutions were not obtain 
$166. 
Vide equation (18) of the present paper. 
1921 
as a tendency to ‘whirl.’ The two factors in this result are far more comparable in 
importance, and the criterion of instability (fortunately simple) must be used in its 
complete form.” The question then arises—when end thrust, twist and rotation act 
simultaneously. as will often occur in practice, are we justified by Greenhill’s result in 
neglecting the influence of twist, or must we allow for a tendency on the part of the 
Te 
pes | bp | WSEX Th b ttached to 
ISO IINGISERN contours are values cf 4 
Ye NG N Circles denote 
: calculated points. 
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70.5 
Values of +8 
Tre 
Fie. 17. 
Totation to combine with the twist on more or less equal terms, as it combined with 
he thrust in the second of the problems just described ? 
$2. An answer to this question is afforded by the accompanying diagrams. 
Instability will occur when a certain relation obtains between the thrust P, the rota- 
Honal speed w and the torque T, and the stability criterion is exhibited graphically 
y means of curves which connect critical values of the first two factors for a series 
f values of the third. The contours given in the figures are the results of calcula- 
if required, by cross-plotting 
Fig. 17 relates to a shaft of which the ends are ‘simply supported, and fig. 18 to 
an * encastré’ shaft, in which change of direction, as well as displacement, is prevented 
b . 
_ *' Proc, Inst. Mech. Eng., 1883, pp. 182-225. The case of clamped ends was also 
treated, with results which do not agree with the calculations of this paper: the ex- 
nation appears to be that the condition of zero displacement at either end was 
not realised. Some discussion was also given of the general equations employed in 
ed. 
* An investigation of this problem is given by A. Morley, ‘ Strength of Materials,’ 
It was briefly discussed by Greenhill in p. 208 of the paper referred to. 
BB 
