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COMPLEX STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS. 347 
algebraic formule of the type (2), besides applying over a wider range; but 
the establishment of these formulz enables us to assert that the relative importance 
of end thrust, rotation and twist is of the same order, when the three actions exist 
simultaneously, as it has been shown to be when they are taken as acting in pairs : 
that is to say, any torque which a shaft of normal dimensions can sustain without 
damage to the material will be negligible in its effect on the stability of that shaft. 
$4. Equations of Neutral Hquilibrium.—tn considering the stability of the 
system, we begin by assuming that the shaft is initially straight and centrally 
loaded, but that conditions of neutral stability are eventually reached under which 
the shaft can be held in equilibrium in a configuration of slight distortion, the 
centre-line taking a curved form, either plane or tortuous. The.influence of gravity 
is neglected. We take the origin of co-ordinates at the central section of the shaft, 
the axis Oz coinciding with the axis of the shaft in the unstrained configuration ; 
and we choose axes Ow, Oy, which rotate with the shaft, perpendicular to each other 
and to Oz. We define the distorted centre-line by the co-ordinates x, y and gz, 
relative to these axes, of a point P on the centre-line which was originally distant s 
from the origin 0. 
The bending of the shaft at any section P is in the osculating plane, at P, to the 
curved centre-line ; and it is produced by a couple, M, of which the axis is parallel 
to the binormal at P. The direction-cosines of the binormal are 
- "2 dz ay 
ds ° ds? ds ds? )’ | 
i (Z ‘ Gx dz ; 3) 
ds dst das" ds \ 
and p (= : 
where p, the radius of curvature, = EI/M ; 
d*y_ dy a) 
ds? ds ° ds? }’ | 
and the couple M can therefore be resolved into component couples, with axes 
parallel to Ox, Oy, Oz respectively, of magnitudes 
dy @z dz dy 
El( —2 . aa OY 
ds ds? ds ds? ) 
dz @e da d% 
HG * tee POR esis 
(i ds? ds i) 
dx dy dy dx 
and ni(> os Ti asf az) 
Now, in considering the stability of the straight shaft, we are concerned only 
with a configuration of infinitesimal distortion, and we may therefore regard a and 
y as small quantities of the first order. Neglecting small quantities of the second 
order, we may take dz/ds as unity, and reduce the expressions for the component 
couples to the simpler forms 
2 2 
—EI%Y, n1% ana 0, 
ds® ds* 
Again, the direction-cosines of the tangent to the strained centre-line at the 
point P are (if we again neglect small quantities of the second order) 
dx dy 
me ae and 1, 
Thus, if T’ is the torsional couple acting on the shaft at the section P, it may 
be resolved into components, with axes parallel to Oz, Oy, Oz respectively, of 
magnitudes - : 
ee ep Ae ' 
de’ Wea and T’, 
