82 APPLIED MECHANICS 
intensity of the shear stress along AB. Let perpendiculars to the sector 
be erected all over its surface to represent the intensity of the stress at 
- each point, and consider for the moment that the stress is perpendicular 
to the plane of the sector, These perpendiculars will be enveloped by a 
pyramid OABFH, in which AH and BF are each equal to 7, The 
volume of this pyramid will be the magnitude of the resultant R of the 
stress over the sector, and this resultant will act through G, the centre of 
gravity of the pyramid, and its line of action will be perpendicular to 
OAB, meeting the latter at g. The real line of action of Ris in the plane 
of OAB and perpendicular to Og, as shown at (a). The moment of 
resistance of the sector to torsion is Rx Og. But R is equal to the 
volume of the pyramid OABFH=AB x fx ir, and Og=#r. Therefore 
Rx Og=ABx4}/r*,, The moment of resistance of the whole section will 
be R x Og multiplied by the number of times that the circle contains the 
sector OAB, that is, by the number of times that the circumference of 
the circle contains the are AB, which is acts Therefore 
AB’ 
: 2ar Ts. 
M=AB a. = fa Of, 
x ffr xX AR af 16? 
If the first method adopted in this Article for finding the moment of 
resistance of a solid circular shaft to torsion be applied to a hollow 
circular shaft (Fig. 102), it follows that the moment 
of resistance of the hollow shaft is ps multiplied by 
the polar moment of inertia of the section, 2.¢. 
- x 5(R — 7), or Be Gas 
KS 
16\ D Yj 
The moment of resistance of the hollow shaft ' 
may, however, be deduced directly from the moment | “~ &- aa 
of resistance of the solid shaft as follows. The mo- ~ Bsr Ae 
ment of resistance of a solid shaft of diameter D is Fia. 102. 
ig The moment of resistance of a solid shaft of diameter d when it 
forms the central portion of the other is 16% where /, is the shear stress 
at radius 7 (Fig. 102), but 4, = fe = fe Hence the moment of resist- 
f the hollow shaft is ™ D8¢— ieee Cae, 
ance of the eeow shat is Ue 160"D ig\D ¥ 
97. Formule for Shafts subjected to Simple Torsion.—It will be 
convenient to collect here the formule which have been proved in the 
three preceding Articles, and give several additional formule easily 
deduced from them. 
T = torque or twisting moment on shaft in inch-pounds. 
N=number of revolutions of shaft per minute. 
H = horse-power transmitted by shaft. 
Z=length of shaft in inches, 
