For a solid or hollow shft of uniform circular cross-section throughout its length, the
theory of pure torsion states that
R = L
where Tis the applied external torque, constant over length L;
J is the polar second moment of area of shaft cross-section
x(D4 - d 4,
for a hollow shaft;
for a solid shaft and
D is the outside diameter; R is the outside radius;
d is the inside diameter;
T is the shear stress at radius R and is the maximum value for both solid and hollow
G is the modulus of rigidity (shear modulus); and
8 is the angle of twist in radians on a length L.
For very thin-walled hollow shafts
J = 2nr3t, where T is the mean radius of the shaft wall and t is the thickness.
Shear stress and shear strain are related to the angle of twist thus:
T = - R = G ~
Strain energy in torsion is given by
x volume for solid shafis
For a circular shaft subjected to combined bending and torsion the equivalent bending
Me = i [ M + J ( M z +T ')I
and the equivalent torque is
where M and T are the applied bending moment and torque respectively.
The p a ~ e r
tansmitted by a shaft carrying torque Tat o rad/s = To.
T, = +J( M +T 2,
8.1. Simple torsion theory
When a uniform circular shaft is subjected to a torque it can be ShOWn that every sectiOn of
the shaft is subjected to a state of pure shear (Fig. 8.1 ), the moment of resistance developed by
the shear stresses being everywhere equal to the magnitude, and opposite in sense, to the
applied torque. For the purposes of deriving a simple theory to describe the behaviour of
shafts subjected to torque it is necessary to make the following basic assumptionS:
(1) The material is homogeneous, i.e. of uniform elastic properties throughout.
(2) The material is elastic, following Hooke's law with shear stress proportional
(3) The stress does nOt exceed the elastic limit or limit of proportionality.
(4) Circular SectiO