Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

6.5 Torsional Deformations
If the shear stresses in a shaft are below the proportional limit of the shaft material,
then Hooke’s law, τ = Gγ, relates shear stress and shear strain in the torsion member. The
relationship between the shear stress in a shaft at any radial coordinate ρ and the internal
torque T is given by Equation (6.6):
τ
p
ρ
= T J
The shear strain is related to the angle of twist per unit length by Equation (6.1):
d
γ = ρ φ
dx
Equations (6.6) and (6.1) can be substituted into Hooke’s law,
τ
ρ
Tρ
= Gγ
∴ = Gρ
dφ
J dx
The resulting equation expresses the angle of twist per unit length in terms of the torque T:
dφ T = (6.11)
dx JG
mecmovies 6.2 presents an
animated derivation of the
angle-of-twist relationship.
To obtain the angle of twist for a specific shaft segment, Equation (6.11) can be integrated
with respect to the longitudinal coordinate x over the length L of the segment:
T
∫dφ = ∫
JG dx
L
If the shaft is homogeneous (i.e., G is constant) and prismatic (meaning that the diameter
is constant and, in turn, J is constant), and if the shaft has a constant internal torque T, then
the angle of twist φ in the shaft can be expressed as
TL
φ = (6.12)
JG
The units of φ are radians in both SI and the U.S. customary system.
Alternatively, Hooke’s law and Equations (6.1), (6.2), (6.5), and (6.6) can be combined
to give the following additional angle-of-twist relationships:
γ ρL
τρL
τ max L
φ = = =
(6.13)
ρ ρG
cG
These relationships are often useful in dual-specification problems, such as those in which
limiting values of φ and τ are both specified.
To reiterate, Equations (6.12) and (6.13) may be used to compute the angle of twist φ
only if the torsional member
142
• is homogeneous (i.e., G is constant),
•
is prismatic (i.e., the diameter is constant and, in turn, J is constant), and
• has a constant internal torque T.