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

the longitudinal axis of the shaft. The resultant shear force dF acting on the small element is
given by the product of the shear stress τ ρ and the area dA. The force dF produces a moment
dM = ρ dF = ρ(τ ρ dA) about the shaft centerline O. The resultant moment produced by the
shear stress about the shaft centerline is found by integrating dM over the cross-sectional
area:
∫
dM
= ∫ ρτ ρ A
If Equation (6.4) is substituted into this equation, the result is
∫dM
= ∫ ρdA
= ∫
A
ρ τ
c
dA
τ
c
max max 2
A
ρ dA
Since τ max and c do not vary with dA, these terms can be moved outside of the integral.
Furthermore, the sum of all elemental moments dM must equal the torque T to satisfy equilibrium;
therefore,
τ max 2
T = ∫ dM = dA
c
∫ ρ
(a)
A
139
TORSIONAL SHEAR STRESS
dF = τ ρ dA
O
dA
FIGURE 6.7 Calculating the
resultant moment produced by
torsion shear stress.
ρ
T
The integral in Equation (a) is called the polar moment of inertia, J:
J
= ∫A
ρ2
Substituting Equation (b) into Equation (a) gives a relationship between the torque T and the
maximum shear stress τ max :
T
τ max
dA
(b)
= J
(c)
c
Alternatively, solving equation (c) for the maximum shear stress gives
Tc
τ max = (6.5)
J
If Equation (6.4) is substituted into Equation (6.5), a more general relationship can be obtained
for the shear stress τ ρ at any radial distance ρ from the shaft centerline:
Tρ
τ ρ =
(6.6)
J
Equation (6.6), for which Equation (6.5) is a special case, is known as the elastic torsion
formula. In general, the internal torque T in a shaft or shaft segment is obtained from a
free-body diagram and an equilibrium equation. Note: Equations (6.5) and (6.6) apply only
to linearly elastic action in homogeneous and isotropic materials.
polar moment of Inertia, J
The polar moment of inertia for a solid circular shaft is given by
π
4
π
4
J = r = d
(6.7)
2 32
where r is the radius of the shaft and d is the diameter. For a hollow circular shaft, the polar
moment of inertia is given by
π
J R r D d
2 [ 4 4 ] π
32 [ 4 4
= − = − ]
(6.8)
where R is the outside radius, r is the inside radius, D is the outside diameter, and d is the
inside diameter, of the shaft.
mecmovies 6.2 presents an
animated derivation of the elastic
torsion formula.
The polar moment of inertia is
also known as the polar second
moment of area.
Typically, J has units of in. 4
in the U.S. Customary System
and mm 4 in SI.