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

6.4 Stresses on oblique planes
The elastic torsion formula [Equation (6.6)] can be used to calculate the maximum shear
stress produced on a transverse section in a circular shaft by a torque. It is necessary to establish
whether the transverse section is a plane of maximum shear stress and whether there
are other significant stresses induced by torsion. For this study, the stresses at point A in the
shaft of Figure 6.8a will be analyzed. Figure 6.8b shows a differential element taken from
the shaft at A, as well as the shear stresses acting on transverse and longitudinal planes. The
stress τ xy may be determined by means of the elastic torsion formula and the equation τ yx = τ xy
(see Section 1.6). If the equations of equilibrium are applied to the free-body diagram of
Figure 6.8c, the following result is obtained:
We then have
Σ F = τ dA − τ ( dAcos θ)cos θ + τ ( dAsin θ)sinθ
= 0
t nt xy yx
2 2
τ = τ (cos θ − sin θ) = τ cos2θ
nt xy xy
(6.9)
and
from which it follows that
Σ F = σ dA − τ ( dAcos θ)sin θ − τ ( dAsin θ)cosθ
= 0
n n xy yx
σn = 2τxysinθcosθ = τxy
sin 2θ
(6.10)
These results are shown in the graph of Figure 6.9, from which it is apparent that the
maximum shear stress occurs on transverse and longitudinal diametral planes (i.e., longitudinal
planes that include the centerline of the shaft). The graph also shows that the
maximum normal stresses occur on planes oriented at 45° with the axis of the shaft and
perpendicular to the surface of the shaft. On one of these planes (θ = 45° in Figure 6.8b),
the normal stress is tensile, and on the other (θ = 135°), the normal stress is compressive.
Furthermore, the maximum magnitudes for both σ and τ are equal. Therefore, the
maximum shear stress given by the elastic torsion formula is also numerically equal
to the maximum normal stress that occurs at a point in a circular shaft subjected to
pure torsion.
T
A
y
FIGURE 6.8a Shaft subjected to
pure torsion.
140
x
T
τ xy
y
θ
τ yx
τ yx
τ xy
FIGURE 6.8b Differential
element at point A on the shaft.
x
t
y
τ
θ
n
x
xy dA cos
θ
dA cos
σn dA
θ
τnt dA
τyx dA sin θ
FIGURE 6.8c Free-body diagram of a wedgeshaped
portion of the differential element.
dA
dA sin θ
θθ