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

with respect to an adjacent cross section. Figures 6.2c and 6.2d show a square rubber shaft
before and after an external torque T is applied to its ends. Plane cross sections in Figure 6.2c
before the torque is applied do not remain plane after T is applied (Figure 6.2d). The behavior
exhibited by the square shaft is characteristic of all but circular sections; therefore, the analysis
that follows is valid only for solid or hollow circular shafts.
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TORSIONAL SHEAR STRAIN
6.2 Torsional Shear Strain
Consider a long, slender shaft of length L and radius c that is fixed at one end, as shown
in Figure 6.3a. When an external torque T is applied to the free end of the shaft at B, the
shaft deforms as shown in Figure 6.3b. All cross sections of the shaft are subjected to the
same internal torque T; therefore, the shaft is said to be in pure torsion. Longitudinal
lines in Figure 6.3a are twisted into helixes as the free end of the shaft rotates through an
angle φ. This angle of rotation is known as the angle of twist. The angle of twist changes
along the length L of the shaft. For a prismatic shaft, the angle of twist will vary linearly
between the ends of the shaft. The twisting deformation does not distort cross sections
of the shaft in any way, and the overall shaft length remains constant. As discussed in
Section 6.1, the following assumptions can be applied to torsion of shafts that have
circular—either solid or hollow—cross sections:
• A plane section before twisting remains plane after twisting. In other words, circular
cross sections do not warp as they twist.
• Cross sections rotate about, and remain perpendicular to, the longitudinal axis of
the shaft.
• Each cross section remains undistorted as it rotates relative to neighboring cross
sections. In other words, the cross section remains circular and there is no strain in
the plane of the cross section. Radial lines remain straight and radial as the cross
section rotates.
• The distances between cross sections remain constant during the twisting
deformation. In other words, no axial strain occurs in a round shaft as it twists.
To help us investigate the deformations that occur during twisting, a short segment ∆x of
the shaft shown in Figure 6.3 is isolated in Figure 6.4a. The shaft radius is c; however, for
more generality, an interior cylindrical portion at the core of the shaft will be examined
(Figure 6.4b). The radius of this core portion is denoted by ρ, where 0 < ρ ≤ c. As the shaft
twists, the two cross sections of the segment rotate about the x axis and line element CD on
the undeformed shaft is twisted into helix C′D′. The angular difference between the rotations
of the two cross sections is equal to ∆φ. This angular difference creates a shear strain
γ in the shaft. The shear strain γ is equal to the angle between line elements C′D′ and C′D″,
as shown in Figure 6.4b. The value of the angle γ is given by
DD ′ ′′
tanγ =
∆x
The distance D′D″ can also be expressed by the arc length ρ∆φ, giving
tanγ = ρ ∆ φ
∆ x
A
A
L
B
(a) Undeformed shaft
∆x
(b) Deformed shaft in response to
torque T
FIGURE 6.3 Prismatic shaft
subjected to pure torsion.
∆x
FIGURE 6.4a Shaft segment
of length ∆x.
C
C′
γ
D
D″
D′
∆x
FIGURE 6.4b Torsional
deformation of shaft segment.
c
O
∆φ
φ
ρ
B
T
x
x
c
x
x