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

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TORSION
Electromagnetic
forces
A
(1)
Armature
a
b
Flange
coupling
B
a
b
Bolt
forces
FIGURE 6.1 Modified free-body diagram of a typical electric motor shaft.
Torsion of noncircular shapes
produces warping, in which
planar cross sections before
application of the loading
become nonplanar, or warped,
after a torque is applied.
In 1784, C. A. Coulomb, a French engineer, experimentally developed the relationship
between the applied torque and the angle of twist for circular bars. 1 A. Duleau, another
French engineer, in a paper published in 1820, analytically derived the same relationship
by making the assumptions that a plane section before twisting remains plane after twisting
and that a radial line on the cross section remains plane after twisting. Visual examination
of twisted models indicates that these assumptions are apparently correct for either
solid or hollow circular sections (provided that the hollow section is circular and symmetrical
with respect to the axis of the shaft), but incorrect for any other shape. For example,
compare the distortions evident in the two prismatic rubber shaft models shown
in Figure 6.2. Figures 6.2a and 6.2b show a circular rubber shaft before and after an external
torque T is applied to its ends. When torque T is applied to the end of the round
shaft, the circular cross sections and longitudinal grid lines marked on the shaft deform
into the pattern shown in Figure 6.2b. Each longitudinal grid line is twisted into a helix
that intersects the circular cross sections at equal angles. The length of the shaft and its
radius remain unchanged. Each cross section remains plane and undistorted as it rotates
T
(a)
T
(b)
T
(c)
T
FIGURE 6.2 Torsional deformations illustrated by rubber models with circular (a, b) and square
(c, d) cross sections.
(d)
1
From S. P. Timoshenko, History of Strength of Materials (New York: McGraw-Hill, 1953).