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

m6.23 A composite shaft consists of a hollow stainless
steel [G = 86 GPa] shaft (1) connected to a solid bronze [G =
38 GPa] shaft (2) at flange B. The outside diameter of shaft
(1) is 75 mm, and the inside diameter is 55 mm. The outside
diameter of shaft (2) is 75 mm. A concentrated torque T will
be applied to the composite shaft at flange B. Determine
(a) the maximum magnitude of the concentrated torque T if
the angle of rotation at flange B cannot exceed 3°.
(b) the maximum shear stresses τ 1 and τ 2 in each shaft.
ExAmpLE 6.10
y
A
D
T 2
(1)
(3)
60 teeth
24 in.
(2)
C
B
E
18 kip.in.
40 teeth
(2)
x′
36 in.
C
18 kip.in.
x
A torque of 18 kip · in. acts on gear C of the assembly shown.
Shafts (1) and (2) are solid 2.00 in. diameter steel shafts, and
shaft (3) is a solid 2.50 in. diameter steel shaft. Assume that
G = 12,000 ksi for all shafts. The bearings shown allow free rotation
of the shafts. Determine
(a) the maximum shear stress magnitudes in shafts (1),
(2), and (3).
(b) the rotation angle of gear E.
(c) the rotation angle of gear C.
Plan the Solution
A torque of 18 kip · in. is applied to gear C. This torque is transmitted
by shaft (2) to gear B, causing it to rotate and, in turn, twist
shaft (1). The rotation of gear B also causes gear E to rotate, which causes shaft (3) to
twist. Therefore, the torque of 18 kip · in. on gear C will produce torques in all three shafts.
The rotation angle of gear B will be dictated by the angle of twist in shaft (1). Similarly,
the rotation angle of gear C will be dictated by the angle of twist in shaft (3). Furthermore,
the relative rotation of gears B and E will be a function of the gear ratio. These relationships
will be considered in analyzing the internal torques produced in the three shafts.
Once the internal torques are known, the maximum shear stresses, twist angles, and
rotation angles can be determined.
x
SOLUTION
Step 1 — Equilibrium Equations: Consider a free-body diagram that cuts
through shaft (2) and includes gear C. A positive internal torque will be assumed
in shaft (2). From this free-body diagram, a moment equilibrium equation about
the x axis can be written to determine the internal torque T 2 in shaft (2).
Σ M = 18 kip⋅in. − T = 0 ∴ T = 18kip⋅ in.
(a)
x 2 2
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