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

Rotation Angle at C
The angles of twists in segments (1) and (2) can be expressed, respectively, as
φ
1
TL 1 1
= and φ2
=
JG
1 1
TL
JG
2 2
2 2
The rotation angle at C is the sum of these two angles of twist:
TL 1 1
φC = φ1 + φ2
= +
JG
Consequently, since T 1 = T 2 = T, it follows that
1 1
TL
JG
2 2
2 2
⎡ L1
φ C = T ⎢ +
⎣ JG 1 1
Solving for the external torque T gives
L2
⎤
JG
⎥
2 2⎦
T
φC
≤
L1
L2
+
JG JG
1 1
2 2
(2 ° )( π rad/180 ° )
≤
16 in.
25 in.
+
4 4
(0.684563 in. )(4,000,000 psi) (0.141510 in. )(11, 000,000 psi)
= 1,593.6 lb⋅in.
(c)
External Torque T
Compare the three torque limits obtained in Equations (a), (b), and (c). On the basis of
these results, the maximum external torque that can be applied to the shaft at C is
T = 1,594 lb⋅ in. = 132.8 lb⋅ ft
Ans.
ExAmpLE 6.4
y
A
(1)
900 N.m
B
0.85 m 1.00 m 0.70 m
A solid steel [G = 80 GPa] shaft of variable diameter is subjected to the torques shown. Segment
(1) of the shaft has a diameter of 36 mm, segment (2) has a diameter of 30 mm, and
segment (3) has a diameter of 25 mm. The bearing shown allows the shaft to turn freely.
Additional bearings have been omitted for clarity.
600 N.m
(2) (3)
C
250 N.m
D
x
(a) Determine the internal torque in segments (1), (2), and (3)
of the shaft. Plot a diagram showing the internal torques in
all segments of the shaft. Use the sign convention presented
in Section 6.6.
(b) Compute the magnitude of the maximum shear stress in
each segment of the shaft.
(c) Determine the rotation angles along the shaft measured
at gears B, C, and D relative to flange A. Plot a diagram
showing the rotation angles at all points on the shaft.
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