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Steel
Aluminum 60 mm FigureP5.21 100 mm Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm 5.22 An aluminum shaft (Gal= 28 GPa) and a steel shaft (GS=82 GPa) are securely fastened to form composite shaft with a cross section
shown in Figure P5.21. If the maximum torsional shear stress in aluminum is 21 MPa, determine the maximum torsional shear stress in steel.
5.23 Determine the direction of torsional shear stress at points A and B in Figure P5.23 (a) by inspection; (b) by using the sign convention
for internal torque and the subscripts. Report your answer as a positive or negative τxy.
B
x
y
T Figure P5.23 A
x January, 2010 M. Vable Mechanics of Materials: Torsion of Shafts 5 232 5.24 Determine the direction of torsional shear stress at points A and B in Figure P5.24 (a) by inspection; (b) by using the sign convention
for internal torque and the subscripts. Report your answer as a positive or negative τxy
A
x
y
T B Figure P5.24
x 5.25 Determine the direction of torsional shear stress at points A and B in Figure P5.25 (a) by inspection; (b) by using the sign convention
for internal torque and the subscripts. Report your answer as a positive or negative τxy.
x A T y x
B Figure P5.25 5.26 Determine the direction of torsional shear stress at points A and B in Figure P5.26 (a) by inspection; (b) by using the sign convention
for internal torque and the subscripts. Report your answer as a positive or negative τxy.
x T A
y x Figure P5.26 B 5.27 Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm The two shafts shown in Figure P5.27 have the same cross sectional areas A. Show that the ratio of the polar moment of inertia of the
hollow shaft to that of the solid shaft is given by the equation below.:
2
J hollow
α +1
 = 2
J solid
α –1 Figure P5.27 5.28 RH
RH RS 3 Show that for a thin tube of thickness t and centerline radius R the polar moment of inertia can be approximated by J = 2 π R t . By
thin tube we imply t < R ⁄ 10 . 5.29 (a) Draw the torque diagram in Figure P5.29. (b) Check the values of internal torque by making imaginary cuts and drawing freebody diagrams. (c) Determine the rotation of the rigid wheel D with respect to the rigid wheel A if the torsional rigidity of the shaft is 90,000
kips·in.2. January, 2010 M. Vable Mechanics of Materials: Torsion of Shafts 5 233 10 in kips
kips
60 in kips
kips
36 Figure P5.29 30 in 5.30 (a) Draw the torque diagram in Figure P5.30. (b) Check the values of internal torque by making imaginary cuts and drawing freebody diagrams. (c) Determine the rotation of the rigid wheel D with respect to the rigid wheel A if the torsional rigidity of the shaft is 1270
kN·m2.
20 kN m
18 kN
12 kN m
10 kN
1.0 m Figure P5.30 0.5 m 5.31 The shaft in Figure P5.31 is made of steel (G = 80 GPa) and has a diameter of 150 mm. Determine (a) the rotation of the rigid wheel
D; (b) the magnitude of the torsional shear stress at point E and show it on a stress cube (Point E is on the top surface of the shaft.); (c) the
magnitude of maximum torsional shear strain in the shaft.
Nm
E 90 k N m
70 kN m 0.25 m
0.5 m Figure P5.31 0.3 m 5.32 The shaft in Figure P5.32 is made of aluminum (G = 4000 ksi) and has a diameter of 4 in. Determine (a) the rotation of the rigid
wheel D; (b) the magnitude of the torsional shear stress at point E and show it on a stress cube (Point E is on the bottom surface of the
shaft.); (c) the magnitude of maximum torsional shear strain in the shaft.
80 in
40 in kips
15 in kips 20 in
n Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm Figure P5.32 25 i n 5.33 Two circular steel shafts (G =12,000 ksi) of diameter 2 in. are securely connected to an aluminum shaft (G =4,000 ksi) of diameter
1.5 in. as shown in Figure P5.33. Determine (a) the rotation of section at D with respect to the wall, and (b) the maximum shear stress in the
shaft.
12 in.kips A Figure P5.33 January, 2010 steel 40 in. B 25 in.kips C
aluminum steel 15 in. 25 in. 15 in.kips D M. Vable Mechanics of Materials: Torsion of Shafts 5 234 A solid circular steel shaft BC (Gs = 12,000 ksi) is securely attached to two hollow steel shafts AB and CD, as shown in Figure
P5.34. Determine (a) the angle of rotation of the section at D with respect to the section at A; (b) the magnitude of maximum torsional
shear stress in the shaft; (c) the torsional shear stress at point E and show it on a stress cube. (Point E is on the inside bottom surface of CD.) 5.34 120 in kips 420 in kips 200 in kips 100 in kips
2 in Figure P5.34 24 in 36 in 24 in A steel shaft (G = 80 GPa) is subjected to the torques shown in Figure P5.35. Determine (a) the rotation of section A with respect to
the noload position; (b) the torsional shear stress at point E and show it on a stress cube. (Point E is on the surface of the shaft.) 5.35 160 kN m
80 kN m 120 kN m A Figure P5.35 2.5 m 2.0 m Tapered shafts
5.36 The radius of the tapered circular shaft shown in Figure P5.36 varies from 2...
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 Spring '10
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 Torsion

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