Torsion.pdf

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CHAPTER 5

TO R S I O N

5–53. The 20-mm-diameter A-36 steel shaft is subjected to the torques shown. Determine the angle of twist of D, C and B.

A

*5–56. The A-36 steel axle is made from tubes AB and CD and a solid section BC. It is supported on smooth bearings that allow it to rotate freely. If the gears, fixed to its ends, are subjected to 85@N # m torques, determine the angle of twist of the end B of the solid section relative to end C. The tubes have an outer diameter of 30 mm and an inner diameter of 20 mm. The solid section has a diameter of 40 mm. 400 mm

D C B

20 N!m

250 mm

200 mm

30 N!m 600 mm

C

400 mm

800 mm

D 85 N!m

B

80 N!m

A

Prob. 5–53 85 N!m

5 5–54. The shaft is made of A992 steel with the allowable shear stress of tallow = 75 MPa. If gear B supplies 15 kW of power, while gears A, C and D withdraw 6 kW, 4 kW and 5 kW, respectively, determine the required minimum diameter d of the shaft to the nearest millimeter. Also, find the corresponding angle of twist of gear A relative to gear D. The shaft is rotating at 600 rpm. 5–55. Gear B supplies 15 kW of power, while gears A, C and D withdraw 6 kW, 4 kW and 5 kW, respectively. If the shaft is made of steel with the allowable shear stress of tallow = 75 MPa, and the relative angle of twist between any two gears cannot exceed 0.05 rad, determine the required minimum diameter d of the shaft to the nearest millimeter. The shaft is rotating at 600 rpm.

Prob. 5–56 5–57. The turbine develops 150 kW of power, which is transmitted to the gears such that C receives 70% and D receives 30%. If the rotation of the 100-mm-diameter A-36 steel shaft is v = 800 rev>min., determine the absolute maximum shear stress in the shaft and the angle of twist of end E of the shaft relative to B. The journal bearing at E allows the shaft to turn freely about its axis. 5–58. The turbine develops 150 kW of power, which is transmitted to the gears such that both C and D receive an equal amount. If the rotation of the 100-mm-diameter A-36 steel shaft is v = 500 rev>min., determine the absolute maximum shear stress in the shaft and the rotation of end B of the shaft relative to E. The journal bearing at E allows the shaft to turn freely about its axis.

A

B

v

B C

600 mm

600 mm

Probs. 5–54/55

D

3m D

600 mm

C

E

4m 2m

Probs. 5–57/58

Average Mechanical Properties of Typical Engineering Materialsa (SI Units)

Density R (Mg/m3)

Moduls of Elasticity E (GPa)

Modulus of Rigidity G (GPa)

Ultimate Strength (MPa) Yield Strength (MPa) Su SY Tens. Comp.b Shear Tens. Comp.b Shear

2014-T6 6061-T6

2.79

73.1

27

414

414

172

469

469

2.71

68.9

26

255

255

131

290

290

Gray ASTM 20 Malleable ASTM A-197

7.19

67.0

27

–

–

–

179

7.28

172

68

–

–

–

8.74

101

37

70.0

70.0

8.83

103

38

345

1.83

44.7

18

Structural A-36

7.85

200

Structural A992

7.85

200

Stainless 304

7.86

Tool L2

%Elongation in 50 mm specimen

Poisson’s Ratio N

Coef. of Therm. Expansion A (10–6)/°C

290

10

0.35

23

186

12

0.35

24

669

–

0.6

0.28

12

276

572

–

5

0.28

12

–

241

241

–

35

0.35

18

345

–

655

655

–

20

0.34

17

152

152

–

276

276

152

1

0.30

26

75

250

250

–

400

400

–

30

0.32

12

75

345

345

–

450

450

–

30

0.32

12

193

75

207

207

–

517

517

–

40

0.27

17

8.16

200

75

703

703

–

800

800

–

22

0.32

12

4.43

120

44

924

924

–

1,000

1,000

–

16

0.36

9.4

Low Strength

2.38

22.1

–

–

–

12

–

–

–

–

0.15

11

High Strength

2.37

29.0

–

–

–

38

–

–

–

–

0.15

11

Plastic

Kevlar 49

1.45

131

–

–

–

–

717

483

20.3

2.8

0.34

–

Reinforced

30% Glass

1.45

72.4

–

–

–

–

90

131

–

–

0.34

–

26d

6.2d

–

0.29e

–

36d

6.7d

–

0.31e

–

Materials Metallic Aluminum Wrought Alloys Cast Iron Alloys Copper Alloys

Red Brass C83400 Bronze C86100

Magnesium Alloy Steel Alloys Titanium Alloy

[Am 1004-T61]

[Ti-6Al-4V] Nonmetallic

Concrete

Wood Select Structural Grade a

Douglas Fir

0.47

13.1

–

–

–

–

2.1c

White Spruce

3.60

9.65

–

–

–

–

2.5c

Specific values may vary for a particular material due to alloy or mineral composition,mechanical working of the specimen,or heat treatment. For a more exact value reference books for the material should be consulted. b The yield and ultimate strengths for ductile materials can be assumed equal for both tension and compression. c Measured perpendicular to the grain. d Measured parallel to the grain. e Deformation measured perpendicular to the grain when the load is applied along the grain.

Average Mechanical Properties of Typical Engineering Materialsa (U.S. Customary Units) Specific Weight (lb/in3)

Moduls of Elasticity E (103) ksi

Modulus of Rigidity G (103) ksi

2014-T6

0.101

10.6

3.9

60

60

25

68

68

6061-T6

0.098

10.0

3.7

37

37

19

42

42

Gray ASTM 20

0.260

10.0

3.9

–

–

–

26

Malleable ASTM A-197

Materials

Yield Strength (ksi) Ultimate Strength (ksi) SY Su Tens. Comp.b Shear Tens. Comp.b Shear

%Elongation in 2 in. specimen

Poisson’s Ratio N

Coef. of Therm. Expansion A (10–6)/°F

42

10

0.35

12.8

27

12

0.35

13.1

96

–

0.6

0.28

6.70

Metallic Aluminum Wrought Alloys Cast Iron Alloys

0.263

25.0

9.8

–

–

–

40

83

–

5

0.28

6.60

Red Brass C83400

0.316

14.6

5.4

11.4

11.4

–

35

35

–

35

0.35

9.80

Bronze C86100

0.319

15.0

5.6

50

50

–

35

35

–

20

0.34

9.60

0.066

6.48

2.5

22

22

–

40

40

22

1

0.30

14.3

Structural A-36

0.284

29.0

11.0

36

36

–

58

58

–

30

0.32

6.60

Structural A992

0.284

29.0

11.0

50

50

–

65

65

–

30

0.32

6.60

Stainless 304

0.284

28.0

11.0

30

30

–

75

75

–

40

0.27

9.60

Tool L2

0.295

29.0

11.0

102

102

–

116

116

–

22

0.32

6.50

0.160

17.4

6.4

134

134

–

145

145

–

16

0.36

5.20

Low Strength

0.086

3.20

–

–

–

1.8

–

–

–

–

0.15

6.0

High Strength

0.086

4.20

–

–

–

5.5

–

–

–

–

0.15

6.0

Plastic

Kevlar 49

0.0524

19.0

–

–

–

–

104

70

10.2

2.8

0.34

–

Reinforced

30% Glass

0.0524

10.5

–

–

–

–

13

19

–

–

0.34

–

3.78d

0.90d

–

0.29e

–

5.18d

0.97d

–

0.31e

–

Copper Alloys

Magnesium Alloy Steel Alloys Titanium Alloy

[Am 1004-T61]

[Ti-6Al-4V] Nonmetallic

Concrete

Wood Select Structural Grade

Douglas Fir

0.017

1.90

–

–

–

–

0.30c

White Spruce

0.130

1.40

–

–

–

–

0.36c

a Specific

values may vary for a particular material due to alloy or mineral composition,mechanical working of the specimen,or heat treatment. For a more exact value reference books for the material should be consulted.

b The

yield and ultimate strengths for ductile materials can be assumed equal for both tension and compression.

c Measured

perpendicular to the grain.

d Measured

parallel to the grain.

e Deformation

measured perpendicular to the grain when the load is applied along the grain.