Theory

A direct tensile stress is induced in a bolt when it carries a load exerted along its axis. This load must be taken by the section of the bolt at the bottom of the thread. If the area at the root of the thread is πd1 2 / 4, and if S is the allowable stress per square inch, then the internal resistance of the bolt is Sπd1 2 / 4. Equating the external load to the internal strength we have:

W = Sπd1 2 / 4 (98)

For bolts which are used to clamp two machine parts together so that they will not separate under the action of an applied load, the initial tension of the bolt must be at least equal to the applied load. If the applied load is W, then the parts are just about to separate when I = W. Therefore the above relation for strength is applicable. As the initial tension to prevent separation should be a little greater than W, a value of S should be chosen so that there will be a margin of safety. For ordinary wrought iron and steel, S may be taken at 6,000 to 8,000.

If, however, the joints must be sach that there is no leakage between the surfaces, as in the case of a steam cylinder head, and supposing that elastic packings are placed in the joints, then a much larger margin should be made, for the maximum load which may come on the bolt is I + W, where W is the proportional share of the internal pressure carried by the bolt. In such cases S = 3,000 to 5,000, using the lower value for bolts of less than -inch diameter.

The table given on page 154 will be found very useful in proportioning bolts with U. S. standard thread for any desired fiber stress.

To find the initial tension due to screwing up the nut, we may assume the length of the handle of an ordinary wrench, measured from the center of the bolt, as about 16 times the diameter of the bolt. For one turn of the wrench a force F at the handle would pass over a distance 2nl, and the work done is equal to the product of the force and space, or F X 2nl. At the same time the axial load P would be moved a distance p along the axle. Assuming that there is no friction, the equation for the equality of the work at the handle and at the screw is;

Theory 10071

Fig. 56a.

F2nl = Pp. (99)

Friction, however, is always present; hence the ratio of the useful work (Pp) to the work applied (F2nl) is not unity as above relationa assume. From numerous experiments on the friction of screws and nuts, it Las been found that the efficiency may be as low as 10 per cent. Introducing the efficiency in above equation, it may be written:

Table For Strength Of Bolts. U. S. Standard Thread

BOLT

DIAMETERS.

AREAS.

APPROXIMATE TENSILE STRENGTH.

AT Bottom OF THREAD.

(In HUNDREDS OF POUNDS).

Approximate Shearing Strength.

Full Bolt Diameter.

(In Hundreds of Pounds).

Diameter. Inches.

Thread per inch.

Bottom of Thread.

Drill.

Bolt

Body.

Bottom of

4,000 lbs.

5.000 lbs. per sq. in.

per sq. in.

4.000 lbs. per sq,. in.

5.000 lbs.

6,000 lbs.

7.000 lbs.

20

.18

4/16

.05

.03

1.08

1.35

1.60

1.88

2.69

1.96

2.45

2.95

3.43

5/16

18

.24

.08

.04

1.82

2.27

2.72

3.18

4.54

3.04

3.83

4.60

5.37

16

.29

5/14

.11

.07

2.71

3.39

4.07

4.75

6.78

4.40

5.52

6.62

7.73

7/16

14

.34

22/24

.15

.09

2.73

4.67

5.60

6.53

9.33

6.00

7.51

9.02

11.00

13

.40

13/22

.20

.13

5.00

6.25

7.50

8.75

12.00

7.84

9.81

12.00

14

5/16

12

.45

15/22

.25

.16

6.48

8.10

9.62

11.00

16

9.92

12.00

15

17

4/8

11

.51

17/32

.31

.20

8.04

10.00

12.00

14

20

12 .00

15

18

21

10

.62

5/8

.44

.30

12 .00

15

18

21

30

18

22

26

31

9

.73

.60

.42

17

24

25

29

42

24

30

36

42

1

8

.84

2/32

.78

.55

22

27

33

38

55

31

39

47

55

1⅛

7

.94

21/12

.99

.69

28

34

41

48

69

40

50

60

70

1

7

1.06

1 3/32

1.23

.89

31

39

47

55

78

49

61

74

86

1⅜

6

1.16

1 3/18

1.48

1.05

42

53

64

73

106

59

74

89

104

1

6

1.28

1 3/32

1.77

l.29

51

64

77

90

128

71

88

106

124

1⅝

5

1.39

1⅝

2.07

1.51

61

78

92

109

153

83

104

124

145

1

5

1.49

1⅛

2.40

1.74

70

88

106

123

176

96

120

144

168

1⅞

5

1.61

1⅝

2.76

2.05

81

101

122

142

203

110

138

166

193

2

4

1.71

1

3.14

2.30

92

115

138

161

230

126

157

186

220

2

4

1.96

1 /21/22

3.98

3.02

125

156

187

218

312

159

199

238

278

2

4

2.17

23/16

4.91

3.72

148

185

222

259

370

196

245

294

344

2

4

2.42

27/16

5.94

4.62

184

230

276

322

460

237

297

356

416

3

3

2.63

2⅜

7.07

5.43

218

272

326

381

544

283

353

424

495

3

3

2.88

210/20

8.29

6.51

264

330

396

462

660

332

415

498

581

3

3

3.10

31/20

9.62

7.55

302

376

452

528

754

385

481

577

673

3

3

3.32

311/22

11.04

8.64

344

430

516

602

860

442

552

663

773

4

3

3.57

310/20

12.57

9.99

396

496

594

693

990

503

628

754

880

4

2⅞

3.80

313/16

14.19

11.33

452

565

678

791

1,130

567

709

851

993

4

2

4.03

41/20

15.90

12.74

507

634

760

888

1,268

636

795

951

1,113

4

2⅛

4.25

43/22

17.72

14.22

567

709

851

1,993

1,420

709

886

1,063

1,240

5

2

4.48

4⅛

19.63

15.76

630

788

946

1,103

1,676

785

982

1,178

1,374

5

2

4.95

421/32

23.76

19.27

770

962

1,154.

1,347

1,924

950

1,188.

1,425

1,663

6

2

5.42

5 7/16

28.27

23.09

923

1,159

1,384

1,617

2,307

1,128

1,414

1,696

1,979

Pp / F2πl = 1/10 (100).

Assuming that 50 pounds is exerted by a workman in tightening up the nut on a 1-inch bolt, the equation above shows that P = 4,021 pounds; or the initial tension is somewhat loss than the tabular safe load shown for a 1-inch bolt, with S assumed at 10,000 pounds per sq. inch.

Table For Strength Of Bolts U S Standard Thread 10072

Fig. 58.

For shearing stresses the bolt should be fittea so that the body of the bolt, not the threads, resists the force tending to shear off the bolt perpendicular to its axis. The internal strength of the bolt to resist shear is the allowable stress S times the area of the bolt in shear, or Sπd3 / 4 . If W represents the external force tending to shear the bolt the equality of the external force to the internal strength is : w = Snd2 / 4. (101)

Table For Strength of Bolts. U. S. Standard Thread

Bolt.

DlAMETERS.

AREAS.

APPROXIMATE TENSILE StRENgth.

At Bottom of Thread.

(In Hundreds or Pounds).

Approximate Shearing Strength.

Full Bolt Diameter.

(In Hundreds or Pounds).

Inches.

Threads per inch.

Bottom of Thread.

Tap Drill.

Bolt Body.

Bottom of Thread.

At

4,000 lbs. per sq. in.

At

5,000 lbs.

per sq. in.

At

6,000 lbt.

per sq. in.

At

7,000 lbn.

per sq. in.

At 10,000 lbn.

per. sq. in.

At

4,000 lbs. per sq. in.

At

5,000 lbs.

per sq. in.

At 6,000 lbs.

per sq. in.

At 7,000 lbs. per sq. in.

20

.18

3/16

.05

.03

1.08

1.35

1.60

1.88

3.69

1.96

3.45

3.95

3.48

8/16

18

.24

X

.08

.04

1.83

3.37

2.72

3.18

4.54

3.04

3.88

4.60

5.87

3/8

16

.29

A

.11

.07

2.71

3.39

4.07

4.75

6.78

4.40

5.53

6.63

7.71

7/16

14

.34

a

.15

.09

3.73

4.67

5.60

6.58

9.38

6.00

7.51

9.03

11.00

13

.40

ii

.20

.13

5.00

6.25

7.60

8.75

13.00

7.84

9.81

13.00

14

9/16

12

.45

4?

.25

.16

6.48

8.10

9.68

11.00

16

9.93

13.00

15

17

96

11

.51

17/22;

.31

.30

8.04

10.00

12.00

14

30

13.00

15

18

21

10

.62

1/3

.44

.30

12.00

15

18

21

so

18

33

26

31

3/8

9

.73

.60

42

17

34

35

29

43

34

so

86

42

1

8

.84

27

.78

.55

22

37

S3

38

55

31

89

47

55

1

7

.94

31/32

.99

.69

28

34

41

48

69

40

50

60

70

1

7

1.06

13/32

1.23

.89

31

39

47

55

78

49

61

74

86

1 3/8

6

1.16

13/16

1.48

1.05

43

53

64

73

106

59

74

89

104

1

6

1.28

1/32

1.77

1.29

51

64

77

90

128

71

88

106

124

1

5

1.39

1 13/33

2.07

! 1.51

61

78

92

109

158

88

104

124

145

1

5

1.49

14

2.40

1.74

70

88

106

123

176

96

130

144

168

17/8

5

1.61

1

2 76

! 2.05

81

101

123

142

308

110

138

166

198

2

44

1 71

1

3.14

2.30

93

115

138

161

230

126

157

186

220

2

44

1.96

l21/32

3. 98

3.02

125

156

187

218

312

159

199

238

378

24

2.17

. 4.91

3.72

148

185

222

259

870

196

345

294

844

2

4

2.42

2ft

5.94

4.62

184

230

276

323

460

287

297

856

416

3

34

2 63

2S

7 07

5.43

218

272

326

381

544

283

853

424

495

3

34

2.88

2/35

8.29

6.51

264

330

396

462

660

332

415

498

581

3

3

3.10

31/32,

9.62

7 55

302

376

452

528

754

385

481

577

678

3

3

3.32

3 1/22

11.04

8.64

344

430

516

602

860

443

552

663

778

4

3

3.57

311/21

! 12.57

9.99

396

495

594

693

990

503

628

754

880

4

27/4

3.80

3

14.19

11.33

453

565

678

791

1.180

567

709

851

998

44

2

4.03

48/12

15.90

12.74

507

634

760

888

1.268

636

795

951

1.113

4

2

4 25

4/2/32

17.72

14.22

567

709

851

1.993

1.430

709

886

1.068

1.240

5

2

4 48

44

19 63

15.76

630

788

946

1.103

1.676

785

982

1,178

1.374

54

2

1.95

4/31/22

23 76

| 19.27

770

963

1.154

1.347

1.924

950

1.188

1.425

1,663

6

2

5 42

5ft

28.27

23 09

933

1.159

1.384

1 617

2.307

1.128

1.414

1.696

1,979

Reference to the table on page 154 for the shearing strength of bolts, may be made to save the labor of calculations.

Let Fig. 58 represent a square thread screw for the transmission of motion. The surface on which the axial pressure bears, if n is the number of threads in the nut, is π/4 (d2- d1 2) n. Suppose that a pressure of k pounds per square inch is allowed on the surface of the thread. Then the greatest permissible axial load P must not exceed the allowable pressure; or, equating,

P = k π/4 (d2 - d1 2)n. (I02)

The value of k varies with the service required. If the motion be slow and the lubrication very good, h may be as high as 900. For rapid motion and doubtful lubrication, k may not be over 200. Between these two extremes the designer must use his judgment, remembering that the higher the speed the lower is the allowable bearing pressure.