### 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 ¾-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;

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. BoltBody. 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. ¼ 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 ¼ .08 .04 1.82 2.27 2.72 3.18 4.54 3.04 3.83 4.60 5.37 ⅜ 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 ½ 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 ¾ 10 .62 5/8 .44 .30 12 .00 15 18 21 30 18 22 26 31 ⅞ 9 .73 ¾ .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 1¼ 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 1½ 6 1.28 1 3/32 1.77 l.29 51 64 77 90 128 71 88 106 124 1⅝ 5½ 1.39 1⅝ 2.07 1.51 61 78 92 109 153 83 104 124 145 1¾ 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 4½ 1.71 1¾ 3.14 2.30 92 115 138 161 230 126 157 186 220 2¼ 4½ 1.96 1 /21/22 3.98 3.02 125 156 187 218 312 159 199 238 278 2½ 4 2.17 23/16 4.91 3.72 148 185 222 259 370 196 245 294 344 2¾ 4 2.42 27/16 5.94 4.62 184 230 276 322 460 237 297 356 416 3 3½ 2.63 2⅜ 7.07 5.43 218 272 326 381 544 283 353 424 495 3¼ 3½ 2.88 210/20 8.29 6.51 264 330 396 462 660 332 415 498 581 3½ 3¼ 3.10 31/20 9.62 7.55 302 376 452 528 754 385 481 577 673 3¾ 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 4¼ 2⅞ 3.80 313/16 14.19 11.33 452 565 678 791 1,130 567 709 851 993 4½ 2¼ 4.03 41/20 15.90 12.74 507 634 760 888 1,268 636 795 951 1,113 4¾ 2⅛ 4.25 43/22 17.72 14.22 567 709 851 1,993 1,420 709 886 1,063 1,240 5 2½ 4.48 4⅛ 19.63 15.76 630 788 946 1,103 1,676 785 982 1,178 1,374 5½ 2¾ 4.95 421/32 23.76 19.27 770 962 1,154. 1,347 1,924 950 1,188. 1,425 1,663 6 2¼ 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.

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. At4,000 lbs. per sq. in. At5,000 lbs.per sq. in. At6,000 lbt.per sq. in. At7,000 lbn.per sq. in. At 10,000 lbn.per. sq. in. At4,000 lbs. per sq. in. At5,000 lbs.per sq. in. At 6,000 lbs.per sq. in. At 7,000 lbs. per sq. in. ½ 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 ½ 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 ¾ 10 .62 1/3 .44 .30 12.00 15 18 21 so 18 33 26 31 3/8 9 .73 ¾ .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 1¼ 7 .94 31/32 .99 .69 28 34 41 48 69 40 50 60 70 1¼ 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 1½ 6 1.28 1/32 1.77 1.29 51 64 77 90 128 71 88 106 124 1½ 5½ 1.39 1 13/33 2.07 ! 1.51 61 78 92 109 158 88 104 124 145 1¾ 5 1.49 14 2.40 1.74 70 88 106 123 176 96 130 144 168 17/8 5 1.61 1½ 2 76 ! 2.05 81 101 123 142 308 110 138 166 198 2 44 1 71 1¾ 3.14 2.30 93 115 138 161 230 126 157 186 220 2¼ 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 2¼ 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 3¼ 34 2.88 2/35 8.29 6.51 264 330 396 462 660 332 415 498 581 3½ 3¼ 3.10 31/32, 9.62 7 55 302 376 452 528 754 385 481 577 678 3¼ 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 4¼ 27/4 3.80 3½ 14.19 11.33 453 565 678 791 1.180 567 709 851 998 44 2½ 4.03 48/12 15.90 12.74 507 634 760 888 1.268 636 795 951 1.113 4¾ 2½ 4 25 4/2/32 17.72 14.22 567 709 851 1.993 1.430 709 886 1.068 1.240 5 2¾ 4 48 44 19 63 15.76 630 788 946 1.103 1.676 785 982 1,178 1.374 54 2¾ 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 2¼ 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.