THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES GIFT OF John S.Prell ' MACHINE DESIGN PART I. FASTENINOS BY WILLIAM LED YARD CATHCART ADJUNCT PROFESSOR OF MECHANICAL ENGINEERING, COLUMBIA UNIVERSITY; MEMBER AMERICAN SOCIETY OP MECHANICAL ENGINEERS; MEMBER OP THE AMERICAN SOCIETY OF NAVAL ENGINEERS; MEMBER OF THE SOCIETY OF NAVAL ARCHITECTS AND MARINE ENGINEERS. NEW YORK D. VAN NOSTRAND COMPANY 23 MURRAY AND 27 WARREN STS. 1903 JOHN S. PRELL Civil & Mechanical Engineer. SAN FRANCISCO, CAL. COPYRIGHT, 1903, BY D. VAN NOSTRAND COMPANY Eagiwerug Library TJ Y.I PREFACE. THE main purpose of this book is to present, in compact lorm for the use of the student and designer, modern American data from the best practice in the branch of Machine Design to which the work refers. The theoretical treatment of the subject has also been given fully ; but this has been done for completeness only, since that field has been covered exhaustively by able writers- Scientific analysis and the records of practice are both essential to success in the design of machine members, but neither alone is trustworthy. The former predicts only those stresses which pre- vail under normal conditions arid ignores the overload, the rough handling, or the slight accident which the machine may meet and against which it should not fail. Practical data, on the other hand, show only the proportions which constructors have given in specific cases of stress and service and empirical formulae founded upon them may give results wide of the mark, if the inherent limitations of these formulae be exceeded. The problem of design is one whose many elements vary continually in num- ber, character, and magnitude, and, for its solution, theoretical analysis, precedent, and the ripened judgment of the designer are required. Elsewhere acknowledgment has been made of the courtesy of the many officials and companies who have furnished information. ^ The author's thanks are due especially to Rear Admiral George W. Melville, Engineer-in-Chief, U. S. Navy ; Professor Philip R. Alger, U. S. Navy ; Professor J. Irvin Chaffee ; Leo Morgan, Esq.; J. M. Allen, Esq., President the Hartford Steam Boiler Inspection and Insurance Company ; C. C. Schneider, Esq., Vice President the American Bridge Company ; Messrs. William Sellers and a Company ; the Baldwin Locomotive Works ; and the Newport News Shipbuilding and Dry Dock Company. The author desires 3 also to express his deep indebtedness to Stevenson Taylor, Esq., Ci President of Webb's Academy and Vice President of the W. and A. Fletcher Co., whose examination of, and additions to, the text have added materially to the value of this work. COLUMBIA UNIVERSITY, NEW YORK, IO February, 1903. 733412 CONTENTS. CHAPTER I. PACK. SHRINKAGE AND PRESSURE JOINTS :...... i i. General formulae. 2. Proportions of the joint. 3. Metals. 4. Forcing pressures. 5. Shrinkage temperatures. 6. Shrinkage vs. pressure fits. 7. Stationary engines, data from practice. 8. Marine engines, data from practice. 9. Railway work, data from practice. 10. Shrinkage in gun construction. CHAPTER II. SCREW FASTENINGS : . . . . . . . .42 ii. Triangular vs. square threads. 12. Requirements of the screw-thread. 13. Elements of the screw-thread. 14. The U. S. standard (Sellers) thread. 15. Modifications of the Sellers system. 16. The sharp V thread. 17. The Whitworth thread. 18. The sharp V, Sellers, and Whitworth threads. 19. The French Stand- ard thread. 20. The International Standard thread. 21. The British Association Standard thread. 22. The square thread. 23. The |--V thread. 24. Special threads. 25. Machine and wood screws. 26. Pipe threads. 27. Stresses in screw-bolts. 28. Stresses in nuts. 29. Efficiency of the screw. 30. Types of screw fastenings. 31. Methods of manufacture. 32. Materials. 33. Nut -locks. 34. Wrenche s. CHAPTER III. RIVETED JOINTS : THEORY AND FORMULAE : . . . .127 35. Rivets. 36. Proportions of rivets. 37. Rivet and plate metals. 38. Rivet-holes. 39. Boiler-seams : longitudinal, cir- cumferential, and helical. 40. Forms of riveted joints. 41. The elements of a riveted joint. 42. The theoretical strength of riveted joints. 43. General formulae for boiler-joints. 44. The thickness of shell sheets. 45. The stresses in riveted joints. 46. The fric- tion of riveted joints. v vi CONTENTS. CHAPTER IV. RIVETED JOINTS : TESTS AND DATA FROM PRACTICE : . . .192 47. Tests of multiple-riveted, double-strapped butt joints. 48. Riveting machines. 49. Riveted joints, marine boilers. 50. Riv- eted joints, locomotive boilers. 51. Riveted joints, stationary boilers. 52. Riveted joints, structural work. 53. Riveted joints, hull plating. CHAPTER V. KEYED JOINTS: PIN-JOINTS: . 251 54. Forms of keys. 55. Proportions of keys. 56. Stresses on keys. 57. Through-keys: forms. 58. Through-keys: stresses. 59. Pin-joints. TABLES. I. Shrinkage vs. Pressure Fits (Wilmore) . . - . 15 II. Pressure Fits (Lane and Bodley Co.) B'.. . . . 17 III. Shrinkage and Pressure Fits (Russell Engine Co.) . 17 IV. Pressure Fits, Stationary Engines . . . . .18 V. Shrinkage and Pressure Fits (Buffalo Forge Co.) . .18 VI. " " " " (B. F. Sturtevant Co.) . . 18 VII. " " " " Summary of Practice . . 19 VIII. " " " " (Union Iron Works) . . 23 IX. " Fits (Am. Railway Master Mechanics' Asso'n) 25 X. U. S. Standard (Sellers) Bolts and Nuts . . . .50 XL Standard Bolts and Nuts, U. S. Navy (Bureau of Steam Engineering) . . . . . . . 52 XII. Manufacturers' Standard Dimensions of Bolt-heads (Am. Iron and Steel M'f g Co.) 53 XIII. Manufacturers' Standard Dimensions of Hot-pressed Nuts (Am. Iron and Steel M'f'g Co.) . . . . -53 XIV. Round Slotted Nuts (Newport News Shipbuilding and Dry Dock Co.) 54 XV. Whitworth System, Bolts and Nuts 55 XVI. French Standard Screw Threads 58 XVII. International Standard Screw Threads . . . -59 XVIII. British Association Standard Thread 60 XIX. Standard Square Threads (William Sellers and Co.) . . 61 XX. Standard Square Threads (Newport News S. B. and D. D. Co.) .62 XXI. X- v Screw Thread (William Sellers & Co.) ... 63 XXII. Standard Bastard Screw Threads (Newport News S. B. & D. D. Co.) 64 XXIII. Acme Standard (29) Screw Thread 65 XXIV. Proportions of Armor Bolts, U. S. Navy . . . .66 XXV. Machine Screws (Tyler) 69 XXVI. Wrought Iron Welded Tubes (Briggs' Standard) . .71 XXVII. Ratio of Bearing Pressure to Tensile Stress (Sellers Threads) ......... 74. XXVIII. " Grooved " Specimens (Howard) 76 XXIX. Threaded and " Grooved " Specimens (Martens) . . 77 XXX. Coefficients of Friction for Square Threads (Kingsbury) . 88 Coefficients of Friction With Various Lubricants (Kings- bury) . , . . . . . . .88 vii viii TABLES. XXXI. Steel Studs for Cylinder Covers (U. S. Navy) ... 94 XXXII. Approximate Efficiencies of Square Threaded Screws (Good- man) .......... 97 XXXIII. Safe Loads for U. S. Standard Bolts (Williams) . . .103 XXXIV. Tap-bolts and Set-screws (Newport News S. B. and D. D. Co.) 105 XXXV. Eye-bolts (Union Iron Works) 107 Dimensions and Conditions of Stay-Bolts (Sprague and Tower) 108 XXXVI. Collar Nuts with Locking Screws (Union Iron Works). . 121 XXXVII. Engineers' Wrenches, Single Head (J. H. Williams & Co.) 124 XXXVIII. Check-Nut Wrenches (J. H. Williams & Co.) . . .125 Wrenches, International Standard Nuts . . . .126 XXXIX. Proportions of Rivet-Heads (Am. Iron and Steel M'f g Co.) 128 XL. Proportions of Rivet-Heads (Champion Rivet Co.) . .129 Tests of Drilled and Punched Plates (Kirkaldy) . . .137 Efficiencies of Butt Joints, Double-Strapped (Traill) . .163 XLI. Tests of Multiple Riveted, Double-Strapped, Butt-joints (U. S. Navy) . 196 XLII. Boiler Rivets (U. S. Navy) ... . . .207 Proportions of Joints, Cylindrical Boilers (U. S. Navy) . 208 Weight of Boiler Rivets (U. S. Navy) . . . .209 XLIII. Locomotive Boilers, Single Riveted Longitudinal Seams (Baldwin Locomotive Works). . . . . .211 XLIV. Locomotive Boilers, Double Riveted Seams (Baldwin Loco- motive Works) 212 XLV. Locomotive Boilers, Quadruple Butt-joint Seams with Welded Ends (Baldwin Locomotive Works). . . . .212 XLVI. Sextuple Butt-joint Seams with Welded Ends (Baldwin Loco- motive Works) . . .213 Locomotive Boilers, Location and Proportions of Seams (Baldwin Locomotive Works). . . . . .214 XLVH., XLVIII., XLIX., L. Stationary Boilers, Riveted Joints (Hart- ford Steam Boiler Inspection and Insurance Co.). 218, 219 LI. Proportions of Rivet-Heads (American Bridge Co.) . . 220 Weight of Rivets, Structural Work 221 LII. Staggering of Rivets (American Bridge Co.) . . . 222 LIII. Rivet Spacing in Angles (Am. Bridge Co.) . . . 223 LIV. Shearing and Bearing Values of Rivets (Am. Bridge Co.) . 225 LV. Angles, Sectional Area (Am. Bridge Co.) . ' . . .230 LVL, LVII. Riveted vs. Bolted Joints (Berlin Iron Bridge Co.) . 234 LVII-A, LVIII. Proportions of Seams and Rivets, Torpedo-boat and Ship-work (U. S. N.) ..... 240, 241 LIX. Diameter of Rivers, Hull-work (U. S. N.) . . . . 242 LX. Allowance for Rivet-Points, Hull-work (U. S. N.) . . 243 TABLES. ix LXI. Breadth of Laps and Straps, Hull-work (U. S. N.) . . 243 LXII. Spacing of Rivets, Hull-work (U. S. N.) . . . .244 LXIII. Minimum Thickness of Outside Plating and Flat Plate Keel (American Bureau of Shipping) 246 LXIV. Diameter of Rivets, Breadth of Laps, Lapped Butts, Width of Butt-straps, and Breadth of Edge Strips on Plate Seams, Hull-work (American Bureau of Shipping) . . . 247 Plating and Transverse Seams (Am. Bu. Shipping) . . 250 LXV. Square Keys (Richards) . . . . . . .257 LXVI. Flat " " 257 LXVII. Feather " " 257 LXVIII. Keys for Shafting (William Sellers and Co.) . . .258 LXIX. " Machine Tools (William Sellers and Co.) . . 258 LXX. Key-ways for Milling Cutters (Brown and Sharpe M'f'g Co.) 259 LXXI. Stationary Engines, Crank and Wheel Keys . . -259 LXXII. Marine Engines, Keys and Key-ways (Newport News S. B. and D. D. Co.) 260 Taper-Pins (Morse Twist Drill and Machine Co.) . . 268 LXXIII. Stationary Engines, Connecting Rod Ends, Bolted Strap . 269 LXXIV. Maximum Bending Moments on Pins (American Bridge Co.) 280 LXXV. Pins with Lomas Nuts (Am. Bridge Co.) .... 282 LXXVI. " Cotters " "".... 283 LXXVII. Eye-Bars (Am. Bridge Co.) 284 AUTHORITIES QUOTED. Alger, Prof. Philip R., U. S. N., 29 Allen, J. M., 134, 192 American Boiler Manufacturers' Asso- ciation, 215 " Bridge Company, 219, 220, 222, 223, 225, to 284, inc. 230, 280 Bureau of Shipping, 245 to 250, inc. " Engineer and Railroad Jour- nal, 134, 266 Iron and Steel M'fg Co., 53, 78, 128 Machinist, 104, 105, 203 " Railway Master Mechanics' Asso'n, 25 Bach, Prof. C., 181, 187, 188, 189, 191 Bailey, F. H., Lieut. Com'd'r, U. S. N., 261 Baldwin Locomotive Works, 192, 210 to 214, inc. Barr, Prof. John H., 100 Bauschinger, Prof., 133 Berlin Iron Bridge Co., 234, 235 Birnie, Major Rogers, U. S. A., 28, 29 Bond, Geo. M., 50, 68 Box, Thos., 78 Briggs, Robert, 69, 70 Broomall, 14 Brown and Sharpe M'fg Co., 258, 259 Bryan, C. W., C.E., 227, 233 Buffalo Forge Co., 16, 18 Bulletin Soc. d'Encour, 59 Bureau of Construction and Repair, Burr 20, 51, 73, 94, 114, 196, 205, 207, 277 Prof. W. H., 4, 13, 227 134. U. S. N., 66, 109, 237 to 244, inc. Bureau of Ordnance, U. S. N., 66, 67 McBride, Jas., 99 " Steam Engineering, U. S. N., i Meier, E. D., 215 x Canaga, Com'd'r A. B., U. S. N., 148 Champion Rivet Co., 129, 132 Chief of Ordnance, U. S. A., 36, 67, 192 Cramp, Edwin S., 236 Wm. S. and E. B. Co., 192 Clavarino, 6 Colby, A. L., 131 Cotterill, Prof. J. H., 4, 277 Goodman, Prof. John, 97 Harlan and Hollingsworth Co., 24 Hartford Steam Boiler Insp. and Ins. Co., 216 to 219, inc., 275 Howard, Jas. E., 75 Johnson, Prof. J. B., 76, 133, 227 Jones, Prof. F. R., 255 Kennedy, Prof., 165 King, Major W. R., U. S. A., 78 Kingsburg, Prof. Albert, 87, 88 Kirkaldy, 75, 78, 137 Lame, 6 Lane and Bodley Co. , 17 Lanza, Prof. G., 101, 105, 106 Lewis, Wilfred, 89, 98 Lineham, Prof. W. J., 14 Linnard, Naval Constructor J. H., U. S. N., 236 Marks, W. D., C.E., 264 Martens, Prof. A., 76, 79, 91 AUTHORITIES QUOTED. Melville, Rear Admiral Geo. W., U. S. N., 112 Merriman, Prof. M., 6, 79 Midvale Steel Co., 22 Morse Twist Drill and Machine Co., 268 New York Shipbuilding Co. , 24 Newport News Shipbuilding and Dry Dock Co., 54, 62, 64, 105, 260 Niles Tool Works Co., 25 Porter, H. F. J., 19 Rankine, Prof. W. J. M., 4, 182 Reuleaux, Prof. F. , 6, 13, 74, 166, 168, 254 Richards, John, 257, 263 Rivet-Dock Co., 113 Russell Engine Co., 17 Schell, Lieut. Com'd'r, U. S. N., 171 Seaton, A. E. , 170 Seaton & Rounthwaite, 102 Sellers, William, & Co., 61, 63, 2-58 Smith, Prof. A. W., 82 Sprague, Chief Eng'r Jas. W., U. S. N., 108, 277 Sternbergh, J. H., 114, 128 Stoney, B. B., 127, 130, 165, 181 Stromeyer, C. E., 186 Sturtevant, B. F., Co., 17, 1 8 Sweet, Prof. J. E., 82 Thurston, Prof. R. H., 14 Thury, Prof., 59 Tower, Chief Eng'r Geo. E., U. S. N., 108, 277 Townsend, David, 112, 136 Traill, Thos. W., F. E. R. N., 134, 163, 170, 171, 210 Union Iron Works, 23, 107, 121 Unwin, Prof. W. C., 86, 145, 185 U. S. Board of Supervising Inspectors of Steam Vessels, 209 Weisbach, Dr. Julius, 74, 118, 119 Whitney M'f g Co., 253 Williams, H. D., 101, 102 Williams, J. H., & Co., 124, 125 Wilmore, Prof., 150 Wood, R. D., & Co., 202 JOHN S. PRELL Civil & Mechanical Engineer. SAN FRANCISCO, CAL. MACHINE DESIGN. CHAPTER I. SHRINKAGE AND PRESSURE JOINTS. Rigid connections of this character between members of a ma- chine or structure are of frequent application. The inner member of the pair to be united is made cylindrical or slightly conical in form ; the corresponding portion of the outer member is bored so that it is of the same shape, but less in diameter throughout. When, therefore, the latter is made to encircle the former, the re- sulting radial pressure, acting at the contact-surfaces, produces a frictional resistance to relative motion of the parts. In a Shrinkage Fit or joint, the outer member is expanded by heating, slipped in place, and held therein by the subsequent contraction in cooling. In a Pressure ("Press" or "Forced") Fit, the parts are driven together by hydraulic pressure. Joints of the latter type are, as a rule, restricted to members of moderate size crank-pins, cranks, and the wheels and axles of engines and cars being familiar examples. The shrinkage fit is applied, not only in the union of large mem- bers in which maximum resistance to relative motion is desired, as in the crank-shafts of engines of high power ; but, as well, in modern ordnance, where results of extreme accuracy are es- sential in order to obtain the desired inward pressure required to withstand the outward force of the gases generated in the powder- chamber. i. General Formulae. The final diameter of a joint made by shrinkage or pressure is intermediate between those of the parts before union, i. e., the inner member has been compressed and the outer expanded. These changes and the elasticity of the metal produce a radial compressive stress acting upon both members at the contact-sur- faces and a consequent circumferential stress or " hoop-tension " MACHINE DESIGN. fyS SHRINKAGE AND PRESSURE JOINTS. 3 within the outer member. The latter stress is a maximum at the joint and decreases rapidly toward the exterior. i. THIN BANDS. When the outer member is thin, as a band or tire, and the inner is, relatively, of large diameter, the compres- sion of the latter is so small as, frequently, to be negligible in practice. The stress of the shrinkage or forcing may then be considered as expended wholly in the expansion of the band. Assume then, as in Fig. I, an incompressible hub upon which is shrunk such a band, the stress upon the latter being within the elastic limit. Let: R = original radius of interior of band ; R = radius of hub ; t = tensile unit stress within band ; e t = unit elongation due to t ; E = modulus of elasticity of band-metal = ; / = unit radial pressure ; b = breadth (axial) of band ; T= thickness (radial) of band, expanded ; /= coefficient of friction. Then: Increase in length, interior of band = 2r:(R R^) ; Original length, interior of band = 27rR () ; r> 7p Elongation per unit of length = e t = ~ ; -"-o /? /? Unit tensile stress = t = Ee t = E ^ -. (l) -^o This tensile stress, t, acts throughout the band, tending to re- sist rupture of the latter on any diametral plane, as A-B. The total resistance opposed thus at A and B = 2(6 x Tx t). (2) The unit radial pressure, /, acts outward, equally at all points upon the band. The latter is, therefore, virtually in the condition of a thin cylinder, of length b and thickness T, subjected to in- ternal fluid pressure. In Fig. I the vertical component of the pressure / is that which tends to part the band on the horizontal 4 MACHINE DESIGN. plane A-B. For an elementary strip of the band, of length Rdd, and of breadth b, we have : Radial force on elementary strip = Rdd x b x p ; Parting force, elementary, on plane, A-B = Rdd x b x / sin 6; Parting force, total, on band = bpR I sin 6dd = 2bpR. (3) Equating (2) and (3) : Tt (R-R^T P ~ R- RR The resistance to movement at the contact-surface is equal to the product of the area of that surface, the radial pressure, and the coefficient of friction, /. e. : r> rj Resistance to slip = E -= - 2~bTf. (5) 2. THICK CYLINDERS. The method, as above, disregards the compression of the inner member, assumes the stress of forcing or shrinkage as expended wholly in expanding the band, and con- siders the unit-stress within the latter as uniform throughout the cross-section. The inner member cannot be incompressible and, therefore, the circumferential stress given by (i) is greater than that which would exist. The method is hence applicable only within the limits noted. In an outer member whose walls are rel- atively thick, the stresses at various radial distances differ widely in intensity ; and, for the determination of their magnitude, recourse must be had to the complex formulae deduced for the investigation of thick, hollow cylinders, subjected to internal fluid pressure. Of such formulae, those founded on the method of Lame * give, with- out the assumptions of Barlow or Brix, the character and intensity of the various stresses at any point within the cylinder walls. Thus, consider, as in Figs. 2 and 3, a horizontal hollow cylin- der, open at the ends, the latter being faced off in a plane normal the axis. Let this cylinder be filled with fluid, which is forced inward by two expanding plungers A, A, the result being the production of a fluid pressure upon the internal surface of the wall. From the construction and operation it is clear that, as the ends are free, the cylinder will remain a cylinder under stress ; *Rankine, "Applied Mechanics," 1869, p. 290. Burr, " Elasticity and Resist- ance," etc., 1897, p. 36. Cotterill, " Applied Mechanics," 1895, p. 408. SHRINKAGE AND PRESSURE JOINTS. 5 that a transverse section, taken normal to the axis "when at rest, remains thus normal under stress ; and that, on such a section, the resultant longitudinal stress is zero, both over the whole area and at every point thereof. Assume that the material is isotropic and that no stress, at any point, exceeds the elastic limit. Consider any point O within the cylinder wall. Let : Ry and R l = inner and outer radii of cylinder ; P and P l = inner and outer pressures upon cylinder ; t = circumferential stress at point ; p = radial pressure at point ; / = longitudinal stress = zero at point O ; r = radius of point 0. Then, from the deduction in 10: (23) . R? - R* R? - R* ' r* f~~ < X-*>' + K?-V It will be observed that the circumferential stress / varies in- versely as r 2 and is therefore a maximum at the cylinder-bore. This condition prescribes the useful limit of thickness for cylinders which are not under exterior compression. No such cylinder can be made sufficiently thick to withstand an internal pressure per sq. in. greater than the ultimate tensile strength per sq. in. of the metal, as is shown by equation (19). Since the working pressure of modern ordnance exceeds considerably the elastic limit in ten- sion of the material used, the necessity for the " built-up " system is apparent. With regard to formulae (23), it will be observed also that t may be either tensile or compressive, as the relations of the radii and pressures determine ; that / is always compres- sive ; and that both p and t are " apparent " and not " true " stresses, since the factor of lateral contraction has not been intro- duced with respect to them. Considering this factor : True Circumferential Stress = /_J/_(_ J^). (6) In a gun, the layer in which the breech-plug houses is under a di- rect longitudinal stress /, arising from the pressure upon the plug. This stress is a maximum at the face of the plug and diminishes rapidly toward the muzzle. If the apparent values of /, / and / 6 MACHINE DESIGN. be substituted in (6), the working equation for true circumferential stress will be obtained, which equation is Clavarino's principal formula for the investigation and design of guns.* 3. THICK HUBS. Professor Reuleaux, in The Constructor,^ gives, largely without deduction, certain working formulas, based upon those of Lame as above, which are especially applicable to the shrinkage or cold forcing of large machine members, such as cranks and wheel-hubs. Thus, consider Fig. 4, which represents a shaft or pin A, forced into a ring or hub B. The deduction applies, theoretically, to either shrinkage or forcing. The notation is : S v = radial compressive stress at r ; S 2 = circumferential tensile stress at r ; p = unit radial pressure upon contact-surfaces = S l ; E^ = modulus of elasticity, inner member ; E 2 = modulus of elasticity, outer member ; r^ = radius of pin before forcing ; r 2 = radius of hole before forcing ; r = radius of fit ; / = length of fit ; d = thickness, outer member, after forcing ; 3 , _ r i~ r 2 _ S i r ' r z ^2' Q = maximum forcing pressure required ; f = coefficient of friction = 0.2. Under the conditions shown in Fig. 4, the notation of equation (23) giving the value of t, when translated into that of The Constructor should be changed thus : ttoS 2 ', P to 5, ; P l to zero ; 7? n to r ; R^ to r -f o ; r remains as r. (a) Stresses and Allowances. Transforming the equation for t, in accordance with the above : (7) *Merriman, " Mechanics of Materials," 1899, p. 318. f Suplee's Translation, 1895, pp. 17, 18, 45-47. SHRINKAGE AND PRESSURE JOINTS. In Fig. 4 : Unit deformation (strain}, inner member = r r. Unit deformation (strain), outer member = ; -. From the definition of the modulus of elasticity : S, r.-r S z r r z p = and T=r = -. (9) Adding : r S + r^ = r-r (10) Whence : By definition and from (10) : From (n) and (12) : vSj and vS 2 are mutually dependent, their relation being expressed by (7) and (8). In view of this and by definition : Si-Sf. (6 4 ,O From (36, C~) and (64, ") : s, S L s s* The second term of each denominator is so small as to be negli- gible. Hence : * For convenience of reference, numbered formulae from The Constructor are given the same numeral, with " C" added. MACHINE DESIGN. If the value of the ratio -, be assumed or known, it may be sub- stituted in (7), thus giving that of the ratio, -- = p, i. e. : If: d = 0.500, i.ooo, 1.500, 2.000, 3.000 ; then p = 0.385, 0.600, 0.724, 0.800, 0.882. Since E v 2 and the allowable value of S 2 are known quantities, the values of

in (38, C), there will be obtained the total allowance for the prescribed diameter. 12 MACHINE DESIGN. 4. FORM. With regard to the form of the contact-surfaces, a slightly tapering hole and corresponding inner member have ad- vantages over the plain cylindrical shape, in that, with the latter, the entrance of the hole must withstand the strain of abrading and compressing the pin or shaft throughout the length of the fit. The tapered member, on the contrary, enters without contact for a considerable distance and is thus well guided ; the compression, upon engagement, is distributed over a greater area ; the parts are separated readily when a renewal of the fit is desired ; and the drawings may be marked : " Fit pin inches from the end of the hole," which is the most trustworthy way of measuring the allow- ance. The disadvantage of this form lies in the difficulty of secur- ing, with the accuracy required, the same taper in both members. 3. Metals. From (9) it will be seen that the radial stress of the inner mem- ber and the circumferential stress within the outer, depend directly upon the modulus of elasticity E of each material so stressed. This follows since E is a measure of the stiffness of a metal, i. e. y the stiffer the latter, the less will be the deformation (strain) under a given stress and the larger the modulus. The following are general values : ELASTIC LIMIT. Cast Iron. Wrought Iron. Steel. Tension 6,000 25,000 50,000 Compression 20,000 25.000 50,000 MODULUS OF ELASTICITY. Cast Iron. Wrought Iron. Steel. Tension 15,000,000 25,000,000 30,000,000 Compression . 15,000,000 25,000,000 30,000,000 The circumferential stress of the outer member is the important element, especially when that member is of cast iron, a metal which has, in tension, a very low elastic limit, as compared with that, in compression, of the steel or wrought iron of the inner member. Cast iron is also, in tension especially, a very uncertain metal, owing to differences in composition, in the size and form of the SHRINKAGE AND PRESSURE JOINTS. 13 casting, and in the intensity of the original shrinkage strains. Professor J. B. Johnson gives E for cast iron as varying from " 10,000,000 to 30,000,000 ; but, for ordinary foundry iron, it may be taken at from 12,000,000 to 15,000,000. * * * The modulus of cast iron is approximately the same in tension, compression and cross-bending." * Professor Burr, in commenting upon certain tensile tests of cast iron, says : "The metal is seen to be very irregular and unreliable in its elastic behavior. A large portion of the material can scarcely be said to have an elastic limit, although no apparent permanent set takes place under a considerable intensity of stress. In other words, although perhaps all tested specimens resume their original shape and dimen- sions for small intensities of stress, yet the ratio between stress and strain is seldom constant for essentially any range of stress."! 4. Forcing Pressures. The pressure required, at any given time during the process, of making the joint, depends, approximately, upon the radial stress, the character and area of the surfaces in contact, and the coeffi- cient of friction. 1. CHARACTER OF SURFACES. This will vary with different metals and with the standard of workmanship. If the surfaces are smooth but not accurately of the same form, the radial and forcing pressures will be irregular in intensity. With rough surfaces the frictional resistance will be increased ; and, in extreme cases, longi- tudinal cutting, uneven bearing, and lessened grip may follow. 2. COEFFICIENTS OF FRICTION. In a pressure fit there is not only surface abrasion but the material of the outer member must be forced aside by the forward part of the advancing inner mem- ber ; and, if the elastic limit of the softer metal be exceeded, some flow of the latter occurs. The resistance is not, therefore, purely frictional and the usual coefficients of friction do not give an ac- curate measure of its amount. In discussing shrinkage and pres- sure fits, Reuleaux takes /= 0.2 which is the value used by Wei s- bach for the usual metals in a dry state. The results of experiments presented in Table I. show, as a rule, much lower values of/ than that quoted above. On the other hand, Rennie, from ex- periments upon solids usually unlubricated, gives, for pressures * " Materials of Construction," 1898, p. 476. f " Elasticity and Resistance of Materials of Engineering," 1897, p. 279. 14 MACHINE DESIGN. per sq. in. ranging from i86| to 560 Ibs., results, for the coeffi- cient of rest, as follows : * Wrought iron on wrought iron, _/"= 0.25 to 0.41 ; Wrought iron on cast iron, /= 0.28 to 0.37 ; Steel on cast iron, /= 0.30 to 0.36. Abrasion occurred in the first case at 672 Ibs. pressure ; and, in the latter case, at 784 Ibs. Broomall f gives, for static friction, as above : Cast iron on cast iron, dry, f= 0.3 1 14; Steel on cast iron, dry, /= 0.2303 ; Steel on steel, dry, 7=0.4408. Since the value of the coefficient is affected by conditions as to motion and rest, temperature, lubrication, and speed of rubbing, reported results vary considerably. Both shrinkage and forced fits have higher radial pressures than those which prevail in the usual friction tests ; the resistance in forming a pressure fit is not purely frictional ; the force required to break such a joint may be less than that of making, if the elastic limit has been exceeded ; and pressure fits may be lubricated only to the extent of wiping the surface with oiled waste, although a lubricant of white-lead and oil, mixed to the consistency of paint, is frequently used to prevent cutting. In view of these conditions the application to these joints of the usual coefficients for unlubricated metals, is inadvisable. 5. Shrinkage Temperatures. Let e= unit diametral or circumferential deformation ; = coeffi- cient of linear expansion for a change of one degree F.; /= number of degrees of change. Assume an outer member of steel with an allowance of o.ooi in. per inch of diameter of fit. Then (Fig. 4) : e= r ^ = a.xt- t= -. (20) Substituting : 0.0000065 i. e., a raise in temperature of this amount would give the mem- bers the same diameter. The usual shrinkage-temperature of wrought iron and steel is about 600, the increase providing for greater allowance, for clearance in assembling, or for both. The value of a for cast iron is 0.0000062 per degree F. *Thurston, "Friction and Lost Work," 1898, p. 215. fLineham, " Mechanical Engineering," 1898, p. 868. SHRINKAGE AND PRESSURE JOINTS. 6. Shrinkage vs. Pressure Fits. Table I. gives the results of comparative tests made under the supervision of Professor Wilmore * upon cast-iron discs which were either forced or shrunk upon steel spindles, the latter being pulled from the discs in the " tension " tests or twisted in the holes in measuring the grip in torsion. TABLE I. No. Fit. Test. Q * Si s, f I P Tension 1,000 O.OOI 9,700 10,116 0-033 2 S " 5,320 " " " 0.170 3 S " 5,820 " " a 0.190 4 S Torsion 2,200 " II a 0.072 5 P Tension 2,150 0.0015 14,516 15,275 0.047 6 P Torsion 2,200 0.048 7 P " 2,800 " II a 0.061 9 S " 9,800 " 'I a O.2IO 10 P Tension 2,570 O.OO2 19,355 20,366 0.042 ii S " 7,500 " O.I 2O 12 S 8,100 " " 0.130 13 P Torsion 4,200 i 0.069 14 P Tension 4,000 O.OO25 24,^194 25,458 0.053 15 S " 9,340 " O.I 20 16 s 9,710 II " ii O.I3O 17 ! P Torsion 4,600 II " a O.06 1 18 S " 13,800 II 'I " O.igO 19 S " 17.000 0.003 29,000 30,550 0.190 The discs were 6 in. in diameter and I in. thick, with, on one side, a boss 2 in. in diameter, projecting ^ in., giving a bore i^ in. long and I in. in diameter. The spindles of machinery steel were \\ in. in diameter, turned at the contact-surface to I in. plus allowance for a length of i^ in., which length was reduced by a taper at the extremity and a shallow groove at the top, each ^ in. long, making the bearing surface I in. in length. The number of spindles tested was 19. The diameter of the various sets differed by 5 ten-thousandths of an inch, the finished dimensions being i.ooi in., 1.0015 in., 1.002 in., 1.0025 in. and 1.003 m - The pressure fits were made without lubrication, other than that from wiping the surfaces with oiled waste. The spindles and holes were found to be in good condition after the tests. The maximum force required to move each spindle is given as Q in the table. After movement had occurred, a less force was required to continue it or begin it anew. Columns Nos. I to 5, inclusive, * American Machinist, Feb. 16, 1899. 16 MACHINE DESIGN. of the table were taken from the data of the tests ; the values in the remaining columns were computed from formulae (15), (38, C), (62, C) and (64, C). Accuracy in calculating the intensities of the stresses ^ and S 2 , and the coefficient f, is to some extent prevented by the boss, groove, and taper described above. The approximation given should be, however, sufficiently close for service. The value of was made = = 5, whence/-* = 0.946. Since both the length and diameter of the contact surface = I in., (/> = allowance in each case. The coefficients E l and E v were taken as 30,000,000 and 15,000,000 respectively. Shrinkage and pressure fits are marked respectively " S" and " P," in the second column of the table. The calculated results show very low coefficients of resistance and very high circumferential stresses. Since the ultimate tensile strength of cast-iron ranges between 15,000 and 35,000 Ibs. per sq. in. and the discs were of good quality, rupture of the inner layer of the bore did not occur ; but the elastic limit, in the ma- jority of the tests, was exceeded. The superiority of the shrinkage fit is marked, as is also that of both types in torsion. Excluding tests Nos. 4 and 8, the results give average ratios of strength, as follows : Tension: Shrinkage to Pressure = 3.66; Torsion: Shrinkage to Pressure = 3.20; Shrinkage: Torsion to Tension =1.50; Pressure : Torsion to Tension - 1.30. 7. Stationary Engines : Data from Practice. Prevailing practice, with regard to diametral allowances in shrinkage and forced fits and the pressures required for the latter, varies considerably, owing to differences in the sizes of the mem- bers, the qualities of the metals, the workmanship upon, and lubri- cation of, the contact-surfaces, etc. There are given below, in tabular form, through the courtesy of leading manufacturers of stationary engines and similar machinery, records of allowances as follows : Table II., the Lane and Bodley Company ; Table III., the Russell Engine Company ; Table IV., a prominent stationary en- gine building company ; Table V., the Buffalo Forge Company; Table VI., the B. F. Sturtevant Company ; Table VII., summary of Tables II. to VI. SHRINKAGE AND PRESSURE JOINTS. TABLE II.* 2 a.^ O^S l>~ 8 Length of Fit un.). Mean Diameter of Hole (in.). Total Allowance S. j! < 1 . & ft Volume within Fitted Surface (cu. in.). Pressure to Enter Pin (tons). Pressure at Mid-position (tons). Maximum Pres- sure (tons). ___ 1.8798 6.125 1.8767 .0031 .0017 36 I6. 7 2 10 2O 2 1.8819 6.I2S 1.877 .0042 .0022 36 I6. 7 2 15 23 3 1.8774 4-375 1.8764 .001 .00052 24.4 13-7 /2 I 4 2-7455 4-5 2.7387 .0068 .00247 38.7 26.5 3 12 25 2.7465 4-5 2.7437 .0028 .001 38.7 26.5 5 12 23 6 3.261 5 3-2542 .0068 .0021 51 41-5 5 2O 45 7 3.2625 5 3-2555 .007 .002 51 41-5 5 IS 30 8 3.267 5 3.261 .006 .0018 51 41-5 5 15 20 9 4-2505 6 4.2402 .0103 .OO24 79.8 85.1 5 22 44 io 4.2388 6.625 4.2478 .0091 .OO2I 78.1 93-4 12 30 60 ii 4 2303 6.5 4.2224 .co79 .OOI9 95-8 91 10 60 125 12 5-9343 4.0625 5.9216 .0127 .OO22 75-7 II2.2 6 16 25 13 5-9381 4 5-9252 .0129 .OO22 74-4 IIO.4 3 18 35 14 5-9294 4-125 5-9I94 .01 .OOI7 76.7 II3.8 5 15 25 15 6.8829 5-125 6.8697 .0132 .002 110.7 I9O.I 8 20 42 16 6.889 5 6.8785 .0105 .0015 108 185.9 5 22 45 X? 6.8692 4-875 6.855 .0142 .O02I 104.8 180.4 5 35 65 18 7.8884 5-5 7.873 .0154 .CO2 135-9 267.3 5 32 64 >9 7.8715 6-5 78575 .014 .0018 160.5 3*5-9 5 25 50 20 7.862 5-625 7.846 .016 .OO2 138.2 272.8 8 40 80 21 8.924 6.125 8.905 .019 .OO2I 170.8 378.9 20 45 68 22 8.9 6-75 8.8848 .0152 .OOI7 188.4 419.9 5 47 96 23 8.878 6-5 8.8669 .OII2 .0013 180.7 401 IO 45 92 TABLE III.* CAST-IRON CRANKS. Total Allowance, In. In. Shrinkage. 1'ressure. 4 to 5 0.0045 o.o' 90 5 ' 7^ 0.0030 O.O060 7/2 ' 9 0.0027 0.0055 10 ' 12 0.0025 O.0050 12 ' l6 0.0020 o. 0040 16 18 O.OOI5 0.0030 The practice of the B. F. Sturtevant Company is as follows : () Shaft couplings are bored 0.003 i n - l ess th an the shaft. The forcing pres- sure ranges from 6 tons for a 2 l I 3 g -in. shaft to 12 tons for a 5~in. shaft. (3) Crank-pins for cast-steel crank-plates are turned 0.005 m - large. The forc- ing pressure ranges from 25 to 28 tons for a 5-in. pin to 10-15 tons for small pins. (c) Crank-pins for cast-iron crank-plates are turned 0.009 m - too.on in. large. The forcing pressure is as in (/>). (J) Cast-iron Counter-balance Plates shrunk on Steel Crank- Discs. For diameters of 9 in. to II in., the total allowance is 0.007 m - With increased diameters, this allowance decreases, i. and, assuming /=/ / (r), this gives, SHRINKAGE AND PRESSURE JOINTS. whence f(r)=-pr- and so, t=f'(r)=-p-r^. Thus, we have t p = k and / -f p = r -r* whence dp the integration of which gives 2p + k = -\ t where k v is a constant fcZ of integration. Combining with t p = k, we have t+p=^\. These, then, are the fundamental equations which express the relation between circumferential tension and radial pressure at all points within the cylinder : t-p=k=T -P G =T l -P l I (/ + py = ** = (T O + />x* = ( T . + p iW 1 Eliminating 7j between the last parts of these equations, we have : and substituting this in the first parts of the same equations, we have, after combining : , -R 2 (23) Substituting these values in the first part of (21), we have, for the tangential strains at the inner and outer surfaces, where r = R and r = R v respectively : (24) Suppose now the pressure P l to be caused by a second cylinder (radii R l and /?,) embracing the first and itself under the external MACHINE DESIGN. pressure P y Let the circumferential tension at its inner surface be designated as Zj' (to distinguish it from T v the tension of the outer surface of the inner cylinder, which is under the same radial pressure P v but not at the same tension as the surface in contact with it) and that at its outer surface as T 2 . Then, applying formula (24) to this second cylinder, we have, for the circumferen- tial strains at the inner and outer surfaces : (25) Finally, assuming P 2 to be caused by a third cylinder (radii, R 2 and R^) whose outer surface is under no pressure, we have, for the circumferential strain at its inner surface : t r . (26) Now let -, -. , and ^ be the values fixed for the maximum strains of the three cylinders respectively, when under the action of the system of pressure P , /*, and P y Substituting these values for e To , e Tl ,, and e Tz , in (24), (25), and (26), we have (27) 2 2 -f 2R? the last of which equations gives the internal pressure which the built-up cylinder will stand, if its parts have been so assembled that the inner surface of each reaches at the same instant the con- dition of maximum circumferential strain assigned to it. This, of course, implies a definite shrinkage for each cylinder, which shrink- age remains to be determined. (&) Relative Shrinkages. Observe now that equations (24), (25) and (26) give the tangential strains resulting from the pressures P , SHRINKAGE AND PRESSURE JOINTS. 33 P v and P 2 , and that if we substitute for these pressures any simul- taneous changes in their values as p Q , p v and / 2 , the same equations will give the corresponding changes of strain. But the surfaces of contact of the cylinders must contract and expand together and so the change of strain at the outer surface of each cylinder must equal that simultaneously occurring at the inner surface of the cylinder embracing it. Hence equating the second part of (24) to the first part of (25) and the second part of (25) to (26), after replacing P , P v and P 2 by p v /, and p v we have : - R* (R* - R*) fi + R>(R> - R^ = 1 R? (R? - Rfip, - R; (X* - R?}p z =o J ' the first of which gives the relation between simultaneously occur- ring changes in the pressures at the radii, R ti , R v and R v and the second, the relations between such changes at the radii, R^ and R r If, now, in the first equation of (28), we make p = P and P 2 = P 2 , we find : R? (R* - X*) P 9 + R* (X* - Rf).Pj l ~ ^i 2 W-^o 2 ) and this is the change of pressure at the radius R lt which would result from the simultaneous removals of the outer cylinder which causes P 2 and of the internal pressure P itself. There- fore, substituting this value of p { for P l and P 2 for P 2 in the second equation of (25), we have, for the change of outer diameter of the middle cylinder, due to removing the outer cylinder and suppressing the internal pressure, the expression : But, by hypothesis, the strain at the inner surface of the outer /i cylinder, before the change just referred to, was -, and, there- fore, the relative shrinkage of the outer cylinder must have been : To find

l . (c) The Method of Procedure, then, is to calculate P v P 1 and P by formulae (27) and then determine the shrinkages by formulae (29) and (30). It may be, however, that the shrinkages thus found would cause excessive compression of the bore of the inner cyl- inder, when at rest ; and, if so, smaller values of 6 l and 6 2 must be used. To ascertain whether this is the case, eliminate p 2 be- tween the parts of equation (28) which gives : 'A J and, making / = P in this, the resulting value of p^ is the change of pressure at the outer surface of the inner cylinder due to the suppression of P Q . Therefore, p v + P l must be the pressure on that outer surface when the system is at rest ; and this must not exceed since, if it does, the tangential compression of the bore will ex- ceed . As a matter of fact, however, experience seems to show that there is no objection to compressing the bore beyond the elastic limit of the material under tension, presumably because the elas- tic resistance to compression is really considerably greater than that so-called elastic limit of tension. It is also to be noted that no account has been taken of the fact that the radial strain at the inner surface of a cylinder may, and SHRINKAGE AND PRESSURE JOINTS. 35 indeed sometimes does, exceed the tangential strain, while our formulae assume that it is only the latter which must not exceed a fixed limit. This, too, can only be justified by the assumption that the material really has a higher limit of elasticity under com- pression than under tension. In assembling U. S. naval guns with shrinkages calculated by the foregoing formulas, # , d l and 2 were taken as the lowest elastic limit given by any specimen from the particular forging considered, excepting where the resulting compression of bore considerably exceeded # , in which case t and 2 were some- what reduced. The formulae as given herein are, of course, easily extended to cover cases where there are more than three layers. The tangential strain is really the change of length per unit length of the circumference and, so also, the change of length per unit length of diameter. An alternative nomenclature of the strains is as follows: Take a circle of radius 'r in the cylinder walls when at rest and suppose that, when the pressures act, each point of the circle moves outwardly Jrand axially Ah, then the tan- Jr dAr gential strain is , the radial strain is -j-, and the longitudinal strain is --,, , these strains being what have been called e t , e r and e v (d} Radii. If only the tangential resistance to internal pressure is to be considered, the maximum value of P will be obtained by making the radii increase in geometrical progression from that of the chamber outward, provided the several cylinders have the same elastic strength and the same modulus of elasticity. Thus, for the case of one cylinder superimposed upon another, make P , formula (27), a function of R l (R and R 2 being constant and 6 l = Q ), differentiate, and make -^ = o. After cancellation, we have R* = R R 2 , showing that the maximum value of elastic re- sistance for a given total thickness of a given material occurs when the radius of the common surface is a mean proportional between the inner and outer radii. For example, with the 6-inch gun of 4-inch chamber-radius and 8-inch thickness of chamber-wall, the maximum resistance against tangential bursting stress would be secured by making R = 4-inch ; /? t = 4^3 ; R 2 = 4^9 ; and ^3=4^27= 12. 36 MACHINE DESIGN. In practice, however, other considerations than tangential stress prevent complete conformity with theory. In the first place, it is necessary to make that layer which takes the longitu- dinal strain of sufficient cross-section. In United States guns, the breech-block houses in the jacket or second layer and the area ~(R T^ 2 ) must be adequate, being, in naval guns, about three times that of the rear end of the chamber, so that the longitudinal stress on the jacket, if uniformly distributed, is one third of the chamber pressures. In French guns, the breech-block usually houses in the tube or inner layer, thus making R l much greater than is necessary for resistance to the maximum tangential stress. Again, the tube thickness over the enlarged chamber should not be too small to prevent lining the bore with a thin tube, after the erosion of the powder gases has cut away the rifling and rendered the gun inaccurate. Finally, the necessity for keeping down weight, which prescribes a decreasing exterior diameter to correspond with the diminishing pressure toward the muzzle, together with the need for avoiding sudden or great changes of section in the various forgings, sometimes dictates dimensions not otherwise desirable. 2. GUN CONSTRUCTION. The 1 6-inch Breech-loading Rifle (Type, Model 1895), completed except as to the final boring, rifling, and the hoops engaging the mount during the year 1900 by the Ordnance Department, U. S. A., at the Watervliet Arsenal, N. Y., is not only the most powerful gun yet built, but is also the largest construction ever assembled by shrinkage. The general data * are as follows : Weight of gun 126 tons (252,000 Ibs.), of armor-piercing pro- jectile, 2,400 Ibs., of powder-charge (smokeless), 576 Ibs.; powder- pressure, 37,000 to 38,000 pounds per sq. in.; muzzle-velocity, 2,300 ft. per second ; muzzle-energy, 88,000 ft.-tons ; penetration in steel at muzzle (De Marre's formula, normal impact), 42.3 in. ; range, 20,978 miles; height of trajectory, 30,51 6 ft. (about 5^ miles) ; length of projectile, 5 ft. 4 in. ; cost per round, powder and shot, $1,000. (a) Description. The gun is shown in section in Fig. 19. Its total length is 590.9 in.; external diameter at rear, 60 in., at muzzle, 28 in.; length of main bore, 448.5 in., diameter, 16 in. ; rifling, 96 lands, 96 grooves ; depth of groove, 0.06 in. ; the * Ordnance Department, U. S. A., "Notes on the Construction of Ordnance," No. 78. SHRINKAGE AND PRESSURE JOINTS. 37 38 MACHINE DESIGN. rifling curve is a semi-cubic parabola, ranging from one turn in 50 calibers to one in 25 at the muzzle. The cylindrical part of the powder-chamber is 90.7 in. long, and 18.9 in. diameter, and is con- nected with the bore by a conical slope 24 in. long. The volume of the chamber is 29,385 cu. in. The recess for breech-block is 24.4 in. long, with a diameter at top of thread of 24.86 in. The breech-mechanism is after the " Stockett System." The gun is built up of parts, as follows : The tube, 566.5 in. long, with a maximum outside diameter of 29.3 in.; two C-hoops shrunk upon the tube from the forward end of the jacket to the muzzle ; the jacket, 304.65 in. long, shrunk upon rear of tube, and overhanging the latter by 24.4 in. to form the breech-recess; the D-koop, 144.5 m - l n g> encircling forward end of jacket and rear of (7-hoop, and having two locking shoulders in its bore which engage corresponding projections on jacket and T-hoop, thus preventing any sliding backward of the former or forward of the latter, from the shock of firing ; three A-hoops, A-i covering the joint between the Z>-hoop and the jacket, and A-2, A-j, being shrunk over the outer surface of the latter ; four B-hoops, encircling the ^4-hoops. Weights (Ibs.). Rough. Finished. Tube with (7-hoops. Jacket. Hoop D. ' A-i. ' A-2. ' A- 3 . ' *B, *B-i, B- 2 , B-S. 124,351 90,058 26,965 19,859 16,137 20,163 58,620 100,260 73,900 23,900 14,910 15,120 19,940 The tube and jacket were each made from a nickel-steel ingot, not fluid-compressed, and octagon in section. After removing the discards, a longitudinal, axial hole was bored through the remain- ing block and the tube or jacket was then forged hollow on a mandrel under a hydraulic press. The completed forging was then rough-turned, bored, tempered in oil, and annealed. The hoops were made of fluid-compressed steel containing no nickel. Excepting that the ingots were round, the general process was similar to that for the tube and jacket. The hoop-metal was the harder, i. e., having the greater elastic limit and tensile strength. * Awaiting decision as to carriage. SHRINKAGE AND PRESSURE JOINTS. 39 All forgings were of sufficient total length to provide test- metal. The specimens for tube and jacket were 0.564 in. diameter and 3 in. long. The average physical qualities obtained in all tests are : Tube. Jacket. Hoops. Elastic limit, Ibs. per square inch. 5^,375 52,250 57,125 Tensile strength, Ibs. per s quare inch. 84,350 87,800 107,050 Elongation, per cent. 20.38 22.16 IQ.28 Contraction, " " 41-93 48.32 45-52 () Shrinkage Furnace. The furnace used in expanding the parts for assemblage is shown in Fig. 20. It consists of a wrought-iron "cage" or frame-work A, surrounding immediately a cylindrical wall B of fire-brick, the whole resting upon solid rock, at the 3O-ft. level, in a corner of the shrinkage-pit (Fig. 21). The thickness of the wall is 13 in. and its internal diameter is 8 ft. 4 in. A cylindrical muffle C, built of ^2 -in. boiler steel, sur- rounds the hoop to be heated. The outer diameter of the muffle is 6 ft. 6 in., there being, thus, an annular space, 1 1 in. wide, which forms a combustion-chamber for the burning gases. The furnace is 27 ft. 9 in. high ; its top is 2 ft. 3 in. below the floor-level ; it is closed by a removable cover D, which confines the steam and gases ; and the products of combustion are drawn off through a flue connecting the top of the chamber with the main chimney. Fuel oil is supplied through a 3 -in. pipe from a 5,ooo-gallon tank and enters the furnaces through 20 burner-openings E, set in five tiers F, of four burners each. The burner consists of an internal steam-pipe of /^-in. bore, the latter being reduced at the end to -Jg in. Surrounding this is a J^-'m. oil-pipe, the forward end of which is plugged and a y^ -in. hole drilled therein, opposite the Y6"^ n - P enm g m the internal pipe. The steam issuing at high velocity through the latter opening, carries the oil with it as a spray ; and its oxygen, combining with the oil, gives an intensely hot flame. The burners are so directed that the flame strikes the muffle at a tangent approximately, thus giving a rapid spiral move- ment to the gases. The muffle transmits the heat to the hoop and the circulation of air within it tends to make the temperature equal at all points of the hoop. The furnace-temperature is governed by a damper in the flue, by the number of jets burning, and by the amounts of oil and steam admitted. Each burner is surmounted by an observation opening, closed by a mica door. MACHINE DESIGN. Uniformity of heating is secured by the tangential direction of the gases and by the intervention of the muffle, the latter keeping the flames from impinging directly upon the hoop and thus causing local heating in excess. ( a-c, there is, other things equal, greater friction with the triangular thread. 2. STRENGTH. In the triangular thread, the section at the root is the full length of the nut, while, in the square form, the sec- tion is but one half this length. Against shearing and flexure at the root, the latter thread is, therefore, proportionately the weaker. FIG. 23. SCREW FASTENINGS. 45 3. NUT. As noted, the triangular type has a bursting action upon the nut, which action, disregarding friction, does not exist with the square thread. In general, the triangular form is more suitable for screw-fasten- ings, owing to its greater strength, its increased frictional holding power which prevents backing off under load, and the finer pitches permissible by the full section at the base of the thread. On the other hand, the square thread is better adapted for power-trans- mission, since it has less friction and its bursting effect upon the nut is so small as to be negligible. 12. Requirements of the Screw-Thread. The screw is used as a detachable fastening in joining the mem- bers of a structure or machine; in producing pressure or tension, as in the screw-jack and testing-machine ; and for the transmission of power and conversion of motion, as in the worm-gear and screw propeller. Its requirements for these uses are : 1. POWER. This depends upon the pitch and form. The effect of the latter upon the strength and power of thread has been dis- cussed. With a given applied force, the less the pitch, the greater the axial load may be, since the pitch fixes the angle of the inclined plane upon which the load virtually moves. 2. STRENGTH. This is governed by the pitch, form and depth of the thread. With constant load, the steeper the pitch, the greater must be the applied power and the consequent normal pressure upon the thread. For the same load and nominal diam- eter, the deeper the thread, the less its mean bearing-pressure will be ; but the moment of the load upon the root will be larger and the effective diameter of the bolt to resist tension, will be reduced. 3. DURABILITY. The most durable thread is one whose form produces the least friction, whose depth gives minimum bearing pressure, and which is most accurately fitted. 13. Elements of the Screw-Thread. The requirements of the screw-thread make its elements inter- dependent. Consider : i. EFFECTIVE DIAMETER. This depends upon the axial load and the torsional stress produced by friction between the threads in setting up the nut. The magnitude of the latter stress is gov- 46 MACHINE DESIGN. erned by the applied power, and that of the power by the axial load and pitch. 2. PITCH. The relations between pitch and diameter in the prevailing systems of screw-threads are the outcome less of log- ical analysis than of long experience. For screw-fastenings, the limit in one direction lies in the fact that, with an excessively coarse pitch, the depth will be too great and the effective diameter will be reduced unduly. Again, that component of the pressure which is parallel to the thread -surfaces will exceed the force of friction between the latter, and, owing to this excess, the nut will back off. On the other hand, with an unduly small pitch-angle, the surface-friction will form too large a proportion of the total work of setting up the nut, the torsional action upon the bolt will be excessive, and the latter may be sheared. In general, fine pitches are unsuitable for soft metals and coarse pitches for shal- low holes. 3. FORM. As stated, the square thread is the form best adapted for power-transmission. For large fastenings requiring to be read- ily and frequently removed and which are strained heavily, but in one direction only, as the breech-block of a gun, the trapezoidal thread (Fig. 30) is most suitable. This thread has the acting face normal to the axis, the rear face at an angle thereto, and combines the greatest strength and least friction attainable. For screw-fastenings in general, the triangular thread, with blunt top, straight sides, and filled-in base-angle, was adopted through various considerations with regard to strength, friction, durability, ease of manufacture, and conformity with general prac- tice. Thus, in strength and frictional holding power, this form is superior ; its straight sides give even wear and maximum bearing surface ; the angle between them is fixed, in the various systems, by compromises between the conditions as to strength, friction, bursting action upon the nut, and facility of verification and pro- duction ; the flat or rounded top reduces the liability to injury ; and the filling in of the reentrant base angle increases the effec- tive diameter of the bolt and, in the Seller's system, the resilience of the latter also. 4. NUT. The nut may yield either by the shearing or rup- ture of its threads or by bursting from the action of the outward component of the pressure upon the thread. The latter, both on bolt and nut, acts as a cantilever beam, fixed at the root and loaded SCREW FASTENINGS. 47 uniformly over the bearing surface. When worn, the area of the latter is reduced, the bearing becomes irregular, the load is practically concentrated, and the bending moment at the root may be increased. If the nut is of a metal materially weaker than that of the bolt, its depth should be greater than the normal. In any event, this depth should be sufficient to give ample strength against flexure and shear at the root of the thread, to provide sufficient bearing surface to prevent abrasion, and to afford a good hold for the wrench. 5. MULTIPLE THREADS. In power-transmission screws of large pitch, a single thread will provide adequate bearing surface only by having a depth so great as to give an unduly small effective diameter of bolt. When the pitch is sufficient to permit it, the use of two or more parallel threads of usual proportions will secure the required surface with a normal effective diameter. Such threads are usually of square or trapezoidal form. 14. The United States Standard (Sellers) Thread. It would be difficult to overestimate the services to English- speaking engineers of Mr. William Sellers and of his predecessor in the same field, the late Sir Joseph Whitworth, in the investiga- tions and efforts which led to the wide adoption of the respective systems of screw threads which bear their names. The two sys- tems are in essentials almost identical. That of Sellers was orig- inally presented by him before, and recommended by, the Frank- lin Institute in 1864. It was adopted later, with trifling modifica- tion, by the U. S. Navy and War Departments and by the Master Mechanics' and Master Car Builders' Associations and is now known as the U. S. Standard System of Screw Threads. The thread, as shown in Fig. 24, is triangular with flat sides in- clined at an angle of 60, the apex being cut off and the base filled in to a radial distance in each case of one eighth the height of the primitive triangle making "flats,"/", at these points each one eighth of the pitch, />, in length. The Sellers system provides dimensions for bolts from one fourth inch to six inches nominal diameter. The notation and formulae are : D = nominal (outside) diameter of bolt, inches ; d= effective diam., ins. = D 2s = D \.-ip = D ; 3r n 48 MACHINE DESIGN. D-d s depth of thread, ins. = p x 0.65 ; (31) p = pitch of thread, ins. = 0.24 VD + 0.625 0.175 ; (32) n = number of threads per inch = - ; /= width of flat = ^ ; ( 3 3 ) H = depth of nut, rough = D ; h = depth of head, rough = ^ d h ; d n = short diam., hex. or square nut, rough = ^ D -\- -|" ; d h = short diam. of head, rough = |- D + -| /r ; The equation for the pitch, as above, is an empirical formula con- structed to cover diameters within the scope of the system. To avoid impracticable fractions, the number of threads, as thus de- duced, is modified to secure a convenient aliquot value. Thus, for a 2 -in. bolt : / = 0.241/2 -f 0.625 0.175 = 0.2138 in.; 1/0.2138 == 4.68 = n = say, 4.5 threads per in. The depth of the thread is obtained from the equation : s = \p cos 30 = 0.65^, deduced from the diagram, Fig. 24. The formula for the short diameter, d n , of the nut is empirical and was derived from success- ful practice. The values of the depths, H and //, of the nut and head respectively were based upon considerations as to adequate bearing surface, shearing stress, and provision for an efficient hold for the wrench. The long diameters of hexagon and square fig- ures may be obtained by multiplying the corresponding short diameters by 1.155 and MH, respectively. The finished dimen- sions for the depths and short diameters are : H finished = Z? The U. S. Navy Department adopted the Sellers system with the single exception that no difference was made in the size SCREW FASTENINGS. 49 - J> -J G- O. 6SJO f- /*- ' o.aeejtr. fnternat'L y. o.ejo MACHINE DESIGN. of finished and unfinished bolt-heads and nuts, in order that the same wrench might be used for both. The size adopted was that given by Sellers for rough work. The formula for " the exact diameter of the tap-drill with no allowance for clearance is : 1.2990381 n d=D- " The usual allowance (for clearance) above exact bottom diam- eter is from 0.004 f r /^ mcn to o.oio for 2-inch taps." * TABLE X. U. S. STANDARD (SELLERS) BOLTS AND NUTS. Bolt. Nut. Head. Nut and Head. Diameter. Area. Threads. Depth. Depth. t fr p w If' |f i -! l|l A 0.185 0.049 0.240 0.077 0.027 0.045 20 18 0.0063 0.0069 I | A 1 0.294 O.I 10 0.068 16 0.0078 i i tt A 0-345 0.150 0.093 ii 0.0089 ? 1 i \ 0.400 0.196 0.126 13 0.0096 " s i A 0-454 0.249 0.162 12 0.0104 ft I H 0.507 0.307 0.2O2 II 0.0114 i i i ft j 0.620 0.442 0.731 0.60 1 0.302 0.420 10 9 0.0125 0.0139 i i! 0.838 0.785 0.550 8 0.0156 i fl JA 0-939 0.994 0.694 7 0.0179 H it 1.064 1 1-227 0.893 7 0.0179 T i 2 I-I59 1.284 1.485 1.767 1-057 1.295 6 6 0.0208 if 0.0208 1 1 i 2f 1.389 2.074 I.5I5 5i 0.0227 if 2 T 9 5 1.490 2.405 1.746 5 0.0250 1 1 | 2 f I.6I5 2.761 2.051 5 0.0250 ' 1 1 2 r| I.7II 3.142 2.302 4 \ 0.0278 2 I T 9 5 3 I.96l 3-023 4 \ 0.0278 ~2\ I* 3 2.175 4.909 3.7I9 4 0.0313 2 IT! 3 2.425 5-940 4.620 4 0.0313 2f 2 i 4 3 2.629 7.069 5428 3 \ 0.0357 j 3 2ft 4 3* 2.879 8.296 6.510 3 ] 0.0357 3* 2 ^ 5 32 3.100 9.621 7.548 Si 0.0385 3' 2 ri 5: 3l 3-3I7 11.045 8.641 3 0.0417 i 3 1 5: 4 3.567 12.566 9.963 3 0.0417 4 3ft 6 4 3.798 ! 14.186 11.329 2 r 0.0435 A 3 \ 6 4 4.028 15.904 12.753 0.0455 ! 4* 3rV 6 4 4- 2 55 17.721 14.226 0.0476 41 1 3t 7 5 4.480 19.635 15.763 0.0500 5 ! 3if 7 5 4-73 21.648 17.572 0.0500 5 4 8 5 4-953 23.758 19.267 0.0526 si- ! 4ft 84 I 5-203 5423 25.967 28.274 21.262 23.098 0.0526 s| 4f 8f 0.0556 6 4ft 9! * "Standards of Length," G. M. Bond, 1887, p. 169. SCREW FASTENINGS. 5 1 The Sellers system was investigated exhaustively by a Board of U. S. Naval Engineer officers in 1868. This Board * found as to i. Pitch. The relations of pitch and diameter did not differ materially from the average proportions dictated by good practice. 2. Form. The thread, as compared with that of ordinary V form, gave with equal pitches a greater effective diameter and was less liable to injury. Further, in the most unfavorable case that of the one-fourth-inch bolt where the inclination of the thread and the torsional stress are maxima, the tendencies of the bolt to yield to tension or torsion are, with clean and well-lubricated surfaces, about equal. 3. Nut. To resist shearing (stripping) of the thread, the depth, H = D, gives a marked excess of strength for perfect threads, since, for the latter, but 0.357^ is required. With regard to bearing surface for fastenings, the depth, H, provides as much or more than nuts were given ordinarily. The diameter, d n , was found to give ample security against bursting action, since, neglect- ing the resistance of the thread and taking the entire section of the bolt as effective, the required diameter, d n = i^D. 4. Head. The depth, h, was sufficient to provide fully against shearing and to afford an efficient hold for the wrench. The proportions of the Sellers system are given in Table X. 15. Modifications of the Sellers System. Experience with the proportions of this system has resulted in modifications as to : 1. PITCH AND DIAMETER. For nominal diameters ranging from 2^ ins. to 6 ins., equation (32) gives the corresponding numbers of threads per inch as ranging from 4 at 2^ in. to 2^ at 6 in. These proportions, theoretically, should be such as will give a bolt equally strong in all respects. In naval practice and in that of many large companies, it is now customary to make the number of threads per inch 4 for all diameters from 2 y 2 in. to 6 in., inclusive, thus increasing materially the effective diameter of the bolt. The proportions of bolts and nuts now prescribed by the Bureau of Steam Engineering, U. S. Navy, are given in Table XI. 2. BOLT-HEADS AND NUTS. The proportions of nuts and bolt-heads, as given in the Sellers system, require odd sizes of bar-metal, not usually rolled by the mills, for the nuts and addi- * " Report of Board to Recommend a Standard Gauge for Bolts, Nuts, and Screw- Threads for U. S. Navy." May, 1868. MACHINE DESIGN. tional upsets in order to obtain sufficient metal for the standard head. Tables XII. and XIII. give dimensions which are without these disadvantages. 3. CIRCULAR NUTS. The Sellers system gives the dimensions of hexagonal and square nuts only. The former are lighter, their long diameter is less, and, where the movement of the wrench is restricted, they are more readily screwed home. The circular, grooved nut is a form applicable for use in a confined space and is of especial value where very large sizes are required as, for ex- ample, on the end of a propeller shaft. The outside diameter of the circular nut is equal to the short diameter of the other types, TABLE XI. STANDARD DIMENSIONS OF BOLTS AND NUTS FOR U. S. NAVY. (BUREAU OF STEAM ENGINEERING.) Dia IB. Eff. Diam. Threads Long Diam. Short De 3th. L > B D d. Per Inch. Hex. Sq. Diam. Head. N , 1 t' 065? .072 20 18 if If V A" c - .081 16 II B 1 || t f 093 14 if II II ( J f .100 13 I 1 JL f J 1 .108 12 1 I f H 1? 5 k .118 II ft I iA n 1 \ .130 IO fi I | i ^ - 144 9 fi 2 TV IT\ ft r .162 8 l 2JL i f 13. .186 7 A 2 T \ iH II .186 7 T 5 . 2| 2 i " .217 6 H 3A 2 T 3 ^ 1 3 3 * .217 6 1 3 2j J tV .236 Si H I T 9 J i. .260 .260 5 5 3^ 4A 2|" 2 r? I* i .289 31! 4H 3* I T 9 jJ r .289 4? 4A 4rw 3* I f I . .325 4 4|f 5H 3* IT! ; | 4 6 4 2 i i 3 4 sH 6H 4 2 f\ 3 3 4 5tt 7A 5 2 5 3 r 3 4 6^ 7H 5 2 T5 3 ; 3 4 8i 5 3 f 4 4 7A SH 6 4 4- ; 4 9r 3 i; 6 3i 4 | 4 7H 9lt 6 3A 4 4 4 8f 10 i 7 31 4 5 4 8$ lOff 71 3 5 1 4 4 9l 9 "A S? 8 8} 4A 5 5 5 6 4 4 I0 ?f io{j I2ff y 4| 6 E SCREW FASTENINGS. 53 plus twice the depth of the groove. In large sizes, this diameter is less than the long diameter of the hexagonal form. Good pro- portions for circular nuts are given in Table XIV. TABLE XII. MANUFACTURERS' STANDARD DIMENSIONS OF BOLT HEADS. (AMERICAN IRON AND STEEL MANUFACTURING COMPANY.) Diameter, Bolt. Square and Hexagon Heads, Diameter, Bolt. Square and He; Heads, cagon Width and Thickness. Width and Thickness. A i XA MX i A X A I I] I 1 I* X i IUX IT X i F i MX f IXA fix i 8x III i iiX f 2}fX 1 i iAX f 2 3 Xii TABLE XIII. MANUFACTURERS' STANDARD DIMENSIONS OF HOT-PRESSED NUTS. (AMERICAN IRON AND STEEL MANUFACTURING COMPANY.) SQUARE. HEXAGON. A Short Dia. Thickness. Hole. Size, Bolt 54 MACHINE DESIGN. TABLE XIV. ROUND SLOTTED NUTS. (NEWPORT NEWS SHIPBUILDING AND DRY DOCK COMPANY.) Diameter of Bolt. Diameter of Bolt. r 16. The Sharp "V" Thread. This thread has been superseded very largely in the United States by that of Sellers. As shown in Fig. 25, the sides are in- clined to each other at an angle of 60 and have a sharp apex and base. A section of the thread forms, therefore, an equilateral tri- angle, each side of which is equal to the pitch of the screw. Using previous notation : s=p cos 30 = 0.866/; d= D 2s = D 1.732/5 The pitch is usually that of the Sellers system. SCREW FASTENINGS. 55 17. The Whitworth System of Screw-Threads. In 1841 the late Sir Joseph Whitworth brought forward, in a communication to the Institution of Civil Engineers, the system of screw-threads which bears his name. This system, modified slightly in 1857 and 1861, has met universal adoption in Great Britain and extended use upon the continent of Europe. The range of diameters was originally, as in the Sellers system, from one quarter inch to six inches. TABLE XV. WHITWORTH SYSTEM. BOLTS AND NUTS. Bolt. Hexagon. Diameter. Area. Pitch. Threads. Head. Nut and Head. Nominal, D. Effective, d. Effective. P- Per Inch, Depth, Short Diam., d n and The proportions of this thread are given in Table XXII. TABLE XXII. STANDARD BASTARD SCREW-THREADS. (NEWPORT NEWS SHIPBUILDING AND DRY DOCK COMPANY.) Diameter, Nominal, D. Diameter, Effective, d. Area, Effective, ltd* -r- 4. Threads Per In., ft. Width of Flat, Nut, Depth, I// 0-3333" 0.0870 6 0.0420 K | 0.4250 O.I4I8 5 0.0500 i I 0.5500 0.2376 5 0.0500 | 0.6530 0-3349 4-5 0.0560 0.7500 0.4418 4 0.0625 0.8750 0.6013 4 0.0625 0.9640 0.7300 3-5 0.0714 1.0900 3-5 0.0714 If 1.1670 1.0700 3 0.0833 2 1.2900 1.3070 3 0.0833 2\ I.4I70 1-5770 3 0.0833 2f 1.4750 1.7090 2-5 O.IOOO 2 2 1. 6000 2.0106 2.5 O.IOOO 2| L 1.8500 2.6880 2-5 O.IOOO 3 \ 2.0000 3.1416 2 0.1250 3} | 2.2500 3.9760 2 0.1250 3f 3 2.5000 4.9087 2 0.1250 4 3. ACME STANDARD (29) THREAD. This form has the same depth as the similar square thread and its sides are at the same inclination as is now adopted generally in cutting worm gears. The formulae are : Angle of sides = 14.5 = 29 included angle ; SCREW FASTENINGS. Number of threads per inch = n ; Width of flat at top, B = ' 37 7 ; Depth of thread, s = h o.oi ; 2n Nominal diameter = D ; Effective diameter = D i + 0.02 J. TABLE XXIII. ACME STANDARD (29) SCREW-THREAD. k-. A + 4 .8)/, (34) where D is the actual external diameter of the tube throughout its parallel length and is expressed in inches. "Further back, beyond the perfect threads, come two having the same taper at the bottom but imperfect at the top. The remaining imperfect portion of the screw-thread, furthest back from the extremity of the tube, is not essential in any way to this system of joint and its imperfection is simply incidental to the process of cutting the thread at a single operation. From the foregoing, it follows that, at the very extremity of the tube, the diameter at the bottom of the thread is : " The thickness of iron below the bottom of the thread, at the tube extremity, is taken empirically to be : o.oi75Z> + 0.025. (3 6 ) " Hence, the actual internal diameter, J, of any tube is found to be in inches : .9)/ 2 (o.oi75Z> -f 0.025) o.o5Z>/ i. 9/ 0.05." (37) The proportions of the Briggs thread are given in Table XXVI. As compared with the Sellers system, the depth of the thread is SCREW FASTENINGS. measured by a greater fraction of the pitch ; but the latter is much finer for a given outside diameter and the thread is therefore shal- lower and more suitable for the thin walls of a tube. TABLE XXVI. WROUGHT-IRON WELDED TUBES. ( Briggs Standard. ) TAPER OF CONICAL TUBE END % INCH PER FOOT, OR i IN 32 TO Axis OF TUBE. Diameter of Tube. Screwed Ends. Thickness of Length Diameter Diameter Nominal Actual Actual Metal Number of Perfect of Bottom of Top of Inside, Inside, Outside, Inches. of Threads Thread of Thread Thread at Inches. Inches. Inches. per Inch. at Bottom, at End of End of Pipe, Inches. Pipe, Inches Inches. P i 0.270 0.405 0.068 2 7 0.19 0-334 0-393 f 0.364 0.540 0.088 18 0.29 0-433 0.522 g 0.494 0.675 0.091 18 0.30 0.567 0.656 0.623 0.840 0.109 14 0-39 0.701 0.815 | 0.824 1.050 O.II3 14 0.40 0.911 1.025 I 1.048 I-3I5 0.134 ii J 0.51 1.144 1-283 l|- 1.380 1. 660 0.140 n| 0-54 1.488 1.627 l| I.6IO 1.900 0.145 n|- 0.55 1.727 1.866 2 2.067 2-375 0.154 lli 0.58 2.200 2-339 2V 2.468 2.875 0.204 8 0.89 2.62O 2.820 3" 3.067 3-500 0.217 8 0-95 3-24I 3-441 3* 3.548 4.OOO 0.226 8 1. 00 3.738 3.938 4 4.026 4.500 0.237 8 1.05 4-235 4-435 4.508 5.000 0.246 8 1. 10 4-732 4-932 5" 5-045 5.563 0.259 8 1.16 5-29I 5-491 6 6.065 6.625 0.28o 8 1.26 6.346 6.546 7 7.023 7.625 0.301 8 1-36 7-340 7-540 8 7.982 8.625 0.322 8 1.46 8-334 8-534 9 8-937 9.625* 0.344 8 i-57 9-328 9-528 10 10.019 10.750 0.366 8 1.68 10.445 10.645 27. Stresses in Screw-Bolts. The body of a screw-bolt may be regarded as a cylindrical bar, subjected in various services either to simple tension or compres- sion or to one of these stresses combined with torsion, or, as in the flanged coupling, to tension and cross-shear. The thread may be considered as a cantilever beam whose section is that cut by a plane passing through the axis, as O-A, Fig. 34. The length of this assumed beam is the depth of the thread, s; its depth at the support is p f, where p = pitch and/~= width of flat at root; and its breadth at the root is the developed distance through which the axial section B-C-E extends. This distance, for one convolution = ltd and for the threads engaged by a nut of depth H ins. and' * Originally, 9.688. 72 MACHINE DESIGN. having n = Up threads per inch = nd x Hn. Let the total axial load on the bolt = W\ the load per convolution engaged = W/Hn = w ; and the permissible tensile and shearing stresses per sq. in.= S t and S, o.SS t , respectively. Consider the assumed beam with regard to : i. SHEARING OF THE THREAD, i. e., "stripping" at the root. The shearing force = W and is opposed by the section of metal at the support or root. The area of this section = breadth x depth of beam . . Resistance to Stripping = W= irdffn( p /),. (3 8) Equating this, for equal strength throughout, to the tensile resist- ance of the bolt : Tensile Resistance = W= T --d' L S = ~dHn( p-f)S. (39) 4 c J $pd "^ = 7^77 T\ (4) In the Sellers system, / = //8. Substituting: ^=0.357^. 2. RUPTURE OF THREAD by bending at the root. Theoretically, the load is uniformly distributed over the surface, which assump- SCREW FASTENINGS. 73 tion could be true only of new and perfect threads ; practically it may be considered as concentrated at the mean thread diameter. Therefore : s Moment of Load = W x - = M= S//c ; Section-Modulus at Root = - = - 5 5 Resistance to Rupture = W = ^ - - ' x -. (41) Equating (41) and (39) : pds ti -*u=rr (42) which expression assumes the tensile stress in the bolt and that in the thread due to flexure to be of the same intensity. Substi- tuting the values for the Sellers system : y/= 0.637^. 3. BEARING PRESSURE UPON THE THREAD. The allowable pressure upon the area of the engaged threads, as projected upon a plane normal to the axis, depends upon the service of the screw, being much greater with fastenings than with screws for the trans- mission of power, since, with the latter, friction and wear should be as small as possible. The projected area of the threads engag- ing the nut is (Fig. 34) : _ (& _ d*) x Hn. 4 Letting S b = permissible bearing stress per sq. in. : Permissible Load in Bearing = W = - (LP d 2 ) HnS b . (43) Equating (43) and (39) : (a) Fastenings. Letting a = effective area of bolt and A = ag- gregate projected area of engaged threads : aS, = AS, and = -. S t A This stress-ratio, the reciprocal of that in (44), is given for the Sellers system (H = D) in Table XXVII. * It will be seen that, * " Report of the Board to Recommend a Standard Gauge for Bolts, Nuts and Screw- Threads for the U. S. Navy," May, 1868. 74 MACHINE DESIGN. in this system, as the nominal diameter increases, there is an in- crease also in the bearing pressure, the latter varying from 0.242 to 0.331 of the permissible tensile stress per sq. in. Thus, for a 2-in. bolt of metal whose ultimate tensile stress is 60,000 Ibs. per sq. in., the permissible tensile stress, allowing for torsion = S t = 7,000 Ibs. per sq. in. From the table, S b / S t = 0.3046, whence S b = 2132.2 Ibs. per sq. in., which pressure is about the maximum allowable for fastenings. TABLE XXVII. RATIO OF BEARING PRESSURE TO TENSILE STRESS. (SELLERS SYSTEM.) a ^ "3 r 1 3 "3 II I 'o 81 V It i* Jl || i$ *k R 51 pi 11 < II 1 1 h N 11 12 S W IE i II * i fc i i n. .02688 .11105 .242 2 in. 2.3019 7-5573 3046 A .04524 .17696 .2556 2 i 3.0232 9.6471 .3134 * " .06789 09347 .25536 .34826 .266 .2684 2 | 3.7188 4.6224 11.8990 14.4881 3125 .3190 H .12566 45949 .273 3 " 5.4283 17.2221 3150 /,. " .16189 58461 .2769 3i " 6.5009 20.4158 .3188 f " .20174 .30190 .72222 1.0470 .2795 .288 3l " 8^416 23-5849 27.0337 .3200 3196 1 " .41969 1.4303 .293 4 9.9929 30.8820 3235 " 55024 1.8813 .3112 11.328 34.9236 3244 ij 11 69399 2.2877 3033 !< 12.743 40.3586 3157 " .89082 2.94245 .3027 14.250 43-2728 .3288 i " 1.0568 3-5310 .2993 5 | 15.763 48.4000 .3260 " 1.2948 4-2507 3051 17.572 53-4950 .3290 I-5I52 4.9925 3035 5| | 19.267 58.6676 .3280 " 1.7460 5-7750 .3023 21.262 62.7850 -3286 " 2.0510 6.6572 .3081 6 ' 23.098 69.8540 3310 (<) Screws for Transmitting Power In such screws, the bear- ing pressure varies within fairly wide limits, being determined by the character and duration of the work. Reuleaux gives 700 pounds per square inch of projected area for square and trape- zoidal threads, which pressure is given also by Weisbach for square threads. Unwin states that for screws constantly in motion this pressure should not exceed 200 pounds, and that with no power-screw should it be more than 1,000 pounds. 4. TENSION UNDER STATIC LOAD. Under this stress, the body of a screw-bolt has a higher elastic limit and a greater ultimate SCREW FASTENINGS. 75 strength than a cylindrical bar of the same metal and of diameter equal to that at the root of the thread. These gains are due to : (a) The Reinforcing Action of the Thread. When a cylindri- cal bar is subjected to simple tension only, it is increased in length and contracted in sectional area. The contraction is gradual, ex- tends over a considerable portion of the specimen, and reaches a minimum at the point where rupture occurs finally. To permit the gradual tapering of the specimen in unrestricted contraction, the bar should be originally of the same diameter throughout the section subject to elongation. If, now, there be turned in the bar one or a series of parallel grooves of any form but of the same depth, the tensile stress and the tendency to elongation and to contraction of area will be greater in the portions of lessened diameter. This reduced sec- tion is, however, insufficient in length to permit considerable con- traction ; and, further, the latter is resisted by the metal under less stress in the ridges of the grooves. In other words, in addi- tion to the lessened distance of least diameter through which stretching occurs, the ridges oppose the contraction of area and the consequent elongation of the reduced section and therefore add to the strength of the latter. As a result, the " grooved specimen " is stronger under static tensile load than a cylindrical bar having the same diameter as that at the base of the grooves. Mr. Kirkaldy * was the first to emphasize the effect of the form of a specimen upon its ultimate strength. In his report upon the Essen and Yorkshire iron plates, he says : "When the breadth of a specimen is reduced to a minimum at one point, a greater resistance is offered to its stretching than when formed parallel for some distance ; and, as the stretching is checked, so will also be the contraction of area and with it will be an increase in the ultimate stress." Table XXVIII. gives the results of tests made by Mr. James E. Howard f upon six specimens from the same i^-in. steel bar, to illustrate the effect of turning a reduced section or " stem," 0.798 in. in diameter on each specimen. Nos. I, 2 and 3 had cylindrical stems, I in., 0.5 in., 0.25 in. long, respectively, connected by full fillets to the body ; in specimens Nos. 4 and 5, the stems were semicircular grooves of 0.4 in. and 0.125 in. radius, respectively; a V-shaped groove was formed in specimen No. 6. *" Experiments on Wrought Iron and Steel," 1862, p. 74. t "Proceedings International Engineering Congress," 1893. MACHINE DESIGN. TABLE XXVIII. GROOVED SPECIMENS. No. Elastic Limit, Pounds Per Sq. In. Tensile Strength, Pounds Per Sq. In. Contraction of Area, Per Cent. 2 3 4 ! 64,900 65,320 68,000 75,000 86,000, about. 90,000 " 94,400 97,800 102,420 116,380 134,960 117,000 49-0 43-4 39-6 31-6 23.0 Indeterminate. In Table XXIX. there are given the results of tests made by Professor Martens which show that a screw-bolt under static ten- sile load is practically equivalent to a specimen with grooves turned in it of the same form as the thread-groove and also that there was an average increase of 14 per cent, in strength for the specimens tested over that of the cylindrical bar having the same diameter as that of the root of the thread. The table and the following particulars are taken from Professor J. B. Johnson's abstract of Professor Martens' paper : * "Two grades of mild steel were used for these bolts, all of which were cut from round bars originally 35 mm. (1.4 in. ) in diameter. The softer material, having a tensile strength of 53,500 Ibs. per sq. in., was used for screw-bolts approximately i in. in diameter, and the harder material having a tensile strength of 62,000 Ibs. per sq. in. was used for the bolts which were reduced to approximately in. in diameter. Four such bolts were made of each of these sizes of the four styles of thread (sharp V, angle 55; Whitworth ; Sellers, and German Society of Engineers. The latter having an angle of 53 8' with flats whose height is one eighth that of the primitive triangle), making in all 32 bolts with screw-threads which were tested. Two of each of these sets were tested in plain tension, the pulling force being applied to the inner face of the nut at one end and increased until rupture occurred. " The other two bolts of each set were tested also in tension, but under a torsional action resulting from the continuous turning of the nut as the load increased to rupture. In this case the distortion resulting from the permanent elongation of the bolt was nearly all taken up by the movement of the testing machine, the distortion taken up by the turning of the nut being the least possible to maintain a continuous torsional action at this point. "The same bars were also tested as plain tension-test specimens with cylindrical bodies and again with grooves turned into them of the same shape as the corresponding screw-threads, leaving the same diameter at the bottom of the groove as obtained at the base of the threads." The ratio f sa -*-f g , given in Table XXIX., is practically unity showing that the grooved and threaded specimens are equal in strength. The ratio, f sa -7- test bar, ranges between no and 119 and averages 114, giving thus a mean excess of strength of 14 * Zeits, d. Ver. Deuts. Ing., April 27, 1895. Abstract by Professor Johnson in " Digest of Physical Tests," July, 1896. SCREW FASTENINGS. ^ a X > X 8 I! Ml O C^ I-* HH ^ S^ Nfr O ON ro ^> I ^ ^ H M M ^vO O "5 Too" < 10^ t^L\O t^v> q ts <>vq M' ^> CO CO t^CO t^ rtCO ON M M o" vo v>^5 vO : . .0 77 ;S MACHINE DESIGN. per cent, for the threaded rods as compared with cylindrical bars of the same net area of cross-section. These results apply only to static or gradually applied loads. It will be noted that the tensile load upon the cross-section of a bolt at the root of the thread is not uniform throughout, since the metal of the latter opposes the elongation of the section imme- diately adjacent at the root, thus increasing its stress beyond that existing at the axis. It is apparent that, other things equal, the finer the pitch the more equable will be the distribution of the stress upon the minimum cross-section and the greater the resili- ence or internal work of the bolt before final yielding. Thus, Major W. R. King, U. S. A., in experimenting with gradually applied loads upon wrought-iron bolts of one and one half inch nominal diameter, U. S. standard, but of varying pitch, obtained results as follows : * Threads per Inch, 6 12 18 Relative Tensile Strength, I 1.21 1.23 Elongation, 0.025 0.06 0.08 Relative Internal Work, I 2.9 4 The U. S. standard pitch for one and one half inch nominal diam- eter gives 6 threads per inch. The bolts with 1 8 threads per inch were the stronger. They yielded finally, neither by stripping nor by fracture at the root, but by lateral contraction, so that the threads of bolt and nut became disengaged. (&) Increased Density of Threaded Section. Mr. Kirkaldy f found that, when the thread was cut with new dies, the strength of the threaded section averaged 72.5 per cent, of that of a cylin- drical bar whose diameter was that of the outside of the thread. When, however, old and worn dies were used, the average strength was increased to 85 per cent. In the latter case the tendency of the tool is to force aside and compress the metal rather than to remove it by clean cutting, thus increasing the density and strength of the thread and adjacent parts. Again, in bolts threaded by the "cold-pressed" method, no metal is removed but the thread is raised or spun above the body of the bolt so that the diameter of the shank is intermediate be- tween those of the top and root of the thread. In frequent tests \ of mild steel bolts threaded by this method, fracture, under * 'Trans. Am. Inst. Mining Engineers, 1885. f Box : " Strength of Materials, " 1883, p. 12. J Catalogue Am. Iron and Steel Mfg. Co., 1899. SCREW FASTENINGS. 79 a gradually applied tensile load, occurred in every instance in the shank, leaving the threaded portion intact. The normal reinforcing action of the thread is, by this process, aided doubtless through the compression and increased density of the thread and adjacent metal. (c) Resume. The experiments of Professor Martens show that, for static or gradually applied loads, the ultimate strength of the section at the root of the thread is 14 per cent, greater than that of a cylindrical bar of the same metal and cross-section. This increase in strength is due to the reinforcing action of the thread, and, in some degree, to the greater density of the metal. Under sudden and repeated stresses, however, the results would probably be less favorable, owing to the appreciable concentration of the stress about the bottom of the groove which would produce fracture at the reentrant angle. The increase in strength of the screw from these causes is, therefore, not considered in designing bolts. 5. TENSION UNDER SUDDEN LOADS OR IMPACT-. In both ma- chinery and structures a bolt may be required to withstand not only the tensile stress of a gradually applied or static load but also that produced by a suddenly applied load or by impact. Examples of such requirements may be found in bridge work, in marine machinery, in rock drills, etc. Let the static or gradually applied load, P, produce in the bolt a total stress, P, and elongation, /. Then, the same load, if sud- denly applied, will produce a maximum, momentary, total stress, 2P, and elongation, 2X, which, after a series of axial oscillations of the bolt, will be reduced, when the latter comes to rest, to the final stress, P, and elongation, ),, due to P as a static load. In impact, the load, P, is assumed to act as if it were not only sud- denly applied but in motion with a velocity, v, such as would be acquired by fall through a height, h. Under these conditions, P produces a maximum, momentary, total stress, Q, and elonga- tion, y, which, when -the bolt after oscillation comes to rest, are reduced to P and ^, respectively. Disregarding the weight and consequent inertia of the bolt, we have : * y*l\i + < r~* \ sl 2 i +I > (45) (46) *Merriman, "Mechanics of Mechanics," 1900. Art. 93. 80 MACHINE DESIGN. When h = o, these formulae become : Q=2P and y = 2)., i. e., the values for a load suddenly applied but without im- pact. In the three cases cited, the total final stress is P. For this stress, the absolute requirement is that the area, a, of the mini- mum cross-section of the bolt shall be such that the unit stress, Pja, shall not exceed the working stress of the metal. The strength of this minimum section is therefore practically the measure of the resistance of the bolt to safe static stress. Work is the product of a resistance by the distance through which the latter is overcome. The external work of impact, P(1i-\-y), is resisted by the elastic resilience or internal work, YZ Q X y, of the bolt. The same internal work may be the prod- uct of a high average, total stress, y 2 Q, and a small elongation y, or, conversely, of a low stress and a large elongation. Under the conditions given, it is apparent that the elastic resilience is the measure of the resistance of the bolt to sudden or impul- sive stress. In order to secure maximum total elongation under sudden load and therefore the least value of <2,the sectional area of the unthreaded portion of the bolt should be the minimum permissible, i. c., that at the root of the thread, which minimum area is determined by the static load. The minimum section should extend through as great a portion of the bolt as possible, since the total elonga- tion depends upon its length. When the area at any point is greater than the minimum, the unit stress over that area is less than over the latter and the elongation of that part and therefore of the bolt will be reduced proportionately and there will be an increase in the average stress. Equating the external and internal work, we have for a bar of sectional area A, length L, and maximum total and unit stresses, Q and q, respectively : 2 K=\ q E -AL, (47) on which K is the internal work or elastic resilience and E is the modulus of elasticity for tension. Consider two bolts of the same total length, length of shank, and area, a, at the root of the thread. In : SCREW FASTENINGS. 81 Bolt No. i : Let the length of threaded portion be / and its minimum sectional area and maximum unit stress be a and q, re- spectively. Let the length of shank be kl and its sectional area be na. Then, the maximum unit stress in the shank will be : . . ? _z. an n Bolt No. 2 : As before, total length = / + kl = I (i + k\ Let the uniform sectional area throughout screw and shank (disregard- ing thread ridges) =a and the maximum unit stress throughout=^ r The elastic resilience of each bolt will be the sum of the internal work of its threaded portion and shank. From (47), we have for : Bolt No. i : 2EK n Bolt No. 2 : ( h K= y ^ (48) (49) 2EK al i -f k \2hK n = \ a/ ' T nk' Assuming the total work, K, as the same in each case, it will be seen that q l <.g, i- c., that, by making the shank of the same sectional area as that at the root of the thread, the maximum unit stress upon the bolt has been re- duced. The equations dis- regard the increase of area due to the thread ridges, which increase, for accur- acy, should be included. When there is no impuls- ive load and a rigid connec- E FIG. 35. tion is required, there is no advantage, possibly the reverse, in increasing the elastic resilience of the bolt by decreasing the cross- section of the shank. 82 MACHINE DESIGN. In reducing its section, the shank may be turned down on the outside to the diameter at the root of the thread or it may be drilled axially from the head inward to the point where the thread begins, both as in Fig. 35. The latter method is preferable, since it leaves a section which is the stronger of the two in torsion. The shearing stress at any point of a section varies directly as the distance of that point from the axis, but the resisting moment of that stress with respect to the axis varies directly as the square of that distance. Therefore, a given area of section is most economically used with regard to torsion by so disposing it that its fibres shall be remote from the axis. Professor Sweet,* in testing solid and drilled bolts, i^ in. nom- inal diameter and 12 ins. long, found that, under gradually ap- plied load, the undrilled bolt broke in the thread with an elonga- tion of ^ in., while the drilled bolt was fractured in the shank after a total elongation of 2\ ins. Assuming the same mean load in each case, the ultimate resilience of the drilled bolt was 9 times that of the other. " Drop tests," i. e., those producing tensile shock, gave similar results. 6. FRICTION OF THE SCREW. The screw-thread is essentially an inclined plane wrapped around a cylinder, as on the bolt, or within a hollow cylinder, as in the nut. If the bolt be vertical, the wrench engages the nut in a horizontal plane and the axial load upon the bolt may be assumed as raised vertically by move- ment along the inclined plane of the nut-thread, the force acting horizontally. The efficiency of the screw, per se, and that of the inclined plane are the same. Sliding friction is generated between the bolt and nut threads as they move upon each other. The resistance or force of this friction acts along the contact-surfaces in opposition to the direction of relative motion of the latter. The magnitude of this force is measured by the product of the coeffi- cient of friction and the total normal pressure between the surfaces. Thread -friction not only reduces the useful work and efficiency of the screw, but also adds to the torsional stress within the body of the bolt produced by the component of the load which is nor- mal to the axis. Therefore, the bolt is subjected, in screwing up, to torsion due to the nut and to tension or compression from the axial load. The combined stress thus developed, exceeds mate- rially the simple axial stress when the nut is screwed home and at *A. W. Smith, "Machine Design," 1895, P- 1 35- SCREW FASTENINGS. 83 rest. This torsional action is of especial importance in small screws, which may readily be sheared by excessive force upon the wrench. In addition to the friction of the threads, the efficiency of the screw is reduced further by the friction of the rotating member of the pair the nut or screw, as the case may be upon its sup- port. Again if, as is usual, the turning moment is applied at one side only and not as a couple, there is a lateral thrust upon the support with a frictional resistance similar to that of a journal. (a) Torsion due to Thread Friction. The pressure upon the FIG. 36. threads in computations respecting friction, may be taken as con- centrated upon the mean helix or the circumference of the mean thread-diameter, d^ of pitch-angle, <5 (Figs. 22 and 36). Each element of the thread-surface is regarded as sustaining an equal elementary portion of the total axial load or stress, W, and each element has, therefore, a frictional resistance of the same magnitude. 84 MACHINE DESIGN. Since the conditions for all elements are thus identical, the total ex- ternal forces and thread-resistances may be assumed to be concen- trated at a single point upon the circumference of diameter, d y In Fig. 36, taking the nut as the turning member, let A-B-C be the inclined plane formed by developing one convolution of the nut thread of diameter, d^. Let A-B be that thread and E-G a portion of the bolt-thread. The base of the plane is xd , the height is the pitch, /, and the pitch-angle, o = B-A-C. Consider the external rotating force as applied in a plane normal to the axis and as tangent to the mean thread-circumference. Let : W = total axial load or tension in bolt ; P = external force to raise W without friction ; P= external force to raise Wwith friction ; P l = external force to lower W with friction ; N = direction of thread -pressure, without friction ; R = direction of thread-pressure, raising, with friction ; R l = direction of thread-pressure, lowering, with friction ; JJL = coefficient of thread-friction = tan

. The resultant of N and F l is R v making the angle

3 , the angle between R^ and the axis is

sec#' pace fad, -fi ^- r ^ + ^sec^' These are the equations for the raising and lowering forces, P and P v respectively, which, considering friction, require to be ap- plied tangentially to the mean thread circumference of a triangular- threaded screw. The form of the equations is that given by Unwin. In the Sellers system, ft = 30 and sec ft = 1.15. Substituting : p=w .^J^ K- i.is &' P i=W .^^~P, (56) 7r^ -f 1.15^ Thus, the ^-in. bolt has, in this system, the maximum inclination of the thread and hence the greatest tendency to be sheared by torsion. For this bolt, D -(- d 0.25 + 0.185 p = 0.05 and d = = -= 0.2175. Taking // = o. 1 24 : P=0.22W. With an ultimate tensile strength of bolt-metal of 60,000 Ibs.: red' 1 W = x 6o,OOO = 1,613 Ibs.; 4 P= 1613 x 0.22 = 355 Ibs., i. e., a force of 355 Ibs. applied, under the conditions as above, to a J-in. screw-fastening will rupture the latter by tensile stress. SCREW FASTENINGS. 8/ The assumed value of ft is suitable only for accurately fitting, well- lubricated threads. Owing to viscidity of the lubricant, the pres- ence of foreign matter, or rough surfaces from abrasion, the coeffi- cient will be usually much higher with a corresponding increase in friction and torsional stress. () Coefficients of Friction for Screw -Threads. In average cases, the value of // is taken as 0.15. This assumes fair conditions of surface and lubrication. Under other circumstances the coefficient may reach 0.40 or more. Professor Albert Kingsbury * has con- tributed to the meager knowledge available upon this question, the results of valuable experiments conducted by him and apply- ing especially to slow-moving power-screws. The tests were made upon a set of square-threaded screws and nuts of materials as given in Table XXX. and of dimensions as follows : Outside Diameter of Screw ... . . 1.426 inches. Inside Diameter of Nut 1.278 " " Mean Diameter " of Thread I -35 2 " Pitch of Thread 333 " Depth (effective) of Nut 1.062 " The nuts fitted the screws very loosely, so that all friction was excluded except that on the faces of the threads directly supporting the load. Four sets of tests were made. The maximum total load was 14,000 pounds in all tests excepting No. 4, in which it was 4,000 pounds. Readings were taken at pressures given in the table. The total bearing area of thread was approximately one square inch, so that the total axial load was equal to the pressure per square inch upon the thread. The lubricants were a purely mineral " Heavy Machinery Oil " of specific gravity, 0.912, and "Winter Lard Oil" of sp. gr., 0.919. The former, in test No. 3, was mixed, in equal volumes, with graphite, the brand being Dixon's " Perfect Lubricator." The screws and nuts were flooded with lubricant immediately before the tests. The threads were carefully cut in the lathe and had been worn down to good condition by previous trials. Screw No. 5 was not quite so smooth as the others. The speed was very slow, being about one revolution in two minutes and the motion, in tightening especially, was also somewhat irregular, so that the action between * Trans. Am. Soc. Meek. Engs., Vol. XVII. 88 MACHINE DESIGN. screw and nut was quite similar to that occurring when machine- bolts are set up in comparatively unyielding material. The re- sults are given in Table XXX. Each figure in test No. i is the average of eight readings ; in the remaining tests, of four readings. TABLE XXX. COEFFICIENTS OF FRICTION FOR SQUARE THREADS. Screws. Nuts. 6 Mild Steel. Wr/ught 8 Cast Iron. cast Brass. I. Mild Steel. 2. Wrought Iron. 3. Cast Iron. 4. Cast Bronze. 5. Mild Steel, Case Hardened. O.I4I 0.139 0.125 0.124 0.133 0.16 0.14 0.139 0.135 0.143 0.136 0.138 O.II9 0.172 O.I 3 0.136 0.147 O.I7I 0.132 0.193 TEST No. i. Heavy Machinery oil. Pressure, 10,000 Ibs. per sq. in. 2. 3- 4- 5- 0.12 O.II25 0.10 0.115 0.1175 0.105 0.1075 O.IO 0.10 0.0975 O.IO O.IO 0.095 O.I I 0.105 O.II 0.12 O.II 0.1325 0.1375 TEST No. 2. Lard oil. Pressure, 1 0,000 Ibs. per sq. in. 2. 3- 4- 5- O.I 1 1 0.089 0.1075 0.071 0.1275 0.0675 0.07 0.071 0.045 0.055 0.065 0.075 0.105 0.044 0.07 0.04 0.055 0.059 0.036 0.035 TEST No. 3. Heavy Mach'y oil and Graphite. Pressure, 10,000 Ibs. per sq. in. 2. 3- 4- 5- 0.147 0.15 0.15 0.127 0.155 0.156 0.16 0.157 0.13 0.1775 0.132 0.15 0.14 0.13 0.1675 0.127 O.II7 O.I2 0.14 0.1325 TEST No. 4. Heavy Machinery oil. Pressure, 3,000 Ibs. per sq. in. Professor Kingsbury's conclusions are : " That, for metallic screws in good condition, turning at extremely slow speeds, under any pressure up to 14,000 Ibs. per square inch of bearing surface and freely lubricated before application of the pressure, the following coefficients of friction may be used : COEFFICIENTS OF FRICTION. Lubricant. Minimum. Maximum. Mean. Lard Oil, 0.09 0.25 O.II Heavy Machinery Oil (Mineral), O.II 0.19 0.143 " and I graphite in equal volumes, / 0.03 0.15 0.07 With regard to the value of the coefficient to be used in design- ing power-screws, Professor Kingsbury says : " That (the value) depends upon the object of the design. If the screw is to be made so that it could not overhaul under the most favorable conditions, with either lard oil or SCREW FASTENINGS. 89 FIG. 38. heavy machinery oil, probably 8 per cent, would be the highest allowable coefficient ; and, for a certain margin of safety, a somewhat lower figure. If the driving mechan- ism is to be designed with a view to making the screw turn, even if perfectly dry, prob- ably 30 or 40 per cent, would be the figure. If the amount of power likely to be lost in the long run is what is wanted, probably 15 per cent, would be a safe coefficient for everyday work. This might be reduced to 10 per cent, with lard oil under the best conditions and at the speeds used in these experiments. ' ' Mr. Wilfred Lewis states that, " for feed screws which turn slowly, [JL = o. 1 5 may be taken as a good gen- eral average." (c) Friction of the Support. The thrust of a power-screw may be taken by the end of the screw itself upon a plane step-bearing whose maximum diameter is equal to the effective diameter, d, of the screw or the thrust may be borne by an annulus forming a collar-bearing at the end of the threaded portion. Both types of bearing are indicated in Fig. 38. In fastenings, the thrust and force of friction act between the under surface of the nut and the washer, the leverage of the force being about two thirds the nominal diameter, D, of the bolt. Let : W = total axial load ; p.' = coefficient of friction ; Wfjf = force of friction j r = radius of plane step bearing of diameter, d\ R l and R 2 = outer and inner radii, respectively, of collar-bearing ; R = 2^ D = leverage of Wfi! in nut. Then, the moment of the friction in the : Step Bearing = W/JL'- %r; (57) E> 3 r> 3 Collar Bearing = Wa'- % - ~ ^ ; (58) K \ K i Nut Wp r % D. (59) The reduction of the moment by the use of a step-bearing is ap- parent. This form, however, produces the most uneven wear and usually the greatest unit pressure. In addition to the vertical load there is usually a sidewise thrust on the screw-support, since the power is generally applied as a single force and not as a couple. This produces lateral pres- 90 MACHINE DESIGN. sure and friction between the threads or shank of the screw and the support or nut and connected parts. The action resembles that of a shaft journal. Views as to the distribution of friction in the latter are somewhat conflicting. In practice, the total pressure is assumed to be divided uniformly over the projected area of the bearing surface. 7. COMBINED TORSIONAL AND TENSILE OR COMPRESSIVE STRESSES. The axial load upon a screw produces a tensile or compressive stress and the external force applied to the nut in order to raise the load, develops a shearing stress. Disregard- ing the reinforcing action of the thread, both stresses may be assumed as acting upon the effective area only of the bolt. Then, the unit tensile stress will be equal to the total load divided by the effective area and the unit shearing stress at the outer.circum- ference of the area where that stress is a maximum will be equal to the twisting moment divided by the polar modulus of the section. Referring to Fig. 36, the twisting moment is P x 4/2. Then : Unit tensile stress = W -- - = S ; (60) 4 Unit shearing stress = P^.~ = ~^ = S t . (6 1 ) These stresses coexist and combine to produce a maximum, unit tensile stress upon a plane whose angle with the axis depends upon their relative magnitude. Similarly, they combine to pro- duce a maximum unit shearing stress upon a plane whose angle differs from that of the first but is governed by similar conditions. Evidently, the required effective area will depend upon the inten- sity of these resultant stresses, the formulae for which are : Maximum tensile unit stress = |- S t + V -S* + ^^ 2 = S t max.; (62) Maximum shearing unit stress = y S* + ^S t 2 = S s max. (63) When a screw which is so short that it may be treated as a strut, is under compression, the maximum compressive and shearing unit stresses may be found by replacing S t in (62) and (63) by the unit compressive stress. In designing a screw for a given load, the maximum stresses, as above, must not exceed the elastic strength of the metal. The usual practice, as given in 29, is to assign a reduced working stress to the material as the diameter decreases. SCREW FASTENINGS. The experiments of Professor Martens the results of which are given in Table XXIX. show the weakening of the effective section of the bolt to axial tensile load which results from the torsional action of the nut. His conclusions, from these tests, are : " The weakening effect of the turning of the nut under stress at rupture, is much less than might have been predicted, when the distortion of the screw below the nut by per- manent elongation is taken into consideration. The tests indicate, for this case, a strength of the I -in. bolts about 20 per cent, less than that of the plain bars and of the ^-in. bolts about 15 per cent, less than that of the plain bars. In general, it may be said that the turning of the nut upon the bolt at rupture reduces the strength of the nut section of the bolt by about 30 per cent." 8. CROSS SHEAR. In the flange coupling shown by Fig. 39, the bolts transmit the torsional stress from one section of the shaft to the next, and, if accurately fitted to the bolt-holes, are exposed practically to cross shear only, there being no bending stress and the tensile load, due to drawing the flanges together, being rela- tively slight. The usual method of design is to assume FIG. 39. the diameter of the bolt circle and equate the resistances to shear- ing of the shaft and bolts, the result being an equation in terms of the diameter and number of the latter. Let : R = radius of centre of bolt-holes ; D = diameter of shaft ; d = diameter of bolts ; n = number of bolts ; T. M. = maximum twisting moment on shaft ; (force, 7!jP.) R. M. = resisting moment of shaft ; T.'M.' = twisting moment at bolt centres ; (force, T.'F.') R.S. = aggregate resistance of bolts to shearing. The resisting moment to shearing of a circular section is equal to the product of the shearing stress, S f , at its periphery by the polar modulus of the section, ~d 3 /i6, where d is the diameter. T.F. is expressed in terms of the unit radius and will be to T.'F.' inversely as their respective radii. We have : 92 MACHINE DESIGN. T.M. = R.M. = r ~ - S ' ID T.'F' : T.F. :: i : R .: T.'F.' = = ~ D S ; K IDA. -d 2 R,S. = -- x n x S f Equating the values of T.'F.' and R.S.: To allow for inaccurate fitting and, therefore, for slight bending, the shearing stress on the bolts is usually made three fourths of that on the shaft. Introducing this fraction : R is usually 0.75 to O.8 times D. The number and diameter of the bolts are interdependent. If it be desired that the outside diameter of the coupling shall be as small as possible, n should be increased and d decreased, n is usually a multiple of the num- ber of duplicate sections of the crank-shaft. The bolts may be either headless taper bolts or " body-bound " and cylindrical with heads, as shown in Fig. 39. With the former type the weight of the head is saved and a rigid joint ensured. The objections to it are the accurate fit required, and, owing to the tapering hole, the impossibility of making the sections of a crank-shaft interchangeable. It will be noted that the analysis assumes the shearing stress to be distributed uniformly over the cross section of the bolt. While this assumption has sufficient practical accuracy, the stress upon the bolt-section varies in intensity, being greatest upon that side of the section which is most remote from the centre of the shaft. 9. STRESS IN CYLINDER-HEAD STUDS. The stress in bolts used in securing steam-cylinder covers and in other joints requiring to be tight against fluid pressure, is affected by somewhat complex conditions. The joint may be made metal to metal and ground or a gasket may be interposed between the flanges. The ma- terial of the latter depends upon the steam pressure and the cor- responding temperature. Rubber and sheet asbestos, plain or in combination, and copper in corrugated sheets, wire, or wire-gauze, are used for this purpose. SCREW FASTENINGS. 93 w The bolts, the flanges, and the gasket (if any), are all more or less elastic. The bolts are set up with an initial tension which is opposed by the force due to the compression of the flanges and gasket. Later, steam is admitted to the cylinder placing an addi- tional tensile load upon the bolts, which load elongates the latter still further and thus reduces the compressive force, as above. Referring to Fig. 40, let : S t = initial unit stress in bolt ; S c = initial unit force on bolt due to compression of gasket and flanges. Then : S, = S . FIG. 40. When the steam enters the cylin- der, the forces acting unon the bolts are the maximum load, W, due to the steam and the reduced compressive force between the flanges. These forces are opposed by the tensile stress within the bolt. Let : S w = unit force on bolt corresponding with external load, W\ S c ' = unit force on bolt corresponding with reduced compres- sion between flanges ; S t f = unit tensile stress in bolt when load, W, is applied. Then : S/ = S w + S e r . If the bolt stretches by an amount equal to the initial compres- sion of the other members, S t ' = S v , and the joint will open. On the other hand, with a short, rigid bolt, connecting ground flanges without gasket, the elongation will be relatively small and, with high initial stress, the value of S t ' approaches S w , plus the initial compressive force, S e . In any event, for a tight joint, the intensity of S t f must exceed S^ and S c f must be greater than zero. In Table XXXI., there are given the numbers, diameters, working stresses, and ultimate unit strengths of the cylinder-head studs for the high-pressure cylinders of some of the later vessels of the U. S. Navy. The area under load includes that of the cylinder and counterbore plus, in some cases, a portion of that over the ports. When a cylinder liner is used, the counterbore may be only ^ inch deep. 94 MACHINE DESIGN. TABLE XXXI. STEEL STUDS FOR CYLINDER COVERS. U. S. NAVY. H. P. Cylinder. Studs. Diameter, Ins. Initial Press. Gauge, Ibs. per sq. in. Total Area Inside of Flange, Total Load at Initial Press., Ibs. Number. Diameter. Stress per sq. in. of Eff. Area at Initial Material of Tensile Strength (Minimum) sq. ins. Press., Ibs. Ibs. per sq. in. H 250 153 38,250 18 : 7036 8o,OOO 20* 250 342.25 85,562 28 7240 8o,OOO 30 200 921.3 184,260 24 I 7264 8o,000 35 2 5 1484 371,000 38 I 7539 75,000 38* 250 1 1720 430,000 38 I 7469 75,000 28. Stresses in Nuts. i. SHEARING, RUPTURE, AND BEARING PRESSURE upon the thread. The conditions as to these stresses are similar to those which exist with the bolt-thread, excepting that, as the diameter at the root of the nut-thread is the nominal diameter, D, plus the clearance spaces, the total section at the root to resist shearing and rupture and the projected area of the thread are slightly greater than those of the bolt. 2. BURSTING STRESS. In Fig. 41,* let: - W= axial load upon the bolt ; * ' ' Report of Board to Recommend a Standard Gauge for Bolts, Nuts, and Screw- Threads, U. S. Navy," May, 1 868. SCREW FASTENINGS. 95 N normal pressure upon one half the thread, resolved in a direction perpendicular to any single element of its heli- coidal surface ; B = component of N acting in a direction perpendicular to the axis of bolt ; /5 = base-angle of thread ;

= 6, o = 42, and =o.8i. Good practical reasons make it undesirable to use so large an angle. Multiple threaded screws, however, owing to their ample bearing surfaces, permit relatively steep pitches. For the friction of the support, we have, for a screw whose thrust-collar has a mean friction-diameter, D' , a work of collar- friction per revolution equal to the force of friction multiplied by its circumferential path = Wp'.nD' = W 'tan' ' = tan

l D> I +-y- In the table relating to square-threaded screws which follows, the efficiencies have been calculated, but, in several cases, they have been checked by experiment and found to be fair average values. The efficiency of any screw will, of course, vary widely with the amount of lubrication. The same coefficient of friction p= 0.15, (f> 8 30' is taken for both thread and thrust- collar. The diameter of the latter is assumed to be that of the thread. E = the efficiency per cent, when there is no friction be- tween the thrust-collar and its bearing; E' = the efficiency per cent, allowing for thrust-collar friction. TABLE XXXIL* APPROXIMATE EFFICIENCIES OF SQUARE THREADED SCREWS. Angle of Thread, S E ^ 2 19 ii 3 26 14 4 32 17 5 36 21 10 55 36 20 67 4 8 45 -| 79 52 The efficiency of a square-threaded screw in lowering IV may be found from the values of the useful and total work by a process similar to that given for the efficiency in raising the weight. * Goodman : " Mechanics Applied to Engineering," 1899, p. 204. 98 MACHINE DESIGN. (&) Triangular Threads. From equation (66), we have for the efficiency of a square thread : tan d + tan y Replacing tan

963 2,485 3.712 5,135 6,903 7,854 9,940 12,270 13,520 16,210 19,150 22,340 -| inch ; threads per inch, 1 2 ; spacing, centre to centre, 4 inches. The stress at root of thread should not exceed 6,000 Ibs. per sq. in. A " detector " hole at which leakage will show when the bolt is broken is drilled or punched, preferably the former, from the outer end of the bolt inward to the beginning of the shank. Flexibility is a most important requirement of these bolts. In some cases, various combinations of the ball-and-socket joint have been applied at one end. In the ordinary type, this quality de- pends upon the material, the reduced shank, and the form and method of driving the heads. As material, the best grade of wrought iron is preferred. The Falls Hollow Staybolt is rolled with a central hole through- out, thus avoiding later drilling or punching. The bolt is also threaded through its full length with, therefore, uniform strength at all points. The size of the hole is usually -| inch or y 3 ^ inch. It serves not only as a " detector " but also, if desired, as an inlet for the admission of air to aid combustion. The data and results of tests of these bolts at McGill University are : io8 MACHINE DESIGN. Material, double-refined charcoal stay-bolt iron, i inch diameter, j 3 g inch hole; length, 25^ inch ; mean diameter, outside, 1.014 inch ; yield-point, 32,000 Ibs. per sq. in. ; ultimate tensile strength, 49,300 Ibs. per sq. in.; equivalent elongation in 8 inches, per cent. ; reduction of area, 45.7 per cent. Chief Engineers Sprague and Tower, U. S. Navy, in 1879, exhaustive experiments upon the strength of boiler-bracing. From their report * the following data are taken with regard to the re- sistance of screw stay-bolts in flat surfaces : "In reference to iron and low steel bolts, andiron and low steel plates, and copper plates and iron bolts, after a careful examination of the results of these experiments in particular, we are satisfied that the following formulae will correctly and safely repre- sent the working strength of good material in flat surfaces, supported by screw stay-bolts with riveted button-shaped heads or with nuts, when the thickness of the plates forming said surfaces and the screw stay-bolts are made in accordance with the dimensions and conditions given in Table Y. W= safe-working pressure ; T= thickness of plate ; */= distance from centre to centre of stay-bolt : T 2 For iron plates and iron bolts W= 24000 T l For low steel plates and iron bolts W 25000 For low steel plates and low steel bolts. For iron plates and iron bolts, with nuts '= 28 '= 40000 Z? For copper plates and iron bolts = 14500 " To obtain the ultimate bursting pressure, multiply the results of the above formulae by 8, which is the factor of safety used. TABLE Y. DIMENSIONS AND CONDITIONS FOR MAKING IRON AND Low STEEL SCREW STAY- BOLTS FOR FLAT SURFACES SUBJECT TO INTERNAL PRESSURE FOR DIS- TANCES RANGING FROM FOUR TO EIGHT INCHES (INCLUSIVE) FROM CENTRE TO CENTRE OF STAY-BOLT. Nuts. ills m 5 ( 155 ; K *" Experiments in Boiler Bracing," U. S. Navy Dep't, 1879. SCREW FASTENINGS. I0 9 " The rivet-heads to be a segment of a sphere, formed by first upsetting the end of the bolt with a few quick, sharp blows of the hammer, then finished to shape with the ham- mer and button-head set. Where nuts can be used instead of riveted heads, they should be of the standard size, suited to trie diameter of the bolt, faced on the side bearing on the plate, and dished out so as to form an annular bearing surface of as large a diameter as the nut will allow, aud of a breadth and depth given in the table. Before securing the nut in place the dished portion should be filled with red-lead putty made stiff with fine iron borings." The regulations (January, 1901) of the U. S. Board of Supervis- ing Inspectors of Steam Vessels, prescribe for plates, ^ inch thick and under, used in boilers as " flat surfaces fitted with screw stay- bolts riveted over, screw stay-bolts and nuts, or plain bolt with single nut and socket, or riveted head and socket," a working pressure determined by the formula : *-*. 04 where P =. working pressure in Ibs., C 112, / = number of six- teenths in plate thickness (i. e., for -j^-inch plate, /= 7), and d = distance between stays in inches. For plates above -j^-inch thick C ' = 1 20. The pressures, as above, refer to fire-box plates. Also, FIG. 45- " on other flat surfaces there may be used stay-bolts with ends threaded, having nuts on same, both on the outside and inside of plates." For these surfaces, formula (76) is used with C = 140. 5.. ARMOR BOLTS. The proportions of threads for these bolts, as used in the U. S. Navy, have been given in 24. The method of their application with side, diagonal, and belt-armor, is illustrated in Fig. 45. The armor-plate is fitted snugly to a backing of teak, the latter being secured to the backing plates of the hull by bolts countersunk in the wood. After the armor-bolt is screwed down no MACHINE DESIGN. to a bearing in the plate, the space around the shank is calked solidly with oakum and the nut is screwed up against a lead washer until it embeds itself in the latter, thus causing the lead to flow into the thread. As an additional precaution against leakage, the backing plates and washer are coated with red lead, all inter- stices in the backing are filled with red lead under pressure, and the joint between the backing plates is calked. Turret-armor is secured by similar bolts which have, however, a solid head instead of a nut. The spacing, in all cases, is such as to provide one bolt for each 5 sq. ft. of armor surface. 3i. Methods of Manufacture. Bolts are headed hot from round stock ; then threaded and pointed. Nuts are pressed or forged hot, or pressed and punched cold, and tapped. i. BOLT-BLANKS. The round stock is sheared into lengths containing enough material for shank and head. Each blank is then heated and the head formed in a forging ma- chine. Figs. 46 and 460. give a view, plan, and details of the I 3^ -inch Heading and Forging Machine built by the Acme Machinery Co., Cleveland, O., and illus- trated herein through the courtesy of that FIG. 46. company. " There are two sets of tools : the stationary or gripping dies, A, which hold and re- lease the blank and the heading die, B, and finishing punch, C, which form the head and are carried by a tool-holder fixed to a reciprocating plunger. The latter is driven from the shaft which is actuated by a fly-wheel with clutch-connection controlled by a pedal. The plunging or upsetting mechanism is omitted from the plan ; it moves on the line marked "centre of heading slide." "The dies, A, are divided and open vertically on the centre-line of the lower cylin- drical groove, Z>, and the upper groove, E, also cylindrical but having a square or hexagonal recess for the bolt-head. The opening and closing of the dies is done by the toggle-joint mechanism shown. The latter is operated, through an intervening spring, by an adjustable connecting rod driven from the shaft. The bolt-blank is upset while in groove, Z>, by the heading die, B. It is then shifted to E where the head is finished by punch, C. The grooves, D and E, are concentric respectively with die, B, and SCREW FASTENINGS. I I I punch, C. The latter die holds a die-plug and the punch has a head, both suitably shaped for upsetting square, or with other forms, hexagonal heads. " In forging a bolt-head, the operator places a heated blank in groove, D, and touches the pedal. The machine makes a "plunge" and the gripping dies close, remaining thus while die, , advances and forms the head and until the plunger has travelled about 3 inches. When the machine has passed its forward centre, the plunger has re- ceded about $ inch and the gripping dies open. The operator now removes the bolt to the upper groove, E, and again touches the pedal, upon which the finishing punch enters the die at E, presses against the head, and removes the slight draught formed during the first stroke. At the same time, the side pressure of the dies drives all ' fins ' back into the head. The bolt is really made during the first stroke, while the heated metal is at its best for working. The second stroke simply removes the slight taper of the head and smooths the sides of the latter. " The toggle-joint gives maximum pressure when the gripping dies are closed. Until its joints are in line, it is acted upon by an elastic force in the spring, so that if the dies become obstructed, the mechanism will yield and the machine will not meet undue FIG. 460. strain. In addition to its action as an automatic relief, the spring forms also, with the connecting rod of adjustable length, a device to regulate the time and duration of closure of the gripping dies with regard to the advance of the heading dies on the plunger, as may be required for various sizes of work." The machine described above is of the " grip-and-plunge " type. In the " hammer-header" form of heading machine, there are, for a square head, five hammers, one striking on the top and one on each of the four faces of the head, simultaneously. In this ma- chine, the head is molded by a succession of relatively light blows while it is cooling. An objection urged against this form is that 112 MACHINE DESIGN. the -bond between the head and shank may be destroyed by a " cold-shut " at the point of juncture. 2. NUT- BLANKS are made by several processes. In the " hot pressed " machine, the nut is formed in a die, pierced, and crowned, and is then placed in the holder of a " burring machine " in which revolving cutters remove the rough edges. In " hot forging ma- chines," the nut is forged smooth by hammers automatically oper- ated ; and, in " cold pressing," the flat bar is fed between the rolls of the machine, cut into blanks, and a nut made complete at each revolution. Finally, if desired, the nut is faced and chamfered in a facing machine. While the cold-punched nut meets extensive service in struc- tural and other work, the rigid specifications of the Bureau of Steam Engineering, U. S. Navy, permit the use of hot-pressed nuts only. With regard to this question, the Engineer-in-chief says : " In making a cold-punched nut, either of wrought iron or steel, the fibre of the metal is injured and its full strength can be restored only by bringing the nut to a welding heat and finishing it under the hammer, as with the hand-made forged nut. The hot- pressed nut, on the contrary, although not so perfect as that made by hand forging, ap preaches the latter so nearly that it can be reamed, tapped, finished, and used with fair degree of safety." The injurious effects upon boiler-plate of punching rivet-holes will be discussed in the succeeding chapter. In 1878, Mr. David Townsend * made some experiments which show the flow of metal in nuts punched cold under the conditions of his test. He found that both the top (nearest the punch) and bottom faces of the nut were depressed ; that the lower diameter was increased, making the sides tapering ; and that a portion of the blank punched from the hole had flowed into the body of the nut throughout a zone nearly half as deep as the nut and beginning almost at the top face of the latter. The original depth of the nut was 1.75 in. ; that of the core removed was 1 .063 in. The density of the latter was found to be the same as that of the metal before punching. Therefore, a volume of metal, whose sectional area was that of the core and whose length was 1.75 1.063 = 0.687 ins. was forced into the body of the nut. It is apparent that the stress was severe. 3. THREADING AND TAPPING. Bolts are threaded in the lathe, or by hand-operated dies, or in the bolt-cutter, the latter being * Jour. Franklin Institute, March, 1878. SCREW FASTENINGS. 113 practically but a set of revolving dies into which the bolt-blank is fed at the required axial speed. The bolt-cutter produces usually a full thread at one cut with, in consequence, greater stress in the bolt metal and greater pressure upon the lead-screw than in the lathe where the same thread would be made in several cuts. Square threads or those requiring unusual accuracy of workman- ship require lathe-work. The merits of the bolt-cutter lie in the rapidity and cheapness of execution and the fact that its product is sufficiently accurate for all ordinary purposes. Fig. 47 shows a threading tool which is illustrated herein through the courtesy of the Rivet-Dock* Company, Boston, Mass. The dis- advantages of the single-point thread tool used in lathe work are : the difficulties of keeping the exact angle in grinding, of setting with FIG the small thread-gauge, the suc- cession of cuts, necessarily light, to prevent burning the point, and the repeated stops to test with a limit-gauge or master-nut. The thread-cutter shown, is a simple disc of tool steel having ten teeth, each of the latter being longer radially than the one pre- ceding. In operation, a cut is run with each tooth. There are thus, in effect, ten cutting tools, the leading ones suitably shaped for roughing out and the final tooth proportioned for finishing with accuracy. The single-point tool both roughs and finishes, while the final tooth of the cutter does finishing work only. The cutter is mounted on a steel slide, the latter having a movement to and from the work by means of an eccentric stud in the hub of the lever. The lever, in moving the slide, engages the pawl and rotates the cutter one tooth for the next cut. The heel of the tooth in action rests upon a stop, which takes the strain of cutting. The stud extends through the lever-hub and is secured on the back by an arm with pin-stop engaging ten holes so spaced that changing the stop from one hole to another moves the slide and cutter a fraction of a thousandth of an inch forward, thus giving the necessary adjustment for fine fits and provision for exact dupli- cation. MACHINE DESIGN. Bolt-threads are produced also by cold rolling. For the de- scription of this process which follows, acknowledgment is due to J. H. Sternbergh, Esq., President of the American Iron and Steel Manufacturing Company. " The machine is horizontal and of simple construction. It has a stationary die with threads cut on the face of the latter at a certain angle. Another die, having threads cut on its face also, is held in a reciprocating cross-head. The bolt-blank is placed perpendicularly between the two dies and the thread is produced by compression in rolling the blank between the latter. The distance between the apices of the dies is the same as the diameter of the bolt at the root of the thread. For a bolt of, say, |^-inch diameter, the dies are about ten inches long and the blank is rolled throughout nearly the whole length of the die, one operation producing the thread. A portion of the latter is actually raised above the external circumference of the bolt and no metal whatever is cut away. Great accuracy, however, is required as to the diameter of the blank bolt in order to produce uniform and perfect threads." There are various types of machines for threading nuts. In one well-known automatic nut-tapper, the blank nuts are placed in a receptacle on the top, from which they are conveyed to the taps by means of guide-ways. After being threaded, the nuts are ejected automatically. It is stated that one operator can attend ten machines and produce about 1 80,000 nuts per day. 32. Materials. The specifications (1901) of the Bureau of Steam Engineering, U. S. Navy, for bolts and nuts of steel and iron are as follows : RODS FOR BOLTS, STUDS, AND RIVETS. I. The physical and chemical characteristics of rods for bolts, studs, and rivets are to be in accordance with the following table : Class. Material. Minimum Tensile Strength. Minimum Elastic Limit. Minimum Elongation. Maximum Amount of P. S. Lbs.fer Lbs.per Per cent. sq. in. sf. in. in 8 Inches. Class A. Open-hearth 75,000 40,000 23 .04 03 Cold and quench nickel or bend about an carbon inner diameter steel. equal to the thickness of the test piece in each case. Quenching temperature 80 to 90 F. Class B. Open-hearth 58,000 30,000 28 .04 .03 I Inner diameter carbon : equal to one steel. j half the thick- 1 ness. SCREW FASTENINGS. 115 If the contractor desires, and so states on his orders, the Bureau will direct that the inspection of the rods be made at the place of manufacture of the bolts, studs or rivets instead of at the place where the rods are rolled. 2. Kind of Material. The steel shall be made by the open-hearth process, shall contain not more than four one-hundredths of I per cent, of phosphorus, nor more than three one-hundredths of I per cent, of sulphur. 3. Surface and other Defects. The rods must be true to form, free from seams, hard spots, brittleness, injurious sand or scale marks, and injurious defects generally. 4. Test Pieces. If the total weight of rods, all of the same diameter, and rolled from the same heat, amounts to more than 6 tons, the inspector shall select at random six tensile test pieces, three cold-bending test pieces and three quench-bending pieces ; but if the weight is less than 6 tons, one half of that number of test-pieces will suffice. If, however, the rods in one heat are not of the same diameter, then the inspector will take such additional test pieces as he may consider necessary according to the number of different sizes of rods in the heat. All of the test pieces shall be taken from rods finished in the rolls and, when practicable, but one piece will be cut from each rod selected for test. Should any test piece be found too large in diameter for the testing machine, the piece may be prepared for test in the manner prescribed for forgings. The tensile tests for rounds fy inch in diameter and less, shall be made on the largest sizes available and the elongation measured on a length equal to eight times the diam- eter. 5. Bending Tests. The cold and quench test pieces of Class Ai rods shall stand bending through an angle of 1 80 around a curve, the inner diameter of which is equal to the diameter of the rod. The cold and quench bends of Class A2 rods shall stand bending through an angle of 1 80 around a curve, the inner diameter of which is equal to one half the diameter of the rod. The quench test piece shall be heated to a dark cherry red in daylight, and plunged into fresh clean water at a temperature between 80 and 90 F. No bending test will be satisfactory if any cracks are to be seen on the outside of the bent portion. FINISHED BOLTS, STUDS, AND RIVETS, CLASSES A AND B. After the rods to be made up into bolts, studs, and rivets have been tested as pre- viously described, the finished articles shall be tested by lots of 500 pounds or fraction thereof, one piece being taken to represent the lot. The failure of 10 per cent, of the lots of 500 pounds to stand the specified tests in a satisfactory manner will render the whole of any delivery liable to rejection. Salts and Studs. When the bolts or studs are of sufficient length in the plain part to admit of being bent cold, they must stand bending double to a curve of which the inner radius is equal to the radius of the bolt or stud, without fracture. When bolts or studs are not of sufficient length in the plain part to admit of being bent cold, the threaded part must stand bending cold without fracture as follows : If of l / 2 inch diameter or less 35 If above % inch diameter and under I inch 3 If I inch diameter or over . 2 5 Where the bending tests can not be applied, the two following hammer tests must be substituted : (a) The test piece to stand flattening out cold to a thickness equal to one half its original diameter without showing cracks. (t>) The test piece to stand flattening out, while heated to a cherry-red heat, to a thickness equal to one third its original diameter without showing cracks. n6 MACHINE DESIGN. Surface Inspection. ( i) All bolts and studs shall be free from surface defects. (2) All bolts are to be headed hot, and the heads made in accordance with the U. S. standard proportions unless otherwise specified. The head must be concentric with the body of the bolt. (3) The threads must be of the U. S. standard unless otherwise specified, and must be clean and sharp. The threads of Classes A and B bolts may be either chased or cut with a die, but the threads of body-bound bolts must be chased and must extend far enough down so that when the nut is screwed home there will be not more than one and one half threads under it. The plain part of body-bound bolts must be turned in a lathe to fit accurately in the bolt hole. STEEL AND IRON NUTS. ( To be used with class A and B bolts and studs. ) 1. One tensile and one bending test bar from each lot of 1,000 pounds of material or less from which nuts are to be made shall be selected by the inspector for test. 2. The material (whether steel or iron) shall show a tensile strength of at least 48,000 pounds per square inch and an elongation of at least 25 per cent, in 8 inches. A bar y 2 inch square or % inch in diameter shall bend back cold through an angle of 180 without showing signs of fracture. 3. The nuts must be free from surface defects and the threads clean, sharp, and well fitting. 4. The dimensions of threads must be in conformity with the United States standard unless otherwise specified. 5. The nuts must be hot-pressed and reamed before threading, the holes to be cen- tral and square with the faces. All nuts must fit on the bolts without shake. FORCINGS. i . The physical and chemical characteristics are to be in accordance with the fol- lowing table : Class. Material. Treatment. Minimum' Minimum Tensile Elastic Strength, j Limit. Minimum Elonga- tion. Maximum Amount of Cold Bend About an Inner Diam eter of P. S. Lbs.per */. in. Lbs.per sq. in. Per crnt. in a in. High Grade, Open-hearth nickel steel. Annealed and oil 95,000 65,000 21 .06 .04 One inch through tempered. 180. Class A. Open-hearth Annealed. 80,000 50,000 25 .06 .04 One inch either nickel Oil tem- through or carbon pered op- 180. steel. tional. Class B. Open-hearth Annealed. 60,000 30,000 30 .06 .04 Half inch carbon steel. through 180. 6. Nuts to be used about machinery must fit so tight that it will be necessary to use a wrench to turn them. All other nuts must be at least thumb tight. 7. For the purpose of test all nuts which fulfill the preceding requirements will b< divided into lots of 500 pounds or less, and two nuts from each lot selected by the in specter for test as follows : SCREW FASTENINGS. 117 (a) One of the two shall stand flattening out cold to a thickness equal to one half its original thickness without showing cracks. (/>) The other shall stand flattening out (when heated to a cherry red in daylight), to a thickness equal to one third of its original thickness without showing cracks. 8. The failure to stand these tests will subject the lot represented by them to rejec- tion. The failure of 10 per cent, of the lot to pass the tests will render the whole order liable to rejection. For bolts requiring unusual strength, the metals described under " Forgings " are specified. Thus, connecting rod bolts are made from " High Grade " forgings as above. For wrought iron and various alloys and bronzes, the maximum tensile strength per sq. in. of cross section is taken as : Wrought Iron 50,000 Ibs. Alloy: Cu88%, Sn 10%, Zn 2% 2 o,ooo '< Phosphor Bronze, rolled 40,000 " Muntz Metal, rolled 40,000 " Manganese Bronze, rolled 50,000 " Tobin Bronze 50,000 " Naval Brass .... 50,000 " The specifications (1900), of a prominent railroad company fix requirements for stay-bolt iron, as follows : " The material desired is fagoted iron, free from admixture of steel and preferably box piled, the rilling of the box being small rods. It shall show when nicked on either side and then broken, a fracture with long fiber with sound welds. The iron must be smoothly rolled, free from slivers and depressions, and shall be truly round within .01 of one inch. It shall not be more than .005 of one inch above and not more than .010 of an inch below nominal size. This to insure freedom from jamming in the thread- ing dies. " Sample bars will be required to meet the following physical test: They shall show when tested in full size as rolled, a tensile strength of not less than 48,000 pounds per square inch, with an elongation of not less than 25 per cent, in 8 inches. One piece from each of the two sample bars shall be subjected to tensile test and one piece from each of them shall be threaded in dies with a sharp "V" thread 12 to one inch and firmly screwed through two holders, having a clear space between them of 5 inches. One of the holders shall be of such form and length that the bolt shall be rigidly held, so as to prevent rocking. This holder will be rigidly secured to the bed of a suitable machine and the holder at the other end will be vibrated in a direction at right angles to the axis over a space of } of an inch, so that the end of the specimen shall be deflected alternately ^ of an inch on each side of the center line. When thus tested acceptable iron should show not less than 2,200 double vibrations before breakage. " If the test of either of the bars shows a tensile strength of less than 48,000 pounds per square inch in an original section or an elongation less than 25 per cent, in a sec- tion originally 8 inches long, or if either bar stands less than 1,700 double vibrations, or if the two give an average of less than 1,900 double vibrations before breakage, the pile represented by such two bars will be rejected and returned to the maker. In addition, those bars which fail to meet the requirements as to rolling will also be rejected and returned." MACHINE DESIGN. 33. Nut-Locks. As has been shown previously, the pitch-angle of screw-fasten- ings is so small that the screw cannot possibly "overhaul," i. e., no static axial load, however great, will cause the nut to back off. On the other hand, on such a screw, when exposed to shock or to repeated, even though small, vibrations, the nut will loosen inevitably. Dr. Weisbach * has discussed fully and clearly the effect upon the nut of these external forces. When the joint is subjected to shock or vibration, work is done upon all of its parts. The work trans- ferred to the nut, expends itself in producing elastic oscillations in the material of the latter, with corresponding stresses of tension or compression, and, therefore, at any instant, a resultant stress. When the moment of this resultant is equal to the moment of nut- friction, any further shock or vibration will cause the nut to yield. It is evident, therefore, that the force with which the nut is screwed home, fixes the magnitude of the shock which will loosen it. Hence, nuts which can be set up with but moderate pressure, as on shaft bearings, especially need locking arrangements. Again, the effect of small vibrations, if they follow each other with sufficient frequency, seems to be cumu- lative, so that, even when nuts are set up with the greatest per- missible force on solid supports, as the fish-plates of rails, they will, if unlocked, back off under these conditions. The usual de- vices for locking a fastening nut are the check-nut, set-screws, spring-washers, and lock-plates. The nut itself is sometimes made elastic or the thread self- locking. i. CHECK-NUTS. A check- nut is essentially a friction -brake for the fastening nut. Assume, as in Fig. 48, a bolt, A, with fastening and check nuts, B and C, respectively. Let the lower nut be held and the upper be screwed against it as tightly as *" Mechanics of Engineering," Vol. III., Parti., Sec. II., 1896, p. 605. FIG. 48. SCREW FASTENINGS. 1 1 9 the strength of the bolt permits. There will be developed a pressure, P, between the adjoining nut-faces, and the nuts B and C will transmit this pressure to the lower and upper surfaces, re- spectively, of the bolt threads. Hence, a unit tensile stress,/, will exist within the bolt section, D-E, included between the limits of action of the nuts upon the bolt. This stress will not be present elsewhere. If now the bolt be subjected to shock, the fastening nut, B, cannot back off unless either its own thread-friction and that of the lower nut be overcome and the two withdraw together, or the nut, B, have a sufficient impulse to move independently, de- spite the friction between the nut-faces. In either case, the check- nut acts as a brake upon the fastening nut. Again, assume, as in the left-hand half of the figure, that the bolt is used for securing the cap, F, of a shaft-bearing. Let the lower nut, C, be first screwed down until the required pressure, P v is produced upon the cap and then the upper nut, B, be screwed home as before, developing the pressure, P, between the nut-faces. The nut, C, is now subjected to a downward force, P, and an up- ward force, P v with a resultant, P P v acting on the bolt-threads. There are three possible cases : (a) If P l > P, the resultant force is upward, the lower nut bears on the lower surfaces of the bolt-threads and aids in sustaining the axial load, P v upon the bolt. (b) If P l = P, the resultant force is zero. Hence the lower nut is unloaded and has no pressure on either the upper or lower sur- faces of the bolt threads. (c) If P l < P, the resultant force is downward, the lower nut bears on the upper surfaces of the bolt-threads, does not aid in sustaining the axial load, and produces an additional tensile stress, as at D-E. In both (a) and (c), thread-friction exists with the lower nut and the latter acts as a brake ; in (a) only this nut aids in sustaining the axial load, P v upon the bolt, which load in (&) and (c) is borne wholly by the upper nut. The fastening and check-nuts are frequently of different thicknesses. The discussion as above adapted largely from Weisbach shows that the upper nut may bear the entire load and should be the thicker of the two, although, in the absence of a thin wrench, the reverse is often the case. In practice, the lower nut is screwed down and home and the upper nut almost entirely so. Then, the latter is held with the wrench 120 MACHINE DESIGN. and the nut, C, is forced backward through a slight angle until it binds on the upper nut. FIG. 49. FIG. 50. FIG. 51. 2. SET-SCREWS. The set-screw, bearing upon a cylindrical prolongation of the nut, is the most effective locking device for heavy nuts requiring to be frequently removed as, for example, those on the connecting rods and main bearing caps of marine engines. Fig. 35 shows a bolt for the former which is fitted with a "collar-nut" and two set-screws one for locking the nut, the other for holding the bolt when backing off the nut. Table XXXVI. gives the proportions of such collar nuts and of the dowelled stop-ring into which the set-screw is tapped. Fig. 49 shows a similar nut, omitting the stop-ring and groove, the pro- portions for typical sizes of which, in inches, are : Diameter Least Value of H for of Bolt. A B C D E F G Wrought Iron or Brass. Cast Iron. 1 2 4 f 4 1 I If 1 i I Ij I ft ! i I \ 6 9 6 2 If I* I Ij- if if 3. ELASTIC NUTS. Fig. 50 shows the nut made by the Na- tional Elastic Nut Company. The blank is cut from a flat steel bar, bent into a ring with a lap on the side, pressed in a die into the shape of a finished nut, and finally tapped with special minus taps, -jj-g- under size. When screwed on the bolt, the split side is forced open about -^-^ of an inch, giving the nut a constant grip. The Wiles lock-nut, shown in Fig. 51, has a slot milled half way through of a width equal to the pitch. When the nut is in place, the walls of the slot are brought slightly together by a set- screw, thus gripping the bolt-thread. SCREW FASTENINGS. 121 TABLE XXXVI. COLLAR-NUTS WITH LOCKING SCREWS. (UNION IRON WORKS.) Nut Set-Screw. Dowel. 4. SELF-LOCKING THREADS. In the " Harvey Grip " thread, the bolt has a ratchet-thread, under cut on the bearing side at about 5 degrees less than a right angle to the axis of the bolt and the apex of the thread is cut to a knife-edge. The nut also has a ratchet-thread, the bearing side of which is about 5 degrees greater than a right angle to the axis of the nut. There is thus a cavity 122 MACHINE DESIGN. of about 10 degrees between the bolt and nut-threads ; and, when the nut is screwed home, the axial pressure upon it forces the thin bolt-threads out into the nut-threads, thus filling the cavity and locking the nut. In another locking device of this class, the thread is triangular with the V cut off at % of its height from the top and filled in at * d}t S t cos 6. The unit shearing stress, S s , on the net section along A-B will be equal to the total shearing load on that section, divided by the area of the section, i. e., FIG. 75. * Commander A. B. Canaga, U.S.N.,/or. Am. Soc. Naval Engrs., VIII., 2. RIVETED JOINTS. 149 Similarly, the unit tensile stress, Sf, on the net section along A-B will be equal to the total tensile load on that section, di- vided by the area of the section, i. e., For steel plates, the unit shearing stress on any section should not exceed T 8 -g- of the unit tensile stress. Hence : S t = 0.8 S/, (p-d)t-S t -smO _ R (p-d)t-S t -cosd (p d -d)t -*' (p d -d)t ; sin 6 = T 8 7 cos 0; tan 6 = 0.8 .-. = 38 40'. (82) Also : tan 6= V-- = 0.8, .-. V= 0.4 /. (83) The value of V depends thus upon that of 6 and the latter is determined by the assumption that S t = 0.8 S t f . As a general rule, Traill takes the available resistance of the metal along the diagonal pitch, for tension normal to the longitudinal pitch, as | of that of the same section along the latter pitch. 6. TRANSVERSE PITCH. The value, V, of this pitch has been calculated in the preceding section for staggered riveting with no rivets omitted in the outer row (Fig. 64 c] and for chain riveting with alternate rivets omitted in the outer row (Fig. 64 d~]. When, in staggered riveting, alternate rivets are omitted in the outer of several rows, the values of Ffor the outer and the next rows are different, since, as shown in Fig. 73, rupture may occur along the pitch, A-D, or along two diagonal pitches and a semi-pitch, as A-B-C-D. The calculation for the value of Fmust be based, therefore, on an equality of strength in these two directions. The method will be given later in the deduction of Traill's formulae. In simple chain riveting, the minimum value of V\s fixed by the same considerations which govern the minimum pitch, /. e., to pre- vent cracking the plate and to provide room for making the rivet- point, V, minimum, should not be less than 2d and is preferably 2.5^/in boiler-joints and ^d in structural work. 7. MARGIN AND LAP. To avoid cracking the plate in punch- ing or riveting, the distance from the nearest edge of the nearest rivet-hole to the edge of a plate or strap should not be less, as 150 MACHINE DESIGN. experience has shown, than the diameter of the rivet-hole. The margin is measured from the centre of the hole. The least value of the margin is therefore : E=i.$d. (84) The width of the lap depends upon the form of the joint. Thus, in Fig. 63 a, it is 2E\ in Fig. 63 b, 2E -f V\ in Fig. 64 k, 2E -f 2 V r As noted previously, the rivet tends to rupture the margin, as in Fig. 70, and to shear it, as in Fig. 71. In designing the margin for rupturing stress, the portion of the plate included between the rivet and the edge may be taken, approximately, as a rectangular beam, fixed at the ends and loaded in the middle, since the rivet is slightly less in diameter than the hole and has, theoretically, but a line bearing on the walls of the latter. For such a beam, the general formulae give : M=\-W-l=S'- and - = ^-, ' c c 6 in which M= maximum bending moment, W= total load, / = span, /= gravity moment of inertia of the cross-section of the beam, c = distance of most remote fibre of the cross-section from the neutral axis, b = breadth, and d= depth of beam. In the assumed beam : Span = diameter of rivet = d ; Breadth = thickness of plate = / ; Depth = E-dl2 (Fig. 70) ; Distance c=\ depth = \ (E dj 2); ^ _ C ~ O Let F= factor of safety and S t -^F= allowable working stress. Substituting : It is assumed necessarily, but practically without warrant that the load on any pitch-section, as m-n-o-r, Fig. 64 d, is divided RIVETED JOINTS. I 5 ! equally among the rivets in that section. Thus, in the figure re- ferred to, the total permissible load on the section is : and, since there are three rivets in the section, the load per rivet is : 3 * In general, let n = the number of rivets in the section. Then : (p-d}t S t n ' F' Substituting in (85): n F S~ F' 6 which equation applies to both lap and butt joints. In a good design, it gives a close approximation to E = i.$d, as in (84). The shearing and crushing of the portion of the plate between the edge and the rivet, as in Fig. 71, remain to be considered. For equal strength of margin with regard to these stresses, we have : Resistance to shearing = 2 ( )/ , ', Resistance to crushing = d- 1- S c . Assuming ^, = f